Practical Foundations for Programming Languages


Practical Foundations for Programming Languages Robert Harper Carnegie Mellon University [Version 1.32 of 05.15.2012.] Copyright c 2012 by Robert Harper. All Rights Reserved. The electronic version of this work is licensed under the Cre- ative Commons Attribution-Noncommercial-No Derivative Works 3.0 United States License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/3.0/us/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA. Preface Types are the central organizing principle of the theory of programming languages. Language features are manifestations of type structure. The syntax of a language is governed by the constructs that define its types, and its semantics is determined by the interactions among those constructs. The soundness of a language design—the absence of ill-defined programs— follows naturally. The purpose of this book is to explain this remark. A variety of pro- gramming language features are analyzed in the unifying framework of type theory. A language feature is defined by its statics, the rules govern- ing the use of the feature in a program, and its dynamics, the rules defining how programs using this feature are to be executed. The concept of safety emerges as the coherence of the statics and the dynamics of a language. In this way we establish a foundation for the study of programming languages. But why these particular methods? The main justification is provided by the book itself. The methods we use are both precise and in- tuitive, providing a uniform framework for explaining programming lan- guage concepts. Importantly, these methods scale to a wide range of pro- gramming language concepts, supporting rigorous analysis of their prop- erties. Although it would require another book in itself to justify this as- sertion, these methods are also practical in that they are directly applicable to implementation and uniquely effective as a basis for mechanized reasoning. No other framework offers as much. Being a consolidation and distillation of decades of research, this book does not provide an exhaustive account of the history of the ideas that in- form it. Suffice it to say that much of the development is not original, but rather is largely a reformulation of what has gone before. The notes at the end of each chapter signpost the major developments, but are not intended as a complete guide to the literature. For further information and alterna- tive perspectives, the reader is referred to such excellent sources as Con- stable(1986), Constable(1998), Girard(1989), Martin-L ¨of(1984), Mitchell (1996), Pierce(2002, 2004), and Reynolds(1998). The book is divided into parts that are, in the main, independent of one another. Parts I and II, however, provide the foundation for the rest of the book, and must therefore be considered prior to all other parts. On first reading it may be best to skim Part I, and begin in earnest with Part II, returning to Part I for clarification of the logical framework in which the rest of the book is cast. Numerous people have read and commented on earlier editions of this book, and have suggested corrections and improvements to it. I am particu- larly grateful to Andrew Appel, Iliano Cervesato, Lin Chase, Derek Dreyer, Zhong Shao, and Todd Wilson for their extensive efforts in reading and criticizing the book. I also thank the following people for their sugges- tions: Arbob Ahmad, Zena Ariola, Eric Bergstrome, Guy Blelloch, William Byrd, Luis Caires, Luca Cardelli, Manuel Chakravarti, Richard C. Cobbe, Karl Crary, Yi Dai, Daniel Dantas, Anupam Datta, Jake Donham, Favo- nia, Matthias Felleisen, Kathleen Fisher, Dan Friedman, Maia Ginsburg, Byron Hawkins, Kevin Hely, Justin Hsu, Cao Jing, Salil Joshi, Gabriele Keller, Scott Kilpatrick, Danielle Kramer, Akiva Leffert, Ruy Ley-Wild, Dan Licata, Karen Liu, Dave MacQueen, Chris Martens, Greg Morrisett, Tom Murphy, Aleksandar Nanevski, Georg Neis, David Neville, Doug Perkins, Frank Pfenning, Jean Pichon, Benjamin Pierce, Andrew M. Pitts, Gordon Plotkin, David Renshaw, John Reynolds, Carter Schonwald, Dale Schu- macher, Dana Scott, Robert Simmons, Pawel Sobocinski, Daniel Spoon- hower, Paulo Tanimoto, Peter Thiemann, Bernardo Toninho, Michael Tschantz, Kami Vaniea, Carsten Varming, David Walker, Dan Wang, Jack Wileden, Roger Wolff, Omer Zach, Luke Zarko, Yu Zhang. I am grateful to the stu- dents of 15–312 and 15–814 at Carnegie Mellon who have provided the impetus for the preparation of this book and who have endured the many revisions to it over the last ten years. I thank the Max Planck Institute for Software Systems in Germany for its hospitality and support. I also thank Espresso a Mano in Pittsburgh, CB2 Cafe in Cambridge, and Thonet Cafe in Saarbr¨ucken for providing a steady supply of coffee and a conducive atmosphere for writing. This material is, in part, based on work supported by the National Sci- ence Foundation under Grant Nos. 0702381 and 0716469. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Robert Harper Pittsburgh March, 2012 Contents Preface iii I Judgments and Rules1 1 Syntactic Objects3 1.1 Abstract Syntax Trees.......................4 1.2 Abstract Binding Trees......................7 1.3 Notes................................ 12 2 Inductive Definitions 15 2.1 Judgments............................. 15 2.2 Inference Rules.......................... 16 2.3 Derivations............................. 17 2.4 Rule Induction........................... 19 2.5 Iterated and Simultaneous Inductive Definitions....... 21 2.6 Defining Functions by Rules................... 22 2.7 Modes............................... 24 2.8 Notes................................ 25 3 Hypothetical and General Judgments 27 3.1 Hypothetical Judgments..................... 27 3.1.1 Derivability........................ 27 3.1.2 Admissibility....................... 29 3.2 Hypothetical Inductive Definitions............... 31 3.3 General Judgments........................ 33 3.4 Generic Inductive Definitions.................. 34 3.5 Notes................................ 36 viii CONTENTS II Statics and Dynamics 37 4 Statics 39 4.1 Syntax............................... 39 4.2 Type System............................ 40 4.3 Structural Properties....................... 42 4.4 Notes................................ 44 5 Dynamics 45 5.1 Transition Systems........................ 45 5.2 Structural Dynamics....................... 47 5.3 Contextual Dynamics....................... 49 5.4 Equational Dynamics....................... 51 5.5 Notes................................ 54 6 Type Safety 55 6.1 Preservation............................ 56 6.2 Progress.............................. 56 6.3 Run-Time Errors.......................... 58 6.4 Notes................................ 59 7 Evaluation Dynamics 61 7.1 Evaluation Dynamics....................... 61 7.2 Relating Structural and Evaluation Dynamics......... 63 7.3 Type Safety, Revisited....................... 64 7.4 Cost Dynamics.......................... 65 7.5 Notes................................ 66 III Function Types 67 8 Function Definitions and Values 69 8.1 First-Order Functions....................... 70 8.2 Higher-Order Functions..................... 71 8.3 Evaluation Dynamics and Definitional Equality....... 73 8.4 Dynamic Scope.......................... 74 8.5 Notes................................ 76 VERSION 1.32 REVISED 05.15.2012 CONTENTS ix 9 G¨odel’s T 77 9.1 Statics................................ 78 9.2 Dynamics............................. 79 9.3 Definability............................ 80 9.4 Undefinability........................... 82 9.5 Notes................................ 84 10 Plotkin’s PCF 85 10.1 Statics................................ 87 10.2 Dynamics............................. 88 10.3 Definability............................ 90 10.4 Notes................................ 92 IV Finite Data Types 93 11 Product Types 95 11.1 Nullary and Binary Products.................. 96 11.2 Finite Products.......................... 97 11.3 Primitive and Mutual Recursion................ 98 11.4 Notes................................ 100 12 Sum Types 101 12.1 Nullary and Binary Sums.................... 101 12.2 Finite Sums............................ 103 12.3 Applications of Sum Types.................... 104 12.3.1 Void and Unit....................... 104 12.3.2 Booleans.......................... 105 12.3.3 Enumerations....................... 106 12.3.4 Options.......................... 106 12.4 Notes................................ 108 13 Pattern Matching 109 13.1 A Pattern Language........................ 110 13.2 Statics................................ 110 13.3 Dynamics............................. 112 13.4 Exhaustiveness and Redundancy................ 114 13.4.1 Match Constraints.................... 114 13.4.2 Enforcing Exhaustiveness and Redundancy...... 116 13.4.3 Checking Exhaustiveness and Redundancy...... 117 REVISED 05.15.2012 VERSION 1.32 x CONTENTS 13.5 Notes................................ 119 14 Generic Programming 121 14.1 Introduction............................ 121 14.2 Type Operators.......................... 122 14.3 Generic Extension......................... 122 14.4 Notes................................ 125 V Infinite Data Types 127 15 Inductive and Co-Inductive Types 129 15.1 Motivating Examples....................... 129 15.2 Statics................................ 133 15.2.1 Types............................ 133 15.2.2 Expressions........................ 134 15.3 Dynamics............................. 134 15.4 Notes................................ 135 16 Recursive Types 137 16.1 Solving Type Isomorphisms................... 138 16.2 Recursive Data Structures.................... 139 16.3 Self-Reference........................... 141 16.4 The Origin of State........................ 144 16.5 Notes................................ 145 VI Dynamic Types 147 17 The Untyped λ-Calculus 149 17.1 The λ-Calculus.......................... 149 17.2 Definability............................ 150 17.3 Scott’s Theorem.......................... 153 17.4 Untyped Means Uni-Typed................... 155 17.5 Notes................................ 156 18 Dynamic Typing 159 18.1 Dynamically Typed PCF..................... 159 18.2 Variations and Extensions.................... 163 18.3 Critique of Dynamic Typing................... 166 18.4 Notes................................ 167 VERSION 1.32 REVISED 05.15.2012 CONTENTS xi 19 Hybrid Typing 169 19.1 A Hybrid Language........................ 169 19.2 Dynamic as Static Typing.................... 171 19.3 Optimization of Dynamic Typing................ 172 19.4 Static Versus Dynamic Typing.................. 175 19.5 Notes................................ 176 VII Variable Types 177 20 Girard’s System F 179 20.1 System F.............................. 180 20.2 Polymorphic Definability.................... 183 20.2.1 Products and Sums.................... 183 20.2.2 Natural Numbers..................... 184 20.3 Parametricity Overview..................... 185 20.4 Restricted Forms of Polymorphism............... 186 20.4.1 Predicative Fragment................... 186 20.4.2 Prenex Fragment..................... 187 20.4.3 Rank-Restricted Fragments............... 189 20.5 Notes................................ 190 21 Abstract Types 191 21.1 Existential Types......................... 192 21.1.1 Statics........................... 192 21.1.2 Dynamics......................... 193 21.1.3 Safety............................ 194 21.2 Data Abstraction Via Existentials................ 194 21.3 Definability of Existentials.................... 196 21.4 Representation Independence.................. 197 21.5 Notes................................ 200 22 Constructors and Kinds 201 22.1 Statics................................ 202 22.2 Higher Kinds........................... 204 22.3 Canonizing Substitution..................... 206 22.4 Canonization........................... 209 22.5 Notes................................ 211 REVISED 05.15.2012 VERSION 1.32 xii CONTENTS VIII Subtyping 213 23 Subtyping 215 23.1 Subsumption............................ 216 23.2 Varieties of Subtyping...................... 216 23.2.1 Numeric Types...................... 216 23.2.2 Product Types....................... 217 23.2.3 Sum Types......................... 218 23.3 Variance.............................. 218 23.3.1 Product and Sum Types................. 219 23.3.2 Function Types...................... 219 23.3.3 Quantified Types..................... 220 23.3.4 Recursive Types...................... 221 23.4 Safety................................ 223 23.5 Notes................................ 225 24 Singleton Kinds 227 24.1 Overview.............................. 228 24.2 Singletons............................. 229 24.3 Dependent Kinds......................... 231 24.4 Higher Singletons......................... 235 24.5 Notes................................ 237 IX Classes and Methods 239 25 Dynamic Dispatch 241 25.1 The Dispatch Matrix....................... 242 25.2 Class-Based Organization.................... 244 25.3 Method-Based Organization................... 245 25.4 Self-Reference........................... 247 25.5 Notes................................ 249 26 Inheritance 251 26.1 Class and Method Extension................... 251 26.2 Class-Based Inheritance..................... 253 26.3 Method-Based Inheritance.................... 254 26.4 Notes................................ 255 VERSION 1.32 REVISED 05.15.2012 CONTENTS xiii X Exceptions and Continuations 257 27 Control Stacks 259 27.1 Machine Definition........................ 259 27.2 Safety................................ 261 27.3 Correctness of the Control Machine............... 262 27.3.1 Completeness....................... 264 27.3.2 Soundness......................... 264 27.4 Notes................................ 266 28 Exceptions 267 28.1 Failures............................... 267 28.2 Exceptions............................. 269 28.3 Exception Type.......................... 271 28.4 Encapsulation of Exceptions................... 272 28.5 Notes................................ 275 29 Continuations 277 29.1 Informal Overview........................ 277 29.2 Semantics of Continuations................... 279 29.3 Coroutines............................. 281 29.4 Notes................................ 285 XI Types and Propositions 287 30 Constructive Logic 289 30.1 Constructive Semantics...................... 290 30.2 Constructive Logic........................ 291 30.2.1 Provability......................... 292 30.2.2 Proof Terms........................ 293 30.3 Proof Dynamics.......................... 295 30.4 Propositions as Types....................... 296 30.5 Notes................................ 297 31 Classical Logic 299 31.1 Classical Logic........................... 300 31.1.1 Provability and Refutability............... 300 31.1.2 Proofs and Refutations.................. 302 31.2 Deriving Elimination Forms................... 305 31.3 Proof Dynamics.......................... 306 REVISED 05.15.2012 VERSION 1.32 xiv CONTENTS 31.4 Law of the Excluded Middle................... 308 31.5 The Double-Negation Translation................ 310 31.6 Notes................................ 311 XII Symbols 313 32 Symbols 315 32.1 Symbol Declaration........................ 316 32.1.1 Scoped Dynamics..................... 317 32.1.2 Scope-Free Dynamics.................. 318 32.2 Symbolic References....................... 319 32.2.1 Statics........................... 319 32.2.2 Dynamics......................... 320 32.2.3 Safety............................ 320 32.3 Notes................................ 321 33 Fluid Binding 323 33.1 Statics................................ 323 33.2 Dynamics............................. 324 33.3 Type Safety............................. 325 33.4 Some Subtleties.......................... 326 33.5 Fluid References.......................... 328 33.6 Notes................................ 330 34 Dynamic Classification 331 34.1 Dynamic Classes......................... 332 34.1.1 Statics........................... 332 34.1.2 Dynamics......................... 333 34.1.3 Safety............................ 334 34.2 Class References.......................... 334 34.3 Definability of Dynamic Classes................. 335 34.4 Classifying Secrets........................ 336 34.5 Notes................................ 337 XIII State 339 35 Modernized Algol 341 35.1 Basic Commands......................... 341 35.1.1 Statics........................... 342 VERSION 1.32 REVISED 05.15.2012 CONTENTS xv 35.1.2 Dynamics......................... 343 35.1.3 Safety............................ 345 35.2 Some Programming Idioms................... 346 35.3 Typed Commands and Typed Assignables........... 348 35.4 Notes................................ 351 36 Assignable References 353 36.1 Capabilities............................ 354 36.2 Scoped Assignables........................ 355 36.3 Free Assignables.......................... 357 36.4 Safety for Free Assignables.................... 360 36.5 Benign Effects........................... 362 36.6 Notes................................ 364 XIV Laziness 365 37 Lazy Evaluation 367 37.1 By-Need Dynamics........................ 368 37.2 Safety................................ 372 37.3 Lazy Data Structures....................... 374 37.4 Suspensions............................ 375 37.5 Notes................................ 377 38 Polarization 379 38.1 Positive and Negative Types................... 380 38.2 Focusing.............................. 381 38.3 Statics................................ 382 38.4 Dynamics............................. 384 38.5 Safety................................ 385 38.6 Notes................................ 386 XV Parallelism 387 39 Nested Parallelism 389 39.1 Binary Fork-Join.......................... 390 39.2 Cost Dynamics.......................... 393 39.3 Multiple Fork-Join........................ 396 39.4 Provably Efficient Implementations............... 398 39.5 Notes................................ 402 REVISED 05.15.2012 VERSION 1.32 xvi CONTENTS 40 Futures and Speculations 403 40.1 Futures............................... 404 40.1.1 Statics........................... 404 40.1.2 Sequential Dynamics................... 404 40.2 Speculations............................ 405 40.2.1 Statics........................... 405 40.2.2 Sequential Dynamics................... 405 40.3 Parallel Dynamics......................... 406 40.4 Applications of Futures...................... 408 40.5 Notes................................ 411 XVI Concurrency 413 41 Process Calculus 415 41.1 Actions and Events........................ 415 41.2 Interaction............................. 417 41.3 Replication............................. 419 41.4 Allocating Channels....................... 421 41.5 Communication.......................... 424 41.6 Channel Passing.......................... 427 41.7 Universality............................ 430 41.8 Notes................................ 432 42 Concurrent Algol 435 42.1 Concurrent Algol......................... 436 42.2 Broadcast Communication.................... 438 42.3 Selective Communication.................... 441 42.4 Free Assignables as Processes.................. 444 42.5 Notes................................ 446 43 Distributed Algol 447 43.1 Statics................................ 448 43.2 Dynamics............................. 450 43.3 Safety................................ 451 43.4 Situated Types........................... 452 43.5 Notes................................ 456 VERSION 1.32 REVISED 05.15.2012 CONTENTS xvii XVII Modularity 457 44 Components and Linking 459 44.1 Simple Units and Linking.................... 460 44.2 Initialization and Effects..................... 461 44.3 Notes................................ 463 45 Type Abstractions and Type Classes 465 45.1 Type Abstraction......................... 467 45.2 Type Classes............................ 468 45.3 A Module Language....................... 472 45.4 First- and Second-Class...................... 476 45.5 Notes................................ 478 46 Hierarchy and Parameterization 481 46.1 Hierarchy............................. 481 46.2 Parameterizaton.......................... 485 46.3 Extending Modules with Hierarchies and Parameterization. 488 46.4 Applicative Functors....................... 491 46.5 Notes................................ 493 XVIII Equational Reasoning 495 47 Equational Reasoning for T 497 47.1 Observational Equivalence.................... 498 47.2 Logical Equivalence........................ 502 47.3 Logical and Observational Equivalence Coincide....... 503 47.4 Some Laws of Equality...................... 506 47.4.1 General Laws....................... 506 47.4.2 Equality Laws....................... 507 47.4.3 Induction Law...................... 507 47.5 Notes................................ 508 48 Equational Reasoning for PCF 509 48.1 Observational Equivalence.................... 509 48.2 Logical Equivalence........................ 510 48.3 Logical and Observational Equivalence Coincide....... 511 48.4 Compactness............................ 514 48.5 Co-Natural Numbers....................... 517 48.6 Notes................................ 519 REVISED 05.15.2012 VERSION 1.32 xviii CONTENTS 49 Parametricity 521 49.1 Overview.............................. 521 49.2 Observational Equivalence.................... 522 49.3 Logical Equivalence........................ 524 49.4 Parametricity Properties..................... 530 49.5 Representation Independence, Revisited............ 533 49.6 Notes................................ 534 50 Process Equivalence 537 50.1 Process Calculus.......................... 537 50.2 Strong Equivalence........................ 540 50.3 Weak Equivalence......................... 543 50.4 Notes................................ 545 XIX Appendices 547 A Finite Sets and Finite Functions 549 VERSION 1.32 REVISED 05.15.2012 Part I Judgments and Rules Chapter 1 Syntactic Objects Programming languages are languages, a means of expressing computa- tions in a form comprehensible to both people and machines. The syntax of a language specifies the means by which various sorts of phrases (expres- sions, commands, declarations, and so forth) may be combined to form programs. But what sort of thing are these phrases? What is a program made of? The informal concept of syntax may be seen to involve several distinct concepts. The surface, or concrete, syntax is concerned with how phrases are entered and displayed on a computer. The surface syntax is usually thought of as given by strings of characters from some alphabet (say, ASCII or Unicode). The structural, or abstract, syntax is concerned with the struc- ture of phrases, specifically how they are composed from other phrases. At this level a phrase is a tree, called an abstract syntax tree, whose nodes are operators that combine several phrases to form another phrase. The binding structure of syntax is concerned with the introduction and use of identifiers: how they are declared, and how declared identifiers are to be used. At this level phrases are abstract binding trees, which enrich abstract syntax trees with the concepts of binding and scope. We will not concern ourselves in this book with matters of concrete syntax, but will instead work at the level of abstract syntax. To prepare the ground for the rest of the book, we begin in this chapter by definin- ing abstract syntax trees and abstract binding trees and some functions and relations associated with them. The definitions are a bit technical, but are absolutely fundamental to what follows. It is probably best to skim this chapter on first reading, returning to it only as the need arises. 4 1.1 Abstract Syntax Trees 1.1 Abstract Syntax Trees An abstract syntax tree, or ast for short, is an ordered tree whose leaves are variables, and whose interior nodes are operators whose arguments are its children. Ast’s are classified into a variety of sorts corresponding to differ- ent forms of syntax. A variable stands for an unspecified, or generic, piece of syntax of a specified sort. Ast’s may be combined by an operator, which has both a sort and an arity, a finite sequence of sorts specifying the num- ber and sorts of its arguments. An operator of sort s and arity s1,..., sn combines n ≥ 0 ast’s of sort s1,..., sn, respectively, into a compound ast of sort s. As a matter of terminology, a nullary operator is one that takes no arguments, a unary operator takes one, a binary operator two, and so forth. The concept of a variable is central, and therefore deserves special em- phasis. As in mathematics a variable is an unknown object drawn from some domain, its range of signficance. In school mathematics the (often implied) range of significance is the set of real numbers. Here variables range over ast’s of a specified sort. Being an unknown, the meaning of a variable is given by substitution, the process of “plugging in” an object from the do- main for the variable in a formula. So, in school, we might plug in π for x in a polynomial, and calculate the result. Here we would plug in an ast of the appropriate sort for a variable in an ast to obtain another ast. The process of substitution is easily understood for ast’s, because it amounts to a “physical” replacement of the variable by an ast within another ast. We will shortly consider a generalization of the concept of ast for which the substitution process is somewhat more complex, but the essential idea is the same, and bears repeating: a variable is given meaning by substitution. For example, consider a simple language of expressions built from num- bers, addition, and multiplication. The abstract syntax of such a language would consist of a single sort, Exp, and an infinite collection of operators that generate the forms of expression: num[n] is a nullary operator of sort Exp whenever n ∈ N; plus and times are binary operators of sort Exp whose arguments are both of sort Exp. The expression 2 + (3 × x), which involves a variable, x, would be represented by the ast plus(num[2]; times(num[3]; x)) of sort Exp, under the assumption that x is also of this sort. Because, say, num[4], is an ast of sort Exp, we may plug it in for x in the above ast to obtain the ast plus(num[2]; times(num[3]; num[4])), VERSION 1.32 REVISED 05.15.2012 1.1 Abstract Syntax Trees 5 which is written informally as 2 + (3 × 4). We may, of course, plug in more complex ast’s of sort Exp for x to obtain other ast’s as result. The tree structure of ast’s supports a very useful principle of reasoning, called structural induction. Suppose that we wish to prove that some prop- erty, P(a), holds of all ast’s, a, of a given sort. To show this it is enough to consider all the ways in which a may be generated, and show that the property holds in each case, under the assumption that it holds for each of its constituent ast’s (if any). So, in the case of the sort Exp just described, we must show 1. The property holds for any variable, x, of sort Exp:P(x). 2. The property holds for any number, num[n]: for every n ∈ N,P(num[n]). 3. Assuming that the property holds for a1 and a2, show that it holds for plus(a1; a2) and times(a1; a2): if P(a1) and P(a2), then P(plus(a1; a2)) and P(times(a1; a2)). Because these cases exhaust all possibilities for the formation of a, we are assured that P(a) holds for any ast a of sort Exp. For the sake of precision, and to prepare the ground for further develop- ments, we will now give precise definitions of the foregoing concepts. Let S be a finite set of sorts. Let {Os }s∈S be a sort-indexed family of operators, o, of sort s with arity ar(o) = (s1,..., sn). Let {Xs }s∈S be a sort-indexed family of variables, x, of each sort s. The family A[X] = {A[X]s }s∈S of ast’s of sort s is defined as follows: 1. A variable of sort s is an ast of sort s: if x ∈ Xs, then x ∈ A[X]s. 2. Operators combine ast’s: if o is an operator of sort s such that ar(o) = (s1,..., sn), and if a1 ∈ A[X]s1 ,..., an ∈ A[X]sn , then o(a1;...;an) ∈ A[X]s. It follows from this definition that the principle of structural induction may be used to prove that some property, P, holds of every ast. To show P(a) holds for every a ∈ A[X], it is enough to show: 1. If x ∈ Xs, then Ps(x). 2. If o ∈ Os and ar(o) = (s1,..., sn), then if Ps1 (a1) and . . . and Psn (an), then Ps(o(a1;...;an)). REVISED 05.15.2012 VERSION 1.32 6 1.1 Abstract Syntax Trees For example, it is easy to prove by structural induction that if X ⊆ Y, then A[X] ⊆ A[Y]. If X is a sort-indexed family of variables, we write X, x, where x is a variable of sort s such that x /∈ Xs, to stand for the family of sets Y such that Ys = Xs ∪ { x } and Ys0 = Xs0 for all s0 6= s. The family X, x, where x is a variable of sort s, is said to be the family obtained by adjoining the variable x to the family X. Variables are given meaning by substitution. If x is a variable of sort s, a ∈ A[X, x]s0 , and b ∈ A[X]s, then [b/x]a ∈ A[X]s0 is defined to be the result of substituting b for every occurrence of x in a. The ast a is called the target, and x is called the subject, of the substitution. Substitution is defined by the following equations: 1. [b/x]x = b and [b/x]y = y if x 6= y. 2. [b/x]o(a1;...;an) = o([b/x]a1;...;[b/x]an). For example, we may check that [num[2]/x]plus(x; num[3]) = plus(num[2]; num[3]). We may prove by structural induction that substitution on ast’s is well- defined. Theorem 1.1. If a ∈ A[X, x], then for every b ∈ A[X] there exists a unique c ∈ A[X] such that [b/x]a = c Proof. By structural induction on a. If a = x, then c = b by definition, otherwise if a = y 6= x, then c = y, also by definition. Otherwise, a = o(a1,..., an), and we have by induction unique c1,..., cn such that [b/x]a1 = c1 and . . . [b/x]an = cn, and so c is c = o(c1;...;cn), by definition of sub- stitution. In most cases it is possible to enumerate all of the operators that gen- erate the ast’s of a sort up front, as we have done in the foregoing exam- ples. However, in some situations this is not possible—certain operators are available only within certain contexts. In such cases we cannot fix the collection of operators, O, in advance, but rather must allow it to be exten- sible. This is achieved by considering families of operators that are indexed by symbolic parameters that serve as “names” for the instances. For exam- ple, in Chapter 34 we will consider a family of nullary operators, cls[u], where u is a symbolic parameter drawn from the set of active parameters. It VERSION 1.32 REVISED 05.15.2012 1.2 Abstract Binding Trees 7 is essential that distinct parameters determine distinct operators: if u and v are active parameters, and u 6= v, then cls[u] and cls[v] are different operators. Extensibility is achieved by introducing new active parameters. So, if u is not active, then cls[u] makes no sense, but if u becomes active, then cls[u] is a nullary operator. Parameters are easily confused with variables, but they are fundamen- tally different concepts. As we remarked earlier, a variable stands for an unknown ast of its sort, but a parameter does not stand for anything. It is a purely symbolic identifier whose only significance is whether it is the same or different as another parameter. Whereas variables are given meaning by substitution, it is not possible, or sensible, to substitute for a parameter. As a consequence, disequality of parameters is preserved by substitution, whereas disequality of variables is not (because the same ast may be sub- stituted for two distinct variables). To account for the set of active parameters, we will write A[U;X] for the set of ast’s with variables drawn from X and with parameters drawn from U. Certain operators, such as cls[u], are parameterized by parameters, u, of a given sort. The parameters are distinguished from the arguments by the square brackets around them. Instances of such operators are permitted only for parameters drawn from the active set, U. So, for example, if u ∈ U, then cls[u] is a nullary operator, but if u /∈ U, then cls[u] is not a valid operator. In the next section we will introduce the means of extending U to make operators available within that context. 1.2 Abstract Binding Trees Abstract binding trees, or abt’s, enrich ast’s with the means to introduce new variables and parameters, called a binding, with a specified range of sig- nificance, called its scope. The scope of a binding is an abt within which the bound identifier may be used, either as a placeholder (in the case of a variable declaration) or as the index of some operator (in the case of a pa- rameter declaration). Thus the set of active identifiers may be larger within a subtree of an abt than it is within the surrounding tree. Moreover, dif- ferent subtrees may introduce identifiers with disjoint scopes. The crucial principle is that any use of an identifier should be understood as a refer- ence, or abstract pointer, to its binding. One consequence is that the choice of identifiers is immaterial, so long as we can always associate a unique binding with each use of an identifier. As a motivating example, consider the expression let x be a1 in a2, which REVISED 05.15.2012 VERSION 1.32 8 1.2 Abstract Binding Trees introduces a variable, x, for use within the expression a2 to stand for the ex- pression a1. The variable x is bound by the let expression for use within a2; any use of x within a1 refers to a different variable that happens to have the same name. For example, in the expression let x be 7 in x + x occurrences of x in the addition refer to the variable introduced by the let. On the other hand in the expression let x be x ∗ x in x + x, occurrences of x within the multiplication refer to a different variable than those occurring within the addition. The latter occurrences refer to the binding introduced by the let, whereas the former refer to some outer binding not displayed here. The names of bound variables are immaterial insofar as they determine the same binding. So, for example, the expression let x be x ∗ x in x + x could just as well have been written let y be x ∗ x in y + y without chang- ing its meaning. In the former case the variable x is bound within the ad- dition, and in the latter it is the variable y, but the “pointer structure” re- mains the same. On the other hand the expression let x be y ∗ y in x + x has a different meaning to these two expressions, because now the vari- able y within the multiplication refers to a different surrounding variable. Renaming of bound variables is constrained to the extent that it must not alter the reference structure of the expression. For example, the expression let x be 2 in let y be 3 in x + x has a different meaning than the expression let y be 2 in let y be 3 in y + y, because the y in the expression y + y in the second case refers to the inner declaration, not the outer one as before. The concept of an ast may be enriched to account for binding and scope of a variable. These enriched ast’s are called abstract binding trees, or abt’s for short. Abt’s generalize ast’s by allowing an operator to bind any finite number (possibly zero) of variables in each argument position. An argu- ment to an operator is called an abstractor, and has the form x1,..., xk.a. The sequence of variables x1,..., xk are bound within the abt a. (When k is zero, we elide the distinction between .a and a itself.) Written in the form of an abt, the expression let x be a1 in a2 has the form let(a1; x.a2), which more clearly specifies that the variable x is bound within a2, and not within a1. We often write ~x to stand for a finite sequence x1,..., xn of distinct vari- ables, and write ~x.a to mean x1,..., xn.a. To account for binding, the arity of an operator is generalized to con- sist of a finite sequence of valences. The length of the sequence determines the number of arguments, and each valence determines the sort of the ar- gument and the number and sorts of the variables that are bound within it. A valence of the form (s1,..., sk)s specifies an argument of sort s that binds k variables of sorts s1,..., sk within it. We often write ~s for a finite VERSION 1.32 REVISED 05.15.2012 1.2 Abstract Binding Trees 9 sequence s1,..., sn of sorts, and we say that ~x is of sort~s to mean that the two sequences have the same length and that each xi is of sort si. Thus, for example, the arity of the operator let is (Exp,(Exp)Exp), which indicates that it takes two arguments described as follows: 1. The first argument is of sort Exp and binds no variables. 2. The second argument is of sort Exp and binds one variable of sort Exp. The definition expression let x be 2 + 2 in x × x is represented by the abt let(plus(num[2]; num[2]); x.times(x; x)). Let O be a sort-indexed family of operators, o, with arities, ar(o). For a given sort-indexed family, X, of variables, the sort-indexed family of abt’s, B[X], is defined similarly to A[X], except that the set of active variables changes for each argument according to which variables are bound within it. A first cut at the definition is as follows: 1. If x ∈ Xs, then x ∈ B[X]s. 2. If ar(o) = ((~s1)s1,...,(~sn)sn), and if, for each 1 ≤ i ≤ n, ~xi is of sort~si and ai ∈ B[X,~xi]si , then o(~x1.a1;...;~xn.an) ∈ B[X]s. The bound variables are adjoined to the set of active variables within each argument, with the sort of each variable determined by the valence of the operator. This definition is almost correct, but fails to properly account for the behavior of bound variables. An abt of the form let(a1; x.let(a2; x.a3)) is ill-formed according to this definition, because the first binding adjoins x to X, which implies that the second cannot also adjoin x to X, x without causing confusion. The solution is to ensure that each of the arguments is well-formed regardless of the choice of bound variable names. This is achieved by altering the second clause of the definition using renaming as follows:1 If ar(o) = ((~x1)s1,...,(~xn)sn), and if, for each 1 ≤ i ≤ n and for each renaming πi : ~xi ↔ ~x0 i, where ~x0 i /∈ X , we have πi · ai ∈ B[X,~x0 i], then o(~x1.a1;...;~xn.an) ∈ B[X]s. 1The action of a renaming extends to abt’s in the obvious way by replacing every occur- rence of x by π(x), including any occurrences in the variable list of an abstractor as well as within its body. REVISED 05.15.2012 VERSION 1.32 10 1.2 Abstract Binding Trees The renaming ensures that when we encounter nested binders we avoid collisions. This is called the freshness condition on binders because it ensures that all bound variables are “fresh” relative to the surrounding context. The principle of structural induction extends to abt’s, and is called struc- tural induction modulo renaming. It states that to show that P(a)[X] holds for every a ∈ B[X], it is enough to show the following: 1. if x ∈ Xs, then P[X]s(x). 2. For every o of sort s and arity ((~s1)s1,...,(~sn)sn), and if for each 1 ≤ i ≤ n, we have P[X,~x0 i]si (πi · ai) for every renaming πi : ~xi ↔ ~x0 i, then P[X]s(o(~x1.a1;...;~xn.an)). The renaming in the second condition ensures that the inductive hypothe- sis holds for all fresh choices of bound variable names, and not just the ones actually given in the abt. As an example let us define the judgment x ∈ a, where a ∈ B[X, x], to mean that x occurs free in a. Informally, this means that x is bound some- where outside of a, rather than within a itself. If x is bound within a, then those occurrences of x are different from those occurring outside the bind- ing. The following definition ensures that this is the case: 1. x ∈ x. 2. x ∈ o(~x1.a1;...;~xn.an) if there exists 1 ≤ i ≤ n such that for every fresh renaming π : ~xi ↔ ~zi we have x ∈ π · ai. The first condition states that x is free in x, but not free in y for any vari- able y other than x. The second condition states that if x is free in some argument, independently of the choice of bound variable names in that ar- gument, then it is free in the overall abt. This implies, in particular, that x is not free in let(zero; x.x). The relation a =α b of α-equivalence (so-called for historical reasons), is defined to mean that a and b are identical up to the choice of bound variable names. This relation is defined to be the strongest congruence containing the following two conditions: 1. x =α x. 2. o(~x1.a1;...;~xn.an) =α o(~x0 1.a0 1;...;~x0 n.a0 n) if for every 1 ≤ i ≤ n, πi · ai =α π0 i · a0 i for all fresh renamings πi : ~xi ↔ ~zi and π0 i : ~x0 i ↔ ~zi. VERSION 1.32 REVISED 05.15.2012 1.2 Abstract Binding Trees 11 The idea is that we rename ~xi and ~x0 i consistently, avoiding confusion, and check that ai and a0 i are α-equivalent. If a =α b, then a and b are said to be α-variants of each other. Some care is required in the definition of substitution of an abt b of sort s for free occurrences of a variable x of sort s in some abt a of some sort, writ- ten [b/x]a. Substitution is partially defined by the following conditions: 1. [b/x]x = b, and [b/x]y = y if x 6= y. 2. [b/x]o(~x1.a1;...;~xn.an) = o(~x1.a0 1;...;~xn.a0 n), where, for each 1 ≤ i ≤ n, we require that ~xi 6∈ b, and we set a0 i = [b/x]ai if x /∈ ~xi, and a0 i = ai otherwise. If x is bound in some argument to an operator, then substitution does not descend into its scope, for to do so would be to confuse two distinct vari- ables. For this reason we must take care to define a0 i in the second equation according to whether or not x ∈ ~xi. The requirement that ~xi 6∈ b in the second equation is called capture avoidance. If some xi,j occurred free in b, then the result of the substitution [b/x]ai would in general contain xi,j free as well, but then forming ~xi.[b/x]ai would incur capture by changing the referent of xi,j to be the jth bound variable of the ith argument. In such cases substitution is undefined because we cannot replace x by b in ai with- out incurring capture. One way around this is to alter the definition of substitution so that the bound variables in the result are chosen fresh by substitution. By the prin- ciple of structural induction we know inductively that, for any renaming πi : ~xi ↔ ~x0 i with ~x0 i fresh, the substitution [b/x](πi · ai) is well-defined. Hence we may define [b/x]o(~x1.a1;...;~xn.an) = o(~x0 1.[b/x](π1 · a1);...;~x0 n.[b/x](πn · an)) for some particular choice of fresh bound variable names (any choice will do). There is no longer any need to take care that x /∈ ~xi in each argument, because the freshness condition on binders ensures that this cannot occur, the variable x already being active. Noting that o(~x1.a1;...;~xn.an) =α o(~x0 1.π1 · a1;...;~x0 n.πn · an), another way to avoid undefined substitutions is to first choose an α-variant of the target of the substitution whose binders avoid any free variables in the substituting abt, and then perform substitution without fear of incur- ring capture. In other words substitution is totally defined on α-equivalence classes of abt’s. REVISED 05.15.2012 VERSION 1.32 12 1.3 Notes To avoid all the bureaucracy of binding, we adopt the following identi- fication convention throughout this book: Abstract binding trees are always to be identified up to α-equivalence. That is, we implicitly work with α-equivalence classes of abt’s, rather than abt’s themselves. We tacitly assert that all operations and relations on abt’s respect α-equivalence, so that they are properly defined on α-equivalence classes of abt’s. Whenever we examine an abt, we are choosing a repre- sentative of its α-equivalence class, and we have no control over how the bound variable names are chosen. On the other hand experience shows that any operation or property of interest respects α-equivalence, so there is no obstacle to achieving it. Indeed, we might say that a property or operation is legitimate exactly insofar as it respects α-equivalence! Parameters, as well as variables, may be bound within an argument of an operator. Such binders introduce a “new,” or “fresh,” parameter within the scope of the binder wherein it may be used to form further abt’s. To allow for parameter declaration, the valence of an argument is generalized to indicate the sorts of the parameters bound within it, as well as the sorts of the variables, by writing (~s1; ~s2)s, where ~s1 specifies the sorts of the pa- rameters and ~s2 specifies the sorts of the variables. The sort-indexed family B[U;X] is the set of abt’s determined by a fixed set of operators using the parameters, U, and the variables, X. We rely on naming conventions to distinguish parameters from variables, reserving u and v for parameters, and x and y for variables. 1.3 Notes The concept of abstract syntax has its orgins in the pioneering work of Church, Turing, and G¨odel, who first considered the possibility of writing programs that act on representations of programs. Originally programs were represented by natural numbers, using encodings, now called G¨odel- numberings, based on the prime factorization theorem. Any standard text on mathematical logic, such as Kleene(1952), contains a thorough account of such representations. The Lisp language (McCarthy, 1965; Allen, 1978) introduced a much more practical and direct representation of syntax as symbolic expressions. These ideas were developed further in the language ML (Gordon et al., 1979), which featured a type system capable of express- ing abstract syntax trees. The AUTOMATH project (Nederpelt et al., 1994) introduced the idea of using Church’s λ notation (Church, 1941) to account VERSION 1.32 REVISED 05.15.2012 1.3 Notes 13 for the binding and scope of variables. These ideas were developed further in LF (Harper et al., 1993). REVISED 05.15.2012 VERSION 1.32 14 1.3 Notes VERSION 1.32 REVISED 05.15.2012 Chapter 2 Inductive Definitions Inductive definitions are an indispensable tool in the study of program- ming languages. In this chapter we will develop the basic framework of inductive definitions, and give some examples of their use. An inductive definition consists of a set of rules for deriving judgments, or assertions, of a variety of forms. Judgments are statements about one or more syntactic objects of a specified sort. The rules specify necessary and sufficient condi- tions for the validity of a judgment, and hence fully determine its meaning. 2.1 Judgments We start with the notion of a judgment, or assertion, about a syntactic object. We shall make use of many forms of judgment, including examples such as these: n nat n is a natural number n = n1 + n2 n is the sum of n1 and n2 τ type τ is a type e : τ expression e has type τ e ⇓ v expression e has value v A judgment states that one or more syntactic objects have a property or stand in some relation to one another. The property or relation itself is called a judgment form, and the judgment that an object or objects have that property or stand in that relation is said to be an instance of that judgment form. A judgment form is also called a predicate, and the objects constituting an instance are its subjects. We write a J for the judgment asserting that J holds of a. When it is not important to stress the subject of the judgment, 16 2.2 Inference Rules we write J to stand for an unspecified judgment. For particular judgment forms, we freely use prefix, infix, or mixfix notation, as illustrated by the above examples, in order to enhance readability. 2.2 Inference Rules An inductive definition of a judgment form consists of a collection of rules of the form J1 ...Jk J(2.1) in which J and J1,...,Jk are all judgments of the form being defined. The judgments above the horizontal line are called the premises of the rule, and the judgment below the line is called its conclusion. If a rule has no premises (that is, when k is zero), the rule is called an axiom; otherwise it is called a proper rule. An inference rule may be read as stating that the premises are suffi- cient for the conclusion: to show J, it is enough to show J1,...,Jk. When k is zero, a rule states that its conclusion holds unconditionally. Bear in mind that there may be, in general, many rules with the same conclusion, each specifying sufficient conditions for the conclusion. Consequently, if the conclusion of a rule holds, then it is not necessary that the premises hold, for it might have been derived by another rule. For example, the following rules constitute an inductive definition of the judgment a nat: zero nat (2.2a) a nat succ(a) nat (2.2b) These rules specify that a nat holds whenever either a is zero, or a is succ(b) where b nat for some b. Taking these rules to be exhaustive, it follows that a nat iff a is a natural number. Similarly, the following rules constitute an inductive definition of the judgment a tree: empty tree (2.3a) a1 tree a2 tree node(a1; a2) tree (2.3b) These rules specify that a tree holds if either a is empty, or a is node(a1; a2), where a1 tree and a2 tree. Taking these to be exhaustive, these rules state VERSION 1.32 REVISED 05.15.2012 2.3 Derivations 17 that a is a binary tree, which is to say it is either empty, or a node consisting of two children, each of which is also a binary tree. The judgment a = b nat defining equality of a nat and b nat is induc- tively defined by the following rules: zero = zero nat (2.4a) a = b nat succ(a) = succ(b) nat (2.4b) In each of the preceding examples we have made use of a notational convention for specifying an infinite family of rules by a finite number of patterns, or rule schemes. For example, Rule (2.2b) is a rule scheme that determines one rule, called an instance of the rule scheme, for each choice of object a in the rule. We will rely on context to determine whether a rule is stated for a specific object, a, or is instead intended as a rule scheme specifying a rule for each choice of objects in the rule. A collection of rules is considered to define the strongest judgment that is closed under, or respects, those rules. To be closed under the rules sim- ply means that the rules are sufficient to show the validity of a judgment: J holds if there is a way to obtain it using the given rules. To be the strongest judgment closed under the rules means that the rules are also necessary:J holds only if there is a way to obtain it by applying the rules. The suffi- ciency of the rules means that we may show that J holds by deriving it by composing rules. Their necessity means that we may reason about it using rule induction. 2.3 Derivations To show that an inductively defined judgment holds, it is enough to exhibit a derivation of it. A derivation of a judgment is a finite composition of rules, starting with axioms and ending with that judgment. It may be thought of as a tree in which each node is a rule whose children are derivations of its premises. We sometimes say that a derivation of J is evidence for the validity of an inductively defined judgment J. We usually depict derivations as trees with the conclusion at the bot- tom, and with the children of a node corresponding to a rule appearing above it as evidence for the premises of that rule. Thus, if J1 ...Jk J REVISED 05.15.2012 VERSION 1.32 18 2.3 Derivations is an inference rule and ∇1,..., ∇k are derivations of its premises, then ∇1 ... ∇k J is a derivation of its conclusion. In particular, if k = 0, then the node has no children. For example, this is a derivation of succ(succ(succ(zero))) nat: zero nat succ(zero) nat succ(succ(zero)) nat succ(succ(succ(zero))) nat . (2.5) Similarly, here is a derivation of node(node(empty; empty); empty) tree: empty tree empty tree node(empty; empty) tree empty tree node(node(empty; empty); empty) tree . (2.6) To show that an inductively defined judgment is derivable we need only find a derivation for it. There are two main methods for finding derivations, called forward chaining, or bottom-up construction, and backward chaining, or top-down construction. Forward chaining starts with the axioms and works forward towards the desired conclusion, whereas backward chaining starts with the desired conclusion and works backwards towards the axioms. More precisely, forward chaining search maintains a set of derivable judgments, and continually extends this set by adding to it the conclusion of any rule all of whose premises are in that set. Initially, the set is empty; the process terminates when the desired judgment occurs in the set. As- suming that all rules are considered at every stage, forward chaining will eventually find a derivation of any derivable judgment, but it is impossible (in general) to decide algorithmically when to stop extending the set and conclude that the desired judgment is not derivable. We may go on and on adding more judgments to the derivable set without ever achieving the intended goal. It is a matter of understanding the global properties of the rules to determine that a given judgment is not derivable. Forward chaining is undirected in the sense that it does not take account of the end goal when deciding how to proceed at each step. In contrast, VERSION 1.32 REVISED 05.15.2012 2.4 Rule Induction 19 backward chaining is goal-directed. Backward chaining search maintains a queue of current goals, judgments whose derivations are to be sought. Initially, this set consists solely of the judgment we wish to derive. At each stage, we remove a judgment from the queue, and consider all rules whose conclusion is that judgment. For each such rule, we add the premises of that rule to the back of the queue, and continue. If there is more than one such rule, this process must be repeated, with the same starting queue, for each candidate rule. The process terminates whenever the queue is empty, all goals having been achieved; any pending consideration of candidate rules along the way may be discarded. As with forward chaining, back- ward chaining will eventually find a derivation of any derivable judgment, but there is, in general, no algorithmic method for determining in general whether the current goal is derivable. If it is not, we may futilely add more and more judgments to the goal set, never reaching a point at which all goals have been satisfied. 2.4 Rule Induction Because an inductive definition specifies the strongest judgment closed un- der a collection of rules, we may reason about them by rule induction. The principle of rule induction states that to show that a property P holds of a judgment J whenever J is derivable, it is enough to show that P is closed un- der, or respects, the rules defining J. Writing P(J) to mean that the property P holds of the judgment J, we say that P respects the rule J1 ...Jk J if P(J) holds whenever P(J1),...,P(Jk). The assumptions P(J1),...,P(Jk) are called the inductive hypotheses, and P(J) is called the inductive conclusion of the inference. The principle of rule induction is simply the expression of the definition of an inductively defined judgment form as the strongest judgment form closed under the rules comprising the definition. This means that the judg- ment form defined by a set of rules is both (a) closed under those rules, and (b) sufficient for any other property also closed under those rules. The for- mer means that a derivation is evidence for the validity of a judgment; the latter means that we may reason about an inductively defined judgment form by rule induction. REVISED 05.15.2012 VERSION 1.32 20 2.4 Rule Induction When specialized to Rules (2.2), the principle of rule induction states that to show P(a nat) whenever a nat, it is enough to show: 1. P(zero nat). 2. for every a, if a nat and P(a nat), then (succ(a) nat and) P(succ(a) nat). This is just the familiar principle of mathematical induction arising as a spe- cial case of rule induction. Similarly, rule induction for Rules (2.3) states that to show P(a tree) whenever a tree, it is enough to show 1. P(empty tree). 2. for every a1 and a2, if a1 tree and P(a1 tree), and if a2 tree and P(a2 tree), then (node(a1; a2) tree and) P(node(a1; a2) tree). This is called the principle of tree induction, and is once again an instance of rule induction. We may also show by rule induction that the predecessor of a natural number is also a natural number. Although this may seem self-evident, the point of the example is to show how to derive this from first principles. Lemma 2.1. If succ(a) nat, then a nat. Proof. It suffices to show that the property, P(a nat) stating that a nat and that a = succ(b) implies b nat is closed under Rules (2.2). Rule (2.2a) Clearly zero nat, and the second condition holds vacuously, because zero is not of the form succ(−). Rule (2.2b) Inductively we know that a nat and that if a is of the form succ(b), then b nat. We are to show that succ(a) nat, which is imme- diate, and that if succ(a) is of the form succ(b), then b nat, and we have b nat by the inductive hypothesis. This completes the proof. Using rule induction we may show that equality, as defined by Rules (2.4) is reflexive. Lemma 2.2. If a nat, then a = a nat. Proof. By rule induction on Rules (2.2): VERSION 1.32 REVISED 05.15.2012 2.5 Iterated and Simultaneous Inductive Definitions 21 Rule (2.2a) Applying Rule (2.4a) we obtain zero = zero nat. Rule (2.2b) Assume that a = a nat. It follows that succ(a) = succ(a) nat by an application of Rule (2.4b). Similarly, we may show that the successor operation is injective. Lemma 2.3. If succ(a1) = succ(a2) nat, then a1 = a2 nat. Proof. Similar to the proof of Lemma 2.1. 2.5 Iterated and Simultaneous Inductive Definitions Inductive definitions are often iterated, meaning that one inductive defi- nition builds on top of another. In an iterated inductive definition the premises of a rule J1 ...Jk J may be instances of either a previously defined judgment form, or the judg- ment form being defined. For example, the following rules define the judg- ment a list stating that a is a list of natural numbers. nil list (2.7a) a nat b list cons(a; b) list (2.7b) The first premise of Rule (2.7b) is an instance of the judgment form a nat, which was defined previously, whereas the premise b list is an instance of the judgment form being defined by these rules. Frequently two or more judgments are defined at once by a simultane- ous inductive definition. A simultaneous inductive definition consists of a set of rules for deriving instances of several different judgment forms, any of which may appear as the premise of any rule. Because the rules defining each judgment form may involve any of the others, none of the judgment forms may be taken to be defined prior to the others. Instead we must un- derstand that all of the judgment forms are being defined at once by the entire collection of rules. The judgment forms defined by these rules are, as REVISED 05.15.2012 VERSION 1.32 22 2.6 Defining Functions by Rules before, the strongest judgment forms that are closed under the rules. There- fore the principle of proof by rule induction continues to apply, albeit in a form that requires us to prove a property of each of the defined judgment forms simultaneously. For example, consider the following rules, which constitute a simulta- neous inductive definition of the judgments a even, stating that a is an even natural number, and a odd, stating that a is an odd natural number: zero even (2.8a) a odd succ(a) even (2.8b) a even succ(a) odd (2.8c) The principle of rule induction for these rules states that to show simul- taneously that P(a even) whenever a even and P(a odd) whenever a odd, it is enough to show the following: 1. P(zero even); 2. if P(a odd), then P(succ(a) even); 3. if P(a even), then P(succ(a) odd). As a simple example, we may use simultaneous rule induction to prove that (1) if a even, then a nat, and (2) if a odd, then a nat. That is, we define the property P by (1) P(a even) iff a nat, and (2) P(a odd) iff a nat. The principle of rule induction for Rules (2.8) states that it is sufficient to show the following facts: 1. zero nat, which is derivable by Rule (2.2a). 2. If a nat, then succ(a) nat, which is derivable by Rule (2.2b). 3. If a nat, then succ(a) nat, which is also derivable by Rule (2.2b). 2.6 Defining Functions by Rules A common use of inductive definitions is to define a function by giving an inductive definition of its graph relating inputs to outputs, and then show- ing that the relation uniquely determines the outputs for given inputs. For example, we may define the addition function on natural numbers as the VERSION 1.32 REVISED 05.15.2012 2.6 Defining Functions by Rules 23 relation sum(a; b; c), with the intended meaning that c is the sum of a and b, as follows: b nat sum(zero; b; b)(2.9a) sum(a; b; c) sum(succ(a); b; succ(c))(2.9b) The rules define a ternary (three-place) relation, sum(a; b; c), among natural numbers a, b, and c. We may show that c is determined by a and b in this relation. Theorem 2.4. For every a nat and b nat, there exists a unique c nat such that sum(a; b; c). Proof. The proof decomposes into two parts: 1. (Existence) If a nat and b nat, then there exists c nat such that sum(a; b; c). 2. (Uniqueness) If sum(a; b; c), and sum(a; b; c0), then c = c0 nat. For existence, let P(a nat) be the proposition if b nat then there exists c nat such that sum(a; b; c). We prove that if a nat then P(a nat) by rule induction on Rules (2.2). We have two cases to consider: Rule (2.2a) We are to show P(zero nat). Assuming b nat and taking c to be b, we obtain sum(zero; b; c) by Rule (2.9a). Rule (2.2b) Assuming P(a nat), we are to show P(succ(a) nat). That is, we assume that if b nat then there exists c such that sum(a; b; c), and are to show that if b0 nat, then there exists c0 such that sum(succ(a); b0; c0). To this end, suppose that b0 nat. Then by induction there exists c such that sum(a; b0; c). Taking c0 = succ(c), and applying Rule (2.9b), we obtain sum(succ(a); b0; c0), as required. For uniqueness, we prove that if sum(a; b; c1), then if sum(a; b; c2), then c1 = c2 nat by rule induction based on Rules (2.9). Rule (2.9a) We have a = zero and c1 = b. By an inner induction on the same rules, we may show that if sum(zero; b; c2), then c2 is b. By Lemma 2.2 we obtain b = b nat. Rule (2.9b) We have that a = succ(a0) and c1 = succ(c0 1), where sum(a0; b; c0 1). By an inner induction on the same rules, we may show that if sum(a; b; c2), then c2 = succ(c0 2) nat where sum(a0; b; c0 2). By the outer inductive hy- pothesis c0 1 = c0 2 nat and so c1 = c2 nat. REVISED 05.15.2012 VERSION 1.32 24 2.7 Modes 2.7 Modes The statement that one or more arguments of a judgment is (perhaps uniquely) determined by its other arguments is called a mode specification for that judg- ment. For example, we have shown that every two natural numbers have a sum according to Rules (2.9). This fact may be restated as a mode spec- ification by saying that the judgment sum(a; b; c) has mode (∀, ∀, ∃). The notation arises from the form of the proposition it expresses: for all a nat and for all b nat, there exists c nat such that sum(a; b; c). If we wish to fur- ther specify that c is uniquely determined by a and b, we would say that the judgment sum(a; b; c) has mode (∀, ∀, ∃!), corresponding to the proposition for all a nat and for all b nat, there exists a unique c nat such that sum(a; b; c). If we wish only to specify that the sum is unique, if it exists, then we would say that the addition judgment has mode (∀, ∀, ∃≤1), corresponding to the proposition for all a nat and for all b nat there exists at most one c nat such that sum(a; b; c). As these examples illustrate, a given judgment may satisfy several dif- ferent mode specifications. In general the universally quantified arguments are to be thought of as the inputs of the judgment, and the existentially quantified arguments are to be thought of as its outputs. We usually try to arrange things so that the outputs come after the inputs, but it is not es- sential that we do so. For example, addition also has the mode (∀, ∃≤1, ∀), stating that the sum and the first addend uniquely determine the second addend, if there is any such addend at all. Put in other terms, this says that addition of natural numbers has a (partial) inverse, namely subtraction. We could equally well show that addition has mode (∃≤1, ∀, ∀), which is just another way of stating that addition of natural numbers has a partial inverse. Often there is an intended, or principal, mode of a given judgment, which we often foreshadow by our choice of notation. For example, when giving an inductive definition of a function, we often use equations to indi- cate the intended input and output relationships. For example, we may re-state the inductive definition of addition (given by Rules (2.9)) using equations: a nat a + zero = a nat (2.10a) a + b = c nat a + succ(b) = succ(c) nat (2.10b) When using this notation we tacitly incur the obligation to prove that the mode of the judgment is such that the object on the right-hand side of the VERSION 1.32 REVISED 05.15.2012 2.8 Notes 25 equations is determined as a function of those on the left. Having done so, we abuse notation, writing a + b for the unique c such that a + b = c nat. 2.8 Notes Aczel(1977) provides a thorough account of the theory of inductive defi- nitions. The formulation given here is strongly influenced by Martin-L¨of’s development of the logic of judgments (Martin-L¨of, 1983, 1987). REVISED 05.15.2012 VERSION 1.32 26 2.8 Notes VERSION 1.32 REVISED 05.15.2012 Chapter 3 Hypothetical and General Judgments A hypothetical judgment expresses an entailment between one or more hy- potheses and a conclusion. We will consider two notions of entailment, called derivability and admissibility. Both enjoy the same structural proper- ties, but they differ in that derivability is stable under extension with new rules, admissibility is not. A general judgment expresses the universality, or genericity, of a judgment. There are two forms of general judgment, the generic and the parametric. The generic judgment expresses generality with respect to all substitution instances for variables in a judgment. The para- metric judgment expresses generality with respect to renamings of sym- bols. 3.1 Hypothetical Judgments The hypothetical judgment codifies the rules for expressing the validity of a conclusion conditional on the validity of one or more hypotheses. There are two forms of hypothetical judgment that differ according to the sense in which the conclusion is conditional on the hypotheses. One is stable under extension with additional rules, and the other is not. 3.1.1 Derivability For a given set, R, of rules, we define the derivability judgment, written J1,...,Jk `RK, where each Ji and K are basic judgments, to mean that we may derive K from the expansion R[J1,...,Jk] of the rules R with the 28 3.1 Hypothetical Judgments additional axioms J1 ...Jk . We treat the hypotheses, or antecedents, of the judgment, J1,...,Jn as “tempo- rary axioms”, and derive the conclusion, or consequent, by composing rules in R. Thus, evidence for a hypothetical judgment consists of a derivation of the conclusion from the hypotheses using the rules in R. We use capital Greek letters, frequently Γ or ∆, to stand for a finite col- lection of basic judgments, and write R[Γ] for the expansion of R with an axiom corresponding to each judgment in Γ. The judgment Γ `RK means that K is derivable from rules R[Γ], and the judgment `RΓ means that `RJ for each J in Γ. An equivalent way of defining J1,...,Jn `RJ is to say that the rule J1 ...Jn J(3.1) is derivable from R, which means that there is a derivation of J composed of the rules in R augmented by treating J1,...,Jn as axioms. For example, consider the derivability judgment a nat `(2.2) succ(succ(a)) nat (3.2) relative to Rules (2.2). This judgment is valid for any choice of object a, as evidenced by the derivation a nat succ(a) nat succ(succ(a)) nat (3.3) which composes Rules (2.2), starting with a nat as an axiom, and ending with succ(succ(a)) nat. Equivalently, the validity of (3.2) may also be expressed by stating that the rule a nat succ(succ(a)) nat (3.4) is derivable from Rules (2.2). It follows directly from the definition of derivability that it is stable un- der extension with new rules. Theorem 3.1 (Stability). If Γ `R J, then Γ `R∪R0 J. Proof. Any derivation of J from R[Γ] is also a derivation from (R ∪ R0)[Γ], because any rule in R is also a rule in R ∪ R0. VERSION 1.32 REVISED 05.15.2012 3.1 Hypothetical Judgments 29 Derivability enjoys a number of structural properties that follow from its definition, independently of the rules, R, in question. Reflexivity Every judgment is a consequence of itself: Γ,J `RJ. Each hypothesis justifies itself as conclusion. Weakening If Γ `RJ, then Γ,K `RJ. Entailment is not influenced by unexercised options. Transitivity If Γ,K `RJ and Γ `RK, then Γ `RJ. If we replace an ax- iom by a derivation of it, the result is a derivation of its consequent without that hypothesis. Reflexivity follows directly from the meaning of derivability. Weakening follows directly from the definition of derivability. Transitivity is proved by rule induction on the first premise. 3.1.2 Admissibility Admissibility, written Γ |=RJ, is a weaker form of hypothetical judgment stating that `RΓ implies `RJ. That is, the conclusion J is derivable from rules R whenever the assumptions Γ are all derivable from rules R. In particular if any of the hypotheses are not derivable relative to R, then the judgment is vacuously true. An equivalent way to define the judgment J1,...,Jn |=RJ is to state that the rule J1 ...Jn J(3.5) is admissible relative to the rules in R. This means that given any deriva- tions of J1,...,Jn using the rules in R, we may construct a derivation of J using the rules in R. For example, the admissibility judgment succ(a) nat |=(2.2) a nat (3.6) is valid, because any derivation of succ(a) nat from Rules (2.2) must con- tain a sub-derivation of a nat from the same rules, which justifies the con- clusion. The validity of (3.6) may equivalently be expressed by stating that the rule succ(a) nat a nat (3.7) is admissible for Rules (2.2). REVISED 05.15.2012 VERSION 1.32 30 3.1 Hypothetical Judgments In contrast to derivability the admissibility judgment is not stable under extension to the rules. For example, if we enrich Rules (2.2) with the axiom succ(junk) nat (3.8) (where junk is some object for which junk nat is not derivable), then the admissibility (3.6) is invalid. This is because Rule (3.8) has no premises, and there is no composition of rules deriving junk nat. Admissibility is as sensitive to which rules are absent from an inductive definition as it is to which rules are present in it. The structural properties of derivability ensure that derivability is stronger than admissibility. Theorem 3.2. If Γ `R J, then Γ |=RJ. Proof. Repeated application of the transitivity of derivability shows that if Γ `RJ and `RΓ, then `RJ. To see that the converse fails, observe that there is no composition of rules such that succ(junk) nat `(2.2) junk nat, yet the admissibility judgment succ(junk) nat |=(2.2) junk nat holds vacuously. Evidence for admissibility may be thought of as a mathematical func- tion transforming derivations ∇1,..., ∇n of the hypotheses into a deriva- tion ∇ of the consequent. Therefore, the admissibility judgment enjoys the same structural properties as derivability, and hence is a form of hypothet- ical judgment: Reflexivity If J is derivable from the original rules, then J is derivable from the original rules: J |=RJ. Weakening If J is derivable from the original rules assuming that each of the judgments in Γ are derivable from these rules, then J must also be derivable assuming that Γ and also K are derivable from the original rules: if Γ |=RJ, then Γ,K |=RJ. VERSION 1.32 REVISED 05.15.2012 3.2 Hypothetical Inductive Definitions 31 Transitivity If Γ,K |=RJ and Γ |=RK, then Γ |=RJ. If the judgments in Γ are derivable, so is K, by assumption, and hence so are the judgments in Γ,K, and hence so is J. Theorem 3.3. The admissibility judgment Γ |=R J enjoys the structural proper- ties of entailment. Proof. Follows immediately from the definition of admissibility as stating that if the hypotheses are derivable relative to R, then so is the conclusion. If a rule, r, is admissible with respect to a rule set, R, then `R,r J is equivalent to `RJ. For if `RJ, then obviously `R,r J, by simply disre- garding r. Conversely, if `R,r J, then we may replace any use of r by its expansion in terms of the rules in R. It follows by rule induction on R, r that every derivation from the expanded set of rules, R, r, may be trans- formed into a derivation from R alone. Consequently, if we wish to show that P(J) whenever `R,r J, it is sufficient to show that P is closed under the rules R alone. That is, we need only consider the rules R in a proof by rule induction to derive P(J). 3.2 Hypothetical Inductive Definitions It is useful to enrich the concept of an inductive definition to permit rules with derivability judgments as premises and conclusions. Doing so per- mits us to introduce local hypotheses that apply only in the derivation of a particular premise, and also allows us to constrain inferences based on the global hypotheses in effect at the point where the rule is applied. A hypothetical inductive definition consists of a collection of hypothetical rules of the following form: ΓΓ1 ` J1 ...ΓΓn ` Jn Γ ` J .(3.9) The hypotheses Γ are the global hypotheses of the rule, and the hypotheses Γi are the local hypotheses of the ith premise of the rule. Informally, this rule states that J is a derivable consequence of Γ whenever each Ji is a derivable consequence of Γ, augmented with the additional hypotheses Γi. Thus, one way to show that J is derivable from Γ is to show, in turn, that each Ji is derivable from ΓΓi. The derivation of each premise involves a “context REVISED 05.15.2012 VERSION 1.32 32 3.2 Hypothetical Inductive Definitions switch” in which we extend the global hypotheses with the local hypothe- ses of that premise, establishing a new set of global hypotheses for use within that derivation. In most cases a rule is stated for all choices of global context, in which case it is said to be uniform. A uniform rule may be given in the implicit form Γ1 ` J1 ...Γn ` Jn J(3.10) which stands for the collection of all rules of the form (3.9) in which the global hypotheses have been made explicit. A hypothetical inductive definition is to be regarded as an ordinary in- ductive definition of a formal derivability judgment Γ ` J consisting of a finite set of basic judgments, Γ, and a basic judgment, J. A collection of hypo- thetical rules, R, defines the strongest formal derivability judgment that is structural and closed under rules R. Structurality means that the formal derivability judgment must be closed under the following rules: Γ,J ` J (3.11a) Γ ` J Γ,K ` J(3.11b) Γ ` KΓ,K ` J Γ ` J(3.11c) These rules ensure that formal derivability behaves like a hypothetical judg- ment. By a slight abuse of notation we write Γ `RJ to indicate that Γ ` J is derivable from rules R. The principle of hypothetical rule induction is just the principle of rule induction applied to the formal hypothetical judgment. So to show that P(Γ ` J) whenever Γ `RJ, it is enough to show that P is closed under both the rules of R and under the structural rules. Thus, for each rule of the form (3.10), whether structural or in R, we must show that if P(ΓΓ1 ` J1) and . . . and P(ΓΓn ` Jn), then P(Γ ` J). This is just a restatement of the principle of rule induction given in Chap- ter2, specialized to the formal derivability judgment Γ ` J. In practice we usually dispense with the structural rules by the method described in Section 3.1.2. By proving that the structural rules are admis- sible any proof by rule induction may restrict attention to the rules in R VERSION 1.32 REVISED 05.15.2012 3.3 General Judgments 33 alone. If all of the rules of a hypothetical inductive definition are uniform, the structural rules (3.11b) and (3.11c) are readily seen to be admissible. Usually, Rule (3.11a) must be postulated explictly as a rule, rather than shown to be admissible on the basis of the other rules. 3.3 General Judgments General judgments codify the rules for handling variables in a judgment. As in mathematics in general, a variable is treated as an unknown ranging over a specified collection of objects. A generic judgment expresses that a judgment holds for any choice of objects replacing designated variables in the judgment. Another form of general judgment codifies the handling of parameters. A parametric judgment expresses generality over any choice of fresh renamings of designated parameters of a judgment. To keep track of the active variables and parameters in a derivation, we write Γ `U;X RJ to indicate that J is derivable from Γ according to rules R, with objects consisting of abt’s over parameters U and variables X. Generic derivability judgment is defined by ~x | Γ `X RJ iff ∀π : ~x ↔ ~x0 π ·Γ `X,~x0 R π ·J, where the quantification is restricted to variables ~x0 not already active in X. Evidence for generic derivability consists of a generic derivation, ∇, in- volving the variables ~x such that for every fresh renaming π : ~x ↔ ~x0, the derivation π · ∇ is evidence for π ·Γ `X,~x0 R π ·J. The renaming ensures that the variables in the generic judgment are fresh (not already declared in X), and that the meaning of the judgment is not dependent on the choice of variable names. For example, the generic derivation, ∇, x nat succ(x) nat succ(succ(x)) nat is evidence for the judgment, x | x nat `X (2.2) succ(succ(x)) nat. This is because every fresh renaming of x to y in ∇ results in a valid deriva- tion of the corresponding renaming of the indicated judgment. The generic derivability judgment enjoys the following structural prop- erties governing the behavior of variables: REVISED 05.15.2012 VERSION 1.32 34 3.4 Generic Inductive Definitions Proliferation If ~x | Γ `X RJ, then ~x, x | Γ `X RJ. Renaming If ~x, x | Γ `X RJ, then ~x, x0 | [x ↔ x0]·Γ `X R[x ↔ x0]·J for any x0 /∈ X ,~x. Substitution If ~x, x | Γ `X RJ and a ∈ B[X,~x], then ~x | [a/x]Γ `X R[a/x]J. (It is left implicit in the principle of substitution that sorts are to be re- spected in that the substituting object must be of the same sort as the vari- able which is being substituted.) Proliferation is guaranteed by the inter- pretation of rule schemes as ranging over all expansions of the universe. Renaming is built into the meaning of the generic judgment. Substitution holds as long as the rules themselves are closed under substitution. This need not be the case, but in practice this requirement is usually met. Parametric derivability is defined analogously to generic derivability, albeit by generalizing over parameters, rather than variables. Parametric derivability is defined by ~u;~x | Γ `U;X RJ iff ∀ρ : ~u ↔ ~u0 ∀π : ~x ↔ ~x0 ρ · π ·Γ `U,~u0;X,~x0 R ρ · π ·J. Evidence for parametric derivability consists of a derivation ∇ involving the parameters ~u and variables~x each of whose fresh renamings is a deriva- tion of the corresponding renaming of the underlying hypothetical judg- ment. Recalling from Chapter1 that parameters admit disequality, we cannot expect any substitution principle for parameters to hold of a parametric derivability. It does, however, validate the structural properties of prolif- eration and renaming, because the presence of additional parameters does not affect the formation of an abt, and parametric derivability is defined to respect all fresh renamings of parameters. 3.4 Generic Inductive Definitions A generic inductive definition admits generic hypothetical judgments in the premises of rules, with the effect of augmenting the variables, as well as the rules, within those premises. A generic rule has the form ~x~x1 | ΓΓ1 ` J1 ... ~x~xn | ΓΓn ` Jn ~x | Γ ` J .(3.12) The variables~x are the global variables of the inference, and, for each 1 ≤ i ≤ n, the variables ~xi are the local variables of the ith premise. In most cases a VERSION 1.32 REVISED 05.15.2012 3.4 Generic Inductive Definitions 35 rule is stated for all choices of global variables and global hypotheses. Such rules may be given in implicit form, ~x1 | Γ1 ` J1 ... ~xn | Γn ` Jn J.(3.13) A generic inductive definition is just an ordinary inductive definition of a family of formal generic judgments of the form ~x | Γ ` J. Formal generic judgments are identified up to renaming of variables, so that the latter judg- ment is treated as identical to the judgment ~x0 | π ·Γ ` π ·J for any renam- ing π : ~x ↔ ~x0. If R is a collection of generic rules, we write ~x | Γ `RJ to mean that the formal generic judgment ~x | Γ ` J is derivable from rules R. When specialized to a collection of generic rules, the principle of rule induction states that to show P(~x | Γ ` J) whenever~x | Γ `RJ, it is enough to show that P is closed under the rules R. Specifically, for each rule in R of the form (3.12), we must show that if P(~x~x1 | ΓΓ1 ` J1)...P(~x~xn | ΓΓn ` Jn) then P(~x | Γ ` J). By the identification convention (stated in Chapter1) the property P must respect renamings of the variables in a formal generic judgment. To ensure that the formal generic judgment behaves like a generic judg- ment, we must always ensure that the following structural rules are admis- sible: ~x | Γ,J ` J (3.14a) ~x | Γ ` J ~x | Γ,J0 ` J(3.14b) ~x | Γ ` J ~x, x | Γ ` J(3.14c) ~x, x0 | [x ↔ x0]·Γ ` [x ↔ x0]·J ~x, x | Γ ` J(3.14d) ~x | Γ ` J ~x | Γ,J ` J0 ~x | Γ ` J0 (3.14e) ~x, x | Γ ` J a ∈ B[~x] ~x | [a/x]Γ ` [a/x]J(3.14f) The admissibility of Rule (3.14a) is, in practice, ensured by explicitly in- cluding it. The admissibility of Rules (3.14b) and (3.14c) is assured if each REVISED 05.15.2012 VERSION 1.32 36 3.5 Notes of the generic rules is uniform, because we may assimilate the additional parameter, x, to the global parameters, and the additional hypothesis, J, to the global hypotheses. The admissibility of Rule (3.14d) is ensured by the identification convention for the formal generic judgment. Rule (3.14f) must be verified explicitly for each inductive definition. The concept of a generic inductive definition extends to parametric judg- ments as well. Briefly, rules are defined on formal parametric judgments of the form ~u;~x | Γ ` J, with parameters ~u, as well as variables, ~x. Such formal judgments are identified up to renaming of its variables and its parameters to ensure that the meaning is independent of the choice of variable and parameter names. 3.5 Notes The concepts of entailment and generality are fundamental to logic and programming languages. The formulation given here builds on Martin-L¨of (1983, 1987) and Avron(1991). Hypothetical and general reasoning are con- solidated into a single concept in the AUTOMATH languages (Nederpelt et al., 1994) and in the LF Logical Framework (Harper et al., 1993). These systems permit arbitrarily nested combinations of hypothetical and general judgments, whereas the present account considers only general hypotheti- cal judgments over basic judgment forms. The failure to distinguish parameters from variables is the source of many errors in language design. The crucial distinction is that whereas it makes sense to distinguish cases based on whether two parameters are the same or distinct, it makes no sense to do so for variables, because disequal- ity is not preserved by substitution. Adhering carefully to this distinction avoids much confusion and complication in language design (see, for ex- ample, Chapter 41). VERSION 1.32 REVISED 05.15.2012 Part II Statics and Dynamics Chapter 4 Statics Most programming languages exhibit a phase distinction between the static and dynamic phases of processing. The static phase consists of parsing and type checking to ensure that the program is well-formed; the dynamic phase consists of execution of well-formed programs. A language is said to be safe exactly when well-formed programs are well-behaved when exe- cuted. The static phase is specified by a statics comprising a collection of rules for deriving typing judgments stating that an expression is well-formed of a certain type. Types mediate the interaction between the constituent parts of a program by “predicting” some aspects of the execution behavior of the parts so that we may ensure they fit together properly at run-time. Type safety tells us that these predictions are accurate; if not, the statics is con- sidered to be improperly defined, and the language is deemed unsafe for execution. In this chapter we present the statics of the language L{num str} as an illustration of the methodology that we shall employ throughout this book. 4.1 Syntax When defining a language we shall be primarily concerned with its abstract syntax, specified by a collection of operators and their arities. The abstract syntax provides a systematic, unambiguous account of the hierarchical and binding structure of the language, and is therefore to be considered the official presentation of the language. However, for the sake of clarity, it is also useful to specify minimal concrete syntax conventions, without going through the trouble to set up a fully precise grammar for it. 40 4.2 Type System We will accomplish both of these purposes with a syntax chart, whose meaning is best illustrated by example. The following chart summarizes the abstract and concrete syntax of L{num str}. Typ τ ::= num num numbers str str strings Exp e ::= x x variable num[n] n numeral str[s]”s” literal plus(e1; e2) e1 + e2 addition times(e1; e2) e1 ∗ e2 multiplication cat(e1; e2) e1 ^ e2 concatenation len(e) |e| length let(e1; x.e2) let x be e1 in e2 definition This chart defines two sorts, Typ, ranged over by τ, and Exp, ranged over by e. The chart defines a collection of operators and their arities. For exam- ple, the operator let has arity (Exp,(Exp)Exp), which specifies that it has two arguments of sort Exp, and binds a variable of sort Exp in the second argument. 4.2 Type System The role of a type system is to impose constraints on the formations of phrases that are sensitive to the context in which they occur. For exam- ple, whether or not the expression plus(x; num[n]) is sensible depends on whether or not the variable x is restricted to have type num in the surround- ing context of the expression. This example is, in fact, illustrative of the general case, in that the only information required about the context of an expression is the type of the variables within whose scope the expression lies. Consequently, the statics of L{num str} consists of an inductive defi- nition of generic hypothetical judgments of the form ~x | Γ ` e : τ, where ~x is a finite set of variables, and Γ is a typing context consisting of hypotheses of the form x : τ, one for each x ∈ X . We rely on typographical conventions to determine the set of variables, using the letters x and y for variables that serve as parameters of the typing judgment. We write x /∈ dom(Γ) to indicate that there is no assumption in Γ of the form x : τ for any type τ, in which case we say that the variable x is fresh for Γ. VERSION 1.32 REVISED 05.15.2012 4.2 Type System 41 The rules defining the statics of L{num str} are as follows: Γ, x : τ ` x : τ (4.1a) Γ ` str[s]: str (4.1b) Γ ` num[n]: num (4.1c) Γ ` e1 : num Γ ` e2 : num Γ ` plus(e1; e2): num (4.1d) Γ ` e1 : num Γ ` e2 : num Γ ` times(e1; e2): num (4.1e) Γ ` e1 : str Γ ` e2 : str Γ ` cat(e1; e2): str (4.1f) Γ ` e : str Γ ` len(e): num (4.1g) Γ ` e1 : τ1 Γ, x : τ1 ` e2 : τ2 Γ ` let(e1; x.e2): τ2 (4.1h) In Rule (4.1h) we tacitly assume that the variable, x, is not already declared in Γ. This condition may always be met by choosing a suitable representa- tive of the α-equivalence class of the let expression. It is easy to check that every expression has at most one type by induc- tion on typing, which is rule induction applied to Rules (4.1). Lemma 4.1 (Unicity of Typing). For every typing context Γ and expression e, there exists at most one τ such that Γ ` e : τ. Proof. By rule induction on Rules (4.1), making use of the fact that variables have at most one type in any typing context. The typing rules are syntax-directed in the sense that there is exactly one rule for each form of expression. Consequently it is easy to give necessary conditions for typing an expression that invert the sufficient conditions ex- pressed by the corresponding typing rule. Lemma 4.2 (Inversion for Typing). Suppose that Γ ` e : τ. If e = plus(e1; e2), then τ = num,Γ ` e1 : num, and Γ ` e2 : num, and similarly for the other constructs of the language. Proof. These may all be proved by induction on the derivation of the typing judgment Γ ` e : τ. In richer languages such inversion principles are more difficult to state and to prove. REVISED 05.15.2012 VERSION 1.32 42 4.3 Structural Properties 4.3 Structural Properties The statics enjoys the structural properties of the generic hypothetical judg- ment. Lemma 4.3 (Weakening). If Γ ` e0 : τ0, then Γ, x : τ ` e0 : τ0 for any x /∈ dom(Γ) and any type τ. Proof. By induction on the derivation of Γ ` e0 : τ0. We will give one case here, for rule (4.1h). We have that e0 = let(e1; z.e2), where by the conven- tions on parameters we may assume z is chosen such that z /∈ dom(Γ) and z 6= x. By induction we have 1. Γ, x : τ ` e1 : τ1, 2. Γ, x : τ, z : τ1 ` e2 : τ0, from which the result follows by Rule (4.1h). Lemma 4.4 (Substitution). If Γ, x : τ ` e0 : τ0 and Γ ` e : τ, then Γ ` [e/x]e0 : τ0. Proof. By induction on the derivation of Γ, x : τ ` e0 : τ0. We again con- sider only rule (4.1h). As in the preceding case, e0 = let(e1; z.e2), where z may be chosen so that z 6= x and z /∈ dom(Γ). We have by induction and Lemma 4.3 that 1. Γ ` [e/x]e1 : τ1, 2. Γ, z : τ1 ` [e/x]e2 : τ0. By the choice of z we have [e/x]let(e1; z.e2) = let([e/x]e1; z.[e/x]e2). It follows by Rule (4.1h) that Γ ` [e/x]let(e1; z.e2): τ, as desired. From a programming point of view, Lemma 4.3 allows us to use an ex- pression in any context that binds its free variables: if e is well-typed in a context Γ, then we may “import” it into any context that includes the assumptions Γ. In other words the introduction of new variables beyond those required by an expression, e, does not invalidate e itself; it remains VERSION 1.32 REVISED 05.15.2012 4.3 Structural Properties 43 well-formed, with the same type.1 More significantly, Lemma 4.4 expresses the concepts of modularity and linking. We may think of the expressions e and e0 as two components of a larger system in which the component e0 is to be thought of as a client of the implementation e. The client declares a variable specifying the type of the implementation, and is type checked knowing only this information. The implementation must be of the spec- ified type in order to satisfy the assumptions of the client. If so, then we may link them to form the composite system, [e/x]e0. This may itself be the client of another component, represented by a variable, y, that is re- placed by that component during linking. When all such variables have been implemented, the result is a closed expression that is ready for execu- tion (evaluation). The converse of Lemma 4.4 is called decomposition. It states that any (large) expression may be decomposed into a client and implementor by introducing a variable to mediate their interaction. Lemma 4.5 (Decomposition). If Γ ` [e/x]e0 : τ0, then for every type τ such that Γ ` e : τ, we have Γ, x : τ ` e0 : τ0. Proof. The typing of [e/x]e0 depends only on the type of e wherever it oc- curs, if at all. This lemma tells us that any sub-expression may be isolated as a sepa- rate module of a larger system. This is especially useful when the variable x occurs more than once in e0, because then one copy of e suffices for all occurrences of x in e0. The statics of L{num str} given by Rules (4.1) exemplifies a recurrent pattern. The constructs of a language are classified into one of two forms, the introductory and the eliminatory. The introductory forms for a type de- termine the values, or canonical forms, of that type. The eliminatory forms determine how to manipulate the values of a type to form a computation of another (possibly the same) type. In L{num str} the introductory forms for the type num are the numerals, and those for the type str are the literals. The eliminatory forms for the type num are addition and multiplication, and those for the type str are concatenation and length. The importance of this classification will become apparent once we have defined the dynamics of the language in Chapter5. Then we will see that 1This may seem so obvious as to be not worthy of mention, but, suprisingly, there are useful type systems that lack this property. Because they do not validate the structural principle of weakening, they are called sub-structural type systems. REVISED 05.15.2012 VERSION 1.32 44 4.4 Notes the eliminatory forms are inverse to the introductory forms in that they “take apart” what the introductory forms have “put together.” The coher- ence of the statics and dynamics of a language expresses the concept of type safety, the subject of Chapter6. 4.4 Notes The concept of the static semantics of a programming language was histori- cally slow to develop, perhaps because the earliest languages had relatively few features and only very weak type systems. The concept of a static se- mantics in the sense considered here was introduced in the definition of the Standard ML programming language (Milner et al., 1997), building on much earlier work by Church and others on the typed λ-calculus (Baren- dregt, 1992). The concept of introduction and elimination, and the asso- ciated inversion principle, was introduced by Gentzen in his pioneering work on natural deduction (Gentzen, 1969). These principles were applied to the structure of programming languages by Martin-L¨of(1984, 1980). VERSION 1.32 REVISED 05.15.2012 Chapter 5 Dynamics The dynamics of a language is a description of how programs are to be ex- ecuted. The most important way to define the dynamics of a language is by the method of structural dynamics, which defines a transition system that inductively specifies the step-by-step process of executing a program. An- other method for presenting dynamics, called contextual dynamics, is a vari- ation of structural dynamics in which the transition rules are specified in a slightly different manner. An equational dynamics presents the dynamics of a language equationally by a collection of rules for deducing when one program is definitionally equal to another. 5.1 Transition Systems A transition system is specified by the following four forms of judgment: 1. s state, asserting that s is a state of the transition system. 2. s final, where s state, asserting that s is a final state. 3. s initial, where s state, asserting that s is an initial state. 4. s 7→ s0, where s state and s0 state, asserting that state s may transition to state s0. In practice we always arrange things so that no transition is possible from a final state: if s final, then there is no s0 state such that s 7→ s0. A state from which no transition is possible is sometimes said to be stuck. Whereas all final states are, by convention, stuck, there may be stuck states in a tran- sition system that are not final. A transition system is deterministic iff for 46 5.1 Transition Systems every state s there exists at most one state s0 such that s 7→ s0, otherwise it is non-deterministic. A transition sequence is a sequence of states s0,..., sn such that s0 initial, and si 7→ si+1 for every 0 ≤ i < n. A transition sequence is maximal iff there is no s such that sn 7→ s, and it is complete iff it is maximal and, in addition, sn final. Thus every complete transition sequence is maximal, but maximal sequences are not necessarily complete. The judgment s ↓ means that there is a complete transition sequence starting from s, which is to say that there exists s0 final such that s 7→∗ s0. The iteration of transition judgment, s 7→∗ s0, is inductively defined by the following rules: s 7→∗ s (5.1a) s 7→ s0 s0 7→∗ s00 s 7→∗ s00 (5.1b) When applied to the definition of iterated transition, the principle of rule induction states that to show that P(s, s0) holds whenever s 7→∗ s0, it is enough to show these two properties of P: 1. P(s, s). 2. if s 7→ s0 and P(s0, s00), then P(s, s00). The first requirement is to show that P is reflexive. The second is to show that P is closed under head expansion, or closed under inverse evaluation. Using this principle, it is easy to prove that 7→∗ is reflexive and transitive. The n-times iterated transition judgment, s 7→n s0, where n ≥ 0, is induc- tively defined by the following rules. s 7→0 s (5.2a) s 7→ s0 s0 7→n s00 s 7→n+1 s00 (5.2b) Theorem 5.1. For all states s and s0, s 7→∗ s0 iff s 7→k s0 for some k ≥ 0. Proof. From left to right, by induction on the definition of multi-step tran- sition. From right to left, by mathematical induction on k ≥ 0. VERSION 1.32 REVISED 05.15.2012 5.2 Structural Dynamics 47 5.2 Structural Dynamics A structural dynamics for L{num str} is given by a transition system whose states are closed expressions. All states are initial. The final states are the (closed) values, which represent the completed computations. The judgment e val, which states that e is a value, is inductively defined by the following rules: num[n] val (5.3a) str[s] val (5.3b) The transition judgment, e 7→ e0, between states is inductively defined by the following rules: n1 + n2 = n nat plus(num[n1]; num[n2]) 7→ num[n](5.4a) e1 7→ e0 1 plus(e1; e2) 7→ plus(e0 1; e2)(5.4b) e1 val e2 7→ e0 2 plus(e1; e2) 7→ plus(e1; e0 2)(5.4c) s1 ˆ s2 = s str cat(str[s1]; str[s2]) 7→ str[s](5.4d) e1 7→ e0 1 cat(e1; e2) 7→ cat(e0 1; e2)(5.4e) e1 val e2 7→ e0 2 cat(e1; e2) 7→ cat(e1; e0 2)(5.4f)  e1 7→ e0 1 let(e1; x.e2) 7→ let(e0 1; x.e2)  (5.4g) [e1 val] let(e1; x.e2) 7→ [e1/x]e2 (5.4h) We have omitted rules for multiplication and computing the length of a string, which follow a similar pattern. Rules (5.4a), (5.4d), and (5.4h) are in- struction transitions, because they correspond to the primitive steps of eval- uation. The remaining rules are search transitions that determine the order in which instructions are executed. The bracketed rule, Rule (5.4g), and bracketed premise on Rule (5.4h), are to be included for a by-value interpretation of let, and omitted for a REVISED 05.15.2012 VERSION 1.32 48 5.2 Structural Dynamics by-name interpretation. The by-value intepretation evaluates an expression before binding it to the defined variable, whereas the by-name interpreta- tion binds it in unevaluated form. The by-value interpretation saves work if the defined variable is used more than once, but wastes work if it is not used at all. Conversely, the by-name interpretation saves work if the de- fined variable is not used, and wastes work if it is used more than once. A derivation sequence in a structural dynamics has a two-dimensional structure, with the number of steps in the sequence being its “width” and the derivation tree for each step being its “height.” For example, consider the following evaluation sequence. let(plus(num[1]; num[2]); x.plus(plus(x; num[3]); num[4])) 7→ let(num[3]; x.plus(plus(x; num[3]); num[4])) 7→ plus(plus(num[3]; num[3]); num[4]) 7→ plus(num[6]; num[4]) 7→ num[10] Each step in this sequence of transitions is justified by a derivation accord- ing to Rules (5.4). For example, the third transition in the preceding exam- ple is justified by the following derivation: plus(num[3]; num[3]) 7→ num[6](5.4a) plus(plus(num[3]; num[3]); num[4]) 7→ plus(num[6]; num[4]) (5.4b) The other steps are similarly justified by a composition of rules. The principle of rule induction for the structural dynamics of L{num str} states that to show P(e 7→ e0) whenever e 7→ e0, it is sufficient to show that P is closed under Rules (5.4). For example, we may show by rule induction that structural dynamics of L{num str} is determinate, which means that an expression may make a transition to at most one other expression. The proof requires a simple lemma relating transition to values. Lemma 5.2 (Finality of Values). For no expression e do we have both e val and e 7→ e0 for some e0. Proof. By rule induction on Rules (5.3) and (5.4). Lemma 5.3 (Determinacy). If e 7→ e0 and e 7→ e00, then e0 and e00 are α- equivalent. VERSION 1.32 REVISED 05.15.2012 5.3 Contextual Dynamics 49 Proof. By rule induction on the premises e 7→ e0 and e 7→ e00, carried out either simultaneously or in either order. It is assumed that the primitive operators, such as addition, have a unique value when applied to values. Rules (5.4) exemplify the inversion principle of language design, which states that the eliminatory forms are inverse to the introductory forms of a language. The search rules determine the principal arguments of each elimi- natory form, and the instruction rules specify how to evaluate an elimina- tory form when all of its principal arguments are in introductory form. For example, Rules (5.4) specify that both arguments of addition are principal, and specify how to evaluate an addition once its principal arguments are evaluated to numerals. The inversion principle is central to ensuring that a programming language is properly defined, the exact statement of which is given in Chapter6. 5.3 Contextual Dynamics A variant of structural dynamics, called contextual dynamics, is sometimes useful. There is no fundamental difference between contextual and struc- tural dynamics, rather one of style. The main idea is to isolate instruction steps as a special form of judgment, called instruction transition, and to for- malize the process of locating the next instruction using a device called an evaluation context. The judgment, e val, defining whether an expression is a value, remains unchanged. The instruction transition judgment, e1 → e2, for L{num str} is de- fined by the following rules, together with similar rules for multiplication of numbers and the length of a string. m + n = p nat plus(num[m]; num[n]) → num[p](5.5a) s ˆ t = u str cat(str[s]; str[t]) → str[u](5.5b) let(e1; x.e2) → [e1/x]e2 (5.5c) The judgment E ectxt determines the location of the next instruction to execute in a larger expression. The position of the next instruction step is specified by a “hole”, written ◦, into which the next instruction is placed, as REVISED 05.15.2012 VERSION 1.32 50 5.3 Contextual Dynamics we shall detail shortly. (The rules for multiplication and length are omitted for concision, as they are handled similarly.) ◦ ectxt (5.6a) E1 ectxt plus(E1; e2) ectxt (5.6b) e1 val E2 ectxt plus(e1;E2) ectxt (5.6c) The first rule for evaluation contexts specifies that the next instruction may occur “here”, at the point of the occurrence of the hole. The remaining rules correspond one-for-one to the search rules of the structural dynamics. For example, Rule (5.6c) states that in an expression plus(e1; e2), if the first argument, e1, is a value, then the next instruction step, if any, lies at or within the second argument, e2. An evaluation context is to be thought of as a template that is instanti- ated by replacing the hole with an instruction to be executed. The judgment e0 = E{e} states that the expression e0 is the result of filling the hole in the evaluation context E with the expression e. It is inductively defined by the following rules: e = ◦{e}(5.7a) e1 = E1{e} plus(e1; e2) = plus(E1; e2){e}(5.7b) e1 val e2 = E2{e} plus(e1; e2) = plus(e1;E2){e}(5.7c) There is one rule for each form of evaluation context. Filling the hole with e results in e; otherwise we proceed inductively over the structure of the evaluation context. Finally, the contextual dynamics for L{num str} is defined by a single rule: e = E{e0} e0 → e0 0 e0 = E{e0 0} e 7→ e0 (5.8) Thus, a transition from e to e0 consists of (1) decomposing e into an evalua- tion context and an instruction, (2) execution of that instruction, and (3) re- placing the instruction by the result of its execution in the same spot within e to obtain e0. The structural and contextual dynamics define the same transition re- lation. For the sake of the proof, let us write e 7→s e0 for the transition VERSION 1.32 REVISED 05.15.2012 5.4 Equational Dynamics 51 relation defined by the structural dynamics (Rules (5.4)), and e 7→c e0 for the transition relation defined by the contextual dynamics (Rules (5.8)). Theorem 5.4. e 7→s e0 if, and only if, e 7→c e0. Proof. From left to right, proceed by rule induction on Rules (5.4). It is enough in each case to exhibit an evaluation context E such that e = E{e0}, e0 = E{e0 0}, and e0 → e0 0. For example, for Rule (5.4a), take E = ◦, and observe that e → e0. For Rule (5.4b), we have by induction that there exists an evaluation context E1 such that e1 = E1{e0}, e0 1 = E1{e0 0}, and e0 → e0 0. Take E = plus(E1; e2), and observe that e = plus(E1; e2){e0} and e0 = plus(E1; e2){e0 0} with e0 → e0 0. From right to left, observe that if e 7→c e0, then there exists an evaluation context E such that e = E{e0}, e0 = E{e0 0}, and e0 → e0 0. We prove by induc- tion on Rules (5.7) that e 7→s e0. For example, for Rule (5.7a), e0 is e, e0 0 is e0, and e → e0. Hence e 7→s e0. For Rule (5.7b), we have that E = plus(E1; e2), e1 = E1{e0}, e0 1 = E1{e0 0}, and e1 7→s e0 1. Therefore e is plus(e1; e2), e0 is plus(e0 1; e2), and therefore by Rule (5.4b), e 7→s e0. Because the two transition judgments coincide, contextual dynamics may be seen as an alternative way of presenting a structural dynamics. It has two advantages over structural dynamics, one relatively superficial, one rather less so. The superficial advantage stems from writing Rule (5.8) in the simpler form e0 → e0 0 E{e0} 7→ E{e0 0}.(5.9) This formulation is superficially simpler in that it does not make explicit how an expression is to be decomposed into an evaluation context and a reducible expression. The deeper advantage of contextual dynamics is that all transitions are between complete programs. One need never con- sider a transition between expressions of any type other than the ultimate observable type. This simplifies certain arguments, notably the proof of Lemma 48.16. 5.4 Equational Dynamics Another formulation of the dynamics of a language is based on regard- ing computation as a form of equational deduction, much in the style of REVISED 05.15.2012 VERSION 1.32 52 5.4 Equational Dynamics elementary algebra. For example, in algebra we may show that the polyno- mials x2 + 2 x + 1 and (x + 1)2 are equivalent by a simple process of calcu- lation and re-organization using the familiar laws of addition and multipli- cation. The same laws are sufficient to determine the value of any polyno- mial, given the values of its variables. So, for example, we may plug in 2 for x in the polynomial x2 + 2 x + 1 and calculate that 22 + 2 × 2 + 1 = 9, which is indeed (2 + 1)2. This gives rise to a model of computation in which we may determine the value of a polynomial for a given value of its variable by substituting the given value for the variable and proving that the resulting expression is equal to its value. Very similar ideas give rise to the concept of definitional, or computa- tional, equivalence of expressions in L{num str}, which we write as X | Γ ` e ≡ e0 : τ, where Γ consists of one assumption of the form x : τ for each x ∈ X . We only consider definitional equality of well-typed expressions, so that when considering the judgment Γ ` e ≡ e0 : τ, we tacitly assume that Γ ` e : τ and Γ ` e0 : τ. Here, as usual, we omit explicit mention of the parameters, X, when they can be determined from the forms of the assumptions Γ. Definitional equality of expressions in L{num str} under the by-name interpretation of let is inductively defined by the following rules: Γ ` e ≡ e : τ (5.10a) Γ ` e0 ≡ e : τ Γ ` e ≡ e0 : τ (5.10b) Γ ` e ≡ e0 : τ Γ ` e0 ≡ e00 : τ Γ ` e ≡ e00 : τ (5.10c) Γ ` e1 ≡ e0 1 : num Γ ` e2 ≡ e0 2 : num Γ ` plus(e1; e2) ≡ plus(e0 1; e0 2): num (5.10d) Γ ` e1 ≡ e0 1 : str Γ ` e2 ≡ e0 2 : str Γ ` cat(e1; e2) ≡ cat(e0 1; e0 2): str (5.10e) Γ ` e1 ≡ e0 1 : τ1 Γ, x : τ1 ` e2 ≡ e0 2 : τ2 Γ ` let(e1; x.e2) ≡ let(e0 1; x.e0 2): τ2 (5.10f) n1 + n2 = n nat Γ ` plus(num[n1]; num[n2]) ≡ num[n]: num (5.10g) s1 ˆ s2 = s str Γ ` cat(str[s1]; str[s2]) ≡ str[s]: str (5.10h) Γ ` let(e1; x.e2) ≡ [e1/x]e2 : τ (5.10i) VERSION 1.32 REVISED 05.15.2012 5.4 Equational Dynamics 53 Rules (5.10a) through (5.10c) state that definitional equality is an equiva- lence relation. Rules (5.10d) through (5.10f) state that it is a congruence re- lation, which means that it is compatible with all expression-forming con- structs in the language. Rules (5.10g) through (5.10i) specify the mean- ings of the primitive constructs of L{num str}. For the sake of concision, Rules (5.10) may be characterized as defining the strongest congruence closed under Rules (5.10g), (5.10h), and (5.10i). Rules (5.10) are sufficient to allow us to calculate the value of an expres- sion by an equational deduction similar to that used in high school algebra. For example, we may derive the equation let x be 1 + 2 in x + 3 + 4 ≡ 10 : num by applying Rules (5.10). Here, as in general, there may be many different ways to derive the same equation, but we need find only one derivation in order to carry out an evaluation. Definitional equality is rather weak in that many equivalences that we might intuitively think are true are not derivable from Rules (5.10). A pro- totypical example is the putative equivalence x : num, y : num ` x1 + x2 ≡ x2 + x1 : num, (5.11) which, intuitively, expresses the commutativity of addition. Although we shall not prove this here, this equivalence is not derivable from Rules (5.10). And yet we may derive all of its closed instances, n1 + n2 ≡ n2 + n1 : num, (5.12) where n1 nat and n2 nat are particular numbers. The “gap” between a general law, such as Equation (5.11), and all of its instances, given by Equation (5.12), may be filled by enriching the notion of equivalence to include a principle of proof by mathematical induction. Such a notion of equivalence is sometimes called semantic equivalence, be- cause it expresses relationships that hold by virtue of the dynamics of the expressions involved. (Semantic equivalence is developed rigorously for a related language in Chapter 47.) Definitional equality is sometimes called symbolic evaluation, because it allows any subexpression to be replaced by the result of evaluating it ac- cording to the rules of the dynamics of the language. Theorem 5.5. e ≡ e0 : τ iff there exists e0 val such that e 7→∗ e0 and e0 7→∗ e0. REVISED 05.15.2012 VERSION 1.32 54 5.5 Notes Proof. The proof from right to left is direct, because every transition step is a valid equation. The converse follows from the following, more general, proposition. If x1 : τ1,..., xn : τn ` e ≡ e0 : τ, then whenever e1 : τ1,..., en : τn, if [e1,..., en/x1,..., xn]e ≡ [e1,..., en/x1,..., xn]e0 : τ, then there exists e0 val such that [e1,..., en/x1,..., xn]e 7→∗ e0 and [e1,..., en/x1,..., xn]e0 7→∗ e0. This is proved by rule induction on Rules (5.10). 