Functional Programming in Scala Licensed to Emre Sevinc Licensed to Emre Sevinc Functional Programming in Scala PAUL CHIUSANO RÚNAR BJARNASON MANNING SHELTER ISLAND Licensed to Emre Sevinc For online information and ordering of this and other Manning books, please visit www.manning.com. The publisher offers discounts on this book when ordered in quantity. For more information, please contact Special Sales Department Manning Publications Co. 20 Baldwin Road PO Box 761 Shelter Island, NY 11964 Email: orders@manning.com ©2015 by Manning Publications Co. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by means electronic, mechanical, photocopying, or otherwise, without prior written permission of the publisher. Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in the book, and Manning Publications was aware of a trademark claim, the designations have been printed in initial caps or all caps. Recognizing the importance of preserving what has been written, it is Manning’s policy to have the books we publish printed on acid-free paper, and we exert our best efforts to that end. Recognizing also our responsibility to conserve the resources of our planet, Manning books are printed on paper that is at least 15 percent recycled and processed without the use of elemental chlorine. Development editor: Jeff Bleiel Manning Publications Co. Copyeditor: Benjamin Berg 20 Baldwin Road Proofreader: Katie Tennant PO Box 261 Project editor: Janet Vail Shelter Island, NY 11964 Typesetter: Dottie Marsico Illustrator: Chuck Larson Cover designer: Irene Scala ISBN 9781617290657 Printed in the United States of America 12345678910–EBM–191817161514 Licensed to Emre Sevinc v brief contents PART 1INTRODUCTION TO FUNCTIONAL PROGRAMMING ...........1 1 ■ What is functional programming? 3 2 ■ Getting started with functional programming in Scala 14 3 ■ Functional data structures 29 4 ■ Handling errors without exceptions 48 5 ■ Strictness and laziness 64 6 ■ Purely functional state 78 PART 2FUNCTIONAL DESIGN AND COMBINATOR LIBRARIES ......93 7 ■ Purely functional parallelism 95 8 ■ Property-based testing 124 9 ■ Parser combinators 146 PART 3COMMON STRUCTURES IN FUNCTIONAL DESIGN .........173 10 ■ Monoids 175 11 ■ Monads 187 12 ■ Applicative and traversable functors 205 PART 4EFFECTS AND I/O .....................................................227 13 ■ External effects and I/O 229 14 ■ Local effects and mutable state 254 15 ■ Stream processing and incremental I/O 268 Licensed to Emre Sevinc Licensed to Emre Sevinc vii contents foreword xiii preface xv acknowledgments xvi about this book xvii PART 1INTRODUCTION TO FUNCTIONAL PROGRAMMING .......1 1 What is functional programming? 3 1.1 The benefits of FP: a simple example 4 A program with side effects 4 ■ A functional solution: removing the side effects 6 1.2 Exactly what is a (pure) function? 9 1.3 Referential transparency, purity, and the substitution model 10 1.4 Summary 13 2 Getting started with functional programming in Scala 14 2.1 Introducing Scala the language: an example 15 2.2 Running our program 17 2.3 Modules, objects, and namespaces 18 2.4 Higher-order functions: passing functions to functions 19 A short detour: writing loops functionally 20 ■ Writing our first higher-order function 21 Licensed to Emre Sevinc CONTENTSviii 2.5 Polymorphic functions: abstracting over types 22 An example of a polymorphic function 23 ■ Calling HOFs with anonymous functions 24 2.6 Following types to implementations 25 2.7 Summary 28 3 Functional data structures 29 3.1 Defining functional data structures 29 3.2 Pattern matching 32 3.3 Data sharing in functional data structures 35 The efficiency of data sharing 36 ■ Improving type inference for higher-order functions 37 3.4 Recursion over lists and generalizing to higher-order functions 38 More functions for working with lists 41 ■ Loss of efficiency when assembling list functions from simpler components 44 3.5 Trees 44 3.6 Summary 47 4 Handling errors without exceptions 48 4.1 The good and bad aspects of exceptions 48 4.2 Possible alternatives to exceptions 50 4.3 The Option data type 52 Usage patterns for Option 53 ■ Option composition, lifting, and wrapping exception-oriented APIs 56 4.4 The Either data type 60 4.5 Summary 63 5 Strictness and laziness 64 5.1 Strict and non-strict functions 65 5.2 An extended example: lazy lists 68 Memoizing streams and avoiding recomputation 69 ■ Helper functions for inspecting streams 69 5.3 Separating program description from evaluation 70 5.4 Infinite streams and corecursion 73 5.5 Summary 77 Licensed to Emre Sevinc CONTENTS ix 6 Purely functional state 78 6.1 Generating random numbers using side effects 78 6.2 Purely functional random number generation 80 6.3 Making stateful APIs pure 81 6.4 A better API for state actions 84 Combining state actions 85 ■ Nesting state actions 86 6.5 A general state action data type 87 6.6 Purely functional imperative programming 88 6.7 Summary 91 PART 2FUNCTIONAL DESIGN AND COMBINATOR LIBRARIES...93 7 Purely functional parallelism 95 7.1 Choosing data types and functions 96 A data type for parallel computations 97 ■ Combining parallel computations 100 ■ Explicit forking 102 7.2 Picking a representation 104 7.3 Refining the API 105 7.4 The algebra of an API 110 The law of mapping 110 ■ The law of forking 112 Breaking the law: a subtle bug 113 ■ A fully non-blocking Par implementation using actors 115 7.5 Refining combinators to their most general form 120 7.6 Summary 123 8 Property-based testing 124 8.1 A brief tour of property-based testing 124 8.2 Choosing data types and functions 127 Initial snippets of an API 127 ■ The meaning and API of properties 128 ■ The meaning and API of generators 130 Generators that depend on generated values 131 ■ Refining the Prop data type 132 8.3 Test case minimization 134 8.4 Using the library and improving its usability 136 Some simple examples 137 ■ Writing a test suite for parallel computations 138 8.5 Testing higher-order functions and future directions 142 Licensed to Emre Sevinc CONTENTSx 8.6 The laws of generators 144 8.7 Summary 144 9 Parser combinators 146 9.1 Designing an algebra, first 147 9.2 A possible algebra 152 Slicing and nonempty repetition 154 9.3 Handling context sensitivity 156 9.4 Writing a JSON parser 158 The JSON format 158 ■ A JSON parser 159 9.5 Error reporting 160 A possible design 161 ■ Error nesting 162 Controlling branching and backtracking 163 9.6 Implementing the algebra 165 One possible implementation 166 ■ Sequencing parsers 166 Labeling parsers 167 ■ Failover and backtracking 168 Context-sensitive parsing 169 9.7 Summary 171 PART 3COMMON STRUCTURES IN FUNCTIONAL DESIGN......173 10 Monoids 175 10.1 What is a monoid? 175 10.2 Folding lists with monoids 178 10.3 Associativity and parallelism 179 10.4 Example: Parallel parsing 181 10.5 Foldable data structures 183 10.6 Composing monoids 184 Assembling more complex monoids 185 ■ Using composed monoids to fuse traversals 186 10.7 Summary 186 11 Monads 187 11.1 Functors: generalizing the map function 187 Functor laws 189 Licensed to Emre Sevinc CONTENTS xi 11.2 Monads: generalizing the flatMap and unit functions 190 The Monad trait 191 11.3 Monadic combinators 193 11.4 Monad laws 194 The associative law 194 ■ Proving the associative law for a specific monad 196 ■ The identity laws 197 11.5 Just what is a monad? 198 The identity monad 199 ■ The State monad and partial type application 200 11.6 Summary 204 12 Applicative and traversable functors 205 12.1 Generalizing monads 205 12.2 The Applicative trait 206 12.3 The difference between monads and applicative functors 208 The Option applicative versus the Option monad 209 The Parser applicative versus the Parser monad 210 12.4 The advantages of applicative functors 211 Not all applicative functors are monads 211 12.5 The applicative laws 214 Left and right identity 214 ■ Associativity 215 Naturality of product 216 12.6 Traversable functors 218 12.7 Uses of Traverse 219 From monoids to applicative functors 220 ■ Traversals with State 221 ■ Combining traversable structures 223 ■ Traversal fusion 224 ■ Nested traversals 224 ■ Monad composition 225 12.8 Summary 226 PART 4EFFECTS AND I/O .................................................227 13 External effects and I/O 229 13.1 Factoring effects 229 Licensed to Emre Sevinc CONTENTSxii 13.2 A simple IO type 231 Handling input effects 232 ■ Benefits and drawbacks of the simple IO type 235 13.3 Avoiding the StackOverflowError 237 Reifying control flow as data constructors 237 Trampolining: a general solution to stack overflow 239 13.4 A more nuanced IO type 241 Reasonably priced monads 242 ■ A monad that supports only console I/O 243 ■ Pure interpreters 246 13.5 Non-blocking and asynchronous I/O 247 13.6 A general-purpose IO type 250 The main program at the end of the universe 250 13.7 Why the IO type is insufficient for streaming I/O 251 13.8 Summary 253 14 Local effects and mutable state 254 14.1 Purely functional mutable state 254 14.2 A data type to enforce scoping of side effects 256 A little language for scoped mutation 256 ■ An algebra of mutable references 258 ■ Running mutable state actions 259 Mutable arrays 262 ■ A purely functional in-place quicksort 263 14.3 Purity is contextual 264 What counts as a side effect? 266 14.4 Summary 267 15 Stream processing and incremental I/O 268 15.1 Problems with imperative I/O: an example 268 15.2 Simple stream transducers 271 Creating processes 272 ■ Composing and appending processes 275 ■ Processing files 278 15.3 An extensible process type 278 Sources 281 ■ Ensuring resource safety 283 ■ Single-input processes 285 ■ Multiple input streams 287 ■ Sinks 290 Effectful channels 291 ■ Dynamic resource allocation 291 15.4 Applications 292 15.5 Summary 293 index 295 Licensed to Emre Sevinc xiii foreword Functional Programming in Scala is an intriguing title. After all, Scala is generally called a functional programming language and there are dozens of books about Scala on the market. Are all these other books missing the functional aspects of the language? To answer the question it’s instructive to dig a bit deeper. What is functional programming? For me, it’s simply an alias for “programming with functions,” that is, a programming style that puts the focus on the functions in a program. What are func- tions? Here, we find a larger spectrum of definitions. While one definition often admits functions that may have side effects in addition to returning a result, pure functional programming restricts functions to be as they are in mathematics: binary relations that map arguments to results. Scala is an impure functional programming language in that it admits impure as well as pure functions, and in that it does not try to distinguish between these catego- ries by using different syntax or giving them different types. It shares this property with most other functional languages. It would be nice if we could distinguish pure and impure functions in Scala, but I believe we have not yet found a way to do so that is lightweight and flexible enough to be added to Scala without hesitation. To be sure, Scala programmers are generally encouraged to use pure functions. Side effects such as mutation, I/O, or use of exceptions are not ruled out, and they can indeed come in quite handy sometimes, be it for reasons of interoperability, effi- ciency, or convenience. But overusing side effects is generally not considered good style by experts. Nevertheless, since impure programs are possible and even conve- nient to write in Scala, there is a temptation for programmers coming from a more imperative background to keep their style and not make the necessary effort to adapt to the functional mindset. In fact, it’s quite possible to write Scala as if it were Java without the semicolons. So to properly learn functional programming in Scala, should one make a detour via a pure functional language such as Haskell? Any argument in favor of this Licensed to Emre Sevinc approach has been severely weakened by the appearance of Functional Programming in Scala. What Paul and Rúnar do, put simply, is treat Scala as a pure functional program- ming language. Mutable variables, exceptions, classical input/output, and all other traces of impurity are eliminated. If you wonder how one can write useful programs without any of these conveniences, you need to read the book. Building up from first principles and extending all the way to incremental input and output, they demon- strate that, indeed, one can express every concept using only pure functions. And they show that it is not only possible, but that it also leads to beautiful code and deep insights into the nature of computation. The book is challenging, both because it demands attention to detail and because it might challenge the way you think about programming. By reading the book and doing the recommended exercises, you will develop a better appreciation of what pure functional programming is, what it can express, and what its benefits are. What I particularly liked about the book is that it is self-contained. It starts with the simplest possible expressions and every abstraction is explained in detail before fur- ther abstractions are built on them in turn. In a sense, the book develops an alterna- tive Scala universe, where mutable state does not exist and all functions are pure. Commonly used Scala libraries tend to deviate a bit from this ideal; often they are based on a partly imperative implementation with a (mostly) functional interface. That Scala allows the encapsulation of mutable state in a functional interface is, in my opinion, one of its strengths. But it is a capability that is also often misused. If you find yourself using it too often, Functional Programming in Scala is a powerful antidote. MARTIN ODERSKY CREATOR OF SCALA Licensed to Emre Sevinc xv preface Writing good software is hard. After years of struggling with other approaches, both of us discovered and fell in love with functional programming (FP). Though the FP approach is different, we came to appreciate how the discipline leads to a coherent, composable, and beautiful way of writing programs. Both of us participated in the Boston Area Scala Enthusiasts, a group that met reg- ularly in Cambridge. When the group first started, it mainly consisted of Java pro- grammers who were looking for something better. Many expressed frustration that there wasn’t a clear way to learn how to take advantage of FP in Scala. We could empa- thize—we had both learned FP somewhat haphazardly, by writing lots of functional code, talking to and learning from other Scala and Haskell programmers, and reading a patchwork of different articles, blog posts, and books. It felt like there should be an easier way. In April 2010 one of the group’s organizers, Nermin Šerifovic´, suggested that we write a book specifically on the topic of FP in Scala. Based on our learning experiences, we had a clear idea of the kind of book we wanted to write, and we thought it would be quick and easy. More than four years later, we think we have cre- ated a good book. It’s the book we wish had existed when we were learning functional programming. We hope to convey in this book some of the excitement that we felt when we were first discovering FP. Licensed to Emre Sevinc xvi acknowledgments We would like to thank the many people who participated in the creation of this book. To Nermin Šerifovic´, our friend from the Boston Scala group, thank you for first planting the seed of this book in our minds. To the amazing team at Capital IQ, thank you for your support and for bravely helping beta test the first version of the book’s curriculum. We would like to especially acknowledge Tony Morris for embarking on this jour- ney with us. His work on the early stages of the book remains invaluable, as does his larger contribution to the practice of functional programming in Scala. Martin, thank you for your wonderful foreword, and of course for creating this powerful language that is helping to reshape our industry. During the book-writing process, we were grateful for the encouragement from the enthusiastic community of Scala users interested in functional programming. To our reviewers, MEAP readers, and everyone else who provided feedback or submitted bug reports and pull requests, thank you! This book would not be what it is today with- out your help. Special thanks to our development editor Jeff Bleiel, our graphics editor Ben Kovitz, our technical proofreader Tom Lockney, and everyone else at Manning who helped make this a better book, including the following reviewers who read the manuscript at various stages of its development: Ashton Hepburn, Bennett Andrews, Chris Marshall, Chris Nauroth, Cody Koeninger, Dave Cleaver, Douglas Alan, Eric Torreborre, Erich W. Schreiner, Fernando Dobladez, Ionut, G. Stan, Jeton Bacaj, Kai Gellien, Luc Duponcheel, Mark Janssen, Michael Smolyak, Ravindra Jaju, Rintcius Blok, Rod Hilton, Sebastien Nichele, Sukant Hajra, Thad Meyer, Will Hayworth, and William E. Wheeler. Lastly, Sunita and Allison, thank you so much for your support throughout our multi-year odyssey to make this book a reality. Licensed to Emre Sevinc xvii about this book This is not a book about Scala. This book is an introduction to functional programming (FP), a radical, principled approach to writing software. We use Scala as the vehicle to get there, but you can apply the lessons herein to programming in any language. As you work through this book, our goal is for you to gain a firm grasp of functional pro- gramming concepts, become comfortable writing purely functional programs, and be able to absorb new material on the subject, beyond what we cover here. How this book is structured The book is organized into four parts. In part 1, we talk about exactly what functional programming is and introduce some core concepts. The chapters in part 1 give an overview of fundamental techniques like how to organize small functional programs, define purely functional data structures, handle errors, and deal with state. Building on this foundation, part 2 is a series of tutorials on functional design. We work through some examples of practical functional libraries, laying bare the thought process that goes into designing them. While working through the libraries in part 2, it will become clear to you that these libraries follow certain patterns and contain some duplication. This will highlight the need for new and higher abstractions for writing more generalized libraries, and we introduce those abstractions in part 3. These are very powerful tools for reasoning about your code. Once you master them, they hold the promise of making you extraordinarily productive as a programmer. Part 4 then bridges the remainder of the gap towards writing real-world applica- tions that perform I/O (like working with databases, files, or video displays) and make use of mutable state, all in a purely functional way. Throughout the book, we rely heavily on programming exercises, carefully sequenced to help you internalize the material. To understand functional program- ming, it’s not enough to learn the theory abstractly. You have to fire up your text editor Licensed to Emre Sevinc ABOUT THIS BOOKxviii and write some code. You have to take the theory that you have learned and put it into practice in your work. We’ve also provided online notes for all the chapters. Each chapter has a section with discussion related to that chapter, along with links to further material. These chapter notes are meant to be expanded by the community of readers, and are avail- able as an editable wiki at https://github.com/fpinscala/fpinscala/wiki. Audience This book is intended for readers with at least some programming experience. We had a particular kind of reader in mind while writing the book—an intermediate-level Java or C programmer who is curious about functional programming. But we believe this book is well suited for programmers coming from any language, at any level of experience. Prior expertise is not as important as motivation and curiosity. Functional pro- gramming is a lot of fun, but it’s a challenging topic. It may be especially challenging for the most experienced programmers, because it requires such a different way of thinking than they might be used to. No matter how long you have been program- ming, you must come prepared to be a beginner once again. This book does not require any prior experience with Scala, but we won’t spend a lot of time and space discussing Scala’s syntax and language features. Instead we will introduce them as we go, with minimal ceremony, mostly as a consequence of cover- ing other material. These introductions to Scala should be enough to get you started with the exercises. If you have further questions about the Scala language, you should supplement your reading with another book on Scala (http://scala-lang.org/ documentation/books.html) or look up specific questions in the Scala language doc- umentation (http://scala-lang.org/documentation/). How to read this book Although the book can be read sequentially straight through, the sequencing of the four parts is designed so that you can comfortably break between them, apply what you have learned to your own work, and then come back later for the next part. For example, the material in part 4 will make the most sense after you have a strong famil- iarity with the functional style of programming developed in parts 1, 2, and 3. After part 3, it may be a good idea to take a break and try getting more practice writing functional programs beyond the exercises we work on in the chapters. Of course, this is ultimately up to you. Most chapters in this book have a similar structure. We introduce some new idea or technique, explain it with an example, and then work through a number of exercises. We strongly suggest that you download the exercise source code and do the exercises as you go through each chapter. Exercises, hints, and answers are all available at https:// github.com/fpinscala/fpinscla. We also encourage you to visit the scala-functional Licensed to Emre Sevinc ABOUT THIS BOOK xix Google group (https://groups.google.com/forum/#!topic/scala-functional/) and the #fp-in-scala IRC channel on irc.freenode.net for questions and discussion. Exercises are marked for both their difficulty and importance. We will mark exer- cises that we think are hard or that we consider to be optional to understanding the material. The hard designation is our effort to give you some idea of what to expect— it is only our guess and you may find some unmarked questions difficult and some questions marked hard to be quite easy. The optional designation is for exercises that are informative but can be skipped without impeding your ability to follow further material. The exercises have the following icons in front of them to denote whether or not they are optional: EXERCISE 1 A filled-in square next to an exercise means the exercise is critical. EXERCISE 2 An open square means the exercise is optional. Examples are presented throughout the