5.5 Notes The use of transition systems to specify the behavior of programs goes back to the early work of Church and Turing on computability. Turing’s ap- proach emphasized the concept of an abstract machine consisting of a finite program together with unbounded memory. Computation proceeds by changing the memory in accordance with the instructions in the program. Much early work on the operational semantics of programming languages, such as the SECD machine (Landin, 1965), emphasized machine models. Church’s approach emphasized the language for expressing computations, and defined execution in terms of the programs themselves, rather than in terms of auxiliary concepts such as memories or tapes. Plotkin’s elegant formulation of structural operational semantics (Plotkin, 1981), which we use heavily throughout this book, was inspired by Church’s and Landin’s ideas (Plotkin, 2004). Contextual semantics, which was introduced by Felleisen and Hieb(1992), may be seen as an alternative formulation of structural se- mantics in which “search rules” are replaced by “context matching”. Com- putation viewed as equational deduction goes back to the early work of Herbrand, G¨odel, and Church. VERSION 1.32 REVISED 05.15.2012 Chapter 6 Type Safety Most contemporary programming languages are safe (or, type safe, or strongly typed). Informally, this means that certain kinds of mismatches cannot arise during execution. For example, type safety for L{num str} states that it will never arise that a number is to be added to a string, or that two numbers are to be concatenated, neither of which is meaningful. In general type safety expresses the coherence between the statics and the dynamics. The statics may be seen as predicting that the value of an expression will have a certain form so that the dynamics of that expression is well-defined. Consequently, evaluation cannot “get stuck” in a state for which no transition is possible, corresponding in implementation terms to the absence of “illegal instruction” errors at execution time. This is proved by showing that each step of transition preserves typability and by showing that typable states are well-defined. Consequently, evaluation can never “go off into the weeds,” and hence can never encounter an illegal instruc- tion. More precisely, type safety for L{num str} may be stated as follows: Theorem 6.1 (Type Safety). 1. If e : τ and e 7→ e0, then e0 : τ. 2. If e : τ, then either e val, or there exists e0 such that e 7→ e0. The first part, called preservation, says that the steps of evaluation pre- serve typing; the second, called progress, ensures that well-typed expres- sions are either values or can be further evaluated. Safety is the conjunction of preservation and progress. We say that an expression, e, is stuck iff it is not a value, yet there is no e0 such that e 7→ e0. It follows from the safety theorem that a stuck state is 56 6.1 Preservation necessarily ill-typed. Or, putting it the other way around, that well-typed states do not get stuck. 6.1 Preservation The preservation theorem for L{num str} defined in Chapters4 and5 is proved by rule induction on the transition system (rules (5.4)). Theorem 6.2 (Preservation). If e : τ and e 7→ e0, then e0 : τ. Proof. We will consider two cases, leaving the rest to the reader. Consider rule (5.4b), e1 7→ e0 1 plus(e1; e2) 7→ plus(e0 1; e2) . Assume that plus(e1; e2): τ. By inversion for typing, we have that τ = num, e1 : num, and e2 : num. By induction we have that e0 1 : num, and hence plus(e0 1; e2): num. The case for concatenation is handled similarly. Now consider rule (5.4h), let(e1; x.e2) 7→ [e1/x]e2 . Assume that let(e1; x.e2): τ2. By the inversion lemma 4.2, e1 : τ1 for some τ1 such that x : τ1 ` e2 : τ2. By the substitution lemma 4.4 [e1/x]e2 : τ2, as desired. It is easy to check that the primitive operations are all type-preserving; for example, if a nat and b nat and a + b = c nat, then c nat. The proof of preservation is naturally structured as an induction on the transition judgment, because the argument hinges on examining all possi- ble transitions from a given expression. In some cases we may manage to carry out a proof by structural induction on e, or by an induction on typ- ing, but experience shows that this often leads to awkward arguments, or, in some cases, cannot be made to work at all. 6.2 Progress The progress theorem captures the idea that well-typed programs cannot “get stuck”. The proof depends crucially on the following lemma, which characterizes the values of each type. VERSION 1.32 REVISED 05.15.2012 6.2 Progress 57 Lemma 6.3 (Canonical Forms). If e val and e : τ, then 1. If τ = num, then e = num[n] for some number n. 2. If τ = str, then e = str[s] for some string s. Proof. By induction on rules (4.1) and (5.3). Progress is proved by rule induction on rules (4.1) defining the statics of the language. Theorem 6.4 (Progress). If e : τ, then either e val, or there exists e0 such that e 7→ e0. Proof. The proof proceeds by induction on the typing derivation. We will consider only one case, for rule (4.1d), e1 : num e2 : num plus(e1; e2): num , where the context is empty because we are considering only closed terms. By induction we have that either e1 val, or there exists e0 1 such that e1 7→ e0 1. In the latter case it follows that plus(e1; e2) 7→ plus(e0 1; e2), as required. In the former we also have by induction that either e2 val, or there exists e0 2 such that e2 7→ e0 2. In the latter case we have that plus(e1; e2) 7→ plus(e1; e0 2), as required. In the former, we have, by the Canonical Forms Lemma 6.3, e1 = num[n1] and e2 = num[n2], and hence plus(num[n1]; num[n2]) 7→ num[n1 + n2]. Because the typing rules for expressions are syntax-directed, the progress theorem could equally well be proved by induction on the structure of e, appealing to the inversion theorem at each step to characterize the types of the parts of e. But this approach breaks down when the typing rules are not syntax-directed, that is, when there may be more than one rule for a given expression form. No difficulty arises if the proof proceeds by induction on the typing rules. Summing up, the combination of preservation and progress together constitute the proof of safety. The progress theorem ensures that well-typed expressions do not “get stuck” in an ill-defined state, and the preservation theorem ensures that if a step is taken, the result remains well-typed (with the same type). Thus the two parts work hand-in-hand to ensure that the statics and dynamics are coherent, and that no ill-defined states can ever be encountered while evaluating a well-typed expression. REVISED 05.15.2012 VERSION 1.32 58 6.3 Run-Time Errors 6.3 Run-Time Errors Suppose that we wish to extend L{num str} with, say, a quotient operation that is undefined for a zero divisor. The natural typing rule for quotients is given by the following rule: e1 : num e2 : num div(e1; e2): num . But the expression div(num[3]; num[0]) is well-typed, yet stuck! We have two options to correct this situation: 1. Enhance the type system, so that no well-typed program may divide by zero. 2. Add dynamic checks, so that division by zero signals an error as the outcome of evaluation. Either option is, in principle, viable, but the most common approach is the second. The first requires that the type checker prove that an expression be non-zero before permitting it to be used in the denominator of a quotient. It is difficult to do this without ruling out too many programs as ill-formed. This is because we cannot reliably predict statically whether an expression will turn out to be non-zero when executed (because this is an undecidable property). We therefore consider the second approach, which is typical of current practice. The general idea is to distinguish checked from unchecked errors. An unchecked error is one that is ruled out by the type system. No run-time checking is performed to ensure that such an error does not occur, because the type system rules out the possibility of it arising. For example, the dynamics need not check, when performing an addition, that its two argu- ments are, in fact, numbers, as opposed to strings, because the type system ensures that this is the case. On the other hand the dynamics for quotient must check for a zero divisor, because the type system does not rule out the possibility. One approach to modelling checked errors is to give an inductive def- inition of the judgment e err stating that the expression e incurs a checked run-time error, such as division by zero. Here are some representative rules that would appear in a full inductive definition of this judgment: e1 val div(e1; num[0]) err (6.1a) VERSION 1.32 REVISED 05.15.2012 6.4 Notes 59 e1 err plus(e1; e2) err (6.1b) e1 val e2 err plus(e1; e2) err (6.1c) Rule (6.1a) signals an error condition for division by zero. The other rules propagate this error upwards: if an evaluated sub-expression is a checked error, then so is the overall expression. Once the error judgment is available, we may also consider an expres- sion, error, which forcibly induces an error, with the following static and dynamic semantics: Γ ` error : τ (6.2a) error err (6.2b) The preservation theorem is not affected by the presence of checked er- rors. However, the statement (and proof) of progress is modified to account for checked errors. Theorem 6.5 (Progress With Error). If e : τ, then either e err, or e val, or there exists e0 such that e 7→ e0. Proof. The proof is by induction on typing, and proceeds similarly to the proof given earlier, except that there are now three cases to consider at each point in the proof. 6.4 Notes The concept of type safety as it is understood today was first formulated by Milner(1978), who invented the slogan “well-typed programs do not go wrong.” Whereas Milner relied on an explicit notion of “going wrong” to express the concept of a type error, Wright and Felleisen(1994) observed that we can instead show that ill-defined states cannot arise in a well-typed program, giving rise to the slogan “well-typed programs do not get stuck.” However, their formulation relied on an analysis showing that no stuck state is well-typed. This analysis is replaced by the progress theorem given here, which relies on the concept of canonical forms introduced by Martin- L¨of(1980). REVISED 05.15.2012 VERSION 1.32 60 6.4 Notes VERSION 1.32 REVISED 05.15.2012 Chapter 7 Evaluation Dynamics In Chapter5 we defined the evaluation of expressions in L{num str} using the method of structural dynamics. This approach is useful as a foundation for proving properties of a language, but other methods are often more appropriate for other purposes, such as writing user manuals. Another method, called evaluation dynamics presents the dynamics as a relation be- tween a phrase and its value, without detailing how it is to be determined in a step-by-step manner. Evaluation dynamics suppresses the step-by-step details of determining the value of an expression, and hence does not pro- vide any useful notion of the time complexity of a program. Cost dynamics rectifies this by augmenting evaluation dynamics with a cost measure. Var- ious cost measures may be assigned to an expression. One example is the number of steps in the structural dynamics required for an expression to reach a value. 7.1 Evaluation Dynamics An evaluation dynamics, consists of an inductive definition of the evalua- tion judgment, e ⇓ v, stating that the closed expression, e, evaluates to the value, v. The evaluation dynamics of L{num str} is defined by the follow- ing rules: num[n] ⇓ num[n](7.1a) str[s] ⇓ str[s](7.1b) e1 ⇓ num[n1] e2 ⇓ num[n2] n1 + n2 = n nat plus(e1; e2) ⇓ num[n](7.1c) 62 7.1 Evaluation Dynamics e1 ⇓ str[s1] e2 ⇓ str[s2] s1 ˆ s2 = s str cat(e1; e2) ⇓ str[s](7.1d) e ⇓ str[s] |s| = n nat len(e) ⇓ num[n](7.1e) [e1/x]e2 ⇓ v2 let(e1; x.e2) ⇓ v2 (7.1f) The value of a let expression is determined by substitution of the bind- ing into the body. The rules are therefore not syntax-directed, because the premise of Rule (7.1f) is not a sub-expression of the expression in the con- clusion of that rule. Rule (7.1f) specifies a by-name interpretation of definitions. For a by- value interpretation the following rule should be used instead: e1 ⇓ v1 [v1/x]e2 ⇓ v2 let(e1; x.e2) ⇓ v2 (7.2) Because the evaluation judgment is inductively defined, we prove prop- erties of it by rule induction. Specifically, to show that the property P(e ⇓ v) holds, it is enough to show that P is closed under Rules (7.1): 1. Show that P(num[n] ⇓ num[n]). 2. Show that P(str[s] ⇓ str[s]). 3. Show that P(plus(e1; e2) ⇓ num[n]), if P(e1 ⇓ num[n1]),P(e2 ⇓ num[n2]), and n1 + n2 = n nat. 4. Show that P(cat(e1; e2) ⇓ str[s]), if P(e1 ⇓ str[s1]),P(e2 ⇓ str[s2]), and s1 ˆ s2 = s str. 5. Show that P(let(e1; x.e2) ⇓ v2), if P([e1/x]e2 ⇓ v2). This induction principle is not the same as structural induction on e itself, because the evaluation rules are not syntax-directed. Lemma 7.1. If e ⇓ v, then v val. Proof. By induction on Rules (7.1). All cases except Rule (7.1f) are imme- diate. For the latter case, the result follows directly by an appeal to the inductive hypothesis for the premise of the evaluation rule. VERSION 1.32 REVISED 05.15.2012 7.2 Relating Structural and Evaluation Dynamics 63 7.2 Relating Structural and Evaluation Dynamics We have given two different forms of dynamics for L{num str}. It is nat- ural to ask whether they are equivalent, but to do so first requires that we consider carefully what we mean by equivalence. The structural dynamics describes a step-by-step process of execution, whereas the evaluation dy- namics suppresses the intermediate states, focusing attention on the initial and final states alone. This suggests that the appropriate correspondence is between complete execution sequences in the structural dynamics and the evaluation judgment in the evaluation dynamics. (We will consider only numeric expressions, but analogous results hold also for string-valued ex- pressions.) Theorem 7.2. For all closed expressions e and values v, e 7→∗ v iff e ⇓ v. How might we prove such a theorem? We will consider each direction separately. We consider the easier case first. Lemma 7.3. If e ⇓ v, then e 7→∗ v. Proof. By induction on the definition of the evaluation judgment. For ex- ample, suppose that plus(e1; e2) ⇓ num[n] by the rule for evaluating addi- tions. By induction we know that e1 7→∗ num[n1] and e2 7→∗ num[n2]. We reason as follows: plus(e1; e2) 7→∗ plus(num[n1]; e2) 7→∗ plus(num[n1]; num[n2]) 7→ num[n1 + n2] Therefore plus(e1; e2) 7→∗ num[n1 + n2], as required. The other cases are handled similarly. For the converse, recall from Chapter5 the definitions of multi-step evaluation and complete evaluation. Because v ⇓ v whenever v val, it suf- fices to show that evaluation is closed under converse evaluation:1 Lemma 7.4. If e 7→ e0 and e0 ⇓ v, then e ⇓ v. Proof. By induction on the definition of the transition judgment. For ex- ample, suppose that plus(e1; e2) 7→ plus(e0 1; e2), where e1 7→ e0 1. Sup- pose further that plus(e0 1; e2) ⇓ v, so that e0 1 ⇓ num[n1], e2 ⇓ num[n2], n1 + n2 = n nat, and v is num[n]. By induction e1 ⇓ num[n1], and hence plus(e1; e2) ⇓ num[n], as required. 1Converse evaluation is also known as head expansion. REVISED 05.15.2012 VERSION 1.32 64 7.3 Type Safety, Revisited 7.3 Type Safety, Revisited Theorem 6.1 states that a language is safe iff it satisfies both preservation and progress. This formulation depends critically on the use of a transition system to specify the dynamics. But what if we had instead specified the dynamics as an evaluation relation, instead of using a transition system? Can we state and prove safety in such a setting? The answer, unfortunately, is that we cannot. Although there is an ana- logue of the preservation property for an evaluation dynamics, there is no clear analogue of the progress property. Preservation may be stated as say- ing that if e ⇓ v and e : τ, then v : τ. This can be readily proved by induction on the evaluation rules. But what is the analogue of progress? We might be tempted to phrase progress as saying that if e : τ, then e ⇓ v for some v. Although this property is true for L{num str}, it demands much more than just progress — it requires that every expression evaluate to a value! If L{num str} were extended to admit operations that may result in an error (as discussed in Section 6.3), or to admit non-terminating expressions, then this property would fail, even though progress would remain valid. One possible attitude towards this situation is to simply conclude that type safety cannot be properly discussed in the context of an evaluation dynamics, but only by reference to a structural dynamics. Another point of view is to instrument the dynamics with explicit checks for dynamic type errors, and to show that any expression with a dynamic type fault must be statically ill-typed. Re-stated in the contrapositive, this means that a stat- ically well-typed program cannot incur a dynamic type error. A difficulty with this point of view is that we must explicitly account for a form of er- ror solely to prove that it cannot arise! Nevertheless, we will press on to show how a semblance of type safety can be established using evaluation dynamics. The main idea is to define a judgment e⇑ stating, in the jargon of the literature, that the expression e goes wrong when executed. The exact defi- nition of “going wrong” is given by a set of rules, but the intention is that it should cover all situations that correspond to type errors. The following rules are representative of the general case: plus(str[s]; e2)⇑ (7.3a) e1 val plus(e1; str[s])⇑ (7.3b) These rules explicitly check for the misapplication of addition to a string; similar rules govern each of the primitive constructs of the language. VERSION 1.32 REVISED 05.15.2012 7.4 Cost Dynamics 65 Theorem 7.5. If e⇑, then there is no τ such that e : τ. Proof. By rule induction on Rules (7.3). For example, for Rule (7.3a), we observe that str[s]: str, and hence plus(str[s]; e2) is ill-typed. Corollary 7.6. If e : τ, then ¬(e⇑). Apart from the inconvenience of having to define the judgment e⇑ only to show that it is irrelevant for well-typed programs, this approach suffers a very significant methodological weakness. If we should omit one or more rules defining the judgment e⇑, the proof of Theorem 7.5 remains valid; there is nothing to ensure that we have included sufficiently many checks for run-time type errors. We can prove that the ones we define cannot arise in a well-typed program, but we cannot prove that we have covered all possible cases. By contrast the structural dynamics does not specify any behavior for ill-typed expressions. Consequently, any ill-typed expression will “get stuck” without our explicit intervention, and the progress theorem rules out all such cases. Moreover, the transition system corresponds more closely to implementation—a compiler need not make any provisions for checking for run-time type errors. Instead, it relies on the statics to ensure that these cannot arise, and assigns no meaning to any ill-typed program. Execution is therefore more efficient, and the language definition is simpler. 7.4 Cost Dynamics A structural dynamics provides a natural notion of time complexity for pro- grams, namely the number of steps required to reach a final state. An eval- uation dynamics, on the other hand, does not provide such a direct no- tion of complexity. Because the individual steps required to complete an evaluation are suppressed, we cannot directly read off the number of steps required to evaluate to a value. Instead we must augment the evaluation relation with a cost measure, resulting in a cost dynamics. Evaluation judgments have the form e ⇓k v, with the meaning that e evaluates to v in k steps. num[n] ⇓0 num[n](7.4a) e1 ⇓k1 num[n1] e2 ⇓k2 num[n2] plus(e1; e2) ⇓k1+k2+1 num[n1 + n2](7.4b) str[s] ⇓0 str[s](7.4c) REVISED 05.15.2012 VERSION 1.32 66 7.5 Notes e1 ⇓k1 s1 e2 ⇓k2 s2 cat(e1; e2) ⇓k1+k2+1 str[s1 ˆ s2](7.4d) [e1/x]e2 ⇓k2 v2 let(e1; x.e2) ⇓k2+1 v2 (7.4e) For a by-value interpretation of let, Rule (7.4e) should be replaced by the following rule: e1 ⇓k1 v1 [v1/x]e2 ⇓k2 v2 let(e1; x.e2) ⇓k1+k2+1 v2 (7.5) Theorem 7.7. For any closed expression e and closed value v of the same type, e ⇓k v iff e 7→k v. Proof. From left to right proceed by rule induction on the definition of the cost dynamics. From right to left proceed by induction on k, with an inner rule induction on the definition of the structural dynamics. 7.5 Notes The structural similarity between evaluation dynamics and typing rules was first developed in the definition of Standard ML (Milner et al., 1997). The advantage of evaluation semantics is that it directly defines the relation of interest, that between a program and its outcome. The disadvantage is that it is not as well-suited to metatheory as structural semantics, precisely because it glosses over the fine structure of computation. The concept of a cost dynamics was introduced by Blelloch and Greiner(1996b) in their study of parallelism (discussed further in Chapter 39). VERSION 1.32 REVISED 05.15.2012 Part III Function Types Chapter 8 Function Definitions and Values In the language L{num str} we may perform calculations such as the dou- bling of a given expression, but we cannot express doubling as a concept in itself. To capture the general pattern of doubling, we abstract away from the particular number being doubled using a variable to stand for a fixed, but unspecified, number, to express the doubling of an arbitrary number. Any particular instance of doubling may then be obtained by substituting a numeric expression for that variable. In general an expression may involve many distinct variables, necessitating that we specify which of several pos- sible variables is varying in a particular context, giving rise to a function of that variable. In this chapter we will consider two extensions of L{num str} with functions. The first, and perhaps most obvious, extension is by adding func- tion definitions to the language. A function is defined by binding a name to an abt with a bound variable that serves as the argument of that function. A function is applied by substituting a particular expression (of suitable type) for the bound variable, obtaining an expression. The domain and range of defined functions are limited to the types nat and str, because these are the only types of expression. Such functions are called first-order functions, in contrast to higher-order functions, which permit functions as arguments and results of other functions. Because the domain and range of a function are types, this requires that we introduce function types whose elements are functions. Consequently, we may form functions of higher type, those whose domain and range may themselves be function types. 70 8.1 First-Order Functions Historically the introduction of higher-order functions was responsible for a mistake in language design that subsequently was re-characterized as a feature, called dynamic binding. Dynamic binding arises from getting the definition of substitution wrong by failing to avoid capture. This makes the names of bound variables important, in violation of the fundamental prin- ciple of binding stating that the names of bound variables are unimportant. 8.1 First-Order Functions The language L{num str fun} is the extension of L{num str} with function definitions and function applications as described by the following gram- mar: Exp e ::= call[f](e) f(e) call fun[τ1; τ2](x1.e2; f.e) fun f(x1:τ1):τ2 = e2 in e definition The expression fun[τ1; τ2](x1.e2; f.e) binds the function name f within e to the pattern x1.e2, which has parameter x1 and definition e2. The do- main and range of the function are, respectively, the types τ1 and τ2. The expression call[f](e) instantiates the binding of f with the argument e. The statics of L{num str fun} defines two forms of judgment: 1. Expression typing, e : τ, stating that e has type τ; 2. Function typing, f(τ1): τ2, stating that f is a function with argument type τ1 and result type τ2. The judgment f(τ1): τ2 is called the function header of f; it specifies the domain type and the range type of a function. The statics of L{num str fun} is defined by the following rules: Γ, x1 : τ1 ` e2 : τ2 Γ, f(τ1): τ2 ` e : τ Γ ` fun[τ1; τ2](x1.e2; f.e): τ (8.1a) Γ ` f(τ1): τ2 Γ ` e : τ1 Γ ` call[f](e): τ2 (8.1b) Function substitution, written [[x.e/ f ]]e0, is defined by induction on the structure of e0 much like the definition of ordinary substitution. However, a function name, f, is not a form of expression, but rather can only occur in VERSION 1.32 REVISED 05.15.2012 8.2 Higher-Order Functions 71 a call of the form call[f](e). Function substitution for such expressions is defined by the following rule: [[x.e/ f ]]call[f](e0) = let([[x.e/ f ]]e0; x.e)(8.2) At call sites to f with argument e0, function substitution yields a let expres- sion that binds x to the result of expanding any further calls to f within e0. Lemma 8.1. If Γ, f(τ1): τ2 ` e : τ and Γ, x1 : τ1 ` e2 : τ2, then Γ ` [[x1.e2/ f ]]e : τ. Proof. By induction on the structure of e. The dynamics of L{num str fun} is defined using function substitution: fun[τ1; τ2](x1.e2; f.e) 7→ [[x1.e2/ f ]]e (8.3) Because function substitution replaces all calls to f by appropriate let ex- pressions, there is no need to give a rule for function calls. The safety of L{num str fun} may, with some effort, be derived from the safety theorem for higher-order functions, which we discuss next. 8.2 Higher-Order Functions The syntactic and semantic similarity between variable definitions and func- tion definitions in L{num str fun} is striking. This suggests that it may be possible to consolidate the two concepts into a single definition mechanism. The gap that must be bridged is the segregation of functions from expres- sions. A function name f is bound to an abstractor x.e specifying a pattern that is instantiated when f is applied. To consolidate function definitions with expression definitions it is sufficient to reify the abstractor into a form of expression, called a λ-abstraction, written lam[τ1](x.e). Correspond- ingly, we must generalize application to have the form ap(e1; e2), where e1 is any expression, and not just a function name. These are, respectively, the introduction and elimination forms for the function type, arr(τ1; τ2), whose elements are functions with domain τ1 and range τ2. The language L{num str →} is the enrichment of L{num str} with func- tion types, as specified by the following grammar: Typ τ ::= arr(τ1; τ2) τ1 → τ2 function Exp e ::= lam[τ](x.e) λ (x:τ) e abstraction ap(e1; e2) e1(e2) application REVISED 05.15.2012 VERSION 1.32 72 8.2 Higher-Order Functions Functions are now “first class” in the sense that a function is an expression of function type. The statics of L{num str →} is given by extending Rules (4.1) with the following rules: Γ, x : τ1 ` e : τ2 Γ ` lam[τ1](x.e): arr(τ1; τ2)(8.4a) Γ ` e1 : arr(τ2; τ)Γ ` e2 : τ2 Γ ` ap(e1; e2): τ (8.4b) Lemma 8.2 (Inversion). Suppose that Γ ` e : τ. 1. If e = lam[τ1](x.e2), then τ = arr(τ1; τ2) and Γ, x : τ1 ` e2 : τ2. 2. If e = ap(e1; e2), then there exists τ2 such that Γ ` e1 : arr(τ2; τ) and Γ ` e2 : τ2. Proof. The proof proceeds by rule induction on the typing rules. Observe that for each rule, exactly one case applies, and that the premises of the rule in question provide the required result. Lemma 8.3 (Substitution). If Γ, x : τ ` e0 : τ0, and Γ ` e : τ, then Γ ` [e/x]e0 : τ0. Proof. By rule induction on the derivation of the first judgment. The dynamics of L{num str →} extends that of L{num str} with the following additional rules: lam[τ](x.e) val (8.5a) e1 7→ e0 1 ap(e1; e2) 7→ ap(e0 1; e2)(8.5b)  e1 val e2 7→ e0 2 ap(e1; e2) 7→ ap(e1; e0 2)  (8.5c) [e2 val] ap(lam[τ2](x.e1); e2) 7→ [e2/x]e1 (8.5d) The bracketed rule and premise are to be included for a call-by-value inter- pretation of function application, and excluded for a call-by-name interpre- tation.1 1Although the term “call-by-value” is accurately descriptive, the origin of the term “call- by-name” remains shrouded in mystery. VERSION 1.32 REVISED 05.15.2012 8.3 Evaluation Dynamics and Definitional Equality 73 Theorem 8.4 (Preservation). If e : τ and e 7→ e0, then e0 : τ. Proof. The proof is by induction on Rules (8.5), which define the dynamics of the language. Consider Rule (8.5d), ap(lam[τ2](x.e1); e2) 7→ [e2/x]e1 . Suppose that ap(lam[τ2](x.e1); e2): τ1. By Lemma 8.2 we have e2 : τ2 and x : τ2 ` e1 : τ1, so by Lemma 8.3 [e2/x]e1 : τ1. The other rules governing application are handled similarly. Lemma 8.5 (Canonical Forms). If e : arr(τ1; τ2) and e val, then e = λ (x:τ1) e2 for some variable x and expression e2 such that x : τ1 ` e2 : τ2. Proof. By induction on the typing rules, using the assumption e val. Theorem 8.6 (Progress). If e : τ, then either e val, or there exists e0 such that e 7→ e0. Proof. The proof is by induction on Rules (8.4). Note that because we con- sider only closed terms, there are no hypotheses on typing derivations. Consider Rule (8.4b) (under the by-name interpretation). By induction either e1 val or e1 7→ e0 1. In the latter case we have ap(e1; e2) 7→ ap(e0 1; e2). In the former case, we have by Lemma 8.5 that e1 = lam[τ2](x.e) for some x and e. But then ap(e1; e2) 7→ [e2/x]e. 8.3 Evaluation Dynamics and Definitional Equality An inductive definition of the evaluation judgment e ⇓ v for L{num str →} is given by the following rules: lam[τ](x.e) ⇓ lam[τ](x.e)(8.6a) e1 ⇓ lam[τ](x.e)[e2/x]e ⇓ v ap(e1; e2) ⇓ v (8.6b) It is easy to check that if e ⇓ v, then v val, and that if e val, then e ⇓ e. Theorem 8.7. e ⇓ v iff e 7→∗ v and v val. REVISED 05.15.2012 VERSION 1.32 74 8.4 Dynamic Scope Proof. In the forward direction we proceed by rule induction on Rules (8.6), following along similar lines as the proof of Theorem 7.2. In the reverse direction we proceed by rule induction on Rules (5.1). The proof relies on an analogue of Lemma 7.4, which states that evalua- tion is closed under converse execution, which is proved by induction on Rules (8.5). Definitional equality for the call-by-name dynamics of L{num str →} is defined by a straightforward extension to Rules (5.10). Γ ` ap(lam[τ](x.e2); e1) ≡ [e1/x]e2 : τ2 (8.7a) Γ ` e1 ≡ e0 1 : τ2 → τ Γ ` e2 ≡ e0 2 : τ2 Γ ` ap(e1; e2) ≡ ap(e0 1; e0 2): τ (8.7b) Γ, x : τ1 ` e2 ≡ e0 2 : τ2 Γ ` lam[τ1](x.e2) ≡ lam[τ1](x.e0 2): τ1 → τ2 (8.7c) Definitional equality for call-by-value requires a small bit of additional machinery. The main idea is to restrict Rule (8.7a) to require that the ar- gument be a value. However, to be fully expressive, we must also widen the concept of a value to include all variables that are in scope, so that Rule (8.7a) would apply even when the argument is a variable. The justifi- cation for this is that in call-by-value, the parameter of a function stands for the value of its argument, and not for the argument itself. The call-by-value definitional equality judgment has the form Γ ` e1 ≡ e2 : τ, where Γ consists of paired hypotheses x : τ, x val stating, for each variable x in scope, its type and that it is a value. We write Γ ` e val to indicate that e is a value under these hypotheses, so that, for example, x : τ, x val ` x val. (The typing hypothesis is irrelevant, but harmless, to the value judgment.) 8.4 Dynamic Scope The dynamics of function application given by Rules (8.5) is defined only for expressions without free variables. When a function is called, the ar- gument is substituted for the function parameter, ensuring that the result remains closed. Moreover, because substitution of closed expressions can VERSION 1.32 REVISED 05.15.2012 8.4 Dynamic Scope 75 never incur capture, the scopes of variables are not disturbed by the dy- namics, ensuring that the principles of binding and scope described in Chapter1 are respected. This treatment of variables is called static scoping, or static binding, to contrast it with an alternative approach that we now describe. Another approach, called dynamic scoping, or dynamic binding, is some- times advocated as an alternative to static binding. Evaluation is defined for expressions that may contain free variables. Evaluation of a variable is undefined; it is an error to ask for the value of an unbound variable. Func- tion call is defined similarly to static binding, except that when a function is called, the argument replaces the parameter in the body, possibly incurring, rather than avoiding, capture of free variables in the argument. (As we will explain shortly, this behavior is considered to be a feature, not a bug!) The difference between replacement and substitution may be illustrated by example. Let e be the expression λ (x:str) y + |x| in which the variable y occurs free, and let e0 be the expression λ (y:str) f(y) with free variable f. If we substitute e for f in e0 we obtain an expression of the form λ (y0:str) (λ (x:str) y + |x|)(y0), where the bound variable, y, in e has been renamed to some fresh variable y0 so as to avoid capture. If we instead replace f by e in e0 we obtain λ (y:str) (λ (x:str) y + |x|)(y) in which y is no longer free: it has been captured during replacement. The implications of this seemingly small change to the dynamics of L{→} are far-reaching. The most obvious implication is that the language is not type safe. In the above example we have that y : nat ` e : str → nat, and that f : str → nat ` e0 : str → nat. It follows that y : nat ` [e/ f ]e0 : str → nat, but it is easy to see that the result of replacing f by e in e0 is ill-typed, regardless of what assumption we make about y. The difficulty, of course, is that the bound occurrence of y in e0 has type str, whereas the free occurrence in e must have type nat in order for e to be well-formed. One way around this difficulty is to ignore types altogether, and rely on run-time checks to ensure that bad things do not happen, despite the evident failure of safety. (See Chapter 18 for a full exploration of this ap- proach.) But even if we ignore worries about safety, we are still left with the serious problem that the names of bound variables matter, and cannot be altered without changing the meaning of a program. So, for example, to use expression e0, we must bear in mind that the parameter, f, occurs REVISED 05.15.2012 VERSION 1.32 76 8.5 Notes within the scope of a binder for y, a fact that is not revealed by the type of e0 (and certainly not if we disregard types entirely!) If we change e0 so that it binds a different variable, say z, then we must correspondingly change e to ensure that it refers to z, and not y, in order to preserve the overall behavior of the system of two expressions. This means that e and e0 must be devel- oped in tandem, violating a basic principle of modular decomposition. (For more on dynamic scope, please see Chapter 33.) 8.5 Notes Nearly all programming languages provide some form of function defini- tion mechanism of the kind illustrated here. The main point of the present account is to demonstrate that a more natural, and more powerful, ap- proach is to separate the generic concept of a definition from the specific concept of a function. Function types codify the general notion in a system- atic manner that encompasses function definitions as a special case, and moreover, admits passing functions as arguments and returning them as results without special provision. The essential contribution of Church’s λ- calculus (Church, 1941) was to take the notion of function as primary, and to demonstrate that nothing more is needed to obtain a fully expressive programming language. VERSION 1.32 REVISED 05.15.2012 Chapter 9 G¨odel’s T The language L{nat →}, better known as G¨odel’s T, is the combination of function types with the type of natural numbers. In contrast to L{num str}, which equips the naturals with some arbitrarily chosen arithmetic primi- tives, the language L{nat →} provides a general mechanism, called prim- itive recursion, from which these primitives may be defined. Primitive re- cursion captures the essential inductive character of the natural numbers, and hence may be seen as an intrinsic termination proof for each program in the language. Consequently, we may only define total functions in the language, those that always return a value for each argument. In essence every program in L{nat →} “comes equipped” with a proof of its termi- nation. While this may seem like a shield against infinite loops, it is also a weapon that can be used to show that some programs cannot be written in L{nat →}. To do so would require a master termination proof for every possible program in the language, something that we shall prove does not exist. 78 9.1 Statics 9.1 Statics The syntax of L{nat →} is given by the following grammar: Typ τ ::= nat nat naturals arr(τ1; τ2) τ1 → τ2 function Exp e ::= x x variable z z zero s(e) s(e) successor rec(e; e0; x.y.e1) rec e {z ⇒ e0 | s(x) with y ⇒ e1} recursion lam[τ](x.e) λ (x:τ) e abstraction ap(e1; e2) e1(e2) application We write n for the expression s(... s(z)), in which the successor is ap- plied n ≥ 0 times to zero. The expression rec(e; e0; x.y.e1) is called primi- tive recursion. It represents the e-fold iteration of the transformation x.y.e1 starting from e0. The bound variable x represents the predecessor and the bound variable y represents the result of the x-fold iteration. The “with” clause in the concrete syntax for the recursor binds the variable y to the result of the recursive call, as will become apparent shortly. Sometimes iteration, written iter(e; e0; y.e1), is considered as an alter- native to primitive recursion. It has essentially the same meaning as prim- itive recursion, except that only the result of the recursive call is bound to y in e1, and no binding is made for the predecessor. Clearly iteration is a special case of primitive recursion, because we can always ignore the predecessor binding. Conversely, primitive recursion is definable from it- eration, provided that we have product types (Chapter 11) at our disposal. To define primitive recursion from iteration we simultaneously compute the predecessor while iterating the specified computation. The statics of L{nat →} is given by the following typing rules: Γ, x : τ ` x : τ (9.1a) Γ ` z : nat (9.1b) Γ ` e : nat Γ ` s(e): nat (9.1c) Γ ` e : nat Γ ` e0 : τ Γ, x : nat, y : τ ` e1 : τ Γ ` rec(e; e0; x.y.e1): τ (9.1d) VERSION 1.32 REVISED 05.15.2012 9.2 Dynamics 79 Γ, x : ρ ` e : τ Γ ` lam[ρ](x.e): arr(ρ; τ)(9.1e) Γ ` e1 : arr(τ2; τ)Γ ` e2 : τ2 Γ ` ap(e1; e2): τ (9.1f) As usual, admissibility of the structural rule of substitution is crucially important. Lemma 9.1. If Γ ` e : τ and Γ, x : τ ` e0 : τ0, then Γ ` [e/x]e0 : τ0. 9.2 Dynamics The closed values of L{nat →} are defined by the following rules: z val (9.2a) [e val] s(e) val (9.2b) lam[τ](x.e) val (9.2c) The premise of Rule (9.2b) is to be included for an eager interpretation of successor, and excluded for a lazy interpretation. The transition rules for the dynamics of L{nat →} are as follows:  e 7→ e0 s(e) 7→ s(e0)  (9.3a) e1 7→ e0 1 ap(e1; e2) 7→ ap(e0 1; e2)(9.3b)  e1 val e2 7→ e0 2 ap(e1; e2) 7→ ap(e1; e0 2)  (9.3c) [e2 val] ap(lam[τ](x.e); e2) 7→ [e2/x]e (9.3d) e 7→ e0 rec(e; e0; x.y.e1) 7→ rec(e0; e0; x.y.e1)(9.3e) rec(z; e0; x.y.e1) 7→ e0 (9.3f) REVISED 05.15.2012 VERSION 1.32 80 9.3 Definability s(e) val rec(s(e); e0; x.y.e1) 7→ [e, rec(e; e0; x.y.e1)/x, y]e1 (9.3g) The bracketed rules and premises are to be included for an eager successor and call-by-value application, and omitted for a lazy successor and call- by-name application. Rules (9.3f) and (9.3g) specify the behavior of the recursor on z and s(e). In the former case the recursor reduces to e0, and in the latter case the variable x is bound to the predecessor, e, and y is bound to the (unevaluated) recursion on e. If the value of y is not required in the rest of the computation, the recursive call will not be evaluated. Lemma 9.2 (Canonical Forms). If e : τ and e val, then 1. If τ = nat, then e = s(s(... z)) for some number n ≥ 0 occurrences of the successor starting with zero. 2. If τ = τ1 → τ2, then e = λ (x:τ1) e2 for some e2. Theorem 9.3 (Safety). 1. If e : τ and e 7→ e0, then e0 : τ. 2. If e : τ, then either e val or e 7→ e0 for some e0 9.3 Definability A mathematical function f :N → N on the natural numbers is definable in L{nat →} iff there exists an expression e f of type nat → nat such that for every n ∈ N, e f (n) ≡ f (n): nat. (9.4) That is, the numeric function f :N → N is definable iff there is an ex- pression e f of type nat → nat such that, when applied to the numeral representing the argument n ∈ N, the application is definitionally equal to the numeral corresponding to f (n) ∈ N. Definitional equality for L{nat →}, written Γ ` e ≡ e0 : τ, is the strongest congruence containing these axioms: Γ ` ap(lam[τ1](x.e2); e1) ≡ [e1/x]e2 : τ (9.5a) Γ ` rec(z; e0; x.y.e1) ≡ e0 : τ (9.5b) VERSION 1.32 REVISED 05.15.2012 9.3 Definability 81 Γ ` rec(s(e); e0; x.y.e1) ≡ [e, rec(e; e0; x.y.e1)/x, y]e1 : τ (9.5c) For example, the doubling function, d(n) = 2 × n, is definable in L{nat →} by the expression ed : nat → nat given by λ (x:nat) rec x {z ⇒ z | s(u) with v ⇒ s(s(v))}. To check that this defines the doubling function, we proceed by induction on n ∈ N. For the basis, it is easy to check that ed(0) ≡ 0 : nat. For the induction, assume that ed(n) ≡ d(n): nat. Then calculate using the rules of definitional equality: ed(n + 1) ≡ s(s(ed(n))) ≡ s(s(2 × n)) = 2 × (n + 1) = d(n + 1). As another example, consider the following function, called Ackermann’s function, defined by the following equations: A(0, n) = n + 1 A(m + 1, 0) = A(m, 1) A(m + 1, n + 1) = A(m,A(m + 1, n)). This function grows very quickly. For example, A(4, 2) ≈ 265,536, which is often cited as being much larger than the number of atoms in the universe! Yet we can show that the Ackermann function is total by a lexicographic induction on the pair of arguments (m, n). On each recursive call, either m decreases, or else m remains the same, and n decreases, so inductively the recursive calls are well-defined, and hence so is A(m, n). A first-order primitive recursive function is a function of type nat → nat that is defined using primitive recursion, but without using any higher or- der functions. Ackermann’s function is defined so that it is not first-order primitive recursive, but is higher-order primitive recursive. The key to REVISED 05.15.2012 VERSION 1.32 82 9.4 Undefinability showing that it is definable in L{nat →} is to observe that A(m + 1, n) it- erates n times the function A(m, −), starting with A(m, 1). As an auxiliary, let us define the higher-order function it :(nat → nat) → nat → nat → nat to be the λ-abstraction λ (f:nat → nat) λ (n:nat) rec n {z ⇒ id | s( ) with g ⇒ f ◦ g}, where id = λ (x:nat) x is the identity, and f ◦ g = λ (x:nat) f(g(x)) is the composition of f and g. It is easy to check that it(f)(n)(m) ≡ f (n)(m): nat, where the latter expression is the n-fold composition of f starting with m. We may then define the Ackermann function ea : nat → nat → nat to be the expression λ (m:nat) rec m {z ⇒ s | s( ) with f ⇒ λ (n:nat) it(f)(n)(f(1))}. It is instructive to check that the following equivalences are valid: ea(0)(n) ≡ s(n)(9.6) ea(m + 1)(0) ≡ ea(m)(1)(9.7) ea(m + 1)(n + 1) ≡ ea(m)(ea(s(m))(n)). (9.8) That is, the Ackermann function is definable in L{nat →}. 9.4 Undefinability It is impossible to define an infinite loop in L{nat →}. Theorem 9.4. If e : τ, then there exists v val such that e ≡ v : τ. Proof. See Corollary 47.15. VERSION 1.32 REVISED 05.15.2012 9.4 Undefinability 83 Consequently, values of function type in L{nat →} behave like mathe- matical functions: if f : ρ → τ and e : ρ, then f(e) evaluates to a value of type τ. Moreover, if e : nat, then there exists a natural number n such that e ≡ n : nat. Using this, we can show, using a technique called diagonalization, that there are functions on the natural numbers that are not definable in L{nat →}. We make use of a technique, called G¨odel-numbering, that assigns a unique natural number to each closed expression of L{nat →}. This allows us to manipulate expressions as data values in L{nat →}, and hence permits L{nat →} to compute with its own programs.1 The essence of G¨odel-numbering is captured by the following simple construction on abstract syntax trees. (The generalization to abstract bind- ing trees is slightly more difficult, the main complication being to ensure that all α-equivalent expressions are assigned the same G¨odel number.) Re- call that a general ast, a, has the form o(a1,..., ak), where o is an operator of arity k. Fix an enumeration of the operators so that every operator has an index i ∈ N, and let m be the index of o in this enumeration. Define the G¨odelnumber paq of a to be the number 2m 3n1 5n2 ... pnk k , where pk is the kth prime number (so that p0 = 2, p1 = 3, and so on), and n1,..., nk are the G¨odel numbers of a1,..., ak, respectively. This assigns a natural number to each ast. Conversely, given a natural number, n, we may apply the prime factorization theorem to “parse” n as a unique abstract syntax tree. (If the factorization is not of the appropriate form, which can only be because the arity of the operator does not match the number of factors, then n does not code any ast.) Now, using this representation, we may define a (mathematical) func- tion funiv :N → N → N such that, for any e : nat → nat, funiv(peq)(m) = n iff e(m) ≡ n : nat.2 The determinacy of the dynamics, together with The- orem 9.4, ensure that funiv is a well-defined function. It is called the universal function for L{nat →} because it specifies the behavior of any expression e of type nat → nat. Using the universal function, let us define an auxil- iary mathematical function, called the diagonal function, d :N → N, by the equation d(m) = funiv(m)(m). This function is chosen so that d(peq) = n iff 1The same technique lies at the heart of the proof of G¨odel’s celebrated incompleteness theorem. The undefinability of certain functions on the natural numbers within L{nat →} may be seen as a form of incompleteness similar to that considered by G¨odel. 2The value of funiv(k)(m) may be chosen arbitrarily to be zero when k is not the code of any expression e. REVISED 05.15.2012 VERSION 1.32 84 9.5 Notes e(peq) ≡ n : nat. (The motivation for this definition will become apparent in a moment.) The function d is not definable in L{nat →}. Suppose that d were de- fined by the expression ed, so that we have ed(peq) ≡ e(peq): nat. Let eD be the expression λ (x:nat) s(ed(x)) of type nat → nat. We then have eD(peDq) ≡ s(ed(peDq)) ≡ s(eD(peDq)). But the termination theorem implies that there exists n such that eD(peDq) ≡ n, and hence we have n ≡ s(n), which is impossible. We say that a language L is universal if it is possible to write an inter- preter for L in L itself. It is intuitively evident that funiv is computable in the sense that we can define it in a sufficiently powerful programming lan- guage. But the preceding argument shows that L{nat →} is not sufficiently powerful for this task. That is, L{nat →} is not universal. By demanding termination we sacrifice expressiveness. The preceding argument shows that this is an inescapable tradeoff. If you want universality, you have to give up termination, and if you want termination, then you must give up universality. There is no alternative. 9.5 Notes L{nat →} was introduced by G¨odel in his study of the consistency of arith- metic (G¨odel, 1980). G¨odel showed how to “compile” proofs in arithmetic into well-typed terms of the language L{nat →}, and to reduce the consis- tency problem for arithmetic to the termination of programs in L{nat →}. This was perhaps the first programming language whose design was di- rectly influenced by the verification (of termination) of its programs. VERSION 1.32 REVISED 05.15.2012 Chapter 10 Plotkin’s PCF The language L{nat *}, also known as Plotkin’s PCF, integrates functions and natural numbers using general recursion, a means of defining self-referential expressions. In contrast to L{nat →} expressions in L{nat *} might not terminate when evaluated: its definable functions are, in general, partial rather than total. Informally, the difference between L{nat *} and L{nat →} is that the former moves the proof of termination for an expression from the expression itself into the mind of the programmer. The type system no longer ensures termination, which permits a wider range of functions to be defined in the system, but at the cost of admitting infinite loops when the termination proof is either incorrect or absent. The crucial concept embodied in L{nat *} is the fixed point characteri- zation of recursive definitions. In ordinary mathematical practice we may define a function f by recursion equations such as these: f (0), 1 f (n + 1),(n + 1) × f (n). These may be viewed as simultaneous equations in the variable, f, ranging over functions on the natural numbers. The function we seek is a solution to these equations—a function f :N → N such that the above conditions are satisfied. We must, of course, show that these equations have a unique so- lution, which is easily shown by mathematical induction on the argument to f. The solution to such a system of equations may be characterized as the fixed point of an associated functional (operator mapping functions to 86 functions). To see this, let us re-write these equations in another form: f (n), (1 if n = 0 n × f (n0) if n = n0 + 1. Re-writing yet again, we seek f given by n 7→ (1 if n = 0 n × f (n0) if n = n0 + 1. Now define the functional F by the equation F( f ) = f 0, where f 0 is given by n 7→ (1 if n = 0 n × f (n0) if n = n0 + 1. Note well that the condition on f 0 is expressed in terms of the argument, f, to the functional F, and not in terms of f 0 itself! The function f we seek is then a fixed point of F, which is a function f :N → N such that f = F( f ). In other words f is defined to the fix(F), where fix is an operator on functionals yielding a fixed point of F. Why does an operator such as F have a fixed point? Informally, a fixed point may be obtained as the limit of a series of approximations of the desired solution obtained by iterating the functional F. This is where partial functions come into the picture. Let us say that a partial func- tion, φ on the natural numbers, is an approximation to a total function, f, if φ(m) = n implies that f (m) = n. Let ⊥:N*N be the totally unde- fined partial function—⊥(n) is undefined for every n ∈ N. Intuitively, this is the “worst” approximation to the desired solution, f, of the recursion equations given above. Given any approximation, φ, of f, we may “im- prove” it by considering φ0 = F(φ). Intuitively, φ0 is defined on 0 and on m + 1 for every m ≥ 0 on which φ is defined. Continuing in this manner, φ00 = F(φ0) = F(F(φ)) is an improvement on φ0, and hence a further im- provement on φ. If we start with ⊥ as the initial approximation to f, then pass to the limit lim i≥0 F(i)(⊥), we will obtain the least approximation to f that is defined for every m ∈ N, and hence is the function f itself. Turning this around, if the limit exists, it must be the solution we seek. This fixed point characterization of recursion equations is taken as a primitive concept in L{nat *}—we may obtain the least fixed point of any VERSION 1.32 REVISED 05.15.2012 10.1 Statics 87 functional definable in the language. Using this we may solve any set of recursion equations we like, with the proviso that there is no guarantee that the solution is a total function. Rather, it is guaranteed to be a partial function that may be undefined on some, all, or no inputs. This is the price we pay for expressive power—we may solve all systems of equations, but the solution may not be as well-behaved as we might like. It is our task as programmers to ensure that the functions defined by recursion are total— all of our loops terminate. 10.1 Statics The abstract binding syntax of L{nat *} is given by the following gram- mar: Typ τ ::= nat nat naturals parr(τ1; τ2) τ1 * τ2 partial function Exp e ::= x x variable z z zero s(e) s(e) successor ifz(e; e0; x.e1) ifz e {z ⇒ e0 | s(x) ⇒ e1} zero test lam[τ](x.e) λ (x:τ) e abstraction ap(e1; e2) e1(e2) application fix[τ](x.e) fix x:τ is e recursion The expression fix[τ](x.e) is called general recursion; it is discussed in more detail below. The expression ifz(e; e0; x.e1) branches according to whether e evaluates to z or not, binding the predecessor to x in the case that it is not. The statics of L{nat *} is inductively defined by the following rules: Γ, x : τ ` x : τ (10.1a) Γ ` z : nat (10.1b) Γ ` e : nat Γ ` s(e): nat (10.1c) Γ ` e : nat Γ ` e0 : τ Γ, x : nat ` e1 : τ Γ ` ifz(e; e0; x.e1): τ (10.1d) REVISED 05.15.2012 VERSION 1.32 88 10.2 Dynamics Γ, x : τ1 ` e : τ2 Γ ` lam[τ1](x.e): parr(τ1; τ2)(10.1e) Γ ` e1 : parr(τ2; τ)Γ ` e2 : τ2 Γ ` ap(e1; e2): τ (10.1f) Γ, x : τ ` e : τ Γ ` fix[τ](x.e): τ (10.1g) Rule (10.1g) reflects the self-referential nature of general recursion. To show that fix[τ](x.e) has type τ, we assume that it is the case by assigning that type to the variable, x, which stands for the recursive expression itself, and checking that the body, e, has type τ under this very assumption. The structural rules, including in particular substitution, are admissible for the static semantics. Lemma 10.1. If Γ, x : τ ` e0 : τ0,Γ ` e : τ, then Γ ` [e/x]e0 : τ0. 10.2 Dynamics The dynamic semantics of L{nat *} is defined by the judgments e val, specifying the closed values, and e 7→ e0, specifying the steps of evaluation. The judgment e val is defined by the following rules: z val (10.2a) [e val] s(e) val (10.2b) lam[τ](x.e) val (10.2c) The bracketed premise on Rule (10.2b) is to be included for the eager inter- pretation of the sucessor operation, and omitted for the lazy interpretation. (See Chapter 37 for a further discussion of laziness.) The transition judgment e 7→ e0 is defined by the following rules:  e 7→ e0 s(e) 7→ s(e0)  (10.3a) e 7→ e0 ifz(e; e0; x.e1) 7→ ifz(e0; e0; x.e1)(10.3b) ifz(z; e0; x.e1) 7→ e0 (10.3c) VERSION 1.32 REVISED 05.15.2012 10.2 Dynamics 89 s(e) val ifz(s(e); e0; x.e1) 7→ [e/x]e1 (10.3d) e1 7→ e0 1 ap(e1; e2) 7→ ap(e0 1; e2)(10.3e)  e1 val e2 7→ e0 2 ap(e1; e2) 7→ ap(e1; e0 2)  (10.3f) [e val] ap(lam[τ](x.e); e2) 7→ [e2/x]e (10.3g) fix[τ](x.e) 7→ [fix[τ](x.e)/x]e (10.3h) The bracketed Rule (10.3a) is to be included for an eager interpretation of the successor, and omitted otherwise. Bracketed Rule (10.3f) and the brack- eted premise on Rule (10.3g) are to be included for a call-by-value interpre- tation, and omitted for a call-by-name interpretation, of function applica- tion. Rule (10.3h) implements self-reference by substituting the recursive expression itself for the variable x in its body; this is called unwinding the recursion. Theorem 10.2 (Safety). 1. If e : τ and e 7→ e0, then e0 : τ. 2. If e : τ, then either e val or there exists e0 such that e 7→ e0. Proof. The proof of preservation is by induction on the derivation of the transition judgment. Consider Rule (10.3h). Suppose that fix[τ](x.e): τ. By inversion and substitution we have [fix[τ](x.e)/x]e : τ, as required. The proof of progress proceeds by induction on the derivation of the typing judgment. For example, for Rule (10.1g) the result follows immediately because we may make progress by unwinding the recursion. It is easy to check directly that if e val, then e is irreducible in that there is no e0 such that e 7→ e0. The safety theorem implies the converse, namely that an irreducible expression is a value, provided that it is closed and well- typed. Definitional equality for the call-by-name variant of L{nat *}, written Γ ` e1 ≡ e2 : τ, is defined to be the strongest congruence containing the following axioms: Γ ` ifz(z; e0; x.e1) ≡ e0 : τ (10.4a) REVISED 05.15.2012 VERSION 1.32 90 10.3 Definability Γ ` ifz(s(e); e0; x.e1) ≡ [e/x]e1 : τ (10.4b) Γ ` fix[τ](x.e) ≡ [fix[τ](x.e)/x]e : τ (10.4c) Γ ` ap(lam[τ1](x.e2); e1) ≡ [e1/x]e2 : τ (10.4d) These rules are sufficient to calculate the value of any closed expression of type nat: if e : nat, then e ≡ n : nat iff e 7→∗ n. 10.3 Definability General recursion is a very flexible programming technique that permits a wide variety of functions to be defined within L{nat *}. The drawback is that, in contrast to primitive recursion, the termination of a recursively defined function is not intrinsic to the program itself, but rather must be proved extrinsically by the programmer. The benefit is a much greater free- dom in writing programs. Let us write fun x(y:τ1):τ2 is e for a recursive function within whose body, e : τ2, are bound two variables, y : τ1 standing for the argument and x : τ1 → τ2 standing for the function itself. The dynamic semantics of this construct is given by the axiom (fun x(y:τ1):τ2 is e)(e1) 7→ [fun x(y:τ1):τ2 is e, e1/x, y]e . That is, to apply a recursive function, we substitute the recursive function itself for x and the argument for y in its body. Recursive functions may be defined in L{nat *} using a combination of recursion and functions, writing fix x:τ1 * τ2 is λ (y:τ1) e for fun x(y:τ1):τ2 is e. It is a good exercise to check that the static and dynamic semantics of recursive functions are derivable from this definition. The primitive recursion construct of L{nat →} is defined in L{nat *} using recursive functions by taking the expression rec e {z ⇒ e0 | s(x) with y ⇒ e1} VERSION 1.32 REVISED 05.15.2012 10.3 Definability 91 to stand for the application, e0(e), where e0 is the general recursive function fun f(u:nat):τ is ifz u {z ⇒ e0 | s(x) ⇒ [ f(x)/y]e1}. The static and dynamic semantics of primitive recursion are derivable in L{nat *} using this expansion. In general, functions definable in L{nat *} are partial in that they may be undefined for some arguments. A partial (mathematical) function, φ : N*N, is definable in L{nat *} iff there is an expression eφ : nat * nat such that φ(m) = n iff eφ(m) ≡ n : nat. So, for example, if φ is the totally undefined function, then eφ is any function that loops without returning whenever it is called. It is informative to classify those partial functions φ that are definable in L{nat *}. These are the so-called partial recursive functions, which are defined to be the primitive recursive functions augmented by the minimiza- tion operation: given φ(m, n), define ψ(n) to be the least m ≥ 0 such that (1) for m0 < m, φ(m0, n) is defined and non-zero, and (2) φ(m, n) = 0. If no such m exists, then ψ(n) is undefined. Theorem 10.3. A partial function φ on the natural numbers is definable in L{nat *} iff it is partial recursive. Proof sketch. Minimization is readily definable in L{nat *}, so it is at least as powerful as the set of partial recursive functions. Conversely, we may, with considerable tedium, define an evaluator for expressions of L{nat *} as a partial recursive function, using G¨odel-numbering to represent expres- sions as numbers. Consequently, L{nat *} does not exceed the power of the set of partial recursive functions. Church’s Law states that the partial recursive functions coincide with the set of effectively computable functions on the natural numbers—those that can be carried out by a program written in any programming language currently available or that will ever be available.1 Therefore L{nat *} is as powerful as any other programming language with respect to the set of definable functions on the natural numbers. The universal function, φuniv, for L{nat *} is the partial function on the natural numbers defined by φuniv(peq)(m) = n iff e(m) ≡ n : nat. 1See Chapter 17 for further discussion of Church’s Law. REVISED 05.15.2012 VERSION 1.32 92 10.4 Notes In contrast to L{nat →}, the universal function φuniv for L{nat *} is par- tial (may be undefined for some inputs). It is, in essence, an interpreter that, given the code peq of a closed expression of type nat * nat, simulates the dynamic semantics to calculate the result, if any, of applying it to the m, obtaining n. Because this process may fail to terminate, the universal function is not defined for all inputs. By Church’s Law the universal function is definable in L{nat *}. In contrast, we proved in Chapter9 that the analogous function is not defin- able in L{nat →} using the technique of diagonalization. It is instructive to examine why that argument does not apply in the present setting. As in Section 9.4, we may derive the equivalence eD(peDq) ≡ s(eD(peDq)) for L{nat *}. The difference, however, is that this equation is not incon- sistent! Rather than being contradictory, it is merely a proof that the expres- sion eD(peDq) does not terminate when evaluated, for if it did, the result would be a number equal to its own successor, which is impossible. 10.4 Notes The language L{nat *} is derived from Plotkin(1977). Plotkin introduced PCF to study the relationship between its operational and denotational se- mantics, but many authors have used PCF as the subject of study for many issues in the design and semantics of languages. In this respect PCF may be thought of as the E. coli of programming languages. VERSION 1.32 REVISED 05.15.2012 Part IV Finite Data Types Chapter 11 Product Types The binary product of two types consists of ordered pairs of values, one from each type in the order specified. The associated eliminatory forms are pro- jections, which select the first and second component of a pair. The nullary product, or unit, type consists solely of the unique “null tuple” of no val- ues, and has no associated eliminatory form. The product type admits both a lazy and an eager dynamics. According to the lazy dynamics, a pair is a value without regard to whether its components are values; they are not evaluated until (if ever) they are accessed and used in another computation. According to the eager dynamics, a pair is a value only if its components are values; they are evaluated when the pair is created. More generally, we may consider the finite product, hτiii∈I, indexed by a finite set of indices,I. The elements of the finite product type are I-indexed tuples whose ith component is an element of the type τi, for each i ∈ I. The components are accessed by I-indexed projection operations, general- izing the binary case. Special cases of the finite product include n-tuples, indexed by sets of the form I = { 0, . . . , n − 1 }, and labeled tuples, or records, indexed by finite sets of symbols. Similarly to binary products, finite prod- ucts admit both an eager and a lazy interpretation. 96 11.1 Nullary and Binary Products 11.1 Nullary and Binary Products The abstract syntax of products is given by the following grammar: Typ τ ::= unit unit nullary product prod(τ1; τ2) τ1 × τ2 binary product Exp e ::= triv hi null tuple pair(e1; e2) he1, e2i ordered pair pr[l](e) e · l left projection pr[r](e) e · r right projection There is no elimination form for the unit type, there being nothing to extract from the null tuple. The statics of product types is given by the following rules. Γ ` hi : unit (11.1a) Γ ` e1 : τ1 Γ ` e2 : τ2 Γ ` he1, e2i : τ1 × τ2 (11.1b) Γ ` e : τ1 × τ2 Γ ` e · l : τ1 (11.1c) Γ ` e : τ1 × τ2 Γ ` e · r : τ2 (11.1d) The dynamics of product types is specified by the following rules: hi val (11.2a) [e1 val][e2 val] he1, e2i val (11.2b)  e1 7→ e0 1 he1, e2i 7→ he0 1, e2i  (11.2c)  e1 val e2 7→ e0 2 he1, e2i 7→ he1, e0 2i  (11.2d) e 7→ e0 e · l 7→ e0 · l (11.2e) e 7→ e0 e · r 7→ e0 · r (11.2f) VERSION 1.32 REVISED 05.15.2012 11.2 Finite Products 97 [e1 val][e2 val] he1, e2i · l 7→ e1 (11.2g) [e1 val][e2 val] he1, e2i · r 7→ e2 (11.2h) The bracketed rules and premises are to be omitted for a lazy dynamics, and included for an eager dynamics of pairing. The safety theorem applies to both the eager and the lazy dynamics, with the proof proceeding along similar lines in each case. Theorem 11.1 (Safety). 1. If e : τ and e 7→ e0, then e0 : τ. 2. If e : τ then either e val or there exists e0 such that e 7→ e0. Proof. Preservation is proved by induction on transition defined by Rules (11.2). Progress is proved by induction on typing defined by Rules (11.1). 11.2 Finite Products The syntax of finite product types is given by the following grammar: Typ τ ::= prod({i ,→ τi}i∈I) hτiii∈I product Exp e ::= tpl({i ,→ ei}i∈I) heiii∈I tuple pr[i](e) e · i projection The variable I stands for a finite index set over which products are formed. The type prod({i ,→ τi}i∈I), or ∏i∈I τi for short, is the type of I-tuples of expressions ei of type τi, one for each i ∈ I. An I-tuple has the form tpl({i ,→ ei}i∈I), or heiii∈I for short, and for each i ∈ I the ith projection from an I-tuple, e, is written pr[i](e), or e · i for short. When I = { i1,..., in }, the I-tuple type may be written in the form hi1 ,→ τ1,..., in ,→ τni in which we make explicit the association of a type to each index i ∈ I. Similarly, we may write hi1 ,→ e1,..., in ,→ eni for the I-tuple whose ith component is ei. Finite products generalize empty and binary products by choosing I to be empty or the two-element set { l, r }, respectively. In practice I is often REVISED 05.15.2012 VERSION 1.32 98 11.3 Primitive and Mutual Recursion chosen to be a finite set of symbols that serve as labels for the components of the tuple so as to enhance readability. The statics of finite products is given by the following rules: Γ ` e1 : τ1 ...Γ ` en : τn Γ ` hi1 ,→ e1,..., in ,→ eni : hi1 ,→ τ1,..., in ,→ τni (11.3a) Γ ` e : hi1 ,→ τ1,..., in ,→ τni (1 ≤ k ≤ n) Γ ` e · ik : τk (11.3b) In Rule (11.3b) the index i ∈ I is a particular element of the index set I, whereas in Rule (11.3a), the index i ranges over the index set I. The dynamics of finite products is given by the following rules: [e1 val ... en val] hi1 ,→ e1,..., in ,→ eni val (11.4a) " e1 val ... ej−1 val ej 7→ e0 j e0 j+1 = ej+1 ... e0 n = en hi1 ,→ e1,..., in ,→ eni 7→ hi1 ,→ e0 1,..., in ,→ e0ni # (11.4b) e 7→ e0 e · i 7→ e0 · i (11.4c) hi1 ,→ e1,..., in ,→ eni val hi1 ,→ e1,..., in ,→ eni · ik 7→ ek (11.4d) As formulated, Rule (11.4b) specifies that the components of a tuple are to be evaluated in some sequential order, without specifying the order in which the components are considered. It is not hard, but a bit technically complicated, to impose an evaluation order by imposing a total ordering on the index set and evaluating components according to this ordering. Theorem 11.2 (Safety). If e : τ, then either e val or there exists e0 such that e0 : τ and e 7→ e0. Proof. The safety theorem may be decomposed into progress and preserva- tion lemmas, which are proved as in Section 11.1. 11.3 Primitive and Mutual Recursion In the presence of products we may simplify the primitive recursion con- struct defined in Chapter9 so that only the result on the predecessor, and not the predecessor itself, is passed to the successor branch. Writing this as VERSION 1.32 REVISED 05.15.2012 11.3 Primitive and Mutual Recursion 99 iter(e; e0; x.e1), we may define primitive recursion in the sense of Chap- ter9 to be the expression e0 · r, where e0 is the expression iter(e; hz, e0i; x.hs(x · l),[x · r/x]e1i). The idea is to compute inductively both the number, n, and the result of the recursive call on n, from which we can compute both n + 1 and the result of an additional recursion using e1. The base case is computed directly as the pair of zero and e0. It is easy to check that the statics and dynamics of the recursor are preserved by this definition. We may also use product types to implement mutual recursion, which allows several mutually recursive computations to be defined simultane- ously. For example, consider the following recursion equations defining two mathematical functions on the natural numbers: E(0) = 1 O(0) = 0 E(n + 1) = O(n) O(n + 1) = E(n) Intuitively, E(n) is non-zero if and only if n is even, and O(n) is non-zero if and only if n is odd. If we wish to define these functions in L{nat *}, we immediately face the problem of how to define two functions simul- taneously. There is a trick available in this special case that takes advan- tage of the fact that E and O have the same type: simply define eo of type nat → nat → nat so that eo(0) represents E and eo(1) represents O. (We leave the details as an exercise for the reader.) A more general solution is to recognize that the definition of two mutu- ally recursive functions may be thought of as the recursive definition of a pair of functions. In the case of the even and odd functions we will define the labeled tuple, eEO, of type, τEO, given by heven ,→ nat → nat, odd ,→ nat → nati. From this we will obtain the required mutually recursive functions as the projections eEO · even and eEO · odd. To effect the mutual recursion the expression eEO is defined to be fix this:τEO is heven ,→ eE, odd ,→ eOi, where eE is the expression λ (x:nat) ifz x {z ⇒ s(z) | s(y) ⇒ this · odd(y)}, REVISED 05.15.2012 VERSION 1.32 100 11.4 Notes and eO is the expression λ (x:nat) ifz x {z ⇒ z | s(y) ⇒ this · even(y)}. The functions eE and eO refer to each other by projecting the appropriate component from the variable this standing for the object itself. The choice of variable name with which to effect the self-reference is, of course, imma- terial, but it is common to use this or self to emphasize its role. 11.4 Notes Product types are the essence of structured data. All languages have some form of product type, but frequently in a form that is combined with other, separable, concepts. Common manifestations of products include: (1) func- tions with “multiple arguments” or “multple results”; (2) “objects” repre- sented as tuples of mutually recursive functions; (3) “structures,” which are tuples with mutable components. There are many papers on finite prod- uct types, which include record types as a special case. Pierce(2002) pro- vides a thorough account of record types, and their subtyping properties (for which, see Chapter 23). Allen et al.(2006) analyzes many of the key ideas in the framework of dependent type theory. VERSION 1.32 REVISED 05.15.2012 Chapter 12 Sum Types Most data structures involve alternatives such as the distinction between a leaf and an interior node in a tree, or a choice in the outermost form of a piece of abstract syntax. Importantly, the choice determines the structure of the value. For example, nodes have children, but leaves do not, and so forth. These concepts are expressed by sum types, specifically the binary sum, which offers a choice of two things, and the nullary sum, which offers a choice of no things. Finite sums generalize nullary and binary sums to permit an arbitrary number of cases indexed by a finite index set. As with products, sums come in both eager and lazy variants, differing in how val- ues of sum type are defined. 12.1 Nullary and Binary Sums The abstract syntax of sums is given by the following grammar: Typ τ ::= void void nullary sum sum(τ1; τ2) τ1 + τ2 binary sum Exp e ::= abort[τ](e) abort(e) abort in[τ1; τ2][l](e) l · e left injection in[τ1; τ2][r](e) r · e right injection case(e; x1.e1; x2.e2) case e {l · x1 ⇒ e1 | r · x2 ⇒ e2} case analysis The nullary sum represents a choice of zero alternatives, and hence admits no introductory form. The eliminatory form, abort(e), aborts the com- putation in the event that e evaluates to a value, which it cannot do. The elements of the binary sum type are labeled to indicate whether they are 102 12.1 Nullary and Binary Sums drawn from the left or the right summand, either l · e or r · e. A value of the sum type is eliminated by case analysis. The statics of sum types is given by the following rules. Γ ` e : void Γ ` abort(e): τ (12.1a) Γ ` e : τ1 Γ ` l · e : τ1 + τ2 (12.1b) Γ ` e : τ2 Γ ` r · e : τ1 + τ2 (12.1c) Γ ` e : τ1 + τ2 Γ, x1 : τ1 ` e1 : τ Γ, x2 : τ2 ` e2 : τ Γ ` case e {l · x1 ⇒ e1 | r · x2 ⇒ e2}: τ (12.1d) For the sake of readability, in Rules (12.1b) and (12.1c) we have written l · e and r · e in place of the abstract syntax in[τ1; τ2][l](e) and in[τ1; τ2][r](e), which includes the types τ1 and τ2 explicitly. In Rule (12.1d) both branches of the case analysis must have the same type. Because a type expresses a static “prediction” on the form of the value of an expression, and because an expression of sum type could evaluate to either form at run-time, we must insist that both branches yield the same type. The dynamics of sums is given by the following rules: e 7→ e0 abort(e) 7→ abort(e0)(12.2a) [e val] l · e val (12.2b) [e val] r · e val (12.2c)  e 7→ e0 l · e 7→ l · e0  (12.2d)  e 7→ e0 r · e 7→ r · e0  (12.2e) e 7→ e0 case e {l · x1 ⇒ e1 | r · x2 ⇒ e2} 7→ case e0 {l · x1 ⇒ e1 | r · x2 ⇒ e2} (12.2f) [e val] case l · e {l · x1 ⇒ e1 | r · x2 ⇒ e2} 7→ [e/x1]e1 (12.2g) VERSION 1.32 REVISED 05.15.2012 12.2 Finite Sums 103 [e val] case r · e {l · x1 ⇒ e1 | r · x2 ⇒ e2} 7→ [e/x2]e2 (12.2h) The bracketed premises and rules are to be included for an eager dynamics, and excluded for a lazy dynamics. The coherence of the statics and dynamics is stated and proved as usual. Theorem 12.1 (Safety). 1. If e : τ and e 7→ e0, then e0 : τ. 2. If e : τ, then either e val or e 7→ e0 for some e0. Proof. The proof proceeds by induction on Rules (12.2) for preservation, and by induction on Rules (12.1) for progress. 12.2 Finite Sums Just as we may generalize nullary and binary products to finite products, so may we also generalize nullary and binary sums to finite sums. The syntax for finite sums is given by the following grammar: Typ τ ::= sum({i ,→ τi}i∈I)[τi]i∈I sum Exp e ::= in[~τ][i](e) i · e injection case(e;{i ,→ xi.ei}i∈I) case e {i · xi ⇒ ei}i∈I case analysis The variable I stands for a finite index set over which sums are formed. The notation ~τ stands for a finite function {i ,→ τi}i∈I for some index set I. The type sum({i ,→ τi}i∈I), or ∑i∈I τi for short, is the type of I-classified values of the form in[I][i](ei), or i · ei for short, where i ∈ I and ei is an expression of type τi. An I-classified value is analyzed by an I-way case analysis of the form case(e;{i ,→ xi.ei}i∈I). When I = { l1,..., ln }, the type of I-classified values may be written [i1 ,→ τ1,..., in ,→ τn] specifying the type associated to each class li ∈ I. Correspondingly, the I-way case analysis has the form case e {i1 · x1 ⇒ e1 | ... | in · xn ⇒ en}. Finite sums generalize empty and binary products by choosing I to be empty or the two-element set { l, r }, respectively. In practice I is often chosen to be a finite set of symbols that serve as symbolic names for the classes so as to enhance readability. REVISED 05.15.2012 VERSION 1.32 104 12.3 Applications of Sum Types The statics of finite sums is defined by the following rules: Γ ` e : τk (1 ≤ k ≤ n) Γ ` ik · e :[i1 ,→ τ1,..., in ,→ τn](12.3a) Γ ` e :[i1 ,→ τ1,..., in ,→ τn]Γ, x1 : τ1 ` e1 : τ ...Γ, xn : τn ` en : τ Γ ` case e {i1 · x1 ⇒ e1 | ... | in · xn ⇒ en}: τ (12.3b) These rules generalize to the finite case the statics for nullary and binary sums given in Section 12.1. The dynamics of finite sums is defined by the following rules: [e val] i · e val (12.4a)  e 7→ e0 i · e 7→ i · e0  (12.4b) e 7→ e0 case e {i · xi ⇒ ei}i∈I 7→ case e0 {i · xi ⇒ ei}i∈I (12.4c) i · e val case i · e {i · xi ⇒ ei}i∈I 7→ [e/xi]ei (12.4d) These again generalize the dynamics of binary sums given in Section 12.1. Theorem 12.2 (Safety). If e : τ, then either e val or there exists e0 : τ such that e 7→ e0. Proof. The proof is similar to that for the binary case, as described in Sec- tion 12.1. 12.3 Applications of Sum Types Sum types have numerous uses, several of which we outline here. More interesting examples arise once we also have recursive types, which are introduced in PartV. 12.3.1 Void and Unit It is instructive to compare the types unit and void, which are often con- fused with one another. The type unit has exactly one element, hi, whereas the type void has no elements at all. Consequently, if e : unit, then if e eval- uates to a value, it must be unit — in other words, e has no interesting value VERSION 1.32 REVISED 05.15.2012 12.3 Applications of Sum Types 105 (but it could diverge). On the other hand, if e : void, then e must not yield a value; if it were to have a value, it would have to be a value of type void, of which there are none. This shows that what is called the void type in many languages is really the type unit because it indicates that an expression has no interesting value, not that it has no value at all! 12.3.2 Booleans Perhaps the simplest example of a sum type is the familiar type of Booleans, whose syntax is given by the following grammar: Typ τ ::= bool bool booleans Exp e ::= true true truth false false falsity if(e; e1; e2) if e then e1 else e2 conditional The expression if(e; e1; e2) branches on the value of e : bool. The statics of Booleans is given by the following typing rules: Γ ` true : bool (12.5a) Γ ` false : bool (12.5b) Γ ` e : bool Γ ` e1 : τ Γ ` e2 : τ Γ ` if e then e1 else e2 : τ (12.5c) The dynamics is given by the following value and transition rules: true val (12.6a) false val (12.6b) if true then e1 else e2 7→ e1 (12.6c) if false then e1 else e2 7→ e2 (12.6d) e 7→ e0 if e then e1 else e2 7→ if e0 then e1 else e2 (12.6e) REVISED 05.15.2012 VERSION 1.32 106 12.3 Applications of Sum Types The type bool is definable in terms of binary sums and nullary prod- ucts: bool = unit + unit (12.7a) true = l · hi (12.7b) false = r · hi (12.7c) if e then e1 else e2 = case e {l · x1 ⇒ e1 | r · x2 ⇒ e2}(12.7d) In the last equation above the variables x1 and x2 are chosen arbitrarily such that x1 /∈ e1 and x2 /∈ e2. It is a simple matter to check that the evident statics and dynamics of the type bool are engendered by these definitions. 12.3.3 Enumerations More generally, sum types may be used to define finite enumeration types, those whose values are one of an explicitly given finite set, and whose elim- ination form is a case analysis on the elements of that set. For example, the type suit, whose elements are ♣, ♦, ♥, and ♠, has as elimination form the case analysis case e {♣ ⇒ e0 | ♦ ⇒ e1 | ♥ ⇒ e2 | ♠ ⇒ e3}, which distinguishes among the four suits. Such finite enumerations are easily representable as sums. For example, we may define suit = [unit] ∈I, where I = { ♣, ♦, ♥, ♠ } and the type family is constant over this set. The case analysis form for a labeled sum is almost literally the desired case anal- ysis for the given enumeration, the only difference being the binding for the uninteresting value associated with each summand, which we may ignore. 12.3.4 Options Another use of sums is to define the option types, which have the following syntax: Typ τ ::= opt(τ) τ opt option Exp e ::= null null nothing just(e) just(e) something ifnull[τ](e; e1; x.e2) check e {null ⇒ e1 | just(x) ⇒ e2} null test The type opt(τ) represents the type of “optional” values of type τ. The introductory forms are null, corresponding to “no value”, and just(e), VERSION 1.32 REVISED 05.15.2012 12.3 Applications of Sum Types 107 corresponding to a specified value of type τ. The elimination form dis- criminates between the two possibilities. The option type is definable from sums and nullary products according to the following equations:1 τ opt = unit + τ (12.8a) null = l · hi (12.8b) just(e) = r · e (12.8c) check e {null ⇒ e1 | just(x2) ⇒ e2} = case e {l · ⇒ e1 | r · x2 ⇒ e2} (12.8d) We leave it to the reader to examine the statics and dynamics implied by these definitions. The option type is the key to understanding a common misconception, the null pointer fallacy. This fallacy, which is particularly common in object- oriented languages, is based on two related errors. The first error is to deem the values of certain types to be mysterious entities called pointers, based on suppositions about how these values might be represented at run-time, rather than on the semantics of the type itself. The second error compounds the first. A particular value of a pointer type is distinguished as the null pointer, which, unlike the other elements of that type, does not designate a value of that type at all, but rather rejects all attempts to use it as such. To help avoid such failures, such languages usually include a function, say null : τ → bool, that yields true if its argument is null, and false otherwise. This allows the programmer to take steps to avoid using null as a value of the type it purports to inhabit. Consequently, programs are riddled with conditionals of the form if null(e) then . . . error . . . else . . . proceed . . .. (12.9) Despite this, “null pointer” exceptions at run-time are rampant, in part be- cause it is quite easy to overlook the need for such a test, and in part be- cause detection of a null pointer leaves little recourse other than abortion of the program. The underlying problem may be traced to the failure to distinguish the type τ from the type τ opt. Rather than think of the elements of type τ as pointers, and thereby have to worry about the null pointer, we instead distinguish between a genuine value of type τ and an optional value of type 1We often write an underscore in place of a bound variable that is not used within its scope. REVISED 05.15.2012 VERSION 1.32 108 12.4 Notes τ. An optional value of type τ may or may not be present, but, if it is, the underlying value is truly a value of type τ (and cannot be null). The elimination form for the option type, check e {null ⇒ eerror | just(x) ⇒ eok}, (12.10) propagates the information that e is present into the non-null branch by binding a genuine value of type τ to the variable x. The case analysis ef- fects a change of type from “optional value of type τ” to “genuine value of type τ”, so that within the non-null branch no further null checks, explicit or implicit, are required. Observe that such a change of type is not achieved by the simple Boolean-valued test exemplified by expression (12.9); the ad- vantage of option types is precisely that it does so. 12.4 Notes Heterogeneous data structures are ubiquitous. Sums codify heterogeneity, yet few languages support them in the form given here. The best approxi- mation in commercial languages is the concept of a class in object-oriented programming. A class is an injection into a sum type, and dispatch is case analysis on the class of the data object. (See Chapter 25 for more on this correspondence.) The absence of sums is the origin of C.A.R. Hoare’s self- described “billion dollar mistake,” the null pointer (Hoare, 2009). Bad lan- guage designs impose the burden of handling “null” values on program- mers, resulting in countless errors that manifest themselves only at run- time. VERSION 1.32 REVISED 05.15.2012 Chapter 13 Pattern Matching Pattern matching is a natural and convenient generalization of the elimina- tion forms for product and sum types. For example, rather than write let x be e in x · l + x · r to add the components of a pair, e, of natural numbers, we may instead write match e {hx1, x2i ⇒ x1 + x2}, using pattern matching to name the components of the pair and refer to them directly. The first argument to the match is called the match value and the second argument consist of a finite sequence of rules, separated by ver- tical bars. In this example there is only one rule, but as we shall see shortly there is, in general, more than one rule in a given match expression. Each rule consists of a pattern, possibly involving variables, and an expression that may involve those variables. The value of the match is determined by considering each rule in the order given to determine the first rule whose pattern matches the match value. If such a rule is found, the value of the match is the value of the expression part of the matching rule, with the variables of the pattern replaced by the corresponding components of the match value. Pattern matching becomes more interesting, and useful, when com- bined with sums. The patterns l · x and r · x match the corresponding val- ues of sum type. These may be used in combination with other patterns to express complex decisions about the structure of a value. For example, the following match expresses the computation that, when given a pair of type (unit + unit) × nat, either doubles or squares its second component 110 13.1 A Pattern Language depending on the form of its first component: match e {hl · hi, xi ⇒ x + x | hr · hi, yi ⇒ y ∗ y}. (13.1) It is an instructive exercise to express the same computation using only the primitives for sums and products given in Chapters 11 and 12. In this chapter we study a simple language, L{pat}, of pattern matching over eager product and sum types. 13.1 A Pattern Language The abstract syntax of L{pat} is defined by the following grammar: Exp e ::= match(e; rs) match e {rs} case analysis Rules rs ::= rules[n](r1;...; rn) r1 | ... | rn (n ≥ 0) Rule r ::= rule[k](p; x1,..., xk.e) p ⇒ e (k ≥ 0) Pat p ::= wild wild card x x variable triv hi unit pair(p1; p2) hp1, p2i pair in[l](p) l · p left injection in[r](p) r · p right injection The operator match has two operands, the expression to match and a series of rules. A sequence of rules is constructed using the operator rules[n], which has n ≥ 0 operands. Each rule is constructed by the operator rule[k], which specifies that it has two operands, binding k variables in the second. 13.2 Statics The statics of L{pat} makes use of a special form of hypothetical judgment, written x1 : τ1,..., xk : τk p : τ, with almost the same meaning as x1 : τ1,..., xk : τk ` p : τ, except that each variable is required to be used at most once in p. When reading the judgment Λ p : τ it is helpful to think of Λ as an output, and VERSION 1.32 REVISED 05.15.2012 13.2 Statics 111 p and τ as inputs. Given p and τ, the rules determine the hypotheses Λ such that Λ p : τ. x : τ x : τ (13.2a) ∅ : τ (13.2b) ∅ hi : unit (13.2c) Λ1 p1 : τ1 Λ2 p2 : τ2 dom(Λ1) ∩ dom(Λ2) = ∅ Λ1 Λ2 hp1, p2i : τ1 × τ2 (13.2d) Λ1 p : τ1 Λ1 l · p : τ1 + τ2 (13.2e) Λ2 p : τ2 Λ2 r · p : τ1 + τ2 (13.2f) Rule (13.2a) states that a variable is a pattern of type τ. Rule (13.2d) states that a pair pattern consists of two patterns with disjoint variables. The typing judgments for a rule, p ⇒ e : τ τ0, and for a sequence of rules, r1 | ... | rn : τ τ0, specify that rules transform a value of type τ into a value of type τ0. These judgments are inductively defined as follows: Λ p : τ ΓΛ ` e : τ0 Γ ` p ⇒ e : τ τ0 (13.3a) Γ ` r1 : τ τ0 ...Γ ` rn : τ τ0 Γ ` r1 | ... | rn : τ τ0 (13.3b) Using the typing judgments for rules, the typing rule for a match ex- pression may be stated quite easily: Γ ` e : τ Γ ` rs : τ τ0 Γ ` match e {rs}: τ0 (13.4) REVISED 05.15.2012 VERSION 1.32 112 13.3 Dynamics 13.3 Dynamics A substitution, θ, is a finite mapping from variables to values. If θ is the sub- stitution {x0 ,→ e0,..., xk−1 ,→ ek−1}, we write ˆθ(e) for [e1,..., ek/x1,..., xk]e. The judgment θ :Λ is inductively defined by the following rules: ∅ : ∅ (13.5a) θ :Λ e : τ θ ⊗ x ,→ e :Λ, x : τ (13.5b) The judgment θ p / e states that the pattern, p, matches the value, e, as witnessed by the substitution, θ, defined on the variables of p. This judgment is inductively defined by the following rules: x ,→ e x / e (13.6a) ∅ / e (13.6b) ∅ hi / hi (13.6c) θ1 p1 / e1 θ2 p2 / e2 dom(θ1) ∩ dom(θ2) = ∅ θ1 ⊗ θ2 hp1, p2i / he1, e2i (13.6d) θ p / e θ l · p / l · e (13.6e) θ p / e θ r · p / r · e (13.6f) These rules simply collect the bindings for the pattern variables required to form a substitution witnessing the success of the matching process. The judgment e ⊥ p states that e does not match the pattern p. It is inductively defined by the following rules: e1 ⊥ p1 he1, e2i ⊥ hp1, p2i (13.7a) e2 ⊥ p2 he1, e2i ⊥ hp1, p2i (13.7b) l · e ⊥ r · p (13.7c) e ⊥ p l · e ⊥ l · p (13.7d) VERSION 1.32 REVISED 05.15.2012 13.3 Dynamics 113 r · e ⊥ l · p (13.7e) e ⊥ p r · e ⊥ r · p (13.7f) Neither a variable nor a wildcard nor a null-tuple can mismatch any value of appropriate type. A pair can only mismatch a pair pattern due to a mis- match in one of its components. An injection into a sum type can mismatch the opposite injection, or it can mismatch the same injection by having its argument mismatch the argument pattern. Theorem 13.1. Suppose that e : τ, e val, and Λ p : τ. Then either there exists θ such that θ :Λ and θ p / e, or e ⊥ p. Proof. By rule induction on Rules (13.2), making use of the canonical forms lemma to characterize the shape of e based on its type. The dynamics of the match expression is given in terms of the pattern match and mismatch judgments as follows: e 7→ e0 match e {rs} 7→ match e0 {rs}(13.8a) e val match e {} err (13.8b) e val θ p0 / e match e {p0 ⇒ e0|rs} 7→ ˆθ(e0)(13.8c) e val e ⊥ p0 match e {rs} 7→ e0 match e {p0 ⇒ e0|rs} 7→ e0 (13.8d) Rule (13.8b) specifies that evaluation results in a checked error once all rules are exhausted. Rules (13.8c) specifies that the rules are to be considered in order. If the match value, e, matches the pattern, p0, of the initial rule in the sequence, then the result is the corresponding instance of e0; otherwise, matching continues by considering the remaining rules. Theorem 13.2 (Preservation). If e 7→ e0 and e : τ, then e0 : τ. Proof. By a straightforward induction on the derivation of e 7→ e0. REVISED 05.15.2012 VERSION 1.32 114 13.4 Exhaustiveness and Redundancy 13.4 Exhaustiveness and Redundancy Although it is possible to state and prove a progress theorem for L{pat} as defined in Section 13.1, it would not have much force, because the statics does not rule out pattern matching failure. What is missing is enforcement of the exhaustiveness of a sequence of rules, which ensures that every value of the domain type of a sequence of rules must match some rule in the sequence. In addition it would be useful to rule out redundancy of rules, which arises when a rule can only match values that are also matched by a preceding rule. Because pattern matching considers rules in the order in which they are written, such a rule can never be executed, and hence can be safely eliminated. 13.4.1 Match Constraints To express exhaustiveness and irredundancy, we introduce a language of match constraints that identify a subset of the closed values of a type. With each rule we associate a constraint that classifies the values that are matched by that rule. A sequence of rules is exhaustive if every value of the domain type of the rules satisfies the match constraint of some rule in the sequence. A rule in a sequence is redundant if every value that satisfies its match con- straint also satisfies the match constraint of some preceding rule. The language of match constraints is defined by the following grammar: Constr ξ ::= all[τ] > truth and(ξ1; ξ2) ξ1 ∧ ξ2 conjunction nothing[τ] ⊥ falsity or(ξ1; ξ2) ξ1 ∨ ξ2 disjunction l · ξ1 l · ξ1 left injection r · ξ2 r · ξ2 right injection triv hi unit pair(ξ1; ξ2) hξ1, ξ2i pair It is easy to define the judgment ξ : τ specifying that the constraint ξ con- strains values of type τ. The De Morgan Dual, ξ, of a match constraint, ξ, is defined by the fol- VERSION 1.32 REVISED 05.15.2012 13.4 Exhaustiveness and Redundancy 115 lowing rules: > =⊥ ξ1 ∧ ξ2 = ξ1 ∨ ξ2 ⊥ = > ξ1 ∨ ξ2 = ξ1 ∧ ξ2 l · ξ1 = l · ξ1 ∨ r · > r · ξ1 = r · ξ1 ∨ l · > hi =⊥ hξ1, ξ2i = hξ1, ξ2i ∨ hξ1, ξ2i ∨ hξ1, ξ2i Intuitively, the dual of a match constraint expresses the negation of that constraint. In the case of the last four rules it is important to keep in mind that these constraints apply only to specific types. The satisfaction judgment, e |= ξ, is defined for values e and constraints ξ of the same type by the following rules: e |= > (13.9a) e |= ξ1 e |= ξ2 e |= ξ1 ∧ ξ2 (13.9b) e |= ξ1 e |= ξ1 ∨ ξ2 (13.9c) e |= ξ2 e |= ξ1 ∨ ξ2 (13.9d) e1 |= ξ1 l · e1 |= l · ξ1 (13.9e) e2 |= ξ2 r · e2 |= r · ξ2 (13.9f) hi |= hi (13.9g) e1 |= ξ1 e2 |= ξ2 he1, e2i |= hξ1, ξ2i (13.9h) The De Morgan dual construction negates a constraint. REVISED 05.15.2012 VERSION 1.32 116 13.4 Exhaustiveness and Redundancy Lemma 13.3. If ξ is a constraint on values of type τ, then e |= ξ if, and only if, e 6|= ξ. We define the entailment of two constraints, ξ1 |= ξ2 to mean that e |= ξ2 whenever e |= ξ1. By Lemma 13.3 we have that ξ1 |= ξ2 iff |= ξ1 ∨ ξ2. We often write ξ1,..., ξn |= ξ for ξ1 ∧ ... ∧ ξn |= ξ so that in particular |= ξ means e |= ξ for every value e : τ. 13.4.2 Enforcing Exhaustiveness and Redundancy To enforce exhaustiveness and irredundancy the statics of pattern match- ing is augmented with constraints that express the set of values matched by a given set of rules. A sequence of rules is exhaustive if every value of suitable type satisfies the associated constraint. A rule is redundant relative to the preceding rules if every value satisfying its constraint satisfies one of the preceding constraints. A sequence of rules is irredundant iff no rule is redundant relative to the rules that precede it in the sequence. The judgment Λ p : τ [ξ] augments the judgment Λ p : τ with a match constraint characterizing the set of values of type τ matched by the pattern p. It is inductively defined by the following rules: x : τ x : τ [>](13.10a) ∅ : τ [>](13.10b) ∅ hi : unit [hi](13.10c) Λ1 p : τ1 [ξ1] Λ1 l · p : τ1 + τ2 [l · ξ1](13.10d) Λ2 p : τ2 [ξ2] Λ2 r · p : τ1 + τ2 [r · ξ2](13.10e) Λ1 p1 : τ1 [ξ1]Λ2 p2 : τ2 [ξ2] dom(Λ1) ∩ dom(Λ2) = ∅ Λ1 Λ2 hp1, p2i : τ1 × τ2 [hξ1, ξ2i](13.10f) Lemma 13.4. Suppose that Λ p : τ [ξ]. For every e : τ such that e val, e |= ξ iff θ p / e for some θ, and e 6|= ξ iff e ⊥ p. The judgment Γ ` r : τ τ0 [ξ] augments the formation judgment for a rule with a match constraint characterizing the pattern component of the rule. The judgment Γ ` rs : τ τ0 [ξ] augments the formation judgment VERSION 1.32 REVISED 05.15.2012 13.4 Exhaustiveness and Redundancy 117 for a sequence of rules with a match constraint characterizing the values matched by some rule in the given rule sequence. Λ p : τ [ξ]ΓΛ ` e : τ0 Γ ` p ⇒ e : τ τ0 [ξ](13.11a) (∀1 ≤ i ≤ n) ξi 6|= ξ1 ∨ ... ∨ ξi−1 Γ ` r1 : τ τ0 [ξ1]...Γ ` rn : τ τ0 [ξn] Γ ` r1 | ... | rn : τ τ0 [ξ1 ∨ ... ∨ ξn] (13.11b) Rule (13.11b) requires that each successive rule not be redundant relative to the preceding rules. The overall constraint associated to the rule sequence specifies that every value of type τ satisfy the constraint associated with some rule. The typing rule for match expressions demands that the rules that com- prise it be exhaustive: Γ ` e : τ Γ ` rs : τ τ0 [ξ] |= ξ Γ ` match e {rs}: τ0 (13.12) Rule (13.11b) ensures that ξ is a disjunction of the match constraints asso- ciated to the constituent rules of the match expression. The requirement that ξ be valid amounts to requiring that every value of type τ satisfies the constraint of at least one rule of the match. Theorem 13.5. If e : τ, then either e val or there exists e0 such that e 7→ e0. Proof. The exhaustiveness check in Rule (13.12) ensures that if e val and e : τ, then e |= ξ. The form of ξ given by Rule (13.11b) ensures that e |= ξi for some constraint ξi corresponding to the ith rule. By Lemma 13.4 the value e must match the ith rule, which is enough to ensure progress. 13.4.3 Checking Exhaustiveness and Redundancy Checking exhaustiveness and redundacy reduces to showing that the con- straint validity judgment |= ξ is decidable. We will prove this by defining a judgment Ξ incon, where Ξ is a finite set of constraints of the same type, with the meaning that no value of this type satisfies all of the constraints in Ξ. We will then show that either Ξ incon or not. The rules defining inconsistency of a finite set, Ξ, of constraints of the same type are as follows: Ξ incon Ξ, > incon (13.13a) REVISED 05.15.2012 VERSION 1.32 118 13.4 Exhaustiveness and Redundancy Ξ, ξ1, ξ2 incon Ξ, ξ1 ∧ ξ2 incon (13.13b) Ξ, ⊥ incon (13.13c) Ξ, ξ1 incon Ξ, ξ2 incon Ξ, ξ1 ∨ ξ2 incon (13.13d) Ξ, l · ξ1, r · ξ2 incon (13.13e) Ξ incon l ·Ξ incon (13.13f) Ξ incon r ·Ξ incon (13.13g) Ξ1 incon hΞ1,Ξ2i incon (13.13h) Ξ2 incon hΞ1,Ξ2i incon (13.13i) In Rule (13.13f) we write l ·Ξ for the finite set of constraints l · ξ1,..., l · ξn, where Ξ = ξ1,..., ξn, and similarly in Rules (13.13g), (13.13h), and (13.13i). Lemma 13.6. It is decidable whether or not Ξ incon. Proof. The premises of each rule involves only constraints that are proper components of the constraints in the conclusion. Consequently, we can simplify Ξ by inverting each of the applicable rules until no rule applies, then determine whether or not the resulting set, Ξ0, is contradictory in the sense that it contains ⊥ or both l · ξ and r · ξ0 for some ξ and ξ0. Lemma 13.7. Ξ incon iff Ξ |= ⊥. Proof. From left to right we proceed by induction on Rules (13.13). From right to left we may show that if Ξ incon is not derivable, then there exists a value e such that e |= Ξ, and hence Ξ 6|= ⊥. VERSION 1.32 REVISED 05.15.2012 13.5 Notes 119 13.5 Notes Pattern-matching against heterogeneous structured data was first explored in the context of logic programming languages, such as Prolog (Kowalski, 1988; Colmerauer and Roussel, 1993), but with an execution model based on proof search. Pattern matching in the form described here is present in the functional languages Miranda (Turner, 1987), Hope (Burstall et al., 1980), Standard ML (Milner et al., 1997), Caml (Cousineau and Mauny, 1998), and Haskell (Jones, 2003). REVISED 05.15.2012 VERSION 1.32 120 13.5 Notes VERSION 1.32 REVISED 05.15.2012 Chapter 14 Generic Programming 14.1 Introduction Many programs can be seen as instances of a general pattern applied to a particular situation. Very often the pattern is determined by the types of the data involved. For example, in Chapter9 the pattern of computing by recursion over a natural number is isolated as the defining characteristic of the type of natural numbers. This concept will itself emerge as an instance of the concept of type-generic, or just generic, programming. Suppose that we have a function, f, of type ρ → ρ0 that transforms values of type ρ into values of type ρ0. For example, f might be the doubling function on natural numbers. We wish to extend f to a transformation from type [ρ/t]τ to type [ρ0/t]τ by applying f to various spots in the input where a value of type ρ occurs to obtain a value of type ρ0, leaving the rest of the data structure alone. For example, τ might be bool × ρ, in which case f could be extended to a function of type bool × ρ → bool × ρ0 that sends the pairs ha, bi to the pair ha, f(b)i. This example glosses over a significant problem of ambiguity of the ex- tension. Given a function f of type ρ → ρ0, it is not obvious in general how to extend it to a function mapping [ρ/t]τ to [ρ0/t]τ. The problem is that it is not clear which of many occurrences of ρ in [ρ/t]τ are to be transformed by f, even if there is only one occurrence of ρ. To avoid ambiguity we need a way to mark which occurrences of ρ in [ρ/t]τ are to be transformed, and which are to be left fixed. This can be achieved by isolating the type operator, t.τ, which is a type expression in which a designated variable, t, marks the spots at which we wish the transformation to occur. Given t.τ and f : ρ → ρ0, we can extend f unambiguously to a function of type 122 14.2 Type Operators [ρ/t]τ → [ρ0/t]τ. The technique of using a type operator to determine the behavior of a piece of code is called generic programming. The power of generic pro- gramming depends on which forms of type operator are considered. The simplest case is that of a polynomial type operator, one constructed from sum and product of types, including their nullary forms. These may be extended to positive type operators, which also permit restricted forms of function types. 14.2 Type Operators A type operator is a type equipped with a designated variable whose oc- currences mark the positions in the type where a transformation is to be applied. A type operator is represented by an abstractor t.τ such that t type ` τ type. An example of a type operator is the abstractor t.unit + (bool × t) in which occurrences of t mark the spots in which a transformation is to be applied. An instance of the type operator t.τ is obtained by substitut- ing a type, ρ, for the variable, t, within the type τ. We sometimes write Map[t.τ](ρ) for the substitution instance [ρ/t]τ. The polynomial type operators are those constructed from the type vari- able, t, the types void and unit, and the product and sum type construc- tors, τ1 × τ2 and τ1 + τ2. It is a straightforward exercise to give inductive definitions of the judgment t.τ poly stating that the operator t.τ is a poly- nomial type operator. 14.3 Generic Extension The generic extension primitive has the form map[t.τ](x.e0; e) with statics given by the following rule: t type ` τ type Γ, x : ρ ` e0 : ρ0 Γ ` e :[ρ/t]τ Γ ` map[t.τ](x.e0; e):[ρ0/t]τ (14.1) The abstractor x.e0 specifies a transformation from type ρ, the type of x, to type ρ0, the type of e0. The expression e of type [ρ/t]τ determines the value VERSION 1.32 REVISED 05.15.2012 14.3 Generic Extension 123 to be transformed to obtain a value of type [ρ0/t]τ. The occurrences of t in τ determine the spots at which the transformation given by x.e is to be performed. The dynamics of generic extension is specified by the following rules. We consider here only polynomial type operators, leaving the extension to positive type operators to be considered later. map[t.t](x.e0; e) 7→ [e/x]e0 (14.2a) map[t.unit](x.e0; e) 7→ hi (14.2b) map[t.τ1 × τ2](x.e0; e) 7→ hmap[t.τ1](x.e0; e · l), map[t.τ2](x.e0; e · r)i (14.2c) map[t.void](x.e0; e) 7→ abort(e)(14.2d) map[t.τ1 + τ2](x.e0; e) 7→ case e {l · x1 ⇒ l · map[t.τ1](x.e0; x1) | r · x2 ⇒ r · map[t.τ2](x.e0; x2)} (14.2e) Rule (14.2a) applies the transformation x.e0 to e itself, because the operator t.t specifies that the transformation is to be perfomed directly. Rule (14.2b) states that the empty tuple is transformed to itself. Rule (14.2c) states that to transform e according to the operator t.τ1 × τ2, the first component of e is transformed according to t.τ1 and the second component of e is trans- formed according to t.τ2. Rule (14.2d) states that the transformation of a value of type void aborts, because there can be no such values. Rule (14.2e) states that to transform e according to t.τ1 + τ2, case analyze e and recon- struct it after transforming the injected value according to t.τ1 or t.τ2. Consider the type operator t.τ given by t.unit + (bool × t). Let x.e be the abstractor x.s(x), which increments a natural number. Using Rules (14.2) we may derive that map[t.τ](x.e; r · htrue, ni) 7→∗ r · htrue, n + 1i. REVISED 05.15.2012 VERSION 1.32 124 14.3 Generic Extension The natural number in the second component of the pair is incremented, because the type variable, t, occurs in that position in the type operator t.τ. Theorem 14.1 (Preservation). If map[t.τ](x.e0; e): τ0 and map[t.τ](x.e0; e) 7→ e00, then e00 : τ0. Proof. By inversion of Rule (14.1) we have 1. t type ` τ type; 2. x : ρ ` e0 : ρ0 for some ρ and ρ0; 3. e :[ρ/t]τ; 4. τ0 is [ρ0/t]τ. We proceed by cases on Rules (14.2). For example, consider Rule (14.2c). It follows from inversion that map[t.τ1](x.e0; e · l) :[ρ0/t]τ1, and similarly that map[t.τ2](x.e0; e · r) :[ρ0/t]τ2. It is easy to check that hmap[t.τ1](x.e0; e · l), map[t.τ2](x.e0; e · r)i has type [ρ0/t]τ1 × τ2, as required. The positive type operators extend the polynomial type operators to ad- mit restricted forms of function type. Specifically, t.τ1 → τ2 is a positive type operator, provided that (1) t does not occur in τ1, and (2) t.τ2 is a pos- itive type operator. In general, any occurrences of a type variable t in the domain a function type are said to be negative occurrences, whereas any oc- currences of t within the range of a function type, or within a product or sum type, are said to be positive occurrences.1 A positive type operator is one for which only positive occurrences of the parameter, t, are permitted. The generic extension according to a positive type operator is defined similarly to the case of a polynomial type operator, with the following ad- ditional rule: map[t.τ1 → τ2](x.e0; e) 7→ λ (x1:τ1) map[t.τ2](x.e0; e(x1)) (14.3) 1The origin of this terminology seems to be that a function type τ1 → τ2 is analogous to the implication φ1 ⊃ φ2, which is classically equivalent to ¬φ1 ∨ φ2, so that occurrences in the domain are under the negation. VERSION 1.32 REVISED 05.15.2012 14.4 Notes 125 Because t is not permitted to occur within the domain type, the type of the result is τ1 → [ρ0/t]τ2, assuming that e is of type τ1 → [ρ/t]τ2. It is easy to verify preservation for the generic extension of a positive type operator. It is interesting to consider what goes wrong if we relax the restric- tion on positive type operators to admit negative, as well as positive, oc- currences of the parameter of a type operator. Consider the type opera- tor t.τ1 → τ2, without restriction on t, and suppose that x : ρ ` e0 : ρ0. The generic extension map[t.τ1 → τ2](x.e0; e) should have type [ρ0/t]τ1 → [ρ0/t]τ2, given that e has type [ρ/t]τ1 → [ρ/t]τ2. The extension should yield a function of the form λ (x1:[ρ0/t]τ1)...(e(...(x1))) in which we apply e to a transformation of x1 and then transform the re- sult. The trouble is that we are given, inductively, that map[t.τ1](x.e0; −) transforms values of type [ρ/t]τ1 into values of type [ρ0/t]τ1, but we need to go the other way around in order to make x1 suitable as an argument for e. Unfortunately, there is no obvious way to obtain the required transforma- tion. One solution to this is to assume that the fundamental transformation x.e0 is invertible so that we may apply the inverse transformation on x1 to get an argument of type suitable for e, then apply the forward transforma- tion on the result, just as in the positive case. Because we cannot invert an arbitrary transformation, we must instead pass both the transformation and its inverse to the generic extension operation so that it can “go back- wards” as necessary to cover negative occurrences of the type parameter. So the generic extension applies only when we are given a type isomorphism (a pair of mutually inverse mappings between two types), and then results in another isomorphism pair. 14.4 Notes The generic extension of a type operator is an example of the concept of a functor in category theory (MacLane, 1998). Generic programming is es- sentially functorial programming, exploiting the functorial action of poly- nomial type operators (Hinze and Jeuring, 2003). REVISED 05.15.2012 VERSION 1.32 126 14.4 Notes VERSION 1.32 REVISED 05.15.2012 Part V Infinite Data Types Chapter 15 Inductive and Co-Inductive Types The inductive and the coinductive types are two important forms of recur- sive type. Inductive types correspond to least, or initial, solutions of certain type isomorphism equations, and coinductive types correspond to their greatest, or final, solutions. Intuitively, the elements of an inductive type are those that may be obtained by a finite composition of its introductory forms. Consequently, if we specify the behavior of a function on each of the introductory forms of an inductive type, then its behavior is determined for all values of that type. Such a function is called a recursor, or catamorphism. Dually, the elements of a coinductive type are those that behave properly in response to a finite composition of its elimination forms. Consequently, if we specify the behavior of an element on each elimination form, then we have fully specified that element as a value of that type. Such an element is called an generator, or anamorphism. 