book, and they are meant to be tried rather than just read. Before you begin, you should have the Scala interpreter running and ready. We encourage you to experiment on your own with variations of what you see in the examples. A good way to understand something is to change it slightly and see how the change affects the outcome. Sometimes we will show a Scala interpreter session to demonstrate the result of running or evaluating some code. This will be marked by lines beginning with the scala> prompt of the interpreter. Code that follows this prompt is to be typed or pasted into the interpreter, and the line just below will show the interpreter’s response, like this: scala> println("Hello, World!") Hello, World! Code conventions and downloads All source code in listings or in text is in a fixed-width font like this to separate it from ordinary text. Key words in Scala are set in bold fixed-width font like this. Code annotations accompany many of the listings, highlighting important concepts. To download the source code for the examples in the book, the exercise code, and the chapter notes, please go to https://github.com/fpinscala/fpinscala or to the pub- lisher’s website at www.manning.com/FunctionalProgramminginScala. Licensed to Emre Sevinc ABOUT THIS BOOKxx Setting expectations Although functional programming has a profound impact on the way we write soft- ware at every level, it takes time to build up to that. It’s an incremental process. Don’t expect to be blown away by how amazing functional programming is right in the first chapter. The principles laid out in the beginning are quite subtle, and may even seem like they’re just common sense. If you think to yourself “that’s something I can already do without knowing FP,” then that’s great! That’s exactly the point. Most programmers are already doing FP to some extent, without even knowing it. Many things that most people consider to be best practices (like making a function have only a single respon- sibility, or making data immutable) are implied by accepting the premise of functional programming. We are simply taking the principles underlying those best practices and carrying them all the way to their logical conclusion. It’s highly likely that in reading this book, you will simultaneously learn both Scala syntax and functional programming. As a result, at first it may seem to you that the code looks very alien, the techniques are unnatural, and the exercises are brain- bending. That is perfectly normal. Do not be deterred. If you get stuck, look at the hints and answers,1 or take your questions to the Google Group (https://groups .google.com/forum/#!topic/scala-functional/) or the IRC channel (#fp-in-scala on irc.freenode.net). Above all, we hope that this book will be a fun and rewarding experience for you, and that functional programming makes your work easier and more enjoyable as it has done for us. This book’s purpose, when all is said and done, is to help you be more pro- ductive in your work. It should make you feel less like the software you are writing is a collection of dirty hacks, and more like you are creating a thing of beauty and utility. Author Online Purchase of Functional Programming in Scala includes free access to a private web forum run by Manning Publications where you can make comments about the book, ask technical questions, and receive help from the authors and other users. To access the forum and subscribe to it, point your web browser to www.manning.com/Functional ProgramminginScala. This Author Online page provides information on how to get on the forum once you’re registered, what kind of help is available, and the rules of conduct on the forum. Manning’s commitment to our readers is to provide a venue where a meaningful dialog among individual readers and between readers and the authors can take place. It’s not a commitment to any specific amount of participation on the part of the authors, whose contribution to the forum remains voluntary (and unpaid). The Author Online forum and the archives of previous discussions will be accessi- ble from the publisher’s website as long as the book is in print. 1 https://github.com/fpinscala/fpinscala. Licensed to Emre Sevinc Part 1 Introduction to functional programming We begin this book with a radical premise—that we will restrict ourselves to constructing programs using only pure functions with no side effects such as reading from files or mutating memory. This idea, of functional programming, leads to a very different way of writing programs than you may be used to. We therefore start from the very beginning, relearning how to write the simplest of programs in a functional way. In the first chapter, we’ll explain exactly what functional programming means and give you some idea of its benefits. The rest of the chapters in part 1 introduce the basic techniques for functional programming in Scala. Chapter 2 introduces Scala the language and covers fundamentals like how to write loops functionally and manipulate functions as ordinary values. Chapter 3 deals with in-memory data structures that may change over time. Chapter 4 talks about handling errors in pure functions, and chapter 5 introduces the notion of non- strictness, which can be used to improve the efficiency and modularity of func- tional code. Finally, chapter 6 introduces modeling stateful programs using pure functions. The intent of this first part of the book is to get you thinking about programs purely in terms of functions from inputs to outputs, and to teach you the tech- niques you’ll need in part 2, when we start writing some practical code. Licensed to Emre Sevinc Licensed to Emre Sevinc 3 What is functional programming? Functional programming (FP) is based on a simple premise with far-reaching impli- cations: we construct our programs using only pure functions—in other words, func- tions that have no side effects. What are side effects? A function has a side effect if it does something other than simply return a result, for example: Modifying a variable Modifying a data structure in place Setting a field on an object Throwing an exception or halting with an error Printing to the console or reading user input Reading from or writing to a file Drawing on the screen We’ll provide a more precise definition of side effects later in this chapter, but con- sider what programming would be like without the ability to do these things, or with significant restrictions on when and how these actions can occur. It may be dif- ficult to imagine. How is it even possible to write useful programs at all? If we can’t reassign variables, how do we write simple programs like loops? What about work- ing with data that changes, or handling errors without throwing exceptions? How can we write programs that must perform I/O, like drawing to the screen or read- ing from a file? The answer is that functional programming is a restriction on how we write pro- grams, but not on what programs we can express. Over the course of this book, we’ll learn how to express all of our programs without side effects, and that includes programs that perform I/O, handle errors, and modify data. We’ll learn Licensed to Emre Sevinc 4 CHAPTER 1 What is functional programming? how following the discipline of FP is tremendously beneficial because of the increase in modularity that we gain from programming with pure functions. Because of their modularity, pure functions are easier to test, reuse, parallelize, generalize, and reason about. Furthermore, pure functions are much less prone to bugs. In this chapter, we’ll look at a simple program with side effects and demonstrate some of the benefits of FP by removing these side effects. We’ll also discuss the bene- fits of FP more generally and define two important concepts—referential transparency and the substitution model. 1.1 The benefits of FP: a simple example Let’s look at an example that demonstrates some of the benefits of programming with pure functions. The point here is just to illustrate some basic ideas that we’ll return to throughout this book. This will also be your first exposure to Scala’s syntax. We’ll talk through Scala’s syntax much more in the next chapter, so don’t worry too much about following every detail. As long as you have a basic idea of what the code is doing, that’s what’s important. 1.1.1 A program with side effects Suppose we’re implementing a program to handle purchases at a coffee shop. We’ll begin with a Scala program that uses side effects in its implementation (also called an impure program). class Cafe { def buyCoffee(cc: CreditCard): Coffee = { val cup = new Coffee() cc.charge(cup.price) cup } } The line cc.charge(cup.price) is an example of a side effect. Charging a credit card involves some interaction with the outside world—suppose it requires contacting the credit card company via some web service, authorizing the transaction, charging the Listing 1.1 A Scala program with side effects The class keyword introduces a class, much like in Java. Its body is contained in curly braces, { and }. A method of a class is introduced by the def keyword. cc: CreditCard defines a parameter named cc of type CreditCard. The Coffee return type of the buyCoffee method is given after the parameter list, and the method body consists of a block within curly braces after an = sign. No semicolons are necessary. Newlines delimit statements in a block. Side effect. Actually charges the credit card. We don’t need to say return. Since cup is the last statement in the block, it is automatically returned. Licensed to Emre Sevinc 5The benefits of FP: a simple example card, and (if successful) persisting some record of the transaction for later reference. But our function merely returns a Coffee and these other actions are happening on the side, hence the term “side effect.” (Again, we’ll define side effects more formally later in this chapter.) As a result of this side effect, the code is difficult to test. We don’t want our tests to actually contact the credit card company and charge the card! This lack of testability is suggesting a design change: arguably, CreditCard shouldn’t have any knowledge baked into it about how to contact the credit card company to actually execute a charge, nor should it have knowledge of how to persist a record of this charge in our internal systems. We can make the code more modular and testable by letting Credit- Card be ignorant of these concerns and passing a Payments object into buyCoffee. class Cafe { def buyCoffee(cc: CreditCard, p: Payments): Coffee = { val cup = new Coffee() p.charge(cc, cup.price) cup } } Though side effects still occur when we call p.charge(cc, cup.price), we have at least regained some testability. Payments can be an interface, and we can write a mock implementation of this interface that is suitable for testing. But that isn’t ideal either. We’re forced to make Payments an interface, when a concrete class may have been fine otherwise, and any mock implementation will be awkward to use. For example, it might contain some internal state that we’ll have to inspect after the call to buy- Coffee, and our test will have to make sure this state has been appropriately modified (mutated) by the call to charge. We can use a mock framework or similar to handle this detail for us, but this all feels like overkill if we just want to test that buyCoffee creates a charge equal to the price of a cup of coffee. Separate from the concern of testing, there’s another problem: it’s difficult to reuse buyCoffee. Suppose a customer, Alice, would like to order 12 cups of coffee. Ideally we could just reuse buyCoffee for this, perhaps calling it 12 times in a loop. But as it is currently implemented, that will involve contacting the payment system 12 times, authorizing 12 separate charges to Alice’s credit card! That adds more process- ing fees and isn’t good for Alice or the coffee shop. What can we do about this? As the figure at the top of page 6 illustrates, we could write a whole new function, buyCoffees, with special logic for batching up the charges.1 Here, that might not be such a big deal, since the logic of buyCoffee is so Listing 1.2 Adding a payments object 1 We could also write a specialized BatchingPayments implementation of the Payments interface, that some- how attempts to batch successive charges to the same credit card. This gets complicated though. How many charges should it try to batch up, and how long should it wait? Do we force buyCoffee to indicate that the batch is finished, perhaps by calling closeBatch? And how would it know when it’s appropriate to do that, anyway? Licensed to Emre Sevinc 6 CHAPTER 1 What is functional programming? simple, but in other cases the logic we need to duplicate may be nontrivial, and we should mourn the loss of code reuse and composition! 1.1.2 A functional solution: removing the side effects The functional solution is to eliminate side effects and have buyCoffee return the charge as a value in addition to returning the Coffee. The concerns of processing the charge by sending it off to the credit card company, persisting a record of it, and so on, will be handled elsewhere. Again, we’ll cover Scala’s syntax more in later chapters, but here’s what a functional solution might look like: class Cafe { def buyCoffee(cc: CreditCard): (Coffee, Charge) = { val cup = new Coffee() (cup, Charge(cc, cup.price)) } } Here we’ve separated the concern of creating a charge from the processing or interpreta- tion of that charge. The buyCoffee function now returns a Charge as a value along with the Coffee. We’ll see shortly how this lets us reuse it more easily to purchase mul- tiple coffees with a single transaction. But what is Charge? It’s a data type we just invented containing a CreditCard and an amount, equipped with a handy function, combine, for combining charges with the same CreditCard: With a side effect A call to buyCoffee Can’t test buyCoffee without credit card server. Can’t combine two transactions into one. Side effect Send transaction Credit card Cup Credit card Cup Credit card server buyCoffee Without a side effect Charge buyCoffee Charge Coalesce List (charge1, charge2, ...) If buyCoffee returns a charge object instead of performing a side effect, a caller can easily combine several charges into one transaction. (and can easily test the buyCoffee function without needing a payment processor). buyCoffee now returns a pair of a Coffee and a Charge, indicated with the type (Coffee, Charge). Whatever system processes payments is not involved at all here.To create a pair, we put the cup and Charge in parentheses separated by a comma. Licensed to Emre Sevinc 7The benefits of FP: a simple example case class Charge(cc: CreditCard, amount: Double) { def combine(other: Charge): Charge = if (cc == other.cc) Charge(cc, amount + other.amount) else throw new Exception("Can't combine charges to different cards") } Now let’s look at buyCoffees, to implement the purchase of n cups of coffee. Unlike before, this can now be implemented in terms of buyCoffee, as we had hoped. class Cafe { def buyCoffee(cc: CreditCard): (Coffee, Charge) = ... def buyCoffees(cc: CreditCard, n: Int): (List[Coffee], Charge) = { val purchases: List[(Coffee, Charge)] = List.fill(n)(buyCoffee(cc)) val (coffees, charges) = purchases.unzip (coffees, charges.reduce((c1,c2) => c1.combine(c2))) } } Overall, this solution is a marked improvement—we’re now able to reuse buyCoffee directly to define the buyCoffees function, and both functions are trivially testable without having to define complicated mock implementations of some Payments inter- face! In fact, the Cafe is now completely ignorant of how the Charge values will be Listing 1.3 Buying multiple cups with buyCoffees A case class has one primary constructor whose argument list comes after the class name (here, Charge). The parameters in this list become public, unmodifiable (immutable) fields of the class and can be accessed using the usual object-oriented dot notation, as in other.cc. An if expression has the same syntax as in Java, but it also returns a value equal to the result of whichever branch is taken. If cc == other.cc, then combine will return Charge(..); otherwise the exception in the else branch will be thrown. A case class can be created without the keyword new. We just use the class name followed by the list of arguments for its primary constructor. The syntax for throwing exceptions is the same as in Java and many other languages. We’ll discuss more functional ways of handling error conditions in a later chapter. List[Coffee] is an immutable singly linked list of Coffee values. We’ll discuss this data type more in chapter 3. List.fill(n)(x) creates a List with n copies of x. We’ll explain this funny function call syntax in a later chapter. unzip splits a list of pairs into a pair of lists. Here we’re destructuring this pair to declare two values (coffees and charges) on one line. charges.reduce reduces the entire list of charges to a single charge, using combine to combine charges two at a time. reduce is an example of a higher-order function, which we’ll properly introduce in the next chapter. Licensed to Emre Sevinc 8 CHAPTER 1 What is functional programming? processed. We can still have a Payments class for actually processing charges, of course, but Cafe doesn’t need to know about it. Making Charge into a first-class value has other benefits we might not have antici- pated: we can more easily assemble business logic for working with these charges. For instance, Alice may bring her laptop to the coffee shop and work there for a few hours, making occasional purchases. It might be nice if the coffee shop could com- bine these purchases Alice makes into a single charge, again saving on credit card pro- cessing fees. Since Charge is first-class, we can write the following function to coalesce any same-card charges in a List[Charge]: def coalesce(charges: List[Charge]): List[Charge] = charges.groupBy(_.cc).values.map(_.reduce(_ combine _)).toList We’re passing functions as values to the groupBy, map, and reduce methods. You’ll learn to read and write one-liners like this over the next several chapters. The _.cc and _ combine _ are syntax for anonymous functions, which we’ll introduce in the next chapter. You may find this kind of code difficult to read because the notation is very com- pact. But as you work through this book, reading and writing Scala code like this will become second nature to you very quickly. This function takes a list of charges, groups them by the credit card used, and then combines them into a single charge per card. It’s perfectly reusable and testable without any additional mock objects or interfaces. Imagine trying to implement the same logic with our first implementation of buyCoffee! This is just a taste of why functional programming has the benefits claimed, and this example is intentionally simple. If the series of refactorings used here seems natu- ral, obvious, unremarkable, or standard practice, that’s good. FP is merely a discipline that takes what many consider a good idea to its logical endpoint, applying the disci- pline even in situations where its applicability is less obvious. As you’ll learn over the course of this book, the consequences of consistently following the discipline of FP are profound and the benefits enormous. FP is a truly radical shift in how programs are organized at every level—from the simplest of loops to high-level program architec- ture. The style that emerges is quite different, but it’s a beautiful and cohesive approach to programming that we hope you come to appreciate. What about the real world? We saw in the case of buyCoffee how we could separate the creation of the Charge from the interpretation or processing of that Charge. In general, we’ll learn how this sort of transformation can be applied to any function with side effects to push these effects to the outer layers of the program. Functional programmers often speak of implementing programs with a pure core and a thin layer on the outside that handles effects. Licensed to Emre Sevinc 9Exactly what is a (pure) function? 1.2 Exactly what is a (pure) function? We said earlier that FP means programming with pure functions, and a pure function is one that lacks side effects. In our discussion of the coffee shop example, we worked off an informal notion of side effects and purity. Here we’ll formalize this notion, to pinpoint more precisely what it means to program functionally. This will also give us additional insight into one of the benefits of functional programming: pure functions are easier to reason about. A function f with input type A and output type B (written in Scala as a single type: A=> B, pronounced “A to B” or “A arrow B”) is a computation that relates every value a of type A to exactly one value b of type B such that b is determined solely by the value of a. Any changing state of an internal or external process is irrelevant to computing the result f(a). For example, a function intToString having type Int => String will take every integer to a corresponding string. Furthermore, if it really is a function, it will do nothing else. In other words, a function has no observable effect on the execution of the pro- gram other than to compute a result given its inputs; we say that it has no side effects. We sometimes qualify such functions as pure functions to make this more explicit, but this is somewhat redundant. Unless we state otherwise, we’ll often use function to imply no side effects.2 You should be familiar with a lot of pure functions already. Consider the addition (+) function on integers. It takes two integer values and returns an integer value. For any two given integer values, it will always return the same integer value. Another example is the length function of a String in Java, Scala, and many other languages where strings can’t be modified (are immutable). For any given string, the same length is always returned and nothing else occurs. We can formalize this idea of pure functions using the concept of referential trans- parency (RT). This is a property of expressions in general and not just functions. For the purposes of our discussion, consider an expression to be any part of a program that can be evaluated to a result—anything that you could type into the Scala interpreter 2 Procedure is often used to refer to some parameterized chunk of code that may have side effects. But even so, surely at some point we must actually have an effect on the world and submit the Charge for processing by some external system. And aren’t there other useful programs that necessitate side effects or mutation? How do we write such pro- grams? As we work through this book, we’ll discover how many programs that seem to necessitate side effects have some functional analogue. In other cases we’ll find ways to structure code so that effects occur but aren’t observable. (For example, we can mutate data that’s declared locally in the body of some function if we ensure that it can’t be referenced outside that function, or we can write to a file as long as no enclosing function can observe this occurring.) Licensed to Emre Sevinc 10 CHAPTER 1 What is functional programming? and get an answer. For example, 2 + 3 is an expression that applies the pure function + to the values 2 and 3 (which are also expressions). This has no side effect. The evalu- ation of this expression results in the same value 5 every time. In fact, if we saw 2 + 3 in a program we could simply replace it with the value 5 and it wouldn’t change a thing about the meaning of our program. This is all it means for an expression to be referentially transparent—in any pro- gram, the expression can be replaced by its result without changing the meaning of the program. And we say that a function is pure if calling it with RT arguments is also RT. We’ll look at some examples next.3 1.3 Referential transparency, purity, and the substitution model Let’s see how the definition of RT applies to our original buyCoffee example: def buyCoffee(cc: CreditCard): Coffee = { val cup = new Coffee() cc.charge(cup.price) cup } Whatever the return type of cc.charge(cup.price) (perhaps it’s Unit, Scala’s equiva- lent of void in other languages), it’s discarded by buyCoffee. Thus, the result of eval- uating buyCoffee(aliceCreditCard) will be merely cup, which is equivalent to a new Coffee(). For buyCoffee to be pure, by our definition of RT, it must be the case that p(buyCoffee(aliceCreditCard)) behaves the same as p(new Coffee()), for any p. This clearly doesn’t hold—the program new Coffee() doesn’t do anything, whereas buyCoffee(aliceCreditCard) will contact the credit card company and authorize a charge. Already we have an observable difference between the two programs. Referential transparency forces the invariant that everything a function does is rep- resented by the value that it returns, according to the result type of the function. This constraint enables a simple and natural mode of reasoning about program evaluation called the substitution model. When expressions are referentially transparent, we can imagine that computation proceeds much like we’d solve an algebraic equation. We fully expand every part of an expression, replacing all variables with their referents, and then reduce it to its simplest form. At each step we replace a term with an 3 There are some subtleties to this definition, and we’ll refine it later in this book. See the chapter notes at our GitHub site (https://github.com/pchiusano/fpinscala; see the preface) for more discussion. Referential transparency and purity An expression e is referentially transparent if, for all programs p, all occurrences of e in p can be replaced by the result of evaluating e without affecting the meaning of p. A function f is pure if the expression f(x) is referentially transparent for all referen- tially transparent x.3 Licensed to Emre Sevinc 11Referential transparency, purity, and the substitution model equivalent one; computation proceeds by substituting equals for equals. In other words, RT enables equational reasoning about programs. Let’s look at two more examples—one where all expressions are RT and can be rea- soned about using the substitution model, and one where some expressions violate RT. There’s nothing complicated here; we’re just formalizing something you likely already understand. Let’s try the following in the Scala interpreter (also known as the Read-Eval-Print- Loop or REPL, pronounced like “ripple,” but with an e instead of an i). Note that in Java and in Scala, strings are immutable. A “modified” string is really a new string and the old string remains intact: scala> val x = "Hello, World" x: java.lang.String = Hello, World scala> val r1 = x.reverse r1: String = dlroW ,olleH scala> val r2 = x.reverse r2: String = dlroW ,olleH Suppose we replace all occurrences of the term x with the expression referenced by x (its definition), as follows: scala> val r1 = "Hello, World".reverse r1: String = dlroW ,olleH scala> val r2 = "Hello, World".reverse r2: String = dlroW ,olleH This transformation doesn’t affect the outcome. The values of r1 and r2 are the same as before, so x was referentially transparent. What’s more, r1 and r2 are referentially transparent as well, so if they appeared in some other part of a larger program, they could in turn be replaced with their values throughout and it would have no effect on the program. Now let’s look at a function that is not referentially transparent. Consider the append function on the java.lang.StringBuilder class. This function operates on the StringBuilder in place. The previous state of the StringBuilder is destroyed after a call to append. Let’s try this out: scala> val x = new StringBuilder("Hello") x: java.lang.StringBuilder = Hello scala> val y = x.append(", World") y: java.lang.StringBuilder = Hello, World scala> val r1 = y.toString r1: java.lang.String = Hello, World scala> val r2 = y.toString r2: java.lang.String = Hello, World r1 and r2 are the same. r1 and r2 are still the same. r1 and r2 are the same. Licensed to Emre Sevinc 12 CHAPTER 1 What is functional programming? So far so good. Now let’s see how this side effect breaks RT. Suppose we substitute the call to append like we did earlier, replacing all occurrences of y with the expression referenced by y: scala> val x = new StringBuilder("Hello") x: java.lang.StringBuilder = Hello scala> val r1 = x.append(", World").toString r1: java.lang.String = Hello, World scala> val r2 = x.append(", World").toString r2: java.lang.String = Hello, World, World This transformation of the program results in a different outcome. We therefore con- clude that StringBuilder.append is not a pure function. What’s going on here is that although r1 and r2 look like they’re the same expression, they are in fact referencing two different values of the same StringBuilder. By the time r2 calls x.append, r1 will have already mutated the object referenced by x. If this seems difficult to think about, that’s because it is. Side effects make reasoning about program behavior more difficult. Conversely, the substitution model is simple to reason about since effects of evalua- tion are purely local (they affect only the expression being evaluated) and we need not mentally simulate sequences of state updates to understand a block of code. Understanding requires only local reasoning. We need not mentally track all the state changes that may occur before or after our function’s execution to understand what our function will do; we simply look at the function’s definition and substitute the arguments into its body. Even if you haven’t used the name “substitution model,” you have certainly used this mode of reasoning when thinking about your code.4 Formalizing the notion of purity this way gives insight into why functional pro- grams are often more modular. Modular programs consist of components that can be understood and reused independently of the whole, such that the meaning of the whole depends only on the meaning of the components and the rules governing their composition; that is, they are composable. A pure function is modular and composable because it separates the logic of the computation itself from “what to do with the result” and “how to obtain the input”; it’s a black box. Input is obtained in exactly one way: via the argument(s) to the function. And the output is simply computed and returned. By keeping each of these concerns separate, the logic of the computation is more reusable; we may reuse the logic wherever we want without worrying about whether the side effect being done with the result or the side effect requesting the input are appropriate in all contexts. We saw this in the buyCoffee example—by elim- inating the side effect of payment processing being done with the output, we were more easily able to reuse the logic of the function, both for purposes of testing and for purposes of further composition (like when we wrote buyCoffees and coalesce). 4 In practice, programmers don’t spend time mechanically applying substitution to determine if code is pure— it will usually be obvious. r1 and r2 are no longer the same. Licensed to Emre Sevinc 13Summary 1.4 Summary In this chapter, we introduced functional programming and explained exactly what FP is and why you might use it. Though the full benefits of the functional style will become more clear over the course of this book, we illustrated some of the benefits of FP using a simple example. We also discussed referential transparency and the substi- tution model and talked about how FP enables simpler reasoning about programs and greater modularity. In this book, you’ll learn the concepts and principles of FP as they apply to every level of programming, starting from the simplest of tasks and building on that founda- tion. In subsequent chapters, we’ll cover some of the fundamentals—how do we write loops in FP? Or implement data structures? How do we deal with errors and excep- tions? We need to learn how to do these things and get comfortable with the low-level idioms of FP. We’ll build on this understanding when we explore functional design techniques in parts 2, 3, and 4. Licensed to Emre Sevinc 14 Getting started with functional programming in Scala Now that we have committed to using only pure functions, a question naturally emerges: how do we write even the simplest of programs? Most of us are used to thinking of programs as sequences of instructions that are executed in order, where each instruction has some kind of effect. In this chapter, we’ll begin learning how to write programs in the Scala language just by combining pure functions. This chapter is mainly intended for those readers who are new to Scala, to func- tional programming, or both. Immersion is an effective method for learning a for- eign language, so we’ll just dive in. The only way for Scala code to look familiar and not foreign is if we look at a lot of Scala code. We’ve already seen some in the first chapter. In this chapter, we’ll start by looking at a small but complete program. We’ll then break it down piece by piece to examine what it does in some detail, in order to understand the basics of the Scala language and its syntax. Our goal in this book is to teach functional programming, but we’ll use Scala as our vehicle, and need to know enough of the Scala language and its syntax to get going. Once we’ve covered some of the basic elements of the Scala language, we’ll then introduce some of the basic techniques for how to write functional programs. We’ll discuss how to write loops using tail recursive functions, and we’ll introduce higher- order functions (HOFs). HOFs are functions that take other functions as arguments and may themselves return functions as their output. We’ll also look at some examples of polymorphic HOFs where we use types to guide us toward an implementation. There’s a lot of new material in this chapter. Some of the material related to HOFs may be brain-bending if you have a lot of experience programming in a lan- guage without the ability to pass functions around like that. Remember, it’s not cru- cial that you internalize every single concept in this chapter, or solve every exercise. Licensed to Emre Sevinc 15Introducing Scala the language: an example We’ll come back to these concepts again from different angles throughout the book, and our goal here is just to give you some initial exposure. 2.1 Introducing Scala the language: an example The following is a complete program listing in Scala, which we’ll talk through. We aren’t introducing any new concepts of functional programming here. Our goal is just to introduce the Scala language and its syntax. // A comment! /* Another comment */ /** A documentation comment */ object MyModule { def abs(n: Int): Int = if (n<0)-n else n private def formatAbs(x: Int) = { val msg = "The absolute value of %d is %d" msg.format(x, abs(x)) } def main(args: Array[String]): Unit = println(formatAbs(-42)) } We declare an object (also known as a module) named MyModule. This is simply to give our code a place to live and a name so we can refer to it later. Scala code has to be in an object or a class, and we use an object here because it’s simpler. We put our code inside the object, between curly braces. We’ll discuss objects and classes in more detail shortly. For now, we’ll just look at this particular object. The MyModule object has three methods, intro- duced with the def keyword: abs, formatAbs, and main. We’ll use the term method to refer to some function or field defined within an object or class using the def keyword. Let’s now go through the methods of MyModule one by one. Listing 2.1 A simple Scala program Declares a singleton object, which simultaneously declares a class and its only instance. abs takes an integer and returns an integer. Returns the negation of n if it’s less than zero. A private method can only be called by other members of MyModule. A string with two placeholders for numbers marked as %d. Replaces the two %d placeholders in the string with x and abs(x) respectively. Unit serves the same purpose as void in languages like Java or C. The object keyword The object keyword creates a new singleton type, which is like a class that only has a sin- gle named instance. If you’re familiar with Java, declaring an object in Scala is a lot like cre- ating a new instance of an anonymous class. Scala has no equivalent to Java’s static keyword, and an object is often used in Scala where you might use a class with static members in Java. Licensed to Emre Sevinc 16 CHAPTER 2 Getting started with functional programming in Scala The abs method is a pure function that takes an integer and returns its absolute value: def abs(n: Int): Int = if (n<0)-n else n The def keyword is followed by the name of the method, which is followed by the parameter list in parentheses. In this case, abs takes only one argument, n of type Int. Following the closing parenthesis of the argument list, an optional type annotation (the : Int) indicates that the type of the result is Int (the colon is pronounced “has type”). The body of the method itself comes after a single equals sign (=). We’ll sometimes refer to the part of a declaration that goes before the equals sign as the left-hand side or signature, and the code that comes after the equals sign as the right-hand side or defini- tion. Note the absence of an explicit return keyword. The value returned from a method is simply whatever value results from evaluating the right-hand side. All expressions, including if expressions, produce a result. Here, the right-hand side is a single expression whose value is either -n or n, depending on whether n < 0. The formatAbs method is another pure function: private def formatAbs(x: Int) = { val msg = "The absolute value of %d is %d." msg.format(x, abs(x)) } Here we’re calling the format method on the msg object, passing in the value of x along with the value of abs applied to x. This results in a new string with the occur- rences of %d in msg replaced with the evaluated results of x and abs(x), respectively. This method is declared private, which means that it can’t be called from any code outside of the MyModule object. This function takes an Int and returns a String, but note that the return type is not declared. Scala is usually able to infer the return types of methods, so they can be omitted, but it’s generally considered good style to explicitly declare the return types of methods that you expect others to use. This method is private to our module, so we can omit the type annotation. The body of the method contains more than one statement, so we put them inside curly braces. A pair of braces containing statements is called a block. Statements are separated by newlines or by semicolons. In this case, we’re using a newline to separate our statements, so a semicolon isn’t necessary. The first statement in the block declares a String named msg using the val key- word. It’s simply there to give a name to the string value so we can refer to it again. A val is an immutable variable, so inside the body of the formatAbs method the name msg will always refer to the same String value. The Scala compiler will complain if you try to reassign msg to a different value in the same block. Remember, a method simply returns the value of its right-hand side, so we don’t need a return keyword. In this case, the right-hand side is a block. In Scala, the value format is a standard library method defined on String Licensed to Emre Sevinc 17Running our program of a multistatement block inside curly braces is the same as the value returned by the last expression in the block. Therefore, the result of the formatAbs method is just the value returned by the call to msg.format(x, abs(x)). Finally, our main method is an outer shell that calls into our purely functional core and prints the answer to the console. We’ll sometimes call such methods procedures (or impure functions) rather than functions, to emphasize the fact that they have side effects: def main(args: Array[String]): Unit = println(formatAbs(-42)) The name main is special because when you run a program, Scala will look for a method named main with a specific signature. It has to take an Array of Strings as its argument, and its return type must be Unit. The args array will contain the arguments that were given at the command line that ran the program. We’re not using them here. Unit serves a similar purpose to void in programming languages like C and Java. In Scala, every method has to return some value as long as it doesn’t crash or hang. But main doesn’t return anything meaningful, so there’s a special type Unit that is the return type of such methods. There’s only one value of this type and the literal syntax for it is (), a pair of empty parentheses, pronounced “unit” just like the type. Usually a return type of Unit is a hint that the method has a side effect. The body of our main method prints to the console the String returned by the call to formatAbs. Note that the return type of println is Unit, which happens to be what we need to return from main. 2.2 Running our program This section discusses the simplest possible way of running your Scala programs, suit- able for short examples. More typically, you’ll build and run your Scala code using sbt, the build tool for Scala, and/or an IDE like IntelliJ or Eclipse. See the book’s source code repo on GitHub (https://github.com/fpinscala/fpinscala) for more information on getting set up with sbt. The simplest way we can run this Scala program (MyModule) is from the command line, by invoking the Scala compiler directly ourselves. We start by putting the code in a file called MyModule.scala or something similar. We can then compile it to Java bytecode using the scalac compiler: > scalac MyModule.scala This will generate some files ending with the .class suffix. These files contain com- piled code that can be run with the Java Virtual Machine (JVM). The code can be exe- cuted using the scala command-line tool: > scala MyModule The absolute value of -42 is 42. Licensed to Emre Sevinc 18 CHAPTER 2 Getting started with functional programming in Scala Actually, it’s not strictly necessary to compile the code first with scalac. A simple pro- gram like the one we’ve written here can be run using just the Scala interpreter by passing it to the scala command-line tool directly: > scala MyModule.scala The absolute value of -42 is 42. This can be handy when using Scala for scripting. The interpreter will look for any object within the file MyModule.scala that has a main method with the appropriate signature, and will then call it. Lastly, an alternative way is to start the Scala interpreter’s interactive mode, the REPL (which stands for read-evaluate-print loop). It’s a great idea to have a REPL win- dow open so you can try things out while you’re programming in Scala. We can load our source file into the REPL and try things out (your actual console output may differ slightly): > scala Welcome to Scala. Type in expressions to have them evaluated. Type :help for more information. scala> :load MyModule.scala Loading MyModule.scala... defined module MyModule scala> MyModule.abs(-42) res0: Int = 42 It’s also possible to copy and paste individual lines of code into the REPL. It even has a paste mode (accessed with the :paste command) designed to paste code that spans multiple lines. It’s a good idea to get familiar with the REPL and its features because it’s a tool that you’ll use a lot as a Scala programmer. 2.3 Modules, objects, and namespaces In this section, we’ll discuss some additional aspects of Scala’s syntax related to mod- ules, objects, and namespaces. In the preceding REPL session, note that in order to refer to our abs method, we had to say MyModule.abs because abs was defined in the MyModule object. We say that MyModule is its namespace. Aside from some technicalities, every value in Scala is what’s called an object,1 and each object may have zero or more members. An object whose primary purpose is giving its members a namespace is some- times called a module. A member can be a method declared with the def keyword, or it can be another object declared with val or object. Objects can also have other kinds of members that we’ll ignore for now. We access the members of objects with the typical object-oriented dot notation, which is a namespace (the name that refers to the object) followed by a dot (the period 1 Unlike Java, values of primitive types like Int are also considered objects for the purposes of this discussion. :load is a command to the REPL to interpret a Scala source file. (Note that unfortunately this won’t work for Scala files with package declarations.) We can type Scala expressions at the prompt. The REPL evaluates our Scala expression and prints the answer. It also gives the answer a name, res0, that we can refer to later, and shows its type, which in this case is Int. Licensed to Emre Sevinc 19Higher-order functions: passing functions to functions character), followed by the name of the member, as in MyModule.abs(-42). To use the toString member on the object 42, we’d use 42.toString. The implementations of members within an object can refer to each other unqualified (without prefixing the object name), but if needed they have access to their enclosing object using a special name: this.2 Note that even an expression like 2 + 1 is just calling a member of an object. In that case, what we’re calling is the + member of the object 2. It’s really syntactic sugar for the expression 2.