15.1 Motivating Examples The most important example of an inductive type is the type of natural numbers as formalized in Chapter9. The type nat is defined to be the least type containing z and closed under s(−). The minimality condition is witnessed by the existence of the recursor, iter e {z ⇒ e0 | s(x) ⇒ e1}, which transforms a natural number into a value of type τ, given its value for zero, and a transformation from its value on a number to its value on the successor of that number. This operation is well-defined precisely because there are no other natural numbers. Put the other way around, the existence 130 15.1 Motivating Examples of this operation expresses the inductive nature of the type nat. With a view towards deriving the type nat as a special case of an in- ductive type, it is useful to consolidate zero and successor into a single introductory form, and to correspondingly consolidate the basis and in- ductive step of the recursor. The following rules specify the statics of this reformulation: Γ ` e : unit + nat Γ ` foldnat(e): nat (15.1a) Γ, x : unit + τ ` e1 : τ Γ ` e2 : nat Γ ` recnat[x.e1](e2): τ (15.1b) The expression foldnat(e) is the unique introductory form of the type nat. Using this, the expression z is defined to be foldnat(l · hi), and s(e) is de- fined to be foldnat(r · e). The recursor, recnat[x.e1](e2), takes as argu- ment the abstractor x.e1 that consolidates the basis and inductive step into a single computation that, given a value of type unit + τ, yields a value of type τ. Intuitively, if x is replaced by the value l · hi, then e1 computes the base case of the recursion, and if x is replaced by the value r · e, then e1 computes the inductive step as a function of the result, e, of the recursive call. The dynamics of the consolidated representation of natural numbers is given by the following rules: foldnat(e) val (15.2a) e2 7→ e0 2 recnat[x.e1](e2) 7→ recnat[x.e1](e0 2)(15.2b) recnat[x.e1](foldnat(e2)) 7→ [map[t.unit + t](y.recnat[x.e1](y); e2)/x]e1 (15.2c) Rule (15.2c) makes use of generic extension (see Chapter 14) to apply the recursor to the predecessor, if any, of a natural number. The idea is that the result of extending the recursor from the type unit + nat to the type unit + τ is substituted into the inductive step, given by the expression e1. If we expand the definition of the generic extension in place, we obtain the VERSION 1.32 REVISED 05.15.2012 15.1 Motivating Examples 131 following reformulation of this rule: recnat[x.e1](foldnat(e2)) 7→ [case e2 {l · ⇒ l · hi | r · y ⇒ r · recnat[x.e1](y)}/x]e1 An illustrative example of a coinductive type is the type of streams of natural numbers. A stream is an infinite sequence of natural numbers such that an element of the stream can be computed only after computing all preceding elements in that stream. That is, the computations of successive elements of the stream are sequentially dependent in that the computation of one element influences the computation of the next. This characteristic of the introductory form for streams is dual to the analogous property of the eliminatory form for natural numbers whereby the result for a number is determined by its result for all preceding numbers. A stream is characterized by its behavior under the elimination forms for the stream type: hd(e) returns the next, or head, element of the stream, and tl(e) returns the tail of the stream, the stream resulting when the head element is removed. A stream is introduced by a generator, the dual of a recursor, that determines the head and the tail of the stream in terms of the current state of the stream, which is represented by a value of some type. The statics of streams is given by the following rules: Γ ` e : stream Γ ` hd(e): nat (15.3a) Γ ` e : stream Γ ` tl(e): stream (15.3b) Γ ` e : τ Γ, x : τ ` e1 : nat Γ, x : τ ` e2 : τ Γ ` strgen e {hd(x) ⇒ e1 | tl(x) ⇒ e2}: stream (15.3c) In Rule (15.3c) the current state of the stream is given by the expression e of some type τ, and the head and tail of the stream are determined by the expressions e1 and e2, respectively, as a function of the current state. The dynamics of streams is given by the following rules: strgen e {hd(x) ⇒ e1 | tl(x) ⇒ e2} val (15.4a) e 7→ e0 hd(e) 7→ hd(e0)(15.4b) REVISED 05.15.2012 VERSION 1.32 132 15.1 Motivating Examples hd(strgen e {hd(x) ⇒ e1 | tl(x) ⇒ e2}) 7→ [e/x]e1 (15.4c) e 7→ e0 tl(e) 7→ tl(e0)(15.4d) tl(strgen e {hd(x) ⇒ e1 | tl(x) ⇒ e2}) 7→ strgen [e/x]e2 {hd(x) ⇒ e1 | tl(x) ⇒ e2} (15.4e) Rules (15.4c) and (15.4e) express the dependency of the head and tail of the stream on its current state. Observe that the tail is obtained by applying the generator to the new state determined by e2 as a function of the current state. To derive streams as a special case of a coinductive type, we consolidate the head and the tail into a single eliminatory form, and reorganize the generator correspondingly. This leads to the following statics: Γ ` e : stream Γ ` unfoldstream(e): nat × stream (15.5a) Γ, x : τ ` e1 : nat × τ Γ ` e2 : τ Γ ` genstream[x.e1](e2): stream (15.5b) Rule (15.5a) states that a stream may be unfolded into a pair consisting of its head, a natural number, and its tail, another stream. The head, hd(e), and tail, tl(e), of a stream, e, are defined to be the projections unfoldstream(e)· l and unfoldstream(e)· r, respectively. Rule (15.5b) states that a stream may be generated from the state element, e2, by an expression e1 that yields the head element and the next state as a function of the current state. The dynamics of streams is given by the following rules: genstream[x.e1](e2) val (15.6a) e 7→ e0 unfoldstream(e) 7→ unfoldstream(e0)(15.6b) unfoldstream(genstream[x.e1](e2)) 7→ map[t.nat × t](y.genstream[x.e1](y);[e2/x]e1) (15.6c) VERSION 1.32 REVISED 05.15.2012 15.2 Statics 133 Rule (15.6c) uses generic extension to generate a new stream whose state is the second component of [e2/x]e1. Expanding the generic extension we obtain the following reformulation of this rule: unfoldstream(genstream[x.e1](e2)) 7→ h([e2/x]e1)· l, genstream[x.e1](([e2/x]e1)· r)i 15.2 Statics We may now give a fully general account of inductive and coinductive types, which are defined in terms of positive type operators. We will con- sider the language, L{µiµf}, with inductive and co-inductive types. 15.2.1 Types The syntax of inductive and coinductive types involves type variables, which are, of course, variables ranging over types. The abstract syntax of induc- tive and coinductive types is given by the following grammar: Typ τ ::= t t self-reference ind(t.τ) µi(t.τ) inductive coi(t.τ) µf(t.τ) coinductive Type formation judgments have the form t1 type,..., tn type ` τ type, where t1,..., tn are type names. We let ∆ range over finite sets of hypothe- ses of the form t type, where t is a type name. The type formation judgment is inductively defined by the following rules: ∆, t type ` t type (15.7a) ∆ ` unit type (15.7b) ∆ ` τ1 type ∆ ` τ2 type ∆ ` prod(τ1; τ2) type (15.7c) ∆ ` void type (15.7d) REVISED 05.15.2012 VERSION 1.32 134 15.3 Dynamics ∆ ` τ1 type ∆ ` τ2 type ∆ ` sum(τ1; τ2) type (15.7e) ∆ ` τ1 type ∆ ` τ2 type ∆ ` arr(τ1; τ2) type (15.7f) ∆, t type ` τ type ∆ ` t.τ pos ∆ ` ind(t.τ) type (15.7g) ∆, t type ` τ type ∆ ` t.τ pos ∆ ` coi(t.τ) type (15.8) 15.2.2 Expressions The abstract syntax of expressions for inductive and coinductive types is given by the following grammar: Exp e ::= fold[t.τ](e) fold(e) constructor rec[t.τ][x.e1](e2) rec[x.e1](e2) recursor unfold[t.τ](e) unfold(e) destructor gen[t.τ][x.e1](e2) gen[x.e1](e2) generator The statics for inductive and coinductive types is given by the following typing rules: Γ ` e :[ind(t.τ)/t]τ Γ ` fold[t.τ](e): ind(t.τ)(15.9a) Γ, x :[ρ/t]τ ` e1 : ρ Γ ` e2 : ind(t.τ) Γ ` rec[t.τ][x.e1](e2): ρ (15.9b) Γ ` e : coi(t.τ) Γ ` unfold[t.τ](e):[coi(t.τ)/t]τ (15.9c) Γ ` e2 : ρ Γ, x : ρ ` e1 :[ρ/t]τ Γ ` gen[t.τ][x.e1](e2): coi(t.τ)(15.9d) 15.3 Dynamics The dynamics of inductive and coinductive types is given in terms of the generic extension operation described in Chapter 14. The following rules specify a lazy dynamics for L{µiµf}: fold(e) val (15.10a) VERSION 1.32 REVISED 05.15.2012 15.4 Notes 135 e2 7→ e0 2 rec[x.e1](e2) 7→ rec[x.e1](e0 2)(15.10b) rec[x.e1](fold(e2)) 7→ [map[t.τ](y.rec[x.e1](y); e2)/x]e1 (15.10c) gen[x.e1](e2) val (15.10d) e 7→ e0 unfold(e) 7→ unfold(e0)(15.10e) unfold(gen[x.e1](e2)) 7→ map[t.τ](y.gen[x.e1](y);[e2/x]e1) (15.10f) Rule (15.10c) states that to evaluate the recursor on a value of recursive type, we inductively apply the recursor as guided by the type operator to the value, and then perform the inductive step on the result. Rule (15.10f) is simply the dual of this rule for coinductive types. Lemma 15.1. If e : τ and e 7→ e0, then e0 : τ. Proof. By rule induction on Rules (15.10). Lemma 15.2. If e : τ, then either e val or there exists e0 such that e 7→ e0. Proof. By rule induction on Rules (15.9). 15.4 Notes The treatment of inductive and coinductive types is derived from Mendler (1987), which is based on the categorial analysis of these concepts (MacLane, 1998; Taylor, 1999). The functorial action of a type constructor (described in Chapter 14) plays a central role. Specifically, inductive types are initial algebras and coinductive types are final coalgebras for a functor given by a composition of type constructors. The positivity requirement imposed on well-formed inductive and coinductive types ensures that the action of the associated type constructor is properly functorial. REVISED 05.15.2012 VERSION 1.32 136 15.4 Notes VERSION 1.32 REVISED 05.15.2012 Chapter 16 Recursive Types Inductive and coinductive types, such as natural numbers and streams, may be seen as examples of fixed points of type operators up to isomorphism. An isomorphism between two types, τ1 and τ2, is given by two expressions 1. x1 : τ1 ` e2 : τ2, and 2. x2 : τ2 ` e1 : τ1 that are mutually inverse to each other.1 For example, the types nat and unit + nat are isomorphic, as witnessed by the following two expressions: 1. x : unit + nat ` case x {l · ⇒ z | r · x2 ⇒ s(x2)}: nat, and 2. x : nat ` ifz x {z ⇒ l · hi | s(x2) ⇒ r · x2}: unit + nat. These are called, respectively, the fold and unfold operations of the iso- morphism nat ∼= unit + nat. Thinking of unit + nat as [nat/t](unit + t), this means that nat is a fixed point of the type operator t.unit + t. In this chapter we study the language L{+×*µ}, which provides so- lutions to all such type equations. The recursive type µt.τ is defined to be a solution to the isomorphism problem µt.τ ∼= [µt.τ/t]τ. This is witnessed by the operations x : µt.τ ` unfold(x):[µt.τ/t]τ 1To make this precise requires a discussion of equivalence of expressions to be taken up in Chapter 47. For now we will rely on an intuitive understanding of when two expressions are equivalent. 138 16.1 Solving Type Isomorphisms and x :[µt.τ/t]τ ` fold(x): µt.τ, which are mutually inverse to each other. Requiring solutions to all type equations may seem suspicious, because we know by Cantor’s Theorem that an isomorphism such as X ∼= (X → 2) is set-theoretically impossible. This negative result tells us not that our re- quirement is untenable, but rather that types are not sets. To permit solution of arbitrary type equations, we must take into account that types describe computations, some of which may not even terminate. Consequently, the function space does not coincide with the set-theoretic function space, but rather is analogous to it (in a precise sense that we shall not go into here). 16.1 Solving Type Isomorphisms The recursive type µt.τ, where t.τ is a type operator, represents a solution for t to the isomorphism t ∼= τ. The solution is witnessed by two oper- ations, fold(e) and unfold(e), that relate the recursive type µt.τ to its unfolding, [µt.τ/t]τ, and serve, respectively, as its introduction and elimi- nation forms. The language L{+×*µ} extends L{*} with recursive types and their associated operations. Typ τ ::= t t self-reference rec(t.τ) µt.τ recursive Exp e ::= fold[t.τ](e) fold(e) constructor unfold(e) unfold(e) destructor The statics of L{+×*µ} consists of two forms of judgment. The first, called type formation, is a general hypothetical judgment of the form ∆ ` τ type, where ∆ has the form t1 type,..., tk type. Type formation is inductively defined by the following rules: ∆, t type ` t type (16.1a) ∆ ` τ1 type ∆ ` τ2 type ∆ ` arr(τ1; τ2) type (16.1b) VERSION 1.32 REVISED 05.15.2012 16.2 Recursive Data Structures 139 ∆, t type ` τ type ∆ ` rec(t.τ) type (16.1c) The second form of judgment of the statics is the typing judgment, which is a hypothetical judgment of the form Γ ` e : τ, where we assume that τ type. Typing for L{+×*µ} is inductively defined by the following rules: Γ ` e :[rec(t.τ)/t]τ Γ ` fold[t.τ](e): rec(t.τ)(16.2a) Γ ` e : rec(t.τ) Γ ` unfold(e):[rec(t.τ)/t]τ (16.2b) The dynamics of L{+×*µ} is specified by one axiom stating that the elimination form is inverse to the introduction form. [e val] fold[t.τ](e) val (16.3a)  e 7→ e0 fold[t.τ](e) 7→ fold[t.τ](e0)  (16.3b) e 7→ e0 unfold(e) 7→ unfold(e0)(16.3c) fold[t.τ](e) val unfold(fold[t.τ](e)) 7→ e (16.3d) The bracketed premise and rule are to be included for an eager interpreta- tion of the introduction form, and omitted for a lazy interpretation. It is a straightforward exercise to prove type safety for L{+×*µ}. Theorem 16.1 (Safety). 1. If e : τ and e 7→ e0, then e0 : τ. 2. If e : τ, then either e val, or there exists e0 such that e 7→ e0. 16.2 Recursive Data Structures One important application of recursive types is to the representation of in- ductive data types such as the type of natural numbers. We may think of the type nat as a solution (up to isomorphism) of the type equation nat ∼= [z ,→ unit, s ,→ nat] REVISED 05.15.2012 VERSION 1.32 140 16.2 Recursive Data Structures According to this isomorphism every natural number is either zero or the successor of another natural number. A solution is given by the recursive type µt.[z ,→ unit, s ,→ t]. (16.4) The introductory forms for the type nat are defined by the following equa- tions: z = fold(z · hi) s(e) = fold(s · e). The conditional branch may then be defined as follows: ifz e {z ⇒ e0 | s(x) ⇒ e1} = case unfold(e){z · ⇒ e0 | s · x ⇒ e1}, where the “underscore” indicates a variable that does not occur free in e0. It is easy to check that these definitions exhibit the expected behavior. As another example, the type list of lists of natural numbers may be represented by the recursive type µt.[n ,→ unit, c ,→ nat × t] so that we have the isomorphism list ∼= [n ,→ unit, c ,→ nat × list]. The list formation operations are represented by the following equations: nil = fold(n · hi) cons(e1; e2) = fold(c · he1, e2i). A conditional branch on the form of the list may be defined by the follow- ing equation: case e {nil ⇒ e0 | cons(x; y) ⇒ e1} = case unfold(e){n · ⇒ e0 | c · hx, yi ⇒ e1}, where we have used an underscore for a “don’t care” variable, and used pattern-matching syntax to bind the components of a pair. As long as sums and products are evaluated eagerly, there is a natural correspondence between this representation of lists and the conventional “blackboard notation” for linked lists. We may think of fold as an abstract VERSION 1.32 REVISED 05.15.2012 16.3 Self-Reference 141 heap-allocated pointer to a tagged cell consisting of either (a) the tag n with no associated data, or (b) the tag c attached to a pair consisting of a natural number and another list, which must be an abstract pointer of the same sort. If sums or products are evaluated lazily, then the blackboard notation breaks down because it is unable to depict the suspended computations that are present in the data structure. In general there is no substitute for the type itself. Drawings can be helpful, but the type determines the semantics. We may also represent coinductive types, such as the type of streams of natural numbers, using recursive types. The representation is particularly natural in the case that fold(−) is evaluated lazily, for then we may define the type stream to be the recursive type µt.nat × t. This states that every stream may be thought of as a computation of a pair consisting of a number and another stream. If fold(−) is evaluated ea- gerly, then we may instead consider the recursive type µt.unit → (nat × t), which expresses the same representation of streams. In either case streams cannot be easily depicted in blackboard notation, not so much because they are infinite, but because there is no accurate way to depict the delayed com- putation other than by an expression in the programming language. Here again we see that pictures can be helpful, but are not adequate for accu- rately defining a data structure. 16.3 Self-Reference In the general recursive expression, fix[τ](x.e), the variable, x, stands for the expression itself. This is ensured by the unrolling transition fix[τ](x.e) 7→ [fix[τ](x.e)/x]e, which substitutes the expression itself for x in its body during execution. It is useful to think of x as an implicit argument to e that is implicitly in- stantiated to itself whenever the expression is used. In many well-known languages this implicit argument has a special name, such as this or self, to emphasize its self-referential interpretation. Using this intuition as a guide, we may derive general recursion from recursive types. This derivation shows that general recursion may, like REVISED 05.15.2012 VERSION 1.32 142 16.3 Self-Reference other language features, be seen as a manifestation of type structure, rather than an ad hoc language feature. The derivation is based on isolating a type of self-referential expressions given by the following grammar: Typ τ ::= self(τ) τ self self-referential type Exp e ::= self[τ](x.e) self x is e self-referential expression unroll(e) unroll(e) unroll self-reference The statics of these constructs is given by the following rules: Γ, x : self(τ) ` e : τ Γ ` self[τ](x.e): self(τ)(16.5a) Γ ` e : self(τ) Γ ` unroll(e): τ (16.5b) The dynamics is given by the following rule for unrolling the self-reference: self[τ](x.e) val (16.6a) e 7→ e0 unroll(e) 7→ unroll(e0)(16.6b) unroll(self[τ](x.e)) 7→ [self[τ](x.e)/x]e (16.6c) The main difference, compared to general recursion, is that we distinguish a type of self-referential expressions, rather than impose self-reference at every type. However, as we shall see shortly, the self-referential type is sufficient to implement general recursion, so the difference is largely one of technique. The type self(τ) is definable from recursive types. As suggested ear- lier, the key is to consider a self-referential expression of type τ to be a func- tion of the expression itself. That is, we seek to define the type self(τ) so that it satisfies the isomorphism self(τ) ∼= self(τ) → τ. This means that we seek a fixed point of the type operator t.t → τ, where t /∈ τ is a type variable standing for the type in question. The required fixed point is just the recursive type rec(t.t → τ), VERSION 1.32 REVISED 05.15.2012 16.3 Self-Reference 143 which we take as the definition of self(τ). The self-referential expression self[τ](x.e) is then defined to be the expression fold(λ (x:self(τ)) e). We may easily check that Rule (16.5a) is derivable according to this defi- nition. The expression unroll(e) is correspondingly defined to be the ex- pression unfold(e)(e). It is easy to check that Rule (16.5b) is derivable from this definition. More- over, we may check that unroll(self[τ](y.e)) 7→∗ [self[τ](y.e)/y]e. This completes the derivation of the type self(τ) of self-referential expres- sions of type τ. One consequence of admitting the self-referential type self(τ) is that we may use it to define general recursion for any type. To be precise, we may define fix[τ](x.e) to stand for the expression unroll(self[τ](y.[unroll(y)/x]e)) in which we have unrolled the recursion at each occurrence of x within e. It is easy to check that this verifies the statics of general recursion given in Chapter 10. Moreover, it also validates the dynamics, as evidenced by the following derivation: fix[τ](x.e) = unroll(self[τ](y.[unroll(y)/x]e)) 7→∗ [unroll(self[τ](y.[unroll(y)/x]e))/x]e = [fix[τ](x.e)/x]e. It follows that recursive types may be used to define a non-terminating expression of every type, namely fix[τ](x.x). Unlike many other type constructs we have considered, recursive types change the meaning of ev- ery type, not just those that involve recursion. Recursive types are there- fore said to be a non-conservative extension of languages such as L{nat →}, which otherwise admits no non-terminating computations. REVISED 05.15.2012 VERSION 1.32 144 16.4 The Origin of State 16.4 The Origin of State The notion of state in a computation—which will be discussed thoroughly in Part XIII—has its origins in the concept of recursion, or self-reference, which, as we have just seen, arises from the concept of recursive types. For example, you may be familiar with the concept of a flip-flop or a latch at the hardware level. These are circuits built from combinational logic elements (typically, nor or nand gates) that have the characteristic that they maintain an alterable state over time. An RS latch, for example, maintains its output at the logical level of zero or one in response to a signal on the R or S inputs, respectively, after a brief settling delay. This behavior is achieved using feedback, which is just a form of self-reference, or recursion: the output of the gate is fed back into its input so as to convey the current state of the gate to the logic that determines its next state. One way to model an RS latch using recursive types is to make explicit the passage of time in the determination of the current output of the gate as a function of its inputs and its previous outputs. An RS latch is a value of type τrsl given by µt.hX,→ bool,Q,→ bool,N,→ ti. The X and Q components of the latch represent its current outputs (of which Q represents the current state of the latch), and the N component represents the next state of the latch. If e is of type τrsl, then we define e @X to mean unfold(e)·X, and define e @Q and e @N similarly. The expressions e @X and e @Q evaluate to the “current” outputs of the latch, e, and e @N evaluates to another latch representing the “next” state determined as a function of the “current” state.2 For given values, r and s, a new latch is computed from an old latch by the recursive function rsl defined as follows: fix rsl is λ (o:τrsl) fix this is ersl, where ersl is given by fold(hX,→ nor(hs, o @Qi),Q,→ nor(hr, o @Xi),N,→ rsl(this)i) 2For simplicity the R and S inputs are fixed, which amounts to requiring that we build a new latch whenever these are changed. It is straightforward to modify the construction so that new R and S inputs may be provided whenever the next state of a latch is computed, allowing for these inputs to change over time. VERSION 1.32 REVISED 05.15.2012 16.5 Notes 145 and nor is the obvious function defined on the booleans.3 The outputs of the latch are computed as a function of the r and s inputs and the ouputs of the previous state of the latch. To get the construction started, we define an initial state of the latch in which the outputs are arbitrarily set to false, and whose next state is determined by applying rsl to the initial state: fix this is fold(hX,→ false,Q,→ false,N,→ rsl(this)i). Selection of the N component causes the outputs to be recalculated based on the current outputs. Notice the essential role of self-reference in main- taining the state of the latch. The foregoing implementation of a latch models time explicitly by pro- viding the N component of the latch to compute the next state from the current one. It is also possible to model time implicitly by treating the latch as a transducer whose inputs and outputs are signals that change over time. A signal may be represented by a stream of booleans (as described in Chap- ter 15 or using general recursive types as described earlier in this chapter), in which case a transducer is a stream transformer that computes the suc- cessive elements of the outputs from the successive elements of the inputs by applying a function to them. This implicit formulation is arguably more natural than the explicit one given above, but it nevertheless relies on recur- sive types and self-reference, just as does the implementation given above. 16.5 Notes The systematic study of recursive types in programming was initiated by Scott(1976, 1982) to provide a mathematical model of the untyped λ-calculus. The derivation of recursion from recursive types is essentially an applica- tion of Scott’s theory to find the interpretation of a fixed point combina- tor in a model of the λ-calculus given by a recursive type. The category- theoretic view of recursive types was developed by Wand(1979) and Smyth and Plotkin(1982). Implementing state using self-reference is fundamental to digital logic. Abadi and Cardelli(1996) and Cook(2009), among others, explore similar ideas to model objects. The account of signals as streams is inspired by the pioneering work of Kahn (MacQueen, 2009). 3It suffices to require that fold be evaluated lazily to ensure that recursion is well- grounded. This assumption is unnecessary if the next state component is abstracted on the R and S inputs, as suggested earlier. REVISED 05.15.2012 VERSION 1.32 146 16.5 Notes VERSION 1.32 REVISED 05.15.2012 Part VI Dynamic Types Chapter 17 The Untyped λ-Calculus Types are the central organizing principle in the study of programming languages. Yet many languages of practical interest are said to be untyped. Have we missed something important? The answer is no. The supposed opposition between typed and untyped languages turns out to be illusory. In fact, untyped languages are special cases of typed languages with a sin- gle, pre-determined recursive type. Far from being untyped, such languages are uni-typed. In this chapter we study the premier example of a uni-typed program- ming language, the (untyped) λ-calculus. This formalism was introduced by Church in the 1930’s as a universal language of computable functions. It is distinctive for its austere elegance. The λ-calculus has but one “feature”, the higher-order function. Everything is a function, hence every expression may be applied to an argument, which must itself be a function, with the result also being a function. To borrow a turn of phrase, in the λ-calculus it’s functions all the way down. 17.1 The λ-Calculus The abstract syntax of L{λ} is given by the following grammar: Exp u ::= x x variable λ(x.u) λ (x) u λ-abstraction ap(u1; u2) u1(u2) application The statics of L{λ} is defined by general hypothetical judgments of the form x1 ok,..., xn ok ` u ok, stating that u is a well-formed expression in- volving the variables x1,..., xn. (As usual, we omit explicit mention of the 150 17.2 Definability parameters when they can be determined from the form of the hypotheses.) This relation is inductively defined by the following rules: Γ, x ok ` x ok (17.1a) Γ ` u1 ok Γ ` u2 ok Γ ` ap(u1; u2) ok (17.1b) Γ, x ok ` u ok Γ ` λ(x.u) ok (17.1c) The dynamics of L{λ} is given equationally, rather than via a transition system. Definitional equality for L{λ} is a judgment of the form Γ ` u ≡ u0, where Γ = x1 ok,..., xn ok for some n ≥ 0, and u and u0 are terms having at most the variables x1,..., xn free. It is inductively defined by the following rules: Γ, u ok ` u ≡ u (17.2a) Γ ` u ≡ u0 Γ ` u0 ≡ u (17.2b) Γ ` u ≡ u0 Γ ` u0 ≡ u00 Γ ` u ≡ u00 (17.2c) Γ ` e1 ≡ e0 1 Γ ` e2 ≡ e0 2 Γ ` ap(e1; e2) ≡ ap(e0 1; e0 2)(17.2d) Γ, x ok ` u ≡ u0 Γ ` λ(x.u) ≡ λ(x.u0)(17.2e) Γ, x ok ` e2 ok Γ ` e1 ok Γ ` ap(λ(x.e2); e1) ≡ [e1/x]e2 (17.2f) We often write just u ≡ u0 when the variables involved need not be empha- sized or are clear from context. 17.2 Definability Interest in the untyped λ-calculus stems from its surprising expressiveness. It is a Turing-complete language in the sense that it has the same capability to express computations on the natural numbers as does any other known programming language. Church’s Law states that any conceivable notion VERSION 1.32 REVISED 05.15.2012 17.2 Definability 151 of computable function on the natural numbers is equivalent to the λ- calculus. This is certainly true for all known means of defining computable functions on the natural numbers. The force of Church’s Law is that it pos- tulates that all future notions of computation will be equivalent in expres- sive power (measured by definability of functions on the natural numbers) to the λ-calculus. Church’s Law is therefore a scientific law in the same sense as, say, Newton’s Law of Universal Gravitation, which makes a prediction about all future measurements of the acceleration in a gravitational field.1 We will sketch a proof that the untyped λ-calculus is as powerful as the language PCF described in Chapter 10. The main idea is to show that the PCF primitives for manipulating the natural numbers are definable in the untyped λ-calculus. This means, in particular, that we must show that the natural numbers are definable as λ-terms in such a way that case analysis, which discriminates between zero and non-zero numbers, is definable. The principal difficulty is with computing the predecessor of a number, which requires a bit of cleverness. Finally, we show how to represent general recursion, completing the proof. The first task is to represent the natural numbers as certain λ-terms, called the Church numerals. 0 , λ (b) λ (s) b (17.3a) n + 1 , λ (b) λ (s) s(n(b)(s)) (17.3b) It follows that n(u1)(u2) ≡ u2(...(u2(u1))), the n-fold application of u2 to u1. That is, n iterates its second argument (the induction step) n times, starting with its first argument (the basis). Using this definition it is not difficult to define the basic functions of arithmetic. For example, successor, addition, and multiplication are de- fined by the following untyped λ-terms: succ , λ (x) λ (b) λ (s) s(x(b)(s)) (17.4) plus , λ (x) λ (y) y(x)(succ) (17.5) times , λ (x) λ (y) y(0)(plus(x)) (17.6) 1Unfortunately, it is common in Computer Science to put forth as “laws” assertions that are not scientific laws at all. For example, Moore’s Law is merely an observation about a near-term trend in microprocessor fabrication that is certainly not valid over the long term, and Amdahl’s Law is but a simple truth of arithmetic. Worse, Church’s Law, which is a proper scientific law, is usually called Church’s Thesis, which, to the author’s ear, suggests something less than the full force of a scientific law. REVISED 05.15.2012 VERSION 1.32 152 17.2 Definability It is easy to check that succ(n) ≡ n + 1, and that similar correctness con- ditions hold for the representations of addition and multiplication. To define ifz(u; u0; x.u1) requires a bit of ingenuity. We wish to find a term pred such that pred(0) ≡ 0 (17.7) pred(n + 1) ≡ n. (17.8) To compute the predecessor using Church numerals, we must show how to compute the result for n + 1 as a function of its value for n. At first glance this seems straightforward—just take the successor—until we consider the base case, in which we define the predecessor of 0 to be 0. This invalidates the obvious strategy of taking successors at inductive steps, and necessi- tates some other approach. What to do? A useful intuition is to think of the computation in terms of a pair of “shift registers” satisfying the invariant that on the nth iteration the registers contain the predecessor of n and n itself, respectively. Given the result for n, namely the pair (n − 1, n), we pass to the result for n + 1 by shifting left and incrementing to obtain (n, n + 1). For the base case, we initialize the registers with (0, 0), reflecting the stipulation that the prede- cessor of zero be zero. To compute the predecessor of n we compute the pair (n − 1, n) by this method, and return the first component. To make this precise, we must first define a Church-style representation of ordered pairs. hu1, u2i , λ (f) f(u1)(u2)(17.9) u · l , u(λ (x) λ (y) x)(17.10) u · r , u(λ (x) λ (y) y)(17.11) It is easy to check that under this encoding hu1, u2i · l ≡ u1, and that a similar equivalence holds for the second projection. We may now define the required representation, up, of the predecessor function: u0 p , λ (x) x(h0, 0i)(λ (y) hy · r, succ (y · r)i)(17.12) up , λ (x) u0 p(x)· l (17.13) It is easy to check that this gives us the required behavior. Finally, we may define ifz(u; u0; x.u1) to be the untyped term u(u0)(λ ()[up(u)/x]u1). VERSION 1.32 REVISED 05.15.2012 17.3 Scott’s Theorem 153 This gives us all the apparatus of PCF, apart from general recursion. But this is also definable using a fixed point combinator. There are many choices of fixed point combinator, of which the best known is the Y combinator: Y, λ (F)(λ (f)F(f(f)))(λ (f)F(f(f))). It is easy to check that Y(F) ≡ F(Y(F)). Using the Y combinator, we may define general recursion by writing Y(λ (x) u), where x stands for the recursive expression itself. Although it is clear that Y as just defined computes a fixed point of its argument, it is probably less clear why it works or how we might have invented it in the first place. The main idea is actually quite simple. If a function is to be recursive, it is given an additional first argument, which is arranged to stand for the function itself. Whenever the function wishes to call itself, it calls the implicit first argument, which is for this reason often called this or self. At each call site to a recursive function, the function is applied to itself before being applied to any other argument. This ensures that the argument called this actually refers to the function itself. With this in mind, it is easy to see how to derive the definition of Y. If F is the function whose fixed point we seek, then the function F0 = λ (f)F(f(f)) is a variant of F in which the self-application convention has been imposed by replacing each use of f in F(f) by f(f). Now ob- serve that F0(F0) ≡ F(F0(F0)), so that F0(F0) is the desired fixed point of F. Expanding the definition of F0, we have derived that the desired fixed point is λ (f)F(f(f))(λ (f)F(f(f))). To finish the derivation, we need only observe that nothing depends on the particular choice of F, which means that we can compute a fixed point for F uniformly in F. That is, we may define a single function, namely Y as defined above, that computes the fixed point of any F. 17.3 Scott’s Theorem Scott’s Theorem states that definitional equality for the untyped λ-calculus is undecidable: there is no algorithm to determine whether or not two un- typed terms are definitionally equal. The proof uses the concept of insepa- rability. Any two properties, A0 and A1, of λ-terms are inseparable if there is no decidable property, B, such that A0 u implies that B u and A1 u implies REVISED 05.15.2012 VERSION 1.32 154 17.3 Scott’s Theorem that it is not the case that B u. We say that a property, A, of untyped terms is behavioral iff whenever u ≡ u0, then A u iff A u0. The proof of Scott’s Theorem decomposes into two parts: 1. For any untyped λ-term u, we may find an untyped term v such that u(pvq) ≡ v, where pvq is the G¨odel number of v, and pvq is its rep- resentation as a Church numeral. (See Chapter9 for a discussion of G¨odel-numbering.) 2. Any two non-trivial2 behavioral properties A0 and A1 of untyped terms are inseparable. Lemma 17.1. For any u there exists v such that u(pvq) ≡ v. Proof Sketch. The proof relies on the definability of the following two oper- ations in the untyped λ-calculus: 1. ap(pu1q)(pu2q) ≡ pu1(u2)q. 2. nm(n) ≡ pnq. Intuitively, the first takes the representations of two untyped terms, and builds the representation of the application of one to the other. The sec- ond takes a numeral for n, and yields the representation of n. Given these, we may find the required term v by defining v , w(pwq), where w , λ (x) u(ap(x)(nm(x))). We have v = w(pwq) ≡ u(ap(pwq)(nm(pwq))) ≡ u(pw(pwq)q) ≡ u(pvq). The definition is very similar to that of Y(u), except that u takes as input the representation of a term, and we find a v such that, when applied to the representation of v, the term u yields v itself. Lemma 17.2. Suppose that A0 and A1 are two non-trivial behavioral properties of untyped terms. Then there is no untyped term w such that 1. For every u either w(puq) ≡ 0 or w(puq) ≡ 1. 2A property of untyped terms is said to be trivial if it either holds for all untyped terms or never holds for any untyped term. VERSION 1.32 REVISED 05.15.2012 17.4 Untyped Means Uni-Typed 155 2. If A0 u, then w(puq) ≡ 0. 3. If A1 u, then w(puq) ≡ 1. Proof. Suppose there is such an untyped term w. Let v be the untyped term λ (x) ifz(w(x); u1;.u0), where A0 u0 and A1 u1. By Lemma 17.1 there is an untyped term t such that v(ptq) ≡ t. If w(ptq) ≡ 0, then t ≡ v(ptq) ≡ u1, and so A1 t, because A1 is behavioral and A1 u1. But then w(ptq) ≡ 1 by the defining properties of w, which is a contradiction. Similarly, if w(ptq) ≡ 1, then A0 t, and hence w(ptq) ≡ 0, again a contradiction. Corollary 17.3. There is no algorithm to decide whether or not u ≡ u0. Proof. For fixed u, the property Eu u0 defined by u0 ≡ u is a non-trivial behavioral property of untyped terms. It is therefore inseparable from its negation, and hence is undecidable. 17.4 Untyped Means Uni-Typed The untyped λ-calculus may be faithfully embedded in a typed language with recursive types. This means that every untyped λ-term has a represen- tation as a typed expression in such a way that execution of the representa- tion of a λ-term corresponds to execution of the term itself. This embedding is not a matter of writing an interpreter for the λ-calculus in L{+×*µ} (which we could surely do), but rather a direct representation of untyped λ-terms as typed expressions in a language with recursive types. The key observation is that the untyped λ-calculus is really the uni-typed λ-calculus. It is not the absence of types that gives it its power, but rather that it has only one type, namely the recursive type D, µt.t → t. A value of type D is of the form fold(e) where e is a value of type D → D — a function whose domain and range are both D. Any such function can be regarded as a value of type D by “rolling”, and any value of type D can be turned into a function by “unrolling”. As usual, a recursive type may be seen as a solution to a type isomorphism equation, which in the present case is the equation D ∼= D → D. This specifies that D is a type that is isomorphic to the space of functions on D itself, something that is impossible in conventional set theory, but is feasible in the computationally-based setting of the λ-calculus. REVISED 05.15.2012 VERSION 1.32 156 17.5 Notes This isomorphism leads to the following translation, of L{λ} into L{+×*µ}: x†, x (17.14a) λ (x) u†, fold(λ (x:D) u†)(17.14b) u1(u2)†, unfold(u† 1)(u† 2)(17.14c) Observe that the embedding of a λ-abstraction is a value, and that the embedding of an application exposes the function being applied by un- rolling the recursive type. Consequently, λ (x) u1(u2)† = unfold(fold(λ (x:D) u† 1))(u† 2) ≡ λ (x:D) u† 1(u† 2) ≡ [u† 2/x]u† 1 = ([u2/x]u1)†. The last step, stating that the embedding commutes with substitution, is easily proved by induction on the structure of u1. Thus β-reduction is faith- fully implemented by evaluation of the embedded terms. Thus we see that the canonical untyped language, L{λ}, which by dint of terminology stands in opposition to typed languages, turns out to be but a typed language after all. Rather than eliminating types, an untyped language consolidates an infinite collection of types into a single recursive type. Doing so renders static type checking trivial, at the expense of incur- ring substantial dynamic overhead to coerce values to and from the recur- sive type. In Chapter 18 we will take this a step further by admitting many different types of data values (not just functions), each of which is a com- ponent of a “master” recursive type. This shows that so-called dynamically typed languages are, in fact, statically typed. Thus this traditional distinction can hardly be considered an opposition, because dynamic languages are but particular forms of static languages in which undue emphasis is placed on a single recursive type. 17.5 Notes The untyped λ-calculus was introduced by Church(1941) as a codifica- tion of the informal concept of a computable function. Unlike the well- known machine models, such as the Turing machine or the random access VERSION 1.32 REVISED 05.15.2012 17.5 Notes 157 machine, the λ-calculus directly codifies mathematical and programming practice. Barendregt(1984) is the definitive reference for all aspects of the untyped λ-calculus; the proof of Scott’s theorem is adapted from Baren- dregt’s account. Scott(1980) gave the first model of the untyped λ-calculus in terms of an elegant theory of recursive types. This construction under- lies Scott’s apt description of the λ-calculus as “unityped”, rather than “un- typed.” REVISED 05.15.2012 VERSION 1.32 158 17.5 Notes VERSION 1.32 REVISED 05.15.2012 Chapter 18 Dynamic Typing We saw in Chapter 17 that an untyped language may be viewed as a uni- typed language in which the so-called untyped terms are terms of a distin- guished recursive type. In the case of the untyped λ-calculus this recursive type has a particularly simple form, expressing that every term is isomor- phic to a function. Consequently, no run-time errors can occur due to the misuse of a value—the only elimination form is application, and its first ar- gument can only be a function. This property breaks down once more than one class of value is permitted into the language. For example, if we add natural numbers as a primitive concept to the untyped λ-calculus (rather than defining them via Church encodings), then it is possible to incur a run-time error arising from attempting to apply a number to an argument, or to add a function to a number. One school of thought in language design is to turn this vice into a virtue by embracing a model of computation that has multiple classes of value of a single type. Such languages are said to be dynamically typed, in purported opposition to statically typed languages. But the supposed opposition is illusory: just as the so-called untyped λ- calculus turns out to be uni-typed, so dynamic languages turn out to be but restricted forms of static language. This remark is so important it bears repeating: every dynamic language is inherently a static language in which we confine ourselves to a (needlessly) restricted type discipline to ensure safety. 18.1 Dynamically Typed PCF To illustrate dynamic typing we formulate a dynamically typed version of L{nat *}, called L{dyn}. The abstract syntax of L{dyn} is given by the 160 18.1 Dynamically Typed PCF following grammar: Exp d ::= x x variable num(n) n numeral zero zero zero succ(d) succ(d) successor ifz(d; d0; x.d1) ifz d {zero ⇒ d0 | succ(x) ⇒ d1} zero test fun(λ (x) d) λ (x) d abstraction ap(d1; d2) d1(d2) application fix(x.d) fix x is d recursion There are two classes of values in L{dyn}, the numbers, which have the form n, and the functions, which have the form λ (x) d. The expressions zero and succ(d) are not values, but rather are operations that evaluate to values. General recursion is definable using a fixed point combinator, but is taken as primitive here to simplify the analysis of the dynamics in Section 18.3. As usual, the abstract syntax of L{dyn} is what matters, but we use the concrete syntax to write examples in a convenient manner. However, it is often the case for dynamic languages, including L{dyn}, that the concrete syntax is deceptive in that it obscures an important detail of the abstract syntax, namely that every value is tagged with a classifier that plays a sig- nificant role at run-time (as we shall see shortly). So although the concrete syntax for a number, n, suggests a “bare” representation, the abstract syn- tax reveals that the number is labelled with the class num to indicate that the value is of the numeric class. This is done to distinguish it from a function value, which concretely has the form λ (x) d, but whose abstract syntax, fun(λ (x) d), indicates that it is to be classified with the tag fun to distin- guish it from a number. As we shall see shortly, this tagging is of prime importance in any dynamic language, so it is important to pay close atten- tion to the abstract form in what follows. The statics of L{dyn} is essentially the same as that of L{λ} given in Chapter 17; it merely checks that there are no free variables in the expres- sion. The judgment x1 ok,... xn ok ` d ok states that d is a well-formed expression with free variables among those in the hypotheses. If the assumptions are empty, then we write just d ok to indicate that d is a closed expression of L{dyn}. VERSION 1.32 REVISED 05.15.2012 18.1 Dynamically Typed PCF 161 The dynamics of L{dyn} must check for errors that would never arise in a language such as L{nat *}. For example, evaluation of a function application must ensure that the value being applied is indeed a function, signaling an error if it is not. Similarly the conditional branch must ensure that its principal argument is a number, signaling an error if it is not. To account for these possibilities, the dynamics is given by several judgment forms, as summarized in the following chart: d val d is a (closed) value d 7→ d0 d evaluates in one step to d0 d err d incurs a run-time error d is num n d is of class num with value n d isnt num d is not of class num d is fun x.d d is of class fun with body x.d d isnt fun d is not of class fun The last four judgment forms implement dynamic class checking. They are only relevant when d has already been determined to be a value. The affir- mative class-checking judgments have a second argument that represents the underlying structure of a value; this argument is not itself a value. The value judgment, d val, states that d is a fully evaluated (closed) expression: num(n) val (18.1a) fun(λ (x) d) val (18.1b) The affirmative class-checking judgments are defined by the following rules: num(n) is num n (18.2a) fun(λ (x) d) is fun x.d (18.2b) The negative class-checking judgments are correspondingly defined by these rules: num( ) isnt fun (18.3a) fun( ) isnt num (18.3b) The transition judgment, d 7→ d0, and the error judgment, d err, are defined simultaneously by the following rules: zero 7→ num(z) (18.4a) d 7→ d0 succ(d) 7→ succ(d0)(18.4b) REVISED 05.15.2012 VERSION 1.32 162 18.1 Dynamically Typed PCF d err succ(d) err (18.4c) d is num n succ(d) 7→ num(s(n)) (18.4d) d isnt num succ(d) err (18.4e) d 7→ d0 ifz(d; d0; x.d1) 7→ ifz(d0; d0; x.d1)(18.4f) d err ifz(d; d0; x.d1) err (18.4g) d is num 0 ifz(d; d0; x.d1) 7→ d0 (18.4h) d is num n + 1 ifz(d; d0; x.d1) 7→ [num(n)/x]d1 (18.4i) d isnt num ifz(d; d0; x.d1) err (18.4j) d1 7→ d0 1 ap(d1; d2) 7→ ap(d0 1; d2)(18.4k) d1 err ap(d1; d2) err (18.4l) d1 is fun x.d ap(d1; d2) 7→ [d2/x]d (18.4m) d1 isnt fun ap(d1; d2) err (18.4n) fix(x.d) 7→ [fix(x.d)/x]d (18.4o) Rule (18.4i) labels the predecessor with the class num to maintain the invari- ant that variables are bound to expressions of L{dyn}. Lemma 18.1 (Class Checking). If d val, then 1. either d is num n for some n, or d isnt num; 2. either d is fun x.d0 for some x and d0, or d isnt fun. Proof. By a straightforward inspection of the rules defining the class-checking judgments. VERSION 1.32 REVISED 05.15.2012 18.2 Variations and Extensions 163 Theorem 18.2 (Progress). If d ok, then either d val, or d err, or there exists d0 such that d 7→ d0. Proof. By induction on the structure of d. For example, if d = succ(d0), then we have by induction either d0 val, d0 err, or d0 7→ d00 for some d00. In last case we have by Rule (18.4b) that succ(d0) 7→ succ(d00), and in the second-to-last case we have by Rule (18.4c) that succ(d0) err. Is d0 val, then by Lemma 18.1, either d0 is num n or d0 isnt num. In the former case succ(d0) 7→ num(n + 1), and in the latter succ(d0) err. The other cases are handled similarly. Lemma 18.3 (Exclusivity). For any d in L{dyn}, exactly one of the following holds: d val, or d err, or d 7→ d0 for some d0. Proof. By induction on the structure of d, making reference to Rules (18.4). 18.2 Variations and Extensions The dynamic language L{dyn} defined in Section 18.1 closely parallels the static language L{nat *} defined in Chapter 10. One discrepancy, how- ever, is in the treatment of natural numbers. Whereas in L{nat *} the zero and successor operations are introductory forms for the type nat, in L{dyn} they are elimination forms that act on separately-defined numer- als. This is done to ensure that there is a single class of numbers, rather than a separate class for zero and successor. An alternative is to treat zero and succ(d) as values of two separate classes, and to introduce the obvious class checking judgments for them. This complicates the error checking rules, and admits problematic values such as succ(λ (x) d), but it allows us to avoid having a class of numbers. When written in this style, the dynamics of the conditional branch is given as follows: d 7→ d0 ifz(d; d0; x.d1) 7→ ifz(d0; d0; x.d1)(18.5a) d is zero ifz(d; d0; x.d1) 7→ d0 (18.5b) d is succ d0 ifz(d; d0; x.d1) 7→ [d0/x]d1 (18.5c) d isnt zero d isnt succ ifz(d; d0; x.d1) err (18.5d) REVISED 05.15.2012 VERSION 1.32 164 18.2 Variations and Extensions Notice that the predecessor of a value of the successor class need not be a number, whereas in the previous formulation this possibility does not arise. Structured data may be added to L{dyn} using similar techniques. The classic example is to introduce a null value, and a constructor for combin- ing two values into one. Exp d ::= nil nil null cons(d1; d2) cons(d1; d2) pair ifnil(d; d0; x, y.d1) ifnil d {nil ⇒ d0 | cons(x; y) ⇒ d1} conditional The expression ifnil(d; d0; x, y.d1) distinguishes the null value from a pair, and signals an error on any other class of value. Lists may be represented using null and pairing. For example, the list consisting of three zeroes is represented by the value cons(zero; cons(zero; cons(zero; nil))). But what to make of this beast? cons(zero; cons(zero; cons(zero; λ (x) x))) This does not correspond to a list, because it does not end with nil. The difficulty with encoding lists using null and pair becomes appar- ent when defining functions that operate on them. For example, here is a possible definition of the function that appends two lists: fix a is λ (x) λ (y) ifnil(x; y; x1, x2.cons(x1; a(x2)(y))) Nothing prevents us from applying this function to any two values, re- gardless of whether they are lists. If the first argument is not a list, then execution aborts with an error. But the function does not traverse its sec- ond argument, it can be any value at all. For example, we may append a list to a function, and obtain the “list” that ends with a λ given above. It might be argued that the conditional branch that distinguishes null from a pair is inappropriate in L{dyn}, because there are more than just these two classes in the language. One approach that avoids this criticism is to abandon the idea of pattern matching on the class of data entirely, replacing it by a general conditional branch that distinguishes null from all other values, and adding to the language predicates1 that test the class of a value and destructors that invert the constructors of each class. 1Predicates evaluate to the null value to indicate that a condition is false, and some non- null value to indicate that it is true. VERSION 1.32 REVISED 05.15.2012 18.2 Variations and Extensions 165 In the present case we would reformulate the extension of L{dyn} with null and pairing as follows: Exp d ::= cond(d; d0; d1) cond(d; d0; d1) conditional nil?(d) nil?(d) nil test cons?(d) cons?(d) pair test car(d) car(d) first projection cdr(d) cdr(d) second projection The conditional cond(d; d0; d1) distinguishes d between nil and all other values. If d is not nil, the conditional evaluates to d0, and otherwise eval- uates to d1. In other words the value nil represents boolean falsehood, and all other values represent boolean truth. The predicates nil?(d) and cons?(d) test the class of their argument, yielding nil if the argument is not of the specified class, and yielding some non-nil if so. The destructors car(d) and cdr(d)2 decompose cons(d1; d2) into d1 and d2, respectively. Written in this form, the append function is given by the expression fix a is λ (x) λ (y) cond(x; cons(car(x); a(cdr(x))(y)); y). The behavior of this formulation of append is no different from the earlier one; the only difference is that instead of dispatching on whether a value is either null or a pair, we instead allow discrimination on any predicate of the value, which includes such checks as special cases. An alternative, which is not widely used, is to enhance, rather than re- strict, the conditional branch so that it includes cases for each possible class of value in the language. So, for example, in a language with numbers, functions, null, and pairing, the conditional would have four branches. The fourth branch, for pairing, would deconstruct the pair into its constituent parts. The difficulty with this approach is that in realistic languages there are many classes of data, and such a conditional would be rather unwieldy. Moreover, even once we have dispatched on the class of a value, it is never- theless necessary for the primitive operations associated with that class to perform run-time checks. For example, we may determine that a value, d, is of the numeric class, but there is no way to propagate this information into the branch of the conditional that then adds d to some other number. The addition operation must still check the class of d, recover the underlying number, and create a new value of numeric class. This is an inherent lim- itation of dynamic languages, which do not permit handling values other than classified values. 2This terminology for the projections is archaic, but firmly established in the literature. REVISED 05.15.2012 VERSION 1.32 166 18.3 Critique of Dynamic Typing 18.3 Critique of Dynamic Typing The safety theorem for L{dyn} is often promoted as an advantage of dy- namic over static typing. Unlike static languages, which rule out some candidate programs as ill-typed, essentially every piece of abstract syntax in L{dyn} is well-formed, and hence, by Theorem 18.2, has a well-defined dynamics. But this can also be seen as a disadvantage, because errors that could be ruled out at compile time by type checking are not signalled until run time in L{dyn}. To make this possible, the dynamics of L{dyn} must enforce conditions that need not be checked in a statically typed language. Consider, for example, the addition function in L{dyn}, whose spec- ification is that, when passed two values of class num, returns their sum, which is also of class num:3 fun(λ (x) fix(p.fun(λ (y) ifz(y; x; y0.succ(p(y0)))))). The addition function may, deceptively, be written in concrete syntax as follows: λ (x) fix p is λ (y) ifz y {zero ⇒ x | succ(y0) ⇒ succ(p(y0))}. It is deceptive, because it obscures the class tags on values, and the opera- tions that check the validity of those tags. Let us now examine the costs of these operations in a bit more detail. First, observe that the body of the fixed point expression is labeled with class fun. The dynamics of the fixed point construct binds p to this function. This means that the dynamic class check incurred by the application of p in the recursive call is guaranteed to succeed. But L{dyn} offers no means of suppressing this redundant check, because it cannot express the invariant that p is always bound to a value of class fun. Second, observe that the result of applying the inner λ-abstraction is either x, the argument of the outer λ-abstraction, or the successor of a re- cursive call to the function itself. The successor operation checks that its argument is of class num, even though this is guaranteed for all but the base case, which returns the given x, which can be of any class at all. In principle we can check that x is of class num once, and observe that it is oth- erwise a loop invariant that the result of applying the inner function is of this class. However, L{dyn} gives us no way to express this invariant; the 3This specification imposes no restrictions on the behavior of addition on arguments that are not classified as numbers, but we could make the further demand that the function abort when applied to arguments that are not classified by num. VERSION 1.32 REVISED 05.15.2012 18.4 Notes 167 repeated, redundant tag checks imposed by the successor operation cannot be avoided. Third, the argument, y, to the inner function is either the original ar- gument to the addition function, or is the predecessor of some earlier re- cursive call. But as long as the original call is to a value of class num, then the dynamics of the conditional will ensure that all recursive calls have this class. And again there is no way to express this invariant in L{dyn}, and hence there is no way to avoid the class check imposed by the conditional branch. Classification is not free—storage is required for the class label, and it takes time to detach the class from a value each time it is used and to attach a class to a value whenever it is created. Although the overhead of classi- fication is not asymptotically significant (it slows down the program only by a constant factor), it is nevertheless non-negligible, and should be elim- inated whenever possible. But this is impossible within L{dyn}, because it cannot enforce the restrictions required to express the required invariants. For that we need a static type system. 18.4 Notes The earliest dynamically typed language is Lisp (McCarthy, 1965), which continues to influence language design a half century after its invention. Dynamic PCF is essentially the core of Lisp, but with a proper treatment of variable binding, correcting what McCarthy himself has described as an error in the original design. Informal discussions of dynamic languages are often confused by the ellision of the dynamic checks that are made ex- plicit here. Although the surface syntax of dynamic PCF is essentially the same as that for PCF, minus the type annotations, the underlying dynam- ics is fundamentally different. It is for this reason that static PCF cannot be properly seen as a restriction of dynamic PCF by the imposition of a type system, contrary to what seems to be a widely held belief. REVISED 05.15.2012 VERSION 1.32 168 18.4 Notes VERSION 1.32 REVISED 05.15.2012 Chapter 19 Hybrid Typing A hybrid language is one that combines static and dynamic typing by en- riching a statically typed language with a distinguished type, dyn, of dy- namic values. The dynamically typed language considered in Chapter 18 may be embedded into the hybrid language by regarding a dynamically typed program as a statically typed program of type dyn. This shows that static and dynamic types are not opposed to one another, but may coexist harmoniously. The notion of a hybrid language, however, is itself illusory, because the type dyn is really a particular recursive type. This shows that there is no need for any special mechanisms to support dynamic typing. Rather, they may be derived from the more general concept of a recursive type. More- over, this shows that dynamic typing is but a mode of use of static typing. The supposed opposition between dynamic and static typing is, therefore, a fallacy: dynamic typing can hardly be opposed to that of which it is but a special case. 19.1 A Hybrid Language Consider the language L{nat dyn *}, which extends L{nat *} with the following additional constructs: Typ τ ::= dyn dyn dynamic Exp e ::= new[l](e) l ! e construct cast[l](e) e ? l destruct Cls l ::= num num number fun fun function 170 19.1 A Hybrid Language The type dyn is the type of dynamically classified values. The new operation attaches a classifier to a value, and the cast operation checks the classifier and returns the associated value. The statics of L{nat dyn *} extends that of L{nat *} with the follow- ing additional rules: Γ ` e : nat Γ ` new[num](e): dyn (19.1a) Γ ` e : dyn * dyn Γ ` new[fun](e): dyn (19.1b) Γ ` e : dyn Γ ` cast[num](e): nat (19.1c) Γ ` e : dyn Γ ` cast[fun](e): dyn * dyn (19.1d) The statics ensures that class labels are applied to objects of the appropriate type, namely num for natural numbers, and fun for functions defined over labeled values. The dynamics of L{nat dyn *} extends that of L{nat *} with the fol- lowing rules: e val new[l](e) val (19.2a) e 7→ e0 new[l](e) 7→ new[l](e0)(19.2b) e 7→ e0 cast[l](e) 7→ cast[l](e0)(19.2c) new[l](e) val cast[l](new[l](e)) 7→ e (19.2d) new[l0](e) val l 6= l0 cast[l](new[l0](e)) err (19.2e) Casting compares the class of the object to the required class, returning the underlying object if these coincide, and signalling an error otherwise. Lemma 19.1 (Canonical Forms). If e : dyn and e val, then e = new[l](e0) for some class l and some e0 val. If l = num, then e0 : nat, and if l = fun, then e0 : dyn * dyn. Proof. By a straightforward rule induction on the statics of L{nat dyn *}. Theorem 19.2 (Safety). The language L{nat dyn *} is safe: VERSION 1.32 REVISED 05.15.2012 19.2 Dynamic as Static Typing 171 1. If e : τ and e 7→ e0, then e0 : τ. 2. If e : τ, then either e val, or e err, or e 7→ e0 for some e0. Proof. Preservation is proved by rule induction on the dynamics, and progress is proved by rule induction on the statics, making use of the canonical forms lemma. The opportunities for run-time errors are the same as those for L{dyn}—a well-typed cast might fail at run-time if the class of the cast does not match the class of the value. The type dyn need not be taken as primitive in any language with sum types and recursive types. Specifically, the type dyn is definable in such a language by the following correspondences:1 dyn , µt.[num ,→ nat, fun ,→ t * t](19.3) new[num](e), fold(num · e)(19.4) new[fun](e), fold(fun · e)(19.5) cast[num](e), case unfold(e){num · x ⇒ x | fun · x ⇒ error}(19.6) cast[fun](e), case unfold(e){num · x ⇒ error | fun · x ⇒ x}. (19.7) Thus there is no need for a primitive notion of dynamic type, provided that sums and recursive types are available. 19.2 Dynamic as Static Typing The language L{dyn} described in Chapter 18 may be embedded into L{nat dyn *} by a simple translation that makes explicit the class checking in the dynam- ics of L{dyn}. Specifically, we may define a translation d† of expressions of L{dyn} into expressions of L{nat dyn *} according to the following static correctness criterion: Theorem 19.3. If x1 ok,..., xn ok ` d ok according to the statics of L{dyn}, then x1 : dyn,..., xn : dyn ` d†: dyn in L{nat dyn *}. 1The expression error aborts the computation with an error; this can be accomplished using exceptions, which are described in Chapter 28. REVISED 05.15.2012 VERSION 1.32 172 19.3 Optimization of Dynamic Typing The proof of Theorem 19.3 is a straightforward induction on the struc- ture of d based on the following translation: x†, x num(n)†, new[num](n) zero†, new[num](z) succ(d)†, new[num](s(cast[num](d†))) ifz(d; d0; x.d1), ifz(cast[num](d†); d† 0; x.[new[num](x)/x]d† 1) (λ (x) d)†, new[fun](λ (x:dyn) d†) (d1(d2))†, cast[fun](d† 1)(d† 2) fix(x.d), fix[dyn](x.d†) Although a rigorous proof requires methods extending those to be devel- oped in Chapter 48, it should be clear that the translation is faithful to the dynamics of L{dyn} given in Chapter 18. 19.3 Optimization of Dynamic Typing The language L{nat dyn *} combines static and dynamic typing by en- riching L{nat *} with the type, dyn, of classified values. It is, for this reason, called a hybrid language. Unlike a purely dynamic type system, a hybrid type system can express invariants that are crucial to the optimiza- tion of programs in L{dyn}. Consider the addition function in L{dyn} given in Section 18.3, which we transcribe here for easy reference: λ (x) fix p is λ (y) ifz y {zero ⇒ x | succ(y0) ⇒ succ(p(y0))}. This function may be regarded as a value of type dyn in L{nat dyn *} given as follows: fun ! λ (x:dyn) fix p:dyn is fun ! λ (y:dyn) ex,p,y, where x : dyn, p : dyn, y : dyn ` ex,p,y : dyn is the expression ifz (y ? num) {zero ⇒ x | succ(y0) ⇒ num !(s((p ? fun)(num ! y0)? num))}. VERSION 1.32 REVISED 05.15.2012 19.3 Optimization of Dynamic Typing 173 The embedding into L{nat dyn *} makes explicit the run-time checks that are implicit in the dynamics of L{dyn}. Careful examination of the embedded formulation of addition reveals a great deal of redundancy and overhead that can be eliminated in the stat- ically typed version. Eliminating this redundancy requires a static type discipline, because the intermediate computations involve values of a type other than dyn. Because a dynamic language can only express values of one type, it is impossible to express the optimized form within a dynamic language. This shows that the superficial freedoms offered by dynamic languages supposedly accruing from the omission of types are, in fact, se- vere restrictions on the expressiveness of the language compared to a static language with a type of dynamic values. The first redundancy arises from the use of recursion in a dynamic lan- guage. In the above example we use recursion to define the inner loop, p, of the computation. The value p is, by definition, a λ-abstraction, which is explicitly tagged as a function. Yet the call to p within the loop checks at run-time whether p is in fact a function before applying it. Because p is an internally defined function, all of its call sites are under the control of the addition function, which means that there is no need for such pessimism at calls to p, provided that we change its type to dyn * dyn, which directly expresses the invariant that p is a function acting on dynamic values. Performing this transformation, we obtain the following reformulation of the addition function that eliminates this redundancy: fun ! λ (x:dyn) fun ! fix p:dyn * dyn is λ (y:dyn) e0 x,p,y, where e0 x,p,y is the expression ifz (y ? num) {zero ⇒ x | succ(y0) ⇒ num !(s(p(num ! y0)? num))}. We have “hoisted” the function class label out of the loop, and suppressed the cast inside the loop. Correspondingly, the type of p has changed to dyn * dyn. Next, observe that the parameter y of type dyn is cast to a number on each iteration of the loop before it is tested for zero. Because this function is recursive, the bindings of y arise in one of two ways: at the initial call to the addition function, and on each recursive call. But the recursive call is made on the predecessor of y, which is a true natural number that is labeled with num at the call site, only to be removed by the class check at the conditional on the next iteration. This suggests that we hoist the check on y outside of the loop, and avoid labeling the argument to the recursive REVISED 05.15.2012 VERSION 1.32 174 19.3 Optimization of Dynamic Typing call. Doing so changes the type of the function, however, from dyn * dyn to nat * dyn. Consequently, further changes are required to ensure that the entire function remains well-typed. Before doing so, let us make another observation. The result of the re- cursive call is checked to ensure that it has class num, and, if so, the under- lying value is incremented and labeled with class num. If the result of the recursive call came from an earlier use of this branch of the conditional, then obviously the class check is redundant, because we know that it must have class num. But what if the result came from the other branch of the conditional? In that case the function returns x, which need not be of class num because it is provided by the caller of the function. However, we may reasonably insist that it is an error to call addition with a non-numeric ar- gument. This can be enforced by replacing x in the zero branch of the con- ditional by x ? num. Combining these optimizations we obtain the inner loop e00 x defined as follows: fix p:nat * nat is λ (y:nat) ifz y {zero ⇒ x ? num | succ(y0) ⇒ s(p(y0))}. This function has type nat * nat, and runs at full speed when applied to a natural number—all checks have been hoisted out of the inner loop. Finally, recall that the overall goal is to define a version of addition that works on values of type dyn. Thus we require a value of type dyn * dyn, but what we have at hand is a function of type nat * nat. This can be converted to the required form by pre-composing with a cast to num and post-composing with a coercion to num: fun ! λ (x:dyn) fun ! λ (y:dyn) num !(e00 x (y ? num)). The innermost λ-abstraction converts the function e00 x from type nat * nat to type dyn * dyn by composing it with a class check that ensures that y is a natural number at the initial call site, and applies a label to the result to restore it to type dyn. The outcome of these transformations is that the inner loop of the com- putation runs at “full speed”, without any manipulation of tags on func- tions or numbers. But the outermost form of addition has been retained as a value of type dyn encapsulating a curried function that takes two ar- guments of type dyn. This preserves the correctness of all calls to addi- tion, which pass and return values of type dyn, while optimizing its execu- tion during the computation. Of course, we could strip the class tags from the addition function, changing its type from dyn to the more descriptive VERSION 1.32 REVISED 05.15.2012 19.4 Static Versus Dynamic Typing 175 dyn * dyn * dyn, but this imposes the requirement on the caller to treat addition not as a value of type dyn, but rather as a function that must be applied to two successive values of type dyn whose class is num. As long as the call sites to addition are under programmer control, there is no obsta- cle to effecting this transformation. It is only when there may be external call sites, not directly under programmer control, that there is any need to package addition as a value of type dyn. Applying this principle gener- ally, we see that dynamic typing is only of marginal utility—that is, is used only at the margins of a system where uncontrolled calls arise. Internally to a system there is no benefit, and considerable drawback, to restricting attention to the type dyn. 19.4 Static Versus Dynamic Typing There have been many attempts by advocates of dynamic typing to distin- guish dynamic from static languages. It is useful to review the supposed distinctions from the present viewpoint. 1. Dynamic languages associate types with values, whereas static languages associate types to variables. But this is nonsense arising from the con- fusion of types with classes. Dynamic languages associate classes, not types, to values by tagging them with identifiers such as num and fun. This form of classification amounts to a use of recursive sum types within a statically typed language, and hence cannot be seen as a dis- tinguishing feature of dynamic languages. Morever, static languages assign types to expressions, not just variables. Because dynamic lan- guages are just particular static languages (with a single type), the same can be said of dynamic languages. 2. Dynamic languages check types at run-time, whereas static language check types at compile time. This, too, is erroneous. Dynamic languages are just as surely statically typed as static languages, albeit for a degen- erate type system with only one type. As we have seen, dynamic languages do perform class checks at run-time, but so too do static languages that admit sum types. The difference is only the extent to which we must use classification: always in a dynamic language, only as necessary in a static language. 3. Dynamic languages support heterogeneous collections, whereas static lan- guages support homogeneous collections. But this, too, is in error. Sum REVISED 05.15.2012 VERSION 1.32 176 19.5 Notes types exist to support heterogeneity, and any static language with sums admits heterogenous data structures. A typical example is a list such as cons(num(1); cons(fun(λ (x) x); nil)). It is sometimes said that such a list is not representable in a static language, because of the disparate nature of its components. In both static and dynamic languages are type homogeneous, but may be class heterogeneous. All elements of the above list are of type dyn; the first is of class num, and the second is of class fun. What, then, are we to make of the supposed distinction between dy- namic and static languages? Rather than being in opposition to each other, it is more accurate to say that dynamic languages are a mode of use of static languages. Every dynamic language is a static language, albeit one with a paucity of types available to the programmer (only one!). But as we have seen above, types express and enforce invariants that are crucial to the cor- rectness and efficiency of programs. 19.5 Notes The concept of a hybrid type system is wholly artificial, serving only as an explanatory bridge between dynamic and static languages. Viewing dynamic languages as static languages with recursive types was first pro- posed by Scott(1980), who also suggested the term “unityped” as a more descriptive alternative to “untyped.” VERSION 1.32 REVISED 05.15.2012 Part VII Variable Types Chapter 20 Girard’s System F The languages we have considered so far are all monomorphic in that every expression has a unique type, given the types of its free variables, if it has a type at all. Yet it is often the case that essentially the same behavior is re- quired, albeit at several different types. For example, in L{nat →} there is a distinct identity function for each type τ, namely λ (x:τ) x, even though the behavior is the same for each choice of τ. Similarly, there is a distinct composition operator for each triple of types, namely ◦τ1,τ2,τ3 = λ (f:τ2 → τ3) λ (g:τ1 → τ2) λ (x:τ1) f(g(x)). Each choice of the three types requires a different program, even though they all exhibit the same behavior when executed. Obviously it would be useful to capture the general pattern once and for all, and to instantiate this pattern each time we need it. The expression patterns codify generic (type-independent) behaviors that are shared by all instances of the pattern. Such generic expressions are said to be polymor- phic. In this chapter we will study a language introduced by Girard under the name System F and by Reynolds under the name polymorphic typed λ- calculus. Although motivated by a simple practical problem (how to avoid writing redundant code), the concept of polymorphism is central to an im- pressive variety of seemingly disparate concepts, including the concept of data abstraction (the subject of Chapter 21), and the definability of product, sum, inductive, and coinductive types considered in the preceding chap- ters. (Only general recursive types extend the expressive power of the lan- guage.) 180 20.1 System F 20.1 System F System F, or the polymorphic λ-calculus, or L{→∀}, is a minimal functional language that illustrates the core concepts of polymorphic typing, and per- mits us to examine its surprising expressive power in isolation from other language features. The syntax of System F is given by the following gram- mar: Typ τ ::= t t variable arr(τ1; τ2) τ1 → τ2 function all(t.τ) ∀(t.τ) polymorphic Exp e ::= x x lam[τ](x.e) λ (x:τ) e abstraction ap(e1; e2) e1(e2) application Lam(t.e)Λ(t.e) type abstraction App[τ](e) e[τ] type application A type abstraction, Lam(t.e), defines a generic, or polymorphic, function with type parameter t standing for an unspecified type within e.A type application, or instantiation, App[τ](e), applies a polymorphic function to a specified type, which is then plugged in for the type parameter to obtain the result. Polymorphic functions are classified by the universal type, all(t.τ), that determines the type, τ, of the result as a function of the argument, t. The statics of L{→∀} consists of two judgment forms, the type formation judgment, ∆ ` τ type, and the typing judgment, ∆ Γ ` e : τ. The hypotheses ∆ have the form t type, where t is a variable of sort Typ, and the hypotheses Γ have the form x : τ, where x is a variable of sort Exp. The rules defining the type formation judgment are as follows: ∆, t type ` t type (20.1a) ∆ ` τ1 type ∆ ` τ2 type ∆ ` arr(τ1; τ2) type (20.1b) ∆, t type ` τ type ∆ ` all(t.τ) type (20.1c) The rules defining the typing judgment are as follows: ∆ Γ, x : τ ` x : τ (20.2a) VERSION 1.32 REVISED 05.15.2012 20.1 System F 181 ∆ ` τ1 type ∆ Γ, x : τ1 ` e : τ2 ∆ Γ ` lam[τ1](x.e): arr(τ1; τ2)(20.2b) ∆ Γ ` e1 : arr(τ2; τ) ∆ Γ ` e2 : τ2 ∆ Γ ` ap(e1; e2): τ (20.2c) ∆, t type Γ ` e : τ ∆ Γ ` Lam(t.e): all(t.τ)(20.2d) ∆ Γ ` e : all(t.τ0) ∆ ` τ type ∆ Γ ` App[τ](e):[τ/t]τ0 (20.2e) Lemma 20.1 (Regularity). If ∆ Γ ` e : τ, and if ∆ ` τi type for each assumption xi : τi in Γ, then ∆ ` τ type. Proof. By induction on Rules (20.2). The statics admits the structural rules for a general hypothetical judg- ment. In particular, we have the following critical substitution property for type formation and expression typing. Lemma 20.2 (Substitution). 1. If ∆, t type ` τ0 type and ∆ ` τ type, then ∆ ` [τ/t]τ0 type. 2. If ∆, t type Γ ` e0 : τ0 and ∆ ` τ type, then ∆ [τ/t]Γ ` [τ/t]e0 :[τ/t]τ0. 3. If ∆ Γ, x : τ ` e0 : τ0 and ∆ Γ ` e : τ, then ∆ Γ ` [e/x]e0 : τ0. The second part of the lemma requires substitution into the context, Γ, as well as into the term and its type, because the type variable t may occur freely in any of these positions. Returning to the motivating examples from the introduction, the poly- morphic identity function, I, is written Λ(t.λ (x:t) x); it has the polymorphic type ∀(t.t → t). Instances of the polymorphic identity are written I[τ], where τ is some type, and have the type τ → τ. Similarly, the polymorphic composition function, C, is written Λ(t1.Λ(t2.Λ(t3.λ (f:t2 → t3) λ (g:t1 → t2) λ (x:t1) f(g(x))))). REVISED 05.15.2012 VERSION 1.32 182 20.1 System F The function C has the polymorphic type ∀(t1.∀(t2.∀(t3.(t2 → t3) → (t1 → t2) → (t1 → t3)))). Instances of C are obtained by applying it to a triple of types, written C[τ1][τ2][τ3]. Each such instance has the type (τ2 → τ3) → (τ1 → τ2) → (τ1 → τ3). Dynamics The dynamics of L{→∀} is given as follows: lam[τ](x.e) val (20.3a) Lam(t.e) val (20.3b) [e2 val] ap(lam[τ1](x.e); e2) 7→ [e2/x]e (20.3c) e1 7→ e0 1 ap(e1; e2) 7→ ap(e0 1; e2)(20.3d)  e1 val e2 7→ e0 2 ap(e1; e2) 7→ ap(e1; e0 2)  (20.3e) App[τ](Lam(t.e)) 7→ [τ/t]e (20.3f) e 7→ e0 App[τ](e) 7→ App[τ](e0)(20.3g) The bracketed premises and rule are to be included for a call-by-value in- terpretation, and omitted for a call-by-name interpretation of L{→∀}. It is a simple matter to prove safety for L{→∀}, using familiar methods. Lemma 20.3 (Canonical Forms). Suppose that e : τ and e val, then 1. If τ = arr(τ1; τ2), then e = lam[τ1](x.e2) with x : τ1 ` e2 : τ2. 2. If τ = all(t.τ0), then e = Lam(t.e0) with t type ` e0 : τ0. Proof. By rule induction on the statics. VERSION 1.32 REVISED 05.15.2012 20.2 Polymorphic Definability 183 Theorem 20.4 (Preservation). If e : τ and e 7→ e0, then e0 : τ. Proof. By rule induction on the dynamics. Theorem 20.5 (Progress). If e : τ, then either e val or there exists e0 such that e 7→ e0. Proof. By rule induction on the statics. 20.2 Polymorphic Definability The language L{→∀} is astonishingly expressive. Not only are all finite products and sums definable in the language, but so are all inductive and coinductive types. This is most naturally expressed using definitional equal- ity, which is defined to be the least congruence containing the following two axioms: ∆ Γ, x : τ1 ` e2 : τ2 ∆ Γ ` e1 : τ1 ∆ Γ ` λ (x:τ) e2(e1) ≡ [e1/x]e2 : τ2 (20.4a) ∆, t type Γ ` e : τ ∆ ` ρ type ∆ Γ ` Λ(t.e)[ρ] ≡ [ρ/t]e :[ρ/t]τ (20.4b) In addition there are rules omitted here specifying that definitional equality is a congruence relation (that is, an equivalence relation respected by all expression-forming operations). 20.2.1 Products and Sums The nullary product, or unit, type is definable in L{→∀} as follows: unit , ∀(r.r → r) hi ,Λ(r.λ (x:r) x) The identity function plays the role of the null tuple, because it is the only closed value of this type. Binary products are definable in L{→∀} by using encoding tricks sim- ilar to those described in Chapter 17 for the untyped λ-calculus: τ1 × τ2 , ∀(r.(τ1 → τ2 → r) → r) he1, e2i ,Λ(r.λ (x:τ1 → τ2 → r) x(e1)(e2)) e · l , e[τ1](λ (x:τ1) λ (y:τ2) x) e · r , e[τ2](λ (x:τ1) λ (y:τ2) y) REVISED 05.15.2012 VERSION 1.32 184 20.2 Polymorphic Definability The statics given in Chapter 11 is derivable according to these definitions. Moreover, the following definitional equalities are derivable in L{→∀} from these definitions: he1, e2i · l ≡ e1 : τ1 and he1, e2i · r ≡ e2 : τ2. The nullary sum, or void, type is definable in L{→∀}: void , ∀(r.r) abort[ρ](e), e[ρ] There is no definitional equality to be checked, there being no introductory rule for the void type. Binary sums are also definable in L{→∀}: τ1 + τ2 , ∀(r.(τ1 → r) → (τ2 → r) → r) l · e ,Λ(r.λ (x:τ1 → r) λ (y:τ2 → r) x(e)) r · e ,Λ(r.λ (x:τ1 → r) λ (y:τ2 → r) y(e)) case e {l · x1 ⇒ e1 | r · x2 ⇒ e2}, e[ρ](λ (x1:τ1) e1)(λ (x2:τ2) e2) provided that the types make sense. It is easy to check that the following equivalences are derivable in L{→∀}: case l · d1 {l · x1 ⇒ e1 | r · x2 ⇒ e2} ≡ [d1/x1]e1 : ρ and case r · d2 {l · x1 ⇒ e1 | r · x2 ⇒ e2} ≡ [d2/x2]e2 : ρ. Thus the dynamic behavior specified in Chapter 12 is correctly implemented by these definitions. 20.2.2 Natural Numbers As we remarked above, the natural numbers (under a lazy interpretation) are also definable in L{→∀}. The key is the representation of the iterator, whose typing rule we recall here for reference: e0 : nat e1 : τ x : τ ` e2 : τ iter(e0; e1; x.e2): τ . VERSION 1.32 REVISED 05.15.2012 20.3 Parametricity Overview 185 Because the result type τ is arbitrary, this means that if we have an iterator, then it can be used to define a function of type nat → ∀(t.t → (t → t) → t). This function, when applied to an argument n, yields a polymorphic func- tion that, for any result type, t, given the initial result for z and a trans- formation from the result for x into the result for s(x), yields the result of iterating the transformation n times, starting with the initial result. Because the only operation we can perform on a natural number is to iterate up to it in this manner, we may simply identify a natural number, n, with the polymorphic iterate-up-to-n function just described. This means that we may define the type of natural numbers in L{→∀} by the following equations: nat , ∀(t.t → (t → t) → t) z ,Λ(t.λ (z:t) λ (s:t → t) z) s(e),Λ(t.λ (z:t) λ (s:t → t) s(e[t](z)(s))) iter(e0; e1; x.e2), e0[τ](e1)(λ (x:τ) e2) It is easy to check that the statics and dynamics of the natural numbers type given in Chapter9 are derivable in L{→∀} under these definitions. This shows that L{→∀} is at least as expressive as L{nat →}. But is it more expressive? Yes! It is possible to show that the evaluation function for L{nat →} is definable in L{→∀}, even though it is not definable in L{nat →} itself. However, the same diagonal argument given in Chap- ter9 applies here, showing that the evaluation function for L{→∀} is not definable in L{→∀}. We may enrich L{→∀} a bit more to define the eval- uator for L{→∀}, but as long as all programs in the enriched language terminate, we will once again have an undefinable function, the evaluation function for that extension. 20.3 Parametricity Overview A remarkable property of L{→∀} is that polymorphic types severely con- strain the behavior of their elements. We may prove useful theorems about an expression knowing only its type—that is, without ever looking at the code. For example, if i is any expression of type ∀(t.t → t), then it must be the identity function. Informally, when i is applied to a type, τ, and REVISED 05.15.2012 VERSION 1.32 186 20.4 Restricted Forms of Polymorphism an argument of type τ, it must return a value of type τ. But because τ is not specified until i is called, the function has no choice but to return its argument, which is to say that it is essentially the identity function. Sim- ilarly, if b is any expression of type ∀(t.t → t → t), then b must be either Λ(t.λ (x:t) λ (y:t) x) or Λ(t.λ (x:t) λ (y:t) y). For when b is applied to two arguments of some type, its only choice to return a value of that type is to return one of the two. What is remarkable is that these properties of i and b have been de- rived without knowing anything about the expressions themselves, but only their types. The theory of parametricity implies that we are able to derive the- orems about the behavior of a program knowing only its type. Such the- orems are sometimes called free theorems because they come “for free” as a consequence of typing, and require no program analysis or verification to derive. These theorems underpin the remarkable experience with poly- morphic languages that well-typed programs tend to behave as expected when executed. That is, satisfying the type checker is sufficient condition for correctness. Parametricity so constrains the behavior of a program that there are relatively few programs of the same type that exhibit unintended behavior, ruling out a large class of mistakes that commonly arise when writing code. Parametricity also guarantees representation independence for abstract types, a topic that is discussed further in Chapter 21. 20.4 Restricted Forms of Polymorphism In this section we briefly examine some restricted forms of polymorphism with less than the full expressive power of L{→∀}. These are obtained in one of two ways: 1. Restricting type quantification to unquantified types. 2. Restricting the occurrence of quantifiers within types. 20.4.1 Predicative Fragment The remarkable expressive power of the language L{→∀} may be traced to the ability to instantiate a polymorphic type with another polymorphic type. For example, if we let τ be the type ∀(t.t → t), and, assuming that e : τ, we may apply e to its own type, obtaining the expression e[τ] of type τ → τ. Written out in full, this is the type ∀(t.t → t) → ∀(t.t → t), VERSION 1.32 REVISED 05.15.2012 20.4 Restricted Forms of Polymorphism 187 which is larger (both textually, and when measured by the number of oc- currences of quantified types) than the type of e itself. In fact, this type is large enough that we can go ahead and apply e[τ] to e again, obtaining the expression e[τ](e), which is again of type τ — the very type of e. This property of L{→∀} is called impredicativity1; the language L{→∀} is said to permit impredicative (type) quantification. The distinguishing char- acteristic of impredicative polymorphism is that it involves a kind of cir- cularity in that the meaning of a quantified type is given in terms of its instances, including the quantified type itself. This quasi-circularity is re- sponsible for the surprising expressive power of L{→∀}, and is corre- spondingly the prime source of complexity when reasoning about it (for example, in the proof that all expressions of L{→∀} terminate). Contrast this with L{→}, in which the type of an application of a func- tion is evidently smaller than the type of the function itself. For if e : τ1 → τ2, and e1 : τ1, then we have e(e1): τ2, a smaller type than the type of e. This situation extends to polymorphism, provided that we impose the re- striction that a quantified type can only be instantiated by an un-quantified type. For in that case passage from ∀(t.τ) to [ρ/t]τ decreases the num- ber of quantifiers (even if the size of the type expression viewed as a tree grows). For example, the type ∀(t.t → t) may be instantiated with the type u → u to obtain the type (u → u) → (u → u). This type has more symbols in it than τ, but is smaller in that it has fewer quantifiers. The re- striction to quantification only over unquantified types is called predicative2 polymorphism. The predicative fragment is significantly less expressive than the full impredicative language. In particular, the natural numbers are no longer definable in it. 20.4.2 Prenex Fragment A rather more restricted form of polymorphism, called the prenex fragment, further restricts polymorphism to occur only at the outermost level — not only is quantification predicative, but quantifiers are not permitted to occur within the arguments to any other type constructors. This restriction, called prenex quantification, is often imposed for the sake of type inference, which permits type annotations to be omitted entirely in the knowledge that they can be recovered from the way the expression is used. We will not discuss type inference here, but we will give a formulation of the prenex fragment 1pronounced im-PRED-ic-a-tiv-it-y 2pronounced PRED-i-ca-tive REVISED 05.15.2012 VERSION 1.32 188 20.4 Restricted Forms of Polymorphism of L{→∀}, because it plays an important role in the design of practical polymorphic languages. The prenex fragment of L{→∀} is designated L1{→∀}, for reasons that will become clear in the next subsection. It is defined by stratifying types into two sorts, the monotypes (or rank-0 types) and the polytypes (or rank-1 types). The monotypes are those that do not involve any quantifica- tion, and may be used to instantiate the polymorphic quantifier. The poly- types include the monotypes, but also permit quantification over mono- types. These classifications are expressed by the judgments ∆ ` τ mono and ∆ ` τ poly, where ∆ is a finite set of hypotheses of the form t mono, where t is a type variable not otherwise declared in ∆. The rules for deriv- ing these judgments are as follows: ∆, t mono ` t mono (20.5a) ∆ ` τ1 mono ∆ ` τ2 mono ∆ ` arr(τ1; τ2) mono (20.5b) ∆ ` τ mono ∆ ` τ poly (20.5c) ∆, t mono ` τ poly ∆ ` all(t.τ) poly (20.5d) Base types, such as nat (as a primitive), or other type constructors, such as sums and products, would be added to the language as monotypes. The statics of L1{→∀} is given by rules for deriving hypothetical judg- ments of the form ∆ Γ ` e : ρ, where ∆ consists of hypotheses of the form t mono, and Γ consists of hypotheses of the form x : ρ, where ∆ ` ρ poly. The rules defining this judgment are as follows: ∆ Γ, x : τ ` x : τ (20.6a) ∆ ` τ1 mono ∆ Γ, x : τ1 ` e2 : τ2 ∆ Γ ` lam[τ1](x.e2): arr(τ1; τ2)(20.6b) ∆ Γ ` e1 : arr(τ2; τ) ∆ Γ ` e2 : τ2 ∆ Γ ` ap(e1; e2): τ (20.6c) ∆, t mono Γ ` e : τ ∆ Γ ` Lam(t.e): all(t.τ)(20.6d) ∆ ` τ mono ∆ Γ ` e : all(t.τ0) ∆ Γ ` App[τ](e):[τ/t]τ0 (20.6e) VERSION 1.32 REVISED 05.15.2012 20.4 Restricted Forms of Polymorphism 189 We tacitly exploit the inclusion of monotypes as polytypes so that all typing judgments have the form e : ρ for some expression e and polytype ρ. The restriction on the domain of a λ-abstraction to be a monotype means that a fully general let construct is no longer definable—there is no means of binding an expression of polymorphic type to a variable. For this reason it is usual to augment L1{→∀} with a primitive let construct whose statics is as follows: ∆ ` τ1 poly ∆ Γ ` e1 : τ1 ∆ Γ, x : τ1 ` e2 : τ2 ∆ Γ ` let[τ1](e1; x.e2): τ2 .(20.7) For example, the expression let I:∀(t.t → t) be Λ(t.λ (x:t) x) in I[τ → τ](I[τ]) has type τ → τ for any polytype τ. 20.4.3 Rank-Restricted Fragments The binary distinction between monomorphic and polymorphic types in L1{→∀} may be generalized to form a hierarchy of languages in which the occurrences of polymorphic types are restricted in relation to function types. The key feature of the prenex fragment is that quantified types are not permitted to occur in the domain of a function type. The prenex frag- ment also prohibits polymorphic types from the range of a function type, but it would be harmless to admit it, there being no significant difference between the type ρ → ∀(t.τ) and the type ∀(t.ρ → τ)(where t /∈ ρ). This motivates the definition of a hierarchy of fragments of L{→∀} that subsumes the prenex fragment as a special case. We will define a judgment of the form τ type [k], where k ≥ 0, to mean that τ is a type of rank k. Informally, types of rank 0 have no quantification, and types of rank k + 1 may involve quantification, but the domains of function types are restricted to be of rank k. Thus, in the terminology of Section 20.4.2, a monotype is a type of rank 0 and a polytype is a type of rank 1. The definition of the types of rank k is defined simultaneously for all k by the following rules. These rules involve hypothetical judgments of the form ∆ ` τ type [k], where ∆ is a finite set of hypotheses of the form ti type [ki] for some pairwise distinct set of type variables ti. The rules defin- ing these judgments are as follows: ∆, t type [k] ` t type [k](20.8a) REVISED 05.15.2012 VERSION 1.32 190 20.5 Notes ∆ ` τ1 type [0] ∆ ` τ2 type [0] ∆ ` arr(τ1; τ2) type [0](20.8b) ∆ ` τ1 type [k] ∆ ` τ2 type [k + 1] ∆ ` arr(τ1; τ2) type [k + 1](20.8c) ∆ ` τ type [k] ∆ ` τ type [k + 1](20.8d) ∆, t type [k] ` τ type [k + 1] ∆ ` all(t.τ) type [k + 1](20.8e) With these restrictions in mind, it is a good exercise to define the statics of Lk{→∀}, the restriction of L{→∀} to types of rank k (or less). It is most convenient to consider judgments of the form e : τ [k] specifying simulta- neously that e : τ and τ type [k]. For example, the rank-limited rules for λ-abstractions is phrased as follows: ∆ ` τ1 type [0] ∆ Γ, x : τ1 [0] ` e2 : τ2 [0] ∆ Γ ` lam[τ1](x.e2): arr(τ1; τ2)[0](20.9a) ∆ ` τ1 type [k] ∆ Γ, x : τ1 [k] ` e2 : τ2 [k + 1] ∆ Γ ` lam[τ1](x.e2): arr(τ1; τ2)[k + 1](20.9b) The remaining rules follow a similar pattern. The rank-limited languages Lk{→∀} clarify the need for a primitive (as opposed to derived) definition mechanism in L1{→∀}. The prenex frag- ment of L{→∀} corresponds to the rank-one fragment L1{→∀}. The let construct for rank-one types is definable in L2{→∀} from λ-abstraction and application. This definition only makes sense at rank two, because it abstracts over a rank-one polymorphic type, and is therefore not available at lesser rank. 20.5 Notes System F was introduced by Girard(1972) in the context of proof theory and by Reynolds(1974) in the context of programming languages. The concept of parametricity was originally isolated by Strachey, but was not fully developed until the work of Reynolds(1983). The description of para- metricity as providing “theorems for free” was popularized by Wadler(1989). VERSION 1.32 REVISED 05.15.2012 Chapter 21 Abstract Types Data abstraction is perhaps the most important technique for structuring programs. The main idea is to introduce an interface that serves as a contract between the client and the implementor of an abstract type. The interface specifies what the client may rely on for its own work, and, simultaneously, what the implementor must provide to satisfy the contract. The interface serves to isolate the client from the implementor so that each may be devel- oped in isolation from the other. In particular one implementation may be replaced by another without affecting the behavior of the client, provided that the two implementations meet the same interface and are, in a sense to be made precise below, suitably related to one another. (Roughly, each simulates the other with respect to the operations in the interface.) This property is called representation independence for an abstract type. Data abstraction may be formalized by extending the language L{→∀} with existential types. Interfaces are modelled as existential types that pro- vide a collection of operations acting on an unspecified, or abstract, type. Implementations are modelled as packages, the introductory form for exis- tentials, and clients are modelled as uses of the corresponding elimination form. It is remarkable that the programming concept of data abstraction is modelled so naturally and directly by the logical concept of existential type quantification. Existential types are closely connected with universal types, and hence are often treated together. The superficial reason is that both are forms of type quantification, and hence both require the machin- ery of type variables. The deeper reason is that existentials are definable from universals — surprisingly, data abstraction is actually just a form of polymorphism! One consequence of this observation is that representation independence is just a use of the parametricity properties of polymorphic 192 21.1 Existential Types functions discussed in Chapter 20. 21.1 Existential Types The syntax of L{→∀∃} is the extension of L{→∀} with the following con- structs: Typ τ ::= some(t.τ) ∃(t.τ) interface Exp e ::= pack[t.τ][ρ](e) pack ρ with e as ∃(t.τ) implementation open[t.τ][ρ](e1; t, x.e2) open e1 as t with x:τ in e2 client The introductory form for the existential type ∃(t.τ) is a package of the form pack ρ with e as ∃(t.τ), where ρ is a type and e is an expression of type [ρ/t]τ. The type ρ is called the representation type of the package, and the expression e is called the implementation of the package. The elimina- tory form for existentials is the expression open e1 as t with x:τ in e2, which opens the package e1 for use within the client e2 by binding its representa- tion type to t and its implementation to x for use within e2. Crucially, the typing rules ensure that the client is type-correct independently of the ac- tual representation type used by the implementor, so that it may be varied without affecting the type correctness of the client. The abstract syntax of the open construct specifies that the type variable, t, and the expression variable, x, are bound within the client. They may be renamed at will by α-equivalence without affecting the meaning of the con- struct, provided, of course, that the names are chosen so as not to conflict with any others that may be in scope. In other words the type, t, may be thought of as a “new” type, one that is distinct from all other types, when it is introduced. This is sometimes called generativity of abstract types: the use of an abstract type by a client “generates” a “new” type within that client. This behavior is simply a consequence of identifying terms up to α-equivalence, and is not particularly tied to data abstraction. 21.1.1 Statics The statics of existential types is specified by rules defining when an exis- tential is well-formed, and by giving typing rules for the associated intro- ductory and eliminatory forms. ∆, t type ` τ type ∆ ` some(t.τ) type (21.1a) VERSION 1.32 REVISED 05.15.2012 21.1 Existential Types 193 ∆ ` ρ type ∆, t type ` τ type ∆ Γ ` e :[ρ/t]τ ∆ Γ ` pack[t.τ][ρ](e): some(t.τ)(21.1b) ∆ Γ ` e1 : some(t.τ) ∆, t type Γ, x : τ ` e2 : τ2 ∆ ` τ2 type ∆ Γ ` open[t.τ][τ2](e1; t, x.e2): τ2 (21.1c) Rule (21.1c) is complex, so study it carefully! There are two important things to notice: 1. The type of the client, τ2, must not involve the abstract type t. This restriction prevents the client from attempting to export a value of the abstract type outside of the scope of its definition. 2. The body of the client, e2, is type checked without knowledge of the representation type, t. The client is, in effect, polymorphic in the type variable t. Lemma 21.1 (Regularity). Suppose that ∆ Γ ` e : τ. If ∆ ` τi type for each xi : τi in Γ, then ∆ ` τ type. Proof. By induction on Rules (21.1). 21.1.2 Dynamics The dynamics of existential types is specified by the following rules (in- cluding the bracketed material for an eager interpretation, and omitting it for a lazy interpretation): [e val] pack[t.τ][ρ](e) val (21.2a)  e 7→ e0 pack[t.τ][ρ](e) 7→ pack[t.τ][ρ](e0)  (21.2b) e1 7→ e0 1 open[t.τ][τ2](e1; t, x.e2) 7→ open[t.τ][τ2](e0 1; t, x.e2)(21.2c) [e val] open[t.τ][τ2](pack[t.τ][ρ](e); t, x.e2) 7→ [ρ, e/t, x]e2 (21.2d) It is important to observe that, according to these rules, there are no abstract types at run time! The representation type is propagated to the client by sub- stitution when the package is opened, thereby eliminating the abstraction boundary between the client and the implementor. Thus, data abstraction is a compile-time discipline that leaves no traces of its presence at execution time. REVISED 05.15.2012 VERSION 1.32 194 21.2 Data Abstraction Via Existentials 21.1.3 Safety The safety of the extension is stated and proved as usual. The argument is a simple extension of that used for L{→∀} to the new constructs. Theorem 21.2 (Preservation). If e : τ and e 7→ e0, then e0 : τ. Proof. By rule induction on e 7→ e0, making use of substitution for both expression- and type variables. Lemma 21.3 (Canonical Forms). If e : some(t.τ) and e val, then e = pack[t.τ][ρ](e0) for some type ρ and some e0 such that e0 :[ρ/t]τ. Proof. By rule induction on the statics, making use of the definition of closed values. Theorem 21.4 (Progress). If e : τ then either e val or there exists e0 such that e 7→ e0. Proof. By rule induction on e : τ, making use of the canonical forms lemma. 21.2 Data Abstraction Via Existentials To illustrate the use of existentials for data abstraction, we consider an ab- stract type of queues of natural numbers supporting three operations: 1. Forming the empty queue. 2. Inserting an element at the tail of the queue. 3. Removing the head of the queue, which is assumed to be non-empty. This is clearly a bare-bones interface, but is sufficient to illustrate the main ideas of data abstraction. Queue elements may be taken to be of any type, τ, of our choosing; we will not be specific about this choice, because nothing depends on it. The crucial property of this description is that nowhere do we specify what queues actually are, only what we can do with them. This is captured by the following existential type, ∃(t.τ), which serves as the interface of the queue abstraction: ∃(t.hemp ,→ t, ins ,→ nat × t → t, rem ,→ t → nat × ti). VERSION 1.32 REVISED 05.15.2012 21.2 Data Abstraction Via Existentials 195 The representation type, t, of queues is abstract — all that is specified about it is that it supports the operations emp, ins, and rem, with the specified types. An implementation of queues consists of a package specifying the rep- resentation type, together with the implementation of the associated op- erations in terms of that representation. Internally to the implementation, the representation of queues is known and relied upon by the operations. Here is a very simple implementation, el, in which queues are represented as lists: pack list with hemp ,→ nil, ins ,→ ei, rem ,→ eri as ∃(t.τ), where ei : nat × list → list = λ (x:nat × list) e0 i, and er : list → nat × list = λ (x:list) e0 r. Here the expression e0 i conses the first component of x, the element, onto the second component of x, the queue. Correspondingly, the expression e0 r re- verses its argument, and returns the head element paired with the reversal of the tail. These operations “know” that queues are represented as values of type list, and are programmed accordingly. It is also possible to give another implementation, ep, of the same inter- face, ∃(t.τ), but in which queues are represented as pairs of lists, consist- ing of the “back half” of the queue paired with the reversal of the “front half”. This representation avoids the need for reversals on each call, and, as a result, achieves amortized constant-time behavior: pack list × list with hemp ,→ hnil, nili, ins ,→ ei, rem ,→ eri as ∃(t.τ). In this case ei has type nat × (list × list) → (list × list), and er has type (list × list) → nat × (list × list). These operations “know” that queues are represented as values of type list × list, and are implemented accordingly. The important point is that the same client type checks regardless of which implementation of queues we choose. This is because the represen- tation type is hidden, or held abstract, from the client during type checking. REVISED 05.15.2012 VERSION 1.32 196 21.3 Definability of Existentials Consequently, it cannot rely on whether it is list or list × list or some other type. That is, the client is independent of the representation of the abstract type. 21.3 Definability of Existentials It turns out that it is not necessary to extend L{→∀} with existential types to model data abstraction, because they are already definable using only universal types! Before giving the details, let us consider why this should be possible. The key is to observe that the client of an abstract type is poly- morphic in the representation type. The typing rule for open e1 as t with x:τ in e2 : τ2, where e1 : ∃(t.τ), specifies that e2 : τ2 under the assumptions t type and x : τ. In essence, the client is a polymorphic function of type ∀(t.τ → τ2), where t may occur in τ (the type of the operations), but not in τ2 (the type of the result). This suggests the following encoding of existential types: ∃(t.τ), ∀(u.∀(t.τ → u) → u) pack ρ with e as ∃(t.τ),Λ(u.λ (x:∀(t.τ → u)) x[ρ](e)) open e1 as t with x:τ in e2 , e1[τ2](Λ(t.λ (x:τ) e2)) An existential is encoded as a polymorphic function taking the overall re- sult type, u, as argument, followed by a polymorphic function representing the client with result type u, and yielding a value of type u as overall re- sult. Consequently, the open construct simply packages the client as such a polymorphic function, instantiates the existential at the result type, τ, and applies it to the polymorphic client. (The translation therefore depends on knowing the overall result type, τ, of the open construct.) Finally, a package consisting of a representation type ρ and an implementation e is a polymorphic function that, when given the result type, t, and the client, x, instantiates x with ρ and passes to it the implementation e. It is then a straightforward exercise to show that this translation cor- rectly reflects the statics and dynamics of existential types. VERSION 1.32 REVISED 05.15.2012 21.4 Representation Independence 197 21.4 Representation Independence An important consequence of parametricity is that it ensures that clients are insensitive to the representations of abstract types. More precisely, there is a criterion, called bisimilarity, for relating two implementations of an ab- stract type such that the behavior of a client is unaffected by swapping one implementation by another that is bisimilar to it. This leads to a sim- ple methodology for proving the correctness of candidate implementation of an abstract type, which is to show that it is bisimilar to an obviously correct reference implementation of it. Because the candidate and the ref- erence implementations are bisimilar, no client may distinguish them from one another, and hence if the client behaves properly with the reference implementation, then it must also behave properly with the candidate. To derive the definition of bisimilarity of implementations, it is helpful to examine the definition of existentials in terms of universals given in Sec- tion 21.3. It is an immediate consequence of the definition that the client of an abstract type is polymorphic in the representation of the abstract type. A client, c, of an abstract type ∃(t.τ) has type ∀(t.τ → τ2), where t does not occur free in τ2 (but may, of course, occur in τ). Applying the parametricity property described informally in Chapter 20 (and developed rigorously in Chapter 49), this says that if R is a bisimulation relation between any two implementations of the abstract type, then the client behaves identically on both of them. The fact that t does not occur in the result type ensures that the behavior of the client is independent of the choice of relation between the implementations, provided that this relation is preserved by the opera- tions that implement it. To see what this means requires that we specify what is meant by a bisimulation. This is best done by example. Consider the existential type ∃(t.τ), where τ is the labeled tuple type hemp ,→ t, ins ,→ nat × t → t, rem ,→ t → (nat × t) opti. This specifies an abstract type of queues. The operations emp, ins, and rem specify, respectively, the empty queue, an insert operation, and a remove operation. For the sake of simplicity the element type is chosen to be the natural numbers. The result of removal is an optional pair, according to whether the queue is empty or not. Theorem 49.12 ensures that if ρ and ρ0 are any two closed types, R is a relation between expressions of these two types, then if any of the im- plementations e :[ρ/x]τ and e0 :[ρ0/x]τ respect R, then c[ρ]e behaves the REVISED 05.15.2012 VERSION 1.32 198 21.4 Representation Independence same as c[ρ0]e0. It remains to define when two implementations respect the relation R. Let e , hemp ,→ em, ins ,→ ei, rem ,→ eri and e0 , hemp ,→ e0 m, ins ,→ e0 i, rem ,→ e0 ri. For these implementations to respect R means that the following three con- ditions hold: 1. The empty queues are related: R(em, e0 m). 2. Inserting the same element on each of two related queues yields re- lated queues: if d : τ and R(q, q0), then R(ei(d)(q), e0 i(d)(q0)). 3. If two queues are related, then either they are both empty, or their front elements are the same and their back elements are related: if R(q, q0), then either (a) er(q) ∼= null ∼= e0 r(q0), or (b) er(q) ∼= just(hd, ri) and e0 r(q0) ∼= just(hd0, r0i), with d ∼= d0 and R(r, r0). If such a relation R exists, then the implementations e and e0 are said to be bisimilar. The terminology stems from the requirement that the operations of the abstract type preserve the relation: if it holds before an operation is performed, then it must also hold afterwards, and the relation must hold for the initial state of the queue. Thus each implementation simulates the other up to the relationship specified by R. To see how this works in practice, let us consider informally two im- plementations of the abstract type of queues specified above. For the refer- ence implementation we choose ρ to be the type list, and define the empty queue to be the empty list, define insert to add the specified element to the head of the list, and define remove to remove the last element of the list. The code is as follows: t , list emp , nil ins , λ (x:nat) λ (q:t) cons(x; q) rem , λ (q:t) case rev(q){nil ⇒ null | cons(f; qr) ⇒ just(h f, rev(qr)i)}. Removing an element takes time linear in the length of the list, because of the reversal. VERSION 1.32 REVISED 05.15.2012 21.4 Representation Independence 199 For the candidate implementation we choose ρ0 to be the type list × list of pairs of lists hb, f i in which b is the “back half” of the queue, and f is the reversal of the “front half” of the queue. For this representation we de- fine the empty queue to be a pair of empty lists, define insert to extend the back with that element at the head, and define remove based on whether the front is empty or not. If it is non-empty, the head element is removed from it, and returned along with the pair consisting of the back and the tail of the front. If it is empty, and the back is not, then we reverse the back, remove the head element, and return the pair consisting of the empty list and the tail of the now-reversed back. The code is as follows: t , list × list emp , hnil, nili ins , λ (x:nat) λ (hbs, f si:t) hcons(x; bs), f si rem , λ (hbs, f si:t) case f s {nil ⇒ e | cons(f; f s0) ⇒ hbs, f s0i}, where e , case rev(bs){nil ⇒ null | cons(b; bs0) ⇒ just(hb, hnil, bs0ii)}. The cost of the occasional reversal may be amortized across the sequence of inserts and removes to show that each operation in a sequence costs unit time overall. To show that the candidate implementation is correct, we show that it is bisimilar to the reference implementation. This reduces to specifying a re- lation, R, between the types list and list × list such that the three sim- ulation conditions given above are satisfied by the two implementations just described. The relation in question states that R(l, hb, f i) iff the list l is the list app(b)(rev(f)), where app is the evident append function on lists. That is, thinking of l as the reference representation of the queue, the candidate must maintain that the elements of b followed by the elements of f in reverse order form precisely the list l. It is easy to check that the implementations just described preserve this relation. Having done so, we are assured that the client, c, behaves the same regardless of whether we use the reference or the candidate. Because the reference implementation is obviously correct (albeit inefficient), the candidate must also be correct in that the behavior of any client is unaffected by using it instead of the reference. REVISED 05.15.2012 VERSION 1.32 200 21.5 Notes 21.5 Notes The connection between abstract types in programming languages and ex- istential types in logic was made by Mitchell and Plotkin(1988). Closely related ideas were already present in Reynolds(1974), but the connection with existential types was not explicitly drawn there. The account of rep- resentation independence given here is derived from Mitchell(1986). VERSION 1.32 REVISED 05.15.2012 Chapter 22 Constructors and Kinds The types nat → nat and nat list may be thought of as being built from other types by the application of a type constructor, or type operator. These two examples differ from each other in that the function space type con- structor takes two arguments, whereas the list type constructor takes only one. We may, for the sake of uniformity, think of types such as nat as be- ing built by a type constructor of no arguments. More subtly, we may even think of the types ∀(t.