+(1), which passes 1 as an argument to the method + on the object 2. Scala has no special notion of operators. It’s simply the case that + is a valid method name in Scala. Any method name can be used infix like that (omitting the dot and parentheses) when calling it with a single argument. For example, instead of MyModule.abs(42) we can say MyModule abs 42 and get the same result. You can use whichever you find more pleasing in any given case. We can bring an object’s member into scope by importing it, which allows us to call it unqualified from then on: scala> import MyModule.abs import MyModule.abs scala> abs(-42) res0: 42 We can bring all of an object’s (nonprivate) members into scope by using the under- score syntax: import MyModule._ 2.4 Higher-order functions: passing functions to functions Now that we’ve covered the basics of Scala’s syntax, we’ll move on to covering some of the basics of writing functional programs. The first new idea is this: functions are values. And just like values of other types—such as integers, strings, and lists—func- tions can be assigned to variables, stored in data structures, and passed as arguments to functions. When writing purely functional programs, we’ll often find it useful to write a func- tion that accepts other functions as arguments. This is called a higher-order function (HOF), and we’ll look next at some simple examples to illustrate. In later chapters, we’ll see how useful this capability really is, and how it permeates the functional pro- gramming style. But to start, suppose we wanted to adapt our program to print out both the absolute value of a number and the factorial of another number. Here’s a sample run of such a program: The absolute value of -42 is 42 The factorial of 7 is 5040 2 Note that in this book, we’ll use the term function to refer more generally to either so-called standalone functions like sqrt or abs, or members of some class, including methods. When it’s clear from the context, we’ll also use the terms method and function interchangeably, since what matters is not the syntax of invocation (obj.method(12) vs. method(obj, 12), but the fact that we’re talking about some parameterized block of code. Licensed to Emre Sevinc 20 CHAPTER 2 Getting started with functional programming in Scala 2.4.1 A short detour: writing loops functionally First, let’s write factorial: def factorial(n: Int): Int = { def go(n: Int, acc: Int): Int = if (n <= 0) acc else go(n-1, n*acc) go(n, 1) } The way we write loops functionally, without mutating a loop variable, is with a recur- sive function. Here we’re defining a recursive helper function inside the body of the factorial function. Such a helper function is often called go or loop by convention. In Scala, we can define functions inside any block, including within another function definition. Since it’s local, the go function can only be referred to from within the body of the factorial function, just like a local variable would. The definition of factorial finally just consists of a call to go with the initial conditions for the loop. The arguments to go are the state for the loop. In this case, they’re the remaining value n, and the current accumulated factorial acc. To advance to the next iteration, we simply call go recursively with the new loop state (here, go(n-1, n*acc)), and to exit from the loop, we return a value without a recursive call (here, we return acc in the case that n <= 0). Scala detects this sort of self-recursion and compiles it to the same sort of bytecode as would be emitted for a while loop,3 so long as the recursive call is in tail position. See the sidebar for the technical details on this, but the basic idea is that this optimization4 (called tail call elimination) is applied when there’s no addi- tional work left to do after the recursive call returns. 3 We can write while loops by hand in Scala, but it’s rarely necessary and considered bad form since it hinders good compositional style. 4 The term optimization is not really appropriate here. An optimization usually connotes some nonessential per- formance improvement, but when we use tail calls to write loops, we generally rely on their being compiled as iterative loops that don’t consume a call stack frame for each iteration (which would result in a StackOverflowError for large inputs). An inner function, or local definition. It’s common in Scala to write functions that are local to the body of another function. In functional programming, we shouldn’t consider this a bigger deal than local integers or strings. Tail calls in Scala A call is said to be in tail position if the caller does nothing other than return the value of the recursive call. For example, the recursive call to go(n-1,n*acc) we discussed earlier is in tail position, since the method returns the value of this recursive call directly and does nothing else with it. On the other hand, if we said 1 + go(n-1,n*acc), go would no longer be in tail position, since the method would still have work to do when go returned its result (namely, adding 1 to it). If all recursive calls made by a function are in tail position, Scala automatically com- piles the recursion to iterative loops that don’t consume call stack frames for each iteration. By default, Scala doesn’t tell us if tail call elimination was successful, but if we’re expecting this to occur for a recursive function we write, we can tell the Scala Licensed to Emre Sevinc 21Higher-order functions: passing functions to functions NOTE See the preface for information on the exercises. EXERCISE 2.1 Write a recursive function to get the nth Fibonacci number (http://mng.bz/C29s). The first two Fibonacci numbers are 0 and 1. The nth number is always the sum of the previous two—the sequence begins 0, 1, 1, 2, 3, 5. Your definition should use a local tail-recursive function. def fib(n: Int): Int 2.4.2 Writing our first higher-order function Now that we have factorial, let’s edit our program from before to include it. object MyModule { ... private def formatAbs(x: Int) = { val msg = "The absolute value of %d is %d." msg.format(x, abs(x)) } private def formatFactorial(n: Int) = { val msg = "The factorial of %d is %d." msg.format(n, factorial(n)) } def main(args: Array[String]): Unit = { println(formatAbs(-42)) println(formatFactorial(7)) } } Listing 2.2 A simple program including the factorial function compiler about this assumption using the tailrec annotation (http://mng.bz/ bWT5), so it can give us a compile error if it’s unable to eliminate the tail calls of the function. Here’s the syntax for this: def factorial(n: Int): Int = { @annotation.tailrec def go(n: Int, acc: Int): Int = if (n <= 0) acc else go(n-1, n*acc) go(n, 1) } We won’t talk much more about annotations in this book (you’ll find more information at http://mng.bz/GK8T), but we’ll use @annotation.tailrec extensively. Definitions of abs and factorial go here. Licensed to Emre Sevinc 22 CHAPTER 2 Getting started with functional programming in Scala The two functions, formatAbs and formatFactorial, are almost identical. If we like, we can generalize these to a single function, formatResult, which accepts as an argu- ment the function to apply to its argument: def formatResult(name: String, n: Int, f: Int => Int) = { val msg = "The %s of %d is %d." msg.format(name, n, f(n)) } Our formatResult function is a higher-order function (HOF) that takes another func- tion, called f (see sidebar on variable-naming conventions). We give a type to f, as we would for any other parameter. Its type is Int => Int (pronounced “int to int” or “int arrow int”), which indicates that f expects an integer argument and will also return an integer. Our function abs from before matches that type. It accepts an Int and returns an Int. And likewise, factorial accepts an Int and returns an Int, which also matches the Int => Int type. We can therefore pass abs or factorial as the f argument to formatResult: scala> formatResult("absolute value", -42, abs) res0: String = "The absolute value of -42 is 42." scala> formatResult("factorial", 7, factorial) res1: String = "The factorial of 7 is 5040." 2.5 Polymorphic functions: abstracting over types So far we’ve defined only monomorphic functions, or functions that operate on only one type of data. For example, abs and factorial are specific to arguments of type Int, and the higher-order function formatResult is also fixed to operate on functions that take arguments of type Int. Often, and especially when writing HOFs, we want to write code that works for any type it’s given. These are called polymorphic functions,5 and in the chapters ahead, you’ll get plenty of experience writing such functions. Here we’ll just introduce the idea. 5 We’re using the term polymorphism in a slightly different way than you might be used to if you’re familiar with object-oriented programming, where that term usually connotes some form of subtyping or inheritance rela- tionship. There are no interfaces or subtyping here in this example. The kind of polymorphism we’re using here is sometimes called parametric polymorphism. f is required to be a function from Int to Int. Variable-naming conventions It’s a common convention to use names like f, g, and h for parameters to a higher- order function. In functional programming, we tend to use very short variable names, even one-letter names. This is usually because HOFs are so general that they have no opinion on what the argument should actually do. All they know about the argu- ment is its type. Many functional programmers feel that short names make code eas- ier to read, since it makes the structure of the code easier to see at a glance. Licensed to Emre Sevinc 23Polymorphic functions: abstracting over types 2.5.1 An example of a polymorphic function We can often discover polymorphic functions by observing that several monomorphic functions all share a similar structure. For example, the following monomorphic func- tion, findFirst, returns the first index in an array where the key occurs, or -1 if it’s not found. It’s specialized for searching for a String in an Array of String values. def findFirst(ss: Array[String], key: String): Int = { @annotation.tailrec def loop(n: Int): Int = if (n >= ss.length) -1 else if (ss(n) == key) n else loop(n + 1) loop(0) } The details of the code aren’t too important here. What’s important is that the code for findFirst will look almost identical if we’re searching for a String in an Array[String], an Int in an Array[Int], or an A in an Array[A] for any given type A. We can write findFirst more generally for any type A by accepting a function to use for testing a particular A value. def findFirst[A](as: Array[A], p: A => Boolean): Int = { @annotation.tailrec def loop(n: Int): Int = if (n >= as.length) -1 else if (p(as(n))) n else loop(n + 1) loop(0) } This is an example of a polymorphic function, sometimes called a generic function. We’re abstracting over the type of the array and the function used for searching it. To write a polymorphic function as a method, we introduce a comma-separated list of type parameters, surrounded by square brackets (here, just a single [A]), following the name of the function, in this case findFirst. We can call the type parameters any- thing we want—[Foo, Bar, Baz] and [TheParameter, another_good_one] are valid type parameter declarations—though by convention we typically use short, one-letter, uppercase type parameter names like [A,B,C]. Listing 2.3 Monomorphic function to find a String in an array Listing 2.4 Polymorphic function to find an element in an array If n is past the end of the array, return -1, indicating the key doesn’t exist in the array. ss(n) extracts the nth element of the array ss. If the element at n is equal to the key, return n, indicating that the element appears in the array at that index. Otherwise, increment n and keep looking. Start the loop at the first element of the array. Instead of hardcoding String, take a type A as a parameter. And instead of hardcoding an equality check for a given key, take a function with which to test each element of the array. If the function p matches the current element, we’ve found a match and we return its index in the array. Licensed to Emre Sevinc 24 CHAPTER 2 Getting started with functional programming in Scala The type parameter list introduces type variables that can be referenced in the rest of the type signature (exactly analogous to how variables introduced in the parameter list to a function can be referenced in the body of the function). In findFirst, the type variable A is referenced in two places: the elements of the array are required to have the type A (since it’s an Array[A]), and the p function must accept a value of type A (since it’s a function of type A => Boolean). The fact that the same type variable is referenced in both places in the type signature implies that the type must be the same for both arguments, and the compiler will enforce this fact anywhere we try to call findFirst. If we try to search for a String in an Array[Int], for instance, we’ll get a type mismatch error. EXERCISE 2.2 Implement isSorted, which checks whether an Array[A] is sorted according to a given comparison function: def isSorted[A](as: Array[A], ordered: (A,A) => Boolean): Boolean 2.5.2 Calling HOFs with anonymous functions When using HOFs, it’s often convenient to be able to call these functions with anony- mous functions or function literals, rather than having to supply some existing named function. For instance, we can test the findFirst function in the REPL as follows: scala> findFirst(Array(7, 9, 13), (x: Int) => x == 9) res2: Int = 1 There is some new syntax here. The expression Array(7, 9, 13) is an array literal. It constructs a new array with three integers in it. Note the lack of a keyword like new to construct the array. The syntax (x: Int) => x == 9 is a function literal or anonymous function. Instead of defining this function as a method with a name, we can define it inline using this con- venient syntax. This particular function takes one argument called x of type Int, and it returns a Boolean indicating whether x is equal to 9. In general, the arguments to the function are declared to the left of the => arrow, and we can then use them in the body of the function to the right of the arrow. For example, if we want to write an equality function that takes two integers and checks if they’re equal to each other, we could write that like this: scala> (x: Int, y: Int) => x == y res3: (Int, Int) => Boolean = The notation given by the REPL indicates that the value of res3 is a func- tion that takes two arguments. When the type of the function’s inputs can be inferred Licensed to Emre Sevinc 25Following types to implementations by Scala from the context, the type annotations on the function’s arguments may be elided, for example, (x,y) => x < y. We’ll see an example of this in the next section, and lots more examples throughout this book. 2.6 Following types to implementations As you might have seen when writing isSorted, the universe of possible implementa- tions is significantly reduced when implementing a polymorphic function. If a func- tion is polymorphic in some type A, the only operations that can be performed on that A are those passed into the function as arguments (or that can be defined in terms of these given operations).6 In some cases, you’ll find that the universe of possibilities for a given polymorphic type is constrained such that only one implementation is possible! Let’s look at an example of a function signature that can only be implemented in one way. It’s a higher-order function for performing what’s called partial application. This function, partial1, takes a value and a function of two arguments, and returns a function of one argument as its result. The name comes from the fact that the func- tion is being applied to some but not all of the arguments it requires: def partial1[A,B,C](a: A, f: (A,B) => C): B => C 6 Technically, all values in Scala can be compared for equality (using ==), and turned into strings with toString and integers with hashCode. But this is something of a wart inherited from Java. Functions as values in Scala When we define a function literal, what is actually being defined in Scala is an object with a method called apply. Scala has a special rule for this method name, so that objects that have an apply method can be called as if they were themselves meth- ods. When we define a function literal like (a, b) => a < b, this is really syntactic sugar for object creation: val lessThan = new Function2[Int, Int, Boolean] { def apply(a: Int, b: Int) = a < b } lessThan has type Function2[Int,Int,Boolean], which is usually written (Int,Int) => Boolean. Note that the Function2 interface (known in Scala as a trait) has an apply method. And when we call the lessThan function with less- Than(10, 20), it’s really syntactic sugar for calling its apply method: scala> val b = lessThan.apply(10, 20) b: Boolean = true Function2 is just an ordinary trait (an interface) provided by the standard Scala library (API docs link: http://mng.bz/qFMr) to represent function objects that take two arguments. Also provided are Function1, Function3, and others, taking a num- ber of arguments indicated by the name. Because functions are just ordinary Scala objects, we say that they’re first-class values. We’ll often use “function” to refer to either such a first-class function or a method, depending on context. Licensed to Emre Sevinc 26 CHAPTER 2 Getting started with functional programming in Scala The partial1 function has three type parameters: A, B, and C. It then takes two argu- ments. The argument f is itself a function that takes two arguments of types A and B, respectively, and returns a value of type C. The value returned by partial1 will also be a function, of type B => C. How would we go about implementing this higher-order function? It turns out that there’s only one implementation that compiles, and it follows logically from the type signature. It’s like a fun little logic puzzle.7 Let’s start by looking at the type of thing that we have to return. The return type of partial1 is B => C, so we know that we have to return a function of that type. We can just begin writing a function literal that takes an argument of type B: def partial1[A,B,C](a: A, f: (A,B) => C): B => C = (b: B) => ??? This can be weird at first if you’re not used to writing anonymous functions. Where did that B come from? Well, we’ve just written, “Return a function that takes a value b of type B.” On the right-hand-side of the => arrow (where the question marks are now) comes the body of that anonymous function. We’re free to refer to the value b in there for the same reason that we’re allowed to refer to the value a in the body of partial1.8 Let’s keep going. Now that we’ve asked for a value of type B, what do we want to return from our anonymous function? The type signature says that it has to be a value of type C. And there’s only one way to get such a value. According to the signature, C is the return type of the function f. So the only way to get that C is to pass an A and a B to f. That’s easy: def partial1[A,B,C](a: A, f: (A,B) => C): B => C = (b: B) => f(a, b) And we’re done! The result is a higher-order function that takes a function of two arguments and partially applies it. That is, if we have an A and a function that needs both A and B to produce C, we can get a function that just needs B to produce C (since we already have the A). It’s like saying, “If I can give you a carrot for an apple and a banana, and you already gave me an apple, you just have to give me a banana and I’ll give you a carrot.” Note that the type annotation on b isn’t needed here. Since we told Scala the return type would be B => C, Scala knows the type of b from the context and we could just write b => f(a,b) as the implementation. Generally speaking, we’ll omit the type annotation on a function literal if it can be inferred by Scala. 7 Even though it’s a fun puzzle, this isn’t a purely academic exercise. Functional programming in practice involves a lot of fitting building blocks together in the only way that makes sense. The purpose of this exercise is to get practice using higher-order functions, and using Scala’s type system to guide your programming. 8 Within the body of this inner function, the outer a is still in scope. We sometimes say that the inner function closes over its environment, which includes a. Licensed to Emre Sevinc 27Following types to implementations EXERCISE 2.3 Let’s look at another example, currying,9 which converts a function f of two arguments into a function of one argument that partially applies f. Here again there’s only one implementation that compiles. Write this implementation. def curry[A,B,C](f: (A, B) => C): A => (B => C) EXERCISE 2.4 Implement uncurry, which reverses the transformation of curry. Note that since => associates to the right, A => (B => C) can be written as A => B => C. def uncurry[A,B,C](f: A => B => C): (A, B) => C Let’s look at a final example, function composition, which feeds the output of one func- tion to the input of another function. Again, the implementation of this function is fully determined by its type signature. EXERCISE 2.5 Implement the higher-order function that composes two functions. def compose[A,B,C](f: B => C, g: A => B): A => C This is such a common thing to want to do that Scala’s standard library provides compose as a method on Function1 (the interface for functions that take one argu- ment). To compose two functions f and g, we simply say f compose g.10 It also pro- vides an andThen method. f andThen g is the same as g compose f: scala> val f = (x: Double) => math.Pi/2-x f: Double => Double = scala> val cos = f andThen math.sin cos: Double => Double = It’s all well and good to puzzle together little one-liners like this, but what about pro- gramming with a large real-world code base? In functional programming, it turns out to be exactly the same. Higher-order functions like compose don’t care whether 9 This is named after the mathematician Haskell Curry, who discovered the principle. It was independently dis- covered earlier by Moses Schoenfinkel, but Schoenfinkelization didn’t catch on. 10 Solving the compose exercise by using this library function is considered cheating. Licensed to Emre Sevinc 28 CHAPTER 2 Getting started with functional programming in Scala they’re operating on huge functions backed by millions of lines of code or functions that are simple one-liners. Polymorphic, higher-order functions often end up being extremely widely applicable, precisely because they say nothing about any particular domain and are simply abstracting over a common pattern that occurs in many con- texts. For this reason, programming in the large has much the same flavor as program- ming in the small. We’ll write a lot of widely applicable functions over the course of this book, and the exercises in this chapter are a taste of the style of reasoning you’ll employ when writing such functions. 2.7 Summary In this chapter, we learned enough of the Scala language to get going, and some pre- liminary functional programming concepts. We learned how to define simple func- tions and programs, including how we can express loops using recursion; then we introduced the idea of higher-order functions, and we got some practice writing poly- morphic functions in Scala. We saw how the implementations of polymorphic func- tions are often significantly constrained, such that we can often simply “follow the types” to the correct implementation. This is something we’ll see a lot of in the chap- ters ahead. Although we haven’t yet written any large or complex programs, the principles we’ve discussed here are scalable and apply equally well to programming in the large as they do to programming in the small. Next we’ll look at using pure functions to manipulate data. Licensed to Emre Sevinc 29 Functional data structures We said in the introduction that functional programs don’t update variables or modify mutable data structures. This raises pressing questions: what sort of data structures can we use in functional programming, how do we define them in Scala, and how do we operate on them? In this chapter, we’ll learn the concept of func- tional data structures and how to work with them. We’ll use this as an opportunity to introduce how data types are defined in functional programming, learn about the related technique of pattern matching, and get practice writing and generalizing pure functions. This chapter has a lot of exercises, particularly to help with this last point—writ- ing and generalizing pure functions. Some of these exercises may be challenging. As always, consult the hints or the answers at our GitHub site (https://github.com/ fpinscala/fpinscala; see the preface), or ask for help online if you need to. 3.1 Defining functional data structures A functional data structure is (not surprisingly) operated on using only pure func- tions. Remember, a pure function must not change data in place or perform other side effects. Therefore, functional data structures are by definition immutable. For exam- ple, the empty list (written List() or Nil in Scala) is as eternal and immutable as the integer values 3 or 4. And just as evaluating 3 + 4 results in a new number 7 with- out modifying either 3 or 4, concatenating two lists together (the syntax for this is a++ b for two lists a and b) yields a new list and leaves the two inputs unmodified. Doesn’t this mean we end up doing a lot of extra copying of the data? Perhaps surprisingly, the answer is no, and we’ll talk about exactly why that is. But first let’s examine what’s probably the most ubiquitous functional data structure, the singly linked list. The definition here is identical in spirit to (though simpler than) the Licensed to Emre Sevinc 30 CHAPTER 3 Functional data structures List data type defined in Scala’s standard library. This code listing introduces a lot of new syntax and concepts that we’ll talk through in detail. package fpinscala.datastructures sealed trait List[+A] case object Nil extends List[Nothing] case class Cons[+A](head: A, tail: List[A]) extends List[A] object List { def sum(ints: List[Int]): Int = ints match { case Nil => 0 case Cons(x,xs) => x + sum(xs) } def product(ds: List[Double]): Double = ds match { case Nil => 1.0 case Cons(0.0, _) => 0.0 case Cons(x,xs) => x * product(xs) } def apply[A](as: A*): List[A] = if (as.isEmpty) Nil else Cons(as.head, apply(as.tail: _*)) } Let’s look first at the definition of the data type, which begins with the keywords sealed trait. In general, we introduce a data type with the trait keyword. A trait is an abstract interface that may optionally contain implementations of some meth- ods. Here we’re declaring a trait, called List, with no methods on it. Adding sealed in front means that all implementations of the trait must be declared in this file.1 There are two such implementations, or data constructors, of List (each introduced with the keyword case) declared next, to represent the two possible forms a List can take. As the figure at the top of page 31 shows, a List can be empty, denoted by the data constructor Nil, or it can be nonempty, denoted by the data constructor Cons (traditionally short for construct). A nonempty list consists of an initial element, head, followed by a List (possibly empty) of remaining elements (the tail): case object Nil extends List[Nothing] case class Cons[+A](head: A, tail: List[A]) extends List[A] Listing 3.1 Singly linked lists 1 We could also say abstract class here instead of trait. The distinction between the two is not at all sig- nificant for our purposes right now. See section 5.3 in the Scala Language Specification (http://mng.bz/ R75t) for more on the distinction. List data type, parameterized on a type, A. A List data constructor representing the empty list. Another data constructor, representing nonempty lists. Note that tail is another List[A], which may be Nil or another Cons. List companion object. Contains functions for creating and working with lists. A function that uses pattern matching to add up a list of integers. The sum of the empty list is 0. The sum of a list starting with x is x plus the sum of the rest of the list. Variadic function syntax. Licensed to Emre Sevinc 31Defining functional data structures Just as functions can be polymorphic, data types can be as well, and by adding the type parameter [+A] after sealed trait List and then using that A parameter inside of the Cons data constructor, we declare the List data type to be polymorphic in the type of elements it contains, which means we can use this same definition for a list of Int elements (denoted List[Int]), Double elements (denoted List[Double]), String elements (List[String]), and so on (the + indicates that the type parameter A is cova- riant—see sidebar “More about variance” for more information). A data constructor declaration gives us a function to construct that form of the data type. Here are a few examples: val ex1: List[Double] = Nil val ex2: List[Int] = Cons(1, Nil) val ex3: List[String] = Cons("a", Cons("b", Nil)) The case object Nil lets us write Nil to construct an empty List, and the case class Cons lets us write Cons(1, Nil), Cons("a", Cons("b", Nil)) and so on to build singly linked lists of arbitrary lengths.2 Note that because List is parameterized on a type, A, these are polymorphic functions that can be instantiated with different types for A. Here, ex2 instantiates the A type parameter to Int, while ex3 instantiates it to String. The ex1 example is interesting—Nil is being instantiated with type List[Double], which is allowed because the empty list contains no elements and can be considered a list of whatever type we want! Each data constructor also introduces a pattern that can be used for pattern match- ing, as in the functions sum and product. We’ll examine pattern matching in more detail next. 2 Scala generates a default def toString: String method for any case class or case object, which can be convenient for debugging. You can see the output of this default toString implementation if you exper- iment with List values in the REPL, which uses toString to render the result of each expression. Cons(1,Nil) will be printed as the string "Cons(1, Nil)", for instance. But note that the generated toString will be naively recursive and will cause stack overflow when printing long lists, so you may wish to provide a different implementation. Singly linked lists TailHead Cons "a" TailHead Cons "b" Nil These mean the same thing. They mean this data structure in memory. List("a", "b") Cons("a", Cons("b", Nil)) More about variance In the declaration trait List[+A], the + in front of the type parameter A is a vari- ance annotation that signals that A is a covariant or “positive” parameter of List. This means that, for instance, List[Dog] is considered a subtype of List[Animal], assuming Dog is a subtype of Animal. (More generally, for all types X and Y, if X is a Licensed to Emre Sevinc 32 CHAPTER 3 Functional data structures 3.2 Pattern matching Let’s look in detail at the functions sum and product, which we place in the object List, sometimes called the companion object to List (see sidebar). Both these defini- tions make use of pattern matching: def sum(ints: List[Int]): Int = ints match { case Nil=>0 case Cons(x,xs) => x + sum(xs) } def product(ds: List[Double]): Double = ds match { case Nil=>1.0 case Cons(0.0, _) => 0.0 case Cons(x,xs) => x * product(xs) } As you might expect, the sum function states that the sum of an empty list is 0, and the sum of a nonempty list is the first element, x, plus the sum of the remaining elements, xs.3 Likewise the product definition states that the product of an empty list is 1.0, the product of any list starting with 0.0 is 0.0, and the product of any other nonempty list is the first element multiplied by the product of the remaining elements. Note that these are recursive definitions, which are common when writing functions that oper- ate over recursive data types like List (which refers to itself recursively in its Cons data constructor). Pattern matching works a bit like a fancy switch statement that may descend into the structure of the expression it examines and extract subexpressions of that 3 We could call x and xs anything there, but it’s a common convention to use xs, ys, as, or bs as variable names for a sequence of some sort, and x, y, z, a, or b as the name for a single element of a sequence. Another common naming convention is h for the first element of a list (the head of the list), t for the remaining ele- ments (the tail), and l for an entire list. (continued) subtype of Y, then List[X] is a subtype of List[Y]). We could leave out the + in front of the A, which would make List invariant in that type parameter. But notice now that Nil extends List[Nothing]. Nothing is a subtype of all types, which means that in conjunction with the variance annotation, Nil can be considered a List[Int], a List[Double], and so on, exactly as we want. These concerns about variance aren’t very important for the present discussion and are more of an artifact of how Scala encodes data constructors via subtyping, so don’t worry if this is not completely clear right now. It’s certainly possible to write code without using variance annotations at all, and function signatures are some- times simpler (whereas type inference often gets worse). We’ll use variance annota- tions throughout this book where it’s convenient to do so, but you should feel free to experiment with both approaches. Licensed to Emre Sevinc 33Pattern matching structure. It’s introduced with an expression (the target or scrutinee) like ds, followed by the keyword match, and a {}-wrapped sequence of cases. Each case in the match consists of a pattern (like Cons(x,xs)) to the left of the => and a result (like x * product(xs)) to the right of the =>. If the target matches the pattern in a case (discussed next), the result of that case becomes the result of the entire match expression. If multiple pat- terns match the target, Scala chooses the first matching case. 4 Let’s look at a few more examples of pattern matching: List(1,2,3) match { case _ => 42 } results in 42. Here we’re using a variable pattern, _, which matches any expression. We could say x or foo instead of _, but we usually use _ to indicate a variable whose value we ignore in the result of the case.5 List(1,2,3) match { case Cons(h,_) => h } results in 1. Here we’re using a data constructor pattern in conjunction with variables to capture or bind a subex- pression of the target. List(1,2,3) match { case Cons(_,t) => t } results in List(2,3). List(1,2,3) match { case Nil => 42 } results in a MatchError at runtime. A MatchError indicates that none of the cases in a match expression matched the target. 4 There is some special support for them in the language that isn’t really relevant for our purposes. 5 The _ variable pattern is treated somewhat specially in that it may be mentioned multiple times in the pattern to ignore multiple parts of the target. Companion objects in Scala We’ll often declare a companion object in addition to our data type and its data con- structors. This is just an object with the same name as the data type (in this case List) where we put various convenience functions for creating or working with values of the data type. If, for instance, we wanted a function def fill[A](n: Int, a: A): List[A] that created a List with n copies of the element a, the List companion object would be a good place for it. Companion objects are more of a convention in Scala.4 We could have called this module Foo if we wanted, but calling it List makes it clear that the module contains functions relevant to working with lists. List(1, 2, 3) match {case Cons(h, _) => h} Cons(1, Cons(2, Cons(3, Nil))) The result is 1. Matching a list Equal Licensed to Emre Sevinc 34 CHAPTER 3 Functional data structures What determines if a pattern matches an expression? A pattern may contain literals like 3 or "hi"; variables like x and xs, which match anything, indicated by an identifier start- ing with a lowercase letter or underscore; and data constructors like Cons(x,xs) and Nil, which match only values of the corresponding form. (Nil as a pattern matches only the value Nil, and Cons(h,t) or Cons(x,xs) as a pattern only match Cons values.) These components of a pattern may be nested arbitrarily—Cons(x1, Cons(x2, Nil)) and Cons(y1, Cons(y2, Cons(y3, _))) are valid patterns. A pattern matches the target if there exists an assignment of variables in the pattern to subexpressions of the target that make it structurally equivalent to the target. The resulting expression for a matching case will then have access to these variable assignments in its local scope. EXERCISE 3.1 What will be the result of the following match expression? val x = List(1,2,3,4,5) match { case Cons(x, Cons(2, Cons(4, _))) => x case Nil=>42 case Cons(x, Cons(y, Cons(3, Cons(4, _)))) =>x+y case Cons(h, t) => h + sum(t) case _ => 101 } You’re strongly encouraged to try experimenting with pattern matching in the REPL to get a sense for how it behaves. Variadic functions in Scala The function apply in the object List is a variadic function, meaning it accepts zero or more arguments of type A: def apply[A](as: A*): List[A] = if (as.isEmpty) Nil else Cons(as.head, apply(as.tail: _*)) For data types, it’s a common idiom to have a variadic apply method in the compan- ion object to conveniently construct instances of the data type. By calling this function apply and placing it in the companion object, we can invoke it with syntax like List(1,2,3,4) or List("hi","bye"), with as many values as we want separated by commas (we sometimes call this the list literal or just literal syntax). Variadic functions are just providing a little syntactic sugar for creating and passing a Seq of elements explicitly. Seq is the interface in Scala’s collections library imple- mented by sequence-like data structures such as lists, queues, and vectors. Inside apply, the argument as will be bound to a Seq[A] (documentation at http://mng.bz/ f4k9), which has the functions head (returns the first element) and tail (returns all elements but the first). The special _* type annotation allows us to pass a Seq to a variadic method. Licensed to Emre Sevinc 35Data sharing in functional data structures 3.3 Data sharing in functional data structures When data is immutable, how do we write functions that, for example, add or remove elements from a list? The answer is simple. When we add an element 1 to the front of an existing list, say xs, we return a new list, in this case Cons(1,xs). Since lists are immutable, we don’t need to actually copy xs; we can just reuse it. This is called data sharing. Sharing of immutable data often lets us implement functions more efficiently; we can always return immutable data structures without having to worry about subse- quent code modifying our data. There’s no need to pessimistically make copies to avoid modification or corruption.6 In the same way, to remove an element from the front of a list mylist = Cons(x,xs), we simply return its tail, xs. There’s no real removing going on. The orig- inal list, mylist, is still available, unharmed. We say that functional data structures are persistent, meaning that existing references are never changed by operations on the data structure. Let’s try implementing a few different functions for modifying lists in different ways. You can place this, and other functions that we write, inside the List companion object. EXERCISE 3.2 Implement the function tail for removing the first element of a List. Note that the function takes constant time. What are different choices you could make in your implementation if the List is Nil? We’ll return to this question in the next chapter. 6 Pessimistic copying can become a problem in large programs. When mutable data is passed through a chain of loosely coupled components, each component has to make its own copy of the data because other compo- nents might modify it. Immutable data is always safe to share, so we never have to make copies. We find that in the large, FP can often achieve greater efficiency than approaches that rely on side effects, due to much greater sharing of data and computation. .tail Data sharing "a" "b" "c" "d" List("a", "b", "c", "d") List("b", "c", "d") Both lists share the same data in memory. .tail does not modify the original list, it simply references the tail of the original list. Defensive copying is not needed, because the list is immutable. Licensed to Emre Sevinc 36 CHAPTER 3 Functional data structures EXERCISE 3.3 Using the same idea, implement the function setHead for replacing the first element of a List with a different value. 3.3.1 The efficiency of data sharing Data sharing often lets us implement operations more efficiently. Let’s look at a few examples. EXERCISE 3.4 Generalize tail to the function drop, which removes the first n elements from a list. Note that this function takes time proportional only to the number of elements being dropped—we don’t need to make a copy of the entire List. def drop[A](l: List[A], n: Int): List[A] EXERCISE 3.5 Implement dropWhile, which removes elements from the List prefix as long as they match a predicate. def dropWhile[A](l: List[A], f: A => Boolean): List[A] A more surprising example of data sharing is this function that adds all the elements of one list to the end of another: def append[A](a1: List[A], a2: List[A]): List[A] = a1 match { case Nil => a2 case Cons(h,t) => Cons(h, append(t, a2)) } Note that this definition only copies values until the first list is exhausted, so its run- time and memory usage are determined only by the length of a1. The remaining list then just points to a2. If we were to implement this same function for two arrays, we’d be forced to copy all the elements in both arrays into the result. In this case, the immutable linked list is much more efficient than an array! Licensed to Emre Sevinc 37Data sharing in functional data structures EXERCISE 3.6 Not everything works out so nicely. Implement a function, init, that returns a List consisting of all but the last element of a List. So, given List(1,2,3,4), init will return List(1,2,3). Why can’t this function be implemented in constant time like tail? def init[A](l: List[A]): List[A] Because of the structure of a singly linked list, any time we want to replace the tail of a Cons, even if it’s the last Cons in the list, we must copy all the previous Cons objects. Writing purely functional data structures that support different operations efficiently is all about finding clever ways to exploit data sharing. We’re not going to cover these data structures here; for now, we’re content to use the functional data structures oth- ers have written. As an example of what’s possible, in the Scala standard library there’s a purely functional sequence implementation, Vector (documentation at http:// mng.bz/Xhl8), with constant-time random access, updates, head, tail, init, and con- stant-time additions to either the front or rear of the sequence. See the chapter notes for links to further reading about how to design such data structures. 3.3.2 Improving type inference for higher-order functions Higher-order functions like dropWhile will often be passed anonymous functions. Let’s look at a typical example. Recall the signature of dropWhile: def dropWhile[A](l: List[A], f: A => Boolean): List[A] When we call it with an anonymous function for f, we have to specify the type of its argument, here named x: val xs: List[Int] = List(1,2,3,4,5) val ex1 = dropWhile(xs, (x: Int) =>x<4) The value of ex1 is List(4,5). It’s a little unfortunate that we need to state that the type of x is Int. The first argu- ment to dropWhile is a List[Int], so the function in the second argument must accept an Int. Scala can infer this fact if we group dropWhile into two argument lists: def dropWhile[A](as: List[A])(f: A => Boolean): List[A] = as match { case Cons(h,t) if f(h) => dropWhile(t)(f) case _=>as } Licensed to Emre Sevinc 38 CHAPTER 3 Functional data structures The syntax for calling this version of dropWhile looks like dropWhile(xs)(f). That is, dropWhile(xs) is returning a function, which we then call with the argument f (in other words, dropWhile is curried7). The main reason for grouping the arguments this way is to assist with type inference. We can now use dropWhile without annotations: val xs: List[Int] = List(1,2,3,4,5) val ex1 = dropWhile(xs)(x => x < 4) Note that x is not annotated with its type. More generally, when a function definition contains multiple argument groups, type information flows from left to right across these argument groups. Here, the first argument group fixes the type parameter A of dropWhile to Int, so the annotation on x => x < 4 is not required.8 We’ll often group and order our function arguments into multiple argument lists to maximize type inference. 3.4 Recursion over lists and generalizing to higher-order functions Let’s look again at the implementations of sum and product. We’ve simplified the product implementation slightly, so as not to include the “short-circuiting” logic of checking for 0.0: def sum(ints: List[Int]): Int = ints match { case Nil=>0 case Cons(x,xs) => x + sum(xs) } def product(ds: List[Double]): Double = ds match { case Nil=>1.0 case Cons(x, xs) => x * product(xs) } Note how similar these two definitions are. They’re operating on different types (List[Int] versus List[Double]), but aside from this, the only differences are the value to return in the case that the list is empty (0 in the case of sum, 1.0 in the case of product), and the operation to combine results (+ in the case of sum, * in the case of product). Whenever you encounter duplication like this, you can generalize it away by pulling subexpressions out into function arguments. If a subexpression refers to any local variables (the + operation refers to the local variables x and xs introduced by the pattern, similarly for product), turn the subexpression into a function that accepts these variables as arguments. Let’s do that now. Our function will take as arguments 7 Recall from the previous chapter that a function of two arguments can be represented as a function that accepts one argument and returns another function of one argument. 