τ) and ∃(t.τ) as being built up in the same way by regarding the quantifiers as higher-order type operators. These seemingly disparate cases may be treated uniformly by enrich- ing the syntactic structure of a language with a new layer of constructors. To ensure that constructors are used properly (for example, that the list constructor is given only one argument, and that the function constructor is given two), we classify constructors by kinds. Constructors of a distin- guished kind, T, are types, which may be used to classify expressions. To allow for multi-argument and higher-order constructors, we will also con- sider finite product and function kinds. (Later we shall consider even richer kinds.) The distinction between constructors and kinds on one hand and types and expressions on the other reflects a fundamental separation between the static and dynamic phase of processing of a programming language, called the phase distinction. The static phase implements the statics and the dynamic phase implements the dynamics. Constructors may be seen as a form of static data that is manipulated during the static phase of process- ing. Expressions are a form of dynamic data that is manipulated at run-time. Because the dynamic phase follows the static phase (we only execute well- typed programs), we may also manipulate constructors at run-time. 202 22.1 Statics Adding constructors and kinds to a language introduces more techni- cal complications than might at first be apparent. The main difficulty is that as soon as we enrich the kind structure beyond the distinguished kind of types, it becomes essential to simplify constructors to determine whether they are equivalent. For example, if we admit product kinds, then a pair of constructors is a constructor of product kind, and projections from a con- structor of product kind are also constructors. But what if we form the first projection from the pair consisting of the constructors nat and str? This should be equivalent to nat, because the elimination form is post-inverse to the introduction form. Consequently, any expression (say, a variable) of the one type should also be an expression of the other. That is, typing should respect definitional equality of constructors. There are two main ways to deal with this. One is to introduce a con- cept of definitional equality for constructors, and to demand that the typing judgment for expressions respect definitional equality of constructors of kind T. This means, however, that we must show that definitional equality is decidable if we are to build a complete implementation of the language. The other is to prohibit formation of awkward constructors such as the pro- jection from a pair so that there is never any issue of when two constructors are equivalent (only when they are identical). But this complicates the def- inition of substitution, because a projection from a constructor variable is well-formed, until you substitute a pair for the variable. Both approaches have their benefits, but the second is simplest, and is adopted here. 22.1 Statics The syntax of kinds is given by the following grammar: Kind κ ::= Type T types Unit 1 nullary product Prod(κ1; κ2) κ1 × κ2 binary product Arr(κ1; κ2) κ1 → κ2 function The kinds consist of the kind of types, T, and the unit kind, Unit, and are closed under formation of product and function kinds. The syntax of constructors is divided into two syntactic sorts, the neutral VERSION 1.32 REVISED 05.15.2012 22.1 Statics 203 and the canonical, according to the following grammar: NCon a ::= u u variable proj[l](a) a · l first projection proj[r](a) a · r second projection app(a1; c2) a1[c2] application CCon c ::= atom(a) ba atomic unit hi null tuple pair(c1; c2) hc1,c2i pair lam(u.c) λ u.c abstraction The reason to distinguish neutral from canonical constructors is to en- sure that it is impossible to apply an elimination form to an introduction form, which demands an equation to capture the inversion principle. For example, the putative constructor hc1,c2i · l, which would be definition- ally equal to c1, is ill-formed according to the above syntax chart. This is because the argument to a projection must be neutral, but a pair is only canonical, not neutral. The canonical constructor ba is the inclusion of neutral constructors into canonical constructors. However, the grammar does not capture a crucial property of the statics that ensures that only neutral constructors of kind T may be treated as canonical. This requirement is imposed to limit the forms of canonical constructors of the other kinds. In particular, variables of function, product, or unit kind will turn out not to be canonical, but only neutral. The statics of constructors and kinds is specified by the judgments ∆ ` a ⇑ κ neutral constructor formation ∆ ` c ⇓ κ canonical constructor formation In each of these judgments ∆ is a finite set of hypotheses of the form u1 ⇑ κ1,..., un ⇑ κn for some n ≥ 0. The form of the hypotheses expresses the principle that variables are neutral constructors. The formation judgments are to be un- derstood as generic hypothetical judgments with parameters u1,..., un that are determined by the forms of the hypotheses. The rules for constructor formation are as follows: ∆, u ⇑ κ ` u ⇑ κ (22.1a) REVISED 05.15.2012 VERSION 1.32 204 22.2 Higher Kinds ∆ ` a ⇑ κ1 × κ2 ∆ ` a · l ⇑ κ1 (22.1b) ∆ ` a ⇑ κ1 × κ2 ∆ ` a · r ⇑ κ2 (22.1c) ∆ ` a1 ⇑ κ2 → κ ∆ ` c2 ⇓ κ2 ∆ ` a1[c2] ⇑ κ (22.1d) ∆ ` a ⇑ T ∆ ` ba ⇓ T(22.1e) ∆ ` hi ⇓ 1 (22.1f) ∆ ` c1 ⇓ κ1 ∆ ` c2 ⇓ κ2 ∆ ` hc1,c2i ⇓ κ1 × κ2 (22.1g) ∆, u ⇑ κ1 ` c2 ⇓ κ2 ∆ ` λ u.c2 ⇓ κ1 → κ2 (22.1h) Rule (22.1e) specifies that the only neutral constructors that are canon- ical are those with kind T. This ensures that the language enjoys the fol- lowing canonical forms property, which is easily proved by inspection of Rules (22.1). Lemma 22.1. Suppose that ∆ ` c ⇓ κ. 1. If κ = 1, then c = hi. 2. If κ = κ1 × κ2, then c = hc1,c2i for some c1 and c2 such that ∆ ` ci ⇓ κi for i = 1, 2. 3. If κ = κ1 → κ2, then c = λ u.c2 for some u and c2 such that ∆, u ⇑ κ1 ` c2 ⇓ κ2. 22.2 Higher Kinds To equip a language, L, with constructors and kinds requires that we aug- ment its statics with hypotheses governing constructor variables, and that we relate constructors of kind T(types as static data) to the classifiers of dynamic expressions (types as classifiers). To achieve this the statics of L must be defined to have judgments of the following two forms: ∆ ` τ type type formation ∆ Γ ` e : τ expression formation VERSION 1.32 REVISED 05.15.2012 22.2 Higher Kinds 205 where, as before, Γ is a finite set of hypotheses of the form x1 : τ1,..., xk : τk for some k ≥ 0 such that ∆ ` τi type for each 1 ≤ i ≤ k. As a general principle, every constructor of kind T is a classifier: ∆ ` τ ⇑ T ∆ ` τ type .(22.2) In many cases this is the sole rule of type formation, so that every classifier is a constructor of kind T. However, this need not be the case. In some situations we may wish to have strictly more classifiers than constructors of the distinguished kind. To see how this might arise, let us consider two extensions of L{→∀} from Chapter 20. In both cases we extend the universal quantifier ∀(t.τ) to admit quantification over an arbitrary kind, written ∀ u :: κ.τ, but the two languages differ in what constitutes a constructor of kind T. In one case, the impredicative, we admit quantified types as constructors, and in the other, the predicative, we exclude quantified types from the domain of quantification. The impredicative fragment includes the following two constructor con- stants: ∆ ` → ⇑ T → T → T(22.3a) ∆ ` ∀κ ⇑ (κ → T) → T(22.3b) We regard the classifier τ1 → τ2 to be the application →[τ1][τ2]. Similarly, we regard the classifier ∀ u :: κ.τ to be the application ∀κ[λ u.τ]. The predicative fragment excludes the constant specified by Rule (22.3b) in favor of a separate rule for the formation of universally quantified types: ∆, u ⇑ κ ` τ type ∆ ` ∀ u :: κ.τ type .(22.4) The point is that ∀ u :: κ.τ is a type (as classifier), but is not a constructor of kind type. The significance of this distinction becomes apparent when we con- sider the introduction and elimination forms for the generalized quantifier, which are the same for both fragments: ∆, u ⇑ κ Γ ` e : τ ∆ Γ ` Λ(u::κ.e): ∀ u :: κ.τ (22.5a) REVISED 05.15.2012 VERSION 1.32 206 22.3 Canonizing Substitution ∆ Γ ` e : ∀ u :: κ.τ ∆ ` c ⇓ κ ∆ Γ ` e[c]:[c/u]τ (22.5b) (Rule (22.5b) makes use of substitution, whose definition requires some care. We will return to this point in Section 22.3.) Rule (22.5b) makes clear that a polymorphic abstraction quantifies over the constructors of kind κ. When κ is T this kind may or may not include all of the classifiers of the language, according to whether we are working with the impredicative formulation of quantification (in which the quanti- fiers are distinguished constants for building constructors of kind T) or the predicative formulation (in which quantifiers arise only as classifiers and not as constructors). The main idea is that constructors are static data, so that a constructor abstraction Λ(u::κ.e) of type ∀ u :: κ.τ is a mapping from static data c of kind κ to dynamic data [c/u]e of type [c/u]τ. Rule (22.1e) tells us that every constructor of kind T determines a classifier, but it may or may not be the case that every classifier arises in this manner. 22.3 Canonizing Substitution Rule (22.5b) involves substitution of a canonical constructor, c, of kind κ into a family of types u ⇑ κ ` τ type. This operation is written [c/u]τ, as usual. Although the intended meaning is clear, it is in fact impossible to in- terpret [c/u]τ as the standard concept of substitution defined in Chapter1. The reason is that to do so would risk violating the distinction between neutral and canonical constructors. Consider, for example, the case of the family of types u ⇑ T → T ` u[d] ⇑ T, where d ⇑ T. (It is not important what we choose for d, so we leave it ab- stract.) Now if c ⇓ T → T, then by Lemma 22.1 we have that c is λ u0.c0. Thus, if interpreted conventionally, substitution of c for u in the given fam- ily yields the “constructor” (λ u0.c0)[d], which is not well-formed. The solution is to define a form of canonizing substitution that simplifies such “illegal” combinations as it performs the replacement of a variable by a constructor of the same kind. In the case just sketched this means that we must ensure that [λ u0.c0/u]u[d] = [d/u0]c0. If viewed as a definition this equation is problematic because it switches from substituting for u in the constructor u[d] to substituting for u0 in the VERSION 1.32 REVISED 05.15.2012 22.3 Canonizing Substitution 207 unrelated constructor c0. Why should such a process terminate? The an- swer lies in the observation that the kind of u0 is definitely smaller than the kind of u, because the kind of the former is the domain kind of the lat- ter. In all other cases of substitution (as we shall see shortly) the size of the target of the substitution becomes smaller; in the case just cited the size may increase, but the type of the target variable decreases. Therefore by a lexicographic induction on the type of the target variable and the struc- ture of the target constructor, we may prove that canonizing substitution is well-defined. We now turn to the task of making this precise. We will define simulta- neously two principal forms of substitution, one of which divides into two cases: [c/u : κ]a = a0 canonical into neutral yielding neutral [c/u : κ]a = c0 ⇓ κ0 canonical into neutral yielding canonical and kind [c/u : κ]c0 = c00 canonical into canonical yielding canonical Substitution into a neutral constructor divides into two cases according to whether the substituted variable u occurs in critical position in a sense to be made precise below. These forms of substitution are simultaneously inductively defined by the following rules, which are broken into groups for clarity. The first set of rules defines substitution of a canonical constructor into a canonical constructor; the result is always canonical. [c/u : κ]a0 = a00 [c/u : κ]ba0 = ba00 (22.6a) [c/u : κ]a0 = c00 ⇓ κ00 [c/u : κ]ba0 = c00 (22.6b) [u/hi : κ]=hi (22.6c) [c/u : κ]c0 1 = c00 1 [c/u : κ]c0 2 = c00 2 [c/u : κ]hc0 1,c0 2i = hc00 1 ,c00 2 i (22.6d) [c/u : κ]c0 = c00 (u 6= u0)(u0 /∈ c) [c/u : κ]λ u0.c0 = λ u0.c00 (22.6e) The conditions on variables in Rule (22.6e) may always be met by renaming the bound variable, u0, of the abstraction. REVISED 05.15.2012 VERSION 1.32 208 22.3 Canonizing Substitution The second set of rules defines substitution of a canonical constructor into a neutral constructor, yielding another neutral constructor. (u 6= u0) [c/u : κ]u0 = u0 (22.7a) [c/u : κ]a0 = a00 [c/u : κ]a0 · l = a00 · l (22.7b) [c/u : κ]a0 = a00 [c/u : κ]a0 · r = a00 · r (22.7c) [c/u : κ]a1 = a0 1 [c/u : κ]c2 = c0 2 [c/u : κ]a1[c2] = a0 1(c0 2)(22.7d) Rule (22.7a) pertains to a non-critical variable, which is not the target of sub- stitution. The remaining rules pertain to situations in which the recursive call on a neutral constructor yields a neutral constructor. The third set of rules defines substitution of a canonical constructor into a neutral constructor, yielding a canonical constructor and its kind. [c/u : κ]u = c ⇓ κ (22.8a) [c/u : κ]a0 = hc0 1,c0 2i ⇓ κ0 1 × κ0 2 [c/u : κ]a0 · l = c0 1 ⇓ κ0 1 (22.8b) [c/u : κ]a0 = hc0 1,c0 2i ⇓ κ0 1 × κ0 2 [c/u : κ]a0 · r = c0 2 ⇓ κ0 2 (22.8c) [c/u : κ]a0 1 = λ u0.c0 ⇓ κ0 2 → κ0 [c/u : κ]c0 2 = c00 2 [c00 2 /u0 : κ0 2]c0 = c00 [c/u : κ]a0 1[c0 2] = c00 ⇓ κ0 (22.8d) Rule (22.8a) governs a critical variable, which is the target of substitution. The substitution transforms it from a neutral constructor to a canonical con- structor. This has a knock-on effect in the remaining rules of the group, which analyze the canonical form of the result of the recursive call to de- termine how to proceed. Rule (22.8d) is the most interesting rule. In the third premise, all three arguments to substitution change as we substitute the (substituted) argument of the application for the parameter of the (sub- stituted) function into the body of that function. Here we require the type of the function in order to determine the type of its parameter. VERSION 1.32 REVISED 05.15.2012 22.4 Canonization 209 Theorem 22.2. Suppose that ∆ ` c ⇓ κ, and ∆, u ⇑ κ ` c0 ⇓ κ0, and ∆, u ⇑ κ ` a0 ⇑ κ0. There exists a unique ∆ ` c00 ⇓ κ0 such that [c/u : κ]c0 = c00. Either there exists a unique ∆ ` a00 ⇑ κ0 such that [c/u : κ]a0 = a00, or there exists a unique ∆ ` c00 ⇓ κ0 such that [c/u : κ]a0 = c00, but not both. Proof. Simultaneously by a lexicographic induction with major index be- ing the structure of the kind κ, and with minor index determined by the formation rules for c0 and a0. For all rules except Rule (22.8d) the induc- tive hypothesis applies to the premise(s) of the relevant formation rules. For Rule (22.8d) we appeal to the major inductive hypothesis applied to κ0 2, which is a component of the kind κ0 2 → κ0. 22.4 Canonization With canonizing substitution in hand, it is perfectly possible to confine our attention to constructors in canonical form. However, for some purposes it can be useful to admit a more relaxed syntax in which it is possible to form non-canonical constructors that may be transformed into canonical form. The prototypical example is the constructor (λ u.c2)[c1], which is malformed according to Rules (22.1), because the first argument of an application is required to be in atomic form, whereas the λ-abstraction is in canonical form. However, if c1 and c2 are already canonical, then the malformed application may be transformed into the well-formed canoni- cal form [c1/u]c2, where substitution is as defined in Section 22.3. If c1 or c2 are not already canonical we may, inductively, put them into canonical form before performing the substitution, resulting in the same canonical form. A constructor in general form is one that is well-formed with respect to Rules (22.1), but disregarding the distinction between atomic and canoni- cal forms. We write ∆ ` c :: κ to mean that c is a well-formed construc- tor of kind κ in general form. The difficulty with admitting general form constructors is that they introduce non-trivial equivalences between con- structors. For example, we must ensure that hint,booli · l is equivalent to int wherever the former may occur. With this in mind we will introduce a canonization procedure that allows us to define equivalence of general form constructors, written ∆ ` c1 ≡ c2 :: κ, to mean that c1 and c2 have identical canonical forms (up to α-equivalence). Canonization of general-form constructors is defined by these two judg- ments: REVISED 05.15.2012 VERSION 1.32 210 22.4 Canonization 1. Canonization: ∆ ` c :: κ ⇓ c: transform general-form constructor c of kind κ to canonical form c. 2. Atomization: ∆ ` c ⇑ c :: κ: transform general-form constructor c to obtain atomic form c of kind κ. These two judgments are defined simultaneously by the following rules. The canonization judgment is used to determine the canonical form of a general-form constructor; the atomization judgment is an auxiliary to the first that transforms constructors into atomic form. The canonization judg- ment is to be thought of as having mode (∀, ∀, ∃), whereas the atomization judgment is to be thought of as having mode (∀, ∃, ∃). ∆ ` c ⇑ c ::T ∆ ` c ::T ⇓ bc (22.9a) ∆ ` c :: 1 ⇓ hi (22.9b) ∆ ` c · l :: κ1 ⇓ c1 ∆ ` c · r :: κ2 ⇓ c2 ∆ ` c :: κ1 × κ2 ⇓ hc1,c2i (22.9c) ∆, u ⇑ κ1 ` c[u]:: κ2 ⇓ c2 ∆ ` c :: κ1 → κ2 ⇓ λ u.c2 (22.9d) ∆, u ⇑ κ ` u ⇑ u :: κ (22.9e) ∆ ` c ⇑ c :: κ1 × κ2 ∆ ` c · l ⇑ c · l :: κ1 (22.9f) ∆ ` c ⇑ c :: κ1 × κ2 ∆ ` c · r ⇑ c · r :: κ2 (22.9g) ∆ ` c1 ⇑ c1 :: κ1 → κ2 ∆ ` c2 :: κ1 ⇓ c2 ∆ ` c1[c2] ⇑ c1[c2]:: κ2 (22.9h) The canonization judgment produces canonical forms, and the atom- ization judgment produces atomic forms. Lemma 22.3. 1. If ∆ ` c :: κ ⇓ c, then ∆ ` c ⇓ κ. 2. If ∆ ` c ⇑ c :: κ, then ∆ ` c ⇑ κ. Proof. By induction on Rules (22.9). Theorem 22.4. If Γ ` c :: κ, then there exists c such that ∆ ` c :: κ ⇓ c. Proof. By induction on the formation rules for general-form constructors, making use of an analysis of the general-form constructors of kind T. VERSION 1.32 REVISED 05.15.2012 22.5 Notes 211 22.5 Notes The classical approach is to consider general-form constructors at the out- set, for which substitution is readily defined, and then to test equivalence of general-form constructors by reduction to a common irreducible form. Two main lemmas are required for this approach. First, every constructor must reduce in a finite number of steps to an irreducible form; this is called normalization. Second, the relation “has a common irreducible form” must be shown to be transitive; this is called confluence. Here we have turned the development on its head by considering only canonical constructors in the first place, then defining canonizing substitution introduced by Watkins et al.(2008). It is then straightforward to decide equivalence of general- form constructors by canonization of both sides of a candidate equation. REVISED 05.15.2012 VERSION 1.32 212 22.5 Notes VERSION 1.32 REVISED 05.15.2012 Part VIII Subtyping Chapter 23 Subtyping A subtype relation is a pre-order (reflexive and transitive relation) on types that validates the subsumption principle: if τ0 is a subtype of τ, then a value of type τ0 may be provided whenever a value of type τ is required. The subsumption principle relaxes the strictures of a type system to permit values of one type to be treated as values of another. Experience shows that the subsumption principle, although useful as a general guide, can be tricky to apply correctly in practice. The key to get- ting it right is the principle of introduction and elimination. To determine whether a candidate subtyping relationship is sensible, it suffices to con- sider whether every introductory form of the subtype can be safely manip- ulated by every eliminatory form of the supertype. A subtyping principle makes sense only if it passes this test; the proof of the type safety theorem for a given subtyping relation ensures that this is the case. A good way to get a subtyping principle wrong is to think of a type merely as a set of values (generated by introductory forms), and to consider whether every value of the subtype can also be considered to be a value of the supertype. The intuition behind this approach is to think of subtyping as akin to the subset relation in ordinary mathematics. But, as we shall see, this can lead to serious errors, because it fails to take account of the eliminatory forms that are applicable to the supertype. It is not enough to think only of the introductory forms; subtyping is a matter of behavior, rather than containment. 216 23.1 Subsumption 23.1 Subsumption A subtyping judgment has the form τ0 <: τ, and states that τ0 is a subtype of τ. At a minimum we demand that the following structural rules of subtyp- ing be admissible: τ <: τ (23.1a) τ00 <: τ0 τ0 <: τ τ00 <: τ (23.1b) In practice we either tacitly include these rules as primitive, or prove that they are admissible for a given set of subtyping rules. The point of a subtyping relation is to enlarge the set of well-typed pro- grams, which is achieved by the subsumption rule: Γ ` e : τ0 τ0 <: τ Γ ` e : τ (23.2) In contrast to most other typing rules, the rule of subsumption is not syntax- directed, because it does not constrain the form of e. That is, the subsump- tion rule may be applied to any form of expression. In particular, to show that e : τ, we have two choices: either apply the rule appropriate to the particular form of e, or apply the subsumption rule, checking that e : τ0 and τ0 <: τ. 23.2 Varieties of Subtyping In this section we will informally explore several different forms of subtyp- ing for various extensions of L{*}. In Section 23.4 we will examine some of these in more detail from the point of view of type safety. 23.2.1 Numeric Types For languages with numeric types, our mathematical experience suggests subtyping relationships among them. For example, in a language with types int, rat, and real, representing, respectively, the integers, the ratio- nals, and the reals, it is tempting to postulate the subtyping relationships int <: rat <: real by analogy with the set containments Z ⊆ Q ⊆ R VERSION 1.32 REVISED 05.15.2012 23.2 Varieties of Subtyping 217 familiar from mathematical experience. But are these subtyping relationships sensible? The answer depends on the representations and interpretations of these types! Even in mathe- matics, the containments just mentioned are usually not quite true—or are true only in a somewhat generalized sense. For example, the set of rational numbers may be considered to consist of ordered pairs (m, n), with n 6= 0 and gcd(m, n) = 1, representing the ratio m/n. The set Z of integers may be isomorphically embedded within Q by identifying n ∈ Z with the ratio n/1. Similarly, the real numbers are often represented as convergent se- quences of rationals, so that strictly speaking the rationals are not a subset of the reals, but rather may be embedded in them by choosing a canonical representative (a particular convergent sequence) of each rational. For mathematical purposes it is entirely reasonable to overlook fine dis- tinctions such as that between Z and its embedding within Q. This is jus- tified because the operations on rationals restrict to the embedding in the expected manner: if we add two integers thought of as rationals in the canonical way, then the result is the rational associated with their sum. And similarly for the other operations, provided that we take some care in defining them to ensure that it all works out properly. For the purposes of computing, however, we cannot be quite so cavalier, because we must also take account of algorithmic efficiency and the finiteness of machine representations. Often what are called “real numbers” in a programming language are, in fact, finite precision floating point numbers, a small subset of the rational numbers. Not every rational can be exactly represented as a floating point number, nor does floating point arithmetic restrict to ratio- nal arithmetic, even when its arguments are exactly represented as floating point numbers. 23.2.2 Product Types Product types give rise to a form of subtyping based on the subsumption principle. The only elimination form applicable to a value of product type is a projection. Under mild assumptions about the dynamics of projections, we may consider one product type to be a subtype of another by consid- ering whether the projections applicable to the supertype may be validly applied to values of the subtype. Consider a context in which a value of type τ = hτjij∈J is required. The statics of finite products (Rules (11.3)) ensures that the only operation we may perform on a value of type τ, other than to bind it to a variable, is to take the jth projection from it for some j ∈ J to obtain a value of type τj. REVISED 05.15.2012 VERSION 1.32 218 23.3 Variance Now suppose that e is of type τ0. If the projection e · j is to be well-formed, then τ0 must be a finite product type hτ0 i ii∈I such that j ∈ I. Moreover, for this to be of type τj, it is enough to require that τ0 j = τj. Because j ∈ J is arbitrary, we arrive at the following subtyping rule for finite product types: J ⊆ I ∏i∈I τi <: ∏j∈J τj .(23.3) This rule is sufficient for the required subtyping, but not necessary; we will consider a more liberal form of this rule in Section 23.3. The justification for Rule (23.3) is that we may evaluate e · i regardless of the actual form of e, provided only that it has a field indexed by i ∈ I. 23.2.3 Sum Types By an argument dual to the one given for finite product types we may de- rive a related subtyping rule for finite sum types. If a value of type ∑j∈J τj is required, the statics of sums (Rules (12.3)) ensures that the only non-trivial operation that we may perform on that value is a J-indexed case analysis. If we provide a value of type ∑i∈I τ0 i instead, no difficulty will arise so long as I ⊆ J and each τ0 i is equal to τi. If the containment is strict, some cases cannot arise, but this does not disrupt safety. This leads to the following subtyping rule for finite sums: I ⊆ J ∑i∈I τi <: ∑j∈J τj .(23.4) Note well the reversal of the containment as compared to Rule (23.3). 23.3 Variance In addition to basic subtyping principles such as those considered in Sec- tion 23.2, it is also important to consider the effect of subtyping on type constructors. A type constructor is said to be covariant in an argument if subtyping in that argument is preserved by the constructor. It is said to be contravariant if subtyping in that argument is reversed by the constructor. It is said to be invariant in an argument if subtyping for the constructed type is not affected by subtyping in that argument. VERSION 1.32 REVISED 05.15.2012 23.3 Variance 219 23.3.1 Product and Sum Types Finite product types are covariant in each field. For if e is of type ∏i∈I τ0 i , and the projection e · j is expected to be of type τj, then it is sufficient to require that j ∈ I and τ0 j <: τj. This is summarized by the following rule: (∀i ∈ I) τ0 i <: τi ∏i∈I τ0 i <: ∏i∈I τi (23.5) It is implicit in this rule that the dynamics of projection must not be sen- sitive to the precise type of any of the fields of a value of finite product type. Finite sum types are also covariant, because each branch of a case anal- ysis on a value of the supertype expects a value of the corresponding sum- mand, for which it is sufficient to provide a value of the corresponding subtype summand: (∀i ∈ I) τ0 i <: τi ∑i∈I τ0 i <: ∑i∈I τi (23.6) 23.3.2 Function Types The variance of the function type constructor is a bit more subtle. Let us consider first the variance of the function type in its range. Suppose that e : τ1 → τ0 2. This means that if e1 : τ1, then e(e1): τ0 2. If τ0 2 <: τ2, then e(e1): τ2 as well. This suggests the following covariance principle for function types: τ0 2 <: τ2 τ1 → τ0 2 <: τ1 → τ2 (23.7) Every function that delivers a value of type τ0 2 also delivers a value of type τ2, provided that τ0 2 <: τ2. Thus the function type constructor is covariant in its range. Now let us consider the variance of the function type in its domain. Suppose again that e : τ1 → τ2. This means that e may be applied to any value of type τ1 to obtain a value of type τ2. Hence, by the subsumption principle, it may be applied to any value of a subtype, τ0 1, of τ1, and it will still deliver a value of type τ2. Consequently, we may just as well think of e as having type τ0 1 → τ2. τ0 1 <: τ1 τ1 → τ2 <: τ0 1 → τ2 (23.8) REVISED 05.15.2012 VERSION 1.32 220 23.3 Variance The function type is contravariant in its domain position. Note well the reversal of the subtyping relation in the premise as compared to the con- clusion of the rule! Combining these rules we obtain the following general principle of contra- and co-variance for function types: τ0 1 <: τ1 τ0 2 <: τ2 τ1 → τ0 2 <: τ0 1 → τ2 (23.9) Beware of the reversal of the ordering in the domain! 23.3.3 Quantified Types The extension of subtyping to quantified types requires a judgment of the form ∆ ` τ0 <: τ, where ∆ ` τ0 type and ∆ ` τ type. The variance principles for the quantifiers may then be stated so that both are covariant in the quantified type: ∆, t type ` τ0 <: τ ∆ ` ∀(t.τ0) <: ∀(t.τ)(23.10a) ∆, t type ` τ0 <: τ ∆ ` ∃(t.τ0) <: ∃(t.τ)(23.10b) The judgment ∆ ` τ0 <: τ states that τ0 is a subtype of τ uniformly in the type variables declared in ∆. Consequently, we may derive the principle of substitution: Lemma 23.1. If ∆, t type ` τ0 <: τ, and ∆ ` ρ type, then ∆ ` [ρ/t]τ0 <: [ρ/t]τ. Proof. By induction on the subtyping derivation. It is easy to check that the above variance principles for the quantifiers are consistent with the principle of subsumption. For example, a package of the subtype ∃(t.τ0) consists of a representation type, ρ, and an imple- mentation, e, of type [ρ/t]τ0. But if t type ` τ0 <: τ, we have by substitution that [ρ/t]τ0 <:[ρ/t]τ, and hence e is also an implementation of type [ρ/t]τ. This is sufficient to ensure that the package is also of the supertype. It is natural to extend subtyping to the quantifiers by allowing quantifi- cation over all subtypes of a specified type. This is called bounded quantifica- tion. To express bounded quantification we consider additional hypotheses VERSION 1.32 REVISED 05.15.2012 23.3 Variance 221 of the form t <: ρ, expressing that t is a variable that may only be instanti- ated to subtypes of ρ. ∆, t type, t <: τ ` t <: τ (23.11a) ∆ ` τ ::T ∆ ` τ <: τ (23.11b) ∆ ` τ00 <: τ0 ∆ ` τ0 <: τ ∆ ` τ00 <: τ (23.11c) ∆ ` ρ0 <: ρ ∆, t type, t <: ρ0 ` τ0 <: τ ∆ ` ∀ t <: ρ.τ0 <: ∀ t <: ρ0.τ0 (23.11d) ∆ ` ρ0 <: ρ ∆, t type, t <: ρ0 ` τ0 <: τ ∆ ` ∃ t <: ρ0.τ0 <: ∃ t <: ρ.τ (23.11e) Rule (23.11d) states that the universal quantifier is contravariant in its bound, whereas Rule (23.11e) states that the existential quantifier is covariant in its bound. 23.3.4 Recursive Types The variance principle for recursive types is rather subtle, and has been the source of errors in language design. To gain some intuition, consider the type of labeled binary trees with natural numbers at each node, µt.[empty ,→ unit, binode ,→ hdata ,→ nat, lft ,→ t, rht ,→ ti], and the type of “bare” binary trees, without labels on the nodes, µt.[empty ,→ unit, binode ,→ hlft ,→ t, rht ,→ ti]. Is either a subtype of the other? Intuitively, we might expect the type of labeled binary trees to be a subtype of the type of bare binary trees, because any use of a bare binary tree can simply ignore the presence of the label. Now consider the type of bare “two-three” trees with two sorts of nodes, those with two children, and those with three: µt.[empty ,→ unit, binode ,→ τ2, trinode ,→ τ3], where τ2 , hlft ,→ t, rht ,→ ti, and τ3 , hlft ,→ t, mid ,→ t, rht ,→ ti. REVISED 05.15.2012 VERSION 1.32 222 23.3 Variance What subtype relationships should hold between this type and the preced- ing two tree types? Intuitively the type of bare two-three trees should be a supertype of the type of bare binary trees, because any use of a two-three tree must proceed by three-way case analysis, which covers both forms of binary tree. To capture the pattern illustrated by these examples, we must formulate a subtyping rule for recursive types. It is tempting to consider the following rule: t type ` τ0 <: τ µt.τ0 <: µt.τ ?? (23.12) That is, to determine whether one recursive type is a subtype of the other, we simply compare their bodies, with the bound variable treated as a pa- rameter. Notice that by reflexivity of subtyping, we have t <: t, and hence we may use this fact in the derivation of τ0 <: τ. Rule (23.12) validates the intuitively plausible subtyping between la- beled binary tree and bare binary trees just described. Deriving this re- quires checking that the subtyping relationship hdata ,→ nat, lft ,→ t, rht ,→ ti <: hlft ,→ t, rht ,→ ti, holds generically in t, which is evidently the case. Unfortunately, Rule (23.12) also underwrites incorrect subtyping rela- tionships, as well as some correct ones. As an example of what goes wrong, consider the recursive types τ0 = µt.ha ,→ t → nat, b ,→ t → inti and τ = µt.ha ,→ t → int, b ,→ t → inti. We assume for the sake of the example that nat <: int, so that by using Rule (23.12) we may derive τ0 <: τ, which we will show to be incorrect. Let e : τ0 be the expression fold(ha ,→ λ (x:τ0) 4, b ,→ λ (x:τ0) q((unfold(x)· a)(x))i), where q : nat → nat is the discrete square root function. Because τ0 <: τ, it follows that e : τ as well, and hence unfold(e): ha ,→ τ → int, b ,→ τ → inti. VERSION 1.32 REVISED 05.15.2012 23.4 Safety 223 Now let e0 : τ be the expression fold(ha ,→ λ (x:τ) -4, b ,→ λ (x:τ) 0i). (The important point about e0 is that the a method returns a negative num- ber; the b method is of no significance.) To finish the proof, observe that (unfold(e)· b)(e0) 7→∗ q(-4), which is a stuck state. We have derived a well-typed program that “gets stuck”, refuting type safety! Rule (23.12) is therefore incorrect. But what has gone wrong? The error lies in the choice of a single parameter to stand for both recursive types, which does not correctly model self-reference. In effect we are regarding two distinct recursive types as equal while checking their bodies for a sub- typing relationship. But this is clearly wrong! It fails to take account of the self-referential nature of recursive types. On the left side the bound variable stands for the subtype, whereas on the right the bound variable stands for the super-type. Confusing them leads to the unsoundness just illustrated. As is often the case with self-reference, the solution is to assume what we are trying to prove, and check that this assumption can be maintained by examining the bodies of the recursive types. To do so we make use of bounded quantification to state the rule of subsumption for recursive types: ∆, t type, t0 type, t0 <: t ` τ0 <: τ ∆, t0 type ` τ0 type ∆, t type ` τ type ∆ ` µt0.τ0 <: µt.τ . (23.13) That is, to check whether µt0.τ0 <: µt.τ, we assume that t0 <: t, because t0 and t stand for the respective recursive types, and check that τ0 <: τ under this assumption. It is instructive to check that the unsound subtyping is not derivable using this rule: the subtyping assumption is at odds with the contravariance of the function type in its domain. 23.4 Safety Proving safety for a language with subtyping is considerably more delicate than for languages without. The rule of subsumption means that the static type of an expression reveals only partial information about the underly- ing value. This changes the proof of the preservation and progress theo- rems, and requires some care in stating and proving the auxiliary lemmas required for the proof. REVISED 05.15.2012 VERSION 1.32 224 23.4 Safety As a representative case we will sketch the proof of safety for a language with subtyping for product types. The subtyping relation is defined by Rules (23.3) and (23.5). We assume that the statics includes subsumption, Rule (23.2). Lemma 23.2 (Structurality). 1. The tuple subtyping relation is reflexive and transitive. 2. The typing judgment Γ ` e : τ is closed under weakening and substitution. Proof. 1. Reflexivity is proved by induction on the structure of types. Tran- sitivity is proved by induction on the derivations of the judgments τ00 <: τ0 and τ0 <: τ to obtain a derivation of τ00 <: τ. 2. By induction on Rules (11.3), augmented by Rule (23.2). Lemma 23.3 (Inversion). 1. If e · j : τ, then e : ∏i∈I τi, j ∈ I, and τj <: τ. 2. If heiii∈I: τ, then ∏i∈I τ0 i <: τ where ei : τ0 i for each i ∈ I. 3. If τ0 <: ∏j∈J τj, then τ0 = ∏i∈I τ0 i for some I and some types τ0 i for i ∈ I. 4. If ∏i∈I τ0 i <: ∏j∈J τj, then J ⊆ I and τ0 j <: τj for each j ∈ J. Proof. By induction on the subtyping and typing rules, paying special at- tention to Rule (23.2). Theorem 23.4 (Preservation). If e : τ and e 7→ e0, then e0 : τ. Proof. By induction on Rules (11.4). For example, consider Rule (11.4d), so that e = heiii∈I· k, e0 = ek. By Lemma 23.3 we have that heiii∈I: ∏j∈J τj, k ∈ J, and τk <: τ. By another application of Lemma 23.3 for each i ∈ I there exists τ0 i such that ei : τ0 i and ∏i∈I τ0 i <: ∏j∈J τj. By Lemma 23.3 again, we have J ⊆ I and τ0 j <: τj for each j ∈ J. But then ek : τk, as desired. The remaing cases are similar. Lemma 23.5 (Canonical Forms). If e val and e : ∏j∈J τj, then e is of the form heiii∈I, where J ⊆ I, and ej : τj for each j ∈ J. VERSION 1.32 REVISED 05.15.2012 23.5 Notes 225 Proof. By induction on Rules (11.3) augmented by Rule (23.2). Theorem 23.6 (Progress). If e : τ, then either e val or there exists e0 such that e 7→ e0. Proof. By induction on Rules (11.3) augmented by Rule (23.2). The rule of subsumption is handled by appeal to the inductive hypothesis on the premise of the rule. Rule (11.4d) follows from Lemma 23.5. 23.5 Notes Subtyping is perhaps the most widely misunderstood concept in program- ming languages. Subtyping is principally a convenience, akin to type in- ference, that makes some programs simpler to write. But the subsumption rule cuts both ways. Inasmuch as it allows the implicit passage from τ0 to τ whenever τ0 is a subtype of τ, it also weakens the meaning of a type assertion e : τ to mean that e has some type contained in the type τ. This precludes expressing the requirement that e has exactly the type τ, or that two expressions jointly have the same type. And it is precisely this weak- ness that creates so many of the difficulties with subtyping. Much has been written about subtyping. Standard ML (Milner et al., 1997) is one of the earliest full-scale languages to make essential use of sub- typing. The statics of the ML module system makes use of two subtyping relations, called enrichment and realization, corresponding to product sub- typing and type definitions. The first systematic studies of subtyping in- clude those by Mitchell(1984); Reynolds(1980), and Cardelli(1988). Pierce (2002) gives a thorough account of subtyping, especially of recursive and polymorphic types, and proves that subtyping for bounded impredicative universal quantification is undecidable. REVISED 05.15.2012 VERSION 1.32 226 23.5 Notes VERSION 1.32 REVISED 05.15.2012 Chapter 24 Singleton Kinds The expression let e1:τ be x in e2 is a form of abbreviation mechanism by which we may bind e1 to the variable x for use within e2. In the presence of function types this expression is definable as the application (λ (x:τ) e2)(e1), which accomplishes the same thing. It is natural to consider an analogous form of let expression which permits a type expression to be bound to a type variable within a specified scope. Using def t is τ in e to bind the type variable t to τ within the expression e, we may write expressions such as def t is nat × nat in λ (x:t) s(x · l), which introduces a type abbreviation within an expression. To ensure that this expression is well-typed, the type variable t must be synonymous with the type nat × nat, for otherwise the body of the λ-abstraction is not type correct. Following the pattern of the expression-level let, we might guess that def t is τ in e abbreviates the polymorphic instantiation Λ(t.e)[τ], which binds t to τ within e. This does, indeed, capture the dynamics of type ab- breviation, but it fails to validate the intended statics. The difficulty is that, according to this interpretation of type definitions, the expression e is type- checked in the absence of any knowledge of the binding of t, rather than in the knowledge that t is synonymous with τ. Thus, in the above exam- ple, the expression s(x · l) would fail to type check, unless the binding of t were exposed. The interpretation of type definition in terms of type abstraction and type application fails. Lacking any other idea, we might argue that type abbreviation ought to be considered as a primitive concept, rather than a derived notion. The expression def t is τ in e would be taken as a primitive 228 24.1 Overview form of expression whose statics is given by the following rule: Γ ` [τ/t]e : τ0 Γ ` def t is τ in e : τ0 (24.1) This would address the problem of supporting type abbreviations, but it does so in a rather ad hoc manner. We seek a more principled solution that arises naturally from the type structure of the language. The methodology of identifying language constructs with type struc- ture suggests that we ask not how to support type abbreviations, but rather what form of type structure gives rise to type abbreviations? Thinking along these lines leads to the concept of singleton kinds, which not only ac- count for type abbreviations but also play a crucial role in the design of module systems, as will be discussed in detail in Chapters 45 and 46. 24.1 Overview The central organizing principle of type theory is compositionality. To en- sure that a program may be decomposed into separable parts, we ensure that the composition of a program from constituent parts is mediated by the types of those parts. Put in other terms, the only thing that one portion of a program “knows” about another is its type. For example, the forma- tion rule for addition of natural numbers depends only on the type of its arguments (both must have type nat), and not on their specific form or value. But in the case of a type abbreviation of the form def t is τ in e, the principle of compositionality dictates that the only thing that e “knows” about the type variable t is its kind, namely T, and not its binding, namely τ. This is accurately captured by the proposed representation of type ab- breviation as the combination of type abstraction and type application, but, as we have just seen, this is not the intended meaning of the construct! We could, as suggested in the introduction, abandon the core principles of type theory, and introduce type abbreviations as a primitive notion. But there is no need to do so. Instead we can simply note that what is needed is for the kind of t to capture its identity. This may be achieved through the notion of a singleton kind. Informally, the kind S(τ) is the kind of types that are definitionally equal to τ. That is, up to definitional equality, this kind has only one inhabitant, namely τ. Consequently, if u :: S(τ) is a vari- able of singleton kind, then within its scope, the variable u is synonymous with τ. Thus we may represent def t is τ in e by Λ(t::S(τ).e)[τ], which correctly propagates the identity of t, namely τ, to e during type checking. VERSION 1.32 REVISED 05.15.2012 24.2 Singletons 229 A proper treatment of singleton kinds requires some additional machin- ery at the constructor and kind level. First, we must capture the idea that a constructor of singleton kind is a fortiori a constructor of kind T, and hence is a type. Otherwise, a variable, u, of singleton kind cannot be used as a type, even though it is explicitly defined to be one! This may be captured by introducing a subkinding relation, κ1 :<: κ2, which is analogous to sub- typing, except at the kind level. The fundamental axiom of subkinding is S(τ):<:T, stating that every constructor of singleton kind is a type. Second, we must account for the occurrence of a constructor of kind T within the singleton kind S(τ). This intermixing of the constructor and kind level means that singletons are a form of dependent kind in that a kind may depend on a constructor. Another way to say the same thing is that S(τ) represents a family of kinds indexed by constructors of kind T. This, in turn, implies that we must generalize the product and function kinds to dependent products and dependent functions. The dependent product kind, Σ u::κ1.κ2, classifies pairs hc1,c2i such that c1 :: κ1, as might be expected, and c2 ::[c1/u]κ2, in which the kind of the second component is sensitive to the first component itself, and not just its kind. The dependent function kind, Π u::κ1.κ2 classifies functions that, when applied to a constructor c1 :: κ1, results in a constructor of kind [c1/u]κ2. The important point is that the kind of the result is sensitive to the argument, and not just to its kind. Third, it is useful to consider singletons not just of kind T, but also of higher kinds. To support this we introduce higher singletons, written S(c :: κ), where κ is a kind and c is a constructor of kind k. These are defin- able in terms of the basic form of singleton kinds using dependent function and product kinds. 24.2 Singletons The syntax of singleton kinds is given by the following grammar: Kind κ ::= S(c) S(c) singleton Informally, the singleton kind, S(c), classifies constructors that are equiv- alent (in a sense to be made precise shortly) to c. For the time being we tacitly include the constructors and kinds given in Chapter 22 (but see Sec- tion 24.3). REVISED 05.15.2012 VERSION 1.32 230 24.2 Singletons The following judgment forms comprise the statics of singletons: ∆ ` κ kind kind formation ∆ ` κ1 ≡ κ2 kind equivalence ∆ ` c :: κ constructor formation ∆ ` c1 ≡ c2 :: κ constructor equivalence ∆ ` κ1 :<: κ2 subkinding These judgments are defined simultaneously by a collection of rules includ- ing the following: ∆ ` c :: Type ∆ ` S(c) kind (24.2a) ∆ ` c :: Type ∆ ` c :: S(c)(24.2b) ∆ ` c :: S(d) ∆ ` c ≡ d :: Type (24.2c) ∆ ` c :: κ1 ∆ ` κ1 :<: κ2 ∆ ` c :: κ2 (24.2d) ∆ ` c :: Type ∆ ` S(c):<: Type (24.2e) ∆ ` c ≡ d :: Type ∆ ` S(c) ≡ S(d)(24.2f) ∆ ` κ1 ≡ κ2 ∆ ` κ1 :<: κ2 (24.2g) ∆ ` κ1 :<: κ2 ∆ ` κ2 :<: κ3 ∆ ` κ1 :<: κ3 (24.2h) Omitted for brevity are rules stating that constructor and kind equivalence are reflexive, symmetric, transitive, and preserved by kind and constructor formation. Rule (24.2b) expresses the principle of “self-recognition,” which states that every constructor, c, of kind Type also has the kind S(c). By Rule (24.2c) any constructor of kind S(c) is definitionally equal to c. Consequently, self- recognition is in this sense an expression of the reflexivity of constructor equivalence. Rule (24.2e) is just the subsumption principle re-stated at the level of constructors and kinds. Rule (24.2f) states that the singleton kind respects equivalence of its constructors, so that equivalent constructors de- termine the same singletons. Rules (24.2g) and (24.2h) state that the sub- kinding relation is a pre-order that respects kind equivalence. VERSION 1.32 REVISED 05.15.2012 24.3 Dependent Kinds 231 To see these rules in action let us consider a few illustrative examples. First, consider the behavior of variables of singleton kind. Suppose that ∆ ` u :: S(c) is such a variable. Then by Rule (24.2c) we may deduce that ∆ ` u ≡ c ::T. Thus, the declaration of u with a singleton kind serves to define u to be the constructor (of kind T) specified by its kind. Singletons capture the concept of a type definition discussed in the introduction to this chapter. Taking this a step further, the existential type ∃ u :: S(c).τ is the type of packages whose representation type is (equivalent to) c—it is an abstract type whose identity is revealed by assigning it a singleton kind. By the gen- eral principles of equivalence we have that the type ∃ u :: S(c).τ is equiva- lent to the type ∃ :: S(c).[c/u]τ, wherein we have propagated the equiv- alence of u and c into the type τ. On the other hand we may also “forget” the definition of u, because the subtyping ∃ u :: S(c).τ <: ∃ u :: T.τ is derivable using the following variance rule for existentials over a kind: ∆ ` κ1 :<: κ2 ∆, u :: κ1 ` τ1 <: τ2 ∆ ` ∃ u :: κ1.τ1 <: ∃ u :: κ2.τ2 (24.3) Similarly, we may derive the subtyping ∀ u :: T.τ <: ∀ u :: S(c).τ from the following variance rule for universals over a kind: ∆ ` κ2 :<: κ1 ∆, u :: κ2 ` τ1 <: τ2 ∆ ` ∀ u :: κ1.τ1 <: ∀ u :: κ2.τ2 (24.4) Informally, the displayed subtyping states that a polymorphic function that may be applied to any type is one that may only be applied to a particular type, c. These examples show that singleton kinds express the idea of a scoped definition of a type variable in a way that is not tied to an ad hoc definition mechanism, but rather arises naturally from general principles of binding and scope. We will see in Chapters 45 and 46 more sophisticated uses of singletons to manage the interaction among program modules. 24.3 Dependent Kinds Although it is perfectly possible to add singleton kinds to the framework of higher kinds introduced in Chapter 22, to do so would be to shortchange REVISED 05.15.2012 VERSION 1.32 232 24.3 Dependent Kinds the expressiveness of the language. Using higher kinds we can express the kind of constructors that, when applied to a type, yield a specific type, say int, as result, namely T → S(int). But we cannot express the kind of constructors that, when applied to a type, yield that very type as result, for there is no way for the result kind to refer to the argument of the func- tion. Similarly, using product kinds we can express the kind of pairs whose first component is int and whose second component is an arbitrary type, namely S(int) × T. But we cannot express the kind of pairs whose sec- ond component is equivalent to its first component, for there is no way for the kind of the second component to make reference to the first component itself. To express such concepts requires a generalization of product and func- tion kinds in which the kind of the second component of a pair may men- tion the first component of that pair, or the kind of the result of a function may mention the argument to which it is applied. Such kinds are called dependent kinds because they involve kinds that mention, or depend upon, constructors (of kind T). The syntax of dependent kinds is given by the following grammar: Kind κ ::= S(c) S(c) singleton Σ(κ1; u.κ2)Σ u::κ1.κ2 dependent product Π(κ1; u.κ2)Π u::κ1.κ2 dependent function Con c ::= u u variable pair(c1; c2) hc1,c2i pair proj[l](c) c · l first projection proj[r](c) c · r second projection lam[κ](u.c) λ (u::κ) c abstraction app(c1; c2) c1[c2] application As a notational convenience, when there is no dependency in a kind we write κ1 × κ2 for Σ::κ1.κ2, and κ1 → κ2 for Π::κ1.κ2, where the “blank” stands for an irrelevant variable. The syntax of dependent kinds differs from that given in Chapter 22 for higher kinds in that we do not draw a distinction between atomic and canonical constructors, and consider that substitution is defined conven- tionally, rather than hereditarily. This simplifies the syntax, but at the ex- pense of leaving open the decidability of constructor equivalence. The method of hereditary substitution considered in Chapter 22 may be ex- tended to singleton kinds, but we will not develop this extension here. In- stead we will simply assert without proof that equivalence of well-formed constructors is decidable. VERSION 1.32 REVISED 05.15.2012 24.3 Dependent Kinds 233 The dependent product kind Σ u::κ1.κ2 classifies pairs hc1,c2i of con- structors in which c1 has kind κ1 and c2 has kind [c1/u]κ2. For example, the kind Σ u::T.S(u) classifies pairs hc,ci, where c is a constructor of kind T. More generally, this kind classifies pairs of the form hc1,c2i where c1 and c2 are equivalent, but not necessarily identical, constructors. The dependent function kind Π u::κ1.κ2 classifies constructors c that, when applied to a constructor c1 of kind κ1 yield a constructor of kind [c1/u]κ2. For example, the kind Π u::T.S(u) classifies constructors that, when applied to a con- structor, c, yield a constructor equivalent to c; a constructor of this kind is essentially the identity function. We may, of course, combine these to form kinds such as Π u::T × T.S(u · r) × S(u · l), which classifies functions that swap the components of a pair of types. (Such examples may lead us to surmise that the behavior of any construc- tor may be pinned down precisely using dependent kinds. We shall see in Section 24.4 that this is indeed the case.) The formation, introduction, and elimination rules for the product kind are as follows: ∆ ` κ1 kind ∆, u :: κ1 ` κ2 kind ∆ ` Σ u::κ1.κ2 kind (24.5a) ∆ ` c1 :: κ1 ∆ ` c2 ::[c1/u]κ2 ∆ ` hc1,c2i ::Σ u::κ1.κ2 (24.5b) ∆ ` c ::Σ u::κ1.κ2 ∆ ` c · l :: κ1 (24.5c) ∆ ` c ::Σ u::κ1.κ2 ∆ ` c · r ::[c1/u]κ2 (24.5d) In Rule (24.5a), observe that the variable, u, may occur in the kind κ2 by ap- pearing in a singleton kind. Correspondingly, Rules (24.5b), (24.5c), and (24.5d) substitute a constructor for this variable. Constructor equivalence is defined to be an equivalence relation that is compatible with all forms of constructors and kinds, so that a constructor may always be replaced by an equivalent constructor and the result will be equivalent. The following equivalence axioms govern the constructors associated with the dependent product kind: ∆ ` c1 :: κ1 ∆ ` c2 :: κ2 ∆ ` hc1,c2i · l ≡ c1 :: κ1 (24.6a) ∆ ` c1 :: κ1 ∆ ` c2 :: κ2 ∆ ` hc1,c2i · r ≡ c2 :: κ2 (24.6b) REVISED 05.15.2012 VERSION 1.32 234 24.3 Dependent Kinds The subkinding rule for the dependent product kind specifies that it is covariant in both positions: ∆ ` κ1 :<: κ0 1 ∆, u :: κ1 ` κ2 :<: κ0 2 ∆ ` Σ u::κ1.κ2 :<:Σ u::κ0 1.κ0 2 (24.7) The congruence rule for equivalence of dependent product kinds is for- mally similar: ∆ ` κ1 ≡ κ0 1 ∆, u :: κ1 ` κ2 ≡ κ0 2 ∆ ` Σ u::κ1.κ2 ≡ Σ u::κ0 1.κ0 2 (24.8) Notable consequences of these rules include the subkindings Σ u::S(int).S(u):<:Σ u::T.S(u) and Σ u::T.S(u):<:T × T, and the equivalence Σ u::S(int).S(u) ≡ S(int) × S(int). Subkinding is used to “forget” information about the identity of the com- ponents of a pair, and equivalence is used to propagate such information within a kind. The formation, introduction, and elimination rules for dependent func- tion kinds are as follows: ∆ ` κ1 kind ∆, u :: κ2 ` κ2 kind ∆ ` Π u::κ1.κ2 kind (24.9a) ∆, u :: κ1 ` c :: κ2 ∆ ` λ (u::κ1) c ::Π u::κ1.κ2 (24.9b) ∆ ` c ::Π u::κ1.κ2 ∆ ` c1 :: κ1 ∆ ` c[c1]::[c1/u]κ2 (24.9c) Rule (24.9b) specifies that the result kind of a λ-abstraction depends uni- formly on the argument, u. Correspondingly, Rule (24.9c) specifies that the kind of an application is obtained by substitution of the argument into the result kind of the function itself. The following rule of equivalence governs the constructors associated with the dependent product kind: ∆, u :: κ1 ` c :: κ2 ∆ ` c1 :: κ1 ∆ ` (λ (u::κ1) c)[c1] ≡ [c1/u]c :: κ2 (24.10) VERSION 1.32 REVISED 05.15.2012 24.4 Higher Singletons 235 The subkinding rule for the dependent function kind specifies that it is contravariant in its domain and covariant in its range: ∆ ` κ0 1 :<: κ1 ∆, u :: κ0 1 ` κ2 :<: κ0 2 ∆ ` Π u::κ1.κ2 :<:Π u::κ0 1.κ0 2 (24.11) The equivalence rule is similar, except that the symmetry of equivalence obviates a choice of variance: ∆ ` κ1 ≡ κ0 1 ∆, u :: κ1 ` κ2 ≡ κ0 2 ∆ ` Π u::κ1.κ2 ≡ Π u::κ0 1.κ0 2 (24.12) Rule (24.11) gives rise to the subkinding Π u::T.S(int) :<:Π u::S(int).T, which illustrates the co- and contra-variance of the dependent function kind. In particular a function that takes any type and delivers the type int is also a function that takes the type int and delivers a type. Rule (24.12) gives rise to the equivalence Π u::S(int).S(u) ≡ S(int) → S(int), which propagates information about the argument into the range kind. Combining these two rules we may derive the subkinding Π u::T.S(u):<: S(int) → S(int). Intuitively, a constructor function that yields its argument is, in particular, a constructor function that may only be applied to int, and yields int. Formally, by contravariance we have the subkinding Π u::T.S(u):<:Π u::S(int).S(u), and by sharing propagation we may derive the indicated superkind. 24.4 Higher Singletons Although singletons are restricted to constructors of kind T, we may use dependent product and function kinds to define singletons of every kind. Specifically, we wish to define the kind S(c :: κ), where c is of kind κ, that classifies constructors equivalent to c. When κ = T this is, of course, just REVISED 05.15.2012 VERSION 1.32 236 24.4 Higher Singletons S(c); the problem is to define singletons for the higher kinds Σ u::κ1.κ2 and Π u::κ1.κ2. To see what is involved, suppose that c :: κ1 × κ2. The singleton kind S(c :: κ1 × κ2) should classify constructors equivalent to c. If we assume, inductively, that singletons have been defined for κ1 and κ2, then we need only observe that c is equivalent to hc · l,c · ri. For then the singleton S(c :: κ1 × κ2) may be defined to be S(c · l :: κ1) × S(c · r :: κ2). Similarly, suppose that c :: κ1 → κ2. Using the equivalence of c and λ (u::κ1 → κ2) c[u], we may define S(c :: κ1 → κ2) to be Π u::κ1.S(c[u]:: κ2). In general the kind S(c :: κ) is defined by induction on the structure of κ by the following kind equivalences: ∆ ` c :: S(c0) ∆ ` S(c :: S(c0)) ≡ S(c)(24.13a) ∆ ` c ::Σ u::κ1.κ2 ∆ ` S(c ::Σ u::κ1.κ2) ≡ Σ u::S(c · l :: κ1).S(c · r :: κ2)(24.13b) ∆ ` c ::Π u::κ1.κ2 ∆ ` S(c ::Π u::κ1.κ2) ≡ Π u::κ1.S(c[u]:: κ2)(24.13c) The sensibility of these equations relies on Rule (24.2c) together with the following principles of constructor equivalence, called extensionality princi- ples: ∆ ` c ::Σ u::κ1.κ2 ∆ ` c ≡ hc · l,c · ri ::Σ u::κ1.κ2 (24.14a) ∆ ` c ::Π u::κ1.κ2 ∆ ` c ≡ λ (u::κ1) c[u]::Π u::κ1.κ2 (24.14b) Rule (24.2c) states that the only constructors of kind S(c0) are those equiv- alent to c0, and Rules (24.14a) and (24.14b) state that the only members of the dependent product and function types are, respectively, pairs and λ- abstractions of the appropriate kinds. Finally, the following self-recognition rules are required to ensure that Rule (24.2b) may be extended to higher kinds. ∆ ` c · l :: κ1 ∆ ` c · r ::[c · l/u]κ2 ∆ ` c ::Σ u::κ1.κ2 (24.15a) ∆, u :: κ1 ` c[u]:: κ2 ∆ ` c ::Π u::κ1.κ2 (24.15b) An illustrative case arises when u is a constructor variable of kind Σ v::T.S(v). We may derive that u · l :: S(u · l) using Rule (24.2b). We may also derive VERSION 1.32 REVISED 05.15.2012 24.5 Notes 237 u · r :: S(u · l) using Rule (24.5d). Therefore, by Rule (24.15a), we may de- rive u ::Σ v::S(u · l).S(u · l), which is a subkind of Σ v::T.S(v). This more precise kind is a correct kinding for u, because the first component of u is indeed u · l, and the second component of u is equivalent to the first component, and hence is also u · l. But without Rule (24.15a) it is impossi- ble to derive this fact. The point of introducing higher singletons is to ensure that every con- structor may be classified by a kind that determines it up to definitional equality. Viewed as an extension of singleton types, we would expect that higher singletons enjoy similar properties. This is captured by the follow- ing lemma: Theorem 24.1. If ∆ ` c :: κ, then ∆ ` S(c :: κ):<: κ and ∆ ` c ::S(c :: κ). The proof of this theorem is surprisingly intricate; the reader is referred to the references below for details. 24.5 Notes Singleton kinds were introduced by Stone and Harper(2006) to isolate the concept of type sharing that arises in the ML module system (Milner et al., 1997; Harper and Lillibridge, 1994; Leroy, 1994). Crary(2009) extends the concept of hereditary substitution discussed in Chapter 22 to singleton kinds. REVISED 05.15.2012 VERSION 1.32 238 24.5 Notes VERSION 1.32 REVISED 05.15.2012 Part IX Classes and Methods Chapter 25 Dynamic Dispatch It frequently arises that the values of a type are partitioned into a variety of classes, each classifying data with distinct internal structure. A good exam- ple is provided by the type of points in the plane, which may be classified according to whether they are represented in cartesian or polar form. Both are represented by a pair of real numbers, but in the cartesian case these are the x and y coordinates of the point, whereas in the polar case these are its distance, r, from the origin and its angle, θ, with the polar axis. A classified value is said to be an object, or instance, of its class. The class determines the type of the classified data, which is called the instance type of the class. The classified data itself is called the instance data of the object. Functions that act on classified values are sometimes called methods. The behavior of a method is determined by the class of its argument. The method is said to dispatch on the class of the argument.1 Because it hap- pens at run-time, this is called dynamic dispatch. For example, the squared distance of a point from the origin is calculated differently according to whether the point is represented in cartesian or polar form. In the former case the required distance is x2 + y2, whereas in the latter it is simply r itself. Similarly, the quadrant of a cartesian point may be determined by examining the sign of its x and y coordinates, and the quadrant of a polar point may be calculated by taking the integral part of the angle θ divided by π/2. Dynamic dispatch is often described in terms of a particular implemen- tation strategy, which we will call the class-based organization. In this or- ganization each object is represented by a vector of methods specialized to 1More generally, we may dispatch on the class of multiple arguments simultaneously. We concentrate on single dispatch for the sake of simplicity. 242 25.1 The Dispatch Matrix the class of that object. We may equivalently use a method-based organiza- tion in which each method branches on the class of an object to determine its behavior. Regardless of the organization used, the fundamental idea is that (a) objects are classified, and (b) methods dispatch on the class of an object. The class-based and method-based organizations are interchange- able, and, in fact, related by a natural duality between sum and product types. We elucidate this symmetry by focusing first on the behavior of each method on each object, which is given by a dispatch matrix. From this we derive both a class-based and a method-based organization in such a way that their equivalence is evident. 25.1 The Dispatch Matrix Because each method acts by dispatch on the class of its argument, we may envision the entire system of classes and methods as a matrix, edm, called the dispatch matrix, whose rows are classes, whose columns are methods, and whose (c, d)-entry defines the behavior of method d acting on an argu- ment of class c, expressed as a function of the instance data of the object. Thus, the dispatch matrix has a type of the form ∏ c∈C ∏ d∈D (τc → ρd), where C is the set of class names, D is the set of method names, τc is the instance type associated with class c and ρd is the result type of method d. The instance type is the same for all methods acting on a given class, and the result type is the same for all classes acted on by a given method. As an illustrative example, let us consider the type of points in the plane classified into two classes, cart and pol, corresponding to the cartesian and polar representations. The instance data for a cartesian point has the type τcart = hx ,→ real, y ,→ reali, and the instance data for a polar point has the type τpol = hr ,→ real, th ,→ reali. Consider two methods acting on points, dist and quad, which com- pute, respectively, the squared distance of a point from the origin and the quadrant of a point. The squared distance method is given by the tuple edist = hcart ,→ ecart dist, pol ,→ epol disti of type hcart ,→ τcart → ρdist, pol ,→ τpol → ρdisti, VERSION 1.32 REVISED 05.15.2012 25.1 The Dispatch Matrix 243 where ρdist = real is the result type, ecart dist = λ (u:τcart)(u · x)2 + (u · y)2 is the squared distance computation for a cartesian point, and epol dist = λ (v:τpol)(v · r)2 is the squared distance computation for a polar point. Similarly, the quad- rant method is given by the tuple equad = hcart ,→ ecart quad, pol ,→ epol quadi of type hcart ,→ τcart → ρquad, pol ,→ τpol → ρquadi, where ρquad = [I, II, III, IV] is the type of quadrants, and ecart quad and epol quad are expressions that compute the quadrant of a point in rectangular and polar forms, respectively. Now let C = { cart, pol } and let D = { dist, quad }, and define the dispatch matrix, edm, to be the value of type ∏ c∈C ∏ d∈D (τc → ρd) such that, for each class c and method d, edm · c · d 7→∗ ec d. That is, the entry in the dispatch matrix, edm, for class c and method d de- fines the behavior of that method acting on an object of that class. Dynamic dispatch is an abstraction given by the following components: • A type, obj, of objects, which are classified by the classes on which the methods act. • An operation new[c](e) of type obj that creates an object of the class c with instance data given by the expression e of type τc. • An operation e ⇐ d of type ρd that invokes method d on the object given by the expression e of type obj. These operations are required to satisfy the defining characteristic of dy- namic dispatch, (new[c](e)) ⇐ d 7→∗ ec d(e), which states that invoking method d on an object of class c with instance data e amounts to applying ec d, the code in the dispatch matrix for class c and method d, to the instance data, e. REVISED 05.15.2012 VERSION 1.32 244 25.2 Class-Based Organization There are two main ways to implement this abstraction. One formula- tion, called the class-based organization, defines objects as tuples of meth- ods, and creates objects by specializing the methods to the given instance data. Another formulation, called the method-based organization, creates objects by tagging the instance data with the class, and defines methods by dispatch on the class of the object. These two organizations are isomor- phic to one another, and hence may be interchanged at will. Nevertheless, many languages favor one representation over the other, asymmetrizing an inherently symmetric situation. 25.2 Class-Based Organization The class-based organization starts with the observation that the dispatch matrix may be reorganized to “factor out” the instance data for each method acting on that class to obtain the class vector, ecv, of type τcv , ∏ c∈C (τc → (∏ d∈D ρd)). Each row of the class vector consists of a constructor that determines the result of each of the methods when acting on given instance data. An object has the type ρ = ∏d∈D ρd consisting of the product over the methods of the result types of the methods. For example, in the case of points in the plane, the type ρ is the product type hdist ,→ ρdist, quad ,→ ρquadi. Each component specifies the result of each of the methods acting on that object. The message send operation, e ⇐ d, is just the projection e · d. So, in the case of points in the plane, e ⇐ dist is the projection e · dist, and similarly e ⇐ quad is the projection e · quad. The class-based organization consolidates the implementation of each class into a class vector, ecv, a tuple of type τcv consisting of the constructor of type τc → ρ for each class c ∈ C. The class vector is defined by ecv = hecic∈C, where for each c ∈ C the expression ec is λ (u:τc) hedm · c · d(u)id∈D. For example, the constructor for the class cart is the function ecart given by the expression λ (u:τcart) hdist ,→ edm · cart · dist(u), quad ,→ edm · cart · quad(u)i. VERSION 1.32 REVISED 05.15.2012 25.3 Method-Based Organization 245 Similarly, the constructor for the class pol is the function epol given by the expression λ (u:τpol) hdist ,→ edm · pol · dist(u), quad ,→ edm · pol · quad(u)i. The class vector, ecv, in this case is the tuple hcart ,→ ecart, pol ,→ epoli of type hcart ,→ τcart → ρ, pol ,→ τpol → ρi. An object of a class is obtained by applying the constructor for that class to the instance data: new[c](e), ecv · c(e). For example, a cartesian point is obtained by writing new[cart](hx ,→ x0, y ,→ y0i), which is defined by the expression ecv · cart(hx ,→ x0, y ,→ y0i). Similarly, a polar point is obtained by writing new[pol](r ,→ r0, th ,→ θ0), which is defined by the expression ecv · pol(hr ,→ r0, th ,→ θ0i). It is easy to check for this organization of points that for each class c and method d, we may derive (new[c](e)) ⇐ d 7→∗ (ecv · c(e)) · d 7→∗ edm · c · d(e). That is, the message send evokes the behavior of the given method on the instance data of the given object. 25.3 Method-Based Organization The method-based organization starts with the transpose of the dispatch matrix, which has the type ∏ d∈D ∏ c∈C (τc → ρd). By observing that each row of the transposed dispatch matrix determines a method, we obtain the method vector, emv, of type τmv , ∏ d∈D (∑ c∈C τc) → ρd. REVISED 05.15.2012 VERSION 1.32 246 25.3 Method-Based Organization Each entry of the method vector consists of a dispatcher that determines the result as a function of the instance data associated with a given object. An object is a value of type τ = ∑c∈C τc, the sum over the classes of the instance types. For example, the type of points in the plane is the sum type [cart ,→ τcart, pol ,→ τpol]. Each point is labeled with its class, specifying its representation as having either cartesian or polar form. An object of a class c is just the instance data labeled with its class to form an element of the object type: new[c](e), c · e. For example, a cartesian point with coordinates x0 and y0 is given by the expression new[cart](hx ,→ x0, y ,→ y0i), cart · hx ,→ x0, y ,→ y0i. Similarly, a polar point with distance r0 and angle θ0 is given by the expres- sion new[pol](hr ,→ r0, th ,→ θ0i), pol · hr ,→ r0, th ,→ θ0i. The method-based organization consolidates the implementation of each method into the method vector, emv of type τmv, defined by hedid∈D, where for each d ∈ D the expression ed : τ → ρd is λ (this:τ) case this {c · u ⇒ edm · c · d(u)}c∈C. Each entry in the method vector may be thought of as a dispatch function that determines the action of that method on each class of object. In the case of points in the plane, the method vector has the product type hdist ,→ τ → ρdist, quad ,→ τ → ρquadi. The dispatch function for the dist method has the form λ (this:τ) case this {cart · u ⇒ edm · cart · dist(u) | pol · v ⇒ edm · pol · dist(v)}, and the dispatch function for the quad method has the similar form λ (this:τ) case this {cart · u ⇒ edm · cart · quad(u) | pol · v ⇒ edm · pol · quad(v)}. VERSION 1.32 REVISED 05.15.2012 25.4 Self-Reference 247 The message send operation, e ⇐ d, applies the dispatch function for method d to the object e: e ⇐ d , emv · d(e). Thus we have, for each class, c, and method, d, (new[c](e)) ⇐ d 7→∗ emv · d(c · e) 7→∗ edm · c · d(e) The result is, of course, the same as for the class-based organization. 25.4 Self-Reference It is often useful to allow methods to create new objects or to send messages to objects. This is not possible using the simple dispatch matrix described in Section 25.1, for the simple reason that there is no provision for self- reference within its entries. This deficiency may be remedied by changing the type of the entries of the dispatch matrix to account for sending mes- sages and creating objects, as follows: ∏ c∈C ∏ d∈D ∀(t.τcv → τmv → τc → ρd). The type variable, t, is an abstract type representing the object type. The types τcv and τmv, are, respectively, the type of the class and method vectors, defined in terms of the abstract type t. They are defined by the equations τcv , ∏ c∈C (τc → t) and τmv , ∏ d∈D (t → ρd). The component of the class vector corresponding to a class, c, is a construc- tor that builds a value of the abstract object type, t, from the instance data for c. The component of the method vector corresponding to a method, d, is a dispatcher that yields a result of type ρd when applied to a value of the abstract object type, t. In accordance with the revised type of the dispatch matrix the behavior associated to class c and method d has the form Λ(t.λ (cv:τcv) λ (mv:τmv) λ (u:τc) ec d). REVISED 05.15.2012 VERSION 1.32 248 25.4 Self-Reference The arguments cv and mv are used to create new objects and to send mes- sages to objects. Within the expression ec d an object of class c0 with instance data e0 is created by writing cv · c0(e0), which selects the appropriate con- structor from the class vector, cv, and applies it to the given instance data. The class c0 may well be the class c itself; this is one form of self-reference within ec d. Similarly, within e a method d0 is invoked on e0 by writing mv · d0(e0). The method d0 may well be the method d itself; this is another aspect of self-reference within ec d. To account for self-reference the method vector will be defined to have the self-referential type self([τ/t]τmv) in which the object type, τ, is, as before, the sum of the instance types of the classes, ∑c∈C τc. The method vector is defined by the following expression: self mv is hd ,→ λ (this:τ) case this {c · u ⇒ edm · c · d[τ](e0 cv)(e0 mv)(u)}c∈Cid∈D, where e0 cv , hc ,→ λ (u:τc) c · uic∈C and e0 mv , unroll(mv). Object creation is defined by the equation new[c](e), c · e and message send is defined by the equation e ⇐ d , unroll(emv)· d(e). To account for self-reference in the class-based organization, the class vector will be defined to have the type self([ρ/t]τcv) in which the object type, ρ, is, as before, the product of the result types of the methods, ∏d∈D ρd. The class vector is defined by the following expression: self cv is hc ,→ λ (u:τc) hd ,→ edm · c · d[ρ](e00 cv)(e00 mv)(u)id∈Dic∈C, where e00 cv , unroll(cv) VERSION 1.32 REVISED 05.15.2012 25.5 Notes 249 and e00 mv , hd ,→ λ (this:ρ) this · did∈D. Object creation is defined by the equation new[c](e), unroll(ecv)· c(e) and message send is defined by the equation e ⇐ d , e · d. The symmetries between the two organizations are striking. They are a reflection of the fundamental symmetries between sum and product types. 25.5 Notes The term “object-oriented” means many things to many people, but cer- tainly dynamic dispatch, the association of “methods” with “classes,” is one of its central concepts. According to the present development these characteristic features emerge as instances of the more general concepts of sum-, product-, and function types, which are useful, alone and in combi- nation, in a wide variety of circumstances. A bias towards either a class- or method-based organization of functions defined on sums seems misplaced in view of the inherent symmetries of the situation. Either formulation may be readily combined with recursive types and self-reference as described in Chapter 16 to account for methods that return objects as results. The literature on object-oriented programming, of which dynamic dis- patch is one aspect, is extensive. Abadi and Cardelli(1996) and Pierce (2002) provide a thorough account of the foundations of the subject. REVISED 05.15.2012 VERSION 1.32 250 25.5 Notes VERSION 1.32 REVISED 05.15.2012 Chapter 26 Inheritance In this chapter we build on Chapter 25 and consider the process of defining the dispatch matrix that determines the behavior of each method on each class. A common strategy is to build the dispatch matrix incrementally by adding new classes or methods to an existing dispatch matrix. To add a class requires that we define the behavior of each method on objects of that class, and to define a method requires that we define the behavior of that method on objects of each of the classes. The definition of these behaviors may be given by any means available in the language. However, it is of- ten suggested that a useful means of defining a new class is to inherit the behavior of another class on some methods, and to override its behavior on others, resulting in an amalgam of the old and new behaviors. The new class is often called a subclass of the old class, which is then called the super- class. Similarly, a new method may be defined by inheriting the behavior of another method on some classes, and overriding the behavior on others. By analogy we may call the new method a submethod of a given supermethod. (It is also possible to admit multiple superclasses or multiple supermethods, but we will confine our attention to single, rather than multiple, inheritance.) For simplicity we restrict attention to the simple, non-self-referential case in the following development. 26.1 Class and Method Extension We begin by considering the extension of a given dispatch matrix, edm, of type ∏ c∈C ∏ d∈D (τc → ρd) 252 26.1 Class and Method Extension with a new class, c∗ /∈ C, and a new method, d∗ /∈ D, to obtain a new dispatch matrix, e∗ dm, of type ∏ c∈C∗ ∏ d∈D∗ (τc → ρd), where C∗ = C ∪ { c∗ } and D∗ = D ∪ { d∗ }. To add a new class, c∗, to the dispatch matrix, we must specify the fol- lowing information:1 1. The instance type τc∗ of the new class, c∗. 2. The behavior, ec∗ d , of each method d ∈ D on an object of the new class c∗, a function of type τc∗ → ρd. This determines a new dispatch matrix, e∗ dm, such that the following condi- tions are satisfied: 1. For each c ∈ C and d ∈ D, the behavior e∗ dm · c · d is the same as the behavior edm · c · d. 2. For each d ∈ D, the behavior e∗ dm · c∗ · d is given by ec∗ d . To define c∗ as a subclass of some class c ∈ C means to define the behavior ec∗ d to be ec d for some (perhaps many) d ∈ D. It is sensible to inherit a method d in this manner only if the subtype relationship τc → ρd <: τc∗ → ρd is valid, which will be the case if τc∗ <: τc. This ensures that the inherited behavior may be invoked on the instance data of the new class. Similarly, to add a new method, d∗, to the dispatch matrix, we must specify the following information: 1. The result type, ρd∗ , of the new method, d∗. 2. The behavior, ec d∗ , of the new method, d∗, on an object of each class c ∈ C, a function of type τc → ρd∗ . This determines a new dispatch matrix, e∗ dm, such that the following condi- tions are satisfied: 1. For each c ∈ C and d ∈ D, the behavior e∗ dm · c · d is the same as edm · c · d. 1The extension with a new method will be considered separately for the sake of clarity. VERSION 1.32 REVISED 05.15.2012 26.2 Class-Based Inheritance 253 2. The behavior e∗ dm · c · d∗ is given by ec d∗ . To define d∗ as a submethod of some d ∈ D means to define the behavior ec d∗ to be ec d for some (perhaps many) classes c ∈ C. This is only sensible if the subtype relationship τc → ρd <: τc → ρd∗ holds, which is the case if ρd <: ρd∗ . This ensures that the result of the old behavior is sufficient for the new behavior. We will now consider how inheritance relates to the method- and class- based organizations of dynamic dispatch considered in Chapter 25. 26.2 Class-Based Inheritance Recall that the class-based organization given in Chapter 25 consists of a class vector, ecv, of type τcv , ∏ c∈C (τc → ρ), where the object type, ρ, is the finite product type ∏d∈D ρd. The class vector consists of a tuple of constructors that specialize the methods to a given object of each class. Let us consider the effect of adding a new class, c∗, as described in Sec- tion 26.1. The new class vector, e∗ cv, has type τ∗ cv , ∏ c∈C∗ (τc → ρ). There is an isomorphism, written ()†, between τ∗ cv and the type τcv × (τc∗ → ρ), which may be used to define the new class vector, e∗ cv, as follows: hecv, λ (u:τc∗ ) hd ,→ ec∗ d (u)id∈Di†. This definition makes clear that the old class vector, ecv, is reused intact in the new class vector, which is just an extension of the old class vector with a new constructor. REVISED 05.15.2012 VERSION 1.32 254 26.3 Method-Based Inheritance Although the object type, ρ, is the same both before and after the exten- sion with the new class, the behavior of an object of class c∗ may differ arbi- trarily from that of any other object, even that of the superclass from which it inherits its behavior. So, knowing that c∗ inherits from c tells us nothing about the behavior of its objects, but only about the means by which the class is defined. In short inheritance carries no semantic significance, but is only a record of the history of how a class is defined. Now let us consider the effect of adding a new method, d∗, as described in Section 26.1. The new class vector, e∗ cv, has type τ∗ cv , ∏ c∈C (τc → ρ∗), where ρ∗ is the product type ∏d∈D∗ ρd. There is an isomorphism, written ()‡, between ρ∗ and the type ρ × ρd∗ , where ρ is the old object type. Using this the new class vector, e∗ cv, may be defined by hc ,→ λ (u:τc) hhd ,→ ((ecv · c)(u)) · did∈D, ec d∗ (u)i‡ic∈C. Observe that each constructor must be re-defined to account for the new method, but the definition makes use of the old class vector for the defini- tions of the old methods. By this construction the new object type, ρ∗, is a subtype of the old object type, ρ. This means that any objects with the new method may be used in situations expecting an object without the new method, as might be expected. To avoid the redefinition of old classes when a new method is introduced, we may restrict inheritance so that new methods are only added to new subclasses. Subclasses may then have more methods than superclasses, and objects of the subclass may be provided when an object of the superclass is required. 26.3 Method-Based Inheritance The situation with the method-based organization is dual to that of the class-based organization. Recall that the method-based organization given in Chapter 25 consists of a method vector, emv, of type τmv , ∏ d∈D τ → ρd, where the instance type, τ, is the sum type ∑c∈C τc. The method vector consists of a tuple of functions that dispatch on the class of the object to determine their behavior. VERSION 1.32 REVISED 05.15.2012 26.4 Notes 255 Let us consider the effect of adding a new method, d∗, as described in Section 26.1. The new method vector, e∗ mv, has type τ∗ mv , ∏ d∈D∗ τ → ρd. There is an isomorphism, written ()‡, between τ∗ mv and the type τmv × (τ → ρd∗ ). Using this isomorphism, the new method vector, e∗ mv, may be defined as hemv, λ (this:τ) case this {c · u ⇒ ec d∗ (u)}c∈Ci‡. The old method vector is re-used intact, extended with an additional dis- patch function for the new method. The object type does not change under the extension with a new method, but because ρ∗ <: ρ, there is no difficulty using a new object in a context expecting an old object—the additional method is ignored. Finally, let us consider the effect of adding a new class, c∗, as described in Section 26.1. The new method vector, e∗ mv, has the type τ∗ mv , ∏ d∈D τ∗ → ρd, where τ∗ is the new object type ∑c∈C∗ τc, which is a supertype of the old object type τ. There is an isomorphism, written ()†, between τ∗ and the sum type τ + τc∗ , which we may use to define the new method vector, e∗ mv, as follows: hd ,→ λ (this:τ∗) case this†{l · u ⇒ (emv · d)(u) | r · u ⇒ ec∗ d (u)}id∈D. Every method must be redefined to account for the new class, but the old method vector is reused in this definition. 26.4 Notes Advocates of object-oriented programming differ on the importance of in- heritance. Philosophers tend to stress inheritance, but practitioners com- monly avoid it, or reduce it to vestigial form. The most common restrictions amount to a reformulation of some of the modularity mechanisms that are discussed in Chapters 45 and 46. REVISED 05.15.2012 VERSION 1.32 256 26.4 Notes Abadi and Cardelli(1996) and Pierce(2002) provide thorough accounts of the interaction of inheritance and subtyping. Liskov and Wing(1994) discuss it from a behavioral perspective. They propose the methodological requirement that subclasses respect the behavior of the superclass when- ever inheritance is used. VERSION 1.32 REVISED 05.15.2012 Part X Exceptions and Continuations Chapter 27 Control Stacks The technique of structural dynamics is very useful for theoretical pur- poses, such as proving type safety, but is too high level to be directly usable in an implementation. One reason is that the use of “search rules” requires the traversal and reconstruction of an expression in order to simplify one small part of it. In an implementation we would prefer to use some mech- anism to record “where we are” in the expression so that we may resume from that point after a simplification. This can be achieved by introduc- ing an explicit mechanism, called a control stack, that keeps track of the context of an instruction step for just this purpose. By making the control stack explicit, the transition rules avoid the need for any premises—every rule is an axiom. This is the formal expression of the informal idea that no traversals or reconstructions are required to implement it. In this chapter we introduce an abstract machine, K{nat*}, for the language L{nat *}. The purpose of this machine is to make control flow explicit by introducing a control stack that maintains a record of the pending sub-computations of a computation. We then prove the equivalence of K{nat*} with the struc- tural dynamics of L{nat *}. 27.1 Machine Definition A state, s, of K{nat*} consists of a control stack, k, and a closed expression, e. States may take one of two forms: 1. An evaluation state of the form k . e corresponds to the evaluation of a closed expression, e, relative to a control stack, k. 260 27.1 Machine Definition 2.A return state of the form k / e, where e val, corresponds to the evalu- ation of a stack, k, relative to a closed value, e. As an aid to memory, note that the separator “points to” the focal entity of the state, the expression in an evaluation state and the stack in a return state. The control stack represents the context of evaluation. It records the “current location” of evaluation, the context into which the value of the current expression is to be returned. Formally, a control stack is a list of frames: e stack (27.1a) f frame k stack k;f stack (27.1b) The definition of frame depends on the language we are evaluating. The frames of K{nat*} are inductively defined by the following rules: s(−) frame (27.2a) ifz(−; e1; x.e2) frame (27.2b) ap(−; e2) frame (27.2c) The frames correspond to search rules in the dynamics of L{nat *}. Thus, instead of relying on the structure of the transition derivation to maintain a record of pending computations, we make an explicit record of them in the form of a frame on the control stack. The transition judgment between states of the K{nat*} machine is inductively defined by a set of inference rules. We begin with the rules for natural numbers. k . z 7→ k / z (27.3a) k . s(e) 7→ k;s(−). e (27.3b) k;s(−)/ e 7→ k / s(e)(27.3c) To evaluate z we simply return it. To evaluate s(e), we push a frame on the stack to record the pending successor, and evaluate e; when that returns with e0, we return s(e0) to the stack. Next, we consider the rules for case analysis. k . ifz(e; e1; x.e2) 7→ k;ifz(−; e1; x.e2). e (27.4a) k;ifz(−; e1; x.e2)/ z 7→ k . e1 (27.4b) VERSION 1.32 REVISED 05.15.2012 27.2 Safety 261 k;ifz(−; e1; x.e2)/ s(e) 7→ k .[e/x]e2 (27.4c) First, the test expression is evaluated, recording the pending case analysis on the stack. Once the value of the test expression has been determined, we branch to the appropriate arm of the conditional, substituting the pre- decessor in the case of a positive number. Finally, we consider the rules for functions and recursion. k . lam[τ](x.e) 7→ k / lam[τ](x.e)(27.5a) k . ap(e1; e2) 7→ k;ap(−; e2). e1 (27.5b) k;ap(−; e2)/ lam[τ](x.e) 7→ k .[e2/x]e (27.5c) k . fix[τ](x.e) 7→ k .[fix[τ](x.e)/x]e (27.5d) These rules ensure that the function is evaluated before the argument, ap- plying the function when both have been evaluated. Note that evaluation of general recursion requires no stack space! (But see Chapter 37 for more on evaluation of general recursion.) The initial and final states of the K{nat*} are defined by the following rules: e . e initial (27.6a) e val e / e final (27.6b) 27.2 Safety To define and prove safety for K{nat*} requires that we introduce a new typing judgment, k : τ, which states that the stack k expects a value of type τ. This judgment is inductively defined by the following rules: e : τ (27.7a) k : τ0 f : τ ⇒ τ0 k;f : τ (27.7b) This definition makes use of an auxiliary judgment, f : τ ⇒ τ0, stating that a frame f transforms a value of type τ to a value of type τ0. s(−): nat ⇒ nat (27.8a) e1 : τ x : nat ` e2 : τ ifz(−; e1; x.e2): nat ⇒ τ (27.8b) REVISED 05.15.2012 VERSION 1.32 262 27.3 Correctness of the Control Machine e2 : τ2 ap(−; e2): arr(τ2; τ) ⇒ τ (27.8c) The states of K{nat*} are well-formed if their stack and expression components match: k : τ e : τ k . e ok (27.9a) k : τ e : τ e val k / e ok (27.9b) We leave the proof of safety of K{nat*} as an exercise. Theorem 27.1 (Safety). 1. If s ok and s 7→ s0, then s0 ok. 2. If s ok, then either s final or there exists s0 such that s 7→ s0. 27.3 Correctness of the Control Machine If we evaluate a given expression, e, using K{nat*}, do we get the same result as would be given by L{nat *}, and vice versa? Answering this question decomposes into two conditions relating K{nat*} to L{nat *}: Completeness If e 7→∗ e0, where e0 val, then e . e 7→∗ e / e0. Soundness If e . e 7→∗ e / e0, then e 7→∗ e0 with e0 val. Let us consider, in turn, what is involved in the proof of each part. For completeness a plausible first step is to consider a proof by induc- tion on the definition of multistep transition, which reduces the theorem to the following two lemmas: 1. If e val, then e . e 7→∗ e / e. 2. If e 7→ e0, then, for every v val, if e . e0 7→∗ e / v, then e . e 7→∗ e / v. The first can be proved easily by induction on the structure of e. The second requires an inductive analysis of the derivation of e 7→ e0, giving rise to two complications that must be accounted for in the proof. The first complica- tion is that we cannot restrict attention to the empty stack, for if e is, say, ap(e1; e2), then the first step of the machine is e . ap(e1; e2) 7→ e;ap(−; e2). e1, and so we must consider evaluation of e1 on a non-empty stack. VERSION 1.32 REVISED 05.15.2012 27.3 Correctness of the Control Machine 263 A generalization is to prove that if e 7→ e0 and k . e0 7→∗ k / v, then k . e 7→∗ k / v. Consider again the case e = ap(e1; e2), e0 = ap(e0 1; e2), with e1 7→ e0 1. We are given that k . ap(e0 1; e2) 7→∗ k / v, and we are to show that k . ap(e1; e2) 7→∗ k / v. It is easy to show that the first step of the former derivation is k . ap(e0 1; e2) 7→ k;ap(−; e2). e0 1. We would like to apply induction to the derivation of e1 7→ e0 1, but to do so we must have a v1 such that e0 1 7→∗ v1, which is not immediately at hand. This means that we must consider the ultimate value of each sub-expression of an expression in order to complete the proof. This information is pro- vided by the evaluation dynamics described in Chapter7, which has the property that e ⇓ e0 iff e 7→∗ e0 and e0 val. Lemma 27.2. If e ⇓ v, then for every k stack, k . e 7→∗ k / v. The desired result follows by the analogue of Theorem 7.2 for L{nat *}, which states that e ⇓ v iff e 7→∗ v. For the proof of soundness, it is awkward to reason inductively about the multistep transition from e . e 7→∗ e / v, because the intervening steps may involve alternations of evaluation and return states. Instead we regard each K{nat*} machine state as encoding an expression, and show that K{nat*} transitions are simulated by L{nat *} transitions under this encoding. Specifically, we define a judgment, s # e, stating that state s “unravels to” expression e. It will turn out that for initial states, s = e . e, and final states, s = e / e, we have s # e. Then we show that if s 7→∗ s0, where s0 final, s # e, and s0 # e0, then e0 val and e 7→∗ e0. For this it is enough to show the following two facts: 1. If s # e and s final, then e val. 2. If s 7→ s0, s # e, s0 # e0, and e0 7→∗ v, where v val, then e 7→∗ v. The first is quite simple, we need only observe that the unravelling of a final state is a value. For the second, it is enough to show the following lemma. Lemma 27.3. If s 7→ s0, s # e, and s0 # e0, then e 7→∗ e0. Corollary 27.4. e 7→∗ n iff e . e 7→∗ e / n. The remainder of this section is devoted to the proofs of the soundness and completeness lemmas. REVISED 05.15.2012 VERSION 1.32 264 27.3 Correctness of the Control Machine 27.3.1 Completeness Proof of Lemma 27.2. The proof is by induction on an evaluation dynamics for L{nat *}. Consider the evaluation rule e1 ⇓ lam[τ2](x.e)[e2/x]e ⇓ v ap(e1; e2) ⇓ v (27.10) For an arbitrary control stack, k, we are to show that k . ap(e1; e2) 7→∗ k / v. Applying both of the inductive hypotheses in succession, interleaved with steps of the abstract machine, we obtain k . ap(e1; e2) 7→ k;ap(−; e2). e1 7→∗ k;ap(−; e2)/ lam[τ2](x.e) 7→ k .[e2/x]e 7→∗ k / v. The other cases of the proof are handled similarly. 27.3.2 Soundness The judgment s # e0, where s is either k . e or k / e, is defined in terms of the auxiliary judgment k ./ e = e0 by the following rules: k ./ e = e0 k . e # e0 (27.11a) k ./ e = e0 k / e # e0 (27.11b) In words, to unravel a state we wrap the stack around the expression. The latter relation is inductively defined by the following rules: e ./ e = e (27.12a) k ./ s(e) = e0 k;s(−)./ e = e0 (27.12b) k ./ ifz(e1; e2; x.e3) = e0 k;ifz(−; e2; x.e3)./ e1 = e0 (27.12c) k ./ ap(e1; e2) = e k;ap(−; e2)./ e1 = e (27.12d) These judgments both define total functions. VERSION 1.32 REVISED 05.15.2012 27.3 Correctness of the Control Machine 265 Lemma 27.5. The judgment s # e has mode (∀, ∃!), and the judgment k ./ e = e0 has mode (∀, ∀, ∃!). That is, each state unravels to a unique expression, and the result of wrapping a stack around an expression is uniquely determined. We are therefore justified in writing k ./ e for the unique e0 such that k ./ e = e0. The following lemma is crucial. It states that unravelling preserves the transition relation. Lemma 27.6. If e 7→ e0, k ./ e = d, k ./ e0 = d0, then d 7→ d0. Proof. The proof is by rule induction on the transition e 7→ e0. The inductive cases, in which the transition rule has a premise, follow easily by induction. The base cases, in which the transition is an axiom, are proved by an induc- tive analysis of the stack, k. For an example of an inductive case, suppose that e = ap(e1; e2), e0 = ap(e0 1; e2), and e1 7→ e0 1. We have k ./ e = d and k ./ e0 = d0. It follows from Rules (27.12) that k;ap(−; e2)./ e1 = d and k;ap(−; e2)./ e0 1 = d0. So by induction d 7→ d0, as desired. For an example of a base case, suppose that e = ap(lam[τ2](x.e); e2) and e0 = [e2/x]e with e 7→ e0 directly. Assume that k ./ e = d and k ./ e0 = d0; we are to show that d 7→ d0. We proceed by an inner induction on the structure of k. If k = e, the result follows immediately. Consider, say, the stack k = k0;ap(−; c2). It follows from Rules (27.12) that k0 ./ ap(e; c2) = d and k0 ./ ap(e0; c2) = d0. But by the structural dynamics ap(e; c2) 7→ ap(e0; c2), so by the inner inductive hypothesis we have d 7→ d0, as desired. We are now in a position to complete the proof of Lemma 27.3. Proof of Lemma 27.3. The proof is by case analysis on the transitions of K{nat*}. In each case, after unravelling, the transition will correspond to zero or one transitions of L{nat *}. Suppose that s = k . s(e) and s0 = k;s(−). e. Note that k ./ s(e) = e0 iff k;s(−)./ e = e0, from which the result follows immediately. Suppose that s = k;ap(lam[τ](x.e1); −)/ e2 and s0 = k .[e2/x]e1. Let e0 be such that k;ap(lam[τ](x.e1); −)./ e2 = e0 and let e00 be such that k ./[e2/x]e1 = e00. Observe that k ./ ap(lam[τ](x.e1); e2) = e0. The result follows from Lemma 27.6. REVISED 05.15.2012 VERSION 1.32 266 27.4 Notes 27.4 Notes The abstract machine considered here is typical of a wide class of machines that make control flow explicit in the state. The prototype is the SECD ma- chine (Landin, 1965), which may be seen as a linearization of a structural operational semantics (Plotkin, 1981). An advantage of a machine model is that the explicit treatment of control is required for languages that allow the control state to be explicitly manipulated (see Chapter 29 for a prime ex- ample). A disadvantage is that we are required to make explicit the control state of the computation, rather than leave it implicit as in structural opera- tional semantics. Which is better depends wholly on the situation at hand, though historically there has been greater emphasis on abstract machines than on structural semantics. VERSION 1.32 REVISED 05.15.2012 Chapter 28 Exceptions Exceptions effect a non-local transfer of control from the point at which the exception is raised to an enclosing handler for that exception. This transfer interrupts the normal flow of control in a program in response to unusual conditions. For example, exceptions can be used to signal an error con- dition, or to indicate the need for special handling in certain circumstances that arise only rarely. To be sure, we could use conditionals to check for and process errors or unusual conditions, but using exceptions is often more convenient, particularly because the transfer to the handler is direct and immediate, rather than indirect via a series of explicit checks. 28.1 Failures A failure is a control mechanism that permits a computation to refuse to return a value to the point of its evaluation. A failure is detected by catching it, which means to divert evaluation to a handler that turns the failure into a success (unless the handler itself fails). The following grammar defines the syntax of failures: Exp e ::= fail fail failure catch(e1; e2) catch e1 ow e2 handler The expression fail aborts the current evaluation, and the expression catch(e1; e2) handles any failure in e1 by evaluating e2 instead. The statics of failures is straightforward: Γ ` fail : τ (28.1a) 268 28.1 Failures Γ ` e1 : τ Γ ` e2 : τ Γ ` catch(e1; e2): τ (28.1b) A failure can have any type, because it never returns. The two expressions in a catch expression must have the same type, because either might de- termine the value of that expression. The dynamics of failures may be given using stack unwinding. Evalua- tion of a catch installs a handler on the control stack. Evaluation of a fail unwinds the control stack by popping frames until it reaches the nearest enclosing handler, to which control is passed. The handler is evaluated in the context of the surrounding control stack, so that failures within it prop- agate further up the stack. Stack unwinding can be defined directly using structural dynamics, but we prefer to make use of the stack machine defined in Chapter 27. In ad- dition to states of the form k . e, which evaluates the expression e on the stack k, and k / e, which passes the value e to the stack k, we make use of an additional form of state, k J, which passes a failure up the stack to the nearest enclosing handler. The set of frames defined in Chapter 27 is extended with the additonal form catch(−; e2). The transition rules given in Chapter 27 are extended with the following additional rules: k . fail 7→ k J (28.2a) k . catch(e1; e2) 7→ k;catch(−; e2). e1 (28.2b) k;catch(−; e2)/ v 7→ k / v (28.2c) k;catch(−; e2)J 7→ k . e2 (28.2d) k;f J 7→ k J(28.2e) As a notational convenience, we require that Rule (28.2e) apply only if none of the preceding rules apply. Evaluating fail propagates a failure up the stack. The act of raising an exception may raise an exception. Evaluat- ing catch(e1; e2) consists of pushing the handler onto the control stack and evaluating e1. If a value is propagated to the handler, the handler is re- moved and the value continues to propagate upwards. If a failure is prop- agated to the handler, the stored expression is evaluated with the handler removed from the control stack. All other frames propagate failures. VERSION 1.32 REVISED 05.15.2012 28.2 Exceptions 269 The definition of initial state remains the same as for K{nat*}, but we change the definition of final state to include these two forms: e val e / e final (28.3a) e J final (28.3b) The first of these is as before, corresponding to a normal result with the specified value. The second is new, corresponding to an uncaught excep- tion propagating through the entire program. It is easy to extend the definition of stack typing given in Chapter 27 to account for the new forms of frame, and then to prove safety in the usual way. However, the meaning of progress must be weakened to take account of failure: a well-typed expression is either a value, or may take a step, or may signal failure. Theorem 28.1 (Safety). 1. If s ok and s 7→ s0, then s0 ok. 2. If s ok, then either s final or there exists s0 such that s 7→ s0. 28.2 Exceptions Failures are simplistic in that they do not distinguish different causes, and hence do not permit handlers to react differently to different circumstances. An exception is a generalization of a failure that associates a value with the failure. This value is passed to the handler, allowing it to discriminate be- tween various forms of failures, and to pass data appropriate to that form of failure. The type of values associated with exceptions is discussed in Section 28.3. For now, we simply assume that there is some type, τexn, of values associated with a failure. The syntax of exceptions is given by the following grammar: Exp e ::= raise[τ](e) raise(e) exception handle(e1; x.e2) handle e1 ow x ⇒ e2 handler The argument to raise is evaluated to determine the value passed to the handler. The expression handle(e1; x.e2) binds a variable, x, in the han- dler, e2, to which the associated value of the exception is bound, should an exception be raised during the execution of e1. REVISED 05.15.2012 VERSION 1.32 270 28.2 Exceptions The statics of exceptions extends the statics of failures to account for the type of the value carried with the exception: Γ ` e : τexn Γ ` raise[τ](e): τ (28.4a) Γ ` e1 : τ Γ, x : τexn ` e2 : τ Γ ` handle(e1; x.e2): τ (28.4b) The dynamics of exceptions is a mild generalization of the dynamics of failures in which we generalize the failure state, k J, to the exception state, k J e, which passes a value of type τexn along with the failure. The syntax of stack frames is extended to include raise[τ](−) and handle(−; x.e2). The dynamics of exceptions is specified by the following rules: k . raise[τ](e) 7→ k;raise[τ](−). e (28.5a) k;raise[τ](−)/ e 7→ k J e (28.5b) k;raise[τ](−)J e 7→ k J e (28.5c) k . handle(e1; x.e2) 7→ k;handle(−; x.e2). e1 (28.5d) k;handle(−; x.e2)/ e 7→ k / e (28.5e) k;handle(−; x.e2)J e 7→ k .[e/x]e2 (28.5f) ( f 6= handle(−; x.e2)) k;f J e 7→ k J e (28.5g) It is a straightforward exercise to extend the safety theorem given in Section 28.1 to exceptions. VERSION 1.32 REVISED 05.15.2012 28.3 Exception Type 271 28.3 Exception Type The statics of exceptions is parameterized by a type, τexn, of exception val- ues. There is no restriction on the choice of this type, but it must be one and the same for all exceptions in a program. For otherwise an exception han- dler cannot analyze the value associated with the exception without risking type safety. But how is τexn to be chosen? A very na¨ıve choice would be to take it to be the type, str, of strings. This allows us to associate an “explanation” with an exception. For example, we may write raise "Division by zero error." to signal the obvious arithmetic fault. This is fine as far as it goes, but a handler for such an exception would have to interpret the string if it is to distinguish one exception from another, and this is clearly impractical and inconvenient. Another popular choice is to take τexn to be nat, so that exceptional con- ditions are encoded as error numbers that describe the source of the error.1 By dispatching on the numeric code of the exception the handler can de- termine how to recover from it. But the trouble is that we must establish a globally agreed-upon system of numbering, which is clearly untenable and incompatible with modular decomposition and component reuse. More- over, it is practically impossible to associate meaningful data with an ex- ceptional condition, information that might well be useful to a handler. The latter concern—how to associate data specific to the exceptional condition—can be addressed by taking τexn to be a sum type whose classes are the exceptional conditions. The instance type of the class determines the data associated with the exception. For example, the type τexn might be chosen to be a sum type of the form [div ,→ unit, fnf ,→ string,...]. The class div might represent an arithmetic fault, with no associated data, and the class fnf might represent a “file not found” error, with associated data being the name of the file. Using a sum type for τexn makes it easy for the handler to discriminate on the source of the failure, and to recover the associated data without fear of a type safety violation. For example, we might write 1This convention is used in the Unix operating system, for example. REVISED 05.15.2012 VERSION 1.32 272 28.4 Encapsulation of Exceptions try e1 ow x ⇒ match x { div hi ⇒ ediv | fnf s ⇒ efnf } to handle the exceptions specified by the sum type given in the preceding paragraph. The problem with choosing a sum type for τexn is that it imposes a static classification of the sources of failure in a program. There must be one, glob- ally agreed-upon type that classifies all possible forms of failure, and speci- fies their associated data. Using sums in this manner impedes modular de- velopment and evolution, because all of the modules comprising a system must agree on the one, central type of exception values. A better approach is to use dynamic classification for exception values by choosing τexn to be an extensible sum, one to which new classes may be added at execution time. This allows separate program modules to introduce their own failure clas- sification scheme without worrying about interference with one another; the initialization of the module generates new classes at run-time that are guaranteed to be distinct from all other classes previously or subsequently generated. (See Chapter 34 for more on dynamic classification.) 28.4 Encapsulation of Exceptions It is sometimes useful to distinguish expressions that can fail or raise an exception from those that cannot. An expression is called fallible, or ex- ceptional, if it can fail or raise an exception during its evaluation, and is infallible, or unexceptional, otherwise. The concept of fallibility is intention- ally permissive in that an infallible expression may be considered to be (vacuously) fallible, whereas infallibility is intended to be strict in that an infallible expression cannot fail. Consequently, if e1 and e2 are two infal- lible expressions both of whose values are required in a computation, we may evaluate them in either order without affecting the outcome. If, on the other hand, one or both are fallible, then the outcome of the compu- tation is sensitive to the evaluation order (whichever fails first determines the overall result). To formalize this distinction we distinguish two modes of expression, the fallible and the infallible, linked by a modality classifying the fallible VERSION 1.32 REVISED 05.15.2012 28.4 Encapsulation of Exceptions 273 expressions of a type. Type τ ::= fallible(τ) τ fallible fallible Fall f ::= fail fail failure ok(e) ok(e) success try(e; x.f1; f2) let fall(x) be e in f1 ow f2 handler Infall e ::= x x variable fall(f) fall(f) fallible try(e; x.e1; e2) let fall(x) be e in e1 ow e2 handler The type fallible(τ) is the type of encapsulated fallible expressions of type τ. Fallible expressions include failures, successes (infallible expres- sions thought of as vacuously fallible), and handlers that intercept failures, but which may itself fail. Infallible expressions include variables, encapsu- lated fallible expressions, and handlers that intercept failures, always yield- ing an infallible result. The statics of encapsulated failures consists of two judgment forms, Γ ` e : τ for infallible expressions and Γ ` f ∼ τ for fallible expressions. These judgments are defined by the following rules: Γ, x : τ ` x : τ (28.6a) Γ ` f ∼ τ Γ ` fall(f): fallible(τ)(28.6b) Γ ` e : fallible(τ)Γ, x : τ ` e1 : τ0 Γ ` e2 : τ0 Γ ` try(e; x.e1; e2): τ0 (28.6c) Γ ` fail ∼ τ (28.6d) Γ ` e : τ Γ ` ok(e) ∼ τ (28.6e) Γ ` e : fallible(τ)Γ, x : τ ` f1 ∼ τ0 Γ ` f2 ∼ τ0 Γ ` try(e; x.f1; f2) ∼ τ0 (28.6f) Rule (28.6c) specifies that a handler may be used to turn a fallible expres- sion (encapsulated by e) into an infallible computation, provided that the result is infallible regardless of whether the encapsulated expression suc- ceeds or fails. REVISED 05.15.2012 VERSION 1.32 274 28.4 Encapsulation of Exceptions The dynamics of encapsulated failures is readily derived, though some care must be taken with the elimination form for the modality. fall(f) val (28.7a) k . try(e; x.e1; e2) 7→ k;try(−; x.e1; e2). e (28.7b) k;try(−; x.e1; e2)/ fall(f) 7→ k;try(−; x.e1; e2);fall(−). f (28.7c) k . fail 7→ k J (28.7d) k . ok(e) 7→ k;ok(−). e (28.7e) k;ok(−)/ e 7→ k / ok(e)(28.7f) e val k;try(−; x.e1; e2);fall(−)/ ok(e) 7→ k .[e/x]e1 (28.7g) k;try(−; x.e1; e2);fall(−)J 7→ k . e2 (28.7h) We have omitted the rules for the fallible form of handler; they are sim- ilar to Rules (28.7b) to (28.7c) and (28.7g) to (28.7h), albeit with infallible subexpressions e1 and e2 replaced by fallible subexpressions f1 and f2. An initial state has the form k . e, where e is an infallible expression, and k is a stack of suitable type. Consequently, a fallible expression, f, can only be evaluated on a stack of the form k;try(−; x.e1; e2);fall(−) in which a handler for any failure that may arise from f is present. There- fore, a final state has the form e / e, where e val; no uncaught failure can arise. VERSION 1.32 REVISED 05.15.2012 28.5 Notes 275 28.5 Notes Various forms of exceptions were explored in various dialects of Lisp (for example, Steele(1990)). The original formulation of ML (Gordon et al., 1979) as a metalanguage for mechanized logic made extensive use of ex- ceptions, called “failures,” to implement tactics and tacticals. These days most languages include an exception mechanism of the kind considered here. The essential distinction between the exception mechanism and excep- tion values is often misunderstood. Exception values are often dynamically classified (in the sense of Chapter 34), but dynamic classification has many more uses than just exception values. Another common misconception is to link exceptions erroneously with fluid binding (Chapter 33). REVISED 05.15.2012 VERSION 1.32 276 28.5 Notes VERSION 1.32 REVISED 05.15.2012 Chapter 29 Continuations The semantics of many control constructs (such as exceptions and co-routines) can be expressed in terms of reified control stacks, a representation of a con- trol stack as an ordinary value. This is achieved by allowing a stack to be passed as a value within a program and to be restored at a later point, even if control has long since returned past the point of reification. Rei- fied control stacks of this kind are called continuations; they are values that can be passed and returned at will in a computation. Continuations never “expire”, and it is always sensible to reinstate a continuation without com- promising safety. Thus continuations support unlimited “time travel” — we can go back to a previous point in the computation and then return to some point in its future, at will. Why are continuations useful? Fundamentally, they are representations of the control state of a computation at a given point in time. Using con- tinuations we can “checkpoint” the control state of a program, save it in a data structure, and return to it later. In fact this is precisely what is neces- sary to implement threads (concurrently executing programs) — the thread scheduler must be able to checkpoint a program and save it for later exe- cution, perhaps after a pending event occurs or another thread yields the processor. 29.1 Informal Overview We will extend L{→} with the type cont(τ) of continuations accepting values of type τ. The introduction form for cont(τ) is letcc[τ](x.e), which binds the current continuation (that is, the current control stack) to the variable x, and evaluates the expression e. The corresponding elimination 278 29.1 Informal Overview form is throw[τ](e1; e2), which restores the value of e1 to the control stack that is the value of e2. To illustrate the use of these primitives, consider the problem of mul- tiplying the first n elements of an infinite sequence q of natural numbers, where q is represented by a function of type nat → nat. If zero occurs among the first n elements, we would like to effect an “early return” with the value zero, rather than perform the remaining multiplications. This problem can be solved using exceptions (we leave this as an exercise), but we will give a solution that uses continuations in preparation for what fol- lows. Here is the solution in L{nat *}, without short-cutting: fix ms is λ q : nat * nat. λ n : nat. case n { z ⇒ s(z) | s(n’) ⇒ (q z) × (ms (q ◦ succ) n’) } The recursive call composes q with the successor function to shift the se- quence by one step. Here is the version with short-cutting: λ q : nat * nat. λ n : nat. letcc ret : nat cont in let ms be fix ms is λ q : nat * nat. λ n : nat. case n { z ⇒ s(z) | s(n’) ⇒ case q z { z ⇒ throw z to ret | s(n’’) ⇒ (q z) × (ms (q ◦ succ) n’) } } in ms q n VERSION 1.32 REVISED 05.15.2012 29.2 Semantics of Continuations 279 The letcc binds the return point of the function to the variable ret for use within the main loop of the computation. If zero is encountered, control is thrown to ret, effecting an early return with the value zero. Let’s look at another example: given a continuation k of type τ cont and a function f of type τ0 → τ, return a continuation k0 of type τ0 cont with the following behavior: throwing a value v0 of type τ0 to k0 throws the value f (v0) to k. This is called composition of a function with a continuation. We wish to fill in the following template: fun compose(f:τ0 → τ,k:τ cont):τ0 cont = .... The first problem is to obtain the continuation we wish to return. The second problem is how to return it. The continuation we seek is the one in effect at the point of the ellipsis in the expression throw f(...) to k. This is the continuation that, when given a value v0, applies f to it, and throws the result to k. We can seize this continuation using letcc, writing throw f(letcc x:τ0 cont in ...) to k At the point of the ellipsis the variable x is bound to the continuation we wish to return. How can we return it? By using the same trick as we used for short-circuiting evaluation above! We don’t want to actually throw a value to this continuation (yet), instead we wish to abort it and return it as the result. Here’s the final code: fun compose (f:τ0 → τ, k:τ cont):τ0 cont = letcc ret:τ0 cont cont in throw (f (letcc r in throw r to ret)) to k The type of ret is that of a continuation-expecting continuation. 29.2 Semantics of Continuations We extend the language of L{→} expressions with these additional forms: Type τ ::= cont(τ) τ cont continuation Expr e ::= letcc[τ](x.e) letcc x in e mark throw[τ](e1; e2) throw e1 to e2 goto cont(k) cont(k) continuation The expression cont(k) is a reified control stack, which arises during eval- uation. REVISED 05.15.2012 VERSION 1.32 280 29.2 Semantics of Continuations The statics of this extension is defined by the following rules: Γ, x : cont(τ) ` e : τ Γ ` letcc[τ](x.e): τ (29.1a) Γ ` e1 : τ1 Γ ` e2 : cont(τ1) Γ ` throw[τ0](e1; e2): τ0 (29.1b) The result type of a throw expression is arbitrary because it does not return to the point of the call. The statics of continuation values is given by the following rule: k : τ Γ ` cont(k): cont(τ)(29.2) A continuation value cont(k) has type cont(τ) exactly if it is a stack ac- cepting values of type τ. To define the dynamics we extend K{nat*} stacks with two new forms of frame: throw[τ](−; e2) frame (29.3a) e1 val throw[τ](e1; −) frame (29.3b) Every reified control stack is a value: k stack cont(k) val (29.4) The transition rules for the continuation constructs are as follows: k . letcc[τ](x.e) 7→ k .[cont(k)/x]e (29.5a) k . throw[τ](e1; e2) 7→ k;throw[τ](−; e2). e1 (29.5b) e1 val k;throw[τ](−; e2)/ e1 7→ k;throw[τ](e1; −). e2 (29.5c) k;throw[τ](v; −)/ cont(k0) 7→ k0 / v (29.5d) Evaluation of a letcc expression duplicates the control stack; evaluation of a throw expression destroys the current control stack. The safety of this extension of L{→} may be established by a simple extension to the safety proof for K{nat*} given in Chapter 27. VERSION 1.32 REVISED 05.15.2012 29.3 Coroutines 281 We need only add typing rules for the two new forms of frame, which are as follows: e2 : cont(τ) throw[τ0](−; e2): τ ⇒ τ0 (29.6a) e1 : τ e1 val throw[τ0](e1; −): cont(τ) ⇒ τ0 (29.6b) The rest of the definitions remain as in Chapter 27. Lemma 29.1 (Canonical Forms). If e : cont(τ) and e val, then e = cont(k) for some k such that k : τ. Theorem 29.2 (Safety). 1. If s ok and s 7→ s0, then s0 ok. 2. If s ok, then either s final or there exists s0 such that s 7→ s0. 29.3 Coroutines The distinction between a routine and a subroutine is the distinction be- tween a manager and a worker. The routine calls upon the subroutine to accomplish a piece of work, and the subroutine returns to the routine when its work is done. The relationship is asymmetric in that there is a clear dis- tinction between the caller, the main routine, and the callee, the subroutine. Often it is useful to consider a symmetric situation in which two routines each call the other to help accomplish a task. Such a pair of routines are called coroutines; their relationship to one another is symmetric rather than hierarchical. The key to implementing a subroutine is for the caller to pass to the callee a continuation representing the return point of the subroutine call. When the subroutine is finished, it calls the continuation passed to it by the calling routine. Because the subroutine is finished at that point, there is no need for the callee to pass a continuation back to the caller. The key to implementing coroutines is to have each routine treat the other as a subrou- tine of itself. In particular, whenever a coroutine cedes control to its caller, it provides a continuation that the caller may use to cede control back to the callee, in the process providing a continuation for itself. (This raises an interesting question of how the whole process gets started. We’ll return to this shortly.) To see how a pair of coroutines is implemented, let us consider the type of each routine in the pair. A routine is a continuation accepting two ar- guments, a datum to be passed to that routine when it is resumed, and REVISED 05.15.2012 VERSION 1.32 282 29.3 Coroutines a continuation to be resumed when the routine has finished its task. The datum represents the state of the computation, and the continuation is a coroutine that accepts arguments of the same form. Thus, the type of a coroutine must satisfy the type isomorphism τ coro ∼= (τ × τ coro) cont. So we may take τ coro to be the recursive type τ coro , µt.(τ × t) cont. Up to isomorphism, the type τ coro is the type of continuations that accept a value of type τ, representing the state of the coroutine, and the partner coroutine, a value of the same type. A coroutine, r, passes control to another coroutine, r0, by evaluating the expression resume(hs, r0i), where s is the current state of the computation. Doing so creates a new coroutine whose entry point is the return point (calling site) of the application of resume. Therefore the type of resume is τ × τ coro → τ × τ coro. The definition of resume is as follows: λ (hs, r0i:τ × τ coro) letcc k in throw hs, fold(k)i to unfold(r0) When applied, resume seizes the current continuation, and passes the state, s, and the seized continuation (packaged as a coroutine) to the called corou- tine. But how do we create a system of coroutines in the first place? Because the state is explicitly passed from one routine to the other, a coroutine may be defined as a state transformation function that, when activated with the current state, determines the next state of the computation. A system of coroutines is created by establishing a joint exit point to which the result of the system is thrown, and creating a pair of coroutines that iteratively transform the state and pass control to the partner routine. If either rou- tine wishes to terminate the computation, it does so by throwing a result value to their common exit point. Thus, a coroutine may be specified by a function of type (ρ,τ) rout , ρ cont → τ → τ, where ρ is the result type and τ is the state type of the system of coroutines. To set up a system of coroutines we define a function run that, given two routines, creates a function of type τ → ρ that, when applied to the VERSION 1.32 REVISED 05.15.2012 29.3 Coroutines 283 initial state, computes a result of type ρ. The computation consists of a cooperating pair of routines that share a common exit point. The definition of run begins as follows: λ (hr1, r2i) λ (s0) letcc x0 in let r0 1 be r1(x0) in let r0 2 be r2(x0) in ... Given two routines, run establishes their common exit point, and passes this continuation to both routines. By throwing to this continuation either routine may terminate the computation with a result of type ρ. The body of the run function continues as follows: rep(r0 2)(letcc k in rep(r0 1)(hs0, fold(k)i)) The auxiliary function rep creates an infinite loop that transforms the state and passes control to the other routine: λ (t) fix l is λ (hs, ri) l(resume(ht(s), ri)). The system is initialized by starting routine r1 with the initial state, and arranging that, when it cedes control to its partner, it starts routine r2 with the resulting state. At that point the system is bootstrapped: each routine will resume the other on each iteration of the loop. A good example of coroutining arises whenever we wish to interleave input and output in a computation. We may achieve this using a coroutine between a producer routine and a consumer routine. The producer emits the next element of the input, if any, and passes control to the consumer with that element removed from the input. The consumer processes the next data item, and returns control to the producer, with the result of processing attached to the output. The input and output are modeled as lists of type τi list and τo list, respectively, which are passed back and forth between the routines.1 The routines exchange messages according to the following protocol. The message OK(hi, oi) is sent from the consumer to producer to acknowledge receipt of the previous message, and to pass back the cur- rent state of the input and output channels. The message EMIT(hv, hi, oii), where v is a value of type τi opt, is sent from the producer to the consumer to emit the next value (if any) from the input, and to pass the current state of the input and output channels to the consumer. This leads to the following implementation of the producer/consumer model. The type τ of the state maintained by the routines is the labeled 1In practice the input and output state are implicit, but we prefer to make them explicit for the sake of clarity. REVISED 05.15.2012 VERSION 1.32 284 29.3 Coroutines sum type [OK ,→ τi list × τo list, EMIT ,→ τi opt × (τi list × τo list)]. This type specifies the message protocol between the producer and the con- sumer described in the preceding paragraph. The producer, P, is defined by the expression λ (x0) λ (msg) case msg {b1 | b2 | b3}, where the first branch, b1, is OK · hnil, osi ⇒ EMIT · hnull, hnil, osii and the second branch, b2, is OK · hcons(i; is), osi ⇒ EMIT · hjust(i), his, osii, and the third branch, b3, is EMIT · ⇒ error. In words, if the input is exhausted, the producer emits the value null, along with the current channel state. Otherwise, it emits just(i), where i is the first remaining input, and removes that element from the passed channel state. The producer cannot see an EMIT message, and signals an error if it should occur. The consumer, C, is defined by the expression λ (x0) λ (msg) case msg {b0 1 | b0 2 | b0 3}, where the first branch, b0 1, is EMIT · hnull, h , osii ⇒ throw os to x0, the second branch, b0 2, is EMIT · hjust(i), his, osii ⇒ OK · his, cons(f(i); os)i, and the third branch, b0 3, is OK · ⇒ error. The consumer dispatches on the emitted datum. If it is absent, the output channel state is passed to x0 as the ultimate value of the computation. If VERSION 1.32 REVISED 05.15.2012 29.4 Notes 285 it is present, the function f (unspecified here) of type τi → τo is applied to transform the input to the output, and the result is added to the output channel. If the message OK is received, the consumer signals an error, as the producer never produces such a message. The initial state, s0, has the form OK · his, osi, where is and os are the initial input and output channel state, respectively. The computation is created by the expression run(hP,Ci)(s0), which sets up the coroutines as described earlier. Although it is relatively easy to visualize and implement coroutines in- volving only two partners, it is more complex, and less useful, to consider a similar pattern of control among n ≥ 2 participants. In such cases it is more common to structure the interaction as a collection of n routines, each of which is a coroutine of a central scheduler. When a routine resumes its partner, it passes control to the scheduler, which determines which routine to execute next, again as a coroutine of itself. When structured as corou- tines of a scheduler, the individual routines are called threads. A thread yields control by resuming its partner, the scheduler, which then determines which thread to execute next as a coroutine of itself. This pattern of control is called cooperative multi-threading, because it is based on explicit yields, rather than implicit yields imposed by asynchronous events such as timer interrupts. 29.4 Notes Continuations are a ubiquitous notion in programming languages. Reynolds (1993) provides an excellent account of the multiple discoveries of continu- ations. The formulation given here is inspired by Felleisen and Hieb(1992), who pioneered the development of linguistic theories of control and state. REVISED 05.15.2012 VERSION 1.32 286 29.4 Notes VERSION 1.32 REVISED 05.15.2012 Part XI Types and Propositions Chapter 30 Constructive Logic Constructive logic codifies the principles of mathematical reasoning as it is actually practiced. In mathematics a proposition may be judged to be true exactly when it has a proof, and may be judged to be false exactly when it has a refutation. Because there are, and always will be, unsolved problems, we cannot expect in general that a proposition is either true or false, for in most cases we have neither a proof nor a refutation of it. Constructive logic may be described as logic as if people matter, as distinct from classical logic, which may be described as the logic of the mind of god. From a constructive viewpoint the judgment “φ true” means that “there is a proof of φ.” What constitutes a proof is a social construct, an agreement among peo- ple as to what is a valid argument. The rules of logic codify a set of prin- ciples of reasoning that may be used in a valid proof. The valid forms of proof are determined by the outermost structure of the proposition whose truth is asserted. For example, a proof of a conjunction consists of a proof of each of its conjuncts, and a proof of an implication consists of a trans- formation of a proof of its antecedent to a proof of its consequent. When spelled out in full, the forms of proof are seen to correspond exactly to the forms of expression of a programming language. To each proposition is associated the type of its proofs; a proof is an expression of the associated type. This association between programs and proofs induces a dynamics on proofs, for they are but programs of some type. In this way proofs in constructive logic have computational content, which is to say that they may be interpreted as executable programs of the associated type. Conversely, programs have mathematical content as proofs of the proposition associated to their type. This unification of logic and programming is called the propositions as 290 30.1 Constructive Semantics types principle. It is the central organizing principle of the theory of pro- gramming languages. Propositions are identified with types, and proofs are identified with programs. A programming technique corresponds to a method of proof; a proof technique corresponds to a method of program- ming. Viewing types as behavioral specifications of programs, propositions may be seen as problem statements whose proofs are solutions that imple- ment the specification. 30.1 Constructive Semantics Constructive logic is concerned with two judgments, φ prop, stating that φ expresses a proposition, and φ true, stating that φ is a true proposi- tion. What distinguishes constructive from non-constructive logic is that a proposition is not conceived of as merely a truth value, but instead as a problem statement whose solution, if it has one, is given by a proof. A propo- sition is said to be true exactly when it has a proof, in keeping with ordinary mathematical practice. In practice there is no other criterion of truth than the existence of a proof. This principle has important, possibly surprising, consequences, the most important of which is that we cannot say, in general, that a propo- sition is either true or false. If for a proposition to be true means to have a proof of it, what does it mean for a proposition to be false? It means that we have a refutation of it, showing that it cannot be proved. That is, a proposition is false if we can show that the assumption that it is true (has a proof) contradicts known facts. In this sense constructive logic is a logic of positive, or affirmative, information — we must have explicit evidence in the form of a proof in order to affirm the truth or falsity of a proposition. In light of this it should be clear that not every proposition is either true or false. For if φ expresses an unsolved problem, such as the famous P?= NP problem, then we have neither a proof nor a refutation of it (the mere absence of a proof not being a refutation). Such a problem is undecided, precisely because it is unsolved. Because there will always be unsolved problems (there being infinitely many propositions, but only finitely many proofs at a given point in the evolution of our knowledge), we cannot say that every proposition is decidable, that is, either true or false. Of course, some propositions are decidable, and hence may be consid- ered to be either true or false. For example, if φ expresses an inequality be- tween natural numbers, then φ is decidable, because we can always work out, for given natural numbers m and n, whether m ≤ n or m 6≤ n — we can VERSION 1.32 REVISED 05.15.2012 30.2 Constructive Logic 291 either prove or refute the given inequality. This argument does not extend to the real numbers. To get an idea of why not, consider the presentation of a real number by its decimal expansion. At any finite time we will have ex- plored only a finite initial segment of the expansion, which is not enough to determine if it is, say, less than 1. For if we have determined the expansion to be 0.99 . . . 9, we cannot decide at any time, short of infinity, whether or not the number is 1. (This argument is not a proof, because we may won- der whether there is some other representation of real numbers that admits such a decision to be made finitely, but it turns out that this is not the case.) The constructive attitude is simply to accept the situation as inevitable, and make our peace with that. When faced with a problem we have no choice but to roll up our sleeves and try to prove it or refute it. There is no guarantee of success! Life is hard, but we muddle through somehow. 30.2 Constructive Logic The judgments φ prop and φ true of constructive logic are rarely of interest by themselves, but rather in the context of a hypothetical judgment of the form φ1 true,..., φn true ` φ true. This judgment expresses that the proposition φ is true (has a proof), under the assumptions that each of φ1,..., φn are also true (have proofs). Of course, when n = 0 this is just the same as the judgment φ true. The structural properties of the hypothetical judgment, when special- ized to constructive logic, define what we mean by reasoning under hy- potheses: Γ, φ true ` φ true (30.1a) Γ ` φ1 true Γ, φ1 true ` φ2 true Γ ` φ2 true (30.1b) Γ ` φ2 true Γ, φ1 true ` φ2 true (30.1c) Γ, φ1 true, φ1 true ` φ2 true Γ, φ1 true ` φ2 true (30.1d) Γ1, φ2 true, φ1 true,Γ2 ` φ true Γ1, φ1 true, φ2 true,Γ2 ` φ true (30.1e) REVISED 05.15.2012 VERSION 1.32 292 30.2 Constructive Logic The last two rules are implicit in that we regard Γ as a set of hypotheses, so that two “copies” are as good as one, and the order of hypotheses does not matter. 30.2.1 Provability The syntax of propositional logic is given by the following grammar: Prop φ ::= > > truth ⊥ ⊥ falsity ∧(φ1; φ2) φ1 ∧ φ2 conjunction ∨(φ1; φ2) φ1 ∨ φ2 disjunction ⊃(φ1; φ2) φ1 ⊃ φ2 implication The connectives of propositional logic are given meaning by rules that de- termine (a) what constitutes a “direct” proof of a proposition formed from a given connective, and (b) how to exploit the existence of such a proof in an “indirect” proof of another proposition. These are called the introduc- tion and elimination rules for the connective. The principle of conservation of proof states that these rules are inverse to one another — the elimination rule cannot extract more information (in the form of a proof) than was put into it by the introduction rule, and the introduction rules can be used to re- construct a proof from the information extracted from it by the elimination rules. Truth Our first proposition is trivially true. No information goes into proving it, and so no information can be obtained from it. Γ ` > true (30.2a) (no elimination rule) (30.2b) Conjunction Conjunction expresses the truth of both of its conjuncts. Γ ` φ1 true Γ ` φ2 true Γ ` φ1 ∧ φ2 true (30.3a) Γ ` φ1 ∧ φ2 true Γ ` φ1 true (30.3b) Γ ` φ1 ∧ φ2 true Γ ` φ2 true (30.3c) VERSION 1.32 REVISED 05.15.2012 30.2 Constructive Logic 293 Implication Implication states the truth of a proposition under an as- sumption. Γ, φ1 true ` φ2 true Γ ` φ1 ⊃ φ2 true (30.4a) Γ ` φ1 ⊃ φ2 true Γ ` φ1 true Γ ` φ2 true (30.4b) Falsehood Falsehood expresses the trivially false (refutable) proposition. (no introduction rule) (30.5a) Γ ` ⊥ true Γ ` φ true (30.5b) Disjunction Disjunction expresses the truth of either (or both) of two propositions. Γ ` φ1 true Γ ` φ1 ∨ φ2 true (30.6a) Γ ` φ2 true Γ ` φ1 ∨ φ2 true (30.6b) Γ ` φ1 ∨ φ2 true Γ, φ1 true ` φ true Γ, φ2 true ` φ true Γ ` φ true (30.6c) Negation The negation, ¬φ, of a proposition, φ, may be defined as the implication φ ⊃⊥. This means that ¬φ true if φ true ` ⊥ true, which is to say that the truth of φ is refutable in that we may derive a proof of falsehood from any purported proof of φ. Because constructive truth is identified with the existence of a proof, the implied semantics of negation is rather strong. In particular, a problem, φ, is open exactly when we can neither affirm nor refute it. This is in contrast to the classical conception of truth, which assigns a fixed truth value to each proposition, so that every proposition is either true or false. 30.2.2 Proof Terms The key to the propositions-as-types principle is to make explict the forms of proof. The basic judgment φ true, which states that φ has a proof, is re- placed by the judgment p : φ, stating that p is a proof of φ. (Sometimes p is REVISED 05.15.2012 VERSION 1.32 294 30.2 Constructive Logic called a “proof term”, but we will simply call p a “proof.”) The hypotheti- cal judgment is modified correspondingly, with variables standing for the presumed, but unknown, proofs: x1 : φ1,..., xn : φn ` p : φ. We again let Γ range over such hypothesis lists, subject to the restriction that no variable occurs more than once. The syntax of proof terms is given by the following grammar: Prf p ::= >I hi truth intro ∧I(p1; p2) hp1, p2i conj. intro ∧E[l](p) p · l conj. elim ∧E[r](p) p · r conj. elim ⊃I[φ](x.p) λ (x:φ) p impl. intro ⊥E(p) abort(p) false elim ∨I[l](p) l · p disj. intro ∨I[r](p) r · p disj. intro ∨E(p; x1.p1; x2.p2) case p {l · x1 ⇒ p1 | r · x2 ⇒ p2} disj. elim The concrete syntax of proof terms is chosen to stress the correspondence between propositions and types discussed in Section 30.4 below. The rules of constructive propositional logic may be restated using proof terms as follows. Γ ` hi : > (30.7a) Γ ` p1 : φ1 Γ ` p2 : φ2 Γ ` hp1, p2i : φ1 ∧ φ2 (30.7b) Γ ` p1 : φ1 ∧ φ2 Γ ` p1 · l : φ1 (30.7c) Γ ` p1 : φ1 ∧ φ2 Γ ` p1 · r : φ2 (30.7d) Γ, x : φ1 ` p2 : φ2 Γ ` λ (x:φ1) p2 : φ1 ⊃ φ2 (30.7e) Γ ` p : φ1 ⊃ φ2 Γ ` p1 : φ1 Γ ` p(p1): φ2 (30.7f) Γ ` p : ⊥ Γ ` abort(p): φ (30.7g) VERSION 1.32 REVISED 05.15.2012 30.3 Proof Dynamics 295 Γ ` p1 : φ1 Γ ` l · p1 : φ1 ∨ φ2 (30.7h) Γ ` p2 : φ2 Γ ` r · p2 : φ1 ∨ φ2 (30.7i) Γ ` p : φ1 ∨ φ2 Γ, x1 : φ1 ` p1 : φ Γ, x2 : φ2 ` p2 : φ Γ ` case p {l · x1 ⇒ p1 | r · x2 ⇒ p2}: φ (30.7j) 30.3 Proof Dynamics Proof terms in constructive logic are equipped with a dynamics by Gentzen’s Principle, which states that the eliminatory forms are to be thought of as inverse to the introductory forms. One aspect of Gentzen’s Principle is the principle of conservation of proof, which states that the information in- troduced into a proof of a proposition may be extracted without loss by elimination. So, for example, we may state that conjunction elimination is post-inverse to conjunction introduction by the definitional equivalences Γ ` p1 : φ1 Γ ` p2 : φ2 Γ ` hp1, p2i · l ≡ p1 : φ1 (30.8a) Γ ` p1 : φ1 Γ ` p2 : φ2 Γ ` hp1, p2i · r ≡ p2 : φ2 (30.8b) Another aspect of Gentzen’s Principle is that principle of reversability of proof, which states that every proof may be reconstructed from the informa- tion that may be extracted from it by elimination. In the case of conjunction this may be stated by the definitional equivalence Γ ` p1 : φ1 Γ ` p2 : φ2 Γ ` hp · l, p · ri ≡ p : φ1 ∧ φ2 (30.9) Similar equivalences may be stated for the other connectives. For exam- ple, the conservation and reversability principles for implication are given by these rules: Γ, x : φ1 ` p2 : φ2 Γ ` p2 : φ2 Γ ` (λ (x:φ1) p2)(p1) ≡ [p1/x]p2 : φ2 (30.10a) Γ ` p : φ1 ⊃ φ2 Γ ` λ (x:φ1)(p(x)) ≡ p : φ1 ⊃ φ2 (30.10b) REVISED 05.15.2012 VERSION 1.32 296 30.4 Propositions as Types The corresponding rules for disjunction and falsehood are given as follows: Γ ` p : φ1 ∨ φ2 Γ, x1 : φ1 ` p1 : ψ Γ, x2 : φ2 ` p2 : ψ Γ ` case l · p {l · x1 ⇒ p1 | r · x2 ⇒ p2} ≡ [p/x1]p1 : ψ (30.11a) Γ ` p : φ1 ∨ φ2 Γ, x1 : φ1 ` p1 : ψ Γ, x2 : φ2 ` p2 : ψ Γ ` case r · p {l · x1 ⇒ p1 | r · x2 ⇒ p2} ≡ [p/x2]p2 : ψ (30.11b) Γ ` p : φ1 ∨ φ2 Γ, x : φ1 ∨ φ2 ` q : ψ Γ ` [p/x]q ≡ case p {l · x1 ⇒ [l · x1/x]q | r · x2 ⇒ [r · x2/x]q}: ψ (30.11c) Γ ` p : ⊥ Γ, x : ⊥ ` q : ψ Γ ` [p/x]q ≡ abort(p): ψ (30.11d) 30.4 Propositions as Types Reviewing the statics and dynamics of proofs in constructive logic reveals a striking similarity to the statics and dynamics of expressions of various types. For example, the introduction rule for conjunction specifies that a proof of a conjunction consists of a pair of proofs, one for each conjunct, and the elimination rule inverts this, allowing us to extract a proof of each conjunct from any proof of a conjunction. There is an obvious analogy with the static semantics of product types, whose introductory form is a pair and whose eliminatory forms are projections. Gentzen’s Principle extends the analogy to the dynamics as well, so that, for example, the elimination forms for conjunction amount to projections that extract the appropriate components from an ordered pair. The correspondence between propositions and types and between proofs and programs, is summarized by the following chart: Prop Type > unit ⊥ void φ1 ∧ φ2 τ1 × τ2 φ1 ⊃ φ2 τ1 → τ2 φ1 ∨ φ2 τ1 + τ2 The correspondence between propositions and types is a cornerstone of the theory of programming languages. It exposes a deep connection between computation and deduction, and serves as a framework for the analysis of language constructs and reasoning principles by relating them to one another. VERSION 1.32 REVISED 05.15.2012 30.5 Notes 297 30.5 Notes The propositions as types principle has its origins in the semantics of in- tuitionistic logic developed by Brouwer, according to which the truth of a proposition is witnessed by a construction providing computable evi- dence for it. The forms of evidence are determined by the form of the proposition, so that, for example, evidence for an implication is a com- putable function transforming evidence for the hypothesis into evidence for the conclusion. An explicit formulation of this semantics was intro- duced by Heyting, and further developed by a number of authors, includ- ing de Bruijn, Curry, Gentzen, Girard, Howard, Kolmogorov, Martin-L¨of, and Tait. The propositions-as-types correspondence is sometimes called the Curry-Howard Isomorphism, but this terminology neglects the crucial contri- butions of the others just mentioned. Moreover, the correspondence is not, in general, an isomorphism; rather, it is an expression of Brouwer’s Dictum that the concept of proof is best explained by the more general concept of construction (program). REVISED 05.15.2012 VERSION 1.32 298 30.5 Notes VERSION 1.32 REVISED 05.15.2012 Chapter 31 Classical Logic In constructive logic a proposition is true exactly when it has a proof, a derivation of it from axioms and assumptions, and is false exactly when it has a refutation, a derivation of a contradiction from the assumption that it is true. Constructive logic is a logic of positive evidence. To affirm or deny a proposition requires a proof, either of the proposition itself, or of a con- tradiction, under the assumption that it has a proof. We are not always in a position to affirm or deny a proposition. An open problem is one for which we have neither a proof nor a refutation—so that, constructively speaking, it is neither true nor false. In contrast classical logic (the one we learned in school) is a logic of perfect information in which every proposition is either true or false. We may say that classical logic corresponds to “god’s view” of the world— there are no open problems, rather all propositions are either true or false. Put another way, to assert that every proposition is either true or false is to weaken the notion of truth to encompass all that is not false, dually to the constructively (and classically) valid interpretation of falsity as all that is not true. The symmetry between truth and falsity is appealing, but there is a price to pay for this: the meanings of the logical connectives are weaker in the classical case than in the constructive. A prime example is provided by the law of the excluded middle, the as- sertion that φ ∨ ¬φ true is valid for all propositions φ. Constructively, this principle is not universally valid, because it would mean that every propo- sition either has a proof or a refutation, which is manifestly not the case. Classically, however, the law of the excluded middle is valid, because ev- ery proposition is either considered to be either false or not false (which is identified with being true in classical logic, in contrast to constructive 300 31.1 Classical Logic logic). Nevertheless, classical logic is consistent with constructive logic in that constructive logic does not refute classical logic. As we have seen, con- structive logic proves that the law of the excluded middle is positively not refuted (its double negation is constructively true). This shows that con- structive logic is stronger (more expressive) than classical logic, because it can express more distinctions (namely, between affirmation and irrefutabil- ity), and because it is consistent with classical logic (the law of the excluded middle can be added without fear of contradiction). Proofs in constructive logic have computational content: they can be ex- ecuted as programs, and their behavior is constrained by their type. Proofs in classical logic also have computational content, but in a weaker sense than in classical logic. Rather than positively affirm a proposition, a proof in classical logic is a computation that cannot be refuted. Computationally, a refutation consists of a continuation, or control stack, that takes a proof of a proposition and derives a contradiction from it. So a proof of a proposi- tion in classical logic is a computation that, when given a refutation of that proposition derives a contradiction, witnessing the impossibility of refut- ing it. In this sense the law of the excluded middle has a proof, precisely because it is irrefutable. 31.1 Classical Logic In constructive logic a connective is defined by giving its introduction and elimination rules. In classical logic a connective is defined by giving its truth and falsity conditions. Its truth rules correspond to introduction, and its falsity rules to elimination. The symmetry between truth and falsity is expressed by the principle of indirect proof. To show that φ true it is enough to show that φ false entails a contradiction, and, conversely, to show that φ false it is enough to show that φ true leads to a contradiction. Although the second of these is constructively valid, the first is fundamentally classi- cal, expressing the principle of indirect proof. 31.1.1 Provability and Refutability There are three basic judgment forms in classical logic: 1. φ true, stating that the proposition φ is provable; 2. φ false, stating that the proposition φ is refutable; 3. #, stating that a contradiction has been derived. VERSION 1.32 REVISED 05.15.2012 31.1 Classical Logic 301 These are extended to hypothetical judgments in which we admit both provability and refutability assumptions: φ1 false,..., φm false ψ1 true,..., ψn true ` J. The hypotheses are divided into two zones, one for falsity assumptions, ∆, and one for truth assumptions, Γ. The rules of classical logic are organized around the symmetry between truth and falsity, which is mediated by the contradiction judgment. The hypothetical judgment is reflexive: ∆, φ false Γ ` φ false (31.1a) ∆ Γ, φ true ` φ true (31.1b) The remaining rules are stated so that the structural properties of weaken- ing, contraction, and transitivity are admissible. A contradiction arises when a proposition is judged to be both true and false. A proposition is true if its falsity is absurd, and is false if its truth is absurd. ∆ Γ ` φ false ∆ Γ ` φ true ∆ Γ ` # (31.1c) ∆, φ false Γ ` # ∆ Γ ` φ true (31.1d) ∆ Γ, φ true ` # ∆ Γ ` φ false (31.1e) Truth is trivially true, and cannot be refuted. ∆ Γ ` > true (31.1f) A conjunction is true if both conjuncts are true, and is false if either conjunct is false. ∆ Γ ` φ1 true ∆ Γ ` φ2 true ∆ Γ ` φ1 ∧ φ2 true (31.1g) ∆ Γ ` φ1 false ∆ Γ ` φ1 ∧ φ2 false (31.1h) ∆ Γ ` φ2 false ∆ Γ ` φ1 ∧ φ2 false (31.1i) REVISED 05.15.2012 VERSION 1.32 302 31.1 Classical Logic Falsity is trivially false, and cannot be proved. ∆ Γ ` ⊥ false (31.1j) A disjunction is true if either disjunct is true, and is false if both dis- juncts are false. ∆ Γ ` φ1 true ∆ Γ ` φ1 ∨ φ2 true (31.1k) ∆ Γ ` φ2 true ∆ Γ ` φ1 ∨ φ2 true (31.1l) ∆ Γ ` φ1 false ∆ Γ ` φ2 false ∆ Γ ` φ1 ∨ φ2 false (31.1m) Negation inverts the sense of each judgment: ∆ Γ ` φ false ∆ Γ ` ¬φ true (31.1n) ∆ Γ ` φ true ∆ Γ ` ¬φ false (31.1o) An implication is true if its conclusion is true whenever the assumption is true, and is false if its conclusion is false yet its assumption is true. ∆ Γ, φ1 true ` φ2 true ∆ Γ ` φ1 ⊃ φ2 true (31.1p) ∆ Γ ` φ1 true ∆ Γ ` φ2 false ∆ Γ ` φ1 ⊃ φ2 false (31.1q) 31.1.2 Proofs and Refutations To explain the dynamics of classical proofs we first introduce an explicit syntax for proofs and refutations. We will define three hypothetical judg- ments for classical logic with explicit derivations: 1. ∆ Γ ` p : φ, stating that p is a proof of φ; 2. ∆ Γ ` k ÷ φ, stating that k is a refutation of φ; 3. ∆ Γ ` k # p, stating that k and p are contradictory. VERSION 1.32 REVISED 05.15.2012 31.1 Classical Logic 303 The falsity assumptions, ∆, are represented by a context of the form u1 ÷ φ1,..., um ÷ φm, where m ≥ 0, in which the variables u1,..., un stand for refutations. The truth assumptions, Γ, are represented by a context of the form x1 : ψ1,..., xn : ψn, where n ≥ 0, in which the variables x1,..., xn stand for proofs. The syntax of proofs and refutations is given by the following grammar: Prf p ::= >T hi truth ∧T(p1; p2) hp1, p2i conjunction ∨T[l](p) l · p disjunction left ∨T[r](p) r · p disjunction right ¬T(k) not(k) negation ⊃T(φ; x.p) λ (x:φ) p implication Ref k ::= ⊥F abort falsehood ∧F[l](k) fst ; k conjunction left ∧F[r](k) snd ; k conjunction right ∨F(k1; k2) case(k1; k2) disjunction ¬F(p) not(p) negation ⊃F(p; k) ap(p); k implication Proofs serve as evidence for truth judgments, and refutations serve as evi- dence for false judgments. Contradictions are witnessed by the juxtaposi- tion of a proof and a refutation. A contradiction arises whenever a proposition is both true and false: ∆ Γ ` k ÷ φ ∆ Γ ` p : φ ∆ Γ ` k # p (31.2a) Truth and falsity are defined symmetrically in terms of contradiction: ∆, u ÷ φ Γ ` k # p ∆ Γ ` ccr(u ÷ φ.k # p): φ (31.2b) ∆ Γ, x : φ ` k # p ∆ Γ ` ccp(x : φ.k # p) ÷ φ (31.2c) Reflexivity corresponds to the use of a variable hypothesis: ∆, u ÷ φ Γ ` u ÷ φ (31.2d) REVISED 05.15.2012 VERSION 1.32 304 31.1 Classical Logic ∆ Γ, x : φ ` x : φ (31.2e) The other structure properties are admissible. Truth is trivially true, and cannot be refuted. ∆ Γ ` hi : > (31.2f) A conjunction is true if both conjuncts are true, and is false if either conjunct is false. ∆ Γ ` p1 : φ1 ∆ Γ ` p2 : φ2 ∆ Γ ` hp1, p2i : φ1 ∧ φ2 (31.2g) ∆ Γ ` k1 ÷ φ1 ∆ Γ ` fst ; k1 ÷ φ1 ∧ φ2 (31.2h) ∆ Γ ` k2 ÷ φ2 ∆ Γ ` snd ; k2 ÷ φ1 ∧ φ2 (31.2i) Falsity is trivially false, and cannot be proved. ∆ Γ ` abort ÷ ⊥ (31.2j) A disjunction is true if either disjunct is true, and is false if both dis- juncts are false. ∆ Γ ` p1 : φ1 ∆ Γ ` l · p1 : φ1 ∨ φ2 (31.2k) ∆ Γ ` p2 : φ2 ∆ Γ ` r · p2 : φ1 ∨ φ2 (31.2l) ∆ Γ ` k1 ÷ φ1 ∆ Γ ` k2 ÷ φ2 ∆ Γ ` case(k1; k2) ÷ φ1 ∨ φ2 (31.2m) Negation inverts the sense of each judgment: ∆ Γ ` k ÷ φ ∆ Γ ` not(k): ¬φ (31.2n) ∆ Γ ` p : φ ∆ Γ ` not(p) ÷ ¬φ (31.2o) An implication is true if its conclusion is true whenever the assumption is true, and is false if its conclusion is false, yet its assumption is true. ∆ Γ, x : φ1 ` p2 : φ2 ∆ Γ ` λ (x:φ1) p2 : φ1 ⊃ φ2 (31.2p) ∆ Γ ` p1 : φ1 ∆ Γ ` k2 ÷ φ2 ∆ Γ ` ap(p1); k2 ÷ φ1 ⊃ φ2 (31.2q) VERSION 1.32 REVISED 05.15.2012 31.2 Deriving Elimination Forms 305 31.2 Deriving Elimination Forms The price of achieving a symmetry between truth and falsity in classical logic is that we must very often rely on the principle of indirect proof: to show that a proposition is true, we often must derive a contradiction from the assumption of its falsity. For example, a proof of (φ ∧ (ψ ∧ θ)) ⊃ (θ ∧ φ) in classical logic has the form λ (w:φ ∧ (ψ ∧ θ)) ccr(u ÷ θ ∧ φ.k # w), where k is the refutation fst ; ccp(x : φ.snd ; ccp(y : ψ ∧ θ.snd ; ccp(z : θ.u # hz, xi)# y)# w). And yet in constructive logic this proposition has a direct proof that avoids the circumlocutions of proof by contradiction: λ (w:φ ∧ (ψ ∧ θ)) hw · r · r, w · li. But this proof cannot be expressed (as is) in classical logic, because classical logic lacks the elimination forms of constructive logic. However, we may package the use of indirect proof into a slightly more palatable form by deriving the elimination rules of constructive logic. For example, the rule ∆ Γ ` φ ∧ ψ true ∆ Γ ` φ true is derivable in classical logic: ∆, φ false Γ ` φ false ∆, φ false Γ ` φ ∧ ψ false ∆ Γ ` φ ∧ ψ true ∆, φ false Γ ` φ ∧ ψ true ∆, φ false Γ ` # ∆ Γ ` φ true The other elimination forms are derivable in a similar manner, in each case relying on indirect proof to construct a proof of the truth of a proposition from a derivation of a contradiction from the assumption of its falsity. REVISED 05.15.2012 VERSION 1.32 306 31.3 Proof Dynamics The derivations of the elimination forms of constructive logic are most easily exhibited using proof and refutation expressions, as follows: abort(p) = ccr(u ÷ φ.abort # p) p · l = ccr(u ÷ φ.fst ; u # p) p · r = ccr(u ÷ ψ.snd ; u # p) p1(p2) = ccr(u ÷ ψ.ap(p2); u # p1) case p1 {l · x ⇒ p2 | r · y ⇒ p} = ccr(u ÷ γ.case(ccp(x : φ.u # p2); ccp(y : ψ.u # p)) # p1) It is straightforward to check that the expected elimination rules hold. For example, the rule ∆ Γ ` p1 : φ ⊃ ψ ∆ Γ ` p2 : φ ∆ Γ ` p1(p2): ψ (31.3) is derivable using the definition of p1(p2) given above. By suppressing proof terms, we may derive the corresponding provability rule ∆ Γ ` φ ⊃ ψ true ∆ Γ ` φ true ∆ Γ ` ψ true .(31.4) 31.3 Proof Dynamics The dynamics of classical logic arises from the simplification of the con- tradiction between a proof and a refutation of a proposition. To make this explicit we will define a transition system whose states are contradictions k # p consisting of a proof, p, and a refutation, k, of the same proposition. The steps of the computation consist of simplifications of the contradictory state based on the form of p and k. The truth and falsity rules for the connectives play off one another in a pleasing manner: fst ; k # hp1, p2i 7→ k # p1 (31.5a) snd ; k # hp1, p2i 7→ k # p2 (31.5b) case(k1; k2)# l · p1 7→ k1 # p1 (31.5c) case(k1; k2)# r · p2 7→ k2 # p2 (31.5d) not(p)# not(k) 7→ k # p (31.5e) ap(p1); k # λ (x:φ) p2 7→ k #[p1/x]p2 (31.5f) VERSION 1.32 REVISED 05.15.2012 31.3 Proof Dynamics 307 The rules of indirect proof give rise to the following transitions: ccp(x : φ.k1 # p1)# p2 7→ [p2/x]k1 #[p2/x]p1 (31.5g) k1 # ccr(u ÷ φ.k2 # p2) 7→ [k1/u]k2 #[k1/u]p2 (31.5h) The first of these defines the behavior of the refutation of φ that proceeds by contradicting the assumption that φ is true. This refutation is activated by presenting it with a proof of φ, which is then substituted for the assump- tion in the new state. Thus, “ccp” stands for “call with current proof.” The second transition defines the behavior of the proof of φ that proceeds by contradicting the assumption that φ is false. This proof is activated by pre- senting it with a refutation of φ, which is then substituted for the assump- tion in the new state. Thus, “ccr” stands for “call with current refutation.” Rules (31.5g) to (31.5h) overlap in that there are two possible transitions for a state of the form ccp(x : φ.k1 # p1)# ccr(u ÷ φ.k2 # p2), one to the state [p/x]k1 #[p/x]p1, where p is ccr(u ÷ φ.k2 # p2), and one to the state [k/u]k2 #[k/u]p2, where k is ccp(x : φ.k1 # p1). The dynam- ics of classical logic is therefore non-deterministic. To avoid this one may impose a priority ordering among the two cases, preferring one transition over the other when there is a choice. Preferring the first corresponds to a “lazy” dynamics for proofs, because we pass the unevaluated proof, p, to the refutation on the left, which is thereby activated. Preferring the sec- ond corresponds to an “eager” dynamics for proofs, in which we pass the unevaluated refutation, k, to the proof, which is thereby activated. Theorem 31.1 (Preservation). If k ÷ φ, p : φ, and k # p 7→ k0 # p0, then there exists φ0 such that k0 ÷ φ0 and p0 : φ0. Proof. By rule induction on the dynamics of classical logic. Theorem 31.2 (Progress). If k ÷ φ and p : φ, then either k # p final or k # p 7→ k0 # p0. Proof. By rule induction on the statics of classical logic. To initiate computation we postulate that halt is a refutation of any proposition. The initial and final states of a computation are defined as follows: halt # p initial (31.6a) REVISED 05.15.2012 VERSION 1.32 308 31.4 Law of the Excluded Middle p canonical halt # p final (31.6b) The judgment p canonical states that p is a canonical proof, which is defined to be any proof other than an indirect proof. 31.4 Law of the Excluded Middle The law of the excluded middle is derivable in classical logic: φ ∨ ¬φ false, φ true ` φ true φ ∨ ¬φ false, φ true ` φ ∨ ¬φ true φ ∨ ¬φ false, φ true ` φ ∨ ¬φ false φ ∨ ¬φ false, φ true ` # φ ∨ ¬φ false ` φ false φ ∨ ¬φ false ` ¬φ true φ ∨ ¬φ false ` φ ∨ ¬φ true φ ∨ ¬φ false ` φ ∨ ¬φ false φ ∨ ¬φ false ` # φ ∨ ¬φ true When written out using explicit proofs and refutations, we obtain the proof term p0 : φ ∨ ¬φ: ccr(u ÷ φ ∨ ¬φ.u # r · not(ccp(x : φ.u # l · x))). To understand the computational meaning of this proof, let us juxtapose it with a refutation, k ÷ φ ∨ ¬φ, and simplify it using the dynamics given in Section 31.3. The first step is the transition k # ccr(u ÷ φ ∨ ¬φ.u # r · not(ccp(x : φ.u # l · x))) 7→ k # r · not(ccp(x : φ.k # l · x)), wherein we have replicated k so that it occurs in two places in the result state. By virtue of its type the refutation k must have the form case(k1; k2), where k1 ÷ φ and k2 ÷ ¬φ. Continuing the reduction, we obtain: case(k1; k2)# r · not(ccp(x : φ.case(k1; k2)# l · x)) 7→ k2 # not(ccp(x : φ.case(k1; k2)# l · x)). VERSION 1.32 REVISED 05.15.2012 31.4 Law of the Excluded Middle 309 By virtue of its type k2 must have the form not(p2), where p2 : φ, and hence the transition proceeds as follows: not(p2)# not(ccp(x : φ.case(k1; k2)# l · x)) 7→ ccp(x : φ.case(k1; k2)# l · x)# p2. Observe that p2 is a valid proof of φ. Proceeding, we obtain ccp(x : φ.case(k1; k2)# l · x)# p2 7→ case(k1; k2)# l · p2 7→ k1 # p2 The first of these two steps is the crux of the matter: the refutation, k = case(k1; k2), which was replicated at the outset of the derivation, is re- used, but with a different argument. At the first use, the refutation, k, which is provided by the context of use of the law of the excluded middle, is pre- sented with a proof r · p1 of φ ∨ ¬φ. That is, the proof behaves as though the right disjunct of the law is true, which is to say that φ is false. If the context is such that it inspects this proof, it can only be by providing the proof, p2, of φ that refutes the claim that φ is false. Should this occur, the proof of the law of the excluded middle “backtracks” the context, providing instead the proof l · p2 to k, which then passes p2 to k1 without further incident. The proof of the law of the excluded middle boldly asserts ¬φ true, regardless of the form of φ. Then, if caught in its lie by the context providing a proof of φ, it “changes its mind” and asserts φ to the original context, k, after all. No further reversion is possible, because the context has itself provided a proof, p2, of φ. The law of the excluded middle illustrates that classical proofs are to be thought of as interactions between proofs and refutations, which is to say interactions between a proof and the context in which it is used. In pro- gramming terms this corresponds to an abstract machine with an explicit control stack, or continuation, representing the context of evaluation of an expression. That expression may access the context (stack, continuation) to effect backtracking as necessary to maintain the perfect symmetry be- tween truth and falsity. The penalty is that a closed proof of a disjunction no longer need reveal which disjunct it proves, for as we have just seen, it may, on further inspection, “change its mind.” REVISED 05.15.2012 VERSION 1.32 310 31.5 The Double-Negation Translation 31.5 The Double-Negation Translation One consequence of the greater expressiveness of constructive logic is that classical proofs may be translated systematically into constructive proofs of a classically equivalent proposition. This means that by systematically re- organizing the classical proof we may, without changing its meaning from a classical perspective, turn it into a constructive proof of a constructively weaker proposition. This shows that there is no loss in adhering to con- structive proofs, because every classical proof can be seen as a constructive proof of a constructively weaker, but classically equivalent, proposition. Moreover, it proves that classical logic is weaker (less expressive) than con- structive logic, contrary to a na¨ıve interpretation which would say that the additional reasoning principles, such as the law of the excluded middle, af- forded by classical logic makes it stronger. In programming language terms adding a “feature” does not necessarily strengthen (improve the expressive power) of your language; on the contrary, it may weaken it. We will define a translation φ∗ of propositions that provides an inter- pretation of classical into constructive logic according to the following cor- respondences: Classical Constructive ∆ Γ ` φ true ¬∆∗ Γ∗ ` ¬¬φ∗ true truth ∆ Γ ` φ false ¬∆∗ Γ∗ ` ¬φ∗ true falsity ∆ Γ ` # ¬∆∗ Γ∗ ` ⊥ true contradiction Classical truth is weakened to constructive irrefutability; classical false- hood is represented as constructive refutability; classical contradiction is represented by constructive falsehood. Falsity assumptions are negated after translation to express their falsehood; truth assumptions are merely translated as is. Because the double negations are classically cancellable, the translation will be easily seen to yield a classically equivalent propo- sition. But because ¬¬φ is constructively weaker than φ, we also see that a proof in classical logic is translated to a constructive proof of a weaker statement. Many choices for the translation of propositions are available; we have chosen one that makes the proof of the correspondence between classical and constructive logic go smoothly: >∗ = > (φ1 ∧ φ2)∗ = φ∗ 1 ∧ φ∗ 2 ⊥∗ =⊥ VERSION 1.32 REVISED 05.15.2012 31.6 Notes 311 (φ1 ∨ φ2)∗ = φ∗ 1 ∨ φ∗ 2 (φ1 ⊃ φ2)∗ = φ∗ 1 ⊃ ¬¬φ∗ 2 (¬φ)∗ = ¬φ∗ It is straightforward to show by induction on the rules of classical logic that the correspondences summarized above hold. Some simple lemmas are required. For example, we must show that the entailment ¬¬φ true ¬¬ψ true ` ¬¬(φ ∧ ψ) true is derivable in constructive logic. 31.6 Notes The computational interpretation of classical logic was first explored by Griffin(1990) and Murthy(1991). The account given here was influenced by Wadler(2003), transposed by Nanevski from sequent calculus to nat- ural deduction using multiple forms of judgment. The terminology is in- spired by Lakatos(1976), an insightful and inspiring analysis of the discov- ery of proofs and refutations of conjectures in mathematics. Versions of the double-negation translation were originally given by G¨odel and Gentzen, and have been extended and modified in numerous other studies. The computational content of the double negation translation was first eluci- dated by Murthy(1991), who established the connection with the continuation- passing transformation used in compilers. REVISED 05.15.2012 VERSION 1.32 312 31.6 Notes VERSION 1.32 REVISED 05.15.2012 Part XII Symbols Chapter 32 Symbols A symbol is an atomic datum with no internal structure. Whereas a variable is given meaning by substitution, a symbol is given meaning by a family of operations indexed by symbols. A symbol is therefore just a name, or index, for an instance of a family of operations. Many different interpre- tations may be given to symbols according to the operations we choose to consider, giving rise to concepts such as fluid binding, dynamic classifi- cation, mutable storage, and communication channels. To each symbol is associated a type whose interpretation depends on the particular applica- tion. The type of a symbol influences the type of its associated operations under each interpretation. For example, in the case of mutable storage, the type of a symbol constrains the contents of the cell named by that symbol to values of that type. It is important to bear in mind that a symbol is not a value of its associated type, but only a constraint on how that symbol may be interpreted by the operations associated with it. In this chapter we consider two constructs for computing with symbols. The first is a means of declaring new symbols for use within a specified scope. The expresssion ν a:ρ in e introduces a “new” symbol, a, with associ- ated type, ρ, for use within e. The declared symbol, a, is “new” in the sense that it is bound by the declaration within e, and so may be renamed at will to ensure that it differs from any finite set of active symbols. Whereas the statics determines the scope of a declared symbol, its range of significance, or extent, is determined by the dynamics. There are two different dynamic interpretations of symbols, the scoped and the free (short for scope-free) dy- namics. The scoped dynamics limits the extent of the symbol to its scope; the lifetime of the symbol is restricted to the evaluation of its scope. Alter- natively, under the free dynamics the extent of a symbol exceeds its scope, 316 32.1 Symbol Declaration extending to the entire computation of which it is a part. We may say that in the free dynamics a symbol “escapes its scope,” but it is more accurate to say that its scope widens to encompass the rest of the computation. The second construct associated with symbols is the concept of a sym- bolic reference, a form of expression whose sole purpose is to refer to a par- ticular symbol. Symbolic references are values of a type, ρ sym, and have the form & a for some symbol, a, with associated type, ρ. The eliminatory form for the type ρ sym is a conditional branch that determines whether or not a symbolic reference refers to a statically specified symbol. Crucially, the statics of the eliminatory form is carefully designed so that, in the pos- itive case, the type associated to the referenced symbol is made manifest, whereas in the negative case, no type information is gleaned because the referenced symbol could be of any type. 32.1 Symbol Declaration The ability to declare a new symbol is shared by all applications of symbols in subsequent chapters. The syntax for symbol declaration is given by the following grammar: Exp e ::= new[τ](a.e) ν a:τ in e generation The statics of symbol declaration makes use of a signature, or symbol context, that associates a type to each of a finite set of symbols. We use the letter Σ to range over signatures, which are finite sets of pairs a ∼ τ, where a is a symbol and τ is a type. The typing judgment Γ `Σ e : τ is parameterized by a signature, Σ, associating types to symbols. The statics of symbol declaration makes use of a judgment, τ mobile, whose definition depends on whether the dynamics is scoped or not. In a scoped dynamics mobility is defined so that the computed value of a mobile type cannot depend on any symbol. By constraining the scope of a declaration to have mobile type, we can, under this interpretation, ensure that the extent of a symbol is confined to its scope. In a free dynamics every type is deemed mobile, because the dynamics ensures that the scope of a symbol is widened to accommodate the possibility that the value returned from the scope of a declaration may depend on the declared symbol. The term “mobile” reflects the informal idea that symbols may or may not be “moved” from the scope of their declaration according to the dynamics given to them. A free dynamics allows symbols to be moved freely, whereas a scoped dynamics limits their range of motion. VERSION 1.32 REVISED 05.15.2012 32.1 Symbol Declaration 317 The statics of symbol declaration itself is given by the following rule: Γ `Σ,a∼ρ e : τ τ mobile Γ `Σ new[ρ](a.e): τ (32.1) As mentioned, the condition on τ is to be chosen so as to ensure that the returned value is meaningful in which dynamics we are using. 32.1.1 Scoped Dynamics The scoped dynamics of symbol declaration is given by a transition judg- ment of the form e 7−→ Σ e0 indexed by a signature, Σ, specifying the active symbols of the transition. Either e or e0 may involve the symbols declared in Σ, but no others. e 7−−−→ Σ,a∼ρ e0 new[ρ](a.e) 7−→ Σ new[ρ](a.e0)(32.2a) e valΣ new[ρ](a.e) 7−→ Σ e (32.2b) Rule (32.2a) specifies that evaluation takes place within the scope of the declaration of a symbol. Rule (32.2b) specifies that the declared symbol is “forgotten” once its scope has been evaluated. The definition of the judgment τ mobile must be chosen to ensure that the following mobility condition is satisfied: If τ mobile, `Σ,a∼ρ e : τ, and e valΣ,a∼ρ, then `Σ e : τ and e valΣ. For example, in the presence of symbolic references (see Section 32.2 be- low), a function type cannot be deemed mobile, because a function may contain a reference to a local symbol. The type nat may only be deemed mobile if the successor is evaluated eagerly, for otherwise a symbolic refer- ence may occur within a value of this type, invalidating the condition. Theorem 32.1 (Preservation). If `Σ e : τ and e 7−→ Σ e0, then `Σ e0 : τ. Proof. By induction on the dynamics of symbol declaration. Rule (32.2a) follows directly by induction, applying Rule (32.1). Rule (32.2b) follows directly from the condition on mobility. Theorem 32.2 (Progress). If `Σ e : τ, then either e 7−→ Σ e0, or e valΣ. REVISED 05.15.2012 VERSION 1.32 318 32.1 Symbol Declaration Proof. There is only one rule to consider, Rule (32.1). By induction we have either e 7−−−→ Σ,a∼ρ e0, in which case Rule (32.2a) applies, or e valΣ,a∼ρ, in which case by the mobility condition we have e valΣ, and hence Rule (32.2b) ap- plies. 32.1.2 Scope-Free Dynamics The scope-free dynamics of symbols is defined by a transition system be- tween states of the form ν Σ{ e }, where Σ is a signature and e is an ex- pression over this signature. The judgment ν Σ{ e } 7→ ν Σ0 { e0 } states that evaluation of e relative to symbols Σ results in the expression e0 in the ex- tension Σ0 of Σ. ν Σ{ new[ρ](a.e)} 7→ ν Σ, a ∼ ρ { e }(32.3) Rule (32.3) specifies that symbol generation enriches the signature with the newly introduced symbol by extending the signature for all future transi- tions. All other rules of the dynamics must be changed accordingly to account for the allocated symbols. For example, the dynamics of function applica- tion cannot simply be inherited from Chapter 10, but must be reformulated as follows: ν Σ{ e1 } 7→ ν Σ0 { e0 1 } ν Σ{ e1(e2)} 7→ ν Σ0 { e0 1(e2)}(32.4a) ν Σ{ λ (x:τ) e(e2)} 7→ ν Σ{[e2/x]e }(32.4b) These rules shuffle around the signature so as to account for symbol decla- rations within the constituent expressions of the application. Similar rules must be given for all other constructs of the language. Theorem 32.3 (Preservation). If ν Σ{ e } 7→ ν Σ0 { e0 } and `Σ e : τ, then Σ0 ⊇ Σ and `Σ0 e0 : τ. Proof. There is only one rule to consider, Rule (32.3), which is easily han- dled by inversion of Rule (32.1). Theorem 32.4 (Progress). If `Σ e : τ, then either e valΣ or ν Σ{ e } 7→ ν Σ0 { e0 } for some Σ0 and e0. Proof. Immediate, by Rule (32.3). VERSION 1.32 REVISED 05.15.2012 32.2 Symbolic References 319 32.2 Symbolic References Symbols are not themselves values, but they may be used to form values. One useful example is provided by the type τ sym of symbolic references.A value of this type has the form & a, where a is a symbol in the signature. To compute with a reference we may branch according to whether it is a reference to a specified symbol or not. The syntax of symbolic references is given by the following chart: Typ τ ::= sym(τ) τ sym symbols Exp e sym[a]& a reference is[a][t.τ](e; e1; e2) if e is a then e1 ow e2 comparison The expression sym[a] is a reference to the symbol a, a value of type sym(τ). The expression is[a][t.τ](e; e1; e2) compares the value of e, which must be a reference to some symbol b, with the given symbol, a. If b is a, the expression evaluates to e1, and otherwise to e2. 32.2.1 Statics The typing rules for symbolic references are as follows: Γ `Σ,a∼ρ sym[a]: sym(ρ)(32.5a) Γ `Σ,a∼ρ e : sym(ρ0)Γ `Σ,a∼ρ e1 :[ρ/t]τ Γ `Σ,a∼ρ e2 :[ρ0/t]τ Γ `Σ,a∼ρ is[a][t.τ](e; e1; e2):[ρ0/t]τ (32.5b) Rule (32.5a) is the introduction rule for the type sym(ρ). It states that if a is a symbol with associated type ρ, then sym[a] is an expression of type sym(ρ). Rule (32.5b) is the elimination rule for the type sym(ρ). The type associated to the given symbol, a, is not required to be the same as the type of the symbol referred to by the expression e. If e evaluates to a reference to a, then these types will coincide, but if it refers to another symbol, b 6= a, then these types may well differ. With this in mind, let us examine carefully Rule (32.5b). A priori there is a discrepancy between the type, ρ, of a and the type, ρ0, of the symbol referred to by e. This discrepancy is mediated by the type operator t.τ.1 Regardless of the outcome of the comparison, the overall type of the ex- pression is [ρ0/t]τ. To ensure safety, we must ensure that this is a valid type 1See Chapter 14 for a discussion of type operators. REVISED 05.15.2012 VERSION 1.32 320 32.2 Symbolic References for the result, regardless of whether the comparison succeeds or fails. If e evaluates to the symbol a, then we “learn” that the types ρ0 and ρ coincide, because the specified and referenced symbol coincide. This is reflected by the type [ρ/t]τ for e1. If e evaluates to some other symbol, a0 6= a, then the comparison evaluates to e2, which is required to have type [ρ0/t]τ; no further information about the type of the symbol is acquired in this branch. 32.2.2 Dynamics The (scoped) dynamics of symbolic references is given by the following rules: sym[a] valΣ,a∼ρ (32.6a) is[a][t.τ](sym[a]; e1; e2) 7−−−→ Σ,a∼ρ e1 (32.6b) (a 6= a0) is[a][t.τ](sym[a0]; e1; e2) 7−−−−−−→ Σ,a∼ρ,a0∼ρ0 e2 (32.6c) e 7−−−→ Σ,a∼ρ e0 is[a][t.τ](e; e1; e2) 7−−−→ Σ,a∼ρ is[a][t.τ](e0; e1; e2)(32.6d) Rules (32.6b) and (32.6c) specify that is[a][t.τ](e; e1; e2) branches accord- ing to whether the value of e is a reference to the symbol, a, or not. 32.2.3 Safety To ensure that the mobility condition is satisfied, it is important that sym- bolic reference types not be deemed mobile. Theorem 32.5 (Preservation). If `Σ e : τ and e 7−→ Σ e0, then `Σ e0 : τ. Proof. By rule induction on Rules (32.6). The most interesting case is Rule (32.6b). When the comparison is positive, the types ρ and ρ0 must be the same, be- cause each symbol has at most one associated type. Therefore, e1, which has type [ρ0/t]τ, also has type [ρ/t]τ, as required. Lemma 32.6 (Canonical Forms). If `Σ e : sym(ρ) and e valΣ, then e = sym[a] for some a such that Σ = Σ0, a ∼ ρ. VERSION 1.32 REVISED 05.15.2012 32.3 Notes 321 Proof. By rule induction on Rules (32.5), taking account of the definition of values. Theorem 32.7 (Progress). Suppose that `Σ e : τ. Then either e valΣ, or there exists e0 such that e 7−→ Σ e0. Proof. By rule induction on Rules (32.5). For example, consider Rule (32.5b), in which we have that is[a][t.τ](e; e1; e2) has some type τ and that e : sym(ρ) for some ρ. By induction either Rule (32.6d) applies, or else we have that e valΣ, in which case we are assured by Lemma 32.6 that e is sym[a] for some symbol b of type ρ declared in Σ. But then progress is assured by Rules (32.6b) and (32.6c), because equality of symbols is decidable (either a is b or it is not). 32.3 Notes The concept of a symbol in a programming language was considered by McCarthy in the original formulation of Lisp (McCarthy, 1965). Unfortu- nately, symbols were, and often continue to be, confused with variables, as they were in the original formulation of Lisp. Although symbols are frequently encountered in dynamically typed languages, the formulation given here makes clear that they are equally sensible in statically typed languages. The present account was influenced by Pitts and Stark(1993) and on the declaration of names in the π-calculus (Milner(1999).) REVISED 05.15.2012 VERSION 1.32 322 32.3 Notes VERSION 1.32 REVISED 05.15.2012 Chapter 33 Fluid Binding In this chapter we return to the concept of dynamic scoping of variables that was criticized in Chapter8. There it was observed that dynamic scop- ing is problematic for at least two reasons: • A bound variable may not always be renamed in an expression with- out changing its meaning. • Because the scope of a variable is resolved dynamically, type safety is compromised. These violations of the expected behavior of variables is intolerable, be- cause it is at variance with mathematical practice and because it compro- mises modularity. It is possible, however, to recover a type-safe analogue of dynamic scop- ing by divorcing it from the concept of a variable, and instead introducing a new mechanism, called fluid, or dynamic, binding of a symbol. Fluid bind- ing associates to a symbol a value of its associated type within a specified scope. Upon exiting that scope, the binding is dropped (or, more accurately, reverted to its binding in the surrounding context). 33.1 Statics The language L{fluid} extends the language L{sym} defined in Chap- ter 32 with the following additional constructs: Exp e ::= put[a](e1; e2) put e1 for a in e2 binding get[a] get a retrieval 324 33.2 Dynamics As in Chapter 32, we use a to stand for some unspecified symbol. The ex- pression get[a] evaluates to the value of the current binding of a, if it has one, and is stuck otherwise. The expression put[a](e1; e2) binds the sym- bol a to the value e1 for the duration of the evaluation of e2, at which point the binding of a reverts to what it was prior to the execution. The symbol a is not bound by the put expression, but is instead a parameter of it. The statics of L{fluid} is defined by judgments of the form Γ `Σ e : τ, where Σ is a finite set of symbol declarations of the form a ∼ τ such that no symbol is declared more than once. The statics of L{fluid} extends that of L{sym}(see Chapter 32) with the following rules: Γ `Σ,a∼τ get[a]: τ (33.1a) Γ `Σ,a∼τ1 e1 : τ1 Γ `Σ,a∼τ1 e2 : τ2 Γ `Σ,a∼τ1 put[a](e1; e2): τ2 (33.1b) Rule (33.1b) specifies that the symbol a is a parameter of the expression that must be declared in Σ. 33.2 Dynamics We assume a stack-like dynamics for symbols, as described in Chapter 32. The dynamics of L{fluid} maintains an association of values to symbols that changes in a stack-like manner during execution. We define a family of transition judgments of the form e µ7−→ Σ e0, where Σ is as in the statics, and µ is a finite function mapping some subset of the symbols declared in Σ to values of appropriate type. If µ is defined for some symbol a, then it has the form µ0 ⊗ a ,→ e for some µ0 and value e. If, on the other hand, µ is undefined for some symbol a, we may regard it as having the form µ0 ⊗ a ,→ •. We will write a ,→ to stand ambiguously for either a ,→ • or a ,→ e for some expression e. The dynamics of L{fluid} is given by the following rules: e valΣ,a∼τ get[a] µ⊗a,→e7−−−−→ Σ,a∼τ e (33.2a) VERSION 1.32 REVISED 05.15.2012 33.3 Type Safety 325 e1 µ7−→ Σ e0 1 put[a](e1; e2) µ7−→ Σ put[a](e0 1; e2) (33.2b) e1 valΣ,a∼τ e2 µ⊗a,→e17−−−−→ Σ,a∼τ e0 2 put[a](e1; e2) µ⊗a,→7−−−−→ Σ,a∼τ put[a](e1; e0 2) (33.2c) e1 valΣ,a∼τ e2 valΣ,a∼τ put[a](e1; e2) µ7−→ Σ e2 (33.2d) Rule (33.2a) specifies that get[a] evaluates to the current binding of a, if any. Rule (33.2b) specifies that the binding for the symbol a is to be evalu- ated before the binding is created. Rule (33.2c) evaluates e2 in an environ- ment in which the symbol a is bound to the value e1, regardless of whether or not a is already bound in the environment. Rule (33.2d) eliminates the fluid binding for a once evaluation of the extent of the binding has com- pleted. According to the dynamics defined by Rules (33.2), there is no transition of the form get[a] µ7−→ Σ e if µ(a) = •. The judgment e unboundΣ states that execution of e will lead to such a “stuck” state, and is inductively defined by the following rules: µ(a) = • get[a] unboundµ (33.3a) e1 unboundµ put[a](e1; e2) unboundµ (33.3b) e1 valΣ e2 unboundµ put[a](e1; e2) unboundµ (33.3c) In a larger language it would also be necessary to include error propagation rules of the sort discussed in Chapter6. 33.3 Type Safety Define the auxiliary judgment µ :Σ by the following rules: ∅ : ∅ (33.4a) REVISED 05.15.2012 VERSION 1.32 326 33.4 Some Subtleties `Σ e : τ µ :Σ µ ⊗ a ,→ e :Σ, a ∼ τ (33.4b) µ :Σ µ ⊗ a ,→ • :Σ, a ∼ τ (33.4c) These rules specify that if a symbol is bound to a value, then that value must be of the type associated to the symbol by Σ. No demand is made in the case that the symbol is unbound (equivalently, bound to a “black hole”). Theorem 33.1 (Preservation). If e µ7−→ Σ e0, where µ :Σ and `Σ e : τ, then `Σ e0 : τ. Proof. By rule induction on Rules (33.2). Rule (33.2a) is handled by the defi- nition of µ :Σ. Rule (33.2b) follows immediately by induction. Rule (33.2d) is handled by inversion of Rules (33.1). Finally, Rule (33.2c) is handled by inversion of Rules (33.1) and induction. Theorem 33.2 (Progress). If `Σ e : τ and µ :Σ, then either e valΣ, or e unboundµ, or there exists e0 such that e µ7−→ Σ e0. Proof. By induction on Rules (33.1). For Rule (33.1a), we have Σ ` a ∼ τ from the premise of the rule, and hence, because µ :Σ, we have either µ(a) = • or µ(a) = e for some e such that `Σ e : τ. In the former case we have e unboundµ, and in the latter we have get[a] µ7−→ Σ e. For Rule (33.1b), we have by induction that either e1 valΣ or e1 unboundµ, or e1 µ7−→ Σ e0 1. In the latter two cases we may apply Rule (33.2b) or Rule (33.3b), respectively. If e1 valΣ, we apply induction to obtain that either e2 valΣ, in which case Rule (33.2d) applies; e2 unboundµ, in which case Rule (33.3c) applies; or e2 µ7−→ Σ e0 2, in which case Rule (33.2c) applies. 33.4 Some Subtleties The value of put e1 for a in e2 is the value of e2, calculated in a context in which a is bound to the value of e1. If e2 is of a basic type, such as nat, then the reversion of the binding of a cannot influence the meaning of the result.1 1As long as the the successor is evaluated eagerly; if not, the following examples may be adapted to situations in which the value of e2 is a lazily evaluated number. VERSION 1.32 REVISED 05.15.2012 33.4 Some Subtleties 327 But what if the type of put e1 for a in e2 is a function type, so that the returned value is a λ-abstraction? The body of the returned λ may refer to the binding of a, which is reverted upon return from the put. For example, consider the expression put 17 for a in λ (x:nat) x + get a, (33.5) which has type nat → nat, given that a is a symbol of type nat. Let us assume, for the sake of discussion, that a is unbound at the point at which this expression is evaluated. Evaluating the put binds a to the number 17, and returns the function λ (x:nat) x + get a. But because a is reverted to its unbound state upon exiting the put, applying this function to an argu- ment will result in an error, unless a binding for a is provided. Thus, if f is bound to the result of evaluating (33.5), then the expression put 21 for a in f(7)(33.6) will evaluate successfully to 28, whereas evaluation of f(7) in the absence of a surrounding binding for a will incur an error. Contrast this with the superficially similar expression let y be 17 in λ (x:nat) x + y, (33.7) in which we have replaced the fluid-bound symbol, a, by a statically bound variable, y. This expression evaluates to λ (x:nat) x + 17, which adds 17 to its argument when applied. There is no possibility of an unbound sym- bol arising at execution time, precisely because variables are interpreted by substitution. One way to think about this situation is to consider that fluid-bound symbols serve as an alternative to passing additional arguments to a func- tion to specialize its value whenever it is called. To see this, let e stand for the value of expression (33.5), a λ-abstraction whose body is dependent on the binding of the symbol a. To use this function safely, it is necessary that the programmer provide a binding for a prior to calling it. For example, the expression put 7 for a in (e(9)) evaluates to 16, and the expression put 8 for a in (e(9)) evaluates to 17. Writing just e(9), without a surrounding binding for a, re- sults in a run-time error attempting to retrieve the binding of the unbound symbol a. REVISED 05.15.2012 VERSION 1.32 328 33.5 Fluid References This behavior may be simulated by adding an additional argument to the function value that will be bound to the current binding of the symbol a at the point where the function is called. Instead of using fluid binding, we would provide an additional argument at each call site, writing e0(7)(9) and e0(8)(9), respectively, where e0 is the λ-abstraction λ (y:nat) λ (x:nat) x + y. Additional arguments can be cumbersome, though, especially when sev- eral call sites provide the same binding for a. Using fluid binding we may write put 7 for a in he(8), e(9)i, whereas using an additional argument we must write he0(7)(8), e0(7)(9)i. However, such redundancy can be mitigated by simply factoring out the common part, writing let f be e0(7) in h f(8), f(9)i. The awkwardness of this simulation is usually taken as an argument in favor of including fluid binding in a language. The drawback, which is often perceived as an advantage, is that nothing in the type of a function reveals its dependency on the binding of a symbol. It is therefore quite easy to forget that such a binding is required, leading to run-time failures that might better be caught at compile time. 33.5 Fluid References The get and put operations for fluid binding are indexed by a symbol that must be given as part of the syntax of the operator. Rather than insist that the target symbol be given statically, it is useful to be able to defer until run- time the choice of fluid on which a get or put acts. This may be achieved by introducing references to fluids, which allow the name of a fluid to be VERSION 1.32 REVISED 05.15.2012 33.5 Fluid References 329 represented as a value. References come equipped with analogues of the get and put primitives, but for a dynamically determined symbol. The syntax of references as an extension to L{fluid} is given by the following grammar: Typ τ ::= fluid(τ) τ fluid fluid Exp e ::= fl[a] fl[a] reference getfl(e) getfl e retrieval putfl(e; e1; e2) putfl e is e1 in e2 binding The expression fl[a] is the symbol a considered as a value of type fluid(τ). The expressions getfl(e) and putfl(e; e1; e2) are analogues of the get and put operations for fluid-bound symbols. The statics of these constructs is given by the following rules: Γ `Σ,a∼τ fl[a]: fluid(τ)(33.8a) Γ `Σ e : fluid(τ) Γ `Σ getfl(e): τ (33.8b) Γ `Σ e : fluid(τ)Γ `Σ e1 : τ Γ `Σ e2 : τ2 Γ `Σ putfl(e; e1; e2): τ2 (33.8c) Because we are using a scoped dynamics, references to fluids cannot be deemed mobile. The dynamics of references consists of resolving the referent and defer- ring to the underlying primitives acting on symbols. fl[a] valΣ,a∼τ (33.9a) e µ7−→ Σ e0 getfl(e) µ7−→ Σ getfl(e0) (33.9b) getfl(fl[a]) µ7−→ Σ get[a](33.9c) e µ7−→ Σ e0 putfl(e; e1; e2) µ7−→ Σ putfl(e0; e1; e2) (33.9d) REVISED 05.15.2012 VERSION 1.32 330 33.6 Notes putfl(fl[a]; e1; e2) µ7−→ Σ put[a](e1; e2)(33.9e) 33.6 Notes The concept of dynamic binding arose from the confusion of variables and symbols in early dialects of Lisp. When properly separated, variables re- tain their substitutive meaning, and symbols give rise to a separate con- cept of fluid binding. Allen(1978) contains a thorough discussion of the implementation of fluid binding. The formulation given here also draws on Nanevski(2003). VERSION 1.32 REVISED 05.15.2012 Chapter 34 Dynamic Classification In Chapters 12 and 25 we investigated the use of sums for the classification of values of disparate type. Every value of a classified type is labeled with a symbol that determines the type of the instance data. A classified value is decomposed by pattern matching against a known class, which reveals the type of the instance data. Under this representation the possible classes of an object are fully de- termined statically by its type. However, it is sometimes useful to allow the possible classes of data value to be determined dynamically. There are many uses for such a capability, some less apparent than others. The most obvious is simply extensibility, when we wish to introduce new classes of data during execution (and, presumably, define how methods act on values of those new classes). A less obvious application exploits the fact that the new class is guar- anteed to be distinct from any other class that has already been introduced. The class itself is a kind of “secret” that can be disclosed only if the compu- tation that creates the class discloses its existence to another computation. In particular, the class is opaque to any computation to which this disclo- sure has not been explicitly made. This capability has a number of practical applications. One application is to use dynamic classification as a “perfect encryp- tion” mechanism that guarantees that a value cannot be determined with- out access to the appropriate “keys”. Keys are, in this scenario, pattern matching functions that are created when the class is defined. No party that lacks access to the matcher for that class can recover its underlying instance data, and so we may think of that value as encrypted.1 This can 1In practice this is implemented using probabilistic techniques to avoid the need for a 332 34.1 Dynamic Classes be useful when building programs that communicate over an insecure net- work: dynamic classification allows us to build private channels between two parties in the computation (see Chapter 42 for more on this applica- tion). Another application is to exception handling within a program. Excep- tion handling may be seen as a communication between two agents, one that may raise an exception, and one that may handle it. We wish to en- sure that an exception can be caught only by a designated handler, without fear that any intervening handler may intercept it. This can be achieved by dynamic class allocation. A new class is declared, with the capability to create an instance given only to the raising agent and the capability to match an instance given only to the handler. The exception value cannot be intercepted by any other handler, because no other handler is capable of matching it. 34.1 Dynamic Classes A dynamic class is a symbol that may be generated at run-time. A classified value consists of a symbol of type τ together with a value of that type. To compute with a classified value, it is compared with a known class. If the value is of this class, the underlying instance data is passed to the positive branch, otherwise the negative branch is taken, where it may be matched against other known classes. 34.1.1 Statics The syntax of the language clsfd of dynamic classification is given by the following grammar: Typ τ ::= clsfd clsfd classified Exp e ::= in[a](e) a · e instance isin[a](e; x.e1; e2) match e as a · x ⇒ e1 ow ⇒ e2 comparison The expression in[a](e) is a classified value with class a and underlying value e. The expression isin[a](e; x.e1; e2) checks whether the class of the value given by e is a. If so, the classified value is passed to e1; if not, the expression e2 is evaluated instead. central arbiter of unicity of symbol names. However, such methods require a source of randomness, which may be seen as just such an arbiter in disguise. There is no free lunch. VERSION 1.32 REVISED 05.15.2012 34.1 Dynamic Classes 333 The statics of clsfd is defined by the following rules: Γ `Σ,a∼ρ e : ρ Γ `Σ,a∼ρ in[a](e): clsfd (34.1a) Γ `Σ,a∼ρ e : clsfd Γ, x : ρ `Σ,a∼ρ e1 : τ Γ `Σ,a∼ρ e2 : τ Γ `Σ,a∼ρ isin[a](e; x.e1; e2): τ (34.1b) The type associated to the symbol in the signature determines the type of the instance data. 34.1.2 Dynamics To maximize the flexibility in the use of dynamic classification, we will give only a free dynamics for symbol generation. Within this framework the dynamics of classification is given by the following rules: e valΣ in[a](e) valΣ (34.2a) ν Σ{ e } 7→ ν Σ0 { e0 } ν Σ{ in[a](e)} 7→ ν Σ0 { in[a](e0)}(34.2b) e valΣ ν Σ{ isin[a](in[a](e); x.e1; e2)} 7→ ν Σ{[e/x]e1 }(34.2c) (a 6= a0) ν Σ{ isin[a](in[a0](e0); x.e1; e2)} 7→ ν Σ{ e2 }(34.2d) ν Σ{ e } 7→ ν Σ0 { e0 } ν Σ{ isin[a](e; x.e1; e2)} 7→ ν Σ0 { isin[a](e0; x.e1; e2)}(34.2e) Throughout, if the states involved are well-formed, then there will be a declaration a ∼ τ for some type τ in Σ. The dynamics of the elimination form for the type clsfd relies on dis- equality of names (specifically, Rule (34.2d)). Because disequality is not preserved under substitution, it is not sensible to consider any language construct whose dynamics relies on such a substitution. To see what goes wrong, consider the expression match b · hi as a · ⇒ true ow ⇒ match b · hi as b · ⇒ false ow ⇒ true. REVISED 05.15.2012 VERSION 1.32 334 34.2 Class References This is easily seen to evaluate to false, because the outer conditional is on the class a, which is a priori different from b. However, if we substitute b for a in this expression we obtain match b · hi as b · ⇒ true ow ⇒ match b · hi as b · ⇒ false ow ⇒ true, which evaluate to true, because now the outer conditional governs the evaluation. 34.1.3 Safety Theorem 34.1 (Safety). 1. If `Σ e : τ and ν Σ{ e } 7→ ν Σ0 { e0 }, then Σ0 ⊇ Σ and `Σ0 e0 : τ. 2. If `Σ e : τ, then either e valΣ or ν Σ{ e } 7→ ν Σ0 { e0 } for some e0 and Σ0. Proof. Similar to the safety proofs given in Chapters 12, 13, and 32. 34.2 Class References The type class(τ) has as values references to classes. Typ τ ::= class(τ) τ class class reference Exp e ::= cls[a]& a reference mk(e1; e2) mk(e1; e2) instance isofcls(e0; e1; x.e2; e3) isofcls(e0; e1; x.e2; e3) dispatch The statics of these constructs is given by the following rules: Γ `Σ,a∼τ cls[a]: class(τ)(34.3a) Γ `Σ e1 : class(τ)Γ `Σ e2 : τ Γ `Σ mk(e1; e2): clsfd (34.3b) Γ `Σ e0 : class(ρ)Γ `Σ e1 : clsfd Γ, x : ρ `Σ e2 : τ Γ `Σ e3 : τ Γ `Σ isofcls(e0; e1; x.e2; e3): τ (34.3c) The corresponding dynamics is given by these rules: ν Σ{ e1 } 7→ ν Σ0 { e0 1 } ν Σ{ mk(e1; e2)} 7→ ν Σ0 { mk(e0 1; e2)}(34.4a) VERSION 1.32 REVISED 05.15.2012 34.3 Definability of Dynamic Classes 335 e1 valΣ ν Σ{ e2 } 7→ ν Σ0 { e0 2 } ν Σ{ mk(e1; e2)} 7→ ν Σ0 { mk(e1; e0 2)}(34.4b) e valΣ ν Σ{ mk(cls[a]; e)} 7→ ν Σ{ in[a](e)}(34.4c) ν Σ{ e0 } 7→ ν Σ0 { e0 0 } ν Σ{ isofcls(e0; e1; x.e2; e3)} 7→ ν Σ0 { isofcls(e0 0; e1; x.e2; e3)}(34.4d) ν Σ{ isofcls(cls[a]; e1; x.e2; e3)} 7→ ν Σ{ isin[a](e1; x.e2; e3)} (34.4e) Rules (34.4d) and (34.4e) specify that the first argument is evaluated to de- termine the target class, which is then used to check whether the second argument, a classified data value, is of the target class. This may be seen as a two-stage pattern matching process in which evaluation of e0 determines the pattern against which to match the classified value of e1. 34.3 Definability of Dynamic Classes The type clsfd may be defined in terms of symbolic references, product types, and existential types by the type expression clsfd , ∃(t.t sym × t). The introductory form, in[a](e), where a is a symbol whose associated type is ρ and e is an expression of type ρ, is defined to be the package pack ρ with h& a, ei as ∃(t.t sym × t). The eliminatory form, isin[a](e; x.e1; e2), is defined in terms of symbol comparison as defined in Chapter 32. Suppose that the overall type of the conditional is τ and that the type associated to the symbol a is ρ. The type of e must be clsfd, defined as above, and the type of the branches e1 and e2 must be τ, with x assumed to be of type ρ in e1. The conditional is defined to be the expression open e as t with hx, yi:t sym × t in (ebody(y)), where ebody is an expression to be defined shortly. The comparison opens the package, e, representing the classified value, and decomposes it into a type, t, a symbol, x, of type t sym, and an underlying value, y, of type t. The REVISED 05.15.2012 VERSION 1.32 336 34.4 Classifying Secrets expression ebody, which is to be defined shortly, will have the type t → τ, so that the application to y is type correct. The expression ebody compares the symbolic reference, x, to the symbol, a, of type ρ, and yields a value of type t → τ regardless of the outcome. It is therefore defined to be the expression is[a][u.u → τ](x; e0 1; e0 2) where, in accordance with Rule (32.5b), e0 1 has type [ρ/u](u → τ) = ρ → τ, and e0 2 has type [t/u](u → τ) = t → τ. The expression e0 1 “knows” that the abstract type, t, is ρ, the type associated to the symbol a, because the comparison has come out positively. On the other hand, e0 2 does not “learn” anything about the identity of t. It remains to choose the expressions e0 1 and e0 2. In the case of a positive comparison, we wish to pass the classified value to the expression e1 by substitution for the variable x. This is accomplished by defining e0 1 to be the expression λ (x:ρ) e1 : ρ → τ. In the case of a negative comparison no value is to be propagated to e2. We therefore define e0 2 to be the expression λ (:t) e2 : t → τ. We may then check that the statics and dynamics given in Section 34.1 are derivable, given the definitions of the type of classified values and its in- troductory and eliminator forms. 34.4 Classifying Secrets Dynamic classification may be used to enforce confidentiality and integrity of data values in a program. A value of type clsfd may only be con- structed by sealing it with some class, a, and may only be deconstructed by a case analysis that includes a branch for a. By controlling which parties in a multi-party interaction have access to the classifier, a, we may control how classified values are created (ensuring their integrity) and how they are inspected (ensuring their confidentiality). Any party that lacks access to a cannot decipher a value classified by a, nor may it create a classified value with this class. Because classes are dynamically generated symbols, VERSION 1.32 REVISED 05.15.2012 34.5 Notes 337 they provide an absolute confidentiality guarantee among parties in a com- putation.2 Consider the following simple protocol for controlling the integrity and confidentiality of data in a program. A fresh symbol, a, is introduced, and we return a pair of functions of type (τ → clsfd) × (clsfd → τ opt), called the constructor and destructor functions for that class, which is accom- plished by writing newsym a:τ in h λ (x:τ) a · x, λ (x:clsfd) match x as a · y ⇒ just(y) ow ⇒ null i. The first function creates a value classified by a, and the second function recovers the instance data of a value classified by a. Outside of the scope of the declaration the symbol a is an absolutely unguessable secret. To enforce the integrity of a value of type τ, it is sufficient to ensure that only trusted parties have access to the constructor. To enforce the confiden- tiality of a value of type τ, it is sufficient to ensure that only trusted parties have access to the destructor. Ensuring the integrity of a value amounts to associating an invariant to it that is maintained by the trusted parties that may create an instance of that class. Ensuring the confidentiality of a value amounts to propagating the invariant to parties that may decipher it. 34.5 Notes Dynamic classification appears in Standard ML (Milner et al., 1997) as the type, exn, of exception values. The utility of this type is obscured by a too-close association with its application to exception values. The use of dynamic classification to control information flow was popularized in the π-calculus (Milner, 1999) in the form of “channel passing.” (See Chapter 42 for more on this correspondence.) 2Of course, this guarantee is for programs written in conformance with the statics given here. If the abstraction imposed by the type system is violated, no guarantees of confiden- tiality can be made. REVISED 05.15.2012 VERSION 1.32 338 34.5 Notes VERSION 1.32 REVISED 05.15.2012 Part XIII State Chapter 35 Modernized Algol Modernized Algol, or L{nat cmd *}, is an imperative, block-structured pro- gramming language based on the classic language Algol. L{nat cmd *} may be seen as an extension to L{nat *} with a new syntactic sort of commands that act on assignables by retrieving and altering their contents. Assignables are introduced by declaring them for use within a specified scope; this is the essence of block structure. Commands may be combined by sequencing, and may be iterated using recursion. L{nat cmd *} maintains a careful separation between pure expressions, whose meaning does not depend on any assignables, and impure commands, whose meaning is given in terms of assignables. This ensures that the eval- uation order for expressions is not constrained by the presence of assignables in the language, and allows for expressions to be manipulated much as in PCF. Commands, on the other hand, have a tightly constrained execution order, because the execution of one may affect the meaning of another. A distinctive feature of L{nat cmd *} is that it adheres to the stack dis- cipline, which means that assignables are allocated on entry to the scope of their declaration, and deallocated on exit, using a conventional stack discipline. This avoids the need for more complex forms of storage man- agement, at the expense of reducing the expressiveness of the language. 35.1 Basic Commands The syntax of the language L{nat cmd *} of modernized Algol distinguishes pure expressions from impure commands. The expressions include those of L{nat *}(as described in Chapter 10), augmented with one additional construct, and the commands are those of a simple imperative program- 342 35.1 Basic Commands ming language based on assignment. The language maintains a sharp dis- tinction between variables and assignables. Variables are introduced by λ- abstraction, and are given meaning by substitution. Assignables are intro- duced by a declaration, and are given meaning by assignment and retrieval of their contents, which is, for the time being, restricted to natural numbers. Expressions evaluate to values, and have no effect on assignables. Com- mands are executed for their effect on assignables, and also return a value. Composition of commands not only sequences their execution order, but also passes the value returned by the first to the second before it is exe- cuted. The returned value of a command is, for the time being, restricted to the natural numbers. (But see Section 35.3 for the general case.) The syntax of L{nat cmd *} is given by the following grammar, from which we have omitted repetition of the expression syntax of L{nat *} for the sake of brevity. Typ τ ::= cmd cmd command Exp e ::= cmd(m) cmd m encapsulation Cmd m ::= ret(e) ret e return bnd(e; x.m) bnd x ← e ; m sequence dcl(e; a.m) dcl a := e in m new assignable get[a]@ a fetch set[a](e) a := e assign The expression cmd(m) consists of the unevaluated command, m, thought of as a value of type cmd. The command, ret(e), returns the value of the expression e without having any effect on the assignables. The command bnd(e; x.m) evaluates e to an encapsulated command, then this command is executed for its effects on assignables, with its value substituted for x in m. The command dcl(e; a.m) introduces a new assignable, a, for use within the command, m, whose initial contents is given by the expression, e. The command get[a] returns the current contents of the assignable, a, and the command set[a](e) changes the contents of the assignable a to the value of e, and returns that value. 35.1.1 Statics The statics of L{nat cmd *} consists of two forms of judgment: 1. Expression typing: Γ `Σ e : τ. 2. Command formation: Γ `Σ m ok. VERSION 1.32 REVISED 05.15.2012 35.1 Basic Commands 343 The context, Γ, specifies the types of variables, as usual, and the signature, Σ, consists of a finite set of assignables. These judgments are inductively defined by the following rules: Γ `Σ m ok Γ `Σ cmd(m): cmd (35.1a) Γ `Σ e : nat Γ `Σ ret(e) ok (35.1b) Γ `Σ e : cmd Γ, x : nat `Σ m ok Γ `Σ bnd(e; x.m) ok (35.1c) Γ `Σ e : nat Γ `Σ,a m ok Γ `Σ dcl(e; a.m) ok (35.1d) Γ `Σ,a get[a] ok (35.1e) Γ `Σ,a e : nat Γ `Σ,a set[a](e) ok (35.1f) Rule (35.1a) is the introductory rule for the type cmd, and Rule (35.1c) is the corresponding eliminatory form. Rule (35.1d) introduces a new assignable for use within a specified command. The name, a, of the assignable is bound by the declaration, and hence may be renamed to satisfy the im- plicit constraint that it not already be present in Σ. Rule (35.1e) states that the command to retrieve the contents of an assignable, a, returns a natu- ral number. Rule (35.1f) states that we may assign a natural number to an assignable. 35.1.2 Dynamics The dynamics of L{nat cmd *} is defined in terms of a memory, µ, a finite function assigning a numeral to each of a finite set of assignables. The dynamics of expressions consists of these two judgment forms: 1. e valΣ, stating that e is a value relative to Σ. 2. e 7−→ Σ e0, stating that the expression e steps to the expression e0. These judgments are inductively defined by the following rules, together with the rules defining the dynamics of L{nat *}(see Chapter 10). It is important, however, that the successor operation be given an eager, rather REVISED 05.15.2012 VERSION 1.32 344 35.1 Basic Commands than lazy, dynamics so that a closed value of type nat is a numeral (for reasons that will be explained in Section 35.3). cmd(m) valΣ (35.2a) Rule (35.2a) states that an encapsulated command is a value. The dynamics of commands is defined in terms of states m k µ, where µ is a memory mapping assignables to values, and m is a command. There are two judgments governing such states: 1. m k µ finalΣ. The state m k µ is fully executed. 2. m k µ 7−→ Σ m0 k µ0. The state m k µ steps to the state m0 k µ0; the set of active assignables is given by the signature Σ. These judgments are inductively defined by the following rules: e valΣ ret(e) k µ finalΣ (35.3a) e 7−→ Σ e0 ret(e) k µ 7−→ Σ ret(e0) k µ (35.3b) e 7−→ Σ e0 bnd(e; x.m) k µ 7−→ Σ bnd(e0; x.m) k µ (35.3c) e valΣ bnd(cmd(ret(e)); x.m) k µ 7−→ Σ [e/x]m k µ (35.3d) m1 k µ 7−→ Σ m0 1 k µ0 bnd(cmd(m1); x.m2) k µ 7−→ Σ bnd(cmd(m0 1); x.m2) k µ0 (35.3e) get[a] k µ ⊗ a ,→ e 7−→ Σ,a ret(e) k µ ⊗ a ,→ e (35.3f) e 7−→ Σ e0 set[a](e) k µ 7−→ Σ set[a](e0) k µ (35.3g) e valΣ set[a](e) k µ ⊗ a ,→ 7−→ Σ ret(e) k µ ⊗ a ,→ e (35.3h) VERSION 1.32 REVISED 05.15.2012 35.1 Basic Commands 345 e 7−→ Σ e0 dcl(e; a.m) k µ 7−→ Σ dcl(e0; a.m) k µ (35.3i) e valΣ m k µ ⊗ a ,→ e 7−→ Σ,a m0 k µ0 ⊗ a ,→ e0 dcl(e; a.m) k µ 7−→ Σ dcl(e0; a.m0) k µ0 (35.3j) e valΣ e0 valΣ,a dcl(e; a.ret(e0)) k µ 7−→ Σ ret(e0) k µ (35.3k) Rule (35.3a) specifies that a ret command is final if its argument is a value. Rules (35.3c) to (35.3e) specify the dynamics of sequential composition. The expression, e, must, by virtue of the type system, evaluate to an encap- sulated command, which is to be executed to determine its return value, which is then substituted into m before executing it. Rules (35.3i) to (35.3k) define the concept of block structure in a pro- gramming language. Declarations adhere to the stack discipline in that an assignable is allocated for the duration of evaluation of the body of the dec- laration, and deallocated after evaluation of the body is complete. There- fore the lifetime of an assignable can be identified with its scope, and hence we may visualize the dynamic lifetimes of assignables as being nested in- side one another, in the same manner as their static scopes are nested inside one another. This stack-like behavior of assignables is a characteristic fea- ture of what are known as Algol-like languages. 35.1.3 Safety The judgment m k µ okΣ is defined by the rule `Σ m ok µ :Σ m k µ okΣ (35.4) where the auxiliary judgment µ :Σ is defined by the rule ∀a ∈ Σ ∃e µ(a) = e and e val∅ and `∅ e : nat µ :Σ(35.5) That is, the memory must bind a number to each location in Σ. Theorem 35.1 (Preservation). 1. If e 7−→ Σ e0 and `Σ e : τ, then `Σ e0 : τ. REVISED 05.15.2012 VERSION 1.32 346 35.2 Some Programming Idioms 2. If m k µ 7−→ Σ m0 k µ0, with `Σ m ok and µ :Σ, then `Σ m0 ok and µ0 :Σ. Proof. Simultaneously, by induction on Rules (35.2) and (35.3). Consider Rule (35.3j). Assume that `Σ dcl(e; a.m) ok and µ :Σ. By inversion of typing we have `Σ e : nat and `Σ,a m ok. Because e valΣ and µ :Σ, we have µ ⊗ a ,→ e :Σ, a. By induction we have `Σ,a m0 ok and µ0 ⊗ a ,→ e :Σ, a, from which the result follows immediately. Consider Rule (35.3k). Assume that `Σ dcl(e; a.ret(e0)) ok and µ :Σ. By inversion we have `Σ e : nat, `Σ,a ret(e0) ok, and hence that `Σ,a e0 : nat. But because e0 valΣ,a, e0 is a numeral, and hence we also have `Σ e0 : nat, as required. Theorem 35.2 (Progress). 1. If `Σ e : τ, then either e valΣ, or there exists e0 such that e 7−→ Σ e0. 2. If `Σ m ok and µ :Σ, then either m k µ finalΣ or m k µ 7−→ Σ m0 k µ0 for some µ0 and m0. Proof. Simultaneously, by induction on Rules (35.1). Consider Rule (35.1d). By the first inductive hypothesis we have either e 7−→ Σ e0 or e valΣ. In the for- mer case Rule (35.3i) applies. In the latter, we have by the second inductive hypothesis, m k µ ⊗ a ,→ e finalΣ,a or m k µ ⊗ a ,→ e 7−→ Σ,a m0 k µ0 ⊗ a ,→ e0. In the former case we apply Rule (35.3k), and in the latter, Rule (35.3j). 35.2 Some Programming Idioms The language L{nat cmd *} is designed to expose the elegant interplay between the execution of an expression for its value and the execution of a command for its effect on assignables. In this section we show how to de- rive several standard idioms of imperative programming in L{nat cmd *}. We define the sequential composition of commands, written {x ← m1 ; m2}, to stand for the command bnd x ← cmd (m1); m2. This generalizes to an n- ary form by defining {x1 ← m1 ;... xn−1 ← mn−1 ; mn}, VERSION 1.32 REVISED 05.15.2012 35.2 Some Programming Idioms 347 to stand for the iterated composition {x1 ← m1 ;...{xn−1 ← mn−1 ; mn}}. We sometimes write just {m1 ; m2} for the composition { ← m1 ; m2} in which the returned value from m1 is ignored; this generalizes in the ob- vious way to an n-ary form. A related idiom, the command do e, executes an encapsulated command and returns its result. By definition do e stands for the command bnd x ← e ; ret x. The conditional command, if (m) m1 else m2, executes either m1 or m2 according to whether the result of executing m is zero or not: {x ← m ; do (ifz x {z ⇒ cmd m1 | s( ) ⇒ cmd m2})}. The returned value of the conditional is the value returned by the selected command. The while loop command, while (m1) m2, repeatedly executes the com- mand m2 while the command m1 yields a non-zero number. It is defined as follows: do (fix loop:cmd is cmd (if (m1){ret z} else {m2 ; do loop})). This commands runs the self-referential encapsulated command that, when executed, first executes m1, branching on the result. If the result is zero, the loop returns zero (arbitrarily). If the result is non-zero, the command m2 is executed and the loop is repeated. A procedure is a function of type τ * cmd that takes an argument of some type, τ, and yields an unexecuted command as result. Many procedures have the form λ (x:τ) cmd m, which we abbreviate to proc (x:τ) m.A pro- cedure call is the composition of a function application with the activation of the resulting command. If e1 is a procedure and e2 is its argument, then the procedure call call e1(e2) is defined to be the command do (e1(e2)), which immediately runs the result of applying e1 to e2. As an example, here is a procedure of type nat * cmd that returns the factorial of its argument: REVISED 05.15.2012 VERSION 1.32 348 35.3 Typed Commands and Typed Assignables proc (x:nat) { dcl r := 1 in dcl a := x in { while ( @ a ) { y ← @ r ; z ← @ a ; r := (x-z+1)× y ; a := z-1 } ; @ r } } The loop maintains the invariant that the contents of r is the factorial of x minus the contents of a. Initialization makes this invariant true, and it is preserved by each iteration of the loop, so that upon completion of the loop the assignable a contains 0 and r contains the factorial of x, as required. 35.3 Typed Commands and Typed Assignables So far we have restricted the type of the returned value of a command, and the contents of an assignable, to be nat. Can this restriction be relaxed, while adhering to the stack discipline? The key to admitting returned and assignable values of other types lies in the details of the proof of Theorem 35.1. The proof of this theorem relies on an eager interpretation of the successor to ensure that the value is well- typed even in the absence of the locally declared assignable, a. The proof breaks down, and indeed the preservation theorem is false, when the return type of a command or the contents type of an assignable is unrestricted. For example, if we may return values of procedure type, then the fol- lowing command violates safety: dcl a := z in {ret (proc (x:nat) {a := x})}. This command, when executed, allocates a new assignable, a, and returns a procedure that, when called, assigns its argument to a. But this makes no sense, because the assignable, a, is deallocated when the body of the declaration returns, but the returned value still refers to it. If the returned procedure is called, execution will get stuck in the attempt to assign to a. VERSION 1.32 REVISED 05.15.2012 35.3 Typed Commands and Typed Assignables 349 A similar example shows that admitting assignables of arbitrary type is also unsound. For example, suppose that b is an assignable whose contents are of type nat → unit, and consider the command dcl a := z in {b := proc (x:nat) {a := x}; ret z}. We assign to b a procedure that uses a locally declared assignable, a, and then leaves the scope of the declaration. If we then call the procedure stored in b, execution will get stuck attempting to assign to the non-existent assignable, a. To admit declarations to return values and to admit assignables of types other than nat, we must rework the statics of L{nat cmd *} to record the returned type of a command and to record the type of the contents of each assignable. First, we generalize the finite set, Σ, of active assignables to assign a type to each active assignable so that Σ has the form of a finite set of assumptions of the form a ∼ τ, where a is an assignable. Second, we replace the judgment Γ `Σ m ok by the more general form Γ `Σ m ∼ τ, stating that m is a well-formed command returning a value of type τ. Third, the type cmd must be generalized to cmd(τ), which is written in examples as τ cmd, to specify the return type of the encapsulated command. The statics given in Section 35.1.1 may be generalized to admit typed commands and typed assignables, as follows: Γ `Σ m ∼ τ Γ `Σ cmd(m): cmd(τ)(35.6a) Γ `Σ e : τ τ mobile Γ `Σ ret(e) ∼ τ (35.6b) Γ `Σ e : cmd(τ)Γ, x : τ `Σ m ∼ τ0 Γ `Σ bnd(e; x.m) ∼ τ0 (35.6c) Γ `Σ e : τ τ mobile Γ `Σ,a∼τ m ∼ τ0 Γ `Σ dcl(e; a.m) ∼ τ0 (35.6d) Γ `Σ,a∼τ get[a] ∼ τ (35.6e) Γ `Σ,a∼τ e : τ Γ `Σ,a∼τ set[a](e) ∼ τ (35.6f) Apart from the generalization to track returned types and content types, the most important change is to require that the types of assignables and of returned values must be mobile. REVISED 05.15.2012 VERSION 1.32 350 35.3 Typed Commands and Typed Assignables As in Chapter 32, these rules make use of the judgment τ mobile, which states that the type τ is mobile. The definition of this judgment is guided by the following mobility condition: if τ mobile, `Σ e : τ and e valΣ, then `∅ e : τ and e val∅. (35.7) That is, a value of mobile type does not depend on any active assignables. Because the successor is evaluated eagerly, the type nat may be deemed mobile: nat mobile (35.8) Because the body of a procedure may involve an assignable, no procedure type may be considered mobile, nor may the type of commands returning a given type, for similar reasons. On the other hand, a product of mobile types may safely be deemed mobile, provided that pairing is evaluated eagerly: τ1 mobile τ2 mobile τ1 × τ2 mobile (35.9) Similarly, sums may be deemed mobile so long as the injections are evalu- ated eagerly: τ1 mobile τ2 mobile τ1 + τ2 mobile (35.10) Laziness defeats mobility, because values may contain suspended compu- tations that depend on an assignable. For example, if the successor oper- ation for the natural numbers were evaluated lazily, then s(e) would be a value for any expression, e, including one that refers to an assignable, a. Similarly, if pairing were lazy, then products may not be deemed mobile, and if injections were evaluated lazily, then sums may not either. What about function types other than procedure types? We may think they are mobile, because a pure expression cannot depend on an assignable. Although this is indeed the case, the mobility condition need not hold. For example, consider the following value of type nat * nat: λ (x:nat) (λ (:cmd) z)(cmd {@ a}). Although the assignable a is not actually needed to compute the result, it nevertheless occurs in the value, in violation of the safety condition. The mobility restriction on the statics of assignable declaration ensures that the type associated to an assignable is always mobile. We may there- fore assume, without loss of generality, that the types associated to the assignables in the signature Σ are mobile. VERSION 1.32 REVISED 05.15.2012 35.4 Notes 351 Theorem 35.3 (Preservation for Typed Commands). 1. If e 7−→ Σ e0 and `Σ e : τ, then `Σ e0 : τ. 2. If m k µ 7−→ Σ m0 k µ0, with `Σ m ∼ τ and µ :Σ, then `Σ m0 ∼ τ and µ0 :Σ. Theorem 35.4 (Progress for Typed Commands). 1. If `Σ e : τ, then either e valΣ, or there exists e0 such that e 7−→ Σ e0. 2. If `Σ m ∼ τ and µ :Σ, then either m k µ finalΣ or m k µ 7−→ Σ m0 k µ0 for some µ0 and m0. The proofs of Theorems 35.3 and 35.4 follows very closely the proof of Theorems 35.1 and 35.2. The main difference is that we appeal to the mobility condition to ensure that returned values and stored values are independent of the active assignables. 35.4 Notes Modernized Algol is essentially a reformulation of Idealized Algol (Reynolds, 1981) in which we have maintained a clearer separation between computa- tions that depend on the store and those that do not. The modal distinction between expressions and commands was present in the original formula- tion of Algol 60. The same separation has been popularized by Haskell, under the rubric “the IO monad.” What are called here assignables are regrettably called variables in the programming language literature. This clash of terminology is the source of considerable confusion and misunderstanding. It is preferable to retain the well-established meaning of a variable as standing for an unspecified object of a specified type, but to do so requires that we invent a new word for the name of a piece of mutable storage. The word assignable seems apt, and equally as convenient as the misappropriated word variable. In Idealized Algol, as in the original, an assignable may be used as a form of expression standing for its current contents. Although syntacti- cally convenient, this convention introduces an unfortunate dependency of expression evaluation on the store that we avoid here. The concept of mobility of a type was introduced in the ML5 language for distributed computing (Murphy et al., 2004), with the similar meaning REVISED 05.15.2012 VERSION 1.32 352 35.4 Notes that a value of a mobile type cannot depend on local resources. Here the mobility restriction is used to ensure that the language adheres to the stack discipline. VERSION 1.32 REVISED 05.15.2012 Chapter 36 Assignable References A reference to an assignable, a, is a value, written & a, of reference type that uniquely determines the assignable, a. A reference to an assignable pro- vides the capability to get or set the contents of that assignable, even if the assignable itself is not in scope at the point at which it is used. Two refer- ences may also be compared for equality to test whether or not they govern the same underlying assignable. If two references are equal, then setting one will affect the result of getting the other; if they are not equal, then setting one cannot influence the result of getting from the other. Two refer- ences that govern the same underlying assignable are said to be aliases. The possibility of aliasing complicates reasoning about the correctness of code that uses references, for we must always consider for any two references whether or not they might be aliases. Reference types are compatible with both a scoped and a scope-free al- location of assignables. When assignables are scoped, the range of signif- icance of a reference type must be limited to the scope of the assignable to which it refers. This may be achieved by declaring that reference types are immobile, so that they cannot be returned from the body of a declara- tion, nor stored in an assignable. Although ensuring adherence to the stack discipline, this restriction precludes the use of references to create muta- ble data structures, those whose structure can be altered during execution. Mutable data structures have a number of applications in programming, in- cluding improving efficiency (often at the expense of expressiveness) and allowing the creation of cyclic (self-referential) structures. Supporting mu- tability requires that assignables be given a scope-free dynamics, so that their lifetime persists beyond the scope of their declaration. Consequently, all types may be regarded as mobile, and hence values of any type may be 354 36.1 Capabilities stored in assignables or returned from commands. 