8 This is an unfortunate restriction of the Scala compiler; other functional languages like Haskell and OCaml provide complete inference, meaning type annotations are almost never required. See the notes for this chapter for more information and links to further reading. Licensed to Emre Sevinc 39Recursion over lists and generalizing to higher-order functions the value to return in the case of the empty list, and the function to add an element to the result in the case of a nonempty list.9 def foldRight[A,B](as: List[A], z: B)(f: (A, B) => B): B = as match { case Nil => z case Cons(x, xs) => f(x, foldRight(xs, z)(f)) } def sum2(ns: List[Int]) = foldRight(ns, 0)((x,y) =>x+y) def product2(ns: List[Double]) = foldRight(ns, 1.0)(_ * _) foldRight is not specific to any one type of element, and we discover while generaliz- ing that the value that’s returned doesn’t have to be of the same type as the elements of the list! One way of describing what foldRight does is that it replaces the construc- tors of the list, Nil and Cons, with z and f, illustrated here: Cons(1, Cons(2, Nil)) f (1, f (2, z )) Let’s look at a complete example. We’ll trace the evaluation of foldRight(Cons(1, Cons(2, Cons(3, Nil))), 0)((x,y) => x + y) by repeatedly substituting the defi- nition of foldRight for its usages. We’ll use program traces like this throughout this book: foldRight(Cons(1, Cons(2, Cons(3, Nil))), 0)((x,y) => x + y) 1 + foldRight(Cons(2, Cons(3, Nil)), 0)((x,y) =>x+y) 1 + (2 + foldRight(Cons(3, Nil), 0)((x,y) =>x+y)) 1 + (2 + (3 + (foldRight(Nil, 0)((x,y) => x + y)))) 1 + (2 + (3 + (0))) 6 Note that foldRight must traverse all the way to the end of the list (pushing frames onto the call stack as it goes) before it can begin collapsing it. Listing 3.2 Right folds and simple uses 9 In the Scala standard library, foldRight is a method on List and its arguments are curried similarly for bet- ter type inference. Again, placing f in its own argument group after as and z lets type inference determine the input types to f. _ * _ is more concise notation for (x,y) => x * y (see following sidebar). Replace foldRight with its definition. Underscore notation for anonymous functions The anonymous function (x,y) => x + y can be written as _ + _ in situations where the types of x and y could be inferred by Scala. This is a useful shorthand in cases where the function parameters are mentioned just once in the body of the function. Each underscore in an anonymous function expression like _ + _ introduces a new (unnamed) function parameter and references it. Arguments are introduced in left-to- right order. Here are a few more examples: Licensed to Emre Sevinc 40 CHAPTER 3 Functional data structures EXERCISE 3.7 Can product, implemented using foldRight, immediately halt the recursion and return 0.0 if it encounters a 0.0? Why or why not? Consider how any short-circuiting might work if you call foldRight with a large list. This is a deeper question that we’ll return to in chapter 5. EXERCISE 3.8 See what happens when you pass Nil and Cons themselves to foldRight, like this: foldRight(List(1,2,3), Nil:List[Int])(Cons(_,_)).10 What do you think this says about the relationship between foldRight and the data constructors of List? EXERCISE 3.9 Compute the length of a list using foldRight. def length[A](as: List[A]): Int EXERCISE 3.10 Our implementation of foldRight is not tail-recursive and will result in a StackOver- flowError for large lists (we say it’s not stack-safe). Convince yourself that this is the case, and then write another general list-recursion function, foldLeft, that is 10 The type annotation Nil:List[Int] is needed here, because otherwise Scala infers the B type parameter in foldRight as List[Nothing]. (continued) _+_ _*2 _.head _ drop _ Use this syntax judiciously. The meaning of this syntax in expressions like foo(_, g(List(_ + 1), _)) can be unclear. There are precise rules about scoping of these underscore-based anonymous functions in the Scala language specification, but if you have to think about it, we recommend just using ordinary named function parameters. (x,y) => x + y x => x * 2 xs => xs.head (xs,n) => xs.drop(n) Licensed to Emre Sevinc 41Recursion over lists and generalizing to higher-order functions tail-recursive, using the techniques we discussed in the previous chapter. Here is its signature:11 def foldLeft[A,B](as: List[A], z: B)(f: (B, A) => B): B EXERCISE 3.11 Write sum, product, and a function to compute the length of a list using foldLeft. EXERCISE 3.12 Write a function that returns the reverse of a list (given List(1,2,3) it returns List(3,2,1)). See if you can write it using a fold. EXERCISE 3.13 Hard: Can you write foldLeft in terms of foldRight? How about the other way around? Implementing foldRight via foldLeft is useful because it lets us implement foldRight tail-recursively, which means it works even for large lists without overflow- ing the stack. EXERCISE 3.14 Implement append in terms of either foldLeft or foldRight. EXERCISE 3.15 Hard: Write a function that concatenates a list of lists into a single list. Its runtime should be linear in the total length of all lists. Try to use functions we have already defined. 3.4.1 More functions for working with lists There are many more useful functions for working with lists. We’ll cover a few more here, to get additional practice with generalizing functions and to get some basic famil- iarity with common patterns when processing lists. After finishing this section, you’re not going to emerge with an automatic sense of when to use each of these functions. 11 Again, foldLeft is defined as a method of List in the Scala standard library, and it is curried similarly for better type inference, so you can write mylist.foldLeft(0.0)(_ + _). Licensed to Emre Sevinc 42 CHAPTER 3 Functional data structures Just get in the habit of looking for possible ways to generalize any explicit recursive functions you write to process lists. If you do this, you’ll (re)discover these functions for yourself and develop an instinct for when you’d use each one. EXERCISE 3.16 Write a function that transforms a list of integers by adding 1 to each element. (Reminder: this should be a pure function that returns a new List!) EXERCISE 3.17 Write a function that turns each value in a List[Double] into a String. You can use the expression d.toString to convert some d: Double to a String. EXERCISE 3.18 Write a function map that generalizes modifying each element in a list while maintain- ing the structure of the list. Here is its signature:12 def map[A,B](as: List[A])(f: A => B): List[B] EXERCISE 3.19 Write a function filter that removes elements from a list unless they satisfy a given predicate. Use it to remove all odd numbers from a List[Int]. def filter[A](as: List[A])(f: A => Boolean): List[A] EXERCISE 3.20 Write a function flatMap that works like map except that the function given will return a list instead of a single result, and that list should be inserted into the final resulting list. Here is its signature: def flatMap[A,B](as: List[A])(f: A => List[B]): List[B] For instance, flatMap(List(1,2,3))(i => List(i,i)) should result in List(1,1,2,2,3,3). 12 In the standard library, map and flatMap are methods of List. Licensed to Emre Sevinc 43Recursion over lists and generalizing to higher-order functions EXERCISE 3.21 Use flatMap to implement filter. EXERCISE 3.22 Write a function that accepts two lists and constructs a new list by adding correspond- ing elements. For example, List(1,2,3) and List(4,5,6) become List(5,7,9). EXERCISE 3.23 Generalize the function you just wrote so that it’s not specific to integers or addition. Name your generalized function zipWith. LISTS IN THE STANDARD LIBRARY List exists in the Scala standard library (API documentation at http://mng.bz/vu45), and we’ll use the standard library version in subsequent chapters. The main difference between the List developed here and the standard library version is that Cons is called ::, which associates to the right,13 so 1 :: 2 :: Nil is equal to 1 :: (2 :: Nil), which is equal to List(1,2). When pattern matching, case Cons(h,t) becomes case h :: t, which avoids having to nest parentheses if writing a pattern like case h :: h2 :: t to extract more than just the first element of the List. There are a number of other useful methods on the standard library lists. You may want to try experimenting with these and other methods in the REPL after reading the API documentation. These are defined as methods on List[A], rather than as stand- alone functions as we’ve done in this chapter: def take(n: Int): List[A]—Returns a list consisting of the first n elements of this def takeWhile(f: A => Boolean): List[A]—Returns a list consisting of the lon- gest valid prefix of this whose elements all pass the predicate f def forall(f: A => Boolean): Boolean—Returns true if and only if all ele- ments of this pass the predicate f def exists(f: A => Boolean): Boolean—Returns true if any element of this passes the predicate f 13 In Scala, all methods whose names end in : are right-associative. That is, the expression x :: xs is actually the method call xs.::(x), which in turn calls the data constructor ::(x,xs). See the Scala language spec- ification for more information. Licensed to Emre Sevinc 44 CHAPTER 3 Functional data structures scanLeft and scanRight—Like foldLeft and foldRight, but they return the List of partial results rather than just the final accumulated value We recommend that you look through the Scala API documentation after finishing this chapter, to see what other functions there are. If you find yourself writing an explicit recursive function for doing some sort of list manipulation, check the List API to see if something like the function you need already exists. 3.4.2 Loss of efficiency when assembling list functions from simpler components One of the problems with List is that, although we can often express operations and algorithms in terms of very general-purpose functions, the resulting implementation isn’t always efficient—we may end up making multiple passes over the same input, or else have to write explicit recursive loops to allow early termination. EXERCISE 3.24 Hard: As an example, implement hasSubsequence for checking whether a List con- tains another List as a subsequence. For instance, List(1,2,3,4) would have List(1,2), List(2,3), and List(4) as subsequences, among others. You may have some difficulty finding a concise purely functional implementation that is also effi- cient. That’s okay. Implement the function however comes most naturally. We’ll return to this implementation in chapter 5 and hopefully improve on it. Note: Any two values x and y can be compared for equality in Scala using the expression x == y. def hasSubsequence[A](sup: List[A], sub: List[A]): Boolean 3.5 Trees List is just one example of what’s called an algebraic data type (ADT). (Somewhat con- fusingly, ADT is sometimes used elsewhere to stand for abstract data type.) An ADT is just a data type defined by one or more data constructors, each of which may contain zero or more arguments. We say that the data type is the sum or union of its data construc- tors, and each data constructor is the product of its arguments, hence the name alge- braic data type.14 14 The naming is not coincidental. There’s a deep connection, beyond the scope of this book, between the “addition” and “multiplication” of types to form an ADT and addition and multiplication of numbers. Licensed to Emre Sevinc 45Trees Algebraic data types can be used to define other data structures. Let’s define a simple binary tree data structure: sealed trait Tree[+A] case class Leaf[A](value: A) extends Tree[A] case class Branch[A](left: Tree[A], right: Tree[A]) extends Tree[A] Pattern matching again provides a convenient way of operating over elements of our ADT. Let’s try writing a few functions. Tuple types in Scala Pairs and tuples of other arities are also algebraic data types. They work just like the ADTs we’ve been writing here, but have special syntax: scala> val p = ("Bob", 42) p: (java.lang.String, Int) = (Bob,42) scala> p._1 res0: java.lang.String = Bob scala> p._2 res1: Int = 42 scala> p match { case (a,b) => b } res2: Int = 42 In this example, ("Bob", 42) is a pair whose type is (String,Int), which is syn- tactic sugar for Tuple2[String,Int] (API link: http://mng.bz/1F2N). We can extract the first or second element of this pair (a Tuple3 will have a method _3, and so on), and we can pattern match on this pair much like any other case class. Higher arity tuples work similarly—try experimenting with them in the REPL if you’re interested. Branch A tree RightLeft RightLeft Value Branch Leaf "a" Value Leaf "b" RightLeft Value Branch Leaf "c" Value Leaf "d" Licensed to Emre Sevinc 46 CHAPTER 3 Functional data structures EXERCISE 3.25 Write a function size that counts the number of nodes (leaves and branches) in a tree. EXERCISE 3.26 Write a function maximum that returns the maximum element in a Tree[Int]. (Note: In Scala, you can use x.max(y) or x max y to compute the maximum of two integers x and y.) EXERCISE 3.27 Write a function depth that returns the maximum path length from the root of a tree to any leaf. EXERCISE 3.28 Write a function map, analogous to the method of the same name on List, that modi- fies each element in a tree with a given function. 15 15 It’s also possible in Scala to expose patterns like Nil and Cons independent of the actual data constructors of the type. ADTs and encapsulation One might object that algebraic data types violate encapsulation by making public the internal representation of a type. In FP, we approach concerns about encapsulation differently—we don’t typically have delicate mutable state which could lead to bugs or violation of invariants if exposed publicly. Exposing the data constructors of a type is often fine, and the decision to do so is approached much like any other decision about what the public API of a data type should be.15 We do typically use ADTs for situations where the set of cases is closed (known to be fixed). For List and Tree, changing the set of data constructors would significantly change what these data types are. List is a singly linked list—that is its nature— and the two cases Nil and Cons form part of its useful public API. We can certainly write code that deals with a more abstract API than List (we’ll see examples of this later in the book), but this sort of information hiding can be handled as a separate layer rather than being baked into List directly. Licensed to Emre Sevinc 47Summary EXERCISE 3.29 Generalize size, maximum, depth, and map, writing a new function fold that abstracts over their similarities. Reimplement them in terms of this more general function. Can you draw an analogy between this fold function and the left and right folds for List? 3.6 Summary In this chapter, we covered a number of important concepts. We introduced algebraic data types and pattern matching, and showed how to implement purely functional data structures, including the singly linked list. We hope that, through the exercises in this chapter, you got more comfortable writing pure functions and generalizing them. We’ll continue to develop these skills in the chapters ahead. Licensed to Emre Sevinc 48 Handling errors without exceptions We noted briefly in chapter 1 that throwing exceptions is a side effect. If exceptions aren’t used in functional code, what is used instead? In this chapter, we’ll learn the basic principles for raising and handling errors functionally. The big idea is that we can represent failures and exceptions with ordinary values, and we can write higher-order functions that abstract out common patterns of error handling and recovery. The functional solution, of returning errors as values, is safer and retains referential transparency, and through the use of higher-order functions, we can preserve the primary benefit of exceptions—consolidation of error-handling logic. We’ll see how this works over the course of this chapter, after we take a closer look at exceptions and discuss some of their problems. For the same reason that we created our own List data type in the previous chapter, we’ll re-create in this chapter two Scala standard library types: Option and Either. The purpose is to enhance your understanding of how these types can be used for handling errors. After completing this chapter, you should feel free to use the Scala standard library version of Option and Either (though you’ll notice that the standard library versions of both types are missing some of the useful functions we define in this chapter). 4.1 The good and bad aspects of exceptions Why do exceptions break referential transparency, and why is that a problem? Let’s look at a simple example. We’ll define a function that throws an exception and call it. Licensed to Emre Sevinc 49The good and bad aspects of exceptions def failingFn(i: Int): Int = { val y: Int = throw new Exception("fail!") try { val x=42+5 x+y } catch { case e: Exception => 43 } } Calling failingFn from the REPL gives the expected error: scala> failingFn(12) java.lang.Exception: fail! at .failingFn(:8) ... We can prove that y is not referentially transparent. Recall that any RT expression may be substituted with the value it refers to, and this substitution should preserve pro- gram meaning. If we substitute throw new Exception("fail!") for y in x + y, it pro- duces a different result, because the exception will now be raised inside a try block that will catch the exception and return 43: def failingFn2(i: Int): Int = { try { val x=42+5 x+((throw new Exception("fail!")): Int) } catch { case e: Exception => 43 } } We can demonstrate this in the REPL: scala> failingFn2(12) res1: Int = 43 Another way of understanding RT is that the meaning of RT expressions does not depend on context and may be reasoned about locally, whereas the meaning of non-RT expres- sions is context-dependent and requires more global reasoning. For instance, the mean- ing of the RT expression 42 + 5 doesn’t depend on the larger expression it’s embedded in—it’s always and forever equal to 47. But the meaning of the expression throw new Exception("fail") is very context-dependent—as we just demonstrated, it takes on different meanings depending on which try block (if any) it’s nested within. There are two main problems with exceptions: As we just discussed, exceptions break RT and introduce context dependence, moving us away from the simple reasoning of the substitution model and making it possible to write confusing exception-based code. This is the source of the folklore advice that exceptions should be used only for error handling, not for control flow. Listing 4.1 Throwing and catching an exception val y: Int = ... declares y as having type Int and sets it equal to the right-hand side of =. A catch block is just a pattern- matching block like the ones we’ve seen. case e: Exception is a pattern that matches any Exception, and it binds this value to the identifier e. The match returns the value 43. A thrown Exception can be given any type; here we’re annotating it with the type Int. Licensed to Emre Sevinc 50 CHAPTER 4 Handling errors without exceptions Exceptions are not type-safe. The type of failingFn, Int => Int tells us nothing about the fact that exceptions may occur, and the compiler will certainly not force callers of failingFn to make a decision about how to handle those excep- tions. If we forget to check for an exception in failingFn, this won’t be detected until runtime. We’d like an alternative to exceptions without these drawbacks, but we don’t want to lose out on the primary benefit of exceptions: they allow us to consolidate and centralize error-handling logic, rather than being forced to distribute this logic throughout our codebase. The technique we use is based on an old idea: instead of throwing an excep- tion, we return a value indicating that an exceptional condition has occurred. This idea might be familiar to anyone who has used return codes in C to handle excep- tions. But instead of using error codes, we introduce a new generic type for these “pos- sibly defined values” and use higher-order functions to encapsulate common patterns of handling and propagating errors. Unlike C-style error codes, the error-handling strategy we use is completely type-safe, and we get full assistance from the type-checker in forcing us to deal with errors, with a minimum of syntactic noise. We’ll see how all of this works shortly. 4.2 Possible alternatives to exceptions Let’s now consider a realistic situation where we might use an exception and look at different approaches we could use instead. Here’s an implementation of a function that computes the mean of a list, which is undefined if the list is empty: def mean(xs: Seq[Double]): Double = if (xs.isEmpty) throw new ArithmeticException("mean of empty list!") else xs.sum / xs.length Checked exceptions Java’s checked exceptions at least force a decision about whether to handle or reraise an error, but they result in significant boilerplate for callers. More importantly, they don’t work for higher-order functions, which can’t possibly be aware of the spe- cific exceptions that could be raised by their arguments. For example, consider the map function we defined for List: def map[A,B](l: List[A])(f: A => B): List[B] This function is clearly useful, highly generic, and at odds with the use of checked exceptions—we can’t have a version of map for every single checked exception that could possibly be thrown by f. Even if we wanted to do this, how would map even know what exceptions were possible? This is why generic code, even in Java, so often resorts to using RuntimeException or some common checked Exception type. Seq is the common interface of various linear sequence-like collections. Check the API docs (http://mng.bz/f4k9) for more information. sum is defined as a method on Seq only if the elements of the sequence are numeric. The standard library accomplishes this trick with implicits, which we won’t go into here. Licensed to Emre Sevinc 51Possible alternatives to exceptions The mean function is an example of what’s called a partial function: it’s not defined for some inputs. A function is typically partial because it makes some assumptions about its inputs that aren’t implied by the input types.1 You may be used to throwing exceptions in this case, but we have a few other options. Let’s look at these for our mean example. The first possibility is to return some sort of bogus value of type Double. We could simply return xs.sum / xs.length in all cases, and have it result in 0.0/0.0 when the input is empty, which is Double.NaN; or we could return some other sentinel value. In other situations, we might return null instead of a value of the needed type. This gen- eral class of approaches is how error handling is often done in languages without exceptions, and we reject this solution for a few reasons: It allows errors to silently propagate—the caller can forget to check this condi- tion and won’t be alerted by the compiler, which might result in subsequent code not working properly. Often the error won’t be detected until much later in the code. Besides being error-prone, it results in a fair amount of boilerplate code at call sites, with explicit if statements to check whether the caller has received a “real” result. This boilerplate is magnified if you happen to be calling several functions, each of which uses error codes that must be checked and aggregated in some way. It’s not applicable to polymorphic code. For some output types, we might not even have a sentinel value of that type even if we wanted to! Consider a function like max, which finds the maximum value in a sequence according to a custom comparison function: def max[A](xs: Seq[A])(greater: (A,A) => Boolean): A. If the input is empty, we can’t invent a value of type A. Nor can null be used here, since null is only valid for non-primitive types, and A may in fact be a primitive like Double or Int. It demands a special policy or calling convention of callers—proper use of the mean function would require that callers do something other than call mean and make use of the result. Giving functions special policies like this makes it difficult to pass them to higher-order functions, which must treat all arguments uniformly. The second possibility is to force the caller to supply an argument that tells us what to do in case we don’t know how to handle the input: def mean_1(xs: IndexedSeq[Double], onEmpty: Double): Double = if (xs.isEmpty) onEmpty else xs.sum / xs.length This makes mean into a total function, but it has drawbacks—it requires that immediate callers have direct knowledge of how to handle the undefined case and limits them to 1 A function may also be partial if it doesn’t terminate for some inputs. We won’t discuss this form of partiality here, since it’s not a recoverable error so there’s no question of how best to handle it. See the chapter notes for more about partiality. Licensed to Emre Sevinc 52 CHAPTER 4 Handling errors without exceptions returning a Double. What if mean is called as part of a larger computation and we’d like to abort that computation if mean is undefined? Or perhaps we’d like to take some completely different branch in the larger computation in this case? Simply passing an onEmpty parameter doesn’t give us this freedom. We need a way to defer the decision of how to handle undefined cases so that they can be dealt with at the most appropriate level. 4.3 The Option data type The solution is to represent explicitly in the return type that a function may not always have an answer. We can think of this as deferring to the caller for the error-handling strategy. We introduce a new type, Option. As we mentioned earlier, this type also exists in the Scala standard library, but we’re re-creating it here for pedagogical purposes: sealed trait Option[+A] case class Some[+A](get: A) extends Option[A] case object None extends Option[Nothing] Option has two cases: it can be defined, in which case it will be a Some, or it can be undefined, in which case it will be None. We can use Option for our definition of mean like so: def mean(xs: Seq[Double]): Option[Double] = if (xs.isEmpty) None else Some(xs.sum / xs.length) The return type now reflects the possibility that the result may not always be defined. We still always return a result of the declared type (now Option[Double]) from our function, so mean is now a total function. It takes each value of the input type to exactly one value of the output type. Using a sentinel value Valid inputs Valid outputs Invalid inputs Sentinel value . . . 5 4 3 2 1 0 –1 –2 . . . .699 .602 .478 .301 0 –9999999.99 Mapping all invalid inputs to a special value of the same type as the valid outputs. Ambiguous, and compiler can’t check that caller handles it correctly. logID: Double => Double logID Using the Option type Valid inputs Valid outputs Invalid inputs Invalid output . . . 5 4 3 2 1 0 –1 –2 . . . Some(.699) Some(.602) Some(.478) Some(.301) Some(0) None Responding to invalid inputs Every valid output is wrapped in Some. Invalid inputs are mapped to None. The compiler forces the caller to deal explicitly with the possibility of failure. logID: Double => Option[Double] logID Licensed to Emre Sevinc 53The Option data type 4.3.1 Usage patterns for Option Partial functions abound in programming, and Option (and the Either data type that we’ll discuss shortly) is typically how this partiality is dealt with in FP. You’ll see Option used throughout the Scala standard library, for instance: Map lookup for a given key (http://mng.bz/ha64) returns Option. headOption and lastOption defined for lists and other iterables (http:// mng.bz/Pz86) return an Option containing the first or last elements of a sequence if it’s nonempty. These aren’t the only examples—we’ll see Option come up in many different situa- tions. What makes Option convenient is that we can factor out common patterns of error handling via higher-order functions, freeing us from writing the usual boiler- plate that comes with exception-handling code. In this section, we’ll cover some of the basic functions for working with Option. Our goal is not for you to attain fluency with all these functions, but just to get you familiar enough that you can revisit this chapter and make progress on your own when you have to write some functional code to deal with errors. BASIC FUNCTIONS ON OPTION Option can be thought of like a List that can contain at most one element, and many of the List functions we saw earlier have analogous functions on Option. Let’s look at some of these functions. We’ll do something slightly different than in chapter 3 where we put all our List functions in the List companion object. Here we’ll place our functions, when possi- ble, inside the body of the Option trait, so they can be called with the syntax obj.fn(arg1) or obj fn arg1 instead of fn(obj, arg1). This is a stylistic choice with no real significance, and we’ll use both styles throughout this book.2 This choice raises one additional complication with regard to variance that we’ll discuss in a moment. Let’s take a look. trait Option[+A] { def map[B](f: A => B): Option[B] def flatMap[B](f: A => Option[B]): Option[B] def getOrElse[B >: A](default: => B): B def orElse[B >: A](ob: => Option[B]): Option[B] def filter(f: A => Boolean): Option[A] } 2 In general, we’ll use this object-oriented style of syntax where possible for functions that have a single, clear operand (like List.map), and the standalone function style otherwise. Listing 4.2 The Option data type Apply f if the Option is not None. Apply f, which may fail, to the Option if not None. The B >: A says that the B type parameter must be a supertype of A. Don’t evaluate ob unless needed. Convert Some to None if the value doesn’t satisfy f. Licensed to Emre Sevinc 54 CHAPTER 4 Handling errors without exceptions There is some new syntax here. The default: => B type annotation in getOrElse (and the similar annotation in orElse) indicates that the argument is of type B, but won’t be evaluated until it’s needed by the function. Don’t worry about this for now—we’ll talk much more about this concept of non-strictness in the next chapter. Also, the B >: A type parameter on the getOrElse and orElse functions indicates that B must be equal to or a supertype of A. It’s needed to convince Scala that it’s still safe to declare Option[+A] as covariant in A. See the chapter notes for more detail—it’s unfortunately somewhat complicated, but a necessary complication in Scala. Fortunately, fully understanding subtyping and variance isn’t essential for our purposes here. EXERCISE 4.1 Implement all of the preceding functions on Option. As you implement each function, try to think about what it means and in what situations you’d use it. We’ll explore when to use each of these functions next. Here are a few hints for solving this exercise: It’s fine to use pattern matching, though you should be able to implement all the functions besides map and getOrElse without resorting to pattern matching. For map and flatMap, the type signature should be enough to determine the implementation. getOrElse returns the result inside the Some case of the Option, or if the Option is None, returns the given default value. orElse returns the first Option if it’s defined; otherwise, it returns the second Option. USAGE SCENARIOS FOR THE BASIC OPTION FUNCTIONS Although we can explicitly pattern match on an Option, we’ll almost always use the above higher-order functions. Here, we’ll try to give some guidance for when to use each one. Fluency with these functions will come with practice, but the objective here is to get some basic familiarity. Next time you try writing some functional code that uses Option, see if you can recognize the patterns these functions encapsulate before you resort to pattern matching. Let’s start with map. The map function can be used to transform the result inside an Option, if it exists. We can think of it as proceeding with a computation on the assumption that an error hasn’t occurred; it’s also a way of deferring the error han- dling to later code: case class Employee(name: String, department: String) def lookupByName(name: String): Option[Employee] = ... val joeDepartment: Option[String] = lookupByName("Joe").map(_.department) Licensed to Emre Sevinc 55The Option data type Here, lookupByName("Joe") returns an Option[Employee], which we transform using map to pull out the Option[String] representing the department. Note that we don’t need to explicitly check the result of lookupByName("Joe"); we simply continue the computation as if no error occurred, inside the argument to map. If employeesBy- Name.get("Joe") returns None, this will abort the rest of the computation and map will not call the _.department function at all. flatMap is similar, except that the function we provide to transform the result can itself fail. EXERCISE 4.2 Implement the variance function in terms of flatMap. If the mean of a sequence is m, the variance is the mean of math.pow(x - m, 2) for each element x in the sequence. See the definition of variance on Wikipedia (http://mng.bz/0Qsr). def variance(xs: Seq[Double]): Option[Double] As the implementation of variance demonstrates, with flatMap we can construct a computation with multiple stages, any of which may fail, and the computation will abort as soon as the first failure is encountered, since None.flatMap(f) will immedi- ately return None, without running f. lookupByName("Joe").map(_.department) Joe’s dept. if Joe is an employee None if Joe is not an employee lookupByName("Joe").flatMap(_.manager) Some(manager) if Joe has a manager None if Joe is not an employee or doesn’t have a manager lookupByName("Joe").map(_.department).getOrElse("Default Dept.") Joe’s department if he has one "Default Dept." if not Licensed to Emre Sevinc 56 CHAPTER 4 Handling errors without exceptions We can use filter to convert successes into failures if the successful values don’t match the given predicate. A common pattern is to transform an Option via calls to map, flatMap, and/or filter, and then use getOrElse to do error handling at the end: val dept: String = lookupByName("Joe"). map(_.dept). filter(_ != "Accounting"). getOrElse("Default Dept") getOrElse is used here to convert from an Option[String] to a String, by providing a default department in case the key "Joe" didn’t exist in the Map or if Joe’s depart- ment was "Accounting". orElse is similar to getOrElse, except that we return another Option if the first is undefined. This is often useful when we need to chain together possibly failing com- putations, trying the second if the first hasn’t succeeded. A common idiom is to do o.getOrElse(throw new Exception("FAIL")) to con- vert the None case of an Option back to an exception. The general rule of thumb is that we use exceptions only if no reasonable program would ever catch the exception; if for some callers the exception might be a recoverable error, we use Option (or Either, discussed later) to give them flexibility. As you can see, returning errors as ordinary values can be convenient and the use of higher-order functions lets us achieve the same sort of consolidation of error- handling logic we would get from using exceptions. Note that we don’t have to check for None at each stage of the computation—we can apply several transformations and then check for and handle None when we’re ready. But we also get additional safety, since Option[A] is a different type than A, and the compiler won’t let us forget to explicitly defer or handle the possibility of None. 4.3.2 Option composition, lifting, and wrapping exception-oriented APIs It may be easy to jump to the conclusion that once we start using Option, it infects our entire code base. One can imagine how any callers of methods that take or return Option will have to be modified to handle either Some or None. But this doesn’t hap- pen, and the reason is that we can lift ordinary functions to become functions that operate on Option. For example, the map function lets us operate on values of type Option[A] using a function of type A => B, returning Option[B]. Another way of looking at this is that map turns a function f of type A => B into a function of type Option[A] => Option[B]. Let’s make this explicit: def lift[A,B](f: A => B): Option[A] => Option[B]=_mapf This tells us that any function that we already have lying around can be transformed (via lift) to operate within the context of a single Option value. Let’s look at an example: val absO: Option[Double] => Option[Double] = lift(math.abs) Licensed to Emre Sevinc 57The Option data type The math object contains various standalone mathematical functions including abs, sqrt, exp, and so on. We didn’t need to rewrite the math.abs function to work with optional values; we just lifted it into the Option context after the fact. We can do this for any function. Let’s look at another example. Suppose we’re implementing the logic for a car insurance company’s website, which contains a page where users can submit a form to request an instant online quote. We’d like to parse the information from this form and ultimately call our rate function: /** * Top secret formula for computing an annual car * insurance premium from two key factors. */ def insuranceRateQuote(age: Int, numberOfSpeedingTickets: Int): Double We want to be able to call this function, but if the user is submitting their age and number of speeding tickets in a web form, these fields will arrive as simple strings that we have to (try to) parse into integers. This parsing may fail; given a string, s, we can attempt to parse it into an Int using s.toInt, which throws a NumberFormat- Exception if the string isn’t a valid integer: scala> "112".toInt res0: Int = 112 scala> "hello".toInt java.lang.NumberFormatException: For input string: "hello" at java.lang.NumberFormatException.forInputString(...) ... Let’s convert the exception-based API of toInt to Option and see if we can implement a function parseInsuranceRateQuote, which takes the age and number of speeding tickets as strings, and attempts calling the insuranceRateQuote function if parsing both values is successful. def parseInsuranceRateQuote( age: String, numberOfSpeedingTickets: String): Option[Double] = { val optAge: Option[Int] = Try(age.toInt) val optTickets: Option[Int] = Try(numberOfSpeedingTickets.toInt) Listing 4.3 Using Option lift(math.abs): Option[Double] => Option[Double] Double => Doublemath.abs: lift(f) returns a function which maps None to None and applies f to the contents of Some. f need not be aware of the Option type at all. Lifting functions The toInt method is available on any String. Licensed to Emre Sevinc 58 CHAPTER 4 Handling errors without exceptions insuranceRateQuote(optAge, optTickets) } def Try[A](a: => A): Option[A] = try Some(a) catch { case e: Exception => None } The Try function is a general-purpose function we can use to convert from an excep- tion-based API to an Option-oriented API. This uses a non-strict or lazy argument, as indicated by the => A as the type of a. We’ll discuss laziness much more in the next chapter. But there’s a problem—after we parse optAge and optTickets into Option[Int], how do we call insuranceRateQuote, which currently takes two Int values? Do we have to rewrite insuranceRateQuote to take Option[Int] values instead? No, and changing insuranceRateQuote would be entangling concerns, forcing it to be aware that a prior computation may have failed, not to mention that we may not have the ability to modify insuranceRateQuote—perhaps it’s defined in a separate module that we don’t have access to. Instead, we’d like to lift insuranceRateQuote to operate in the context of two optional values. We could do this using explicit pattern matching in the body of parseInsuranceRateQuote, but that’s going to be tedious. EXERCISE 4.3 Write a generic function map2 that combines two Option values using a binary func- tion. If either Option value is None, then the return value is too. Here is its signature: def map2[A,B,C](a: Option[A], b: Option[B])(f: (A, B) => C): Option[C] With map2, we can now implement parseInsuranceRateQuote: def parseInsuranceRateQuote( age: String, numberOfSpeedingTickets: String): Option[Double] = { val optAge: Option[Int] = Try { age.toInt } val optTickets: Option[Int] = Try { numberOfSpeedingTickets.toInt } map2(optAge, optTickes)(insuranceRateQuote) } The map2 function means that we never need to modify any existing functions of two arguments to make them “Option-aware.” We can lift them to operate in the context of Option after the fact. Can you already see how you might define map3, map4, and map5? Let’s look at a few other similar cases. Doesn’t type check! See following discussion. We accept the A argument non-strictly, so we can catch any exceptions that occur while evaluating a and convert them to None. Note: This discards information about the error. We’ll improve on this later in the chapter. Functions accepting a single argument may be called with braces instead of parentheses; this is equivalent to Try(age.toInt). If either parse fails, this will immediately return None. Licensed to Emre Sevinc 59The Option data type EXERCISE 4.4 Write a function sequence that combines a list of Options into one Option containing a list of all the Some values in the original list. If the original list contains None even once, the result of the function should be None; otherwise the result should be Some with a list of all the values. Here is its signature:3 def sequence[A](a: List[Option[A]]): Option[List[A]] Sometimes we’ll want to map over a list using a function that might fail, returning None if applying it to any element of the list returns None. For example, what if we have a whole list of String values that we wish to parse to Option[Int]? In that case, we can simply sequence the results of the map: def parseInts(a: List[String]): Option[List[Int]] = sequence(a map (i => Try(i.toInt))) Unfortunately, this is inefficient, since it traverses the list twice, first to convert each String to an Option[Int], and a second pass to combine these Option[Int] values into an Option[List[Int]]. Wanting to sequence the results of a map this way is a common enough occurrence to warrant a new generic function, traverse, with the following signature: def traverse[A, B](a: List[A])(f: A => Option[B]): Option[List[B]] EXERCISE 4.5 Implement this function. It’s straightforward to do using map and sequence, but try for a more efficient implementation that only looks at the list once. In fact, imple- ment sequence in terms of traverse. 3 This is a clear instance where it’s not appropriate to define the function in the OO style. This shouldn’t be a method on List (which shouldn’t need to know anything about Option), and it can’t be a method on Option, so it goes in the Option companion object. For-comprehensions Since lifting functions is so common in Scala, Scala provides a syntactic construct called the for-comprehension that it expands automatically to a series of flatMap and map calls. Let’s look at how map2 could be implemented with for-comprehensions. Licensed to Emre Sevinc 60 CHAPTER 4 Handling errors without exceptions Between map, lift, sequence, traverse, map2, map3, and so on, you should never have to modify any existing functions to work with optional values. 