36.1 Capabilities The commands get[a] and set[a](e) in L{nat cmd *} operate on a stat- ically specified assignable, a. To even write these commands requires that the assignable, a, be in scope at the point where the command occurs. But suppose that we wish to define a procedure that, say, updates an assignable to double its previous value, and returns the previous value. We can easily write such a procedure for any given assignable, a, but what if we wish to write a generic procedure that works for any given assignable? One way to do this is provide the procedure with the capability to get and set the contents of some caller-specified assignable. Such a capability is a pair consisting of a getter and a setter for that assignable. The getter for an assignable, a, is a command that, when executed, returns the contents of a. The setter for an assignable, a, is a procedure that, when applied to a value of suitable type, assigns that value to a. Thus, a capability for an assignable a containing a value of type τ is a value of type τ cmd × (τ * τ cmd) given by the pair hcmd (@ a), proc (x:τ) a := xi Because a capability type is a product of a command type and a procedure type, no capability type is mobile. This means, in particular, that a capa- bility cannot be returned from a command, nor stored into an assignable. This is as it should be, for otherwise we would violate the stack discipline for allocating assignables. Using capabilities, the proposed generic doubling procedure may be programmed as follows: proc (hget, seti:nat cmd × (τ * τ cmd)) {x ← do get ; y ← do (set(x + x)) ; ret x}. The procedure is to be called with the capability to access an assignable, a. When executed, it invokes the getter to obtain the contents of a, and then invokes the setter to assign to a, returning the previous value. Observe that the assignable, a, need not be directly accessible by this procedure; the capability provided by the caller comprises the commands required to get and set a. VERSION 1.32 REVISED 05.15.2012 36.2 Scoped Assignables 355 36.2 Scoped Assignables A weakness of using a capability to provide indirect access to an assignable is that there is no guarantee that a given getter/setter pair are in fact the ca- pability for a particular assignable. For example, we might (deliberately or accidentally) pair the getter for a with the setter for b, leading to unexpected behavior. There is nothing in the type system that prevents the creation of such mismatched pairs. To avoid this we introduce the concept of a reference to an assignable. A reference is a value from which we may obtain the capability to get and set a particular assignable. Moreover, two references may be compared for equality to determine whether or not they act on the same assignable.1 The reference type ref(τ) has as values references to assignables of type τ. The introduction and elimination forms for this type are given by the following syntax chart: Typ τ ::= ref(τ) τ ref assignable Exp e ::= ref[a]& a reference Cmd m ::= getref(e)* e contents setref(e1; e2) e1 := e2 update The statics of reference types is defined by the following rules: Γ `Σ,a∼τ ref[a]: ref(τ)(36.1a) Γ `Σ e : ref(τ) Γ `Σ getref(e) ∼ τ (36.1b) Γ `Σ e1 : ref(τ)Γ `Σ e2 : τ Γ `Σ setref(e1; e2) ∼ τ (36.1c) Rule (36.1a) specifies that the name of any active assignable is an expression of type ref(τ). The dynamics of reference types simply defers to the corresponding operations on assignables, and does not alter the underlying dynamics of assignables: ref[a] valΣ,a (36.2a) 1The getter and setter are not quite sufficient to define equality, because not all types admit a run-time equality test. When they do, and when there are at least two distinct values of the contents type, we can determine whether they are aliases by assigning to one and checking whether the contents of the other is changed. REVISED 05.15.2012 VERSION 1.32 356 36.2 Scoped Assignables e 7−→ Σ e0 getref(e) k µ 7−→ Σ getref(e0) k µ (36.2b) getref(ref[a]) k µ 7−→ Σ get[a] k µ (36.2c) e1 7−→ Σ e0 1 setref(e1; e2) k µ 7−→ Σ setref(e0 1; e2) k µ (36.2d) setref(ref[a]; e) k µ 7−→ Σ set[a](e) k µ (36.2e) A reference to an assignable is a value. The getref and setref operations on references defer to the corresponding operations on assignables once the reference has been determined. Because references give rise to capabilities, the reference type is deemed to be immobile. Consequently, references cannot be stored in assignables or returned from commands. This ensures safety, as may be readily verified by extending the proof given in Chapter 35. As an example of the use of references, the generic doubling procedure discussed in the preceding section may be programmed with references as follows: proc (r:nat ref) {x ← * r ; r := x + x ; ret x}. Because the argument is a reference, rather than a capability, there is no possibility that the getter and setter refer to different assignables. The ability to pass references to procedures comes at a price. When han- dling two or more references at the same time, we must consider the pos- sibility that they are aliases, which is to say that both refer to the same un- derlying assignable. Consider, for example, a procedure that, when given two references, x and y, adds twice the contents of y to the contents of x. One way to write this code creates no complications: λ (x:nat ref) λ (y:nat ref) cmd {x0 ← * x ; y0 ← * y ; x := x0 + y0 + y0}. Even if x and y refer to the same assignable, the effect will be to set the contents of the assignable referenced by x to the sum of its orginal contents and twice the contents of the assignable referenced by y. VERSION 1.32 REVISED 05.15.2012 36.3 Free Assignables 357 But now consider the following apparently equivalent implementation of this procedure: λ (x:nat ref) λ (y:nat ref) cmd {x += y ; x += y}, where x += y is the command {x0 ← * x ; y0 ← * y ; x := x0 + y0} that adds the contents of y to the contents of x. The second implementation works properly provided that x and y do not refer to the same assignable. For if they both reference the same assignable, a, with contents n, the result is that a is to set 4 × n, instead of the intended 3 × n, because the second get of y is influenced by the first assignment to x. In this case it is entirely obvious how to avoid the problem: use the first implementation, rather than the second. But the difficulty is not in fixing the problem once it has been uncovered, but rather noticing the problem in the first place. Wherever references (or capabilities) are used, the problems of interference lurk. Avoiding them requires very careful consideration of all possible aliasing relationships among all of the references in play at a given point of a computation. The problem is that the number of possible aliasing relationships among n references grows quadratically in n, because we must consider all possible pairings. 36.3 Free Assignables Although it is interesting to note that references and capabilities are com- patible with the stack discipline, this is achieved at the expense of their util- ity. Because references are not mobile, it is not possible to build a data struc- ture containing references internally. In particular, this restriction precludes programming mutable data structures (those whose structure changes dur- ing execution). To allow for more flexible uses of references, we must relax the re- quirement that assignables are to be stack-allocated, and instead arrange that the lifetime of an assignable extends beyond the scope of its declara- tion. Such assignables are called scope-free, or just free, assignables. If all assignables are scope-free, then every type may safely be deemed mobile. Consequently, references may be used to implement mutable and cyclic data structures. REVISED 05.15.2012 VERSION 1.32 358 36.3 Free Assignables Supporting free assignables amounts to changing the dynamics so that allocation of assignables persists across transitions. This is achieved by employing transition judgments of the form ν Σ{ m k µ } 7→ ν Σ0 { m0 k µ0 }. Execution of a command may allocate new assignables, may alter the con- tents of existing assignables, and may give rise to a new command to be executed at the next step. The rules defining the dynamics of free assignables are as follows: e valΣ ν Σ{ ret(e) k µ } final (36.3a) e 7−→ Σ e0 ν Σ{ ret(e) k µ } 7→ ν Σ{ ret(e0) k µ } (36.3b) e 7−→ Σ e0 ν Σ{ bnd(e; x.m) k µ } 7→ ν Σ{ bnd(e0; x.m) k µ } (36.3c) e valΣ ν Σ{ bnd(cmd(ret(e)); x.m) k µ } 7→ ν Σ{[e/x]m k µ }(36.3d) ν Σ{ m1 k µ } 7→ ν Σ0 { m0 1 k µ0 } ν Σ{ bnd(cmd(m1); x.m2) k µ } 7→ ν Σ0 { bnd(cmd(m0 1); x.m2) k µ0 } (36.3e) ν Σ, a ∼ τ { get[a] k µ ⊗ a ,→ e } 7→ ν Σ, a ∼ τ { ret(e) k µ ⊗ a ,→ e } (36.3f) e 7−→ Σ e0 ν Σ{ set[a](e) k µ } 7→ ν Σ{ set[a](e0) k µ } (36.3g) e valΣ,a∼τ ν Σ, a ∼ τ { set[a](e) k µ ⊗ a ,→ } 7→ ν Σ, a ∼ τ { ret(e) k µ ⊗ a ,→ e } (36.3h) e 7−→ Σ e0 ν Σ{ dcl(e; a.m) k µ } 7→ ν Σ{ dcl(e0; a.m) k µ } (36.3i) e valΣ ν Σ{ dcl(e; a.m) k µ } 7→ ν Σ, a ∼ τ { m k µ ⊗ a ,→ e }(36.3j) VERSION 1.32 REVISED 05.15.2012 36.3 Free Assignables 359 The language L{nat cmd ref *} extends L{nat cmd *} with references to free assignables. Its dynamics is similar to that of references to scoped assignables given earlier. e 7−→ Σ e0 ν Σ{ getref(e) k µ } 7→ ν Σ{ getref(e0) k µ } (36.4a) ν Σ{ getref(ref[a]) k µ } 7→ ν Σ{ get[a] k µ }(36.4b) e1 7−→ Σ e0 1 ν Σ{ setref(e1; e2) k µ } 7→ ν Σ{ setref(e0 1; e2) k µ } (36.4c) ν Σ{ setref(ref[a]; e2) k µ } 7→ ν Σ{ set[a](e2) k µ }(36.4d) Observe that the evaluation of expressions cannot alter or extend the mem- ory, only commands may do this. As an illustrative example of the use of references to scope-free assignables, consider the command newref[τ](e) defined by dcl a := e in ret (& a). (36.5) This command allocates a fresh assignable, and returns a reference to it. Its static and dynamics may be derived from the foregoing rules, as follows: Γ `Σ e : τ Γ `Σ newref[τ](e) ∼ ref(τ)(36.6) e 7−→ Σ e0 ν Σ{ newref[τ](e) k µ } 7→ ν Σ{ newref[τ](e0) k µ } (36.7a) e valΣ ν Σ{ newref[τ](e) k µ } 7→ ν Σ, a ∼ τ { ret(ref[a]) k µ ⊗ a ,→ e } (36.7b) Oftentimes the command newref[τ](e) is taken as primitive, and the dec- laration command is omitted. In that case all assignables are accessed by reference, and no direct access to assignables is provided. REVISED 05.15.2012 VERSION 1.32 360 36.4 Safety for Free Assignables 36.4 Safety for Free Assignables Although the proof of safety for references to scoped assignables presents few difficulties, the safety for free assignables is suprisingly tricky. The main difficulty is to account for the possibility of cyclic dependencies of data structures in memory. The contents of one assignable may contain a reference to itself, or a reference to another assignable that contains a reference to it, and so forth. For example, consider the following procedure, e, of type nat → nat cmd: proc (x:nat) {if (x) ret (1) else {f ← @ a ; y ← f(x − 1); ret (x ∗ y)}}. Let µ be a memory of the form µ0 ⊗ a ,→ e in which the contents of a con- tains, via the body of the procedure, a reference to a itself. Indeed, if the procedure e is called with a non-zero argument, it will “call itself” by indi- rect reference through a. The possibility of cyclic dependencies means that some care in the def- inition of the judgment µ :Σ is required. The following rule defines the well-formed states: `Σ m ∼ τ `Σ µ :Σ ν Σ{ m k µ } ok (36.8) The first premise of the rule states that the command m is well-formed rel- ative to Σ. The second premise states that the memory, µ, conforms to Σ, relative to the whole of Σ so that cyclic dependencies are permitted. The judg- ment `Σ0 µ :Σ is defined as follows: ∀a ∼ ρ ∈ Σ ∃e µ(a) = e and `Σ0 e : ρ `Σ0 µ :Σ(36.9) Theorem 36.1 (Preservation). 1. If `Σ e : τ and e 7−→ Σ e0, then `Σ e0 : τ. 2. If ν Σ{ m k µ } ok and ν Σ{ m k µ } 7→ ν Σ0 { m0 k µ0 }, then ν Σ0 { m0 k µ0 } ok. Proof. Simultaneously, by induction on transition. We prove the following stronger form of the second statement: If ν Σ{ m k µ } 7→ ν Σ0 { m0 k µ0 }, where `Σ m ∼ τ, `Σ µ :Σ, then Σ0 extends Σ, and `Σ0 m0 ∼ τ, and `Σ0 µ0 :Σ0. VERSION 1.32 REVISED 05.15.2012 36.4 Safety for Free Assignables 361 Consider, for example, the transition ν Σ{ dcl(e; a.m) k µ } 7→ ν Σ, a ∼ ρ { m k µ ⊗ a ,→ e } where e valΣ. By assumption and inversion of Rule (35.6d) we have ρ such that `Σ e : ρ, `Σ,a∼ρ m ∼ τ, and `Σ µ :Σ. But because extension of Σ with a fresh assignable does not affect typing, we also have `Σ,a∼ρ µ :Σ and `Σ,a∼ρ e : ρ, from which it follows by Rule (36.9) that `Σ,a∼ρ µ ⊗ a ,→ e : Σ, a ∼ ρ. The other cases follow a similar pattern, and are left as an exercise for the reader. Theorem 36.2 (Progress). 1. If `Σ e : τ, then either e valΣ or there exists e0 such that e 7−→ Σ e0. 2. If ν Σ{ m k µ } ok then either ν Σ{ m k µ } final or ν Σ{ m k µ } 7→ ν Σ0 { m0 k µ0 } for some Σ0, µ0, and m0. Proof. Simultaneously, by induction on typing. For the second statement we prove the stronger form If `Σ m ∼ τ and `Σ µ :Σ, then either ν Σ{ m k µ } final, or ν Σ{ m k µ } 7→ ν Σ0 { m0 k µ0 } for some Σ0, µ0, and m0. Consider, for example, the typing rule Γ `Σ e : ρ Γ `Σ,a∼ρ m ∼ τ Γ `Σ dcl(e; a.m) ∼ τ We have by the first inductive hypothesis that either e valΣ or e 7−→ Σ e0 for some e0. In the latter case we have by Rule (36.3i) ν Σ{ dcl(e; a.m) k µ } 7→ ν Σ{ dcl(e0; a.m) k µ }. In the former case we have by Rule (36.3j) that ν Σ{ dcl(e; a.m) k µ } 7→ ν Σ, a ∼ ρ { m k µ ⊗ a ,→ e }. As another example, consider the typing rule Γ `Σ,a∼τ get[a] ∼ τ REVISED 05.15.2012 VERSION 1.32 362 36.5 Benign Effects By assumption `Σ,a∼τ µ :Σ, a ∼ τ, and hence there exists e valΣ,a∼τ such that µ = µ0 ⊗ a ,→ e and `Σ,a∼τ e : τ. By Rule (36.3f) ν Σ, a ∼ τ { get[a] k µ0 ⊗ a ,→ e } 7→ ν Σ, a ∼ τ { ret(e) k µ0 ⊗ a ,→ e }, as required. The other cases are handled similarly. 36.5 Benign Effects The modal separation between commands and expressions ensures that the meaning of an expression does not depend on the (ever-changing) contents of assignables. Although this is advantageous in many, perhaps most, sit- uations, it also precludes programming techniques that use storage effects to implement purely functional behavior. A prime example is memoization in a lazy language (which is described in detail in Chapter 37.) Externally, a suspended computation behaves exactly like the underlying computation; internally, an assignable is associated with the computation that stores the result of any evaluation of the computation for future use. Another class of examples are self-adjusting data structures, which use state internally to improve their efficiency without changing their overall purely functional behavior. For example, a splay tree is a binary search tree that uses muta- tion internally to rebalance the tree as elements are inserted, deleted, and retrieved. This ensures that, for example, lookup operations take time pro- portional to the logarithm of the number of elements. These are examples of benign storage effects, uses of mutation in a data structure to improve efficiency without disrupting its functional behavior. Because values are forms of expression, it is essential to relax the strict sep- aration between expressions and commands that characterizes the modal type system for storage effects described in Chapter 35. Although several ad hoc methods have been considered in the literature, the most general approach is to simply do away with the distinction entirely, coalescing ex- pressions and commands into a single syntactic category. The penalty is that the type system no longer ensures that an expression of type τ denotes a value of that type; it might, in addition, engender storage effects dur- ing evaluation. The benefit of this approach is that it is straightforward to implement benign effects that are impossible to implement when a strict modal separation between expressions and commands is maintained. The language L{nat ref *} is a reformulation of L{nat cmd ref *} in which commands are integrated with expressions. For example, the fol- VERSION 1.32 REVISED 05.15.2012 36.5 Benign Effects 363 lowing rules illustrate the structure of the statics of L{nat cmd ref *}: Γ `Σ e1 : τ1 Γ `Σ,a∼τ1 e2 : τ2 Γ `Σ dcl(e1; a.e2): τ2 (36.10a) Γ `Σ,a∼τ get[a]: τ (36.10b) Γ `Σ,a∼τ e : τ Γ `Σ,a∼τ set[a](e): τ (36.10c) Correspondingly, the dynamics of L{nat ref *} is given by transitions of the form ν Σ{ e k µ } 7→ ν Σ{ e0 k µ0 }, where e is an expression, rather than a command. The rules defining the dynamics are very similar to those for L{nat cmd ref *}, but with com- mands and expressions integrated into a single category. To illustrate the concept of a benign effect, consider the technique of backpatching to implement recursion. Here is a formulation of the facto- rial function in L{nat ref *} in which recursive calls are mediated by an assignable containing the function itself: dcl a := λn:nat.0 in { f ← a := λn:nat.ifz(n, 1, n0.n×(@a)(n0)) ; ret(f) } This expression returns a function of type nat * nat that is obtained by (a) allocating a free assignable initialized arbitrarily (and immaterially) with a function of this type, (b) defining a λ-abstraction in which each “recur- sive call” consists of retrieving and applying the function stored in that assignable, (c) assigning this function to the assignable, and (d) returning that function. The result is a function on the natural numbers, even though it uses state internally to its implementation. Backpatching is impossible in L{nat cmd ref *}, because it makes ex- plicit the reliance on state. To see this, let us consider recoding the above example in the language with a command modality: dcl a := proc(n:nat){ret 0} in { f ← a := ... ; ret(f) }, REVISED 05.15.2012 VERSION 1.32 364 36.6 Notes where the elided procedure assigned to a is given by proc(n:nat){if(ret(n)){ret(1)}else{f←@a;x←f(n-1);ret(n×x)}}. The difficulty is that what we have is a command, rather than an expres- sion. Moreover, the result of the command is of procedure type nat → (nat cmd), rather than function type nat → nat. This means that we cannot use the factorial “function” (so implemented) in an expression, but must instead execute it as a command, so that the factorial of n is computed by writing { f ← fact ; x ← f(n); ret(x)}. In short, the use of storage effects is exposed, rather than hidden, as is possible in L{nat ref *}. 36.6 Notes Reynolds(1981) uses capabilities to provide indirect access to assignables; it is a short step from there to references in the form considered here. Often references are considered only for free assignables, but this is not essential. It is perfectly possible to have references to scoped assignables as well, pro- vided that suitable mobility restrictions are imposed to ensure adherence to the stack discipline. The proof of safety of free references given here is in- spired by Wright and Felleisen(1994) and Harper(1994). Benign effects are central to the distinction between Haskell, which is based on an Algol-like separation between commands and expressions, and ML, which is based on the integration of evaluation with execution. Each approach has its advantages and complementary disadvantages; nei- ther is uniformly superior to the other. VERSION 1.32 REVISED 05.15.2012 Part XIV Laziness Chapter 37 Lazy Evaluation Lazy evaluation refers to a variety of concepts that seek to defer evaluation of an expression until it is definitely required, and to share the results of any such evaluation among all instances of a single deferred computation. The net result is that a computation is performed at most once among all of its instances. Laziness manifests itself in a number of ways. One form of laziness is the by-need evaluation strategy for function ap- plication. Recall from Chapter8 that the by-name evaluation order passes the argument to a function in unevaluated form so that it is only evaluated if it is actually used. But because the argument is replicated by substitu- tion, it may be evaluated more than once. By-need evaluation ensures that the argument to a function is evaluated at most once, by ensuring that all copies of an argument share the result of evaluating any one copy. Another form of laziness is the concept of a lazy data structure. As we have seen in Chapters 11, 12, and 16, we may choose to defer evaluation of the components of a data structure until they are actually required, rather than when the data structure is created. But if a component is required more than once, then the same computation will, without further provi- sion, be repeated on each use. To avoid this, the deferred portions of a data structure are shared so an access to one will propagate its result to all occurrences of the same computation. Yet another form of laziness arises from the concept of general recursion considered in Chapter 10. Recall that the dynamics of general recursion is given by unrolling, which replicates the recursive computation on each use. It would be preferable to share the results of such computation across unrollings. A lazy implementation of recursion avoids such replications by sharing the unrollings. 368 37.1 By-Need Dynamics Traditionally, languages are biased towards either eager or lazy eval- uation. Eager languages use a by-value dynamics for function applica- tions, and evaluate the components of data structures when they are cre- ated. Lazy languages adopt the opposite strategy, preferring a by-name dynamics for functions, and a lazy dynamics for data structures. The over- head of laziness is mitigated by managing sharing to avoid redundancy. Experience has shown, however, that the distinction is better drawn at the level of types, rather than at the level of a language design. What is impor- tant is to have available both lazy and eager types, so that the programmer may choose which to use in a given situation, rather than having the choice forced by the language designer. In this chapter we make precise the means by which sharing of com- putations is achieved in the implementation of laziness. We then isolate these mechanisms into a type of suspended computations whose results are shared across all copies of a given suspension. 37.1 By-Need Dynamics By-need evaluation of functions uses memoization to record the value of a computation so that any future use of the same computation may return the previously computed value (or compute it from scratch if there is none). This is achieved by “naming” each deferred computation with a symbol, which is then used to access its value whenever it is used. A memo table records the deferred computation associated to each symbol until such time as it is evaluated, after which it records the value of that computation. Thus naming implements sharing, and the memo table ensures irredundancy. The by-need dynamics for L{nat *} is based on a transition system with states of the form ν Σ{ e k µ }, where Σ is a finite set of hypotheses a1 ∼ τ1,..., an ∼ τn associating types to symbols, e is an expression that may involve the symbols in Σ, and µ maps each symbol declared in Σ to either an expression or a special symbol, •, called the black hole. (The role of the black hole will be made clear below.) As a notational convenience, we em- ploy a bit of legerdemain with the concrete syntax similar to that employed in Chapter 35. Specifically, the concrete syntax for the expression get[a], which fetches the contents of the assignable a, is just @ a, omitting explicit mention of the “get” operation. The by-need dynamics consists of the following two forms of judgment: 1. e valΣ, stating that e is a value that may involve the symbols in Σ. VERSION 1.32 REVISED 05.15.2012 37.1 By-Need Dynamics 369 2. ν Σ{ e k µ } 7→ ν Σ0 { e0 k µ0 }, stating that one step of evaluation of the expression e relative to memo table µ with the symbols declared in Σ results in the expression e0 relative to the memo table µ0 with symbols declared in Σ0. The dynamics is defined so that the collection of active symbols grows monotonically, and so that the type of a symbol never changes. The memo table may be altered destructively during execution to reflect progress in the evaluation of the expression associated with a given symbol. The judgment e valΣ expressing that e is a closed value is defined by the following rules: z valΣ (37.1a) s(a) valΣ,a∼nat (37.1b) λ (x:τ) e valΣ (37.1c) Rules (37.1a) through (37.1c) specify that z is a value, any expression of the form s(a), where a is a symbol, is a value, and that any λ-abstraction, possibly containing symbols, is a value. It is important that symbols them- selves are not values, rather they stand for (possibly unevaluated) expres- sions as specified by the memo table. The expression @ a, which is short for get[a], is not closed. Rather, it must be evaluated to determine, and possibly update, the binding of the symbol a in memory. The initial and final states of evaluation are defined as follows: ν ∅ { e k ∅ } initial (37.2a) e valΣ ν Σ{ e k µ } final (37.2b) Rule (37.2a) specifies that an initial state consists of an expression eval- uated relative to an empty memo table. Rule (37.2b) specifies that a final state has the form ν Σ{ e k µ }, where e is a value relative to Σ. The transition judgment for the by-need dynamics of L{nat *} is de- fined by the following rules: REVISED 05.15.2012 VERSION 1.32 370 37.1 By-Need Dynamics e valΣ,a∼τ ν Σ, a ∼ τ { a k µ ⊗ a ,→ e } 7→ ν Σ, a ∼ τ { e k µ ⊗ a ,→ e }(37.3a) ν Σ, a ∼ τ { e k µ ⊗ a ,→ • } 7→ ν Σ0, a ∼ τ { e0 k µ0 ⊗ a ,→ • } ν Σ, a ∼ τ { a k µ ⊗ a ,→ e } 7→ ν Σ0, a ∼ τ { a k µ0 ⊗ a ,→ e0 }(37.3b) ν Σ{ s(e) k µ } 7→ ν Σ, a ∼ nat { s(a) k µ ⊗ a ,→ e }(37.3c) ν Σ{ e k µ } 7→ ν Σ0 { e0 k µ0 } ν Σ{ ifz e {z ⇒ e0 | s(x) ⇒ e1} k µ } 7→ ν Σ0 { ifz e0 {z ⇒ e0 | s(x) ⇒ e1} k µ0 } (37.3d) ν Σ{ ifz z {z ⇒ e0 | s(x) ⇒ e1} k µ } 7→ ν Σ{ e0 k µ }(37.3e)    ν Σ, a ∼ nat { ifz s(a){z ⇒ e0 | s(x) ⇒ e1} k µ ⊗ a ,→ e } 7→ ν Σ, a ∼ nat {[a/x]e1 k µ ⊗ a ,→ e }    (37.3f) ν Σ{ e1 k µ } 7→ ν Σ0 { e0 1 k µ0 } ν Σ{ e1(e2) k µ } 7→ ν Σ0 { e0 1(e2) k µ0 }(37.3g)    ν Σ{ λ (x:τ) e(e2) k µ } 7→ ν Σ, a ∼ τ {[a/x]e k µ ⊗ a ,→ e2 }    (37.3h) ν Σ{ fix x:τ is e k µ } 7→ ν Σ, a ∼ τ { a k µ ⊗ a ,→ [a/x]e }(37.3i) Rule (37.3a) governs a symbol whose associated expression is a value; the value of the symbol is the value associated to that symbol in the memo VERSION 1.32 REVISED 05.15.2012 37.1 By-Need Dynamics 371 table. Rule (37.3b) specifies that if the expression associated to a symbol is not a value, then it is evaluated “in place” until such time as Rule (37.3a) applies. This is achieved by switching the focus of evaluation to the asso- ciated expression, while at the same time associating the black hole to that symbol. The black hole represents the absence of a value for that symbol, so that any attempt to access it during evaluation of its associated expres- sion cannot make progress. This signals a circular dependency that, if not caught using a black hole, would initiate an infinite regress. We may there- fore think of the black hole as catching a particular form of non-termination that arises when the value of an expression associated to a symbol depends on the symbol itself. Rule (37.3c) specifies that evaluation of s(e) allocates a fresh symbol, a, for the expression e, and yields the value s(a). The value of e is not deter- mined until such time as the predecessor is required in a subsequent com- putation. This implements a lazy dynamics for the successor. Rule (37.3f), which governs a conditional branch on a successor, substitutes the sym- bol, a, for the variable, x, when computing the predecessor of a non-zero number, ensuring that all occurrences of x share the same predecessor com- putation. Rule (37.3g) specifies that the value of the function position of an appli- cation must be determined before the application can be executed. Rule (37.3h) specifies that to evaluate an application of a λ-abstraction we allocate a fresh symbol for the argument, and substitute this symbol for the param- eter of the function. The argument is evaluated only if it is needed in the subsequent computation, and then that value is shared among all occur- rences of the parameter in the body of the function. General recursion is implemented by Rule (37.3i). Recall from Chap- ter 10 that the expression fix x:τ is e stands for the solution of the recur- sion equation x = e. Rule (37.3i) computes this solution by associating a fresh symbol, a, with the body, e, substituting a for x within e to effect the self-reference. It is this substitution that permits a named expression to de- pend on its own name. For example, the expression fix x:τ is x associates the expression a to a in the memo table, and returns a. The next step of evaluation is stuck, because it seeks to evaluate a with a bound to the black hole. In contrast an expression such as fix f:ρ → τ is λ (x:ρ) e does not get stuck, because the self-reference is “hidden” within the λ-abstraction, and hence need not be evaluated to determine the value of the binding. REVISED 05.15.2012 VERSION 1.32 372 37.2 Safety 37.2 Safety We write Γ `Σ e : τ to mean that e has type τ under the assumptions Γ, treating symbols declared in Σ as expressions of their associated type. The rules are as in Chapter 10, with the addition of the following rule for symbols: Γ `Σ,a∼τ a : τ (37.4) This rule amounts to an implicit coercion that turns a symbol into a form of expression. The expression involves a tacit operation to obtain the binding of a symbol. The judgment ν Σ{ e k µ } ok is defined by the following rules: `Σ e : τ `Σ µ :Σ ν Σ{ e k µ } ok (37.5a) ∀a ∼ τ ∈ Σ µ(a) = e 6= • =⇒`Σ0 e : τ `Σ0 µ :Σ(37.5b) Rule (37.5b) permits self-reference through the memo table by allowing the expression associated to a symbol, a, to contain a, or, more generally, to con- tain a symbol whose associated expression contains a, and so on through any finite chain of such dependencies. Moreover, a symbol that is bound to the “black hole” is deemed to be of any type. Theorem 37.1 (Preservation). Suppose that ν Σ{ e k µ } 7→ ν Σ0 { e0 k µ0 }. If ν Σ{ e k µ } ok, then ν Σ0 { e0 k µ0 } ok. Proof. We prove by induction on Rules (37.3) that if ν Σ{ e k µ } 7→ ν Σ0 { e0 k µ0 } and `Σ µ :Σ and `Σ e : τ, then Σ0 ⊇ Σ and `Σ0 µ0 :Σ0 and `Σ0 e0 : τ. Consider Rule (37.3b), for which we have e = e0 = a, µ = µ0 ⊗ a ,→ e0, µ0 = µ0 0 ⊗ a ,→ e0 0, and ν Σ, a ∼ τ { e0 k µ0 ⊗ a ,→ • } 7→ ν Σ0, a ∼ τ { e0 0 k µ0 0 ⊗ a ,→ • }. Assume that `Σ,a∼τ µ :Σ, a ∼ τ. It follows that `Σ,a∼τ e0 : τ and `Σ,a∼τ µ0 : Σ, and hence that `Σ,a∼τ µ0 ⊗ a ,→ • :Σ, a ∼ τ. We have by induction that Σ0 ⊇ Σ and `Σ0,a∼τ e0 0 : τ0 and `Σ0,a∼τ µ0 ⊗ a ,→ • :Σ, a ∼ τ. VERSION 1.32 REVISED 05.15.2012 37.2 Safety 373 But then `Σ0,a∼τ µ0 :Σ0, a ∼ τ, which suffices for the result. Consider Rule (37.3g), so that e is the application e1(e2) and ν Σ{ e1 k µ } 7→ ν Σ0 { e0 1 k µ0 }. Suppose that `Σ µ :Σ and `Σ e : τ. By inversion of typing `Σ e1 : τ2 → τ for some type τ2 such that `Σ e2 : τ2. By induction Σ0 ⊇ Σ and `Σ0 µ0 :Σ0 and `Σ0 e0 1 : τ2 → τ. By weakening we have `Σ0 e2 : τ2, so that `Σ0 e0 1(e2): τ, which is enough for the result. The statement of the progress theorem allows for the possibility of en- countering a black hole, representing a checkable form of non-termination. The judgment ν Σ{ e k µ } loops, stating that e diverges by virtue of encoun- tering the black hole, is defined by the following rules: ν Σ, a ∼ τ { a k µ ⊗ a ,→ • } loops (37.6a) ν Σ, a ∼ τ { e k µ ⊗ a ,→ • } loops ν Σ, a ∼ τ { a k µ ⊗ a ,→ e } loops (37.6b) ν Σ{ e k µ } loops ν Σ{ ifz e {z ⇒ e0 | s(x) ⇒ e1} k µ } loops (37.6c) ν Σ{ e1 k µ } loops ν Σ{ e1(e2) k µ } loops (37.6d) Theorem 37.2 (Progress). If ν Σ{ e k µ } ok, then either ν Σ{ e k µ } final, or ν Σ{ e k µ } loops, or there exists µ0 and e0 such that ν Σ{ e k µ } 7→ ν Σ0 { e0 k µ0 }. Proof. We proceed by induction on the derivations of `Σ e : τ and `Σ µ :Σ implicit in the derivation of ν Σ{ e k µ } ok. Consider Rule (10.1a), where the variable, a, is declared in Σ. Thus Σ = Σ0, a ∼ τ and `Σ µ :Σ. It follows that µ = µ0 ⊗ a ,→ e0 with `Σ µ0 :Σ0 and `Σ e0 : τ. Note that `Σ µ0 ⊗ a ,→ • :Σ. Applying induction to the derivation of `Σ e0 : τ, we consider three cases: 1. ν Σ{ e0 k µ ⊗ a ,→ • } final. By inversion of Rule (37.2b) we have e0 valΣ, and hence by Rule (37.3a) we obtain ν Σ{ a k µ } 7→ ν Σ{ e0 k µ }. REVISED 05.15.2012 VERSION 1.32 374 37.3 Lazy Data Structures 2. ν Σ{ e0 k µ0 ⊗ a ,→ • } loops. By applying Rule (37.6b) we obtain ν Σ{ a k µ } loops. 3. ν Σ{ e0 k µ0 ⊗ a ,→ • } 7→ ν Σ0 { e0 0 k µ0 0 ⊗ a ,→ • }. By applying Rule (37.3b) we obtain ν Σ{ a k µ ⊗ a ,→ e0 } 7→ ν Σ0 { a k µ0 ⊗ a ,→ e0 0 }. 37.3 Lazy Data Structures The by-need dynamics extends to product, sum, and recursive types in a straightforward manner. For example, the by-need dynamics of lazy prod- uct types is given by the following rules: ha1, a2i valΣ,a1∼τ1,a2∼τ2 (37.7a)    ν Σ{ he1, e2i k µ } 7→ ν Σ, a1 ∼ τ1, a2 ∼ τ2 { ha1, a2i k µ ⊗ a1 ,→ e1 ⊗ a2 ,→ e2 }    (37.7b) ν Σ{ e k µ } 7→ ν Σ0 { e0 k µ0 } ν Σ{ e · l k µ } 7→ ν Σ0 { e0 · l k µ0 }(37.7c) ν Σ{ e k µ } loops ν Σ{ e · l k µ } loops (37.7d)    ν Σ, a1 ∼ τ1, a2 ∼ τ2 { ha1, a2i · l k µ } 7→ ν Σ, a1 ∼ τ1, a2 ∼ τ2 { a1 k µ }    (37.7e) ν Σ{ e k µ } 7→ ν Σ0 { e0 k µ0 } ν Σ{ e · r k µ } 7→ ν Σ0 { e0 · r k µ0 }(37.7f) ν Σ{ e k µ } loops ν Σ{ e · r k µ } loops (37.7g) VERSION 1.32 REVISED 05.15.2012 37.4 Suspensions 375    ν Σ, a1 ∼ τ1, a2 ∼ τ2 { ha1, a2i · r k µ } 7→ ν Σ, a1 ∼ τ1, a2 ∼ τ2 { a2 k µ }    (37.7h) A pair is considered a value only if its arguments are symbols (Rule (37.7a)), which are introduced when the pair is created (Rule (37.7b)). The first and second projections evaluate to one or the other symbol in the pair, inducing a demand for the value of that component (Rules (37.7e) and (37.7h)). Similar techniques may be used to give a by-need dynamics to sums and recursive types. 37.4 Suspensions Another way to introduce laziness is to consolidate the machinery of the by-need dynamics into a single type whose values are possibly uneval- uated, memoized computations. The type of suspensions of type τ, writ- ten τ susp, has as introductory form susp x : τ is e representing the sus- pended, possibly self-referential, computation, e, of type τ, and as elimi- natory form the operation force(e) that evaluates the suspended compu- tation presented by e, records the value in a memo table, and returns that value as result. Using suspension types we may construct other lazy types according to our needs in a particular program. For example, the type of lazy pairs with components of type τ1 and τ2 is expressible as the type τ1 susp × τ2 susp and the type of by-need functions with domain τ1 and range τ2 is express- ible as the type τ1 susp → τ2. We may also express more complex combinations of eagerness and lazi- ness, such as the type of “lazy lists” consisting of computations that, when forced, evaluate either to the empty list, or a non-empty list consisting of a natural number and another lazy list: µt.(unit + (nat × t)) susp. This type should be contrasted with the type µt.(unit + (nat × t susp)) REVISED 05.15.2012 VERSION 1.32 376 37.4 Suspensions whose values are the empty list and a pair consisting of a natural number and a computation of another such value. The syntax of suspensions is given by the following grammar: Typ τ ::= susp(τ) τ susp suspension Exp e ::= susp[τ](x.e) susp x : τ is e delay force(e) force(e) force susp[a] susp[a] self-reference Suspensions are self-referential; the bound variable, x, refers to the suspen- sion itself. The expression susp[a] is a reference to the suspension named a. The statics of the suspension type is given using a judgment of the form Γ `Σ e : τ, where Σ assigns types to the names of suspensions. It is defined by the following rules: Γ, x : susp(τ) `Σ e : τ Γ `Σ susp[τ](x.e): susp(τ)(37.8a) Γ `Σ e : susp(τ) Γ `Σ force(e): τ (37.8b) Γ `Σ,a∼τ susp[a]: susp(τ)(37.8c) Rule (37.8a) checks that the expression, e, has type τ under the assumption that x, which stands for the suspension itself, has type susp(τ). The by-need dynamics of suspensions is defined by the following rules: susp[a] valΣ,a∼τ (37.9a)    ν Σ{ susp[τ](x.e) k µ } 7→ ν Σ, a ∼ τ { susp[a] k µ ⊗ a ,→ [a/x]e }    (37.9b) ν Σ{ e k µ } 7→ ν Σ0 { e0 k µ0 } ν Σ{ force(e) k µ } 7→ ν Σ0 { force(e0) k µ0 }(37.9c) e valΣ,a∼τ    ν Σ, a ∼ τ { force(susp[a]) k µ ⊗ a ,→ e } 7→ ν Σ, a ∼ τ { e k µ ⊗ a ,→ e }    (37.9d) VERSION 1.32 REVISED 05.15.2012 37.5 Notes 377 ν Σ, a ∼ τ { e k µ ⊗ a ,→ • } 7→ ν Σ0, a ∼ τ { e0 k µ0 ⊗ a ,→ • }    ν Σ, a ∼ τ { force(susp[a]) k µ ⊗ a ,→ e } 7→ ν Σ0, a ∼ τ { force(susp[a]) k µ0 ⊗ a ,→ e0 }    (37.9e) Rule (37.9a) specifies that a reference to a suspension is a value. Rule (37.9b) specifies that evaluation of a delayed computation consists of allocating a fresh symbol for it in the memo table, and returning a reference to that suspension. Rules (37.9c) to (37.9e) specify that demanding the value of a suspension forces evaluation of the suspended computation, which is then stored in the memo table and returned as the result. 37.5 Notes The by-need dynamics given here is inspired by Ariola and Felleisen(1997), but with the crucial distinction that by-need cells are regarded as assignables, rather than variables. This is in keeping with the principle that a variable is given meaning by substitution. The goal of by-need evaluation is to limit substitution in the interest of avoiding redundant computations. As such it cannot properly be modeled using variables, but rather requires a form of assignable (introduced in Chapter 35) to which at most one assignment is ever performed. REVISED 05.15.2012 VERSION 1.32 378 37.5 Notes VERSION 1.32 REVISED 05.15.2012 Chapter 38 Polarization Up to this point we have frequently encountered arbitrary choices in the dynamics of various language constructs. For example, when specifying the dynamics of pairs, we must choose, rather arbitrarily, between the lazy dynamics, in which all pairs are values regardless of the value status of their components, and the eager dynamics, in which a pair is a value only if its components are both values. We could even consider a half-eager (or, equivalently, half-lazy) dynamics, in which a pair is a value only if, say, the first component is a value, but without regard to the second. Similar questions arise with sums (all injections are values, or only in- jections of values are values), recursive types (all folds are values, or only folds of values are values), and function types (functions should be called by-name or by-value). Whole languages are built around adherence to one policy or another. For example, Haskell decrees that products, sums, and recursive types are to be lazy, and functions are to be called by name, whereas ML decrees the exact opposite policy. Not only are these choices arbitrary, but it is also unclear why they should be linked. For example, we could very sensibly decree that products, sums, and recursive types are lazy, yet impose a call-by-value discipline on functions. Or we could have eager products, sums, and recursive types, yet insist on call-by-name. It is not at all clear which of these points in the space of choices is right; each has its adherents, and each has its detractors. Are we therefore stuck in a tarpit of subjectivity? No! The way out is to recognize that these distinctions should not be imposed by the language designer, but rather are choices that are to be made by the programmer. This may be achieved by recognizing that differences in dynamics reflect fundamental type distinctions that are being obscured by languages that im- 380 38.1 Positive and Negative Types pose one policy or another. We can have both eager and lazy pairs in the same language by simply distinguishing them as two distinct types, and similarly we can have both eager and lazy sums in the same language, and both by-name and by-value function spaces, by providing sufficient type distinctions as to make the choice available to the programmer. Eager and lazy types are distinguished by their polarity, which is either positive or negative according to whether the type is defined by the values that inhabit the type or the behavior of expressions of that type. Positive types are eager, or inductive, in that they are defined by their values. Neg- ative types are lazy, or coinductive, in that they are defined by the behavior of their elements. The polarity of types is made explicit using a technique called focusing. A focused presentation of a programming language dis- tinguishes three general forms of expression, (positive and negative) values, (positive and negative) continuations, and (neutral) computations. 38.1 Positive and Negative Types Polarization consists of distinguishing positive from negative types accord- ing to the following two principles: 1. A positive type is defined by its introduction rules, which specify the values of that type in terms of other values. The elimination rules are inversions that specify a computation by pattern matching on values of that type. 2. A negative type is defined by its elimination rules, which specify the observations that may be performed on elements of that type. The introduction rules specify the values of that type by specifying how they respond to observations. Based on this characterization we can anticipate that the type of natural numbers would be positive, because it is defined by zero and successor, whereas function types would be negative, because they are characterized by their behavior when applied, and not by their internal structure. The language L±{nat*} is a polarized formulation of L{nat *}. The syntax of types in this language is given by the following grammar: PTyp τ+ ::= dn(τ−) ↓ τ− suspension nat nat naturals NTyp τ− ::= up(τ+) ↑ τ+ inclusion parr(τ+ 1 ; τ− 2 ) τ+ 1 * τ− 2 partial function VERSION 1.32 REVISED 05.15.2012 38.2 Focusing 381 The types ↓ τ− and ↑ τ+ effect a polarity shift from negative to positive and positive to negative, respectively. Intuitively, the shifted type ↑ τ+ is just the inclusion of positive into negative values, whereas the shifted type ↓ τ− represents the type of suspended computations of negative type. The domain of the negative function type is required to be positive, but its range is negative. This allows us to form right-iterated function types τ+ 1 *(τ+ 2 *(...(τ+ n−1 * τ− n ))) directly, but to form a left-iterated function type requires shifting, ↓ (τ+ 1 * τ− 2 )* τ−, to turn the negative function type into a positive type. Conversely, shifting is needed to define a function whose range is positive, τ+ 1 * ↑ τ+ 2 . 38.2 Focusing The syntax of L±{nat*} is motivated by the polarization of its types. For each polarity we have a sort of values and a sort of continuations with which we may create (neutral) computations. PVal v+ ::= z z zero s(v+) s(v+) successor del-(e) del-(e) delay PCont k+ ::= ifz(e0; x.e1) ifz(e0; x.e1) conditional force-(k−) force-(k−) evaluate NVal v− ::= lam[τ+](x.e) λ (x:τ+) e abstraction del+(v+) del+(v+) inclusion fix(x.v−) fix x is v− recursion NCont k− ::= ap(v+; k−) ap(v+; k−) application force+(x.e) force+(x.e) evaluate Comp e ::= ret(v−) ret(v−) return cut+(v+; k+) v+ . k+ cut cut-(v−; k−) v− . k− cut The positive values include the numerals, and the negative values include functions. In addition we may delay a computation of a negative value to form a positive value using del-(e), and we may consider a positive value to be a negative value using del+(v+). The positive continuations include REVISED 05.15.2012 VERSION 1.32 382 38.3 Statics the conditional branch, sans argument, and the negative continuations in- clude application sites for functions consisting of a positive argument value and a continuation for the negative result. In addition we include positive continuations to force the computation of a suspended negative value, and to extract an included positive value. Computations, which correspond to machine states, consist of returned negative values (these are final states), states passing a positive value to a positive continuation, and states pass- ing a negative value to a negative continuation. General recursion appears as a form of negative value; the recursion is unrolled when it is made the subject of an observation. 38.3 Statics The statics of L±{nat*} consists of a collection of rules for deriving judg- ments of the following forms: • Positive values: Γ ` v+ : τ+. • Positive continuations: Γ ` k+ : τ+ > γ−. • Negative values: Γ ` v− : τ−. • Negative continuations: Γ ` k− : τ− > γ−. • Computations: Γ ` e : γ−. Throughout Γ is a finite set of hypotheses of the form x1 : τ+ 1 ,..., xn : τ+ n , for some n ≥ 0, and γ− is any negative type. The typing rules for continuations specify both an argument type (on which values they act) and a result type (of the computation resulting from the action on a value). The typing rules for computations specify that the outcome of a computation is a negative type. All typing judgments specify that variables range over positive types. (These restrictions may always be met by appropriate use of shifting.) The statics of positive values consists of the following rules: Γ, x : τ+ ` x : τ+ (38.1a) Γ ` z : nat (38.1b) VERSION 1.32 REVISED 05.15.2012 38.3 Statics 383 Γ ` v+ : nat Γ ` s(v+): nat (38.1c) Γ ` e : τ− Γ ` del-(e): ↓ τ− (38.1d) Rule (38.1a) specifies that variables range over positive values. Rules (38.1b) and (38.1c) specify that the values of type nat are just the numerals. Rule (38.1d) specifies that a suspended computation (necessarily of negative type) is a positive value. The statics of positive continuations consists of the following rules: Γ ` e0 : γ− Γ, x : nat ` e1 : γ− Γ ` ifz(e0; x.e1): nat > γ− (38.2a) Γ ` k− : τ− > γ− Γ ` force-(k−): ↓ τ− > γ− (38.2b) Rule (38.2a) governs the continuation that chooses between two computa- tions according to whether a natural number is zero or non-zero. Rule (38.2b) specifies the continuation that forces a delayed computation with the spec- ified negative continuation. The statics of negative values is defined by these rules: Γ, x : τ+ 1 ` e : τ− 2 Γ ` λ (x:τ+ 1 ) e : τ+ 1 * τ− 2 (38.3a) Γ ` v+ : τ+ Γ ` del+(v+): ↑ τ+ (38.3b) Γ, x : ↓ τ− ` v− : τ− Γ ` fix x is v− : τ− (38.3c) Rule (38.3a) specifies the statics of a λ-abstraction whose argument is a pos- itive value, and whose result is a computation of negative type. Rule (38.3b) specifies the inclusion of positive values as negative values. Rule (38.3c) specifies that negative types admit general recursion. The statics of negative continuations is defined by these rules: Γ ` v+ 1 : τ+ 1 Γ ` k− 2 : τ− 2 > γ− Γ ` ap(v+ 1 ; k− 2 ): τ+ 1 * τ− 2 > γ− (38.4a) Γ, x : τ+ ` e : γ− Γ ` force+(x.e): ↑ τ+ > γ− (38.4b) REVISED 05.15.2012 VERSION 1.32 384 38.4 Dynamics Rule (38.4a) is the continuation representing the application of a function to the positive argument, v+ 1 , and executing the body with negative continua- tion, k− 2 . Rule (38.4b) specifies the continuation that passes a positive value, viewed as a negative value, to a computation. The statics of computations is given by these rules: Γ ` v− : τ− Γ ` ret(v−): τ− (38.5a) Γ ` v+ : τ+ Γ ` k+ : τ+ > γ− Γ ` v+ . k+ : γ− (38.5b) Γ ` v− : τ− Γ ` k− : τ− > γ− Γ ` v− . k− : γ− (38.5c) Rule (38.5a) specifies the basic form of computation that simply returns the negative value v−. Rules (38.5b) and (38.5c) specify computations that pass a value to a continuation of appropriate polarity. 38.4 Dynamics The dynamics of L±{nat*} is given by a transition system e 7→ e0 speci- fying the steps of computation. The rules are all axioms; no premises are required because the continuation is used to manage pending computa- tions. The dynamics consists of the following rules: z . ifz(e0; x.e1) 7→ e0 (38.6a) s(v+). ifz(e0; x.e1) 7→ [v+/x]e1 (38.6b) del-(e). force-(k−) 7→ e ; k− (38.6c) λ (x:τ+) e . ap(v+; k−) 7→ [v+/x]e ; k− (38.6d) del+(v+). force+(x.e) 7→ [v+/x]e (38.6e) fix x is v− . k− 7→ [del-(fix x is v−)/x]v− . k− (38.6f) These rules specify the interaction between values and continuations. Rules (38.6) make use of two forms of substitution, [v+/x]e and [v+/x]v−, which are defined as in Chapter1. They also employ a new form of com- position, written e ; k− 0 , which composes a computation with a continuation VERSION 1.32 REVISED 05.15.2012 38.5 Safety 385 by attaching k− 0 to the end of the computation specified by e. This composi- tion is defined mutually recursive with the compositions k+ ; k− 0 and k− ; k− 0 , which essentially concatenate continuations (stacks). ret(v−); k− 0 = v− . k− 0 (38.7a) k− ; k− 0 = k− 1 (v− . k−); k− 0 = v− . k− 1 (38.7b) k+ ; k− 0 = k+ 1 (v+ . k+); k− 0 = v+ . k+ 1 (38.7c) e0 ; k− = e0 0 x | e1 ; k− = e0 1 ifz(e0; x.e1); k− = ifz(e0 0; x.e0 1)(38.7d) k− ; k− 0 = k− 1 force-(k−); k− 0 = force-(k− 1 )(38.7e) k− ; k− 0 = k1 ap(v+; k−); k− 0 = ap(v+; k− 1 )(38.7f) x | e ; k− 0 = e0 force+(x.e); k− 0 = force+(x.e0)(38.7g) Rules (38.7d) and (38.7g) make use of the generic hypothetical judgment defined in Chapter3 to express that the composition is defined uniformly in the bound variable. 38.5 Safety The proof of preservation for L±{nat*} reduces to the proof of the typing properties of substitution and composition. Lemma 38.1 (Substitution). Suppose that Γ ` v+ : ρ+. 1. If Γ, x : ρ+ ` e : γ−, then Γ ` [v+/x]e : γ−. 2. If Γ, x : ρ+ ` v− : τ−, then Γ ` [v+/x]v− : τ−. 3. If Γ, x : ρ+ ` k+ : τ+ > γ−, then Γ ` [v+/x]k+ : τ+ > γ−. 4. If Γ, x : ρ+ ` v+ 1 : τ+, then Γ ` [v+/x]v+ 1 : τ+. 5. If Γ, x : ρ+ ` k− : τ− > γ−, then Γ ` [v+/x]k− : τ− > γ−. REVISED 05.15.2012 VERSION 1.32 386 38.6 Notes Proof. Simultaneously, by induction on the derivation of the typing of the target of the substitution. Lemma 38.2 (Composition). 1. If Γ ` e : τ− and Γ ` k− : τ− > γ−, then Γ ` e ; k− : τ− > γ−. 2. If Γ ` k+ 0 : τ+ > γ− 0 , and Γ ` k− 1 : γ− 0 > γ− 1 , then Γ ` k+ 0 ; k− 1 : τ+ > γ− 1 . 3. If Γ ` k− 0 : τ− > γ− 0 , and Γ ` k− 1 : γ− 0 > γ− 1 , then Γ ` k− 0 ; k− 1 : τ− > γ− 1 . Proof. Simultaneously, by induction on the derivations of the first premises of each clause of the lemma. Theorem 38.3 (Preservation). If Γ ` e : γ− and e 7→ e0, then Γ ` e0 : γ−. Proof. By induction on transition, appealing to inversion for typing and Lemmas 38.1 and 38.2. The progress theorem reduces to the characterization of the values of each type. Focusing makes the required properties evident, because it de- fines directly the values of each type. Theorem 38.4 (Progress). If Γ ` e : γ−, then either e = ret(v−) for some v−, or there exists e0 such that e 7→ e0. 38.6 Notes The concept of polarization originates with Andreoli(1992), which intro- duced focusing as a technique for proof search in linear logic. The formula- tion given here is inspired by Zeilberger(2008), wherein focusing is related to evaluation order in programming languages. VERSION 1.32 REVISED 05.15.2012 Part XV Parallelism Chapter 39 Nested Parallelism Parallel computation seeks to reduce the running times of programs by al- lowing many computations to be carried out simultaneously. For example, if we wish to add two numbers, each given by a complex computation, we may consider evaluating the addends simultaneously, then computing their sum. The ability to exploit parallelism is limited by the dependencies among parts of a program. Obviously, if one computation depends on the result of another, then we have no choice but to execute them sequentially so that we may propagate the result of the first to the second. Consequently, the fewer dependencies among sub-computations, the greater the opportu- nities for parallelism. This argues for functional models of computation, because the possibility of mutation of shared assignables imposes sequen- tialization constraints on imperative code. In this chapter we discuss nested parallelism in which we nest parallel computations within one another in a hierarchical manner. Nested paral- lelism is sometimes called fork-join parallelism to emphasize the hierarchi- cal structure arising from forking two (or more) parallel computations, then joining these computations to combine their results before proceeding. We will consider two forms of dynamics for nested parallelism. The first is a structural dynamics in which a single transition on a compound expression may involve multiple transitions on its constituent expressions. The second is a cost dynamics (introduced in Chapter7) that focuses attention on the sequential and parallel complexity (also known as the work and depth) of a parallel program by associating a series-parallel graph with each computa- tion. 390 39.1 Binary Fork-Join 39.1 Binary Fork-Join We begin with a parallel language whose sole source of parallelism is the simultaneous evaluation of two variable bindings. This is modelled by a construct of the form par x1 = e1 and x2 = e2 in e, in which we bind two vari- ables, x1 and x2, to two expressions, e1 and e2, respectively, for use within a single expression, e. This represents a simple fork-join primitive in which e1 and e2 may be evaluated independently of one another, with their re- sults combined by the expression e. Some other forms of parallelism may be defined in terms of this primitive. As an example, parallel pairing may be defined as the expression par x1 = e1 and x2 = e2 in hx1, x2i, which evaluates the components of the pair in parallel, then constructs the pair itself from these values. The abstract syntax of the parallel binding construct is given by the ab- stract binding tree par(e1; e2; x1.x2.e), which makes clear that the variables x1 and x2 are bound only within e, and not within their bindings. This ensures that evaluation of e1 is independent of evaluation of e2, and vice versa. The typing rule for an expression of this form is given as follows: Γ ` e1 : τ1 Γ ` e2 : τ2 Γ, x1 : τ1, x2 : τ2 ` e : τ Γ ` par(e1; e2; x1.x2.e): τ (39.1) Although we emphasize the case of binary parallelism, it should be clear that this construct easily generalizes to n-way parallelism for any static value of n. We may also define an n-way parallel let construct from the binary parallel let by cascading binary splits. (For a treatment of n-way parallelism for a dynamic value of n, see Section 39.3.) We will give both a sequential and a parallel dynamics of the parallel let construct. The definition of the sequential dynamics as a transition judgment of the form e 7→seq e0 is entirely straightforward: e1 7→ e0 1 par(e1; e2; x1.x2.e) 7→seq par(e0 1; e2; x1.x2.e)(39.2a) e1 val e2 7→ e0 2 par(e1; e2; x1.x2.e) 7→seq par(e1; e0 2; x1.x2.e)(39.2b) VERSION 1.32 REVISED 05.15.2012 39.1 Binary Fork-Join 391 e1 val e2 val par(e1; e2; x1.x2.e) 7→seq [e1, e2/x1, x2]e (39.2c) The parallel dynamics is given by a transition judgment of the form e 7→par e0, defined as follows: e1 7→par e0 1 e2 7→par e0 2 par(e1; e2; x1.x2.e) 7→par par(e0 1; e0 2; x1.x2.e)(39.3a) e1 7→par e0 1 e2 val par(e1; e2; x1.x2.e) 7→par par(e0 1; e2; x1.x2.e)(39.3b) e1 val e2 7→par e0 2 par(e1; e2; x1.x2.e) 7→par par(e1; e0 2; x1.x2.e)(39.3c) e1 val e2 val par(e1; e2; x1.x2.e) 7→par [e1, e2/x1, x2]e (39.3d) The parallel dynamics is idealized in that it abstracts away from any limi- tations on parallelism that would necessarily be imposed in practice by the availability of computing resources. An important advantage of the present approach is captured by the im- plicit parallelism theorem, which states that the sequential and the parallel dynamics coincide. This means that we need never be concerned with the semantics of a parallel program (its meaning is determined by the sequen- tial dynamics), but only with its efficiency. As a practical matter, this means that a program may be developed on a sequential platform, even if it is in- tended to run on a parallel platform, because the behavior is not affected by whether we execute it using a sequential or a parallel dynamics. Because the sequential dynamics is deterministic (every expression has at most one value), the implicit parallelism theorem implies that the paral- lel dynamics is also deterministic. For this reason the implicit parallelism theorem is also known as the deterministic parallelism theorem. This clearly distinguishes deterministic parallelism, the subject of this chapter, from non- deterministic concurrency, the subject of Chapters 41 and 42. A proof of the implicit parallelism theorem may be given by giving an evaluation dynamics, e ⇓ v, in the style of Chapter7, and showing that e 7→∗ par v iff e ⇓ v iff e 7→∗ seq v (where v is a closed expression such that v val). The crucial rule of the evaluation dynamics is the one governing the parallel let construct: e1 ⇓ v1 e2 ⇓ v2 [v1, v2/x1, x2]e ⇓ v par(e1; e2; x1.x2.e) ⇓ v (39.4) REVISED 05.15.2012 VERSION 1.32 392 39.1 Binary Fork-Join It is easy to show that the sequential dynamics agrees with the evalua- tion dynamics by a straightforward extension of the proof of Theorem 7.2. Lemma 39.1. e 7→∗ seq v iff e ⇓ v. Proof. It suffices to show that if e 7→seq e0 and e0 ⇓ v, then e ⇓ v, and that if e1 7→∗ seq v1 and e2 7→∗ seq v2 and [v1, v2/x1, x2]e 7→∗ seq v, then par x1 = e1 and x2 = e2 in e 7→∗ seq v. By a similar argument we may show that the parallel dynamics also agrees with the evaluation dynamics, and hence with the sequential dy- namics. Lemma 39.2. e 7→∗ par v iff e ⇓ v. Proof. It suffices to show that if e 7→par e0 and e0 ⇓ v, then e ⇓ v, and that if e1 7→∗ par v1 and e2 7→∗ par v2 and [v1, v2/x1, x2]e 7→∗ par v, then par x1 = e1 and x2 = e2 in e 7→∗ par v. The proof of the first is by a straightforward induction on the parallel dy- namics. The proof of the second proceeds by simultaneous induction on the derivations of e1 7→∗ par v1 and e2 7→∗ par v2. If e1 = v1 with v1 val and e2 = v2 with v2 val, then the result follows immediately from the third premise. If e2 = v2 but e1 7→par e0 1 7→∗ par v1, then by induction we have that par x1 = e0 1 and x2 = v2 in e 7→∗ par v, and hence the result follows by an application of Rule (39.3b). The symmetric case follows similarly by an ap- plication of Rule (39.3c), and in the case that both e1 and e2 take a step, the result follows by induction and Rule (39.3a). Theorem 39.3 (Implicit Parallelism). The sequential and parallel dynamics co- incide: for all v val, e 7→∗ seq v iff e 7→∗ par v. Proof. By Lemmas 39.1 and 39.2. The implicit parallelism theorem states that parallelism does not affect the semantics of a program, only the efficiency of its execution. Correctness concerns are factored out, focusing attention on complexity. VERSION 1.32 REVISED 05.15.2012 39.2 Cost Dynamics 393 39.2 Cost Dynamics In this section we define a parallel cost dynamics that assigns a cost graph to the evaluation of an expression. Cost graphs are defined by the following grammar: Cost c ::= 0 zero cost 1 unit cost c1 ⊗ c2 parallel combination c1 ⊕ c2 sequential combination A cost graph is a form of series-parallel directed acyclic graph, with a des- ignated source node and sink node. For 0 the graph consists of one node and no edges, with the source and sink both being the node itself. For 1 the graph consists of two nodes and one edge directed from the source to the sink. For c1 ⊗ c2, if g1 and g2 are the graphs of c1 and c2, respectively, then the graph has two additional nodes, a source node with two edges to the source nodes of g1 and g2, and a sink node, with edges from the sink nodes of g1 and g2 to it. Finally, for c1 ⊕ c2, where g1 and g2 are the graphs of c1 and c2, the graph has as source node the source of g1, as sink node the sink of g2, and an edge from the sink of g1 to the source of g2. The intuition behind a cost graph is that nodes represent subcompu- tations of an overall computation, and edges represent sequentiality con- straints stating that one computation depends on the result of another, and hence cannot be started before the one on which it depends completes. The product of two graphs represents parallelism opportunities in which there are no sequentiality constraints between the two computations. The assign- ment of source and sink nodes reflects the overhead of forking two parallel computations and joining them after they have both completed. We associate with each cost graph two numeric measures, the work, wk(c), and the depth, dp(c). The work is defined by the following equa- tions: wk(c) =    0 if c = 0 1 if c = 1 wk(c1) + wk(c2) if c = c1 ⊗ c2 wk(c1) + wk(c2) if c = c1 ⊕ c2 (39.5) REVISED 05.15.2012 VERSION 1.32 394 39.2 Cost Dynamics The depth is defined by the following equations: dp(c) =    0 if c = 0 1 if c = 1 max(dp(c1), dp(c2)) if c = c1 ⊗ c2 dp(c1) + dp(c2) if c = c1 ⊕ c2 (39.6) Informally, the work of a cost graph determines the total number of com- putation steps represented by the cost graph, and thus corresponds to the sequential complexity of the computation. The depth of the cost graph de- termines the critical path length, the length of the longest dependency chain within the computation, which imposes a lower bound on the parallel com- plexity of a computation. The critical path length is the least number of sequential steps that can be taken, even if we have unlimited parallelism available to us, because of steps that can be taken only after the completion of another. In Chapter7 we introduced cost dynamics as a means of assigning time complexity to evaluation. The proof of Theorem 7.7 shows that e ⇓k v iff e 7→k v. That is, the step complexity of an evaluation of e to a value v is just the number of transitions required to derive e 7→∗ v. Here we use cost graphs as the measure of complexity, then relate these cost graphs to the structural dynamics given in Section 39.1. The judgment e ⇓c v, where e is a closed expression, v is a closed value, and c is a cost graph specifies the cost dynamics. By definition we arrange that e ⇓0 e when e val. The cost assignment for let is given by the following rule: e1 ⇓c1 v1 e2 ⇓c2 v2 [v1, v2/x1, x2]e ⇓c v par(e1; e2; x1.x2.e) ⇓(c1⊗c2)⊕1⊕c v (39.7) The cost assignment specifies that, under ideal conditions, e1 and e2 are to be evaluated in parallel, and that their results are to be propagated to e. The cost of fork and join is implicit in the parallel combination of costs, and assign unit cost to the substitution because we expect it to be implemented in practice by a constant-time mechanism for updating an environment. The cost dynamics of other language constructs is specified in a similar manner, using only sequential combination so as to isolate the source of parallelism to the let construct. Two simple facts about the cost dynamics are important to keep in mind. First, the cost assignment does not influence the outcome. Lemma 39.4. e ⇓ v iff e ⇓c v for some c. VERSION 1.32 REVISED 05.15.2012 39.2 Cost Dynamics 395 Proof. From right to left, erase the cost assignments to obtain an evalua- tion derivation. From left to right, decorate the evaluation derivations with costs as determined by the rules defining the cost dynamics. Second, the cost of evaluating an expression is uniquely determined. Lemma 39.5. If e ⇓c v and e ⇓c0 v, then c is c0. Proof. A routine induction on the derivation of e ⇓c v. The link between the cost dynamics and the structural dynamics given in the preceding section is established by the following theorem, which states that the work cost is the sequential complexity, and the depth cost is the parallel complexity, of the computation. Theorem 39.6. If e ⇓c v, then e 7→wseq v and e 7→dpar v, where w = wk(c) and d = dp(c). Conversely, if e 7→wseq v, then there exists c such that e ⇓c v with wk(c) = w, and if e 7→dpar v0, then there exists c0 such that e ⇓c0 v0 with dp(c0) = d. Proof. The first part is proved by induction on the derivation of e ⇓c v, the interesting case being Rule (39.7). By induction we have e1 7→w1seq v1, e2 7→w2seq v2, and [v1, v2/x1, x2]e 7→wseq v, where w1 = wk(c1), w2 = wk(c2), and w = wk(c). By pasting together derivations we obtain a derivation par(e1; e2; x1.x2.e) 7→w1seq par(v1; e2; x1.x2.e) 7→w2seq par(v1; v2; x1.x2.e) 7→seq [v1, v2/x1, x2]e 7→w seq v. Noting that wk((c1 ⊗ c2) ⊕ 1 ⊕ c) = w1 + w2 + 1 + w completes the proof. Similarly, we have by induction that e1 7→d1par v1, e2 7→d2par v2, and e 7→dpar v, where d1 = dp(c1), d2 = dp(c2), and d = dp(c). Assume, without loss of generality, that d1 ≤ d2 (otherwise simply swap the roles of d1 and d2 in what follows). We may paste together derivations as follows: par(e1; e2; x1.x2.e) 7→d1par par(v1; e0 2; x1.x2.e) 7→d2−d1par par(v1; v2; x1.x2.e) 7→par [v1, v2/x1, x2]e 7→d par v. REVISED 05.15.2012 VERSION 1.32 396 39.3 Multiple Fork-Join Calculating dp((c1 ⊗ c2) ⊕ 1 ⊕ c) = max(d1, d2) + 1 + d completes the proof. Turning to the second part, it suffices to show that if e 7→seq e0 with e0 ⇓c0 v, then e ⇓c v with wk(c) = wk(c0) + 1, and if e 7→par e0 with e0 ⇓c0 v, then e ⇓c v with dp(c) = dp(c0) + 1. Suppose that e = par(e1; e2; x1.x2.e0) with e1 val and e2 val. Then e 7→seq e0, where e = [e1, e2/x1, x2]e0 and there exists c0 such that e0 ⇓c0 v. But then e ⇓c v, where c = (0 ⊗ 0) ⊕ 1 ⊕ c0, and a simple calculation shows that wk(c) = wk(c0) + 1, as required. Similarly, e 7→par e0 for e0 as above, and hence e ⇓c v for some c such that dp(c) = dp(c0) + 1, as required. Suppose that e = par(e1; e2; x1.x2.e0) and e 7→seq e0, where e0 = par(e0 1; e2; x1.x2.e0) and e1 7→seq e0 1. From the assumption that e0 ⇓c0 v, we have by inversion that e0 1 ⇓c0 1 v1, e2 ⇓c0 2 v2, and [v1, v2/x1, x2]e0 ⇓c0 0 v, with c0 = (c0 1 ⊗ c0 2) ⊕ 1 ⊕ c0 0. By induction there exists c1 such that wk(c1) = 1 + wk(c0 1) and e1 ⇓c1 v1. But then e ⇓c v, with c = (c1 ⊗ c0 2) ⊕ 1 ⊕ c0 0. By a similar argument, suppose that e = par(e1; e2; x1.x2.e0) and e 7→par e0, where e0 = par(e0 1; e0 2; x1.x2.e0) and e1 7→par e0 1, e2 7→par e0 2, and e0 ⇓c0 v. Then by inversion e0 1 ⇓c0 1 v1, e0 2 ⇓c0 2 v2,[v1, v2/x1, x2]e0 ⇓c0 v. But then e ⇓c v, where c = (c1 ⊗ c2) ⊕ 1 ⊕ c0, e1 ⇓c1 v1 with dp(c1) = 1 + dp(c0 1), e2 ⇓c2 v2 with dp(c2) = 1 + dp(c0 2), and [v1, v2/x1, x2]e0 ⇓c0 v. Calculating, we obtain dp(c) = max(dp(c0 1) + 1, dp(c0 2) + 1) + 1 + dp(c0) = max(dp(c0 1), dp(c0 2)) + 1 + 1 + dp(c0) = dp((c0 1 ⊗ c0 2) ⊕ 1 ⊕ c0) + 1 = dp(c0) + 1, which completes the proof. Corollary 39.7. If e 7→wseq v and e 7→dpar v0, then v is v0 and e ⇓c v for some c such that wk(c) = w and dp(c) = d. 39.3 Multiple Fork-Join So far we have confined attention to binary fork/join parallelism induced by the parallel let construct. Although technically sufficient for many pur- poses, a more natural programming model admits an unbounded number of parallel tasks to be spawned simultaneously, rather than forcing them to be created by a cascade of binary forks and corresponding joins. Such a model, often called data parallelism, ties the source of parallelism to a data VERSION 1.32 REVISED 05.15.2012 39.3 Multiple Fork-Join 397 structure of unbounded size. The principal example of such a data struc- ture is a sequence of values of a specified type. The primitive operations on sequences provide a natural source of unbounded parallelism. For exam- ple, we may consider a parallel map construct that applies a given function to every element of a sequence simultaneously, forming a sequence of the results. We will consider here a simple language of sequence operations to il- lustrate the main ideas. Typ τ ::= seq(τ) τ seq sequence Exp e ::= seq(e0,...,en−1)[e0,...,en−1] sequence len(e) |e| size sub(e1; e2) e1[e2] element tab(x.e1; e2) tab(x.e1; e2) tabulate map(x.e1; e2)[e1 | x ∈ e2] map cat(e1; e2) cat(e1; e2) concatenate The expression seq(e0,...,en−1) evaluates to an n-sequence whose ele- ments are given by the expressions e0,..., en−1. The operation len(e) re- turns the number of elements in the sequence given by e. The operation sub(e1; e2) retrieves the element of the sequence given by e1 at the index given by e2. The tabulate operation, tab(x.e1; e2), yields the sequence of length given by e2 whose ith element is given by [i/x]e1. The opera- tion map(x.e1; e2) computes the sequence whose ith element is given by [e/x]e1, where e is the ith element of the sequence given by e2. The opera- tion cat(e1; e2) concatenates two sequences of the same type. The statics of these operations is given by the following typing rules: Γ ` e0 : τ ...Γ ` en−1 : τ Γ ` seq(e0,...,en−1): seq(τ)(39.8a) Γ ` e : seq(τ) Γ ` len(e): nat (39.8b) Γ ` e1 : seq(τ)Γ ` e2 : nat Γ ` sub(e1; e2): τ (39.8c) Γ, x : nat ` e1 : τ Γ ` e2 : nat Γ ` tab(x.e1; e2): seq(τ)(39.8d) Γ ` e2 : seq(τ)Γ, x : τ ` e1 : τ0 Γ ` map(x.e1; e2): seq(τ0)(39.8e) Γ ` e1 : seq(τ)Γ ` e2 : seq(τ) Γ ` cat(e1; e2): seq(τ)(39.8f) REVISED 05.15.2012 VERSION 1.32 398 39.4 Provably Efficient Implementations The cost dynamics of these constructs is defined by the following rules: e0 ⇓c0 v0 ... en−1 ⇓cn−1 vn−1 seq(e0,...,en−1) ⇓Nn−1 i=0 ci seq(v0,...,vn−1)(39.9a) e ⇓c seq(v0,...,vn−1) len(e) ⇓c⊕1 num[n](39.9b) e1 ⇓c1 seq(v0,...,vn−1) e2 ⇓c2 num[i](0 ≤ i < n) sub(e1; e2) ⇓c1⊕c2⊕1 vi (39.9c) e2 ⇓c num[n][num[0]/x]e1 ⇓c0 v0 ...[num[n − 1]/x]e1 ⇓cn−1 vn−1 tab(x.e1; e2) ⇓c⊕Nn−1 i=0 ci seq(v0,...,vn−1) (39.9d) e2 ⇓c seq(v0,...,vn−1) [v0/x]e1 ⇓c0 v0 0 ...[vn−1/x]e1 ⇓cn−1 v0 n−1 map(x.e1; e2) ⇓c⊕Nn−1 i=0 ci seq(v0 0,...,v0 n−1) (39.9e) e1 ⇓c1 seq(v0,..., vm−1) e2 ⇓c2 seq(v0 0,..., v0 n−1) cat(e1; e2) ⇓c1⊕c2⊕Nm+n−1 i=0 1 seq(v0,..., vm−1, v0 0,..., v0 n−1) (39.9f) The cost dynamics for sequence operations may be validated by intro- ducing a sequential and parallel cost dynamics and extending the proof of Theorem 39.6 to cover this extension. 39.4 Provably Efficient Implementations Theorem 39.6 states that the cost dynamics accurately models the dynamics of the parallel let construct, whether executed sequentially or in parallel. This validates the cost dynamics from the point of view of the dynamics of the language, and permits us to draw conclusions about the asymptotic complexity of a parallel program that abstracts away from the limitations imposed by a concrete implementation. Chief among these is the restriction to a fixed number, p > 0, of processors on which to schedule the workload. In addition to limiting the available parallelism this also imposes some syn- chronization overhead that must be accounted for in order to make accurate predictions of run-time behavior on a concrete parallel platform. A provably efficient implementation is one for which we may establish an asymptotic bound on the actual execution time once these overheads are taken into account. VERSION 1.32 REVISED 05.15.2012 39.4 Provably Efficient Implementations 399 A provably efficient implementation must take account of the limita- tions and capabilities of the actual hardware on which the program is to be run. Because we are only interested in asymptotic upper bounds, it is convenient to formulate an abstract machine model, and to show that the primitives of the language can be implemented on this model with guar- anteed time (and space) bounds. One popular model is the SMP, or shared- memory multiprocessor, which consists of p > 0 sequential processors co- ordinated by an interconnect network that provides constant-time access to shared memory by each of the processors.1 The multiprocessor is as- sumed to provide a constant-time synchronization primitive with which to control simultaneous access to a memory cell. There are a variety of such primitives, any of which is sufficient to provide a parallel fetch-and-add instruction that allows each processor to obtain the current contents of a memory cell and update it by adding a fixed constant in a single atomic operation—the interconnect serializes any simultaneous accesses by more than one processor. Building a provably efficient implementation of parallelism involves two majors tasks. First, we must show that each of the primitives of the language may be implemented efficiently on the abstract machine model. Second, we must show how to schedule the workload across the proces- sors so as to minimize execution time by maximizing parallelism. When working with a low-level machine model such as an SMP, both tasks in- volve a fair bit of technical detail to show how to use low-level machine instructions, including a synchronization primitive, to implement the lan- guage primitives and to schedule the workload. Collecting these together, we may then give an asymptotic bound on the time complexity of the im- plementation that relates the abstract cost of the computation to cost of implementing the workload on a p-way multiprocessor. The prototypical result of this kind is called Brent’s Theorem. Theorem 39.8. If e ⇓c v with wk(c) = w and dp(c) = d, then e may be evaluated on a p-processor SMP in time O(max(w/p, d)). The theorem tells us that we can never execute a program in fewer steps than its depth, d, and that, at best, we can divide the work up evenly into w/p rounds of execution by the p processors. Observe that if p = 1 then the theorem establishes an upper bound of O(w) steps, the sequential com- plexity of the computation. Moreover, if the work is proportional to the 1A slightly weaker assumption is that each access may require up to lg p time to account for the overhead of synchronization, but we shall neglect this refinement in the present, simplified account. REVISED 05.15.2012 VERSION 1.32 400 39.4 Provably Efficient Implementations depth, then we are unable to exploit parallelism, and the overall time is proportional to the work alone. This motivates the definition of a useful figure of merit, called the par- allelizability ratio, which is the ratio, w/d, of work to depth. If w/d  p, then the program is said to be parallelizable, because then w/p  d, and we may therefore reduce running time by using p processors at each step. If, on the other hand, the parallelizability ratio is a constant, then d will dominate w/p, and we will have little opportunity to exploit parallelism to reduce running time. It is not known, in general, whether a problem admits a parallelizable solution. The best we can say, on present knowledge, is that there are algorithms for some problems that have a high degree of paral- lelizability, and there are problems for which no such algorithm is known. It is a open problem in complexity theory to characterize which problems are parallelizable, and which are not. To illustrate the essential ingredients of the proof of Brent’s Theorem we will consider a dynamics that models the scheduling of work onto p par- allel processors, each of which implements the dynamics of L{nat *} as described in Chapter 10. The parallel dynamics is defined on states ν Σ{ µ } of the form ν a1 ∼ τ1,... an ∼ τn { a1 ,→ e1 ⊗ ... ⊗ an ,→ en }, where n ≥ 1. Such a state represents a computation that has been decom- posed into n parallel tasks. Each task is given a name. The occurrence of a name within a task represents a dependency of that task on the named task. A task is said to be blocked on the tasks on which it depends; a task with no dependencies is said to be ready. There are two forms of transition, the local and the global. The local transitions represent the steps of the individual processors. These consist of the steps of execution of expressions as defined in Chapter 10, augmented with transitions governing parallelism. The global transitions represent the simultaneous execution of local transitions on up to some fixed number, p, of processors. Local transitions have the form ν Σ a ∼ τ { µ ⊗ a ,→ e } 7→loc ν Σ0 a ∼ τ { µ0 ⊗ a ,→ e0 }, where e is ready, and either (a) Σ and Σ0 are empty, and µ and µ0 are empty; or (b) Σ and µ are empty, and Σ0 and µ0 declare the types and bindings of two distinct names; or (c) Σ0 and µ0 are both empty, and Σ and µ declare the types and bindings of two distinct names. These conditions correspond VERSION 1.32 REVISED 05.15.2012 39.4 Provably Efficient Implementations 401 to the three possible outcomes of a local step by a task: (a) the task takes a step of computation in the sense of Chapter 10; (b) the task forks two new tasks, and waits for their completion; (c) the task joins two parallel tasks that have completed execution. ν a ∼ τ { a ,→ (λ (x:τ2) e)(e2)} 7→loc ν a ∼ τ { a ,→ [e2/x]e }(39.10a)    ν a ∼ τ { a ,→ par(e1; e2; x1.x2.e)} 7→loc ν a1 ∼ τ1 a2 ∼ τ2 a ∼ τ { a1 ,→ e1 ⊗ a2 ,→ e2 ⊗ a ,→ par(a1; a2; x1.x2.e)}    (39.10b) e1 val e2 val    ν a1 ∼ τ1 a2 ∼ τ2 a ∼ τ { a1 ,→ e1 ⊗ a2 ,→ e2 ⊗ a ,→ par(a1; a2; x1.x2.e)} 7→loc ν a ∼ τ { a ,→ [e1, e2/x1, x2]e }    (39.10c) Rule (39.10a) illustrates one rule for the dynamics of the non-parallel as- pects of the language; additional rules are required to cover the other cases, allowing for nested uses of parallelism. Rule (39.10b) represents the cre- ation of two parallel tasks on which the executing task depends. The ex- pression par(a1; a2; x1.x2.e) is blocked on tasks a1 and a2, so that no local step applies to it. Rule (39.10c) synchronizes a task with the tasks on which it depends once their execution has completed; those tasks are no longer required, and are therefore eliminated from the state. Each global transition represents the simultaneous execution of one step of computation on each of up to p ≥ 1 processors. ν Σ1a1 ∼ τ1 { µ1 ⊗ a1 ,→ e1 } 7→loc ν Σ0 1a1 ∼ τ1 { µ0 1 ⊗ a1 ,→ e0 1 } ... ν Σnan ∼ τn { µn ⊗ an ,→ en } 7→loc ν Σ0 nan ∼ τn { µ0 n ⊗ an ,→ e0 n }    ν Σ0 Σ1 a1 ∼ τ1 ...Σn an ∼ τn { µ0 ⊗ µ1 ⊗ a1 ,→ e1 ⊗ ... ⊗ µn ⊗ an ,→ en } 7→glo ν Σ0 Σ0 1 a1 ∼ τ1 ...Σ0 n an ∼ τn { µ0 ⊗ µ0 1 ⊗ a1 ,→ e0 1 ⊗ ... ⊗ µ0 n ⊗ an ,→ e0 n }    (39.11) At each global step some number, 1 ≤ n ≤ p, of ready tasks are scheduled REVISED 05.15.2012 VERSION 1.32 402 39.5 Notes for execution.2 Because no two distinct tasks may depend on the same task, we may partition the n tasks so that each scheduled task is grouped with the tasks on which it depends as necessary for any local join step. Any local fork step introduces two fresh tasks that are added to the state as a result of the global transition; any local join step eliminates two tasks whose execution has completed. A subtle point is that it is implicit in our name binding conventions that the names of any created tasks are to be globally unique, even though they are locally created. In implementation terms this requires a synchronization step among the processors to ensure that task names are not acccidentally reused among the parallel processors. The proof of Brent’s Theorem for this high-level dynamics is now obvi- ous, provided only that the global scheduling steps are performed greedily so as to maximize the use of processors at each round. If, at each stage of a computation, there are p ready tasks, then the computation will complete in w/p steps, where w is the work complexity of the program. We may, however, be unable to make full use of all p processors at any given stage. This would only be because the dependencies among computations, which are reflected in the variable occurrences and in the definition of the depth complexity of the computation, inhibits parallelism to the extent that evalu- ation cannot complete in fewer than d rounds. This limitation is significant only to the extent that d is larger than w/p; otherwise, the overall time is bounded by w/p, making maximal use of all p processors. 39.5 Notes Parallelism should not be confused with concurrency. Parallelism is about efficiency, not semantics; the meaning of a program is independent of whether it is executed in parallel or not. Concurrency is about composition, not ef- ficiency; the meaning of a concurrent program is very weakly specified so that we may compose it with other programs without altering its mean- ing. This distinction, and the formulation of it given here, was pioneered by Blelloch(1990). The concept of a cost semantics and the idea of a prov- ably efficient implementation are derived from Blelloch and Greiner(1995, 1996a). 2The rule does not require that n be chosen as large as possible. A scheduler that always chooses the largest possible 1 ≤ n ≤ p is said to be greedy. VERSION 1.32 REVISED 05.15.2012 Chapter 40 Futures and Speculations A future is a computation whose evaluation is initiated in advance of any demand for its value. Like a suspension, a future represents a value that is to be determined later. Unlike a suspension, a future is always evaluated, regardless of whether its value is actually required. In a sequential setting futures are of little interest; a future of type τ is just an expression of type τ. In a parallel setting, however, futures are of interest because they provide a means of initiating a parallel computation whose result is not needed until (presumably) much later, by which time it will have been completed. The prototypical example of the use of futures is to implementing pipelin- ing, a method for overlapping the stages of a multistage computation to the fullest extent possible. This minimizes the latency caused by one stage waiting for the completion of a previous stage by allowing the two stages to proceed in parallel until such time as an explicit dependency is encoun- tered. Ideally, the computation of the result of an earlier stage is completed by the time a later stage requires it. At worst the later stage must be delayed until the earlier stage completes, incurring what is known as a pipeline stall. A speculation is a delayed computation whose result may or may not be needed for the overall computation to finish. The dynamics for specu- lations executes suspended computations in parallel with the main thread of computation, without regard to whether the value of the speculation is actually required by the main thread. If the value of the speculation is re- quired, then such a dynamics pays off, but if not, the effort to compute it is wasted. Futures are work efficient in that the overall work done by a computa- tion involving futures is no more than the work required by a sequential execution. Speculations, in contrast, are work inefficient in that speculative 404 40.1 Futures execution may be in vain—the overall computation may involve more steps than the work required to compute the result. For this reason speculation is a risky strategy for exploiting parallelism. It can make good use of avail- able resources, but perhaps only at the expense of doing more work than necessary! 40.1 Futures The syntax of futures is given by the following grammar: Typ τ ::= fut(τ) τ fut future Exp e ::= fut(e) fut(e) future fsyn(e) fsyn(e) synchronize The type τ fut is the type of futures of type τ. Futures are introduced by the expression fut(e), which schedules e for evaluation and returns a reference to it. Futures are eliminated by the expression fsyn(e), which synchronizes with the future referred to by e, returning its value. 40.1.1 Statics The statics of futures is given by the following rules: Γ ` e : τ Γ ` fut(e): fut(τ)(40.1a) Γ ` e : fut(τ) Γ ` fsyn(e): τ (40.1b) These rules are unsurprising, because futures add no new capabilities to the language beyond providing an opportunity for parallel evaluation. 40.1.2 Sequential Dynamics The sequential dynamics of futures is easily defined. Futures are evaluated eagerly; synchronization returns the value of the future. e val fut(e) val (40.2a) e 7→ e0 fut(e) 7→ fut(e0)(40.2b) VERSION 1.32 REVISED 05.15.2012 40.2 Speculations 405 e 7→ e0 fsyn(e) 7→ fsyn(e0)(40.2c) e val fsyn(fut(e)) 7→ e (40.2d) 40.2 Speculations The syntax of (non-recursive) speculations is given by the following gram- mar:1 Typ τ ::= spec(τ) τ spec speculation Exp e ::= spec(e) spec(e) speculate ssyn(e) ssyn(e) synchronize The type τ spec is the type of speculations of type τ. The introductory form, spec(e), creates a computation that may be speculatively evaluated, and the eliminatory form, ssyn(e), synchronizes with a speculation. 40.2.1 Statics The statics of speculations is given by the following rules: Γ ` e : τ Γ ` spec(e): spec(τ)(40.3a) Γ ` e : spec(τ) Γ ` ssyn(e): τ (40.3b) Thus, the statics for speculations as given by Rules (40.3) is essentially equivalent to the statics for futures given by Rules (40.1). 40.2.2 Sequential Dynamics The definition of the sequential dynamics of speculations is similar to that of futures, except that speculations are values. spec(e) val (40.4a) ssyn(spec(e)) 7→ e (40.4b) The only difference compared to a future is that synchronization with a speculation may proceed even if the speculated computation is not com- pleted. 1We confine ourselves to the non-recursive case to facilitate the comparison with futures. REVISED 05.15.2012 VERSION 1.32 406 40.3 Parallel Dynamics 40.3 Parallel Dynamics Futures are only interesting insofar as they admit a parallel dynamics that allows the computation of the future to proceed concurrently with some other computation. In this section we give a parallel dynamics of futures and speculation in which the creation, execution, and synchronization of tasks is made explicit. Interestingly, the parallel dynamics of futures and speculations is identical, except for the termination condition. Whereas futures require all tasks to be completed before termination, speculations may be abandoned before they are completed. For the sake of concision we will give the parallel dynamics of futures, remarking only where alterations must be made for the parallel dynamics of speculations. The parallel dynamics of futures relies on a modest extension to the language given in Section 40.1 to introduce names for tasks. Let Σ be a finite mapping assigning types to names. The expression fut[a] is a value referring to the outcome of task a. The statics of this expression is given by the following rule:2 Γ `Σ,a∼τ fut[a]: fut(τ)(40.5) Rules (40.1) carry over in the obvious way with Σ recording the types of the task names. States of the parallel dynamics have the form ν Σ{ e k µ }, where e is the focus of evaluation, and µ records the parallel futures (or speculations) that have been activated thus far in the computation. Formally, µ is a finite mapping assigning expressions to the task names declared in Σ. A state is well-formed according to the following rule: `Σ e : τ (∀a ∈ dom(Σ)) `Σ µ(a):Σ(a) ν Σ{ e k µ } ok (40.6) As discussed in Chapter 36 this rule admits self-referential and mutually referential futures. A more refined condition could as well be given that avoids circularities; we leave this as an exercise for the reader. The parallel dynamics is divided into two phases, the local phase, which defines the basic steps of evaluation of an expression, and the global phase, which executes all possible local steps in parallel. The local dynamics of 2A similar rule governs the analogous construct, spec[a], in the case of speculations. VERSION 1.32 REVISED 05.15.2012 40.3 Parallel Dynamics 407 futures is defined by the following rules:3 fut[a] valΣ,a∼τ (40.7a) ν Σ{ fut(e) k µ } 7→loc ν Σ, a ∼ τ { fut[a] k µ ⊗ a ,→ e }(40.7b) ν Σ{ e k µ } 7→loc ν Σ0 { e0 k µ0 } ν Σ{ fsyn(e) k µ } 7→loc ν Σ0 { fsyn(e0) k µ0 }(40.7c) e0 valΣ,a∼τ    ν Σ, a ∼ τ { fsyn(fut[a]) k µ ⊗ a ,→ e0 } 7→loc ν Σ, a ∼ τ { e0 k µ ⊗ a ,→ e0 }    (40.7d) Rule (40.7b) activates a future named a executing the expression e and re- turns a reference to it. Rule (40.7d) synchronizes with a future whose value has been determined. Note that a local transition always has the form ν Σ{ e k µ } 7→loc ν ΣΣ0 { e0 k µ ⊗ µ0 } where Σ0 is either empty or declares the type of a single symbol, and µ0 is either empty or of the form a ,→ e0 for some expression e0. A global step of the parallel dynamics consists of at most one local step for the focal expression and one local step for each of up to p futures, where p > 0 is a fixed parameter representing the number of processors. µ = µ0 ⊗ a1 ,→ e1 ⊗ ... ⊗ an ,→ en µ00 = µ0 ⊗ a1 ,→ e0 1 ⊗ ... ⊗ an ,→ e0 n ν Σ{ e k µ } 7→0,1 loc ν ΣΣ0 { e0 k µ ⊗ µ0 } (∀1 ≤ i ≤ n) ν Σ{ ei k µ } 7→loc ν ΣΣ0 i { e0 i k µ ⊗ µ0 i }    ν Σ{ e k µ } 7→glo ν ΣΣ0 Σ0 1 ...Σ0 n { e0 k µ00 ⊗ µ0 ⊗ µ0 1 ⊗ ... ⊗ µ0 n }    (40.8a) Rule (40.8a) allows the focus expression to take either zero or one steps be- cause it may be blocked awaiting the completion of evaluation of a parallel 3These rules must be augmented by a reformulation of the dynamics of the other con- structs of the language phrased in terms of the present notion of state. REVISED 05.15.2012 VERSION 1.32 408 40.4 Applications of Futures future (or synchronizing with a speculation). The futures allocated by the local steps of execution are consolidated in the result of the global step. We assume without loss of generality that the names of the new futures in each local step are pairwise disjoint so that the combination makes sense. In im- plementation terms satisfying this disjointness assumption means that the processors must synchronize their access to memory. The initial state of a computation, whether for futures or speculations, is defined by the rule ν ∅ { e k ∅ } initial (40.9) Final states differ according to whether we are considering futures or spec- ulations. In the case of futures a state is final iff both the focus and all parallel futures have completed evaluation: e valΣ µ valΣ ν Σ{ e k µ } final (40.10a) (∀a ∈ dom(Σ)) µ(a) valΣ µ valΣ (40.10b) In the case of speculations a state is final iff the focus is a value: e valΣ ν Σ{ e k µ } final (40.11) This corresponds to the speculative nature of the parallel evaluation of speculations whose outcome may not be needed to determine the final out- come of the program. 40.4 Applications of Futures Pipelining provides a good example of the use of parallel futures. Consider a situation in which a producer builds a list whose elements represent units of work, and a consumer traverses the work list and acts on each element of that list. The elements of the work list can be thought of as “instructions” to the consumer, which maps a function over that list to carry out those instructions. An obvious sequential implementation first builds the work list, then traverses it to perform the work indicated by the list. This is fine as long as the elements of the list can be produced quickly, but if each element requires a substantial amount of computation, it would be preferable to VERSION 1.32 REVISED 05.15.2012 40.4 Applications of Futures 409 overlap production of the next list element with execution of the previous unit of work. This can be easily programmed using futures. Let flist be the recursive type µt.unit + (nat × t fut), whose ele- ments are nil, defined to be fold(l · hi), and cons(e1,e2), defined to be fold(r · he1, fut(e2)i). The producer is a recursive function that generates a value of type flist: fix produce : (nat → nat opt) → nat → flist is λ f. λ i. case f(i) { null ⇒ nil | just x ⇒ cons(x, fut (produce f (i+1))) } On each iteration the producer generates a parallel future to produce the tail. This computation proceeds after the producer returns so that it over- laps subsequent computation. The consumer folds an operation over the work list as follows: fix consume : ((nat×nat)→nat) → nat → flist → nat is λ g. λ a. λ xs. case xs { nil ⇒ a | cons (x, xs) ⇒ consume g (g (x, a)) (syn xs) } The consumer synchronizes with the tail of the work list just at the point where it makes a recursive call and hence requires the head element of the tail to continue processing. At this point the consumer will block, if necessary, to await computation of the tail before continuing the recursion. Another application of futures is to provide more control over paral- lelism in a language with lazy suspensions (as described in Chapter 37). Rather than evaluate suspensions speculatively, which is not work efficient, we may instead add futures to the language in addition to suspensions. One application of futures in such a setting is called a spark. A spark is a computation that is executed in parallel with another purely for its effect on suspensions. The spark traverses a data structure, forcing the suspensions within so that their values are computed and stored, but otherwise yielding no useful result. The idea is that the spark forces the suspensions that will be needed by the main computation, but taking advantage of parallelism in the hope that their values will have been computed by the time the main computation requires them. REVISED 05.15.2012 VERSION 1.32 410 40.4 Applications of Futures The sequential dynamics of the spark expression spark(e1; e2) is simply to evaluate e1 before evaluating e2. This is useful in the context of a by- need dynamics for suspensions, because evaluation of e1 will record the values of some suspensions in the memo table for subsequent use by the computation e2. The parallel dynamics specifies, in addition, that e1 and e2 are to be evaluated in parallel. The behavior of sparks is captured by the definition of spark(e1; e2) in terms of futures: let be fut(e1) in e2. Evaluation of e1 commences immediately, but its value, if any, is aban- doned. This encoding does not allow for evaluation of e1 to be abandoned as soon as e2 reaches a value, but this scenario is not expected to arise for the intended mode of use of sparks. The expression e1 should be a quick traversal that does nothing other than force the suspensions in some data structure, exiting as soon as this is complete. Presumably this computation takes less time than it takes for e2 to perform its work before forcing the suspensions that were forced by e2, otherwise there is little to be gained from the use of sparks in the first place! As an example, consider the type strm of streams of numbers defined by the recursive type µt.(unit + (nat × t)) spec. Elements of this type are suspended computations that, when forced, either signals the end of stream, or produces a number and another such stream. Suppose that s is such a stream, and assume that we know, for reasons of its construc- tion, that it is finite. We wish to compute map(f)(s) for some function f, and to overlap this computation with the production of the stream el- ements. We will make use of a function mapforce that forces successive elements of the input stream, but yields no useful output. The compu- tation spark(mapforce(s); map(f)(s)) forces the elements of the stream in parallel with the computation of map(f)(s), with the intention that all suspensions in s are forced before their values are required by the main computation. As another example, we may use futures to encode binary nested par- allelism by defining par(e1; e2; x1.x2.e) to stand for the expression let x0 1 be fut(e1) in let x2 be e2 in let x1 be fsyn(x0 1) in e The order of bindings is important to ensure that evaluation of e2 proceeds in parallel with evaluation of e1. Observe that evaluation of e cannot, in any case, proceed until both are complete. VERSION 1.32 REVISED 05.15.2012 40.5 Notes 411 40.5 Notes Futures were introduced in the MultiLisp language (Halstead, 1985). The same concept was considered by Arvind et al.(1986) under the name “I- structures.” The formulation given here is based on Greiner and Blelloch (1999). REVISED 05.15.2012 VERSION 1.32 412 40.5 Notes VERSION 1.32 REVISED 05.15.2012 Part XVI Concurrency Chapter 41 Process Calculus So far we have mainly studied the statics and dynamics of programs in iso- lation, without regard to their interaction with the world. But to extend this analysis to even the most rudimentary forms of input and output requires that we consider external agents that interact with the program. After all, the whole purpose of a computer is, ultimately, to interact with a person! To extend our investigations to interactive systems, we begin with the study of process calculi, which are abstract formalisms that capture the essence of interaction among independent agents. The development will proceed in stages, starting with simple action models, then extending to interacting concurrent processes, and finally to synchronous and asynchronous com- munication. The calculus consists of two main syntactic categories, pro- cesses and events. The basic form of process is one that awaits the arrival of an event. Processes are closed under parallel composition (the product of processes), replication, and declaration of a channel. The basic forms of event are signaling on a channel and querying a channel; these are later generalized to sending and receiving data on a channel. Events are closed under a finite choice (sum) of events. When enriched with types of mes- sages and channel references, the process calculus may be seen to be uni- versal in that it is at least as powerful as the untyped λ-caclulus. 41.1 Actions and Events Our treatment of concurrent interaction is based on the notion of an event, which specifies the actions that a process is prepared to undertake in con- cert with another process. Two processes interact by undertaking two com- plementary actions, which may be thought of as a signal and a query on a 416 41.1 Actions and Events channel. The processes synchronize when one signals on a channel that the other is querying, after which they both proceed independently to interact with other processes. To begin with we will focus on sequential processes, which simply await the arrival of one of several possible actions, known as an event. Proc P::= await(E) $ E synchronize Evt E::= null 0 null or(E1;E2)E1 + E2 choice que[a](P)?a;P query sig[a](P)!a;P signal The variable a ranges over symbols serving as channel names that mediate communication among the processes. We will not distinguish between events that differ only up to structural congruence, which is defined to be the strongest equivalence relation closed under these rules: E ≡ E0 $ E ≡ $ E0 (41.1a) E1 ≡ E0 1 E2 ≡ E0 2 E1 + E2 ≡ E0 1 + E0 2 (41.1b) P ≡ P0 ?a;P ≡ ?a;P0 (41.1c) P ≡ P0 !a;P ≡ !a;P0 (41.1d) E + 0 ≡ E (41.1e) E1 + E2 ≡ E2 + E1 (41.1f) E1 + (E2 + E3) ≡ (E1 + E2) + E3 (41.1g) Imposing structural congruence on sequential processes enables us to think of an event as having the form !a;P1 + ... + ?a;Q1 + ... consisting of a sum of signal and query events, with the sum of no events being the null event, 0. VERSION 1.32 REVISED 05.15.2012 41.2 Interaction 417 An illustrative example of Milner’s is a simple vending machine that may take in a 2p coin, then optionally either permit selection of a cup of tea, or take another 2p coin, then permit selection of a cup of coffee. V = $ (?2p;$ (!tea;V + ?2p;$ (!cof;V))) As the example indicates, we permit recursive definitions of processes, with the understanding that a defined identifier may always be replaced with its definition wherever it occurs. (Later we will show how to avoid reliance on recursive definitions.) Because the computation occurring within a process is suppressed, se- quential processes have no dynamics on their own, but only through their interaction with other processes. For the vending machine to operate there must be another process (you) who initiates the events expected by the ma- chine, causing both your state (the coins in your pocket) and its state (as just described) to change as a result. 41.2 Interaction Processes become interesting when they are allowed to interact with one another to achieve a common goal. To account for interaction we enrich the language of processes with concurrent composition: Proc P::= await(E) $ E synchronize stop 1 inert par(P1;P2)P1 k P2 composition The process 1 represents the inert process, and the process P1 k P2 represents the concurrent composition of P1 and P2. We may identify 1 with $ 0, the process that awaits the event that will never occur, but we prefer to treat the inert process as a primitive concept. We will identify processes up to structural congruence, which is defined to be the strongest equivalence relation closed under these rules: P k 1 ≡ P (41.2a) P1 k P2 ≡ P2 k P1 (41.2b) P1 k (P2 k P3) ≡ (P1 k P2) k P3 (41.2c) REVISED 05.15.2012 VERSION 1.32 418 41.2 Interaction P1 ≡ P0 1 P2 ≡ P0 2 P1 k P2 ≡ P0 1 k P0 2 (41.2d) Up to structural congruence every process has the form $ E1 k ... k $ En for some n ≥ 0, it being understood that when n = 0 this stands for the null process, 1. Interaction between processes consists of synchronization of two com- plementary actions. The dynamics of interaction is defined by two forms of judgment. The transition judgment P 7→ P0 states that the process P evolves to the process P0 as a result of a single step of computation. The family of transition judgments, P α7−→ P0, where α is an action, states that the process P may evolve to the process P0 provided that the action α is permissible in the context in which the transition occurs (in a sense to be made precise momentarily). As a notational convenience, we often regard the unlabeled transition to be the labeled transition corresponding to the special silent action. The possible actions are given by the following grammar: Act α ::= que[a] a ? query sig[a] a ! signal sil ε silent The query action, a ?, and the signal action, a !, are complementary, and the silent action, ε, is self-complementary. We define the complementary action to α to be the action α given by the equations a ? = a !, a ! = a ?, and ε = ε. $ (!a;P + E) a !7−→ P (41.3a) $ (?a;P + E) a ?7−→ P (41.3b) P1 α7−→ P0 1 P1 k P2 α7−→ P0 1 k P2 (41.3c) P1 α7−→ P0 1 P2 α7−→ P0 2 α 6= ε P1 k P2 7→ P0 1 k P0 2 (41.3d) VERSION 1.32 REVISED 05.15.2012 41.3 Replication 419 Rules (41.3a) and (41.3b) specify that any of the events on which a pro- cess is synchronizing may occur. Rule (41.3d) synchronizes two processes that take complementary actions. As an example, let us consider the interaction of the vending machine, V, with the user process, U, defined as follows: U = $ !2p;$ !2p;$ ?cof;1. Here is a trace of the interaction between V and U: V k U 7→ $ (!tea;V + ?2p;$ !cof;V) k $ !2p;$ ?cof;1 7→ $ !cof;V k $ ?cof;1 7→ V These steps are justified, respectively, by the following pairs of labeled tran- sitions: U 2p !7−→ U0 = $ !2p;$ ?cof;1 V 2p ?7−−→ V0 = $ (!tea;V + ?2p;$ !cof;V) U0 2p !7−→ U00 = $ ?cof;1 V0 2p ?7−−→ V00 = $ !cof;V U00 cof ?7−−→ 1 V00 cof !7−−→ V We have suppressed uses of structural congruence in the above derivations to avoid clutter, but it is important to see its role in managing the non- deterministic choice of events by a process. 41.3 Replication Some presentations of process calculi forego reliance on defining equations for processes in favor of a replication construct, which we write ∗ P. This process stands for as many concurrently executing copies of P as we may require, which may be modeled by the structural congruence ∗ P ≡ P k ∗ P. (41.4) REVISED 05.15.2012 VERSION 1.32 420 41.3 Replication Understood as a principle of structural congruence, this rule hides the steps of process creation, and gives no hint as to how often it can or should be ap- plied. We could alternatively build replication into the dynamics to model the details of replication more closely: ∗ P 7→ P k ∗ P. (41.5) Because the application of this rule is unconstrained, it may be applied at any time to effect a new copy of the replicated process P. So far we have been using recursive process definitions to define pro- cesses that interact repeatedly according to some protocol. Rather than take recursive definition as a primitive notion, we may instead use replication to model repetition. This may be achieved by introducing an “activator” process that is contacted to effect the replication. Consider the recursive definition X = P(X), where P is a process expression that may refer to it- self as X. Such a self-referential process may be simulated by defining the activator process A = ∗ $ (?a;P($ (!a;1))), in which we have replaced occurrences of X within P by an initiator process that signals the event a to the activator. Observe that the activator, A, is structurally congruent to the process A0 k A, where A0 is the process $ (?a;P($ (!a;1))). To start process P we concurrently compose the activator, A, with an initia- tor process, $ (!a;1). Observe that A k $ (!a;1) 7→ A k P(!a;1), which starts the process P while maintaining a running copy of the activa- tor, A. As an example, let us consider Milner’s vending machine written using replication, rather than using recursive process definition: V0 = $ (!v;1)(41.6) V1 = ∗ $ (?v;V2)(41.7) V2 = $ (?2p;$ (!tea;V0 + ?2p;$ (!cof;V0))) (41.8) The process V1 is a replicated server that awaits a signal on channel v to create another instance of the vending machine. The recursive calls are VERSION 1.32 REVISED 05.15.2012 41.4 Allocating Channels 421 replaced by signals along v to re-start the machine. The original machine, V, is simulated by the concurrent composition V0 k V1. This example motivates a commonly-considered restriction on replica- tion that avoids the indeterminacy inherent in choosing when and whether to expand a replication into a parallel composition. To avoid this, we may replace general replication by replicated synchronization, which is governed by the following rules: ∗ $ (!a;P + E) a !7−→ P k ∗ $ (!a;P + E) (41.9a) ∗ $ (?a;P + E) a ?7−→ P k ∗ $ (?a;P + E) (41.9b) The process ∗ $ (E) is to be regarded not as a composition of replication and synchronization, but as the inseparable combination of these two con- structs. The advantage is that the replication occurs only as needed, pre- cisely when a synchronization with another process is possible. This avoids the need to “guess”, either by structural congruence or an explicit step, when to replicate a process. 41.4 Allocating Channels It is often useful (particularly once we have introduced inter-process com- munication) to introduce new channels within a process, rather than as- sume that all channels of interaction are given a priori. To allow for this, the syntax of processes is enriched with a channel declaration primitive: Proc P::= new(a.P) ν a.P new channel The channel, a, is bound within the process P, and hence may be renamed at will (avoiding conflicts) within P. To simplify notation we sometimes write ν a1,..., ak.P for the iterated declaration ν a1.... ν ak.P. Structural congruence is extended with the following rules: P =α P0 P ≡ P0 (41.10a) P ≡ P0 ν a.P ≡ ν a.P0 (41.10b) REVISED 05.15.2012 VERSION 1.32 422 41.4 Allocating Channels a /∈ P2 (ν a.P1) k P2 ≡ ν a.(P1 k P2)(41.10c) (a /∈ P) ν a.P ≡ P(41.10d) Rule (41.10c), called scope extrusion, will be especially important in Sec- tion 41.5. Rule (41.10d) states that channels may be de-allocated once they are no longer in use. To account for the scopes of names (and to prepare for later generaliza- tions) it is useful to introduce a static semantics for processes that ensures that names are properly scoped. A signature,Σ, is, for the time being, a finite set of channels. The judgment `ΣP proc states that a process, P, is well-formed relative to the channels declared in the signature, Σ. `Σ 1 proc (41.11a) `ΣP1 proc `ΣP2 proc `ΣP1 k P2 proc (41.11b) `ΣE event `Σ $ E proc (41.11c) `Σ,a P proc `Σ ν a.P proc (41.11d) The foregoing rules make use of an auxiliary judgment, `ΣE event, stating that E is a well-formed event relative to Σ. `Σ 0 event (41.12a) `Σ,a P proc `Σ,a ?a;P event (41.12b) `Σ,a P proc `Σ,a !a;P event (41.12c) `ΣE1 event `ΣE2 event `ΣE1 + E2 event (41.12d) We shall also have need of the judgment `Σ α action stating that α is a well-formed action relative to Σ: `Σ,a a ? action (41.13a) VERSION 1.32 REVISED 05.15.2012 41.4 Allocating Channels 423 `Σ,a a ! action (41.13b) `Σ ε action (41.13c) The dynamics is correspondingly generalized to keep track of the set of active channels. The judgment P α7−→ Σ P0 states that P transitions to P0 with action α relative to channels Σ. The rules defining the dynamics are indexed forms of those given above, augmented by an additional rule governing the declaration of a channel. We give the complete set of rules here for the sake of clarity. $ (!a;P + E) a !7−→ Σ,a P(41.14a) $ (?a;P + E) a ?7−→ Σ,a P(41.14b) P1 α7−→ Σ P0 1 P1 k P2 α7−→ Σ P0 1 k P2 (41.14c) P1 α7−→ Σ P0 1 P2 α7−→ Σ P0 2 α 6= ε P1 k P2 7−→ Σ P0 1 k P0 2 (41.14d) P α7−→ Σ,a P0 `Σ α action ν a.P α7−→ Σ ν a.P0 (41.14e) Rule (41.14e) states that no process may interact with ν a.P along the locally- allocated channel, a, because to do so would require that a already be de- clared in Σ, which is precluded by the freshness convention on binders. As an example, let us consider again the definition of the vending ma- chine using replication, rather than recursion. The channel, v, used to ini- tialize the machine should be considered private to the machine itself, and not be made available to a user process. This is naturally expressed by the process expression ν v.(V0 k V1), where V0 and V1 are as defined above us- ing the designated channel, v. This process correctly simulates the original REVISED 05.15.2012 VERSION 1.32 424 41.5 Communication machine, V, because it precludes interaction with a user process on channel v. If U is a user process, the interaction begins as follows: (ν v.(V0 k V1)) k U 7−→ Σ (ν v.V2) k U ≡ ν v.(V2 k U). (The processes V0,V1, and V2 are those defined earlier.) The interaction continues as before, albeit within the scope of the binder, provided that v has been chosen (by structural congruence) to be apart from U, ensuring that it is private to the internal workings of the machine. 41.5 Communication Synchronization is the coordination of the execution of two processes that are willing to undertake the complementary actions of signalling and query- ing a common channel. Synchronous communication is a natural generaliza- tion of synchronization to allow more than one bit of data to be communi- cated between two coordinating processes, a sender and a receiver. In prin- ciple any type of data may be communicated from one process to another, and we can give a uniform account of communication that is independent of the type of data communicated between processes. Communication be- comes more interesting in the presence of a type of channel references, which allow access to a communication channel to be propagated from one pro- cess to another, allowing alteration of the interconnection topology among processes during execution. (Channel references will be discussed in Sec- tion 41.6.) To account for interprocess communication we must enrich the lan- guage of processes to include variables, as well as channels, in the formalism. Variables range, as always, over types, and are given meaning by substitu- tion. Channels, on the other hand, are assigned types that classify the data carried on that channel, and are given meaning by send and receive events that generalize the signal and query events considered earlier. The abstract syntax of communication events is given by the following grammar: Evt E::= snd[a](e;P)! a(e;P) send rcv[a](x.P)? a(x.P) receive The event rcv[a](x.P) represents the receipt of a value, x, on the channel a, passing x to the process P. The variable, x, is bound within P, and hence may be chosen freely, subject to the usual restrictions on the choice of names of bound variables. The event snd[a](e;P) represents the transmission of VERSION 1.32 REVISED 05.15.2012 41.5 Communication 425 (the value of) the expression e on channel a, continuing with the process P only once this value has been received. To account for the type of data that may be sent on a channel, the syntax of channel declaration is generalized to associate a type with each channel name. Proc P::= new[τ](a.P) ν a∼τ.P typed channel The process new[τ](a.P) introduces a new channel name, a, with associ- ated type τ for use within the process P. The name, a, is bound within P, and hence may be chosen at will, subject only to avoidance of confusion of distinct names. The statics of communication extends that of synchronization by associ- ating types to channels and by considering variables that range over a type. The judgment Γ `ΣP proc states that P is a well-formed process involving the channels declared in Σ and the variables declared in Γ. It is inductively defined by the following rules, wherein we assume that the typing judg- ment Γ `Σ e : τ is given separately. Γ `Σ 1 proc (41.15a) Γ `ΣP1 proc Γ `ΣP2 proc Γ `ΣP1 k P2 proc (41.15b) Γ `Σ,a∼τ P proc Γ `Σ ν a∼τ.P proc (41.15c) Γ `ΣE event Γ `Σ $ E proc (41.15d) Rules (41.15) make use of the auxiliary judgment Γ `ΣE event, stating that E is a well-formed event relative to Γ and Σ, which is defined as follows: Γ `Σ 0 event (41.16a) Γ `ΣE1 event Γ `ΣE2 event Γ `ΣE1 + E2 event (41.16b) Γ, x : τ `Σ,a∼τ P proc Γ `Σ,a∼τ ? a(x.P) event (41.16c) Γ `Σ,a∼τ e : τ Γ `Σ,a∼τ P proc Γ `Σ,a∼τ ! a(e;P) event (41.16d) REVISED 05.15.2012 VERSION 1.32 426 41.5 Communication Rule (41.16d) makes use of a typing judgment for expressions that ensures that the type of a channel is respected by communication. The dynamics of synchronous communication is similarly an extension of the dynamics of synchronization. Actions are generalized to include the transmitted value, as well as the channel and its orientation: Act α ::= rcv[a](e) a ? e receive snd[a](e) a ! e send sil ε silent Complementarity is defined, essentially as before, to switch the orientation of an action: a ? e = a ! e, a ! e = a ? e, and ε = ε. The statics ensures that the expression associated with these actions is a value of a type suitable for the channel: `Σ,a∼τ e : τ e valΣ,a∼τ `Σ,a∼τ a ! e action (41.17a) `Σ,a∼τ e : τ e valΣ,a∼τ `Σ,a∼τ a ? e action (41.17b) `Σ ε action (41.17c) The dynamics of synchronous communication is defined by replacing Rules (41.14a) and (41.14b) with the following rules: e 7−−−→ Σ,a∼τ e0 $ (! a(e;P) + E) 7−−−→ Σ,a∼τ $ (! a(e0;P) + E)(41.18a) e valΣ,a∼τ $ (! a(e;P) + E) a!e7−−−→ Σ,a∼τ P(41.18b) e valΣ,a∼τ $ (? a(x.P) + E) a?e7−−−→ Σ,a∼τ [e/x]P(41.18c) Rule (41.18c) is non-deterministic in that it “guesses” the value, e, to be re- ceived along channel a. Rules (41.18) make reference to the dynamics of expressions, which is left unspecified here so as to avoid an a priori com- mitment as to the nature of values communicated on a channel. VERSION 1.32 REVISED 05.15.2012 41.6 Channel Passing 427 Using synchronous communication, both the sender and the receiver of a message are blocked until the interaction is completed. This means that the sender must be notified whenever a message is received, and hence there must be an implicit reply channel from the receiver to the sender for the notification. This suggests that synchronous communication may be decomposed into a simpler asynchronous send operation, which transmits a message on a channel without waiting for its receipt, together with channel passing to transmit an acknowledgement channel along with the message data. Asynchronous communication is defined by removing the synchronous send event from the process calculus, and adding a new form of process that simply sends a message on a channel. The syntax of asynchronous send is as follows: Proc P::= asnd[a](e)! a(e) send The process asnd[a](e) sends the message e on channel a, and then termi- nates immediately. Without the synchronous send event, every event is, up to structural congruence, a choice of zero or more read events. The statics of asnychronous send is given by the following rule: Γ `Σ,a∼τ e : τ Γ `Σ,a∼τ ! a(e) proc (41.19) The dynamics is similarly straightforward: e valΣ ! a(e) a!e7−→ Σ 1 (41.20) The rule for interprocess communication given earlier remains unchanged, because the action associated with the asychronous send is the same as in the synchronous case. We may regard a pending asynchronous send as a “buffer” in which the message is held until a receiver is selected. 41.6 Channel Passing An interesting case of interprocess communication arises when one pro- cess passes one channel to another along a common channel. The channel passed by the sending process need not have been known a priori to the receiving process. This allows for new patterns of communication to be REVISED 05.15.2012 VERSION 1.32 428 41.6 Channel Passing established among processes. For example, two processes, P and Q, may share a channel, a, along which they may send and receive messages. If the scope of a is limited to these processes, then no other process, R, may communicate on that channel; it is, in effect, a private channel between P and Q. It frequently arises, however, that P and Q wish to include the process R in their conversation in a controlled manner. This may be accomplished by first expanding the scope of the channel a to encompass R, then send- ing (a reference to) the channel a to R along a pre-arranged channel. Upon receipt of the channel reference, R may communicate with P and Q using send and receive operations that act on channel references. Bearing in mind that channels are not themselves forms of expression, such a scenario can be enacted by introducing a type, τ chan, whose values are references to channels carrying values of type τ. The elimination forms for the chan- nel type are send and receive operations that act on references, rather than explicitly given channels.1 Such a situation may be described schematically by the process expres- sion (ν a∼τ.(P k Q)) k R, in which the process R is initially excluded from the scope of the chan- nel a, whose scope encompasses both the processes P and Q. The type τ represents the type of data communicated along channel a; it may be cho- sen arbitrarily for the sake of this example. The processes P and Q may communicate with each other by sending and receiving along channel a. If these two processes wish to include R in the conversation, then they must communicate the identity of channel a to the process R along some pre- arranged channel, b. If a is a channel carrying values of type τ, then b will be a channel carrying values of type τ chan, which are references to τ-carrying channels. The channel b must be known to at least one of P and Q, and also to channel R. This can be described by the following process expression: ν b∼τ chan.((ν a∼τ.(P k Q)) k R). Suppose that P wishes to include R in the conversation by sending a reference to the channel a along b. The process R correspondingly receives a reference to a channel on the channel b, and commences communication 1It may be helpful to compare channel types with reference types as described in Chap- ters 35 and 36. Channels correspond to assignables, and channel types correspond to refer- ence types. VERSION 1.32 REVISED 05.15.2012 41.6 Channel Passing 429 with P and Q along that channel. Thus P has the form $ (! b(& a;P0)) and R has the form $ (? b(x.R0)). The overall process has the form ν b∼τ chan.(ν a∼τ.($ (! b(& a;P0)) k Q) k $ (? b(x.R0))). Note carefully that the declaration of the channel b specifies that it carries a channel reference, not that it is a channel reference. The process P is prepared to send a reference to the channel a along the channel b, where it may be received by the process R. But the scope of a is limited to processes P and Q, so in order for the communication to succeed, we must first expand its scope to encompass R using the concept of scope extrusion introduced in Section 41.4 to obtain the structurally equivalent process ν b∼τ chan.ν a∼τ.($ (! b(& a;P0)) k Q k $ (? b(x.R0))). The scope of a has been expanded to encompass R, preparing the ground for communication between P and R, which results in the process ν b∼τ chan.ν a∼τ.(P0 k Q k [& a/x]R0). The reference to the channel a has been substituted for the variable x within R0. The process R may now communicate with P and Q by sending and receiving messages along the channel referenced by x. This is accomplished using dynamic forms of send and receive in which the channel on which to communicate is determined by evaluation of an expression, rather than specified statically by an explicit channel name. For example, to send a message e of type τ along the channel referred to by x, the process R0 would have the form $ (!!(x;e;R00)). Similarly, to receive along the referenced channel, the process R0 would have the form $ (??(x;y.R00)). In both cases the dynamic communication forms evolve to the static com- munication forms once the referenced channel has been determined. The syntax of channel types is given by the following grammar: Typ τ ::= chan(τ) τ chan channel type Exp e ::= ch[a]& a reference Evt E::= sndref(e1; e2;P)!!(e1;e2;P) send rcvref(e; x.P)??(e;x.P) receive REVISED 05.15.2012 VERSION 1.32 430 41.7 Universality The events sndref(e1; e2;P) and rcvref(e; x.P) are dynamic versions of the events snd[a](e;P) and rcv[a](x.P) in which the channel is deter- mined dynamically by evaluation of an expression, rather than statically as a fixed parameter of the event. The statics of channel references is given by the following rules: Γ `Σ,a∼τ & a : τ chan (41.21a) Γ `Σ e1 : τ chan Γ `Σ e2 : τ Γ `ΣP proc Γ `Σ!!(e1;e2;P) event (41.21b) Γ `Σ e : τ chan Γ, x : τ `ΣP proc Γ `Σ??(e;x.P) event (41.21c) The introduction of channel references requires that events be evalu- ated to determine the referent of a dynamically determined channel. This is accomplished by adding the following rules for evaluation of a synchro- nizing process: e valΣ $ (!!(& a;e;P) + E) 7−−−→ Σ,a∼τ $ (! a(e;P) + E)(41.22a) $ (??(& a;x.P) + E) 7−−−→ Σ,a∼τ $ (? a(x.P) + E)(41.22b) In addition we require rules for evaluating each of the constituent expres- sions of a dynamically determined event; these rules are omitted here for the sake of concision. 41.7 Universality In the presence of both channel references and recursive types the process calculus with communication is a universal programming language. One way to prove this is to show that it is capable of encoding the untyped λ- calculus with a call-by-name dynamics (see Chapter 17). The main idea of the encoding is to associate with each untyped λ-term, u, a process that represents it. This encoding is defined by induction on the structure of the untyped term, u. For the sake of the induction, the representation is defined relative to a channel reference that represents the context in which the term occurs. Because every term in the untyped λ-calculus is a function, VERSION 1.32 REVISED 05.15.2012 41.7 Universality 431 the context consists of an argument and the continuation for the result of the application. Because of the by-name interpretation of application, variables are represented by references to “servers” that listen on a channel for a channel reference representing a call site, and activate their bindings with that channel reference. We will write u @ z, where u is an untyped λ-term and z is a channel reference representing the continuation of u. The free variables of u will be represented by channels on which we may pass an argument and a contin- uation. Thus, the channel reference z will be a value of type π, and a free variable, x, will be a value of type π chan. The type π is chosen to satisfy the isomorphism π ∼= (π chan × π) chan. That is, a continuation is a channel on which is passed an argument and an- other continuation. An argument, in turn, is a channel on which is passed a continuation. The encoding of untyped λ-terms as processes is given by the following equations: x @ z = !!(x;z) λ (x) u @ z = $ ??(unfold(z);hx, z0i.u @ z0) u1(u2)@ z = ν a1∼π chan × π.(u1 @ fold(& a1)) k ν a∼π.∗ $ ? a(z2.u2 @ z2) k ! a1(h& a, zi) Here we have taken a few liberties with the syntax for the sake of read- ability. We use the asynchronous form of dynamic send operation, because there is no need to be aware of the receipt of the message. Moreover, we use a product pattern, rather than explicit projections, in the dynamic receive to obtain the components of a pair. The use of static and dynamic communication operations in the trans- lation merits careful examination. The call site of a λ-term is determined dynamically; we cannot predict at translation time the continuation of the term. In particular, the binding of a variable may be used at many different call sites, corresponding to the multiple possible uses of that variable. On the other hand the channel associated to an argument is determined stati- cally. The server associated to the variable listens on a statically determined channel for a continuation in which to evaluate its binding, which, as just remarked, is determined dynamically. As a quick check on the correctness of the representation, consider the REVISED 05.15.2012 VERSION 1.32 432 41.8 Notes following derivation: (λ (x) x)(y)@ z 7→∗ ν a1∼τ.($ ? a1(hx, z0i.!!(x;z0))) k ν a∼π.∗ $ ? a(z2.!!(y;z2)) k ! a1(h& a, zi) 7→∗ ν a∼π.∗ $ ? a(z2.!!(y;z2)) k ! a(z) 7→∗ ν a∼π.∗ $ ? a(z2.!!(y;z2)) k !!(y;z) Apart from the idle server process listening on channel a, this is just the translation y @ z. (Using the methods to be developed in detail in Chap- ter 50, we may show that the result of the computation step is “bisimilar” to the translation of y @ z, and hence equivalent to it for all purposes.) 41.8 Notes Process calculi as models of concurrency and interaction were introduced and extensively developed by Hoare(1978) and Milner(1999). Milner’s original formulation, CCS, was introduced to model pure synchronization, whereas Hoare’s, CSP, included value-passing. CCS was subsequently ex- tended to become the π-calculus (Milner, 1999), which includes channel- passing. Dozens upon dozens of variations and extensions of CSP, CCS, and the π-calculus have been considered in the literature, and continue to be a subject of intensive study. (See Engberg and Nielsen(2000) for an ac- count of some of the critical developments in the area.) The process calculus given here is derived from the π-calculus as pre- sented in Milner(1999). (In particular, the vending machine example is adapted from Milner’s monograph.) Unlike Milner’s account (but like re- lated formalisms such as Abadi and Fournet(2001)) we enforce a distinc- tion between variables and names. Variables are given meaning by sub- stitution; a type is the range of significance of a variable, the collection of values that may be substituted for it. Names, on the other hand, are given meaning by the operations associated with them; the type associated with a name is the type of data associated with the operations defined on it. The distinction between variables and names is important, because disequality is well-defined for names, but not for variables. We use the general concept of a reference to pass channel names as data; this is sufficient to ensure the universality of the process calculus. The distinction drawn here between static and dynamic events (that is, those that are given syntactically versus those that arise by evaluation) flows naturally from the prior distinction between variables and names. It VERSION 1.32 REVISED 05.15.2012 41.8 Notes 433 is possible to formulate the process calculus so that all uses of names are suppressed, but then the dynamics cannot be expressed using only the ma- chinery of the calculus itself, but instead must be augmented by an internal concept of names. It seems preferable, in the interest of maintaining a struc- tural operational semantics, to work with a formalism that is closed under its own execution rules. The concept of dynamic events is taken one step further in Concurrent ML (Reppy, 1999), wherein events are values of an event type (see also Chapter 42). REVISED 05.15.2012 VERSION 1.32 434 41.8 Notes VERSION 1.32 REVISED 05.15.2012 Chapter 42 Concurrent Algol In this chapter we integrate concurrency into the framework of Modern- ized Algol described in Chapter 35. The resulting language, called Concur- rent Algol, L{nat cmd * k}, illustrates the integration of the mechanisms of the process calculus described in Chapter 41 into a practical program- ming language. To avoid distracting complications, we drop assignables from Modernized Algol entirely. (There is no loss of generality, however, because free assignables are definable in Concurrent Algol using processes as cells.) The process calculus described in Chapter 41 is intended as a self-standing model of concurrent computation. When viewed in the context of a pro- gramming language, however, it is possible to streamline the machinery to take full advantage of types that are in any case required for other pur- poses. In particular the concept of a channel, which features prominently in Chapter 41, may be identified with the concept of a dynamic class as de- scribed in Chapter 34. More precisely, we take broadcast communication of dynamically classified values as the basic synchronization mechanism of the language. Being dynamically classified, messages consist of a payload tagged with a class, or channel. The type of the channel determines the type of the payload. Importantly, only those processes that have access to the channel may decode the message; all others must treat it as inscrutable data that may be passed around but not examined. In this way we can model not only the mechanisms described in Chapter 41, but also formu- late an abstract account of encryption and decryption in a network using the methods described in Chapter 41. The formulation of Concurrent Algol is based on a modal separation be- tween commands and expressions, much as in Modernized Algol. It is also 436 42.1 Concurrent Algol possible to consolidate these two levels (so as to allow benign concurrency effects), but we do not develop this approach in detail here. 42.1 Concurrent Algol The syntax of L{nat cmd * k} is obtained by stripping out assignables from L{nat cmd *}, and adding a syntactic level of processes: Typ τ ::= cmd(τ) τ cmd commands Exp e ::= cmd(m) cmd m command Cmd m ::= ret e ret e return bnd(e; x.m) bnd x ← e ; m sequence Proc p ::= stop 1 idle proc(m) proc(m) atomic par(p1; p2) p1 k p2 parallel new[τ](a.p) ν a∼τ.p new channel The process proc(m) is an atomic process executing the command, m. The other forms of process are adapted from Chapter 41. If Σ has the form a1 ∼ τ1,..., an ∼ τn, then we sometimes write ν Σ{p} for the iterated form ν a1∼τ1.... ν an∼τn.p. The statics is given by the judgments Γ `Σ e : τ and Γ `Σ m ∼ τ introduced in Chapter 35, augmented by the judgment `Σ p proc stating that p is a well-formed process over the signature Σ. The latter judgment is defined by the following rules: `Σ 1 proc (42.1a) `Σ m ∼ τ `Σ proc(m) proc (42.1b) `Σ p1 proc `Σ p2 proc `Σ p1 k p2 proc (42.1c) `Σ,a∼τ p proc `Σ ν a∼τ.p proc (42.1d) Processes are tacitly identified up to structural equivalence, as described in Chapter 41. The transition judgment p α7−→ Σ p0 states that the process p evolves in one step to the process p0 with associated action α. The particular actions VERSION 1.32 REVISED 05.15.2012 42.1 Concurrent Algol 437 are specified when specific commands are introduced in Section 42.2. As in Chapter 41 we assume that to each action is associated a complementary action, and that the silent action indexes the unlabeled transition judgment. m α=⇒ Σ ν Σ0 { m0 k p } proc(m) α7−→ Σ ν Σ0{proc(m0) k p} (42.2a) e valΣ proc(ret e) 7−→ Σ 1 (42.2b) p1 α7−→ Σ p0 1 p1 k p2 α7−→ Σ p0 1 k p2 (42.2c) p1 α7−→ Σ p0 1 p2 α7−→ Σ p0 2 p1 k p2 7−→ Σ p0 1 k p0 2 (42.2d) p α7−−−→ Σ,a∼τ p0 `Σ α action ν a∼τ.p α7−→ Σ ν a∼τ.p0 (42.2e) Rule (42.2a) states that a step of execution of the atomic process proc(m) consists of a step of execution of the command m, which may result in the allocation of some set, Σ0, of symbols and the creation of a concurrent pro- cess, p. This rule implements scope extrusion for classes (channels) by ex- panding the scope of the declaration of a channel to the context in which the command, m, occurs. Rule (42.2b) states that a completed command evolves to the inert (stopped) process; processes are executed solely for their effect, and not for their value. The remaining rules are those of the process calculus that define the interaction between processes and the allo- cation of symbols within a process. The auxiliary judgment m α=⇒ Σ ν Σ0 { m0 k p0 } defines the execution be- havior of commands. It states that the command, m, transitions to the com- mand, m0, while creating new channels, Σ0, and new processes, p0. The action, α, specifies the interactions of which m is capable when executed. As a notational convenience we drop mention of the new channels or pro- cesses when either are trivial. It is important that the right-hand side of this judgment be construed as a triple consisting of Σ0, m0, and p0, rather than as a process expression comprising these parts. REVISED 05.15.2012 VERSION 1.32 438 42.2 Broadcast Communication The general rules defining this auxiliary judgment are as follows: e 7−→ Σ e0 ret e ε=⇒ Σ ret e0 (42.3a) m1 α=⇒ Σ ν Σ0 { m0 1 k p0 } bnd x ← cmd m1 ; m2 α=⇒ Σ ν Σ0{bnd x ← cmd m0 1 ; m2 k p0} (42.3b) e valΣ bnd x ← cmd ret e ; m2 ε=⇒ Σ [e/x]m2 (42.3c) e1 7−→ Σ e0 1 bnd x ← e1 ; m2 ε=⇒ Σ bnd x ← e0 1 ; m2 (42.3d) These generic rules are supplemented by rules governing commands for communication and synchronization among processes. 42.2 Broadcast Communication In this section we consider a very general form of process synchronization called broadcast. Processes emit and accept messages of type clsfd, the type of dynamically classified values considered in Chapter 34. A message consists of a channel, which is its class, and a payload, which is a value of the type associated with the channel (class). Recipients may pattern match against a message to determine whether it is of a given class, and, if so, recover the associated payload. No process that lacks access to the class of a message may recover the payload of that message. (See Section 34.4 for a discussion of how to enforce confidentiality and integrity restrictions using dynamic classification). The syntax of the commands pertinent to broadcast communication is given by the following grammar: Cmd m ::= spawn(e) spawn(e) spawn emit(e) emit(e) emit message acc acc accept message newch[τ] newch new channel The command spawn(e) spawns a process that executes the encapsulated command given by e. The commands emit(e) and acc emit and accept VERSION 1.32 REVISED 05.15.2012 42.2 Broadcast Communication 439 messages, which are classified values whose class is the channel on which the message is sent. The command newch[τ] returns a reference to a fresh class carrying values of type τ. The statics of broadcast communication is given by the following rules: Γ `Σ e : cmd(unit) Γ `Σ spawn(e) ∼ unit (42.4a) Γ `Σ e : clsfd Γ `Σ emit(e) ∼ unit (42.4b) Γ `Σ acc ∼ clsfd (42.4c) Γ `Σ newch[τ] ∼ class(τ)(42.4d) The execution of commands for broadcast communication is defined by these rules: spawn(cmd(m)) ε=⇒ Σ ret hi k proc(m)(42.5a) e 7−→ Σ e0 spawn(e) ε=⇒ Σ spawn(e0) (42.5b) e valΣ emit(e) e !=⇒ Σ ret hi (42.5c) e 7−→ Σ e0 emit(e) ε=⇒ Σ emit(e0) (42.5d) e valΣ acc e ?=⇒ Σ ret e (42.5e) newch[τ] ε=⇒ Σ ν a∼τ.ret (& a)(42.5f) Rule (42.5c) specifies that emit(e) has the effect of emitting the message e. Correspondingly, Rule (42.5e) specifies that acc may accept (any) message that is being sent. REVISED 05.15.2012 VERSION 1.32 440 42.2 Broadcast Communication As usual, the preservation theorem for L{nat cmd * k} ensures that well-typed programs remain well-typed during execution. The proof of preservation requires a lemma governing the execution of commands. First, let us define the judgment `Σ α action by the following rules: `Σ ε action (42.6a) `Σ e : clsfd `Σ e ! action (42.6b) `Σ e : clsfd `Σ e ? action (42.6c) Lemma 42.1. If m α=⇒ Σ ν Σ0 { m0 k p0 } and `Σ m ∼ τ, then `Σ α action, `ΣΣ0 m0 ∼ τ, and `ΣΣ0 p0 proc. Proof. By induction on Rules (42.3). With this in hand the proof of preservation is straightforward. Theorem 42.2 (Preservation). If `Σ p proc and p 7−→ Σ p0, then `Σ p0 proc. Proof. By induction on transition, appealing to Lemma 42.1 for the crucial steps. Typing does not, however, guarantee progress with respect to unlabeled transition, for the simple reason that there may be no other process with which to communicate. By extending progress to labeled transitions we may state that this is the only way for the execution of a process to get stuck. Theorem 42.3 (Progress). If `Σ p proc, then either p ≡ 1, or there exists p0 and α such that p α7−→ Σ p0. Proof. By induction on Rules (42.1) and (42.4). The assumption that there exists an action rules out degenerate situ- ations in which there are no channels, or all channels carry values of an empty type. VERSION 1.32 REVISED 05.15.2012 42.3 Selective Communication 441 42.3 Selective Communication Broadcast communication provides no means of restricting acceptance to messages of a particular class (that is, of messages on a particular channel). Using broadcast communication we may restrict attention to a particular channel, a, of type, τ, by running the following command: fix loop:τ cmd is {x ← acc ; match x as a · y ⇒ ret y ow ⇒ emit(x); do loop} This command is always capable of receiving a broadcast message. When one arrives, it is examined to determine whether it is classified by the class, a. If so, the underlying value is returned; otherwise the message is re- broadcast to make it available to another process that may be executing a similar command. Polling consists of repeatedly executing the above com- mand until such time as a message of channel a is successfully accepted, if ever. Polling is evidently impractical in most situations. An alternative is to change the language to allow for selective communication. Rather than accept any broadcast message, we may confine attention to messages that are sent on any of several possible channels. This may be accomplished by introducing a type, event(τ), of events consisting of a finite choice of accepts, all of whose associated payload has the type τ. Typ τ ::= event(τ) τ event events Exp e ::= rcv[a]? a select never[τ] never null or(e1; e2) e1 or e2 choice Cmd m ::= sync(e) sync(e) synchronize Events in L{nat cmd * k} correspond directly to those of the asynchronous process calculus described in Chapter 41. One difference is that the se- lect event need not carry with it a continuation, as it does in the process calculus; this is handled by the ambient modal structure of commands. (However, note that all events in a choice share the same continuation, whereas in process calculus a separate continuation is associated to each event in a choice.) Another difference between the two formalisms is that in L{nat cmd * k} events are values of the type τ event, whereas in the process calculus events are not regarded as a form of expression. The statics of event expressions is given by the following rules: Γ `Σ,a∼τ rcv[a]: event(τ)(42.7a) REVISED 05.15.2012 VERSION 1.32 442 42.3 Selective Communication Γ `Σ never[τ]: event(τ)(42.7b) Γ `Σ e1 : event(τ)Γ `Σ e2 : event(τ) Γ `Σ or(e1; e2): event(τ)(42.7c) The corresponding dynamics is defined by these rules: rcv[a] valΣ,a∼τ (42.8a) never[τ] valΣ (42.8b) e1 valΣ e2 valΣ or(e1; e2) valΣ (42.8c) e1 7−→ Σ e0 1 or(e1; e2) 7−→ Σ or(e0 1; e2)(42.8d) e1 valΣ e2 7−→ Σ e0 2 or(e1; e2) 7−→ Σ or(e1; e0 2)(42.8e) Event values are identified up to structural congruence as described in Chapter 41. This ensures that the ordering of events in a choice is immate- rial. Channel references (see Section 34.2) give rise to an additional form of event, rcvref(e), in which the argument, e, is a reference to the channel on which to accept a message. Its statics is given by the rule Γ `Σ e : class(τ) Γ `Σ rcvref(e): event(τ)(42.9) Its dynamics is defined to dereference its argument and evaluate to an ac- cept event for the referenced channel: e 7−→ Σ e0 rcvref(e) 7−→ Σ rcvref(e0)(42.10a) rcvref(& a) 7−−−→ Σ,a∼τ rcv[a](42.10b) VERSION 1.32 REVISED 05.15.2012 42.3 Selective Communication 443 Turning now to the synchronization command, the statics is given by the following rule: Γ `Σ e : event(τ) Γ `Σ sync(e) ∼ τ (42.11a) Its execution is defined by these rules: e 7−→ Σ e0 sync(e) ε=⇒ Σ sync(e0) (42.12a) e α=⇒ Σ m sync(e) α=⇒ Σ m (42.12b) Rule (42.12b) specifies that synchronization may take any action engen- dered by the event given as argument. The possible actions engendered by an event value are defined by the judgment e α=⇒ Σ m, which states that the event value e engenders action α and activates command m. It is defined by the following rules: e valΣ,a∼τ `Σ,a∼τ e : τ rcv[a] a·e ?===⇒ Σ,a∼τ ret(e)(42.13a) e1 α=⇒ Σ m1 or(e1; e2) α=⇒ Σ m1 (42.13b) Rule (42.13a) states that an acceptance on a channel a may synchronize only with messages classified by a. In conjunction with the identification of event values up to structural congruence Rule (42.13b) states that any event among a set of choices may be engender an action. Selective communication and dynamic events may be used together to implement a communication protocol in which a channel reference is passed on a channel in order to establish a communication path with the recipient. Let a be a channel carrying values of type class(τ), and let b be a channel carrying values of type τ, so that & b may be passed as a message along channel a. A process that wishes to accept a channel reference on a and then accept on that channel has the form {x ← sync(? a); y ← sync(?? x);...}. REVISED 05.15.2012 VERSION 1.32 444 42.4 Free Assignables as Processes The event ? a specifies a selective receipt on channel a. Once the value, x, has been accepted, the event ?? x specifies a selective receipt on the channel referenced by x. So, if & b is sent along a, then the event ?? & b evaluates to ? b, which accepts selectively on channel b, even though the receiving process may have no direct access to the channel b itself. Selective communication may be seen as a simple form of pattern match- ing in which patterns are restricted to a · x, where a is a channel carrying values of some type τ, and x is a variable of type τ. The idea is that selective communication filters for messages that match a pattern of this form, and proceeds by returning the associated value, x. From this point of view it is natural to generalize selective communication to allow arbitrary patterns of type clsfd. Because different patterns may bind different variables, it is then natural to associate a separate continuation with each pattern, as in Chapter 13. Basic events are of the form p ⇒ m, where p is a pattern of type clsfd, x1,..., xk are its variables, and m is a command involving these variables. Compound events are compositions of such rules, written r1 | ... | rn, quotiented by structural congruence to ensure that the order of rules is insignificant. The statics of pattern-driven events may be readily derived from the statics of pattern matching given in Chapter 13. The dynamics is defined by the following rule defining the action engendered by an event: e valΣ `Σ e : clsfd θ p / e p ⇒ m|rs e ?=⇒ Σ ˆθ(m)(42.14) This rule states that we may choose any accept action by a value matching the pattern p, continuing with the corresponding instance of the continua- tion of the rule. 42.4 Free Assignables as Processes Scope-free assignables are definable in L{nat cmd * k} by associating to each assignable a server process that sets and gets the contents of the assignable. To each assignable, a, of type ρ is associated a server that selectively accepts a message on channel a with one of two forms: 1. get ·(& b), where b is a channel of type ρ. This message requests that the contents of a be sent on channel b. VERSION 1.32 REVISED 05.15.2012 42.4 Free Assignables as Processes 445 2. set ·(he,& bi), where e is a value of type ρ, and b is a channel of type ρ. This message requests that the contents of a be set to e, and that the new contents be transmitted on channel b. In other words, a is a channel of type τsrvr given by [get ,→ ρ class, set ,→ ρ × ρ class]. The server selectively accepts on channel a, then dispatches on the class of the message to satisfy the request. The server associated with the assignable, a, of type ρ maintains the contents of a using recursion. When called with the current contents of the assignable, the server selectively accepts on channel a, dispatching on the associated request, and calling itself recursively with the (updated, if necessary) contents: λ (u:τsrvr class) fix srvr:ρ → void cmd is λ (x:ρ) cmd {y ← sync(?? u); e(42.16)}. (42.15) The server is a procedure that takes an argument of type ρ, the current contents of the assignable, and yields a command that never terminates, because it restarts the server loop after each request. The server selectively accepts a message on channel a, and dispatches on it as follows: case y {get · z ⇒ e(42.17) | set · hx0, zi ⇒ e(42.18)}. (42.16) A request to get the contents of the assignable a is served as follows: { ← emit(mk(z; x)) ; do srvr(x)}(42.17) A request to set the contents of the assignable a is served as follows: { ← emit(mk(z; x0)) ; do srvr(x0)}(42.18) The type τ ref is defined to be τ class, the type of channels (classes) carrying a value of type τ. A new free assignable is created by the com- mand ref e0, which is defined to be {x ← newch ; ← spawn(e(42.15)(x)(e0)) ; ret x}. (42.19) A channel carrying a value of type τsrvr is allocated to serve as the name of the assignable, and a new server is spawned that accepts requests on that channel, with initial value e0. REVISED 05.15.2012 VERSION 1.32 446 42.5 Notes The commands * e0 and e0 := e1 send a message to the server to get and set the contents of an assignable. The code for * e0 is as follows: {x ← newch ; ← emit(mk(e0; get · x)) ; sync(??(x))}(42.20) A channel is allocated for the return value, the server is contacted with a get message specifying this channel, and the result of receiving on this channel is returned. Similarly, the code for e0 := e1 is as follows: {x ← newch ; ← emit(mk(e0; set · he1, xi)) ; sync(??(x))}(42.21) 42.5 Notes Concurrent Algol is a synthesis of process calculus and Modernized Algol, and may be seen as an “Algol-like” formulation of Concurrent ML (Reppy, 1999) in which interaction is confined to the command modality. The de- sign is influenced by Parallel Algol (Brookes, 2002). The reduction of chan- nels to dynamic classification appears to be new. Most work on concurrent interaction seems to take the notion of communication channel as a central concept (but see Linda (Gelernter, 1985) for an alternative viewpoint, albeit in a unityped setting). VERSION 1.32 REVISED 05.15.2012 Chapter 43 Distributed Algol A distributed computation is one that takes place at many different sites, each of which controls some resources located at that site. For example, the sites might be nodes on a network, and a resource might be a device or sen- sor located at that site, or a database controlled by that site. Only programs that execute at a particular site may access the resources situated at that site. Consequently, command execution always takes place at a particular site, called the locus of execution. Access to resources at a remote site from a local site is achieved by moving the locus of execution to the remote site, running code to access the local resource, and returning a value to the local site. In this chapter we consider the language L{nat cmd * k @}, an exten- sion of Concurrent Algol with a spatial type system that mediates access to located resources on a network. The type safety theorem ensures that all accesses to a resource controlled by a site are through a program executing at that site, even though references to local resources may be freely passed around to other sites on the network. The key idea is that channels and events are located at a particular site, and that synchronization on an event may only occur at the site appropriate to that event. Issues of concurrency, which are to do with non-deterministic composition, are thereby cleanly separated from those of distribution, which are to do with the locality of resources on a network. The concept of location in L{nat cmd * k @} is sufficiently abstract that it admits another useful interpretation that can be useful in computer se- curity settings. The “location” of a computation may also be thought of as the principal on whose behalf the computation is executing. From this point of view, a local resource is one that is accessible to a particular principal, 448 43.1 Statics and a mobile computation is one that may be executed by any principal. Movement from one location to another may then be interpreted as execut- ing a piece of code on behalf of another principal, returning its result to the principal that initiated the transfer. 43.1 Statics The statics of L{nat cmd * k @} is inspired by the possible worlds interpre- tation of modal logic. Under that interpretation the truth of a proposition is relative to a world, which determines the state of affairs described by that proposition. A proposition may be true in one world, and false in another. For example, one may use possible worlds to model counterfactual reason- ing, in which one postulates that certain facts that happen to be true in this, the actual, world, might be otherwise in some other, possible, world. For instance, in the actual world you, the reader, are reading this book, but in a possible world you may never have taken up the study of programming languages at all. Of course not everything is possible: there is no possible world in which 2 + 2 is other than 4, for example. Moreover, once a com- mitment has been made to one counterfactual, others are ruled out. We say that one world is accessible from another when the first is a sensible counterfactual relative to the first. So, for example, one may consider that relative to a possible world in which you are the king, there is no further possible world in which someone else is also the king (there being only one sovereign). In L{nat cmd * k @} we shall interpret possible worlds as sites on a net- work, with accessibility between worlds expressing network connectivity. We postulate that every site is connected to itself (reflexivity); that if one site is reachable from another, then the second is also reachable from the first (symmetry); and that if a site is reachable from a reachable site, then this site is itself reachable from the first (transitivity). From the point of view of modal logics, the type system of L{nat cmd * k @} is derived from the logic S5, for which accessibility is an equivalence relation. The syntax of L{nat cmd * k @} is a modification of that of L{nat cmd * k}. The following grammar summarizes the key changes: Typ τ ::= cmd[w](τ) τ cmd[w] commands chan[w](τ) τ chan[w] channels event[w](τ) τ event[w] events Cmd m ::= at[w](m) at w {m} change site VERSION 1.32 REVISED 05.15.2012 43.1 Statics 449 The command, channel, and event types are indexed by the site, w, to which they pertain. There is a new form of command, at[w](m), that changes the locus of execution from one site to another. A signature, Σ, in L{nat cmd * k @}, consists of a finite set of decla- rations of the form a ∼ ρ @ w, where ρ is a type and w is a site. Such a declaration specifies that a is a channel carrying a payload of type ρ located at the site w. We may think of a signature, Σ, as a family of signatures, Σw, one for each world w, containing the decalarations of the channels located at that world. This partitioning corresponds to the idea that channels are located resources in that they are uniquely associated with a site. They may be handled passively at other sites, but their only active role is at the site at which they are declared. The statics of L{nat cmd * k @} is given by the following two judgment forms: Γ `Σ e : τ expression typing Γ `Σ m ∼ τ @ w command typing The expression typing judgment is independent of the site. This corre- sponds to the idea that the values of a type have a site-independent mean- ing: the number 3 is the number 3, regardless of where it is used. On the other hand commands can only be executed at a particular site, because they depend on the state located at that site. A representative selection of the rules defining the statics of L{nat cmd * k @} is given below: Γ `Σ m ∼ τ @ w Γ `Σ cmd(m): cmd[w](τ)(43.1a) Γ `Σ,a∼ρ@w ch[a]: chan[w](ρ)(43.1b) Γ `Σ never[τ]: event[w](τ)(43.1c) Γ `Σ,a∼ρ@w rcv[a]: event[w](ρ)(43.1d) Γ `Σ e : chan[w](τ) Γ `Σ rcvref(e): event[w](τ)(43.1e) Γ `Σ e1 : event[w](τ)Γ `Σ e2 : event[w](τ) Γ `Σ or(e1; e2): event[w](τ)(43.1f) Γ `Σ e : event[w](τ) Γ `Σ sync(e) ∼ τ @ w (43.1g) REVISED 05.15.2012 VERSION 1.32 450 43.2 Dynamics Γ `Σ m0 ∼ τ0 @ w0 Γ `Σ at[w0](m0) ∼ τ0 @ w (43.1h) Rule (43.1a) states that the type of an encapsulated command records the site at which the command is to be executed. Rules (43.1d) and (43.1e) specify that the type of a (static or dynamic) receive event records the site at which the channel resides. Rules (43.1c) and (43.1f) state that a choice can only be made between events at the same site; there are no cross-site choices. Rule (43.1g) states that the sync command returns a value of the same type as that of the event, and may be executed only at the site to which the given event pertains. Finally, Rule (43.1h) states that to execute a command at a site, w0, requires that the command pertain to that site. The returned value is then passed to the original site. 43.2 Dynamics The dynamics is given by a labeled transition judgment between processes, much as in Chapter 42. The principal difference is that the atomic process consisting of a single command has the form proc[w](m), which specifies the site, w, at which the command, m, is to be executed. The dynamics of processes remains much as in Chapter 42, except for the following rules governing the atomic process: m α==⇒ Σ, w ν Σ0 { m0 k p } proc[w](m) α7−→ Σ ν Σ0 { proc[w](m0) k p } (43.2a) proc[w](ret(hi)) ε7−→ Σ stop (43.2b) The command execution judgment m α==⇒ Σ, w ν Σ0 { m0 k p } states that the command, m, when executed at site, w, may undertake the action, α, and in the process create new channels, Σ0, and a new process, p. (The result of the transition is not a process expression, but rather should be construed as a structure having three parts, the newly allocated channels, the newly created processes, and a new command; we omit any part when it is trivial.) This may be understood as a family of judgments indexed VERSION 1.32 REVISED 05.15.2012 43.3 Safety 451 by signatures, Σ, and sites, w. At each site there is an associated labeled transition system defining concurrent interaction of processes at that site. The command execution judgment is defined by the following rules: spawn(cmd(m)) ε==⇒ Σ, w ret(hi) k proc[w](m)(43.3a) newch[τ] ε==⇒ Σ, w ν a∼τ@w.ret (& a)(43.3b) m α==⇒ Σ, w0 ν Σ0 { m0 k p0 } at[w0](m) α==⇒ Σ, w ν Σ0 { at[w0](m0) k p0 } (43.3c) e valΣ at[w0](ret(e)) ε==⇒ Σ, w ret(e)(43.3d) e α=⇒ Σ m sync(e) α==⇒ Σ, w m (43.3e) Rule (43.3a) states that new processes created at a site remain at that site— the new process executes the given command at the current site. Rules (43.3c) and (43.3d) state that the command at[w0](m) is executed at site w by exe- cuting m at site w0, and returning the result to the site w. Rule (43.3e) states that an action may be undertaken at site w if the given event engenders that action. Notice that no cross-site synchronization is possible. Move- ment between sites is handled separately from synchronization among the processes at a site. 43.3 Safety The safety theorem for L{nat cmd * k @} ensures that synchronization on a channel may only occur at the site on which the channel resides, even though channel references may be propagated from one site to another during a computation. By the time the reference is resolved and synchro- nization is attempted the computation will, as a consequence of typing, be located at the appropriate site. REVISED 05.15.2012 VERSION 1.32 452 43.4 Situated Types The key to the safety proof is the definition of a well-formed process. The judgment `Σ p proc states that the process p is well-formed. Most importantly, the following rule governs the formation of atomic processes: `Σ m ∼ unit @ w `Σ proc[w](m) proc (43.4) That is, an atomic process is well-formed if and only if the command it is executing is well-formed at the site at which the process is located. The proof of preservation relies on a lemma stating the typing proper- ties of the execution judgment. Lemma 43.1 (Execution). Suppose that m α==⇒ Σ, w ν Σ0 { m0 k p }. If `Σ m ∼ τ @ w, then `Σ α action and `Σ ν Σ{proc[w](m0) k p} proc. Proof. By a straightforward induction on Rules (43.3). Theorem 43.2 (Preservation). If p α7−→ Σ p0 and `Σ p proc, then `Σ p0 proc. Proof. By induction on Rules (43.1), appealing to Lemma 43.1 for atomic processes. The progress theorem states that the only impediment to execution of a well-typed program is the possiblity of synchronizing on an event that will never arise. Theorem 43.3 (Progress). If `Σ p proc, then either p ≡ 1 or there exists α and p0 such that p α7−→ Σ p0. 43.4 Situated Types The foregoing formulation of L{nat cmd * k @} relies on indexing com- mand, channel, and event types by the site to which they pertain so that values of these types may be passed around at will without fear of misin- terpretation. The price to pay, however, is that the command, channel, and event types are indexed by the site to which they pertain, leading to repe- tition and redundancy. One way to mitigate this cost is to separate out the skeleton, φ, of a type from its specialization to a particular site, w, which is written φhwi. We will now reformulate the statics of L{nat cmd * k @} using judg- ments of the form Φ `Σ e : φ @ w and Φ `Σ m ∼ φ @ w, where Φ consists of VERSION 1.32 REVISED 05.15.2012 43.4 Situated Types 453 hypotheses of the form xi : φi @ wi. The type of an expression or command is factored into two parts, the skeleton, φ, and a site, w, at which to special- ize it. The meaning of the factored judgments is captured by the following conditions: 1. If Φ `Σ e : φ @ w, then bΦ `Σ e : φhwi. 2. If Φ `Σ m ∼ φ @ w, then bΦ `Σ m ∼ φhwi @ w. If Φ is a context of the form x1 : φ1 @ w1,..., xn : φn @ wn, then bΦ is the context x1 : φ1hw1i,..., xn : φnhwni. The syntax of skeletons is similar to that for types, with the addition of a means of specializing a skeleton to a particular site. Fam φ ::= nat nat numbers arr(φ1; φ2) φ1 → φ2 functions cmd(φ) φ cmd computations chan(φ) φ chan channels event(φ) φ event events at[w](φ) φ at w situated The situated type φ at w fixes the interpretation of φ at the site w. The instantiation of a family, φ, at a site, w, is written φhwi, and is in- ductively defined by the following rules: nathwi = nat (43.5a) φ1hwi = τ1 φ2hwi = τ2 (φ1 → φ2)hwi = τ1 → τ2 (43.5b) φhwi = τ φ cmdhwi = τ cmd[w](43.5c) φhwi = τ φ chanhwi = τ chan[w](43.5d) φhwi = τ φ eventhwi = τ event[w](43.5e) φhw0i = τ0 (φ at w0)hwi = τ0 (43.5f) Crucially, Rule (43.5f) states that the situated family φ at w0 is to be inter- preted at w by the interpretation of φ at w0. Otherwise instantiation serves REVISED 05.15.2012 VERSION 1.32 454 43.4 Situated Types merely to decorate the constituent command, channel, and event skeletons with the site at which they are being interpreted. Any type, τ, of L{nat cmd * k @} may be embedded as a constant fam- ily, exactly(τ), such that exactly(τ)hwi = τ for any site w. The constant family is inductively defined by the following rules: exactly(nat) = nat (43.6a) exactly(τ1) = φ1 exactly(τ2) = φ2 exactly(τ1 → τ2) = φ1 → φ2 (43.6b) exactly(τ) = φ exactly(τ cmd[w]) = φ cmd at w (43.6c) exactly(τ) = φ exactly(τ chan[w]) = φ chan at w (43.6d) exactly(τ) = φ exactly(τ event[w]) = φ event at w (43.6e) It is easy to check that exactly(τ) is a constant family: Lemma 43.4. For any site w, exactly(τ)hwi = τ. The statics of L{nat cmd * k @} may be given in factored form, as is illustrated by the following selection of typing rules: Φ `Σ e : φ @ w Φ `Σ ret e ∼ φ @ w (43.7a) Φ `Σ e1 : φ1 @ w Φ, x : φ1 @ w `Σ m2 ∼ φ2 @ w Φ `Σ bnd x ← e1 ; m2 ∼ φ2 @ w (43.7b) Φ `Σ m ∼ φ @ w Φ `Σ cmd m : φ cmd @ w (43.7c) exactly(ρ) = φ Φ `Σ,a∼ρ@w & a : φ chan @ w (43.7d) Φ `Σ never : φ event @ w (43.7e) exactly(ρ) = φ Φ `Σ,a∼ρ@w ? a : φ event @ w (43.7f) VERSION 1.32 REVISED 05.15.2012 43.4 Situated Types 455 Φ `Σ e : φ chan @ w Φ `Σ?? e : φ event @ w (43.7g) Φ `Σ e1 : φ event @ w Φ `Σ e2 : φ event @ w Φ `Σ e1 or e2 : φ event @ w (43.7h) Φ `Σ e : φ event @ w Φ `Σ sync(e) ∼ φ @ w (43.7i) Φ `Σ m0 ∼ φ0 @ w0 φ0 mobile Φ `Σ at w0 {m0} ∼ φ0 @ w (43.7j) Rule (43.7d) specifies that a reference to a channel carrying a value of type ρ is classified by the constant family yielding the type ρ at each site. Rule (43.7j) is the most interesting rule, because it include a restriction on the family φ0. To see how this arises, inductively we have that bΦ `Σ m0 ∼ φ0hw0i @ w0, which is enough to ensure that bΦ `Σ at w0 {m0} ∼ φ0hw0i @ w. But we are required to show that bΦ `Σ at w0 {m0} ∼ φ0hwi @ w! This will only be the case if φ0hwi = φ0hw0i, which is to say that φ0 is a constant family, whose meaning does not depend on the site at which it is instantiated. The judgment φ mobile states that φ is a mobile family. It is inductively defined by the following rules: nat mobile (43.8a) φ1 mobile φ2 mobile φ1 → φ2 mobile (43.8b) φ at w mobile (43.8c) The remaining families are not mobile, precisely because their instantiation specifies the site of their instances; these do not determine constant fami- lies. Lemma 43.5. 1. If φ mobile, then for every w and w0, φhwi = φhw0i. 2. For any type τ, exactly(τ) mobile. We may then verify that the intended interpretation is valid: Theorem 43.6. REVISED 05.15.2012 VERSION 1.32 456 43.5 Notes 1. If Φ `Σ e : φ @ w, then bΦ `Σ e : φhwi. 2. If Φ `Σ m ∼ φ @ w, then bΦ `Σ m ∼ φhwi @ w. Proof. By induction on Rules (43.7). 43.5 Notes The use of a spatial modality to express locality and mobility constraints in a distributed program was inspired by ML5 (Murphy et al., 2004). The sep- aration of locality concerns from concurrency concerns is expressed here by supporting communication and synchronization within a site, and treating movement between sites separately. The formulation of situated types is based on Licata and Harper(2010). VERSION 1.32 REVISED 05.15.2012 Part XVII Modularity Chapter 44 Components and Linking Modularity is the most important technique for controlling the complex- ity of programs. Programs are decomposed into separate components with precisely specified, and tightly controlled, interactions. The pathways for interaction among components determine dependencies that constrain the process by which the components are integrated, or linked, to form a com- plete system. Different systems may use the same components, and a single system may use multiple instances of a single component. Sharing of com- ponents amortizes the cost of their development across systems, and helps limit errors by limiting coding effort. Modularity is not limited to programming languages. In mathematics the proof of a theorem is decomposed into a collection of definitions and lemmas. Cross-references among lemmas determine a dependency struc- ture that constrains their integration to form a complete proof of the main theorem. Of course, one person’s theorem is another person’s lemma; there is no intrinsic limit on the depth and complexity of the hierarchies of re- sults in mathematics. Mathematical structures are themselves composed of separable parts, as, for example a Lie group is a group structure on a manifold. Modularity arises from the structural properties of the hypothetical and general judgments. Dependencies among components are expressed by free variables whose typing assumptions state the presumed properties of the component. Linking consists of substitution and discharge of the hy- pothesis. 460 44.1 Simple Units and Linking 44.1 Simple Units and Linking Decomposing a program into units amounts to exploiting the transitivity of the hypothetical judgment (see Chapter3). The decomposition may be ex- pressed as an interaction between two parties, the client and the implemen- tor, that is mediated by an agreed-upon contract, called an interface. The client assumes that the implementor upholds the contract, and the imple- mentor guarantees that the contract will be upheld. The assumption made by the client amounts to a declaration of its dependence on the implemen- tor that is discharged by linking the two parties in accordance with their agreed-upon contract. The interface that mediates the interaction between a client and an im- plementor is a type. Linking is nothing other than the implementation of the composite structural rules of substitution and transitivity: Γ ` eimpl : τintf Γ, x : τintf ` eclient : τclient Γ ` [eimpl/x]eclient : τclient (44.1) The type τintf is the interface type. It defines the capabilities to be provided by the implementor, eimpl, that are relied upon by the client, eclient. The free variable, x, expresses the dependency of eclient on eimpl. That is, the client accesses the implementation by using the variable, x. The interface type, τintf, is the contract between the client and the im- plementor. It determines the properties of the implementation on which the client may depend and, at the same time, determines the obligations that the implementor must fulfill. The simplest form of interface type is a finite product type of the form h f1 ,→ τ1,..., fn ,→ τni, specifying a com- ponent with components fi of type τi. Such a type is commonly called an application program interface, or API, because it determines the operations that the client (application) may expect from the implementor. A more so- phisticated form of interface is one that defines an abstract type of the form ∃(t.h f1 ,→ τ1,..., fn ,→ τni), which defines an abstract type, t, representing the internal state of an “abstract machine” whose “instruction set” consists of the operations f1,..., fn whose types may involve t. Being abstract, the type t is not revealed to the client, but is known only to the implementor.1 Conceptually, linking is just substitution, but practically this can be im- plemented in a variety of ways. One method is called separate compilation. The expressions eclient and eimpl, called in this context source modules, are 1See Chapters 21 and 49 for a discussion of type abstraction. VERSION 1.32 REVISED 05.15.2012 44.2 Initialization and Effects 461 translated (compiled) into another, lower-level, language, resulting in ob- ject modules. Linking consists of performing the required substitution at the level of the object language in such a way that the result corresponds to the translation of [eimpl/x]eclient.2 Another method, called separate checking, shifts the requirement for translation to the linker. The client and imple- mentor units are ensured to be type-correct with respect to the interface requirements, but are not translated into lower-level form. Linking then consists of translating the composite program as a whole, often resulting in a more efficient outcome than would be possible when compiling sepa- rately. A more sophisticated, and widely used, implementation of substitu- tion is called dynamic linking. Informally, this means that execution of the client commences before the implementation of the components on which it depends are provided. Rather than link prior to execution, we instead execute and link “on the fly.” At first blush this might seem to be a radical departure from the methodology developed in this book, because we have consistently required that execution be defined only on expressions with no free variables. But looks can be deceiving. What is really going on with dynamic linking is that the client is implemented by a stub that forwards accesses to a stored implementation (typically, in a “file system” or similar data structure). The actual implementation code is not accessed until the client requests it, which may not happen at all. This tends to reduce latency and makes it possible to replace the implementation without recompiling the client. What is important is not how linking is implemented, but rather that the linking principle enables separate development. Once the common interface has been agreed upon, the client and implementor are free to proceed with their work independently of one another. All that is required is that both parties complete their work before the system as a whole can be built. 44.2 Initialization and Effects Linking resolves the dependencies among the components of a program by substitution. This view is valid so long as the components are given by pure expressions, those that evaluate to a value without inducing any effects. For in such cases there is no problem with the replication, or com- plete omission, of a component arising from repeated, or absent, uses of 2The correspondence need not be exact, but must be equivalent for all practical purposes, in the sense discussed in Chapter 48. REVISED 05.15.2012 VERSION 1.32 462 44.2 Initialization and Effects a variable representing it. But what if the expression defining the imple- mentation of a component has an effect when evaluated? At a minimum replication of the component implies replication of its effects. Worse, ef- fects introduce implicit dependencies among components that are not appar- ent from their types. For example, if each of two components mutates a shared assignable, the order in which they are linked with a client program affects the behavior of the whole. This may raise doubts about the treatment of linking as substitution, but on closer inspection it becomes clear that implicit dependencies are naturally managed by paying attention to the modal distinction between expressions and commands introduced in Chapter 35. Specifically, a com- ponent that may have an effect when executed does not have type τintf of implementations of the interface type, but rather the type τintf cmd of en- capsulated commands that, when executed, have effects and yield such an implementation. Being encapsulated, a value of this type is itself free of effects, but it may have effects when evaluated. The distinction between the types τintf and τintf cmd is mediated by the sequentialization command introduced in Chapter 35. For the sake of gen- erality, let us assume that the client is itself an encapsulated command of type τclient cmd, so that it may itself have effects when executed, and may serve as a component of a yet larger system. Assuming that the client refers to the encapsulated implementation by the variable x, the command bnd x ← x ; do eclient first determines the implementation of the interface by running the encap- sulated command, x, then running the client code with the result bound to x. The implicit dependencies of the client on the implementor are made ex- plicit by the sequentialization command, which ensures that the implemen- tor’s effects occur prior to those of the client, precisely because the client depends on the implementor for its execution. More generally, to manage such interactions in a large program it is common to isolate an initialization procedure whose role is to stage the ef- fects engendered by the various components according to some policy or convention. Rather than attempt to survey all possible policies, which are numerous and complex, let us simply observe that the upshot of such con- ventions is that the initialization procedure is a command of the form {x1 ← x1 ;... xn ← xn ; mmain}, where x1,..., xn represent the components of the system and mmain is the main (startup) routine. After linking the initialization procedure has the VERSION 1.32 REVISED 05.15.2012 44.3 Notes 463 form {x1 ← e1 ;... xn ← en ; mmain}, where e1,..., en are the encapsulated implementations of the linked com- ponents. When the initialization procedure is executed, it results in the substitution [v1,..., vn/x1,..., xn]mmain, where the expressions v1,..., vn represent the values resulting from exe- cuting e1,..., en, respectively, and the implicit effects have occurred in the order specified by the initializer. 44.3 Notes The relationship between the structural properties of entailment and the practical problem of separate development was implicit in much early work on programming languages, but became explicit once the correspondence between propositions and types was developed. There are many indica- tions of this correspondence, for example in Proofs and Types (Girard, 1989) and Intuitionistic Type Theory (Martin-L¨of, 1984), but it was first made ex- plicit by Cardelli(1997). REVISED 05.15.2012 VERSION 1.32 464 44.3 Notes VERSION 1.32 REVISED 05.15.2012 Chapter 45 Type Abstractions and Type Classes An interface is a contract that specifies the rights of a client and the respon- sibilities of an implementor. Being a specification of behavior, an interface is a type. In principle any type may serve as an interface, but in practice it is usual to structure code into modules consisting of separable and reusable components. An interface specifies the behavior of a module expected by a client and imposed on the implementor. It is the fulcrum on which is bal- anced the tension between separability and integration. As a rule, a mod- ule should have a well-defined behavior that can be understood separately, but it is equally important that it be easy to combine modules to form an integrated whole. A fundamental question is, what is the type of a module? That is, what form should an interface take? One long-standing idea is for an interface to be a labeled tuple of functions and procedures with specified types. The types of the fields of the tuple are traditionally called function headers, be- cause they summarize the call and return types of each function. Using interfaces of this form is called procedural abstraction, because it limits the dependencies between modules to a specified set of procedures. We may think of the fields of the tuple as being the instruction set of an abstract machine. The client makes use of these instructions in its code, and the implementor agrees to provide their implementations. The problem with procedural abstraction is that it does not provide as much insulation as one might like. For example, a module that implements a dictionary must expose in the types of its operations the exact representa- tion of the tree as, say, a recursive type (or, in more rudimentary languages, 466 a pointer to a structure that itself may contain such pointers). Yet the client really should not depend on this representation: the whole point of ab- straction is to eliminate such dependencies. The solution, as discussed in Chapter 21, is to extend the abstract machine metaphor to allow the internal state of the machine to be hidden from the client. In the case of a dictionary the representation of the dictionary as a binary search tree is hidden by ex- istential quantification. This is called type abstraction, because the type of the underlying data (state of the abstract machine) is hidden. Type abstraction is a powerful method for limiting the dependencies among the modules that constitute a program. It is very useful in many circumstances, but is not universally applicable. It is not always appropri- ate to use abstract types; often it is useful to expose, rather than obscure, type information across a module boundary. A typical example is the im- plementation of a dictionary, which is a mapping from keys to values. To use, say, a binary search tree to implement a dictionary, we require that the key type admit a total ordering with which keys can be compared. The dic- tionary abstraction does not depend on the exact type of the keys, but only requires that the key type be constrained to provide a comparison opera- tion. A type class is a specification of such a requirement. The class of com- parable types, for example, specifies a type, t, together with an operation, leq, of type (t × t) → bool with which to compare them. Superficially, such a specification looks like a type abstraction, because it specifies a type and one or more operations on it, but with the important difference that the type, t, is not hidden from the client. For if it were, the client would only be able to compare keys using leq, but would have no means of obtain- ing keys to compare. A type class, in contrast to a type abstraction, is not intended to be an exhaustive specification of the operations on a type, but rather as a constraint on its behavior expressed by demanding that certain operations, such as comparison, be available, without limiting the other operations that might be defined on it. Type abstractions and type classes are extremal cases of a general con- cept of module type that we shall discuss in detail in this chapter. The crucial idea is the controlled revelation of type information across module boundaries. Type abstractions are opaque; type classes are transparent. These are both instances of translucency, which arises from the combina- tion of existential types (Chapter 21), subtyping (Chapter 23), and singleton kinds and subkinding (Chapter 24). Unlike in Chapter 21, however, we will distinguish the types of modules, which we will called signatures, from the types of ordinary values. The distinction is not essential, but it will be help- ful to keep the two concepts separate at the outset, deferring discussion of VERSION 1.32 REVISED 05.15.2012 45.1 Type Abstraction 467 how to relax the segregation once the basic concepts are in place. 45.1 Type Abstraction Type abstraction is captured by a form of existential type quantification similar to that described in Chapter 21. For example, a dictionary with keys of type τkey and values of type τval implements the signature, σdict, defined as follows: [t :: T;hemp ,→ t,ins ,→ τkey × τval × t → t,fnd ,→ τkey × t → τval opti]. The type variable, t, of kind T is the abstract type of dictionaries on which are defined three operations, emp, ins, and fnd, with the specified types. It is not essential to fix the type τval, because the dictionary operations impose no restrictions on it; we will do so only for the sake of simplicity. However, it is essential, at this stage, that the key type, τkey, be fixed, for reasons that will become clearer as we proceed. An implementation of the signature σdict is a structure,Mdict, of the form [τdict;hemp ,→ ...,ins ,→ ...,fnd ,→ ...i], where the elided parts implement the dictionary operations in terms of the chosen representation type, τdict. For example, τdict might be a recursive type defining a balanced binary search tree, such as a red-black tree. The dictionary operations work on the underlying representation of the dictio- nary as such a tree, just as would a package of existential type discussed in Chapter 21. To ensure that the representation of the dictionary is hidden from a client, the structure Mdict is sealed with the signature σdict to obtain the mod- ule Mdict  σdict. The effect of sealing is to ensure that the only information about Mdict that is propagated to the client is given by σdict. In particular, because σdict only specifies that the type, t, have kind T, no information about the choice of t as τdict in Mdict is made available to the client. A module is a two-phase object consisting of a static part and a dynamic part. The static part is a constructor of a specified kind; the dynamic part is a value of a specified type. There are two elimination forms that extract the static and dynamic parts of a module. These are, respectively, a form of constructor and a form of expression. More precisely, the constructor REVISED 05.15.2012 VERSION 1.32 468 45.2 Type Classes M· s stands for the static part of M, and the expression M· d stands for its dynamic part. According to the inversion principle, if a module, M, has introductory form, then M· s should be equivalent to the static part of M. So, for example, Mdict · s should be equivalent to τdict. But consider the static part of a sealed module, which has the form (Mdict  σdict)· s. Because sealing hides the representation of an abstract type, this constructor should not be equivalent to τdict. If M0 dict is another implementation of σdict, should (Mdict  σdict)· s be equivalent to (M0 dict  σdict)· s? To ensure reflexivity of type equivalence this equation should hold when- ever M and M0 are equivalent modules. But this violates representation independence for abstract types by making equivalence of abstract types sensitive to their implementation. It would seem, then, that there is a fundamental contradiction between two very fundamental concepts, type equivalence and representation in- dependence. The way out of this conundrum is to disallow reference to the static part of a sealed module: the type expression M  σ · s is deemed ill-formed. More generally, we disallow formation of M· s unless M is a module value, whose static part is always manifest. An explicit structure is a module value, as is any module variable (provided that module variables are bound by-value). One effect of this restriction is that sealed modules must be bound to a variable before they are used. Because module variables are bound by- value, doing so has the effect of imposing abstraction at the binding site. In fact, we may think of sealing as a kind of computational effect that “oc- curs” at the binding site, much as the bind operation in Algol discussed in Chapter 35 engenders the effects induced by an encapsulated command. A consequence of this is that two distinct bindings of the same sealed module result in two distinct abstract types. The type system willfully ignores the identity of the two occurrences of the same module in order to ensure that their representations can be changed independently of one another with- out disrupting the behavior of any client code (because the client cannot rely on their identity, it must be prepared for them to be different). 45.2 Type Classes Type abstraction is an essential tool for limiting dependencies among mod- ules in a program. The signature of a type abstraction determines all that is known about a module by a client; no other uses of the values of an ab- stract type are permissible. A complementary tool is to use a signature to VERSION 1.32 REVISED 05.15.2012 45.2 Type Classes 469 partially specify the capabilities of a module. Such a mechanism is called a type class, or a view, in which case an implementation is called an instance of the type class or view. Because the signature of a type class serves only as a constraint specifying the minimum capabilities of an unknown module, some other means of working with values of that type must be available. The key to achieving this is to expose, rather than hide, the identity of the static part of a module. In this sense type classes are the “opposite” of type abstractions, but we shall see below that there is a smooth progression be- tween them, mediated by a subsignature judgment. Let us consider the implementation of dictionaries as a client of the im- plementation of its keys. To implement a dictionary using a binary search tree, for example, the only requirement is that keys come equipped with a total ordering given by a comparison operation. This can be expressed by a signature, σord, given by [t :: T;hleq ,→ (t × t) → booli]. Because a given type may be ordered in many ways, it is essential that the ordering be packaged with the type to determine a type of keys. The implementation of dictionaries as binary search trees takes the form X: σord ` MX bstdict : σX dict, where σX dict is the signature [t :: T;hemp ,→ t,ins ,→ X· s × τval × t → t,fnd ,→ X· s × t → τval opti], and MX bstdict is a structure (not given explicitly here) that implements the dictionary operations using binary search trees.1 Within MX bstdict, the static and dynamic parts of the module X are accessed by writing X· s and X· d, respectively. In particular, the comparison operation on keys is accessed by the projection X· d · leq. The declared signature of the module variable, X, expresses a constraint on the capabilities of a key type by specifying an upper bound on its sig- nature in the subsignature ordering. This implies that any module bound to X must provide a type of keys and a comparison operation on that type, but nothing else is assumed of it. Because this is all we know about the unknown module, X, the dictionary implementation is constrained to rely only on these specified capabilities, and no others. When linking with a 1Here and elsewhere in this chapter and the next, the superscript X serves as a reminder that the module variable, X, may occur free in the annotated module or signature. REVISED 05.15.2012 VERSION 1.32 470 45.2 Type Classes module defining X, the implementation need not be sealed with this sig- nature, but must rather have a signature that is no larger than it in the subsignature relation. Indeed, the signature σord is useless for sealing, as is easily seen by example. Suppose that Mnatord : σord is an instance of the class of ordered types under the usual ordering. If we seal Mnatord with σord by writing Mnatord  σord, the resulting module is useless, because we would then have no way to create values of the key type. We see, then, that a type class amounts to a categorization of a pre- existing type, not a means of introducing a new type. Rather than obscure the identity of the static part of Mnatord, we wish to propagate its identity as nat while specifying a comparison with which to order them. This may be achieved using singleton kinds (Chapter 24). Specifically, the most precise, or principal, signature of a structure is the one that exposes its static part using a singleton kind. In the case of the module Mnatord, the principal signature is the signature, σnatord, given by [t :: S(nat);leq ,→ (t × t) → bool], which, by the rules of equivalence (defined formally in Section 45.3), is equivalent to the signature [:: S(nat);leq ,→ (nat × nat) → bool]. The dictionary implementation, MX bstdict expects a module, X, with sig- nature σord, but the module Mnatord provides the signature σnatord. Applying the rules of subkinding given in Chapter 24, together with the evident co- variance principle for signatures, we obtain the subsignature relationship σnatord <: σord. By the subsumption principle, a module of signature σnatord may be pro- vided whenever a module of signature σord is required. Therefore Mnatord may be linked to X in MX bstdict. The combination of subtyping and sealing provides a smooth gradation between type classes and type abstractions. The principal signature for MX bstdict is the signature ρX dict given by [t :: S(τX bst);hemp ,→ t,ins ,→ X· s × τval × t → t,fnd ,→ X· s × t → τval opti], VERSION 1.32 REVISED 05.15.2012 45.2 Type Classes 471 where τX bst is the type of binary search trees with keys given by the module X of signature σord. This is a subsignature of σX dict given earlier, so that the sealed module MX bstdict  σX dict is well-formed, and has type σX dict, which hides the representation type of the dictionary abstraction. After linking X to Mnatord, the signature of the dictionary is special- ized by propagating the identity of the static part of Mnatord. This, too, is achieved by using the subsignature judgment. As remarked earlier, the dictionary implementation satisfies the typing X: σord ` MX bstdict : σX dict. But because σnatord <: σord, we have, by contravariance, that X: σnatord ` MX bstdict : σX dict. is also a valid typing judgment. If X: σnatord, then X· s is equivalent to nat, because it has kind S(nat), and hence the following typing is also valid: X: σnatord ` MX bstdict : σnatdict. Here σnatdict is the closed signature [t :: T;hemp ,→ t,ins ,→ nat × τval × t → t,fnd ,→ nat × t → τval opti] in which the representation of dictionaries is held abstract, but the repre- sentation of keys as natural numbers is publicized. The dependency on X has been eliminated by replacing all occurrences of X· s within σX dict by the type nat. Having derived this typing we may link X with Mnatord as de- scribed in Chapter 44 to obtain a composite module, Mnatdict, of signature σnatdict, in which keys are natural numbers ordered as specified by Mnatord. It is convenient to exploit subtyping for labeled tuple types to avoid creating an ad hoc module specifying the standard ordering on the natural numbers. Instead we can extract the required module directly from the implementation of the abstract type of numbers using subsumption. As an illustration, let Xnat be a module variable of signature σnat, which has the form [t :: T;hzero ,→ t,succ ,→ t → t,leq ,→ (t × t) → bool, ... i] The fields of the tuple provide all and only the operations that are available on the abstract type of natural numbers. Among them is the comparison REVISED 05.15.2012 VERSION 1.32 472 45.3 A Module Language operation, leq, which is required by the dictionary module. Applying the subtyping rules for labeled tuples given in Chapter 23, together with the covariance of signatures, we obtain the subsignature relationship σnat <: σord, so that by subsumption the variable, Xnat, may be linked to the variable, X, postulated by the dictionary implementation. Subtyping takes care of extracting the required leq field from the abstract type of natural numbers, demonstrating that the natural numbers are an instance of the class of or- dered types. Of course, this approach only works if we wish to order the natural numbers in the natural way provided by the abstract type. If, in- stead, we wish to use another ordering, then we must construct instances of σord “by hand” to define the appropriate ordering. 45.3 A Module Language The language L{mod} is a codification of the ideas outlined in the preced- ing section. The syntax is divided into five levels: expressions classified by types, constructors classified by kinds, and modules classified by signa- tures. The expression and type level consists of various language mecha- nisms described earlier in this book, including at least product, sum, and partial function types. The constructor and kind level is as described in Chapters 22 and 24, with singleton and dependent kinds. The syntax of L{mod} is summarized by the following grammar: Sig σ ::= sig[κ](t.τ)[t :: κ;τ] signature Mod M::= XX variable str(c;e)[c;e] structure seal[σ](M)M  σ seal let[σ](M1;X.M2) (let X be M1 in M2):σ definition Con c ::= stat(M)M· s static part Exp e ::= dyn(M)M· d dynamic part The statics of L{mod} consists of the following forms of judgment, in addition to those governing the kind and type levels: VERSION 1.32 REVISED 05.15.2012 45.3 A Module Language 473 Γ ` σ sig well-formed signature Γ ` σ1 ≡ σ2 equivalent signatures Γ ` σ1 <: σ2 subsignature Γ ` M: σ well-formed module Γ ` M val module value Γ ` e val expression value Rather than segregate hypotheses into zones, we instead admit the follow- ing three forms of hypothesis groups: X: σ,X val module value variable u :: κ constructor variable x : τ, x val expression value variable It is important that module and expression variables are always regarded as values to ensure that type abstraction is properly enforced. Correspond- ingly, each module and expression variable appears in Γ paired with the hypothesis that it is a value. As a notational convenience we will not explic- itly state the value hypotheses associated with module and expression vari- ables, under the convention that all such variables implicitly come paired with such an assumption. The formation, equivalence, and subsignature judgments are defined by the following rules: Γ ` κ kind Γ, u :: κ ` τ type Γ ` [u :: κ;τ] sig (45.1a) Γ ` κ1 ≡ κ2 Γ, u :: κ1 ` τ1 ≡ τ2 Γ ` [u :: κ1;τ1] ≡ [u :: κ2;τ2](45.1b) Γ ` κ1 :<: κ2 Γ, u :: κ1 ` τ1 <: τ2 Γ ` [u :: κ1;τ1] <:[u :: κ2;τ2](45.1c) Most importantly, signatures are covariant in both the kind and type posi- tions: subkinding and subtyping are preserved by the formation of a sig- nature. It is a consequence of Rule (45.1b) that [u :: S(c);τ] ≡ [:: S(c);[c/u]τ] and, further, it is a consequence of Rule (45.1c) that [:: S(c);[c/u]τ] <:[:: T;[c/u]τ] and therefore [u :: S(c);τ] <:[:: T;[c/u]τ]. REVISED 05.15.2012 VERSION 1.32 474 45.3 A Module Language It is also the case that [u :: S(c);τ] <:[u :: T;τ]. But the two supersignatures of [u :: S(c);τ] are incomparable with respect to the subsignature judgment. This fact is important in the statics of module definitions, as will be detailed shortly. The statics of module expressions is given by the following rules: Γ,X: σ ` X: σ (45.2a) Γ ` c :: κ Γ ` e ::[c/u]τ Γ ` [c;e]:[u :: κ;τ](45.2b) Γ ` σ sig Γ ` M: σ Γ ` M  σ : σ (45.2c) Γ ` σ sig Γ ` M1 : σ1 Γ,X: σ1 ` M2 : σ Γ ` (let X be M1 in M2):σ : σ (45.2d) Γ ` M: σ Γ ` σ <: σ0 Γ ` M: σ0 (45.2e) In Rule (45.2b) it is always possible to choose κ to be the most specific kind of c in the subkind ordering, which uniquely determines c up to construc- tor equivalence. For such a choice, the signature [u :: κ;τ] is equivalent to [:: κ;[c/u]τ], which propagates the identity of the static part of the mod- ule expression into the type of its dynamic part. Rule (45.2c) is to be used in conjunction with subsumption (Rule (45.2e)) to ensure that M has the specified signature. The need for a signature annotation on a module definition is a mani- festation of the avoidance problem. Rule (45.2d) would be perfectly sensible were the signature, σ, omitted from the syntax of the definition. How- ever, omitting this information greatly complicates type checking. If σ were omitted from the syntax of the definition, the type checker would be re- quired to find a signature, σ, for the body of the definition that avoids the module variable, X. Inductively, we may suppose that we have found a signature, σ1, for the module M1, and a signature, σ2, for the module M2, under the assumption that X has signature σ1. To find a signature for an unadorned definition, we must find a supersignature, σ, of σ2 that avoids X. To ensure that all possible choices of σ are accounted for, we seek to find the least (most precise) such signature with respect to the subsigna- ture relation; this is called the principal signature of a module. The problem VERSION 1.32 REVISED 05.15.2012 45.3 A Module Language 475 is that there may be no least supersignature of a given signature that avoids a specified variable. (Consider the example above of a signature with two incomparable supersignatures. The example may be chosen so that the supersignatures avoid a variable, X, that occurs in the subsignature.) Con- sequently, modules do not have principal signatures, a significant compli- cation for type checking. To avoid this problem, we insist that the avoiding supersignature, σ, be given by the programmer so that the type checker is not required to find one. In the presence of modules we have a new form of constructor expres- sion, M· s, and a new form of value expression, M· d. These operations, respectively, extract the static and dynamic parts of the module M. Their formation rules are as follows: Γ ` M val Γ ` M:[u :: κ;τ] Γ ` M· s :: κ (45.3a) Γ ` M:[:: κ;τ] Γ ` M· d : τ (45.3b) Rule (45.3a) requires that the module expression, M, be a value in accor- dance with the following rules: Γ,X: σ,X val ` X val (45.4a) Γ ` e val Γ ` [c;e] val (45.4b) (It is not strictly necessary to insist that the dynamic part of a structure be a value in order for the structure itself to be a value, but we impose this requirement to be consistent with the general policy to employ eager eval- uation, and to obtain laziness through types, as described in Chapter 37.) Rule (45.3a) specifies that only structure values have well-defined static parts, and hence precludes reference to the static part of a sealed structure, which is not a value. This ensures representation independence for abstract types, as discussed in Section 45.1. For if M· s were admissible when M is a sealed module, it would be a type whose identity depends on the un- derlying implementation, in violation of the abstraction principle. Module variables are, on the other hand, values, so that if X:[t :: T;τ] is a module variable, then X· s is a well-formed type. What this means in practice is that sealed modules must be bound to variables before they can be used. It is for this reason that we include definitions among module expressions. REVISED 05.15.2012 VERSION 1.32 476 45.4 First- and Second-Class Rule (45.3b) requires that the signature of the module, M, be non-dependent, so that the result type, τ, does not depend on the static part of the module. This may not always be the case. For example, if M is a sealed module, say N  [t :: T;t] for some module N, then projection M· d is ill-formed. For if it were to be well-formed, its type would be M· s, which would vio- late representation independence for abstract types. But if M is a module value, then it is always possible to derive a non-dependent signature for it, provided that we include the following rule of self-recognition: Γ ` M:[u :: κ;τ]Γ ` M val Γ ` M:[u :: S(M· s :: κ);τ](45.5) This rule propagates the identity of the static part of a module value into its signature. The dependency of the type of the dynamic part on the static part is then eliminable by sharing propagation. The following rule of constructor equivalence states that a type projec- tion from a module value is eliminable: Γ ` [c;e]:[t :: κ;τ]Γ ` [c;e] val Γ ` [c;e]· s ≡ c :: κ (45.6) The requirement that the expression, e, be a value, which is implicit in the second premise of the rule, is not strictly necessary, but does no harm. A consequence of this rule is that apparent dependencies of closed construc- tors (or kinds) on modules may always be eliminated. In particular the identity of the constructor [c;e]· s is independent of e, as would be ex- pected if representation independence is to be assured. The dynamics of modules is entirely straightforward: e 7→ e0 [c;e] 7→ [c;e0](45.7a) e val [c;e]· d 7→ e (45.7b) There is no need to evaluate constructors at run-time, because the dynam- ics of expressions does not depend on their types. It is straightforward to prove type safety for this dynamics relative to the foregoing statics. 45.4 First- and Second-Class It is common to draw a distinction between first-class and second-class mod- ules in programming languages. The purported distinction has little force, VERSION 1.32 REVISED 05.15.2012 45.4 First- and Second-Class 477 because it is not precisely defined, but the terminology is chosen to suggest that the former is somehow superior to the latter. As is often the case with such informal concepts, careful analysis reveals that the situation is exactly opposite to what is suggested. To make this precise, we must first give a definition of what is meant by the terms. Simply put, a module system is first-class if signatures are forms of type, and is otherwise not. Here we are using “type” in a precise technical sense as a classifier of expressions, rather than in a loose sense as any form of classifier. If signatures are types in the narrow sense, then modules may be bound to (substituted for) vari- ables, and hence may be passed as arguments to functions and returned as results from them. Moreover, they may be stored in mutable cells, if there are such, and in general may be handled like any other value, precisely be- cause they are classified by types. If, on the other hand, signatures are not types in the narrow sense, then there are limitations to how they may be used that would seem to limit their expressive power, rendering them less useful than they would otherwise be. However, this superficial impression is misleading, for two related rea- sons. First, the apparent restriction to second-class modules allows for more precise distinctions to be drawn than are possible in the purely first- class case. If a module is a value of a certain type, then a module expression may be an arbitrary computation that, in full generality, depends on run- time state as well as the form of the expression itself. For example, we may form a module expression that conditionally branches on the phase of the moon at the time of evaluation, yielding modules with different static components in each case. Because of such a possibility, it is not sensible to track the identity of the static component of a module in the type system, because it quite literally does not have a single static component to track. Consequently, first-class module systems are incompatible with extensions that rely on tracking the identity of the static part of a module. (One exam- ple is the concept of an applicative functor discussed in Chapter 46.) Moreover, a second-class module system is compatible with extensions that permit modules to be handled as first-class values, without requiring that all modules be first-class values. In this important sense it is the second- class modules that are the more expressive, because they allow considera- tion of first-class modules, while retaining the advantages of second-class modules.2 Specifically, we may account for first-class modules within the 2The situation is analogous to that between static and dynamic type systems discussed in Chapter 18. At first glance it sounds as though dynamic typing would be more expressive, but on careful analysis the situation is revealed to be the other way around. REVISED 05.15.2012 VERSION 1.32 478 45.5 Notes language considered in the preceding section by taking the following steps. First, we admit existential types, described in Chapter 21, as types. A “first- class module” is nothing other than a package of existential type, which may be handled just like any other value of any other type. Observe that a module value, M, of signature [t :: κ;τ] may be turned into a value of type ∃ t :: κ.τ by simply forming the package pack M· s with M· d as ∃(t.τ) con- sisting of the static and dynamic parts of M. Second, to allow packages to be treated as modules, we introduce the module open e that “opens” a package as a module according to the following rule: Γ ` e : ∃ t :: κ.τ Γ ` open e :[t :: κ;τ](45.8) Such a module cannot be considered to be a value, because e is an arbi- trary computation, and hence must generally be bound to a module vari- able before it is used. This mimics exactly the elimination form for exis- tential types, which similarly binds the components of a package to vari- ables before they are used. In this manner we may support both first- and second-class modules in a single framework, without having to make a pri- ori commitments to one or the other. 45.5 Notes The use of dependent types to express modularity was first proposed by MacQueen(1986). Subsequent studies extended this proposal to model the phase distinction between compile- and run-time (Harper et al., 1990), and to account for type abstraction as well as type classes (Harper and Lillibridge, 1994; Leroy, 1994). The avoidance problem was first isolated by Castagna and Pierce(1994) and by Harper and Lillibridge(1994). It has come to play a central role in subsequent work on modules, such as Lillibridge(1997) and Dreyer(2005). The self-recognition rule was introduced by Harper and Lillibridge(1994) and by Leroy(1994). It was subsequently identified as a manifestation of higher-order singletons (Stone and Harper, 2006). A con- solidation of these ideas was used as the foundation for a mechanization of the metatheory of modules (Lee et al., 2007). A thorough summary of the main issues in module system design is given in Dreyer(2005). The presentation given here focuses attention on the type structure re- quired to support modularity. An alternative formulation is based on elab- oration, a translation of modularity constructs into more primitive notions such as polymorphism and higher-order functions. The Definition of Stan- dard ML (Milner et al., 1997) pioneered the elaboration approach. Building VERSION 1.32 REVISED 05.15.2012 45.5 Notes 479 on earlier work of Russo, a more rigorous type-theoretic formulation was given by Rossberg et al.(2010). The advantage of the elaboration-based ap- proach is that it can make do with a simpler type theory as the target lan- guage, but at the expense of making the explanation of modularity more complex. It seems clear that some form of elaboration is required (to han- dle identifier scope resolution and type inference, for example), but it is as yet unclear where best to draw the line. REVISED 05.15.2012 VERSION 1.32 480 45.5 Notes VERSION 1.32 REVISED 05.15.2012 Chapter 46 Hierarchy and Parameterization To be adequately expressive it is essential that a module system support the construction of module hierarchies. Hierarchical structure arises natu- rally in programming, both as an organizational device for partitioning of a large program into manageable pieces, and as a localization device that allows one type abstraction or type class to be layered on top of another. In such a scenario the lower layer plays an auxiliary role relative to the upper layer, and we may think of the upper layer as being parameterized by the lower in the sense that any implementation of the lower layer induces an instance of the upper layer corresponding to that instance. The pattern of dependency of one abstraction on another may be captured by an abstrac- tion, or parameterization, mechanism that allows the implementation of one abstraction to be considered a function of the implementation of another. Hierarchies and parameterization work in tandem to provide an expressive language for organizing programs. 46.1 Hierarchy It is common in modular programming to layer a type class or a type ab- straction on top of a type class. For example, the class of equality types, which are those that admit a boolean equivalence test, is described by the signature σeq defined as follows: [t :: T;heq ,→ (t × t) → booli]. 482 46.1 Hierarchy Instances of this class consist of a type together with a binary equality op- eration defined on it. Such instances are modules with a subsignature of σeq, for example σnateq given by [t :: S(nat);heq ,→ (t × t) → booli]. A module value of this signature has the form [nat;heq ,→ ...i], where the elided expression implements an equivalence relation on the nat- ural numbers. All other instance values of the class σeq have a similar form, differing in the choice of type, and/or the choice of comparison operation. The class of ordered types may be considered to be an extension of the class of equality types with an additional binary operation for the (strict) comparison of two elements of that type. One way to formulate this is as the signature [t :: T;heq ,→ (t × t) → bool,lt ,→ (t × t) → booli], which is a subsignature of σeq according to the rules of subtyping given in Chapter 23. This relationship amounts to the requirement that every ordered type is a fortiori an equality type. This is well and good, but it would be even better if there were a way to incrementally extend the equality type class to the ordered type class without having to rewrite the signature as we have done in the foregoing example. Instead, we would like to layer the comparison aspect on top of an equality type class to obtain the ordered type class. This is achieved using a hierarchical signature, σeqord, of the form ∑ X:σeq . σX ord. In this signature we write σX ord for the signature [t :: S(X· s);hlt ,→ (t × t) → booli], which refers to the static part of X, namely the type on which the equality relation is defined. The notation σX ord emphasizes that this signature has a free module variable, X, occurring within it, and hence is only meaningful in a context in which X has been declared. A value of the signature σeqord is a pair of modules, hMeq ;Mordi, in which Meq comprises a type equipped with an equality relation on it, and VERSION 1.32 REVISED 05.15.2012 46.1 Hierarchy 483 the second comprises a type equipped with an ordering relation on it. Cru- cially, the second type is constrained by the singleton kind in σX ord to be the same as the first type. Such a constraint is called a sharing specification. The process of drawing out of the consequences of a sharing specification is called sharing propagation. Sharing propagation is achieved by a combination of subkinding (as described in Chapter 24) and subtyping for signatures. For example, a par- ticular ordering, Mnatord, of the natural numbers is a module with signature ∑ X:σnateq . σX ord. By covariance of the hierarchical signature, this signature is a subsignature of σeqord, so that by subsumption we may regard Mnatord as a module of the latter signature. The static part of the subsignature is a singleton, so we may apply the rules of sharing propagation given in Chapter 24 to show that the subsignature is equivalent to the signature ∑ X:σnateq . σnatord, where σnatord is the closed signature [t :: S(nat);hlt ,→ (t × t) → booli]. Notice that sharing propagation has replaced the type X· s in the signature with nat, eliminating the dependency on the module variable X. After another round of sharing propagation, this signature may be shown to be equivalent to the signature ρnatord given by [:: S(nat);hlt ,→ (nat × nat) → booli]. Here we have replaced both occurrences of t in the type of the comparison operation with nat as a consequence of the kind of t. The net effect is to propagate the identity of the static part of Mnatord to the signature of the second component of Mnatord. Although its value is a pair, which seems symmetric, a module of sig- nature σeqord is asymmetric in that the signature of the second component is dependent on the first component itself. This dependence is made manifest by the occurrence of the module variable, X, in the signature σord. Thus, for hMeq ;Mordi to be a well-formed module of signature σeqord, the first component, Meq, must have signature σeq, which is meaningful indepen- dently of the other component of the pair. On the other hand, the second REVISED 05.15.2012 VERSION 1.32 484 46.1 Hierarchy component, Mord, must have signature σXeq, with the understanding that X stands for the module Meq. In general this signature is not meaningful in- dependently of Meq itself, and hence it may not be possible to handle Mord independently of Meq. Turning this the other way around, if M is any module of signature σeqord, then it is always sensible to project it onto its first coordinate to ob- tain a module, M· 1, of signature σeq. But it is not always sensible to project it onto its second coordinate, because it may not be possible to give a signa- ture to M· 2 in the case that the dependency on the first component cannot be resolved statically. This can happen if, for example, the M· 1 is a sealed module, whose static part cannot be formed in order to ensure representa- tion independence. In such a situation the dependence of the signature σX ord on the module variable, X, cannot be eliminated, and so no signature may be given to the second projection. For this reason the first component of a module hierarchy is called a submodule of the hierarchy, whereas the sec- ond component may or may not be a submodule of it. Put in other terms, the second component of a hierarchy is “projectible” exactly when the de- pendence of its signature on the first component is eliminable by sharing propagation. That is, we may know enough about the first component stat- ically to ensure that an independent type for the second component may be given. In that case the second component may be considered to be a submodule of the pair; otherwise, the second is inseparable from the first, and therefore cannot be projected from the pair. Consider, for example, a module Mnatord of signature σnatord, which, we noted earlier, is a subsignature of σeqord. The first projection Mnatord · 1 is a well-formed module of closed signature σeq and hence is a submodule of Mnatord. The situation is less clear for the second projection, Mnatord · 2, because its signature, σX ord, depends on the first component via the variable X. However, we noted above that the signature σnatord is equivalent to the signature ∑ :σnateq . ρnatord in which the dependency on X has been eliminated by sharing propagation. This, too, is a valid signature for Mnatord, and hence the second projection, Mnatord · 2 is a well-formed module of closed signature ρnatord. If, on the other hand, the only signature available for Mnatord were σeqord, then the second projection would be ill-formed—the second component would not be separable from the first, and hence could not be considered a submodule of the pair. The hierarchical dependency of the signature of the second component VERSION 1.32 REVISED 05.15.2012 46.2 Parameterizaton 485 of a pair on the first component gives rise to a useful alternative interpre- tation of a hierarchical module signature as describing a family of modules given by the second component thought of as being indexed by the first component. In the case at hand, the totality of modules of the signature σeqord gives rise to a family of modules of signature σX ord, where X ranges over σeq. That is, to each choice, Meq, of signature σeq, we associate the col- lection of choices, Mord, coherent with the first choice in accordance with the sharing constraint in σX ord, taking X to be Mord. This collection is called the fibre over Meq, and the totality of modules of signature σeqord is said to be fibred over σeq (by the first projection). The preceding example illustrates the layering of one type class on top of another. It is also useful to layer a type abstraction over a type class. A good example is provided by a dictionary abstraction in which the type of keys is required to be of the class of ordered types, but is otherwise unspec- ified. The signature, σkeydict, of such a dictionary is given as follows: ∑ X:σeqord . σX dict, where σeqord is the signature of ordered equality types (in either of the two forms discussed above), and σX dict is the signature of dictionaries of some type τ given as follows: [t :: T;hemp ,→ t, ins ,→ X· s × τ × t → t, fnd ,→ X· s × t → τ opti]. The ins and fnd operations make use of the type X· s of keys given by the submodule of the dictionary module. We may think of σkeydict as specifying a family of dictionary modules, one for each choice of the ordered type of keys. Regardless of the interpretation, an implementation of the signature σkeydict consists of a two-level hierarchy of the form hM1 ;M2i, where M1 specifies the key type and its ordering, and M2 implements the dictionary for keys of this type in terms of this ordering. 46.2 Parameterizaton The signature σkeydict may be understood as describing a family of dictio- nary modules indexed by a module of ordered keys. The totality of such modules evaluate to pairs consisting of the ordered type of keys together with the dictionary per se, specialized to that choice of keys. Although it is possible that the code of the dictionary operations differs for each choice of keys, it is more often the case that the same implementation may be used for REVISED 05.15.2012 VERSION 1.32 486 46.2 Parameterizaton all choices of keys, the only difference being that references to, say, X· lt refers to a different function for each choice of key module, X. Such a uniform implementation of dictionaries is provided by the con- cept of a parameterized module, or functor. A functor is a module expressed as a function of an unknown module of specified signature. The uniform dictionary module would be expressed as a functor parameterized over the module implemeting keys, which is to say as a λ-abstraction of the form λ Z:σeqord .Mkeydict. Here Mkeydict is the generic implementation of dictionaries in terms of an unspecified module, Z, of signature σeqord. The signature of Z expresses the requirement that the dictionary implementation relies on the keys being an ordered type, but makes no other requirement on it. A functor is a form of module, and hence has a signature as well, called (oddly enough) a functor signature. The signature, σdictfun, of the functor Mkeydict has the form ∏ Z:σeqord . ρZ keydict, which specifies that its domain is the signature, σeqord, of ordered types, and whose range is a signature, ρZ keydict, depends on the module Z. The range, ρZ keydict, of the dictionary functor is defined to be a subsigna- ture of σkeydict in which the key type of the result is constrained to be the same as the key type given as argument. This constraint may be expressed by defining ρZ keydict to be the hierarchical signature ∑ X:ρZ eqord . σX dict, where ρZ eqord is a subsignature of σeqord imposing the desired sharing con- straint. This is itself a hierarchical signature of the form ∑ X:ρZ eq . σX ord, where ρZeq is the subsignature of σeq given by [t :: S(Z· 1 · s);heq ,→ (t × t) → booli]. The singleton kind in ρZeq expresses the required sharing constraint between the key type in the result of the functor and the key type given as its argu- ment. It is evidently rather tedious to write out ρZ eqord separately from σeqord, because it requires repetition of so much that is already present in the latter VERSION 1.32 REVISED 05.15.2012 46.2 Parameterizaton 487 signature. To lighten the notation it is preferable to express more directly the close relationship between the two signatures using signature modifica- tion to impose a type sharing constraint on a given signature. In the present case we may define ρZ eqord to be the modification of σeqord given by Y: σeqord /Y· 1 · s = Z· 1 · s. The modification states that the signature σeqord is to be altered by impos- ing the constraint that the static part of its equality part is to share with the static part of the equality part of Z, which, recall, also has signature σeqord. We may similarly define the range signature, ρZ keydict, of the dictionary func- tor using signature modification as follows: Y: σkeydict /Y· 1 · 1 · s = Z· 1 · s. The left-hand side of the sharing constraint refers to the type of keys in the submodule of the dictionary, and the right-hand side refers to the type of keys given as argument to the functor. The dictionary functor, Mdictfun, defines a generic implementation of dictionaries in terms of an ordered type of keys. An instance of the dic- tionary for a specific choice of keys is obtained by applying, or instantiat- ing, it with a module of its domain signature, σeqord. For example, because Mnatord, the type of natural numbers ordered in the usual way, is such a module, we may form the instance Mkeydict (Mnatord) to obtain a dictionary with numeric keys. By choosing other modules of signature σeqord we may obtain corresponding instances of the dictionary functor. More generally, if M is any module of signature σdictfun, then it is a functor that we may apply it to any module, Mkey, of signature σeqord to obtain the instance M(Mkey). But what is the signature of such an instance, and how may it be de- duced? Recall that the result signature of σdictfun is dependent on the argu- ment itself, and not just its signature. It is therefore not immediately clear what signature to assign to the instance; the dependency on the argument must be resolved in order to obtain a signature that makes sense indepen- dently of the argument. The situation is broadly similar to the problem of computing the signature of the second component of a hiearchical mod- ule, and similar methods are used to resolve the dependencies, namely to exploit subtyping for signatures to obtain a specialization of the result sig- nature appropriate to the argument. This is best illustrated by example. First, we note that by contravariance of subtyping for functor signatures, we may weaken a functor signature by strengthening its domain signature. In the case of the signature σdictfun of the REVISED 05.15.2012 VERSION 1.32 488 46.3 Extending Modules with Hierarchies and . . . dictionary functor, we may obtain a supersignature σnatdictfun by strengthen- ing its domain to require that the key type be the type of natural numbers: ∏ Z:σnatord . ρZ keydict. Fixing Z to be a module variable of the specialized signature σnatord, the range signature, ρZ keydict, is given by the modification Y: σkeydict /Y· 1 · 1 · s = Z· 1 · s. By sharing propagation this is equivalent to the closed signature, ρnatdict, given by Y: σkeydict /Y· 1 · 1 · s = nat, because we may derive the equivalence of Z· 1 · s and nat once the signa- ture of Z is specialized to σnatord. Now by subsumption if M is a module of signature σdictfun, then M is also a module of the supersignature ∏ Z:σnatord . ρZ keydict. We have just shown that the latter signature is equivalent to the non-dependent functor signature ∏ :σnatord . ρnatdict. The range is now given independently of the argument, so we may deduce that if Mnatkey has signature σnatord, then the application M(Mnatkey) has the signature ρnatdict. The crucial point is that the dependence of the range signature on the domain signature is eliminated by propagating knowledge about the type components of the argument itself. Absent this knowledge, the functor ap- plication cannot be regarded as well-formed, much as the second projection from a hierarchy cannot be admitted if the dependency of its signature on the first component cannot be eliminated. If the argument to the functor is a value, then it is always possible to find a signature for it that maximizes the propagation of type sharing information so that the dependency of the range on the argument can always be eliminated. 46.3 Extending Modules with Hierarchies and Param- eterization In this section we sketch the extension of the module language introduced in Chapter 45 to account for module hierarchies and module parameteriza- tion. VERSION 1.32 REVISED 05.15.2012 46.3 Extending Modules with Hierarchies and . . . 489 The syntax of L{mod} is enriched with the following clauses: Sig σ ::= hier(σ1;X.σ2) ∑ X:σ1 . σ2 hierarchy fun(σ1;X.σ2) ∏ X:σ1 . σ2 functor Mod M::= hier(M1;M2) hM1 ;M2i hierarchy fst(M)M· 1 first component snd(M)M· 2 second component fun[σ](X.M) λ X:σ .M functor app(M1;M2)M1 (M2) instance The syntax of signatures is extended to include hierarchies and functors, and the syntax of modules is correspondingly extended with introduction and elimination forms for these signatures. The judgment M projectible states that the module, M, is projectible in the sense that its constituent types may be referenced by compositions of projections, including the static part of a structure. This judgment is induc- tively defined by the following rules: Γ,X: σ ` x projectible (46.1a) Γ ` M1 projectible Γ ` M2 projectible Γ ` hM1 ;M2i projectible (46.1b) Γ ` M projectible Γ ` M· 1 projectible (46.1c) Γ ` M projectible Γ ` M· 2 projectible (46.1d) All module variables are deemed projectible, even though this condition is only relevant for hierarchies of basic structures. Because the purpose of sealing is to hide the representation of an abstract type, no sealed mod- ule is deemed projectible. Furthermore, no functor is projectible, because there is no concept of projection for a functor. More importantly, no func- tor instance is projectible either. This ensures that any two instances of the same functor define distinct abstract types; functors are therefore said to be generative. (See Section 46.4 for a discussion of an alternative treatment of functors.) The signature formation judgment is extended to include these rules: Γ ` σ1 sig Γ,X: σ1 ` σ2 sig Γ ` ∑ X:σ1 . σ2 sig (46.2a) REVISED 05.15.2012 VERSION 1.32 490 46.3 Extending Modules with Hierarchies and . . . Γ ` σ1 sig Γ,X: σ1 ` σ2 sig Γ ` ∏ X:σ1 . σ2 sig (46.2b) Signature equivalence is defined to be compatible with the two new forms of signature: Γ ` σ1 ≡ σ0 1 Γ,X: σ1 ` σ2 ≡ σ0 2 Γ ` ∑ X:σ1 . σ2 ≡ ∑ X:σ0 1 . σ0 2 (46.3a) Γ ` σ1 ≡ σ0 1 Γ,X: σ1 ` σ2 ≡ σ0 2 Γ ` ∏ X:σ1 . σ2 ≡ ∏ X:σ0 1 . σ0 2 (46.3b) The subsignature judgment is augmented with the following rules: Γ ` σ1 <: σ0 1 Γ,X: σ1 ` σ2 <: σ0 2 Γ ` ∑ X:σ1 . σ2 <: ∑ X:σ0 1 . σ0 2 (46.4a) Γ ` σ0 1 <: σ1 Γ,X: σ0 1 ` σ2 <: σ0 2 Γ ` ∏ X:σ1 . σ2 <: ∏ X:σ0 1 . σ0 2 (46.4b) Rule (46.4a) specifies that the hierarchical signature is covariant in both positions, whereas Rule (46.4b) specifies that the functor signature is con- travariant in its domain and covariant in its range. The statics of module expressions is extended by the following rules: Γ ` M1 : σ1 Γ ` M2 : σ2 Γ ` hM1 ;M2i : ∑ :σ1 . σ2 (46.5a) Γ ` M: ∑ X:σ1 . σ2 Γ ` M· 1 : σ1 (46.5b) Γ ` M: ∑ :σ1 . σ2 Γ ` M· 2 : σ2 (46.5c) Γ,X: σ1 ` M2 : σ2 Γ ` λ X:σ1 .M2 : ∏ X:σ1 . σ2 (46.5d) Γ ` M1 : ∏ :σ2 . σ Γ ` M2 : σ2 Γ ` M1 (M2): σ (46.5e) Rule (46.5a) states that an explicit module hierarchy is given a signature in which there is no dependency of the signature of the second compo- nent on the first component (indicated here by the underscore in place of the module variable). A dependent signature may be given to a hierarchy by sealing, which makes it into a non-value, even if the components are VERSION 1.32 REVISED 05.15.2012 46.4 Applicative Functors 491 values. Rule (46.5b) states that the first projection is defined for general hi- erarchical signatures. On the other hand, Rule (46.5c) restricts the second projection to non-dependent hierarchies, as discussed in the preceding sec- tion. Similarly, Rule (46.5e) restricts instantiation to functors whose types are non-dependent, forcing any dependencies to be resolved using the sub- signature relation and sharing propagation prior to application. The self-recognition rules given in Chapter 45 are extended to account for the formation of hierarchical module value by the following rules: Γ ` M projectible Γ ` M: ∑ X:σ1 . σ2 Γ ` M· 1 : σ0 1 Γ ` M: ∑ X:σ0 1 . σ2 (46.6a) Γ ` M projectible Γ ` M: ∑ :σ1 . σ2 Γ ` M· 2 : σ0 2 Γ ` M: ∑ :σ1 . σ0 2 (46.6b) Rules (46.6a) and (46.6b) permit the specialization of the signature of a hier- archical module value to express that its constructor components are equiv- alent to their projections from the module itself. 46.4 Applicative Functors In the module language just described functors are regarded as generative in the sense that any two instances, even with arguments, are considered to “generate” distinct abstract types. This is ensured by treating a functor application, M(M1), to be non-projectible, so that if it defines an abstract type in the result, that type cannot be referenced without first binding the application to a variable. Any two such bindings are necessarily to distinct variables, X and Y, and so the abstract types X· s and Y· s are distinct, regardless of their bindings. The justification for this design decision merits careful consideration. By treating functors as generative, we are ensuring that a client of the func- tor cannot in any way rely on the implementation of that functor. That is, we are extending the principle of representation independence for abstract types to functors in a natural way. One consequence of this policy is that the module language is compatible with extensions such as a conditional module that branches on an arbitrary dynamic condition that might even depend on external conditions such as the phase of the moon! A functor with such an implementation must be considered generative, because the abstract types arising from any instance cannot be regarded as well-defined until the moment when the application is evaluated, which amounts to the REVISED 05.15.2012 VERSION 1.32 492 46.4 Applicative Functors point at which it is bound to a variable. By regarding all functors as gen- erative we are, in effect, maximizing opportunities to exploit changes of representation without disrupting the behavior of clients of the functor, a bedrock principle of modular decomposition. But because the module language considered in the preceding section does not include anything so powerful as a conditional module, we might consider that the restriction to generative functors is too severe, and may be usefully relaxed. One such alternative is the concept of an applicative functor. An applicative functor is one for which instances by values are regarded as projectible:1 M projectible M1 val M(M1) projectible (46.7) It is important to bear in mind that because of this rule applicative func- tors are not compatible with conditional modules. Thus, a module language based on applicative functors is inherently restricted as compared to one based on generative functors. The benefit of regarding a functor instance as projectible is that we may form types such as (M(M1)) · s, which projects the static part of the in- stance. But this raises the question of when two such type expressions are to be deemed equivalent? The difficulty is that the answer to this question depends on the functor argument. For suppose that F is an ap- plicative functor variable, under what conditions should (F(M1)) · s and (F(M2)) · s be regarded as the same type? In the case of generative func- tors we did not have face this question, because the instances are not pro- jectible, but for applicative functors the question cannot be dodged, but must be addressed. We will return to this point in a moment, after consid- ering one further complication that raises a similar issue. The difficulty is that the body of an applicative functor cannot be sealed to impose abstraction, and, according to the rules given in the preceding section, no sealed module is projectible. Because sealing is the only means of imposing abstraction, we must relax this condition and allow sealed pro- jectible modules to be projectible: M projectible M  σ projectible (46.8) 1We may, in addition, regard functor abstractions as projectible, but because all variables are projectible, there is no harm in omitting this and instead insisting that functors be bound to variables before being used. VERSION 1.32 REVISED 05.15.2012 46.5 Notes 493 Thus, we may form type expressions of the form (M  σ)· s, which project the static part of a sealed module. And once again we are faced with the issue that the equivalence of two such types must involve the equivalence of the sealed modules themselves, in apparent violation of representation independence. Summarizing, if we are to deem functors to be applicative, then some compromise of the principle of representation independence for abstract types is required. We must define equivalence for the static parts of sealed modules, and doing so requires at least checking whether the underlying modules are identical. This has two consequences. Because the underlying modules have both static and dynamic parts, this means comparing their executable code for equivalence during type checking. More significantly, because the formation of a client may depend on the equivalence of two modules, we cannot change the representation of a sealed module without fear of disrupting the typing or behavior of the client. This undermines the very purpose of having a module system in the first place! 46.5 Notes Module hierarchies and functors in the form discussed here were intro- duced by Milner et al.(1997), which also employed the reading of a mod- ule hierarchy as an indexed family of modules. The theory of hierarchies and functors was first studied by Harper and Lillibridge(1994) and Leroy (1994), building on earlier work by Mitchell and Plotkin(1988) on existen- tial types. The concept of an applicative functor was introduced by Leroy (1995) and is central to the module system of O’Caml (OCaml). REVISED 05.15.2012 VERSION 1.32 494 46.5 Notes VERSION 1.32 REVISED 05.15.2012 Part XVIII Equational Reasoning Chapter 47 Equational Reasoning for T The beauty of functional programming is that equality of expressions in a functional language corresponds very closely to familiar patterns of math- ematical reasoning. For example, in the language L{nat →} of Chapter9 in which we can express addition as the function plus, the expressions λ (x:nat) λ (y:nat) plus(x)(y) and λ (x:nat) λ (y:nat) plus(y)(x) are equal. In other words, the addition function as programmed in L{nat →} is commutative. This may seem to be obviously true, but why, precisely, is it so? More importantly, what do we even mean for two expressions to be equal in this sense? It is intuitively obvious that these two expressions are not definition- ally equivalent, because they cannot be shown equivalent by symbolic exe- cution. We may say that these two expressions are definitionally inequiva- lent because they describe different algorithms: one proceeds by recursion on x, the other by recursion on y. On the other hand, the two expressions are interchangeable in any complete computation of a natural number, be- cause the only use we can make of them is to apply them to arguments and compute the result. Two functions are logically equivalent if they give equal results for equal arguments—in particular, they agree on all possible argu- ments. Because their behavior on arguments is all that matters for calculat- ing observable results, we may expect that logically equivalent functions are equal in the sense of being interchangeable in all complete programs. Thinking of the programs in which these functions occur as observations of their behavior, these functions are said to be observationally equivalent. The 498 47.1 Observational Equivalence main result of this chapter is that observational and logical equivalence coincide for a variant of L{nat →} in which the successor is evaluated ea- gerly, so that a value of type nat is a numeral. 47.1 Observational Equivalence When are two expressions equal? Whenever we cannot tell them apart! This may seem tautological, but it is not, because it depends on what we consider to be a means of telling expressions apart. What “experiment” are we permitted to perform on expressions in order to distinguish them? What counts as an observation that, if different for two expressions, is a sure sign that they are different? If we permit ourselves to consider the syntactic details of the expres- sions, then very few expressions could be considered equal. For example, if it is deemed significant that an expression contains, say, more than one function application, or that it has an occurrence of λ-abstraction, then very few expressions would come out as equivalent. But such considerations seem silly, because they conflict with the intuition that the significance of an expression lies in its contribution to the outcome of a computation, and not to the process of obtaining that outcome. In short, if two expressions make the same contribution to the outcome of a complete program, then they ought to be regarded as equal. We must fix what we mean by a complete program. Two considerations inform the definition. First, the dynamics of L{nat →} is given only for expressions without free variables, so a complete program should clearly be a closed expression. Second, the outcome of a computation should be observable, so that it is evident whether the outcome of two computations differs or not. We define a complete program to be a closed expression of type nat, and define the observable behavior of the program to be the numeral to which it evaluates. An experiment on, or observation about, an expression is any means of using that expression within a complete program. We define an expression context to be an expression with a “hole” in it serving as a placeholder for another expression. The hole is permitted to occur anywhere, including within the scope of a binder. The bound variables within whose scope the hole lies are said to be exposed to capture by the expression context. These variables may be assumed, without loss of generality, to be distinct from one another. A program context is a closed expression context of type nat— that is, it is a complete program with a hole in it. The meta-variable C VERSION 1.32 REVISED 05.15.2012 47.1 Observational Equivalence 499 stands for any expression context. Replacement is the process of filling a hole in an expression context, C, with an expression, e, which is written C{e}. Importantly, the free vari- ables of e that are exposed by C are captured by replacement (which is why replacement is not a form of substitution, which is defined so as to avoid capture). If C is a program context, then C{e} is a complete program iff all free variables of e are captured by the replacement. For example, if C = λ (x:nat) ◦, and e = x + x, then C{e} = λ (x:nat) x + x. The free occurrences of x in e are captured by the λ-abstraction as a result of the replacement of the hole in C by e. We sometimes write C{◦} to emphasize the occurrence of the hole in C. Expression contexts are closed under composition in that if C1 and C2 are expression contexts, then so is C{◦} ,C1{C2{◦}}, and we have C{e} = C1{C2{e}}. The trivial, or identity, expression context is the “bare hole”, written ◦, for which ◦{e} = e. The statics of expressions of L{nat →} is extended to expression con- texts by defining the typing judgment C:(Γ. τ) (Γ0 . τ0) so that if Γ ` e : τ, then Γ0 ` C{e}: τ0. This judgment may be inductively defined by a collection of rules derived from the statics of L{nat →} (see Rules (9.1)). Some representative rules are as follows: ◦ :(Γ. τ) (Γ. τ)(47.1a) C:(Γ. τ) (Γ0 . nat) s(C):(Γ. τ) (Γ0 . nat)(47.1b) C:(Γ. τ) (Γ0 . nat)Γ0 ` e0 : τ0 Γ0, x : nat, y : τ0 ` e1 : τ0 rec C{z ⇒ e0 | s(x) with y ⇒ e1}:(Γ. τ) (Γ0 . τ0)(47.1c) Γ0 ` e : nat C0 :(Γ. τ) (Γ0 . τ0)Γ0, x : nat, y : τ0 ` e1 : τ0 rec e {z ⇒ C0 | s(x) with y ⇒ e1}:(Γ. τ) (Γ0 . τ0)(47.1d) Γ0 ` e : nat Γ0 ` e0 : τ0 C1 :(Γ. τ) (Γ0, x : nat, y : τ0 . τ0) rec e {z ⇒ e0 | s(x) with y ⇒ C1}:(Γ. τ) (Γ0 . τ0)(47.1e) REVISED 05.15.2012 VERSION 1.32 500 47.1 Observational Equivalence C2 :(Γ. τ) (Γ0, x : τ1 . τ2) λ (x:τ1)C2 :(Γ. τ) (Γ0 . τ1 → τ2)(47.1f) C1 :(Γ. τ) (Γ0 . τ2 → τ0)Γ0 ` e2 : τ2 C1(e2):(Γ. τ) (Γ0 . τ0)(47.1g) Γ0 ` e1 : τ2 → τ0 C2 :(Γ. τ) (Γ0 . τ2) e1(C2):(Γ. τ) (Γ0 . τ0)(47.1h) Lemma 47.1. If C:(Γ. τ) (Γ0 . τ0), then Γ0 ⊆ Γ, and if Γ ` e : τ, then Γ0 ` C{e}: τ0. Contexts are closed under composition, with the trivial context acting as an identity for it. Lemma 47.2. If C:(Γ. τ) (Γ0 . τ0), and C0 :(Γ0 . τ0) (Γ00 . τ00), then C0{C{◦}} :(Γ. τ) (Γ00 . τ00). Lemma 47.3. If C:(Γ. τ) (Γ0 . τ0) and x /∈ dom(Γ), then C:(Γ, x : ρ . τ) (Γ0, x : ρ . τ0). Proof. By induction on Rules (47.1). A complete program is a closed expression of type nat. Definition 47.4. Two complete programs, e and e0, are Kleene equal, written e ' e0, iff there exists n ≥ 0 such that e 7→∗ n and e0 7→∗ n. Kleene equality is evidently reflexive and symmetric; transitivity fol- lows from determinacy of evaluation. Closure under converse evaluation also follows directly from determinacy. It is immediate from the definition that 0 6' 1. Definition 47.5. Suppose that Γ ` e : τ and Γ ` e0 : τ are two expressions of the same type. Two such expressions are observationally equivalent, written Γ ` e ∼= e0 : τ, iff C{e}'C{e0} for every program context C:(Γ. τ) (∅ . nat). In other words, for all possible experiments, the outcome of an experiment on e is the same as the outcome on e0. This is obviously an equivalence relation. For the sake of brevity, we often write e ∼=τ e0 for ∅ ` e ∼= e0 : τ. A family of equivalence relations Γ ` e1 E e2 : τ is a congruence iff it is preserved by all contexts. That is, if Γ ` e E e0 : τ, then Γ0 ` C{e}EC{e0}: τ0 for every expression context C:(Γ. τ) (Γ0 . τ0). Such a family of rela- tions is consistent iff ∅ ` e E e0 : nat implies e ' e0. VERSION 1.32 REVISED 05.15.2012 47.1 Observational Equivalence 501 Theorem 47.6. Observational equivalence is the coarsest consistent congruence on expressions. Proof. Consistency follows directly from the definition by noting that the trivial context is a program context. Observational equivalence is obviously an equivalence relation. To show that it is a congruence, we need only ob- serve that type-correct composition of a program context with an arbitrary expression context is again a program context. Finally, it is the coarsest such equivalence relation, for if Γ ` e E e0 : τ for some consistent congruence E, and if C:(Γ. τ) (∅ . nat), then by congruence ∅ ` C{e}EC{e0}: nat, and hence by consistency C{e}'C{e0}. A closing substitution, γ, for the typing context Γ = x1 : τ1,..., xn : τn is a finite function assigning closed expressions e1 : τ1,..., en : τn to x1,..., xn, respectively. We write ˆγ(e) for the substitution [e1,..., en/x1,..., xn]e, and write γ :Γ to mean that if x : τ occurs in Γ, then there exists a closed expression, e, such that γ(x) = e and e : τ. We write γ ∼=Γ γ0, where γ :Γ and γ0 :Γ, to express that γ(x) ∼=Γ(x) γ0(x) for each x declared in Γ. Lemma 47.7. If Γ ` e ∼= e0 : τ and γ :Γ, then ˆγ(e) ∼=τ ˆγ(e0). Moreover, if γ ∼=Γ γ0, then ˆγ(e) ∼=τ bγ0(e) and ˆγ(e0) ∼=τ bγ0(e0). Proof. Let C:(∅ . τ) (∅ . nat) be a program context; we are to show that C{ ˆγ(e)}'C{ ˆγ(e0)}. Because C has no free variables, this is equivalent to showing that ˆγ(C{e})' ˆγ(C{e0}). Let D be the context λ (x1:τ1)... λ (xn:τn) C{◦}(e1)...(en), where Γ = x1 : τ1,..., xn : τn and γ(x1) = e1,..., γ(xn) = en. By Lemma 47.3 we have C:(Γ. τ) (Γ. nat), from which it follows directly that D: (Γ. τ) (∅ . nat). Because Γ ` e ∼= e0 : τ, we have D{e}'D{e0}. But by construction D{e}' ˆγ(C{e}), and D{e0}' ˆγ(C{e0}), so ˆγ(C{e})' ˆγ(C{e0}). Because C is arbitrary, it follows that ˆγ(e) ∼=τ ˆγ(e0). Defining D0 similarly to D, but based on γ0, rather than γ, we may also show that D0{e}'D0{e0}, and hence bγ0(e) ∼=τ bγ0(e0). Now if γ ∼=Γ γ0, then by congruence we have D{e} ∼=nat D0{e}, and D{e0} ∼=nat D0{e0}. It follows that D{e} ∼=nat D0{e0}, and so, by consistency of observational equivalence, we have D{e}'D0{e0}, which is to say that ˆγ(e) ∼=τ bγ0(e0). Theorem 47.6 licenses the principle of proof by coinduction: to show that Γ ` e ∼= e0 : τ, it is enough to exhibit a consistent congruence, E, such that Γ ` e E e0 : τ. It can be difficult to construct such a relation. In the next section we will provide a general method for doing so that exploits types. REVISED 05.15.2012 VERSION 1.32 502 47.2 Logical Equivalence 47.2 Logical Equivalence The key to simplifying reasoning about observational equivalence is to ex- ploit types. Informally, we may classify the uses of expressions of a type into two broad categories, the passive and the active uses. The passive uses are those that merely manipulate expressions without actually inspecting them. For example, we may pass an expression of type τ to a function that merely returns it. The active uses are those that operate on the expression itself; these are the elimination forms associated with the type of that ex- pression. For the purposes of distinguishing two expressions, it is only the active uses that matter; the passive uses merely manipulate expressions at arm’s length, affording no opportunities to distinguish one from another. This leads to the definition of logical equivalence alluded to in the in- troduction. Definition 47.8. Logical equivalence is a family of relations e ∼τ e0 between closed expressions of type τ. It is defined by induction on τ as follows: e ∼nat e0 iff e ' e0 e ∼τ1→τ2 e0 iff if e1 ∼τ1 e0 1, then e(e1) ∼τ2 e0(e0 1) The definition of logical equivalence at type nat licenses the following principle of proof by nat-induction. To show that E(e, e0) whenever e ∼nat e0, it is enough to show that 1. E(0, 0), and 2. if E(n, n), then E(n + 1, n + 1). This is, of course, justified by mathematical induction on n ≥ 0, where e 7→∗ n and e0 7→∗ n by the definition of Kleene equivalence. Lemma 47.9. Logical equivalence is symmetric and transitive: if e ∼τ e0, then e0 ∼τ e, and if e ∼τ e0 and e0 ∼τ e00, then e ∼τ e00. Proof. Simultaneously, by induction on the structure of τ. If τ = nat, the result is immediate. If τ = τ1 → τ2, then we may assume that logical equiv- alence is symmetric and transitive at types τ1 and τ2. For symmetry, assume that e ∼τ e0; we wish to show e0 ∼τ e. Assume that e0 1 ∼τ1 e1; it suffices to show that e0(e0 1) ∼τ2 e(e1). By induction we have that e1 ∼τ1 e0 1. Therefore by assumption e(e1) ∼τ2 e0(e0 1), and hence by induction e0(e0 1) ∼τ2 e(e1). For transitivity, assume that e ∼τ e0 and e0 ∼τ e00; we are to show e ∼τ e00. VERSION 1.32 REVISED 05.15.2012 47.3 Logical and Observational Equivalence Coincide 503 Suppose that e1 ∼τ1 e00 1 ; it is enough to show that e(e1) ∼τ e00(e00 1 ). By sym- metry and transitivity we have e1 ∼τ1 e1, so by assumption e(e1) ∼τ2 e0(e1). We also have by assumption e0(e1) ∼τ2 e00(e00 1 ). By transitivity we have e0(e1) ∼τ2 e00(e00 1 ), which suffices for the result. Logical equivalence is extended to open terms by substitution of related closed terms to obtain related results. If γ and γ0 are two substitutions for Γ, we define γ ∼Γ γ0 to hold iff γ(x) ∼Γ(x) γ0(x) for every variable, x, such that Γ ` x : τ. Open logical equivalence, written Γ ` e ∼ e0 : τ, is defined to mean that ˆγ(e) ∼τ bγ0(e0) whenever γ ∼Γ γ0. Lemma 47.10. Open logical equivalence is symmetric and transitive. Proof. Follows immediately from Lemma 47.9 and the definition of open logical equivalence. At this point we are “two thirds of the way” to justifying the use of the name “open logical equivalence.” The remaining third, reflexivity, is established in the next section. 47.3 Logical and Observational Equivalence Coincide In this section we prove the coincidence of observational and logical equiv- alence. Lemma 47.11 (Converse Evaluation). Suppose that e ∼τ e0. If d 7→ e, then d ∼τ e0, and if d0 7→ e0, then e ∼τ d0. Proof. By induction on the structure of τ. If τ = nat, then the result follows from the closure of Kleene equivalence under converse evaluation. If τ = τ1 → τ2, then suppose that e ∼τ e0, and d 7→ e. To show that d ∼τ e0, we assume e1 ∼τ1 e0 1 and show d(e1) ∼τ2 e0(e0 1). It follows from the assumption that e(e1) ∼τ2 e0(e0 1). Noting that d(e1) 7→ e(e1), the result follows by induction. Lemma 47.12 (Consistency). If e ∼nat e0, then e ' e0. Proof. Immediate, from Definition 47.8. Theorem 47.13 (Reflexivity). If Γ ` e : τ, then Γ ` e ∼ e : τ. REVISED 05.15.2012 VERSION 1.32 504 47.3 Logical and Observational Equivalence Coincide Proof. We are to show that if Γ ` e : τ and γ ∼Γ γ0, then ˆγ(e) ∼τ bγ0(e). The proof proceeds by induction on typing derivations; we consider two representative cases. Consider the case of Rule (8.4a), in which τ = τ1 → τ2 and e = λ (x:τ1) e2. We are to show that λ (x:τ1) ˆγ(e2) ∼τ1→τ2 λ (x:τ1) bγ0(e2). Assume that e1 ∼τ1 e0 1; by Lemma 47.11, it is enough to show that [e1/x] ˆγ(e2) ∼τ2 [e0 1/x] bγ0(e2). Let γ2 = γ ⊗ x ,→ e1 and γ0 2 = γ0 ⊗ x ,→ e0 1, and observe that γ2 ∼Γ,x:τ1 γ0 2. Therefore, by induction we have ˆγ2(e2) ∼τ2 ˆγ0 2(e2), from which the result follows directly. Now consider the case of Rule (9.1d), for which we are to show that rec( ˆγ(e); ˆγ(e0); x.y. ˆγ(e1)) ∼τ rec( bγ0(e); bγ0(e0); x.y. bγ0(e1)). By the induction hypothesis applied to the first premise of Rule (9.1d), we have ˆγ(e) ∼nat bγ0(e). We proceed by nat-induction. It suffices to show that rec(z; ˆγ(e0); x.y. ˆγ(e1)) ∼τ rec(z; bγ0(e0); x.y. bγ0(e1)), (47.2) and that rec(s(n); ˆγ(e0); x.y. ˆγ(e1)) ∼τ rec(s(n); bγ0(e0); x.y. bγ0(e1)), (47.3) assuming rec(n; ˆγ(e0); x.y. ˆγ(e1)) ∼τ rec(n; bγ0(e0); x.y. bγ0(e1)). (47.4) To show (47.2), by Lemma 47.11 it is enough to show that ˆγ(e0) ∼τ bγ0(e0). This is assured by the outer inductive hypothesis applied to the second premise of Rule (9.1d). To show (47.3), define δ = γ ⊗ x ,→ n ⊗ y ,→ rec(n; ˆγ(e0); x.y. ˆγ(e1)) and δ0 = γ0 ⊗ x ,→ n ⊗ y ,→ rec(n; bγ0(e0); x.y. bγ0(e1)). By (47.4) we have δ ∼Γ,x:nat,y:τ δ0. Consequently, by the outer inductive hypothesis applied to the third premise of Rule (9.1d), and Lemma 47.11, the required follows. VERSION 1.32 REVISED 05.15.2012 47.3 Logical and Observational Equivalence Coincide 505 Corollary 47.14 (Equivalence). Open logical equivalence is an equivalence rela- tion. Corollary 47.15 (Termination). If e : nat, then e 7→∗ e0 for some e0 val. Lemma 47.16 (Congruence). If C0 :(Γ. τ) (Γ0 . τ0), and Γ ` e ∼ e0 : τ, then Γ0 ` C0{e} ∼ C0{e0}: τ0. Proof. By induction on the derivation of the typing of C0. We consider a rep- resentative case in which C0 = λ (x:τ1)C2 so that C0 :(Γ. τ) (Γ0 . τ1 → τ2) and C2 :(Γ. τ) (Γ0, x : τ1 . τ2). Assuming Γ ` e ∼ e0 : τ, we are to show that Γ0 ` C0{e} ∼ C0{e0}: τ1 → τ2, which is to say Γ0 ` λ (x:τ1)C2{e} ∼ λ (x:τ1)C2{e0}: τ1 → τ2. We know, by induction, that Γ0, x : τ1 ` C2{e} ∼ C2{e0}: τ2. Suppose that γ0 ∼Γ0 γ0 0, and that e1 ∼τ1 e0 1. Let γ1 = γ0 ⊗ x ,→ e1, γ0 1 = γ0 0 ⊗ x ,→ e0 1, and observe that γ1 ∼Γ0,x:τ1 γ0 1. By Definition 47.8 it is enough to show that ˆγ1(C2{e}) ∼τ2 ˆγ0 1(C2{e0}), which follows immediately from the inductive hypothesis. Theorem 47.17. If Γ ` e ∼ e0 : τ, then Γ ` e ∼= e0 : τ. Proof. By Lemmas 47.12 and 47.16, and Theorem 47.6. Corollary 47.18. If e : nat, then e ∼=nat n, for some n ≥ 0. Proof. By Theorem 47.13 we have e ∼nat e. Hence for some n ≥ 0, we have e ∼nat n, and so by Theorem 47.17, e ∼=nat n. Lemma 47.19. For closed expressions e : τ and e0 : τ, if e ∼=τ e0, then e ∼τ e0. Proof. We proceed by induction on the structure of τ. If τ = nat, consider the empty context to obtain e ' e0, and hence e ∼nat e0. If τ = τ1 → τ2, then we are to show that whenever e1 ∼τ1 e0 1, we have e(e1) ∼τ2 e0(e0 1). By Theorem 47.17 we have e1 ∼=τ1 e0 1, and hence by congruence of obser- vational equivalence it follows that e(e1) ∼=τ2 e0(e0 1), from which the result follows by induction. REVISED 05.15.2012 VERSION 1.32 506 47.4 Some Laws of Equality Theorem 47.20. If Γ ` e ∼= e0 : τ, then Γ ` e ∼ e0 : τ. Proof. Assume that Γ ` e ∼= e0 : τ, and that γ ∼Γ γ0. By Theorem 47.17 we have γ ∼=Γ γ0, so by Lemma 47.7 ˆγ(e) ∼=τ bγ0(e0). Therefore, by Lemma 47.19, ˆγ(e) ∼τ ˆγ(e0). Corollary 47.21. Γ ` e ∼= e0 : τ iff Γ ` e ∼ e0 : τ. The principle of symbolic evaluation states that definitional equivalence is sufficient for observational equivalence: Theorem 47.22. If Γ ` e ≡ e0 : τ, then Γ ` e ∼ e0 : τ, and hence Γ ` e ∼= e0 : τ. Proof. By an argument similar to that used in the proof of Theorem 47.13 and Lemma 47.16, then appealing to Theorem 47.17. Corollary 47.23. If e ≡ e0 : nat, then there exists n ≥ 0 such that e 7→∗ n and e0 7→∗ n. Proof. By Theorem 47.22 we have e ∼nat e0 and hence e ' e0. 47.4 Some Laws of Equality In this section we summarize some useful principles of observational equiv- alence for L{nat →}. For the most part these may be proved as laws of logical equivalence, and then transferred to observational equivalence by appeal to Corollary 47.21. The laws are presented as inference rules with the meaning that if all of the premises are true judgments about observa- tional equivalence, then so are the conclusions. In other words each rule is admissible as a principle of observational equivalence. 47.4.1 General Laws Logical equivalence is indeed an equivalence relation: it is reflexive, sym- metric, and transitive. Γ ` e ∼= e : τ (47.5a) Γ ` e0 ∼= e : τ Γ ` e ∼= e0 : τ (47.5b) Γ ` e ∼= e0 : τ Γ ` e0 ∼= e00 : τ Γ ` e ∼= e00 : τ (47.5c) VERSION 1.32 REVISED 05.15.2012 47.4 Some Laws of Equality 507 Reflexivity is an instance of a more general principle, that all defini- tional equivalences are observational equivalences. Γ ` e ≡ e0 : τ Γ ` e ∼= e0 : τ (47.6a) This is called the principle of symbolic evaluation. Observational equivalence is a congruence: we may replace equals by equals anywhere in an expression. Γ ` e ∼= e0 : τ C:(Γ. τ) (Γ0 . τ0) Γ0 ` C{e} ∼= C{e0}: τ0 (47.7a) Equivalence is stable under substitution for free variables, and substi- tuting equivalent expressions in an expression gives equivalent results. Γ ` e : τ Γ, x : τ ` e2 ∼= e0 2 : τ0 Γ ` [e/x]e2 ∼= [e/x]e0 2 : τ0 (47.8a) Γ ` e1 ∼= e0 1 : τ Γ, x : τ ` e2 ∼= e0 2 : τ0 Γ ` [e1/x]e2 ∼= [e0 1/x]e0 2 : τ0 (47.8b) 47.4.2 Equality Laws Two functions are equal if they are equal on all arguments. Γ, x : τ1 ` e(x) ∼= e0(x): τ2 Γ ` e ∼= e0 : τ1 → τ2 (47.9) Consequently, every expression of function type is equal to a λ-abstraction: Γ ` e ∼= λ (x:τ1) e(x): τ1 → τ2 (47.10) 47.4.3 Induction Law An equation involving a free variable, x, of type nat can be proved by in- duction on x. Γ ` [n/x]e ∼= [n/x]e0 : τ (for every n ∈ N) Γ, x : nat ` e ∼= e0 : τ (47.11a) To apply the induction rule, we proceed by mathematical induction on n ∈ N, which reduces to showing: 1. Γ ` [z/x]e ∼= [z/x]e0 : τ, and 2. Γ ` [s(n)/x]e ∼= [s(n)/x]e0 : τ, if Γ ` [n/x]e ∼= [n/x]e0 : τ. REVISED 05.15.2012 VERSION 1.32 508 47.5 Notes 47.5 Notes The technique of logical relations interprets types as relations (here, equiv- alence relations) by associating with each type constructor a relational ac- tion that transforms the relation interpreting its arguments to the relation interpreting the constructed type. Logical relations (Statman, 1985) are a fundamental tool in proof theory and provide the foundation for the se- mantics of the NuPRL type theory (Constable, 1986; Allen, 1987; Harper, 1992). The use of logical relations to characterize observational equivalence is essentially an adaptation of the NuPRL semantics to the simpler setting of G¨odel’s System T. VERSION 1.32 REVISED 05.15.2012 Chapter 48 Equational Reasoning for PCF In this Chapter we develop the theory of observational equivalence for L{nat *}, with an eager interpretation of the type of natural numbers. The development proceeds along lines similar to those in Chapter 47, but is complicated by the presence of general recursion. The proof depends on the concept of an admissible relation, one that admits the principle of proof by fixed point induction. 48.1 Observational Equivalence The definition of observational equivalence, along with the auxiliary notion of Kleene equivalence, are defined similarly to Chapter 47, but modified to account for the possibility of non-termination. The collection of well-formed L{nat *} contexts is inductively defined in a manner directly analogous to that in Chapter 47. Specifically, we define the judgment C:(Γ. τ) (Γ0 . τ0) by rules similar to Rules (47.1), mod- ified for L{nat *}. (We leave the precise definition as an exercise for the reader.) When Γ and Γ0 are empty, we write just C: τ τ0. A complete program is a closed expression of type nat. Definition 48.1. We say that two complete programs, e and e0, are Kleene equal, written e ' e0, iff for every n ≥ 0, e 7→∗ n iff e0 7→∗ n. Kleene equality is easily seen to be an equivalence relation and to be closed under converse evaluation. Moreover, 0 6' 1, and, if e and e0 are both divergent, then e ' e0. Observational equivalence is defined just as it is in Chapter 47. 510 48.2 Logical Equivalence Definition 48.2. We say that Γ ` e : τ and Γ ` e0 : τ are observationally, or contextually, equivalent iff for every program context C:(Γ. τ) (∅ . nat), C{e}'C{e0}. Theorem 48.3. Observational equivalence is the coarsest consistent congruence. Proof. See the proof of Theorem 47.6. Lemma 48.4 (Substitution and Functionality). If Γ ` e ∼= e0 : τ and γ :Γ, then ˆγ(e) ∼=τ ˆγ(e0). Moreover, if γ ∼=Γ γ0, then ˆγ(e) ∼=τ ˆγ0(e) and ˆγ(e0) ∼=τ ˆγ0(e0). Proof. See Lemma 47.7. 48.2 Logical Equivalence Definition 48.5. Logical equivalence, e ∼τ e0, between closed expressions of type τ is defined by induction on τ as follows: e ∼nat e0 iff e ' e0 e ∼τ1→τ2 e0 iff e1 ∼τ1 e0 1 implies e(e1) ∼τ2 e0(e0 1) Formally, logical equivalence is defined as in Chapter 47, except that the definition of Kleene equivalence is altered to account for non-termination. Logical equivalence is extended to open terms by substitution. Specifically, we define Γ ` e ∼ e0 : τ to mean that ˆγ(e) ∼τ bγ0(e0) whenever γ ∼Γ γ0. By the same argument as given in the proof of Lemma 47.9 logical equivalence is symmetric and transitive, as is its open extension. Lemma 48.6 (Strictness). If e : τ and e0 : τ are both divergent, then e ∼τ e0. Proof. By induction on the structure of τ. If τ = nat, then the result follows immediately from the definition of Kleene equivalence. If τ = τ1 → τ2, then e(e1) and e0(e0 1) diverge, so by induction e(e1) ∼τ2 e0(e0 1), as required. Lemma 48.7 (Converse Evaluation). Suppose that e ∼τ e0. If d 7→ e, then d ∼τ e0, and if d0 7→ e0, then e ∼τ d0. VERSION 1.32 REVISED 05.15.2012 48.3 Logical and Observational Equivalence Coincide 511 48.3 Logical and Observational Equivalence Coincide The proof of coincidence of logical and observational equivalence relies on the concept of bounded recursion, which we define by induction on m ≥ 0 as follows: fix0 x:τ is e , fix x:τ is x fixm+1 x:τ is e ,[fixm x:τ is e/x]e When m = 0, bounded recursion is defined to be a divergent expression of type τ. When m > 0, bounded recursion is defined by unrolling the re- cursion m times by iterated substitution. Intuitively, the bounded recursive expression fixm x:τ is e is as good as fix x:τ is e for up to m unrollings, after which it is divergent. It is easy to check that the follow rule is derivable for each m ≥ 0: Γ, x : τ ` e : τ Γ ` fixm[τ](x.e): τ .(48.1a) The proof is by induction on m ≥ 0, and amounts to an iteration of the substitution lemma for the statics of L{nat *}. The key property of bounded recursion is the principle of fixed point induction, which permits reasoning about a recursive computation by in- duction on the number of unrollings required to reach a value. The proof relies on compactness, which will be stated and proved in Section 48.4 below. Theorem 48.8 (Fixed Point Induction). Suppose that x : τ ` e : τ. If (∀m ≥ 0) fixm x:τ is e ∼τ fixm x:τ is e0, then fix x:τ is e ∼τ fix x:τ is e0. Proof. Define an applicative context,A, to be either a hole, ◦, or an appli- cation of the form A(e), where A is an applicative context. (The typ- ing judgment A: ρ τ is a special case of the general typing judgment for contexts.) Define logical equivalence of applicative contexts, written A ∼ A0 : ρ τ, by induction on the structure of A as follows: 1. ◦ ∼ ◦ : ρ ρ; 2. if A ∼ A0 : ρ τ2 → τ and e2 ∼τ2 e0 2, then A(e2) ∼ A0(e0 2): ρ τ. REVISED 05.15.2012 VERSION 1.32 512 48.3 Logical and Observational Equivalence Coincide We prove by induction on the structure of τ, if A ∼ A0 : ρ τ and for every m ≥ 0, A{fixm x:ρ is e} ∼τ A0{fixm x:ρ is e0}, (48.2) then A{fix x:ρ is e} ∼τ A0{fix x:ρ is e0}. (48.3) Choosing A = A0 = ◦ (so that ρ = τ) completes the proof. If τ = nat, then assume that A ∼ A0 : ρ nat and (48.2). By Defini- tion 48.5, we are to show A{fix x:ρ is e}'A0{fix x:ρ is e0}. By Corollary 48.17 there exists m ≥ 0 such that A{fix x:ρ is e}'A{fixm x:ρ is e}. By (48.2) we have A{fixm x:ρ is e}'A0{fixm x:ρ is e0}. By Corollary 48.17 A0{fixm x:ρ is e0}'A0{fix x:ρ is e0}. The result follows by transitivity of Kleene equivalence. If τ = τ1 * τ2, then by Definition 48.5, it is enough to show A{fix x:ρ is e}(e1) ∼τ2 A0{fix x:ρ is e0}(e0 1) whenever e1 ∼τ1 e0 1. Let A2 = A(e1) and A0 2 = A0(e0 1). It follows from (48.2) that for every m ≥ 0 A2{fixm x:ρ is e} ∼τ2 A0 2{fixm x:ρ is e0}. Noting that A2 ∼ A0 2 : ρ τ2, we have by induction A2{fix x:ρ is e} ∼τ2 A0 2{fix x:ρ is e0}, as required. Lemma 48.9 (Reflexivity). If Γ ` e : τ, then Γ ` e ∼ e : τ. VERSION 1.32 REVISED 05.15.2012 48.3 Logical and Observational Equivalence Coincide 513 Proof. The proof proceeds along the same lines as the proof of Theorem 47.13. The main difference is the treatment of general recursion, which is proved by fixed point induction. Consider Rule (10.1g). Assuming γ ∼Γ γ0, we are to show that fix x:τ is ˆγ(e) ∼τ fix x:τ is bγ0(e). By Theorem 48.8 it is enough to show that, for every m ≥ 0, fixm x:τ is ˆγ(e) ∼τ fixm x:τ is bγ0(e). We proceed by an inner induction on m. When m = 0 the result is im- mediate, because both sides of the desired equivalence diverge. Assum- ing the result for m, and applying Lemma 48.7, it is enough to show that ˆγ(e1) ∼τ bγ0(e1), where e1 = [fixm x:τ is ˆγ(e)/x] ˆγ(e), and (48.4) e0 1 = [fixm x:τ is bγ0(e)/x] bγ0(e). (48.5) But this follows directly from the inner and outer inductive hypotheses. For by the outer inductive hypothesis, if fixm x:τ is ˆγ(e) ∼τ fixm x:τ is bγ0(e), then [fixm x:τ is ˆγ(e)/x] ˆγ(e) ∼τ [fixm x:τ is bγ0(e)/x] bγ0(e). But the hypothesis holds by the inner inductive hypothesis, from which the result follows. Symmetry and transitivity of eager logical equivalence are easily estab- lished by induction on types, noting that Kleene equivalence is symmetric and transitive. Eager logical equivalence is therefore an equivalence rela- tion. Lemma 48.10 (Congruence). If C0 :(Γ. τ) (Γ0 . τ0), and Γ ` e ∼ e0 : τ, then Γ0 ` C0{e} ∼ C0{e0}: τ0. Proof. By induction on the derivation of the typing of C0, following along similar lines to the proof of Lemma 48.9. Logical equivalence is consistent, by definition. Consequently, it is con- tained in observational equivalence. Theorem 48.11. If Γ ` e ∼ e0 : τ, then Γ ` e ∼= e0 : τ. REVISED 05.15.2012 VERSION 1.32 514 48.4 Compactness Proof. By consistency and congruence of logical equivalence. Lemma 48.12. If e ∼=τ e0, then e ∼τ e0. Proof. By induction on the structure of τ. If τ = nat, then the result is immediate, because the empty expression context is a program context. If τ = τ1 → τ2, then suppose that e1 ∼τ1 e0 1. We are to show that e(e1) ∼τ2 e0(e0 1). By Theorem 48.11 e1 ∼=τ1 e0 1, and hence by Lemma 48.4 e(e1) ∼=τ2 e0(e0 1), from which the result follows by induction. Theorem 48.13. If Γ ` e ∼= e0 : τ, then Γ ` e ∼ e0 : τ. Proof. Assume that Γ ` e ∼= e0 : τ. Suppose that γ ∼Γ γ0. By Theorem 48.11 we have γ ∼=Γ γ0, and so by Lemma 48.4 we have ˆγ(e) ∼=τ ˆγ0(e0). Therefore by Lemma 48.12 we have ˆγ(e) ∼τ ˆγ0(e0). Corollary 48.14. Γ ` e ∼= e0 : τ iff Γ ` e ∼ e0 : τ. 48.4 Compactness The principle of fixed point induction is derived from a critical property of L{nat *}, called compactness. This property states that only finitely many unwindings of a fixed point expression are needed in a complete evaluation of a program. Although intuitively obvious (one cannot complete infinitely many recursive calls in a finite computation), it is rather tricky to state and prove rigorously. The proof of compactness (Theorem 48.16) makes use of the stack ma- chine for L{nat *} defined in Chapter 27, augmented with the following transitions for bounded recursive expressions: k . fix0 x:τ is e 7→ k . fix0 x:τ is e (48.6a) k . fixm+1 x:τ is e 7→ k .[fixm x:τ is e/x]e (48.6b) It is straightforward to extend the proof of correctness of the stack machine (Corollary 27.4) to account for bounded recursion. VERSION 1.32 REVISED 05.15.2012 48.4 Compactness 515 To get a feel for what is involved in the compactness proof, consider first the factorial function, f, in L{nat *}: fix f:nat * nat is λ (x:nat) ifz x {z ⇒ s(z) | s(x0) ⇒ x ∗ f(x0)}. Obviously evaluation of f(n) requires n recursive calls to the function it- self. This means that, for a given input, n, we may place a bound, m, on the recursion that is sufficient to ensure termination of the computation. This can be expressed formally using the m-bounded form of general recursion, fixm f:nat * nat is λ (x:nat) ifz x {z ⇒ s(z) | s(x0) ⇒ x ∗ f(x0)}. Call this expression f (m). It follows from the definition of f that if f(n) 7→∗ p, then f (m)(n) 7→∗ p for some m ≥ 0 (in fact, m = n suffices). When considering expressions of higher type, we cannot expect to get the same result from the bounded recursion as from the unbounded. For example, consider the addition function, a, of type τ = nat *(nat * nat), given by the expression fix p:τ is λ (x:nat) ifz x {z ⇒ id | s(x0) ⇒ s ◦ (p(x0))}, where id = λ (y:nat) y is the identity, e0 ◦ e = λ (x:τ) e0(e(x)) is compo- sition, and s = λ (x:nat) s(x) is the successor function. The application a(n) terminates after three transitions, regardless of the value of n, result- ing in a λ-abstraction. When n is positive, the result contains a residual copy of a itself, which is applied to n − 1 as a recursive call. The m-bounded ver- sion of a, written a(m), is also such that a(m)(n) terminates in three steps, provided that m > 0. But the result is not the same, because the residuals of a appear as a(m−1), rather than as a itself. Turning now to the proof of compactness, it is helpful to introduce some notation. Suppose that x : τ ` ex : τ for some arbitrary abstractor x.ex. Let f (ω) = fix x:τ is ex, and let f (m) = fixm x:τ is ex. Observe that f (ω): τ and f (m): τ for any m ≥ 0. The following technical lemma governing the stack machine permits the bound on occurrences of a recursive expression to be raised without affecting the outcome of evaluation. Lemma 48.15. For every m ≥ 0, if [ f (m)/y]k .[ f (m)/y]e 7→∗ e / n, then [ f (m+1)/y]k .[ f (m+1)/y]e 7→∗ e / n. Proof. By induction on m ≥ 0, and then induction on transition. REVISED 05.15.2012 VERSION 1.32 516 48.4 Compactness Theorem 48.16 (Compactness). Suppose that y : τ ` e : nat where y /∈ f (ω). If [ f (ω)/y]e 7→∗ n, then there exists m ≥ 0 such that [ f (m)/y]e 7→∗ n. Proof. We prove simultaneously the stronger statements that if [ f (ω)/y]k .[ f (ω)/y]e 7→∗ e / n, then for some m ≥ 0, [ f (m)/y]k .[ f (m)/y]e 7→∗ e / n, and if [ f (ω)/y]k /[ f (ω)/y]e 7→∗ e / n then for some m ≥ 0, [ f (m)/y]k /[ f (m)/y]e 7→∗ e / n. (Note that if [ f (ω)/y]e val, then [ f (m)/y]e val for all m ≥ 0.) The result then follows by the correctness of the stack machine (Corollary 27.4). We proceed by induction on transition. Suppose that the initial state is [ f (ω)/y]k . f (ω), which arises when e = y, and the transition sequence is as follows: [ f (ω)/y]k . f (ω) 7→ [ f (ω)/y]k .[ f (ω)/x]ex 7→∗ e / n. Noting that [ f (ω)/x]ex = [ f (ω)/y][y/x]ex, we have by induction that there exists m ≥ 0 such that [ f (m)/y]k .[ f (m)/x]ex 7→∗ e / n. By Lemma 48.15 [ f (m+1)/y]k .[ f (m)/x]ex 7→∗ e / n and we need only recall that [ f (m+1)/y]k . f (m+1) = [ f (m+1)/y]k .[ f (m)/x]ex to complete the proof. If, on the other hand, the initial step is an unrolling, but e 6= y, then we have for some z /∈ f (ω) and z 6= y [ f (ω)/y]k . fix z:τ is dω 7→ [ f (ω)/y]k .[fix z:τ is dω/z]dω 7→∗ e / n. VERSION 1.32 REVISED 05.15.2012 48.5 Co-Natural Numbers 517 where dω = [ f (ω)/y]d. By induction there exists m ≥ 0 such that [ f (m)/y]k .[fix z:τ is dm/z]dm 7→∗ e / n, where dm = [ f (m)/y]d. But then by Lemma 48.15 we have [ f (m+1)/y]k .[fix z:τ is dm+1/z]dm+1 7→∗ e / n, where dm+1 = [ f (m+1)/y]d, from which the result follows directly. Corollary 48.17. There exists m ≥ 0 such that [ f (ω)/y]e '[ f (m)/y]e. Proof. If [ f (ω)/y]e diverges, then taking m to be zero suffices. Otherwise, apply Theorem 48.16 to obtain m, and note that the required Kleene equiv- alence follows. 48.5 Co-Natural Numbers If we change the dynamics of the successor operator in L{nat *} so that s(e) is a value regardless of whether e is a value, then the type nat admits an infinite “number” ω = fix x:nat is s(x). We may think of ω as an infinite stack of successors, hence larger than any finite natural number. Obviously the principle of mathematical induction is not valid for this type, because we may prove by induction that every natural number is finite, whereas ω is infinite. When the successor is evaluated lazily, it is preferable to rename nat to conat, the type of co-natural numbers, which includes ω in addition to all the finite natural numbers. The definition of logical equivalence must be correspondingly altered to account for the conatural numbers. Rather than being defined induc- tively as the strongest relation closed under specified conditions, it is now defined coinductively as the weakest relation consistent with two analogous conditions. We may then show that two expressions are related using the principle of proof by coinduction. The definition of Kleene equivalence must be altered to account for the lazily evaluated successor operation. To account for ω, two computations are compared based solely on the outermost form of their values, if any. We define e ' e0 to hold iff (a) if e 7→∗ z, then e0 7→∗ z, and vice versa; and (b) if e 7→∗ s(e1), then e0 7→∗ s(e0 1), and vice versa. Corollary 48.17 can be proved for the co-natural numbers by essentially the same argument as before. REVISED 05.15.2012 VERSION 1.32 518 48.5 Co-Natural Numbers The definition of logical equivalence at type conat is defined to be the weakest equivalence relation, E, between closed terms of type conat satis- fying the following consistency conditions: if e E e0 : conat, then 1. If e 7→∗ z, then e0 7→∗ z, and vice versa. 2. If e 7→∗ s(e1), then e0 7→∗ s(e0 1) with e1 E e0 1 : conat, and vice versa. It is immediate that if e ∼conat e0, then e ' e0, and so logical equivalence is consistent. It is also strict in that if e and e0 are both divergent expressions of type conat, then e ∼conat e0. The principle of proof by coinduction states that to show e ∼conat e0, it suffices to exhibit a relation, E, such that 1. e E e0 : conat, and 2. E satisfies the above consistency conditions. If these requirements hold, then E is contained in logical equivalence at type conat, and hence e ∼conat e0, as required. As an application of coinduction, let us consider the proof of Theo- rem 48.8. The overall argument remains as before, but the proof for the type conat must be altered as follows. Suppose that A ∼ A0 : ρ conat, and let a = A{fix x:ρ is e} and a0 = A0{fix x:ρ is e0}. Writing a(m) = A{fixm x:ρ is e} and a0(m) = A0{fixm x:ρ is e0}, assume that for every m ≥ 0, a(m) ∼conat a0(m). We are to show that a ∼conat a0. Define the functions pn for n ≥ 0 on closed terms of type conat by the following equations: p0(d) = d p(n+1)(d) = (d0 if pn(d) 7→∗ s(d0) undefined otherwise For n ≥ 0, let an = pn(a) and a0 n = pn(a0). Correspondingly, let a(m) n = pn(a(m)) and a0 n (m) = pn(a0(m)). Define E to be the strongest relation such that an E a0 n : conat for all n ≥ 0. We will show that the relation E satisfies VERSION 1.32 REVISED 05.15.2012 48.6 Notes 519 the consistency conditions, and so it is contained in logical equivalence. Because a E a0 : conat (by construction), the result follows immediately. To show that E is consistent, suppose that an E a0 n : conat for some n ≥ 0. We have by Corollary 48.17 an ' a(m) n , for some m ≥ 0, and hence, by the assumption, an ' a0 n (m), and so by Corollary 48.17 again, a0 n (m)' a0 n. Now if an 7→∗ s(bn), then a(m) n 7→∗ s(b(m) n ) for some b(m) n , and hence there exists b0 n (m) such that a0 n (m) 7→∗ b0 n (m), and so there exists b0 n such that a0 n 7→∗ s(b0 n). But bn = pn+1(a) and b0 n = pn+1(a0), and we have bn E b0 n : conat by construction, as required. 48.6 Notes The use of logical relations to characterize observational equivalence for PCF is inspired by the treatment of partiality in type theory by Constable and Smith(1987) and by the studies of observational equivalence by Pitts (2000). Although the technical details differ, the proof of compactness here is inspired by Pitts’s structurally inductive characterization of termination using an abstract machine. It is critical to restrict attention to transition sys- tems whose states are complete programs (closed expressions of observable type). Structural operational semantics usually does not fulfill this require- ment, thereby requiring a considerably more complex argument than given here. REVISED 05.15.2012 VERSION 1.32 520 48.6 Notes VERSION 1.32 REVISED 05.15.2012 Chapter 49 Parametricity The motivation for introducing polymorphism was to enable more pro- grams to be written — those that are “generic” in one or more types, such as the composition function given in Chapter 20. Then if a program does not depend on the choice of types, we can code it using polymorphism. Moreover, if we wish to insist that a program can not depend on a choice of types, we demand that it be polymorphic. Thus polymorphism can be used both to expand the collection of programs we may write, and also to limit the collection of programs that are permissible in a given context. The restrictions imposed by polymorphic typing give rise to the expe- rience that in a polymorphic functional language, if the types are correct, then the program is correct. Roughly speaking, if a function has a poly- morphic type, then the strictures of type genericity vastly cut down the set of programs with that type. Thus if you have written a program with this type, it is quite likely to be the one you intended! The technical foundation for these remarks is called parametricity. The goal of this chapter is to give an account of parametricity for L{→∀} under a call-by-name interpretation. 49.1 Overview We will begin with an informal discussion of parametricity based on a “seat of the pants” understanding of the set of well-formed programs of a type. Suppose that a function value f has the type ∀(t.t → t). What function could it be? When instantiated at a type τ it should evaluate to a function g of type τ → τ that, when further applied to a value v of type τ returns a value v0 of type τ. Because f is polymorphic, g cannot depend on v, so v0 522 49.2 Observational Equivalence must be v. In other words, g must be the identity function at type τ, and f must therefore be the polymorphic identity. Suppose that f is a function of type ∀(t.t). What function could it be? A moment’s thought reveals that it cannot exist at all. For it must, when instantiated at a type τ, return a value of that type. But not every type has a value (including this one), so this is an impossible assignment. The only conclusion is that ∀(t.t) is an empty type. Let N be the type of polymorphic Church numerals introduced in Chap- ter 20, namely ∀(t.t → (t → t) → t). What are the values of this type? Given any type τ, and values z : τ and s : τ → τ, the expression f[τ](z)(s) must yield a value of type τ. Moreover, it must behave uniformly with respect to the choice of τ. What values could it yield? The only way to build a value of type τ is by using the element z and the function s passed to it. A moment’s thought reveals that the application must amount to the n-fold composition s(s(... s(z)...)). That is, the elements of N are in one-to-one correspondence with the natu- ral numbers. 49.2 Observational Equivalence The definition of observational equivalence given in Chapters 47 and 48 is based on identifying a type of answers that are observable outcomes of com- plete programs. Values of function type are not regarded as answers, but are treated as “black boxes” with no internal structure, only input-output behavior. In L{→∀}, however, there are no (closed) base types. Every type is either a function type or a polymorphic type, and hence no types suitable to serve as observable answers. One way to manage this difficulty is to augment L{→∀} with a base type of answers to serve as the observable outcomes of a computation. The only requirement is that this type have two elements that can be immedi- ately distinguished from each other by evaluation. We may achieve this by enriching L{→∀} with a base type, 2, containing two constants, tt and ff, that serve as possible answers for a complete computation. A complete program is a closed expression of type 2. Kleene equality is defined for complete programs by requiring that e ' e0 iff either (a) e 7→∗ tt and e0 7→∗ tt; or (b) e 7→∗ ff and e0 7→∗ ff. This is VERSION 1.32 REVISED 05.15.2012 49.2 Observational Equivalence 523 obviously an equivalence relation, and it is immediate that tt 6' ff, because these are two distinct constants. As before, we say that a type-indexed fam- ily of equivalence relations between closed expressions of the same type is consistent if it implies Kleene equality at the answer type, 2. To define observational equivalence, we must first define the concept of an expression context for L{→∀} as an expression with a “hole” in it. More precisely, we may give an inductive definition of the judgment C:(∆;Γ. τ) (∆0;Γ0 . τ0), which states that C is an expression context that, when filled with an ex- pression ∆;Γ ` e : τ yields an expression ∆0;Γ0 ` C{e}: τ. (We leave the precise definition of this judgment, and the verification of its properties, as an exercise for the reader.) Definition 49.1. Two expressions of the same type are observationally equiva- lent, ∆;Γ ` e ∼= e0 : τ, iff C{e}'C{e0} whenever C:(∆;Γ. τ) (∅; ∅ . 2). Lemma 49.2. Observational equivalence is the coarsest consistent congruence. Proof. Essentially the same as the the proof of Theorem 47.6. Lemma 49.3. 1. If ∆, t;Γ ` e ∼= e0 : τ and ρ type, then ∆;[ρ/t]Γ ` [ρ/t]e ∼= [ρ/t]e0 : [ρ/t]τ. 2. If ∅;Γ, x : ρ ` e ∼= e0 : τ and d : ρ, then ∅;Γ ` [d/x]e ∼= [d/x]e0 : τ. Moreover, if d ∼=ρ d0, then ∅;Γ ` [d/x]e ∼= [d0/x]e : τ and ∅;Γ ` [d/x]e0 ∼= [d0/x]e0 : τ. Proof. 1. Let C:(∆;[ρ/t]Γ.[ρ/t]τ) (∅ . 2) be a program context. We are to show that C{[ρ/t]e}'C{[ρ/t]e0}. Because C is closed, this is equivalent to [ρ/t]C{e}'[ρ/t]C{e0}. Let C0 be the context Λ(t.C{◦})[ρ], and observe that C0 :(∆, t;Γ. τ) (∅ . 2). Therefore, from the assumption, C0{e}'C0{e0}. But C0{e}'[ρ/t]C{e}, and C0{e0}'[ρ/t]C{e0}, from which the re- sult follows. REVISED 05.15.2012 VERSION 1.32 524 49.3 Logical Equivalence 2. By an argument essentially similar to that for Lemma 47.7. 49.3 Logical Equivalence In this section we introduce a form of logical equivalence that captures the informal concept of parametricity, and also provides a characterization of observational equivalence. This will permit us to derive properties of ob- servational equivalence of polymorphic programs of the kind suggested earlier. The definition of logical equivalence for L{→∀} is somewhat more complex than for L{nat →}. The main idea is to define logical equiva- lence for a polymorphic type, ∀(t.τ) to satisfy a very strong condition that captures the essence of parametricity. As a first approximation, we might say that two expressions, e and e0, of this type should be logically equiva- lent if they are logically equivalent for “all possible” interpretations of the type t. More precisely, we might require that e[ρ] be related to e0[ρ] at type [ρ/t]τ, for any choice of type ρ. But this runs into two problems, one technical, the other conceptual. The same device will be used to solve both problems. The technical problem stems from impredicativity. In Chapter 47 logi- cal equivalence is defined by induction on the structure of types. But when polymorphism is impredicative, the type [ρ/t]τ might well be larger than ∀(t.τ). At the very least we would have to justify the definition of logical equivalence on some other grounds, but no criterion appears to be avail- able. The conceptual problem is that, even if we could make sense of the definition of logical equivalence, it would be too restrictive. For such a def- inition amounts to saying that the unknown type t is to be interpreted as logical equivalence at whatever type it turns out to be when instantiated. To obtain useful parametricity results, we shall ask for much more than this. What we shall do is to consider separately instances of e and e0 by types ρ and ρ0, and treat the type variable t as standing for any relation (of some form) between ρ and ρ0. We may suspect that this is asking too much: per- haps logical equivalence is the empty relation. Surprisingly, this is not the case, and indeed it is this very feature of the definition that we shall exploit to derive parametricity results about the language. To manage both of these problems we will consider a generalization of logical equivalence that is parameterized by a relational interpretation of the free type variables of its classifier. The parameters determine a sepa- VERSION 1.32 REVISED 05.15.2012 49.3 Logical Equivalence 525 rate binding for each free type variable in the classifier for each side of the equation, with the discrepancy being mediated by a specified relation be- tween them. Thus related expressions need not have the same type, with the differences between them mediated by the given relation. We will restrict attention to a certain collection of “admissible” binary relations between closed expressions. The conditions are imposed to ensure that logical equivalence and observational equivalence coincide. Definition 49.4 (Admissibility). A relation R between expressions of types ρ and ρ0 is admissible, written R : ρ ↔ ρ0, iff it satisfies two requirements: 1. Respect for observational equivalence: if R(e, e0) and d ∼=ρ e and d0 ∼=ρ0 e0, then R(d, d0). 2. Closure under converse evaluation: if R(e, e0), then if d 7→ e, then R(d, e0) and if d0 7→ e0, then R(e, d0). Closure under converse evaluation will turn out to be a consequence of respect for observational equivalence, but we are not yet in a position to establish this fact. The judgment δ : ∆ states that δ is a type substitution that assigns a closed type to each type variable t ∈ ∆. A type substitution, δ, induces a substitu- tion function, ˆδ, on types given by the equation ˆδ(τ) = [δ(t1),..., δ(tn)/t1,..., tn]τ, and similarly for expressions. Substitution is extended to contexts point- wise by defining ˆδ(Γ)(x) = ˆδ(Γ(x)) for each x ∈ dom(Γ). Let δ and δ0 be two type substitutions of closed types to the type vari- ables in ∆. An admissible relation assignment, η, between δ and δ0 is an as- signment of an admissible relation η(t): δ(t) ↔ δ0(t) to each t ∈ ∆. The judgment η : δ ↔ δ0 states that η is an admissible relation assignment be- tween δ and δ0. Logical equivalence is defined in terms of its generalization, called para- metric logical equivalence, written e ∼τ e0 [η : δ ↔ δ0], defined as follows. Definition 49.5 (Parametric Logical Equivalence). The relation e ∼τ e0 [η : δ ↔ δ0] is defined by induction on the structure of τ by the following conditions: e ∼t e0 [η : δ ↔ δ0] iff η(t)(e, e0) e ∼2 e0 [η : δ ↔ δ0] iff e ' e0 e ∼τ1→τ2 e0 [η : δ ↔ δ0] iff e1 ∼τ1 e0 1 [η : δ ↔ δ0] implies e(e1) ∼τ2 e0(e0 1)[η : δ ↔ δ0] e ∼∀(t.τ) e0 [η : δ ↔ δ0] iff for every ρ, ρ0, and every admissible R : ρ ↔ ρ0, e[ρ] ∼τ e0[ρ0][η ⊗ t ,→ R: δ ⊗ t ,→ ρ ↔ δ0 ⊗ t ,→ ρ0] REVISED 05.15.2012 VERSION 1.32 526 49.3 Logical Equivalence Logical equivalence is defined in terms of parametric logical equiva- lence by considering all possible interpretations of its free type- and ex- pression variables. An expression substitution, γ, for a context Γ, written γ :Γ, is a substitution of a closed expression γ(x):Γ(x) to each variable x ∈ dom(Γ). An expression substitution, γ :Γ, induces a substitution func- tion, ˆγ, defined by the equation ˆγ(e) = [γ(x1),..., γ(xn)/x1,..., xn]e, where the domain of Γ consists of the variables x1,..., xn. The relation γ ∼Γ γ0 [η : δ ↔ δ0] is defined to hold iff dom(γ) = dom(γ0) = dom(Γ), and γ(x) ∼Γ(x) γ0(x)[η : δ ↔ δ0] for every variable, x, in their common domain. Definition 49.6 (Logical Equivalence). The expressions ∆;Γ ` e : τ and ∆;Γ ` e0 : τ are logically equivalent, written ∆;Γ ` e ∼ e0 : τ iff, for every assignment δ and δ0 of closed types to type variables in ∆, and every admissi- ble relation assignment η : δ ↔ δ0, if γ ∼Γ γ0 [η : δ ↔ δ0], then ˆγ( ˆδ(e)) ∼τ bγ0(bδ0(e0))[η : δ ↔ δ0]. When e, e0, and τ are closed, this definition states that e ∼τ e0 iff e ∼τ e0 [∅ : ∅ ↔ ∅], so that logical equivalence is indeed a special case of its generalization. Lemma 49.7 (Closure under Converse Evaluation). Suppose that e ∼τ e0 [η : δ ↔ δ0]. If d 7→ e, then d ∼τ e0, and if d0 7→ e0, then e ∼τ d0. Proof. By induction on the structure of τ. When τ = t, the result holds by the definition of admissibility. Otherwise the result follows by induction, making use of the definition of the transition relation for applications and type applications. Lemma 49.8 (Respect for Observational Equivalence). Suppose that e ∼τ e0 [η : δ ↔ δ0]. If d ∼= ˆδ(τ) e and d0 ∼=bδ0(τ) e0, then d ∼τ d0 [η : δ ↔ δ0]. Proof. By induction on the structure of τ, relying on the definition of ad- missibility, and the congruence property of observational equivalence. For example, if τ = ∀(t.τ2), then we are to show that for every admissible R: ρ ↔ ρ0, d[ρ] ∼τ2 d0[ρ0][η ⊗ t ,→ R: δ ⊗ t ,→ ρ ↔ δ0 ⊗ t ,→ ρ0]. VERSION 1.32 REVISED 05.15.2012 49.3 Logical Equivalence 527 Because observational equivalence is a congruence, we have d[ρ] ∼=[ρ/t] ˆδ(τ2) e[ρ], and d0[ρ0] ∼=[ρ0/t]bδ0(τ2) e0[ρ]. It follows that e[ρ] ∼τ2 e0[ρ0][η ⊗ t ,→ R: δ ⊗ t ,→ ρ ↔ δ0 ⊗ t ,→ ρ0], from which the result follows by induction. Corollary 49.9. The relation e ∼τ e0 [η : δ ↔ δ0] is an admissible relation be- tween closed types ˆδ(τ) and bδ0(τ). Proof. By Lemmas 49.7 and 49.8. Corollary 49.10. If ∆;Γ ` e ∼ e0 : τ, and ∆;Γ ` d ∼= e : τ and ∆;Γ ` d0 ∼= e0 : τ, then ∆;Γ ` d ∼ d0 : τ. Proof. By Lemma 49.3 and Corollary 49.9. Lemma 49.11 (Compositionality). Let R : ˆδ(ρ) ↔ bδ0(ρ) be the relational inter- pretation of some type ρ, which is to say R(d, d0) holds iff d ∼ρ d0 [η : δ ↔ δ0]. Then e ∼[ρ/t]τ e0 [η : δ ↔ δ0] if, and only if, e ∼τ e0 [η ⊗ t ,→ R: δ ⊗ t ,→ ˆδ(ρ) ↔ δ0 ⊗ t ,→ bδ0(ρ)]. Proof. By induction on the structure of τ. When τ = t, the result is im- mediate from the choice of the relation R. When τ = t0 6= t, the result follows directly from Definition 49.5. When τ = τ1 → τ2, the result follows by induction, using Definition 49.5. Similarly, when or τ = ∀(u.τ1), the result follows by induction, noting that we may assume, without loss of generality, that u 6= t and u /∈ ρ. Despite the strong conditions on polymorphic types, logical equiva- lence is not overly restrictive—every expression satisfies its constraints. This result is sometimes called the parametricity theorem. Theorem 49.12 (Parametricity). If ∆;Γ ` e : τ, then ∆;Γ ` e ∼ e : τ. Proof. By rule induction on the statics of L{→∀} given by Rules (20.2). We consider two representative cases here. Rule (20.2d) Suppose δ : ∆, δ0 : ∆, η : δ ↔ δ0, and γ ∼Γ γ0 [η : δ ↔ δ0]. By induction we have that for all ρ, ρ0, and admissible R: ρ ↔ ρ0, [ρ/t] ˆγ( ˆδ(e)) ∼τ [ρ0/t] bγ0(bδ0(e))[η∗ : δ∗ ↔ δ0 ∗], REVISED 05.15.2012 VERSION 1.32 528 49.3 Logical Equivalence where η∗ = η ⊗ t ,→ R, δ∗ = δ ⊗ t ,→ ρ, and δ0 ∗ = δ0 ⊗ t ,→ ρ0. Because Λ(t. ˆγ( ˆδ(e)))[ρ] 7→∗ [ρ/t] ˆγ( ˆδ(e)) and Λ(t. bγ0(bδ0(e)))[ρ0] 7→∗ [ρ0/t] bγ0(bδ0(e)), the result follows by Lemma 49.7. Rule (20.2e) Suppose δ : ∆, δ0 : ∆, η : δ ↔ δ0, and γ ∼Γ γ0 [η : δ ↔ δ0]. By induction we have ˆγ( ˆδ(e)) ∼∀(t.τ) bγ0(bδ0(e))[η : δ ↔ δ0] Let ˆρ = ˆδ(ρ) and ˆρ0 = bδ0(ρ). Define the relation R: ˆρ ↔ ˆρ0 by R(d, d0) iff d ∼ρ d0 [η : δ ↔ δ0]. By Corollary 49.9, this relation is admissible. By the definition of logical equivalence at polymorphic types, we ob- tain ˆγ( ˆδ(e))[ ˆρ] ∼τ bγ0(bδ0(e))[ ˆρ0][η ⊗ t ,→ R: δ ⊗ t ,→ ˆρ ↔ δ0 ⊗ t ,→ ˆρ0]. By Lemma 49.11 ˆγ( ˆδ(e))[ ˆρ] ∼[ρ/t]τ bγ0(bδ0(e))[ ˆρ0][η : δ ↔ δ0] But ˆγ( ˆδ(e))[ ˆρ] = ˆγ( ˆδ(e))[ ˆδ(ρ)](49.1) = ˆγ( ˆδ(e[ρ])), (49.2) and similarly bγ0(bδ0(e))[ ˆρ0] = bγ0(bδ0(e))[bδ0(ρ)](49.3) = bγ0(bδ0(e[ρ])), (49.4) from which the result follows. Corollary 49.13. If ∆;Γ ` e ∼= e0 : τ, then ∆;Γ ` e ∼ e0 : τ. Proof. By Theorem 49.12 ∆;Γ ` e ∼ e : τ, and hence by Corollary 49.10, ∆;Γ ` e ∼ e0 : τ. VERSION 1.32 REVISED 05.15.2012 49.3 Logical Equivalence 529 Lemma 49.14 (Congruence). If ∆;Γ ` e ∼ e0 : τ and C:(∆;Γ. τ) (∆0;Γ0 . τ0), then ∆0;Γ0 ` C{e} ∼ C{e0}: τ0. Proof. By induction on the structure of C, following along very similar lines to the proof of Theorem 49.12. Lemma 49.15 (Consistency). Logical equivalence is consistent. Proof. Follows immediately from the definition of logical equivalence. Corollary 49.16. If ∆;Γ ` e ∼ e0 : τ, then ∆;Γ ` e ∼= e0 : τ. Proof. By Lemma 49.15 Logical equivalence is consistent, and by Lemma 49.14, it is a congruence, and hence is contained in observational equivalence. Corollary 49.17. Logical and observational equivalence coincide. Proof. By Corollaries 49.13 and 49.16. If d : τ and d 7→ e, then d ∼τ e, and hence by Corollary 49.16, d ∼=τ e. Therefore if a relation respects observational equivalence, it must also be closed under converse evaluation. This shows that the second condition on admissibility is redundant, now that we have established the coincidence of logical and observational equivalence. Corollary 49.18 (Extensionality). 1.e ∼=τ1→τ2 e0 iff for all e1 : τ1, e(e1) ∼=τ2 e0(e1). 2.e ∼=∀(t.τ) e0 iff for all ρ, e[ρ] ∼=[ρ/t]τ e0[ρ]. Proof. The forward direction is immediate in both cases, because observa- tional equivalence is a congruence, by definition. The backward direction is proved similarly in both cases, by appeal to Theorem 49.12. In the first case, by Corollary 49.17 it suffices to show that e ∼τ1→τ2 e0. To this end suppose that e1 ∼τ1 e0 1. We are to show that e(e1) ∼τ2 e0(e0 1). By the as- sumption we have e(e0 1) ∼=τ2 e0(e0 1). By parametricity we have e ∼τ1→τ2 e, and hence e(e1) ∼τ2 e(e0 1). The result then follows by Lemma 49.8. In the second case, by Corollary 49.17 it is sufficient to show that e ∼∀(t.τ) e0. Suppose that R: ρ ↔ ρ0 for some closed types ρ and ρ0. It suffices to show that e[ρ] ∼τ e0[ρ0][η : δ ↔ δ0], where η(t) = R, δ(t) = ρ, and δ0(t) = ρ0. By the assumption we have e[ρ0] ∼=[ρ0/t]τ e0[ρ0]. By parametric- ity e ∼∀(t.τ) e, and hence e[ρ] ∼τ e0[ρ0][η : δ ↔ δ0]. The result then follows by Lemma 49.8. REVISED 05.15.2012 VERSION 1.32 530 49.4 Parametricity Properties Lemma 49.19 (Identity Extension). Let η : δ ↔ δ be such that η(t) is observa- tional equivalence at type δ(t) for each t ∈ dom(δ). Then e ∼τ e0 [η : δ ↔ δ] iff e ∼= ˆδ(τ) e0. Proof. The backward direction follows immediately from Theorem 49.12 and respect for observational equivalence. The forward direction is proved by induction on the structure of τ, appealing to Corollary 49.18 to establish observational equivalence at function and polymorphic types. 49.4 Parametricity Properties The parametricity theorem enables us to deduce properties of expressions of L{→∀} that hold solely because of their type. The stringencies of para- metricity ensure that a polymorphic type has very few inhabitants. For example, we may prove that every expression of type ∀(t.t → t) behaves like the identity function. Theorem 49.20. Let e : ∀(t.t → t) be arbitrary, and let id be Λ(t.λ (x:t) x). Then e ∼=∀(t.t→t) id. Proof. By Corollary 49.17 it is sufficient to show that e ∼∀(t.t→t) id. Let ρ and ρ0 be arbitrary closed types, let R: ρ ↔ ρ0 be an admissible relation, and suppose that e0 R e0 0. We are to show e[ρ](e0) R id[ρ](e0 0), which, given the definition of id and closure under converse evaluation, is to say e[ρ](e0) R e0 0. It suffices to show that e[ρ](e0) ∼=ρ e0, for then the result follows by the admissibility of R and the assumption e0 R e0 0. By Theorem 49.12 we have e ∼∀(t.t→t) e. Let the relation S: ρ ↔ ρ be defined by d S d0 iff d ∼=ρ e0 and d0 ∼=ρ e0. This is clearly admissible, and we have e0 S e0. It follows that e[ρ](e0) S e[ρ](e0), and so, by the definition of the relation S, e[ρ](e0) ∼=ρ e0. VERSION 1.32 REVISED 05.15.2012 49.4 Parametricity Properties 531 In Chapter 20 we showed that product, sum, and natural numbers types are all definable in L{→∀}. The proof of definability in each case consisted of showing that the type and its associated introduction and elimination forms are encodable in L{→∀}. The encodings are correct in the (weak) sense that the dynamics of these constructs as given in the earlier chapters is derivable from the dynamics of L{→∀} via these definitions. By taking advantage of parametricity we may extend these results to obtain a strong correspondence between these types and their encodings. As a first example, let us consider the representation of the unit type, unit, in L{→∀}, as defined in Chapter 20 by the following equations: unit = ∀(r.r → r) hi = Λ(r.λ (x:r) x) It is easy to see that hi : unit according to these definitions. But this merely says that the type unit is inhabited (has an element). What we would like to know is that, up to observational equivalence, the expression hi is the only element of that type. But this is precisely the content of Theorem 49.20. We say that the type unit is strongly definable within L{→∀}. Continuing in this vein, let us examine the definition of the binary prod- uct type in L{→∀}, also given in Chapter 20: τ1 × τ2 = ∀(r.(τ1 → τ2 → r) → r) he1, e2i = Λ(r.λ (x:τ1 → τ2 → r) x(e1)(e2)) e · l = e[τ1](λ (x:τ1) λ (y:τ2) x) e · r = e[τ2](λ (x:τ1) λ (y:τ2) y) It is easy to check that he1, e2i · l ∼=τ1 e1 and he1, e2i · r ∼=τ2 e2 by a direct calculation. We wish to show that the ordered pair, as defined above, is the unique such expression, and hence that Cartesian products are strongly definable in L{→∀}. We will make use of a lemma governing the behavior of the elements of the product type whose proof relies on Theorem 49.12. Lemma 49.21. If e : τ1 × τ2, then e ∼=τ1×τ2 he1, e2i for some e1 : τ1 and e2 : τ2. Proof. Expanding the definitions of pairing and the product type, and ap- plying Corollary 49.17, we let ρ and ρ0 be arbitrary closed types, and let R: ρ ↔ ρ0 be an admissible relation between them. Suppose further that h ∼τ1→τ2→t h0 [η : δ ↔ δ0], REVISED 05.15.2012 VERSION 1.32 532 49.4 Parametricity Properties where η(t) = R, δ(t) = ρ, and δ0(t) = ρ0 (and each is undefined on t0 6= t). We are to show that for some e1 : τ1 and e2 : τ2, e[ρ](h) ∼t h0(e1)(e2)[η : δ ↔ δ0], which is to say e[ρ](h) R h0(e1)(e2). Now by Theorem 49.12 we have e ∼τ1×τ2 e. Define the relation S: ρ ↔ ρ0 by d S d0 iff the following conditions are satisfied: 1. d ∼=ρ h(d1)(d2) for some d1 : τ1 and d2 : τ2; 2. d0 ∼=ρ0 h0(d0 1)(d0 2) for some d0 1 : τ1 and d0 2 : τ2; 3. d R d0. This is clearly an admissible relation. Noting that h ∼τ1→τ2→t h0 [η0 : δ ↔ δ0], where η0(t) = S and η0(t0) is undefined for t0 6= t, we conclude that e[ρ](h) S e[ρ0](h0), and hence e[ρ](h) R h0(d0 1)(d0 2), as required. Now suppose that e : τ1 × τ2 is such that e · l ∼=τ1 e1 and e · r ∼=τ2 e2. We wish to show that e ∼=τ1×τ2 he1, e2i. From Lemma 49.21 it follows that e ∼=τ1×τ2 he · l, e · ri by congruence and direct calculation. Hence, by con- gruence we have e ∼=τ1×τ2 he1, e2i. By a similar line of reasoning we may show that the Church encoding of the natural numbers given in Chapter 20 strongly defines the natural numbers in that the following properties hold: 1. iter z {z ⇒ e0 | s(x) ⇒ e1} ∼=ρ e0. 2. iter s(e){z ⇒ e0 | s(x) ⇒ e1} ∼=ρ [iter e {z ⇒ e0 | s(x) ⇒ e1}/x]e1. 3. Suppose that x : nat ` r(x): ρ. If (a) r(z) ∼=ρ e0, and (b) r(s(e)) ∼=ρ [r(e)/x]e1, VERSION 1.32 REVISED 05.15.2012 49.5 Representation Independence, Revisited 533 then for every e : nat, r(e) ∼=ρ iter e {z ⇒ e0 | s(x) ⇒ e1}. The first two equations, which constitute weak definability, are easily estab- lished by calculation, using the definitions given in Chapter 20. The third property, the unicity of the iterator, is proved using parametricity by show- ing that every closed expression of type nat is observationally equivalent to a numeral n. We then argue for unicity of the iterator by mathematical induction on n ≥ 0. Lemma 49.22. If e : nat, then either e ∼=nat z, or there exists e0 : nat such that e ∼=nat s(e0). Consequently, there exists n ≥ 0 such that e ∼=nat n. Proof. By Theorem 49.12 we have e ∼nat e. Define the relation R: nat ↔ nat to be the strongest relation such that d R d0 iff either d ∼=nat z and d0 ∼=nat z, or d ∼=nat s(d1) and d0 ∼=nat s(d0 1) and d1 R d0 1. It is easy to see that z R z, and if e R e0, then s(e)R s(e0). Letting zero = z and succ = λ (x:nat) s(x), we have e[nat](zero)(succ) R e[nat](zero)(succ). The result follows by the induction principle arising from the definition of R as the strongest relation satisfying its defining conditions. 49.5 Representation Independence, Revisited In Section 21.4 we discussed the property of representation independence for abstract types. This property states that if two implementations of an ab- stract type are “similar”, then the client behavior is not affected by replac- ing one for the other. The crux of the matter is the definition of similarity of two implementations. Informally, two implementations of an abstract type are similar if there is a relation, R, between their representation types that is preserved by the operations of the type. The relation R may be thought of as expressing the “equivalence” of the two representations; checking that each operation preserves R amounts to checking that the result of perform- ing that operation on equivalent representations yields equivalent results. As an example, we argued informally in Section 21.4 that two imple- mentations of a queue abstraction are similar. The two representations of queues are related by a relation, R, such that q R (b, f ) iff q is b followed by the reversal of f. When then argued that the operations preserve this re- lationship, and then claimed, without proof, that the behavior of the client would not be disrupted by changing one implementation to the other. REVISED 05.15.2012 VERSION 1.32 534 49.6 Notes The proof of this claim relies on parametricity, as may be seen by consid- ering the definability of existential types in L{→∀} given in Section 21.3. According to that definition, the client, e, of an abstract type ∃(t.τ) is a polymorphic function of type ∀(t.τ → τ2), where τ2, the result type of the computation, does not involve the type variable t. Being polymorphic, the client enjoys the parametricity property given by Theorem 49.12. Specifi- cally, suppose that ρ1 and ρ2 are two closed representation types and that R: ρ1 ↔ ρ2 is an admissible relation between them. For example, in the case of the queue abstraction, ρ1 is the type of lists of elements of the queue, ρ2 is the type of a pair of lists of elements, and R is the relation given above. Suppose further that e1 :[ρ1/t]τ and e2 :[ρ2/t]τ are two implementations of the operations such that e1 ∼τ e2 [η : δ1 ↔ δ2], (49.5) where η(t) = R, δ1(t) = ρ1, and δ2(t) = ρ2. In the case of the queues exam- ple the expression e1 is the implementation of the queue operations in terms of lists, and the e2 is the implementation in terms of pairs of lists described earlier. Condition (49.5) states that the two implementations are similar in that they preserve the relation R between the representation types. By Theorem 49.12 it follows that the client, e, satisfies e ∼τ2 e [η : δ1 ↔ δ2]. But because τ2 is a closed type (in particular, does not involve t), this is equivalent to e ∼τ2 e [∅ : ∅ ↔ ∅]. But then by Lemma 49.19 we have e[ρ1](e1) ∼=τ2 e[ρ2](e2). That is, the client behavior is not affected by the change of representation. 49.6 Notes The concept of parametricity is latent in the proof of normalization for Sys- tem F(Girard, 1972). Reynolds(1983), though technically flawed due to its reliance on a (non-existent) set-theoretic model of polymorphism, em- phasizes the centrality of logical equivalence for characterizing equality of polymorphic programs. The application of parametricity to representation VERSION 1.32 REVISED 05.15.2012 49.6 Notes 535 independence was suggested by Reynolds, and developed for existential types by Mitchell(1986) and Pitts(1998). The extension of System F with a positive (in the sense of Chapter 38) observable type appears to be needed to even define observational equivalence, but this point seems not to have been made elsewhere in the literature. REVISED 05.15.2012 VERSION 1.32 536 49.6 Notes VERSION 1.32 REVISED 05.15.2012 Chapter 50 Process Equivalence As the name implies a process is an ongoing computation that may interact with other processes by sending and receiving messages. From this point of view a concurrent computation has no definite “final outcome” but rather affords an opportunity for interaction that may well continue indefinitely. The notion of equivalence of processes must therefore be based on their po- tential for interaction, rather than on the “answer” that they may compute. Let P and Q be such that `ΣP proc and `ΣQ proc. We say that P and Q are equivalent, written P ≈ΣQ, iff there is a bisimulation, R, such that PRΣQ. A family of relations R = {RΣ}Σ is a bisimulation iff whenever P may evolve to P0 taking the action α, then Q may also evolve to some process Q0 taking the same action such that P0 RΣQ0, and, conversely, if Q may evolve to Q0 taking action α, then P may evolve to P0 taking the same ac- tion, and P0 RΣQ0. This captures the idea that the two processes afford the same opportunities for interaction in that they each simulate each other’s behavior with respect to their ability to interact with their environment. 50.1 Process Calculus We will consider a process calculus that consolidates the main ideas ex- plored in Chapters 41 and 42. We assume as given an ambient language of expressions that includes the type clsfd of classified values (see Chap- ter 34). Channels are treated as dynamically generated classes with which to build messages, as described in Chapter 42. 538 50.1 Process Calculus The syntax of the process calculus is given by the following grammar: Proc P::= stop 1 inert par(P1;P2)P1 k P2 composition await(E) $ E synchronize new[τ](a.P) ν a∼τ.P allocation emit(e)! e broadcast Evt E::= null 0 null or(E1;E2)E1 + E2 choice acc(x.P)?(x.P) acceptance The statics is given by the judgments Γ `ΣP proc and Γ `ΣE event defined by the following rules. We assume as given a judgment Γ `Σ e : τ for τ a type including the type clsfd of classified values. Γ `Σ 1 proc (50.1a) Γ `ΣP1 proc Γ `ΣP2 proc Γ `ΣP1 k P2 proc (50.1b) Γ `ΣE event Γ `Σ $ E proc (50.1c) Γ `Σ,a∼τ P proc Γ `Σ ν a∼τ.P proc (50.1d) Γ `Σ e : clsfd Γ `Σ! e proc (50.1e) Γ `Σ 0 event (50.1f) Γ `ΣE1 event Γ `ΣE2 event Γ `ΣE1 + E2 event (50.1g) Γ, x : clsfd `ΣP proc Γ `Σ?(x.P) event (50.1h) The dynamics is given by the judgments P α7−→ Σ P0 and E α=⇒ Σ P, defined as in Chapter 41. We assume as given the judgments e 7−→ Σ e0 and e valΣ for ex- pressions. Processes and events are identified up to structural congruence, as described in Chapter 41. P1 α7−→ Σ P0 1 P1 k P2 α7−→ Σ P0 1 k P2 (50.2a) VERSION 1.32 REVISED 05.15.2012 50.1 Process Calculus 539 P1 α7−→ Σ P0 1 P2 α7−→ Σ P0 2 P1 k P2 ε7−→ Σ P0 1 k P0 2 (50.2b) E α=⇒ Σ P $ E α7−→ Σ P (50.2c) P α7−−−→ Σ,a∼τ P0 `Σ α action ν a∼τ.P α7−→ Σ ν a∼τ.P0 (50.2d) e valΣ `Σ e : clsfd ! e e !7−→ Σ 1 (50.2e) E1 α=⇒ Σ P E1 + E2 α=⇒ Σ P (50.2f) e valΣ ?(x.P) e ?=⇒ Σ [e/x]P(50.2g) Assuming that substitution is valid for expressions, it is also valid for processes and events. Lemma 50.1. 1. If Γ, x : τ `ΣP proc and Γ `Σ e : τ, then Γ `Σ[e/x]P proc. 2. If Γ, x : τ `ΣE event and Γ `Σ e : τ, then Γ `Σ[e/x]E event. Transitions preserve well-formedness of processes and events. Lemma 50.2. 1. If `ΣP proc and P α7−→ Σ P0, then `ΣP0 proc. 2. If `ΣE event and E α=⇒ Σ P, then `ΣP proc. REVISED 05.15.2012 VERSION 1.32 540 50.2 Strong Equivalence 50.2 Strong Equivalence Bisimilarity makes precise the informal idea that two processes are equiv- alent if they each can take the same actions and, in doing so, evolve into equivalent processes. A process relation,P, is a family {PΣ} of binary re- lations between processes P and Q such that `ΣP proc and `ΣQ proc, and an event relation,E, is a family {EΣ} of binary relations between events E and F such that `ΣE event and `ΣF event.A(strong) bisimulation is a pair (P,E) consisting of a process relation, P, and an event relation, E, satisfying the following conditions: 1. If PPΣQ, then (a) if P α7−→ Σ P0, then there exists Q0 such that Q α7−→ Σ Q0 with P0 PΣQ0, and (b) if Q α7−→ Σ Q0, then there exists P0 such that P α7−→ Σ P0 with P0 PΣQ0. 2. If EEΣF, then (a) if E α=⇒ Σ P, then there exists Q such that F α=⇒ Σ Q with PPΣQ, and (b) if F α=⇒ Σ Q, then there exists P such that E α=⇒ Σ P with PPΣQ. The qualifier “strong” refers to the fact that the action, α, in the conditions on being a bisimulation include the silent action, ε. (In Section 50.3 we dis- cuss another notion of bisimulation in which the silent actions are treated specially.) (Strong) equivalence is the pair (≈, ≈) of process and event relations such that P ≈ΣQ and E ≈ΣF iff there exists a strong bisimulation (P,E) such that PPΣQ, and EEΣF. Lemma 50.3. Strong equivalence is a strong bisimulation. Proof. Follows immediately from the definition. The definition of strong equivalence gives rise to the principle of proof by coinduction. To show that P ≈ΣQ, it is enough to give a bisimulation (P,E) such that PPΣQ(and similarly for events). An instance of coinduc- tion that arises fairly frequently is to choose (P,E) to be (≈ ∪ P0, ≈ ∪ E0) for some P0 and E0 such that PP0 Q, and show that this expansion is a bisimulation. Because strong equivalence is itself a bisimulation, this re- duces to show that if P0 P0 Q0 and P0 α7−→ Σ P00, then Q0 α7−→ Σ Q00 for some Q00 VERSION 1.32 REVISED 05.15.2012 50.2 Strong Equivalence 541 such that either P00 ≈ΣQ00 or P00 P0 Q00 (and analogously for transitions from Q0, and similarly for event transitions). This proof method amounts to assuming what we are trying to prove and showing that this assumption is tenable. The proof that the expanded relation is a bisimulation may make use of the assumptions P0 and E0; in this sense “circular reasoning” is a perfectly valid method of proof. Lemma 50.4. Strong equivalence is an equivalence relation. Proof. For reflexivity and symmetry, it suffices to note that the identity re- lation is a bisimulation, as is the converse of a bisimulation. For transitivity we need that the composition of two bisimulations is again a bisimulation, which follows directly from the definition. It remains to verify that strong equivalence is a congruence, which means that each of the process- and event-forming constructs respects strong equiv- alence. To show this we require the open extension of strong equivalence to processes and events with free variables. The relation Γ `ΣP ≈ Q is de- fined for processes P and Q such that Γ `ΣP proc and Γ `ΣQ proc to mean that ˆγ(P) ≈Σ ˆγ(Q) for every substitution, γ, of closed values of appropri- ate type for the variables Γ. Lemma 50.5. If Γ, x : clsfd `ΣP ≈ Q, then Γ `Σ?(x.P) ≈ ?(x.Q). Proof. Fix a closing substitution, γ, for Γ, and let ˆP = ˆγ(P) and ˆQ = ˆγ(Q). By assumption we have x : clsfd `Σ ˆP ≈ ˆQ. We are to show that ?(x. ˆP) ≈Σ?(x. ˆQ). The proof is by coinduction, taking P = ≈ and E = ≈ ∪ E0, where E0 = {(?(x.P0),?(x.Q0)) | x : clsfd `ΣP0 ≈ Q0 }. Clearly ? (x. ˆP)E0 ?(x. ˆQ). Suppose that ? (x.P0)E0 ?(x.Q0). By in- spection of Rules (50.2), if ? (x.P0) α=⇒ Σ P00, then α = v ? and P00 = [v/x]P0 for some v valΣ such that `Σ v : clsfd. But ? (x.Q0) v ?=⇒ Σ [v/x]Q0, and we have that [v/x]P0 ≈Σ[v/x]Q0 by the definition of E0, and hence [v/x]P0 E0 [v/x]Q0, as required. The symmetric case follows symmetrically, complet- ing the proof. Lemma 50.6. If Γ `Σ,a∼τ P ≈ Q, then Γ `Σ ν a∼τ.P ≈ ν a∼τ.Q. REVISED 05.15.2012 VERSION 1.32 542 50.2 Strong Equivalence Proof. Fix a closing value substitution, γ, for Γ, and let ˆP = ˆγ(P) and ˆQ = ˆγ(Q). Assuming that ˆP ≈Σ,a∼τ ˆQ, we are to show that ν a∼τ. ˆP ≈Σ ν a∼τ. ˆQ. The proof is by coinduction, taking P = ≈ ∪ P0 and E = ≈, where P0 = {(ν a∼τ.P0, ν a∼τ.Q0) | P0 ≈Σ,a∼τ Q0 }. Clearly ν a∼τ. ˆPP0 ν a∼τ. ˆQ. Suppose that ν a∼τ.P0 P0 ν a∼τ.Q0, and that ν a∼τ.P0 α7−→ Σ P00. By inspection of Rules (50.2), we see that `Σ α action and that P00 = ν a∼τ.P000 for some P000 such that P0 α7−−−→ Σ,a∼τ P000. But by definition of P0 we have P0 ≈Σ,a∼τ Q0, and hence Q0 α7−−−→ Σ,a∼τ Q000 with P000 ≈Σ,a∼τ Q000. Letting Q00 = ν a∼τ.Q0, we have that ν a∼τ.Q0 α7−−−→ Σ,a∼τ Q00 and by definition of P0 we have P00 P0 Q00, as required. The symmetric case is proved symmetrically, completing the proof. Lemmas 50.5 and 50.6 capture two different cases of binding, the former of variables, and the latter of classes. The hypothesis of Lemma 50.5 relates all substitutions for the variable x in the recipient processes, whereas the hypothesis of Lemma 50.6 relates the constituent processes schematically in the class name, a. This makes all the difference, for if we were to consider all substitution instances of a class name by another class name, then a class would no longer be “new” within its scope, because we could identify it with an “old” class by substitution. On the other hand we must consider substitution instances for variables, because the meaning of a variable is given in such terms. This shows that classes and variables must be distinct concepts. (See Chapter 34 for an example of what goes wrong when the two concepts are confused.) Lemma 50.7. If Γ `ΣP1 ≈ Q1 and Γ `ΣP2 ≈ Q2, then Γ `ΣP1 k P2 ≈ Q1 k Q2. Proof. Let γ be a closing value substitution for Γ, and let ˆPi = ˆγ(Pi) and ˆQi = ˆγ(Qi) for i = 1, 2. The proof is by coinduction, considering the rela- tion P = ≈ ∪ P0 and E = ≈, where P0 = {(P0 1 k P0 2,Q0 1 k Q0 2) | P0 1 ≈ΣQ0 1 and P0 2 ≈ΣQ0 2 }. Suppose that P0 1 k P0 2 P0 Q0 1 k Q0 2, and that P0 1 k P0 2 α7−→ Σ P00. There are two cases to consider, the interesting one being Rule (50.2b). In this case we have P00 = P00 1 k P00 2 with P0 1 α7−→ Σ P00 1 and P0 2 α7−→ Σ P00 2 . By definition of P0 we VERSION 1.32 REVISED 05.15.2012 50.3 Weak Equivalence 543 have that Q0 1 α7−→ Σ Q00 1 and Q0 2 α7−→ Σ Q00 2 with P00 1 ≈ΣQ00 1 and P00 2 ≈ΣQ00 2 . Letting Q00 = Q00 1 k Q00 2 , we have that P00 P0 Q00, as required. The symmetric case is handled symmetrically, and Rule (50.2a) is handled similarly. Lemma 50.8. If Γ `ΣE1 ≈ F1 and Γ `ΣE2 ≈ F2, then Γ `ΣE1 + E2 ≈ F1 + F2. Proof. Follows immediately from Rules (50.2) and the definition of bisimu- lation. Lemma 50.9. If Γ `ΣE ≈ F, then Γ `Σ $ E ≈ $ F. Proof. Follows immediately from Rules (50.2) and the definition of bisimu- lation. Lemma 50.10. If Γ `Σ d ∼= e : clsfd, then Γ `Σ! d ≈ ! e. Proof. The process calculus introduces no new observations on expressions, so that d and e remain indistinguishable as actions. Theorem 50.11. Strong equivalence is a congruence. Proof. Follows immediately from the preceding lemmas, which cover each case separately. 50.3 Weak Equivalence Strong equivalence expresses the idea that two processes are equivalent if they simulate each other step-by-step. Every action taken by one process is matched by a corresponding action taken by the other. This seems natural for the non-trivial actions e ! and e ?, but is arguably overly restrictive for the silent action, ε. Silent actions correspond to the actual steps of computation, whereas the send and receive actions express the potential to interact with another process. Silent steps are therefore of a very different flavor than the other forms of action, and therefore might usefully be treated differently from them. Weak equivalence seeks to do just that. Silent actions arise within the process calculus itself (when two pro- cesses communicate), but they play an even more important role when the dynamics of expressions is considered explicitly (as in Chapter 42). For then each step e 7−→ Σ e0 of evaluation of an expression corresponds to a silent transition for any process in which it is embedded. In particular, ! e ε7−→ Σ ! e0 whenever e 7−→ Σ e0. We may also consider atomic processes of REVISED 05.15.2012 VERSION 1.32 544 50.3 Weak Equivalence the form proc(m) consisting of a command to be executed in accordance with the rules of some underlying dynamics. Here again we would expect that each step of command execution induces a silent transition from one atomic process to another. From the point of view of equivalence, it therefore seems sensible to al- low that a silent action by one process may be mimicked by one or more silent actions by another. For example, there appears to be little to be gained by distinguishing, say, proc(ret 3+4) from proc(ret (1+2)+(2+2)) merely because the latter takes more steps to compute the same value than the former! The purpose of weak equivalence is precisely to disregard such trivial distinctions by allowing a transition to be matched by a matching transition, possibly preceded by any number of silent transitions. A weak bisimulation is a pair (P,E) consisting of a process relation, P, and an event relation, E, satisfying the following conditions: 1. If PPΣQ, then (a) if P α7−→ Σ P0, where α 6= ε, then there exists Q00 and Q0 such that Q ε7−→ Σ ∗ Q00 α7−→ Σ Q0 with P0 PΣQ0, and if P ε7−→ Σ P0, then Q ε7−→ Σ ∗ Q0 with P0 PΣQ0; (b) if Q α7−→ Σ Q0, where α 6= ε, then there exists P00 and P0 such that P ε7−→ Σ ∗ P00 α7−→ Σ P0 with P0 PΣQ0, and if Q ε7−→ Σ Q0, then P ε7−→ Σ ∗ P0 with P0 PΣQ0; 2. If EEΣF, then (a) if E α=⇒ Σ P, then there exists Q such that F α=⇒ Σ Q with PPΣQ, and (b) if F α=⇒ Σ Q, then there exists P such that E α=⇒ Σ P with PPΣQ. (The conditions on the event relation are the same as for strong bisimilarity because there are, in this calculus, no silent actions on events.) Weak equivalence is the pair (∼, ∼) of process and event relations defined by P ∼ΣQ and E ∼ΣF iff there exists a weak bisimulation (P,E) such that PPΣQ, and EEΣF. The open extension of weak equivalence, written Γ `ΣP ∼ Q and Γ `ΣE ∼ F, is defined exactly as is the open extension of strong equivalence. Theorem 50.12. Weak equivalence is an equivalence relation and a congruence. Proof. The proof proceeds along similar lines to that of Theorem 50.11. VERSION 1.32 REVISED 05.15.2012 50.4 Notes 545 50.4 Notes The literature on process equivalence is extensive. Numerous variations have been considered for an equally numerous array of formalisms. Mil- ner recounts the history and development of the concept of bisimilarity in his monograph on the π-calculus (Milner, 1999), crediting David Park with its original conception (Park, 1981). The development in this chapter is in- spired by Milner, and by a proof of congruence of strong bisimilarity given by Bernardo Toninho for the process calculus considered in Chapter 41. REVISED 05.15.2012 VERSION 1.32 546 50.4 Notes VERSION 1.32 REVISED 05.15.2012 Part XIX Appendices Appendix A Finite Sets and Finite Functions We make frequent use of the concepts of a finite set of discrete objects and of finite functions between them. A set X is discrete iff equality of its elements is decidable: for every x, y ∈ X, either x = y ∈ X or x 6= y ∈ X. This condition is to be understood constructively as stating that we may effec- tively determine whether any two elements of the set X are equal or not. Perhaps the most basic example of a discrete set is the set, ω, of natural numbers. A set X is countable iff there is a bijection, f :X ∼= ω, between X and the set of natural numbers, and it is finite iff there is a bijection, f :X ∼= { 0, . . . , n − 1 }, where n ∈ ω, between it and some inital segment of the natural numbers. This condition is again to be understood construc- tively in terms of computable mappings, so that countable and finite sets are computably enumerable and, in the finite case, have a computable size. Given countable sets, U and V, a finite function is a computable partial function φ :U → V between them. The domain, dom(φ), of φ is the set { u ∈ U | φ(u) ↓ }, of objects u ∈ U such that φ(u) = v for some v ∈ V. Two finite functions, φ and ψ, between U and V are disjoint iff dom(φ) ∩ dom(ψ) = ∅. The empty finite function, ∅, between U and V is the totally undefined partial function between them. If u ∈ U and v ∈ V, the finite function, u ,→ v, between U and V sends u to v, and is undefined otherwise; its domain is therefore the singleton set { u }. In some situations we write u ∼ v for the finite function u ,→ v. If φ and ψ are two disjoint finite functions from U to V, then φ ⊗ ψ is 550 the finite function from U to V defined by the equation (φ ⊗ ψ)(u) =    φ(u) if u ∈ dom(φ) ψ(v) if v ∈ dom(ψ) undefined otherwise If u1,..., un ∈ U are pairwise distinct, and v1,..., vn ∈ V, then we some- times write u1 ,→ v1,..., un ,→ vn, or u1 ∼ v1,..., un ∼ vn, for the finite func- tion u1 ,→ v1 ⊗ ... ⊗ un ,→ vn. VERSION 1.32 REVISED 05.15.2012 Bibliography Mart´ın Abadi and Luca Cardelli. 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REVISED 05.15.2012 VERSION 1.32 Index abstract binding tree operator index, 12 abstract binding tree,3,7,9 abstractor,8 valence,8 α-equivalence, 10 bound variable, 10 capture, 11 free variable, 10 identification convention, 11 operator,8 arity,8 parameter, 12 structural induction, 10 substitution, 11 abstract syntax tree,3–5 operator,4 arity,4 index,6 parameter,6 structural induction,5 substitution,6 variable,4 abstract types, see existential types, see also signatures abt, see abstract binding tree assignables, see Modernized Algol ast, see abstract syntax tree backpatching, see references benign effects, see references boolean type, 105 call-by-need, see laziness capabilities, 354 channel types, see Concurrent Algol class types, 334 definability, 335 dynamics, 334 statics, 334 classes, see dynamic dispatch classical logic, 299 contradiction, 301–303 derivability of elimination forms, 305 double-negation translation, 310 dynamics, 306 excluded middle, 308 judgments, 300 proof, 302 conjunction, 304 disjunction, 304 implication, 304 negation, 304 truth, 304 variable, 303 provability, 301 conjunction, 301 disjunction, 302 hypothesis, 301 implication, 302 negation, 302 truth, 301 INDEX 561 refutability, 301 conjunction, 301 disjunction, 302 falsehood, 301 hypothesis, 301 implication, 302 negation, 302 refutation, 302 conjunction, 304 disjunction, 304 falsehood, 304 implication, 304 negation, 304 variable, 303 safety, 307 classified type, 331 confidentiality, 336 dynamics, 333 integrity, 336 safety, 334 statics, 332 coinductive types dynamics, 134 statics, 134 streams, 131 command types, see Modernized Al- gol compactness, see equality Concurrent Algol, 435 broadcast communication, 438 dynamics, 439 safety, 439 statics, 439 definability of free assignables, 444 dynamics, 436 selective communication, 441 dynamics, 443 statics, 441, 442 statics, 436 constructive logic, 289 conservation of proof, 295 Gentzen’s Principle, 295 judgment forms, 291 proof, 293 conjunction, 294 disjunction, 295 falsehood, 294 implication, 294 truth, 294 proofs-as-programs, 296 propositions-as-types, 296 provability, 292 conjunction, 292 disjunction, 293 falsehood, 293 implication, 293 negation, 293 truth, 292 reversability of proof, 295 semantics, 290 constructors, 201, 202 canonical, 202, 203 canonization, 209 canonizing substitution, 206 formation rules, 203 general, 209 neutral, 202, 203 contexts seeequality, 499 continuation types, 277 coroutines, 281 dynamics, 280 safety, 280 statics, 279 syntax, 279 contravariance, see subtyping covariance, see subtyping definitional equality, see equality REVISED 05.15.2012 VERSION 1.32 562 INDEX Distributed Algol, 447 dynamics, 450 safety, 451 situated types, 452 mobility, 455 statics, 452, 454 statics, 448 dynamc types destructors, 164 dynamic binding, see fluids dynamic classification, see classified type dynamic dispatch, 241, 242 class-based, 243, 244 class vector, 244 instance, 245 message send, 244 object type, 244 self-reference, 248 dispatch matrix, 242 method-based, 243 message send, 246 method vector, 245, 246 object type, 246 self-reference, 248 self-reference, 247 dynamic types, 159 as static types, 171 class dispatch, 165 cons, 164 critique, 166 dynamics, 160 nil, 164 numeric classes, 163 predicates, 164 safety, 162 statics, 160 dynamic typing vs static typing, 175 dynamics, 39, 45 checked errors, 58 contextual rules, 49 cost rules, 65 definitional equality, 52 determinacy, 48 equational rules, 51 equivalence theorem, 51 evaluation context, 49 evaluation rules, 61 equivalence to transition rules, 63 induction on transition, 46 inversion principle, 49 structural rules, 47 transition system, 45 unchecked errors, 58 enumeration types, 106 equality, 497 admissible relation, 525 coinduction, 501, 517 compactness, 511, 514, 515 congruence, 500 contexts, 499 definitional, 52, 80, 89, 150, 183, 228, 497 equational laws, 506 equivalence candidate, see admis- sible relation fixed point induction, 511 function extensionality, 529 Kleene equality, 509, 522 Kleene equivalence, 500 logical equivalence, 497, 502, 503, 510, 524 closed, 502, 503, 525 compositionality, 527 open, 526 observation, 498 VERSION 1.32 REVISED 05.15.2012 INDEX 563 observational equivalence, 497, 498, 500, 503, 509, 522, 523 parametricity, 521, 527, 530, 533 symbolic evaluation, 53 equivalence, see equality event types, see Concurrent Algol exceptions, 267, 269 dynamics, 270 encapsulation, 272 failures, 267 dynamics, 268 safety, 269 statics, 267 statics, 269 syntax, 269 value type, 269, 271 dynamic classification, 272 static classification, 272 existential types, 192 definability from universals, 196 dynamics, 193 modeling data abstraction, 194 representation independence, 533 representation independence, 197 safety, 194 statics, 192 failures, see exceptions finite function combination, 549 empty, 549 finite function, 549 domain, 549 singleton, 549 finite set, 549 fixed point induction, see equality fluid binding, see fluids fluid types, 328 dynamics, 329 statics, 329 fluids, 323 dynamics, 324 safety, 325 statics, 324 subtleties, 326 freshness condition on binders, 10 function types definitions, 70 dynamic binding, 75 first order, 70 dynamics, 71 safety, 71 statics, 70 higher order, 71 dynamics, 72 safety, 72 statics, 72 static binding, 74 functors, see signatures future types, 403 parallel dynamics, 406 sequential dynamics, 404 sparks, 409 statics, 404 futures types pipelining, 408 G¨odel’s T, 78 definability, 80 definitional equality, 80 dynamics, 79 equality, see equality safety, 80 statics, 78 undefinability, 82 general judgment, 27, 33 generic derivability, 33 proliferation, 33 structurality, 33 substitution, 33 REVISED 05.15.2012 VERSION 1.32 564 INDEX parametric derivability, 34 general recursion, 87 generic inductive definition, 34 formal generic judgment, 35 rule, 34 implicit form, 34 rule induction, 35 structurality, 35 hybrid types, 169 as recursive types, 171 dynamics, 170 optimization of dynamic types, 172 safety, 170 statics, 170 hypothetical inductive definition, 31 formal derivability, 32 rule, 31 uniform, 32 rule induction, 32 hypothetical judgment, 27 admissibility, 29 reflexivity, 30 structurality, 31 transitivity, 30 weakening, 30 derivability, 27 reflexivity, 29 stability, 28 structurality, 29 transitivity, 29 weakening, 29 inductive definition, 15, 16 backward chaining, 18 derivation, 17 forward chaining, 18 function, 22 iterated, 21 rule, 16 admissible, 29 axiom, 16 conclusion, 16 derivable, 28 premise, 16 rule induction, 17, 19 rule scheme, 17 instance, 17 simultaneous, 21 inductive types dynamics, 134 natural numbers, 129 statics, 134 inheritance, 251 class extension, 252 class-based, 253 method extension, 252 method-based, 254 subclass, 251 submethod, 251 superclass, 251 supermethod, 251 interface, see separate compilation iteration, 78 judgment, 15 mode, 24 judgment form, 15 predicate, 15 subject, 15 kinds, 201, 202 dependent, see singleton kinds, Σ kinds, Π kinds function kinds, 202 higher kinds, 204 product kinds, 202 type kind, 202 Kleene equality, see equality VERSION 1.32 REVISED 05.15.2012 INDEX 565 laziness, 367 data structures, 374 dynamics, 368 recursion, 371 safety, 372 suspension types dynamics, 376 statics, 376 suspensions, 375 linking, see separate compilation logical equivalence, see equality methods, see dynamic dispatch mobile types, 349 mobility condition, 349 rules, 349 Modernized Algol, 341 assignables, 341, 355 block structure, 345 command types, 349 commands, 341, 348 expressions, 341 free assignables, 357 free dynamics, 358 idioms conditionals, 347 iteration, 347 procedures, 347 sequential composition, 346 mobile types, see mobile types scoped dynamics, 343 scoped safety, 345 stack discipline, 345 statics, 342, 349 modules, see signatures mutual recursion, 99 nested parallelism, see parallelism null, see option types objects, see dynamic dispatch observational equivalence, see equal- ity option types, 106 parallelism, 389 binary fork-join, 390 Brent’s Theorem, 399 cost dynamics, 393 cost dynamics vs. transition dy- namics, 395 cost graphs, 393 implicit parallelism theorem, 392 multiple fork-join, 396 parallel complexity, 394 parallelizability, 400 provably efficient implementation, 398 sequence types, 397 cost dynamics, 398 statics, 397 sequential and parallel dynam- ics, 390 sequential complexity, 394 task dynamics, 400 work vs. depth, 394 parameterized modules, see signatures parametricity, see equality patterns, 110 constraints, 114 dual, 114 entailment, 116 satisfaction, 115 dynamics, 112 exhaustiveness, 114, 116 redundancy, 114, 116 statics, 110 PCF, 87 bounded recursion, 511 definability, 91 definitional equality, 89 REVISED 05.15.2012 VERSION 1.32 566 INDEX dynamics, 88 equivalence, see equality safety, 89 statics, 87 phase distinction, 39, 201, see also sig- natures Π kinds, 229, 232, 233 elimination rules, 234 equivalence, 234 formation rules, 234 introduction rules, 234 subkinding, 235 polarization, 379 dynamics, 384 focusing, 381 negative types, 380 positive types, 380 safety, 385 statics, 382 polymorphism, see universal types primitive recursion, 78 process calculus, 415, 537 actions, 415 asynchronous communication, 427 bisimilarity, 537 channel types, 427 dynamics, 430 statics, 430 channels, 421, 425 coinduction, see strong and weak bisimilarity concurrent composition, 417 dynamics, 418, 423, 426, 538 equivalence, see bisimilarity events, 415 replication, 419 statics, 422, 425, 538 strong bisimilarity, 540 strong bisumulation, 540 structural congruence, 416, 417 synchronization, 418 synchronous communication, 424 syntax, 537 universality, 430 weak bisimilarity, 543 coinduction, 544 weak bisimulation, 544 process equivalence, see process cal- culus product types, 96 dynamics, 96 finite, 97 safety, 97 statics, 96 recursive types, 138 data structures, 139 dynamics, 139 self-reference, 141 statics, 138 reference types, 353 aliasing, 356 free dynamics, 358 safety, 356, 360 scoped dynamics, 355 statics, 355 references backpatching, 363 benign effects, 362 representation independence, see ex- istential types representation indepndence, see also parametricity safety, 55 canonical forms, 56 checked errors, 59 evaluation rules, 64 preservation, 56 progress, 57 VERSION 1.32 REVISED 05.15.2012 INDEX 567 scoped assignables, see Modernized Algol self types, 141 as recursive types, 142 deriving general recursion, 143 self-reference, 141 unrolling, 141 separate compilation, 459 initialization, 461 interface, 460 linking, 460 units, 460 Σ kinds, 229, 232, 233 elimination rules, 233 equivalence, 233 formation rules, 233 introduction rules, 233 subkinding, 234 signatures, 465 applicative functor, 492 ascription, see sealing avoidance problem, 474 dynamic part, 467 dynamics, 476 first- vs second-class, 476 functors, 486, 487 generative functor, 489 hierarchies, 481, 482 instances, 468 modification, 486 opacity, 466 parameterization, 485 parameterized modules, see func- tors principal signature, 470 revelation, 466 sealing, 467 self-recognition, 476, 491 sharing propagation, 483 sharing specification, 483 static part, 467 statics, 472, 489 structures, 467 submodule, 484 subsignature, 468, 470, 471 syntax, 472, 488 translucency, 466 transparency, 466 type abstractions, 465, 467 type classes, 466, 468 views, 468 singleton kinds, 228, 229 as type definitions, 231 constructor equivalence, 229 higher singletons, 229, 235 kind equivalence, 229 kind formation, 229 subkinding, 229 situated types, see Distributed Algol speculation types, 405 parallel dynamics, 406 sequential dynamics, 405 statics, 405 stack machine, 259 correctness, 262 completeness, 264 soundness, 264 unraveling, 264 dynamics, 260 frame, 260 safety, 261 stack, 260 state, 259 state, 144, see also Modernized Algol,reference types from recursion, 144 from streams, 145 RS latch, 144 statics, 39 canonical forms, 43 REVISED 05.15.2012 VERSION 1.32 568 INDEX decomposition, 43 induction on typing, 41 introduction and elimination, 43 structurality, 42 substitution, 42 type system, 40 unicity, 41 weakening, 42 subkinding, 228 Π kinds, 235 Σ kinds, 234 singleton kinds, 229 submodules, see signatures subtyping, 216 function types, 219 numeric types, 216 product types, 217, 219 quantified types, 220 recursive types, 221 safety, 223 subsumption, 216 sum types, 218, 219 variance, 218 sum types, 101 dynamics, 102 finite, 103 statics, 102 suspension types, see laziness symbol types, 319 dynamics, 320 safety, 320 statics, 319 symbolic reference, see symbol types symbols, 315 mobility, 317 safety, 317, 318 scope-free dynamics, 318 scoped dynamics, 317 statics, 316 syntax,3 abstract,3 binding,3 chart, 39 concrete,3 structural,3 surface,3 System F, see universal types definitional equality, 183 type abstractions, see also existential types,signatures type classes, see signatures type constructors, see constructors type operator, 122 generic extension, 122 polynomial, 122 positive, 124 unit dynamics, 96 statics, 96 unit type, 96 vs void type, 104 units, see separate compilation unityped λ-calculus, 155 as untyped, 155 universal types, 180 definability, 183 natural numbers, 184 products, 183 sums, 184 dynamics, 182 parametricity, 185, see equality predicative fragment, 186 prenex fragment, 187 rank-restricted fragments, 189 safety, 182 statics, 180 untyped λ-calculus, 149 Y combinator, 153 VERSION 1.32 REVISED 05.15.2012 INDEX 569 as unityped, 155 Church numerals, 151 definability, 150 definitional equality, 150 dynamics, 150 Scott’s Theorem, 153 statics, 149 variance, see subtyping void type, 101 vs unit type, 104 dynamics, 102 statics, 102 REVISED 05.15.2012 VERSION 1.32 570 Revision History VERSION 1.32 REVISED 05.15.2012 Revision History Revision Date Author(s) Description 1.0 21.01.11 RH Created 1.1 21.01.11 RH Concurrent and Distributed Algol 1.2 03.02.11 RH Clarified discussion of rule induc- tion. 1.3 06.23.11 RH Revamped notation in equational reasoning chapter; miscellaneous corrections and improvements throughout. 1.4 06.28.2011 RH Minor corrections to inductive defi- nitions and type safety chapters. 1.5 07.19.2011 RH Revamped treatment of syntactic objects. 1.6 07.20.2011 RH Reorganized Part I. 1.7 07.21.2011 RH Moved strings to Part I from Part II. 1.8 07.22.2011 RH Remove exercises, add end notes. 1.9 08.09.2011 RH Revise formulation and presenta- tion of Concurrent and Distributed Algol. 1.10 08.12.2011 RH Added chapter on process equiva- lence. 1.11 08.15.2011 RH Disequality of classes for dynamic classification. 1.12 08.18.2011 RH Revised distributed Algol chapter. 1.13 08.22.2011 RH Units and linking. 1.14 08.23.2011 RH Canonization of constructors. 1.15 08.24.2011 RH Singleton kinds, bounded quantifi- cation. 1.16 08.27.2011 RH Modules. 572 Revision History 1.17 09.13.2011 RH Citations and bibiography. 1.18 09.19.2011 RH Module hierarchies and parameter- ization. 1.19 10.03.2011 RH Polarization, classical logic. 1.20 10.15.2011 RH Revise dynamic typing. 1.21 11.04.2011 RH Index 1.22 12.25.2011 RH Revised modularity chapters. 1.23 01.03.2012 RH Appendix. 1.24 01.18.2012 RH Revised notational conventions for dynamic languages. 1.25 01.30.2012 RH Dynamic and hybrid languages. 1.26 02.06.2012 RH Inheritance. 1.27 02.10.2012 RH Typographical errors. 1.28 02.29.2012 RH Inheritance rewrite. 1.29 03.03.2012 RH Representation independence. 1.30 03.15.2012 RH Recursive types. 1.31 04.30.2012 RH Release. 1.32 05.15.2012 RH Typographical errors corrected. VERSION 1.32 REVISED 05.15.2012
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