4.4 The Either data type The big idea in this chapter is that we can represent failures and exceptions with ordi- nary values, and write functions that abstract out common patterns of error handling and recovery. Option isn’t the only data type we could use for this purpose, and although it gets used frequently, it’s rather simplistic. One thing you may have noticed with Option is that it doesn’t tell us anything about what went wrong in the case of an exceptional condition. All it can do is give us None, indicating that there’s no value to be had. But sometimes we want to know more. For example, we might want a String that gives more information, or if an exception was raised, we might want to know what that error actually was. We can craft a data type that encodes whatever information we want about failures. Sometimes just knowing whether a failure occurred is sufficient, in which case we can use Option; other times we want more information. In this section, we’ll walk through a simple extension to Option, the Either data type, which lets us track a reason for the failure. Let’s look at its definition: sealed trait Either[+E, +A] case class Left[+E](value: E) extends Either[E, Nothing] case class Right[+A](value: A) extends Either[Nothing, A] (continued) Here’s our original version: def map2[A,B,C](a: Option[A], b: Option[B])(f: (A, B) => C): Option[C] = a flatMap (aa => b map (bb => f(aa, bb))) And here’s the exact same code written as a for-comprehension: def map2[A,B,C](a: Option[A], b: Option[B])(f: (A, B) => C): Option[C] = for { aa <- a bb <- b } yield f(aa, bb) A for-comprehension consists of a sequence of bindings, like aa <- a, followed by a yield after the closing brace, where the yield may make use of any of the values on the left side of any previous <- binding. The compiler desugars the bindings to flatMap calls, with the final binding and yield being converted to a call to map. You should feel free to use for-comprehensions in place of explicit calls to flatMap and map. Licensed to Emre Sevinc 61The Either data type Either has only two cases, just like Option. The essential difference is that both cases carry a value. The Either data type represents, in a very general way, values that can be one of two things. We can say that it’s a disjoint union of two types. When we use it to indicate success or failure, by convention the Right constructor is reserved for the success case (a pun on “right,” meaning correct), and Left is used for failure. We’ve given the left type parameter the suggestive name E (for error).4 Let’s look at the mean example again, this time returning a String in case of failure: def mean(xs: IndexedSeq[Double]): Either[String, Double] = if (xs.isEmpty) Left("mean of empty list!") else Right(xs.sum / xs.length) Sometimes we might want to include more information about the error, for example a stack trace showing the location of the error in the source code. In such cases we can simply return the exception in the Left side of an Either: def safeDiv(x: Int, y: Int): Either[Exception, Int] = try Right(x / y) catch { case e: Exception => Left(e) } As we did with Option, we can write a function, Try, which factors out this common pattern of converting thrown exceptions to values: def Try[A](a: => A): Either[Exception, A] = try Right(a) catch { case e: Exception => Left(e) } 4 Either is also often used more generally to encode one of two possibilities in cases where it isn’t worth defin- ing a fresh data type. We’ll see some examples of this throughout the book. Option and Either in the standard library As we mentioned earlier in this chapter, both Option and Either exist in the Scala standard library (Option API is at http://mng.bz/fiJ5; Either API is at http:// mng.bz/106L), and most of the functions we’ve defined here in this chapter exist for the standard library versions. You’re encouraged to read through the API for Option and Either to understand the differences. There are a few missing functions, though, notably sequence, traverse, and map2. And Either doesn’t define a right-biased flatMap directly like we do here. The standard library Either is slightly (but only slightly) more complicated. Read the API for details. Licensed to Emre Sevinc 62 CHAPTER 4 Handling errors without exceptions EXERCISE 4.6 Implement versions of map, flatMap, orElse, and map2 on Either that operate on the Right value. trait Either[+E, +A] { def map[B](f: A => B): Either[E, B] def flatMap[EE >: E, B](f: A => Either[EE, B]): Either[EE, B] def orElse[EE >: E,B >: A](b: => Either[EE, B]): Either[EE, B] def map2[EE >: E, B, C](b: Either[EE, B])(f: (A, B) => C): Either[EE, C] } Note that with these definitions, Either can now be used in for-comprehensions. For instance: def parseInsuranceRateQuote( age: String, numberOfSpeedingTickets: String): Either[Exception,Double] = for { a <- Try { age.toInt } tickets <- Try { numberOfSpeedingTickes.toInt } } yield insuranceRateQuote(a, tickets) Now we get information about the actual exception that occurred, rather than just getting back None in the event of a failure. EXERCISE 4.7 Implement sequence and traverse for Either. These should return the first error that’s encountered, if there is one. def sequence[E, A](es: List[Either[E, A]]): Either[E, List[A]] def traverse[E, A, B](as: List[A])( f: A => Either[E, B]): Either[E, List[B]] As a final example, here’s an application of map2, where the function mkPerson vali- dates both the given name and the given age before constructing a valid Person. case class Person(name: Name, age: Age) sealed class Name(val value: String) sealed class Age(val value: Int) Listing 4.4 Using Either to validate data When mapping over the right side, we must promote the left type parameter to some supertype, to satisfy the +E variance annotation. Similarly for orElse. Licensed to Emre Sevinc 63Summary def mkName(name: String): Either[String, Name] = if (name == "" || name == null) Left("Name is empty.") else Right(new Name(name)) def mkAge(age: Int): Either[String, Age] = if (age < 0) Left("Age is out of range.") else Right(new Age(age)) def mkPerson(name: String, age: Int): Either[String, Person] = mkName(name).map2(mkAge(age))(Person(_, _)) EXERCISE 4.8 In this implementation, map2 is only able to report one error, even if both the name and the age are invalid. What would you need to change in order to report both errors? Would you change map2 or the signature of mkPerson? Or could you create a new data type that captures this requirement better than Either does, with some additional structure? How would orElse, traverse, and sequence behave differently for that data type? 4.5 Summary In this chapter, we noted some of the problems with using exceptions and introduced the basic principles of purely functional error handling. Although we focused on the algebraic data types Option and Either, the bigger idea is that we can represent exceptions as ordinary values and use higher-order functions to encapsulate common patterns of handling and propagating errors. This general idea, of representing effects as values, is something we’ll see again and again throughout this book in vari- ous guises. We don’t expect you to be fluent with all the higher-order functions we wrote in this chapter, but you should now have enough familiarity to get started writing your own functional code complete with error handling. With these new tools in hand, exceptions should be reserved only for truly unrecoverable conditions. Lastly, in this chapter we touched briefly on the notion of a non-strict function (recall the functions orElse, getOrElse, and Try). In the next chapter, we’ll look more closely at why non-strictness is important and how it can buy us greater modular- ity and efficiency in our functional programs. Licensed to Emre Sevinc 64 Strictness and laziness In chapter 3 we talked about purely functional data structures, using singly linked lists as an example. We covered a number of bulk operations on lists—map, filter, foldLeft, foldRight, zipWith, and so on. We noted that each of these operations makes its own pass over the input and constructs a fresh list for the output. Imagine if you had a deck of cards and you were asked to remove the odd- numbered cards and then flip over all the queens. Ideally, you’d make a single pass through the deck, looking for queens and odd-numbered cards at the same time. This is more efficient than removing the odd cards and then looking for queens in the remainder. And yet the latter is what Scala is doing in the following code:1 scala> List(1,2,3,4).map(_ + 10).filter(_%2==0).map(_ * 3) List(36,42) In this expression, map(_ + 10) will produce an intermediate list that then gets passed to filter(_ % 2 == 0), which in turn constructs a list that gets passed to map(_ * 3), which then produces the final list. In other words, each transformation will produce a temporary list that only ever gets used as input to the next transfor- mation and is then immediately discarded. Think about how this program will be evaluated. If we manually produce a trace of its evaluation, the steps would look something like this. List(1,2,3,4).map(_ + 10).filter(_%2==0).map(_ * 3) List(11,12,13,14).filter(_%2==0).map(_ * 3) 1 We’re now using the Scala standard library’s List type here, where map and filter are methods on List rather than standalone functions like the ones we wrote in chapter 3. Listing 5.1 Program trace for List Licensed to Emre Sevinc 65Strict and non-strict functions List(12,14).map(_ * 3) List(36,42) Here we’re showing the result of each substitution performed to evaluate our expres- sion. For example, to go from the first line to the second, we’ve replaced List(1,2,3,4).map(_ + 10) with List(11,12,13,14), based on the definition of map.2 This view makes it clear how the calls to map and filter each perform their own traversal of the input and allocate lists for the output. Wouldn’t it be nice if we could somehow fuse sequences of transformations like this into a single pass and avoid creat- ing temporary data structures? We could rewrite the code into a while loop by hand, but ideally we’d like to have this done automatically while retaining the same high- level compositional style. We want to compose our programs using higher-order func- tions like map and filter instead of writing monolithic loops. It turns out that we can accomplish this kind of automatic loop fusion through the use of non-strictness (or, less formally, laziness). In this chapter, we’ll explain what exactly this means, and we’ll work through the implementation of a lazy list type that fuses sequences of transformations. Although building a “better” list is the motivation for this chapter, we’ll see that non-strictness is a fundamental technique for improving on the efficiency and modularity of functional programs in general. 5.1 Strict and non-strict functions Before we get to our example of lazy lists, we need to cover some basics. What are strictness and non-strictness, and how are these concepts expressed in Scala? Non-strictness is a property of a function. To say a function is non-strict just means that the function may choose not to evaluate one or more of its arguments. In con- trast, a strict function always evaluates its arguments. Strict functions are the norm in most programming languages, and indeed most languages only support functions that expect their arguments fully evaluated. Unless we tell it otherwise, any function definition in Scala will be strict (and all the functions we’ve defined so far have been strict). As an example, consider the following function: def square(x: Double): Double = x * x When you invoke square(41.0 + 1.0), the function square will receive the evaluated value of 42.0 because it’s strict. If you invoke square(sys.error("failure")), you’ll get an exception before square has a chance to do anything, since the sys.error ("failure") expression will be evaluated before entering the body of square. Although we haven’t yet learned the syntax for indicating non-strictness in Scala, you’re almost certainly familiar with the concept. For example, the short-circuiting Boolean functions && and ||, found in many programming languages including Scala, are non-strict. You may be used to thinking of && and || as built-in syntax, part of the 2 With program traces like these, it’s often more illustrative to not fully trace the evaluation of every subexpres- sion. In this case, we’ve omitted the full expansion of List(1,2,3,4).map(_ + 10). We could “enter” the definition of map and trace its execution, but we chose to omit this level of detail here. Licensed to Emre Sevinc 66 CHAPTER 5 Strictness and laziness language, but you can also think of them as functions that may choose not to evaluate their arguments. The function && takes two Boolean arguments, but only evaluates the second argument if the first is true: scala> false && { println("!!"); true } // does not print anything res0: Boolean = false And || only evaluates its second argument if the first is false: scala> true || { println("!!"); false } // doesn't print anything either res1: Boolean = true Another example of non-strictness is the if control construct in Scala: val result = if (input.isEmpty) sys.error("empty input") else input Even though if is a built-in language construct in Scala, it can be thought of as a func- tion accepting three parameters: a condition of type Boolean, an expression of some type A to return in the case that the condition is true, and another expression of the same type A to return if the condition is false. This if function would be non-strict, since it won’t evaluate all of its arguments. To be more precise, we’d say that the if function is strict in its condition parameter, since it’ll always evaluate the condition to determine which branch to take, and non-strict in the two branches for the true and false cases, since it’ll only evaluate one or the other based on the condition. In Scala, we can write non-strict functions by accepting some of our arguments unevaluated. We’ll show how this is done explicitly just to illustrate what’s happening, and then show some nicer syntax for it that’s built into Scala. Here’s a non-strict if function: def if2[A](cond: Boolean, onTrue: () => A, onFalse: () => A): A = if (cond) onTrue() else onFalse() if2(a < 22, () => println("a"), () => println("b") ) The arguments we’d like to pass unevaluated have a () => immediately before their type. A value of type () => A is a function that accepts zero arguments and returns an A.3 In general, the unevaluated form of an expression is called a thunk, and we can force the thunk to evaluate the expression and get a result. We do so by invoking the func- tion, passing an empty argument list, as in onTrue() or onFalse(). Likewise, callers of if2 have to explicitly create thunks, and the syntax follows the same conventions as the function literal syntax we’ve already seen. Overall, this syntax makes it very clear what’s happening—we’re passing a function of no arguments in place of each non-strict parameter, and then explicitly calling this function to obtain a result in the body. But this is such a common case that Scala pro- vides some nicer syntax: 3 In fact, the type () => A is a syntactic alias for the type Function0[A]. The function literal syntax for creating a () => A Licensed to Emre Sevinc 67Strict and non-strict functions def if2[A](cond: Boolean, onTrue: => A, onFalse: => A): A = if (cond) onTrue else onFalse The arguments we’d like to pass unevaluated have an arrow => immediately before their type. In the body of the function, we don’t need to do anything special to evalu- ate an argument annotated with =>. We just reference the identifier as usual. Nor do we have to do anything special to call this function. We just use the normal function call syntax, and Scala takes care of wrapping the expression in a thunk for us: scala> if2(false, sys.error("fail"), 3) res2: Int = 3 With either syntax, an argument that’s passed unevaluated to a function will be evalu- ated once for each place it’s referenced in the body of the function. That is, Scala won’t (by default) cache the result of evaluating an argument: scala> def maybeTwice(b: Boolean, i: => Int) = if (b) i+i else 0 maybeTwice: (b: Boolean, i: => Int)Int scala> val x = maybeTwice(true, { println("hi"); 1+41 }) hi hi x: Int = 84 Here, i is referenced twice in the body of maybeTwice, and we’ve made it particularly obvious that it’s evaluated each time by passing the block {println("hi"); 1+41}, which prints hi as a side effect before returning a result of 42. The expression 1+41 will be computed twice as well. We can cache the value explicitly if we wish to only eval- uate the result once, by using the lazy keyword: scala> def maybeTwice2(b: Boolean, i: => Int) = { | lazy valj=i | if (b) j+j else 0 |} maybeTwice: (b: Boolean, i: => Int)Int scala> val x = maybeTwice2(true, { println("hi"); 1+41 }) hi x: Int = 84 Adding the lazy keyword to a val declaration will cause Scala to delay evaluation of the right-hand side of that lazy val declaration until it’s first referenced. It will also cache the result so that subsequent references to it don’t trigger repeated evaluation. Formal definition of strictness If the evaluation of an expression runs forever or throws an error instead of returning a definite value, we say that the expression doesn’t terminate, or that it evaluates to bottom. A function f is strict if the expression f(x) evaluates to bottom for all x that evaluate to bottom. Licensed to Emre Sevinc 68 CHAPTER 5 Strictness and laziness As a final bit of terminology, we say that a non-strict function in Scala takes its argu- ments by name rather than by value. 5.2 An extended example: lazy lists Let’s now return to the problem posed at the beginning of this chapter. We’ll explore how laziness can be used to improve the efficiency and modularity of functional pro- grams using lazy lists, or streams, as an example. We’ll see how chains of transforma- tions on streams are fused into a single pass through the use of laziness. Here’s a simple Stream definition. There are a few new things here we’ll discuss next. sealed trait Stream[+A] case object Empty extends Stream[Nothing] case class Cons[+A](h: () => A, t: () => Stream[A]) extends Stream[A] object Stream { def cons[A](hd: => A, tl: => Stream[A]): Stream[A] = { lazy val head = hd lazy val tail = tl Cons(() => head, () => tail) } def empty[A]: Stream[A] = Empty def apply[A](as: A*): Stream[A] = if (as.isEmpty) empty else cons(as.head, apply(as.tail: _*)) } This type looks identical to our List type, except that the Cons data constructor takes explicit thunks (() => A and () => Stream[A]) instead of regular strict values. If we wish to examine or traverse the Stream, we need to force these thunks as we did earlier in our definition of if2. For instance, here’s a function to optionally extract the head of a Stream: def headOption: Option[A] = this match { case Empty => None case Cons(h, t) => Some(h()) } Note that we have to force h explicitly via h(), but other than that, the code works the same way as it would for List. But this ability of Stream to evaluate only the portion actually demanded (we don’t evaluate the tail of the Cons) is useful, as we’ll see. Listing 5.2 Simple definition for Stream A nonempty stream consists of a head and a tail, which are both non-strict. Due to technical limitations, these are thunks that must be explicitly forced, rather than by-name parameters.A smart constructor for creating a nonempty stream. We cache the head and tail as lazy values to avoid repeated evaluation. A smart constructor for creating an empty stream of a particular type. A convenient variable-argument method for constructing a Stream from multiple elements. Explicit forcing of the h thunk using h() Licensed to Emre Sevinc 69An extended example: lazy lists 5.2.1 Memoizing streams and avoiding recomputation We typically want to cache the values of a Cons node, once they are forced. If we use the Cons data constructor directly, for instance, this code will actually compute expensive(x) twice: val x = Cons(() => expensive(x), tl) val h1 = x.headOption val h2 = x.headOption We typically avoid this problem by defining smart constructors, which is what we call a function for constructing a data type that ensures some additional invariant or pro- vides a slightly different signature than the “real” constructors used for pattern match- ing. By convention, smart constructors typically lowercase the first letter of the corresponding data constructor. Here, our cons smart constructor takes care of memoizing the by-name arguments for the head and tail of the Cons. This is a com- mon trick, and it ensures that our thunk will only do its work once, when forced for the first time. Subsequent forces will return the cached lazy val: def cons[A](hd: => A, tl: => Stream[A]): Stream[A] = { lazy val head = hd lazy val tail = tl Cons(() => head, () => tail) } The empty smart constructor just returns Empty, but annotates Empty as a Stream[A], which is better for type inference in some cases.4 We can see how both smart construc- tors are used in the Stream.apply function: def apply[A](as: A*): Stream[A] = if (as.isEmpty) empty else cons(as.head, apply(as.tail: _*)) Again, Scala takes care of wrapping the arguments to cons in thunks, so the as.head and apply(as.tail: _*) expressions won’t be evaluated until we force the Stream. 5.2.2 Helper functions for inspecting streams Before continuing, let’s write a few helper functions to make inspecting streams easier. EXERCISE 5.1 Write a function to convert a Stream to a List, which will force its evaluation and let you look at it in the REPL. You can convert to the regular List type in the standard library. You can place this and other functions that operate on a Stream inside the Stream trait. def toList: List[A] 4 Recall that Scala uses subtyping to represent data constructors, but we almost always want to infer Stream as the type, not Cons or Empty. Making smart constructors that return the base type is a common trick. Licensed to Emre Sevinc 70 CHAPTER 5 Strictness and laziness EXERCISE 5.2 Write the function take(n) for returning the first n elements of a Stream, and drop(n) for skipping the first n elements of a Stream. EXERCISE 5.3 Write the function takeWhile for returning all starting elements of a Stream that match the given predicate. def takeWhile(p: A => Boolean): Stream[A] You can use take and toList together to inspect streams in the REPL. For example, try printing Stream(1,2,3).take(2).toList. 5.3 Separating program description from evaluation A major theme in functional programming is separation of concerns. We want to sepa- rate the description of computations from actually running them. We’ve already touched on this theme in previous chapters in different ways. For example, first-class functions capture some computation in their bodies but only execute it once they receive their arguments. And we used Option to capture the fact that an error occurred, where the decision of what to do about it became a separate concern. With Stream, we’re able to build up a computation that produces a sequence of elements without running the steps of that computation until we actually need those elements. More generally speaking, laziness lets us separate the description of an expression from the evaluation of that expression. This gives us a powerful ability—we may choose to describe a “larger” expression than we need, and then evaluate only a por- tion of it. As an example, let’s look at the function exists that checks whether an ele- ment matching a Boolean function exists in this Stream: def exists(p: A => Boolean): Boolean = this match { case Cons(h, t) => p(h()) || t().exists(p) case _=>false } Note that || is non-strict in its second argument. If p(h()) returns true, then exists terminates the traversal early and returns true as well. Remember also that the tail of the stream is a lazy val. So not only does the traversal terminate early, the tail of the stream is never evaluated at all! So whatever code would have generated the tail is never actually executed. Licensed to Emre Sevinc 71Separating program description from evaluation The exists function here is implemented using explicit recursion. But remember that with List in chapter 3, we could implement a general recursion in the form of foldRight. We can do the same thing for Stream, but lazily: def foldRight[B](z: => B)(f: (A, => B) => B): B = this match { case Cons(h,t) => f(h(), t().foldRight(z)(f)) case _=>z } This looks very similar to the foldRight we wrote for List, but note how our combin- ing function f is non-strict in its second parameter. If f chooses not to evaluate its sec- ond parameter, this terminates the traversal early. We can see this by using foldRight to implement exists:5 def exists(p: A => Boolean): Boolean = foldRight(false)((a, b) => p(a) || b) Here b is the unevaluated recursive step that folds the tail of the stream. If p(a) returns true, b will never be evaluated and the computation terminates early. Since foldRight can terminate the traversal early, we can reuse it to implement exists. We can’t do that with a strict version of foldRight. We’d have to write a spe- cialized recursive exists function to handle early termination. Laziness makes our code more reusable. EXERCISE 5.4 Implement forAll, which checks that all elements in the Stream match a given predi- cate. Your implementation should terminate the traversal as soon as it encounters a nonmatching value. def forAll(p: A => Boolean): Boolean EXERCISE 5.5 Use foldRight to implement takeWhile. EXERCISE 5.6 Hard: Implement headOption using foldRight. 5 This definition of exists, though illustrative, isn’t stack-safe if the stream is large and all elements test false. The arrow => in front of the argument type B means that the function f takes its second argument by name and may choose not to evaluate it. If f doesn’t evaluate its second argument, the recursion never occurs. Licensed to Emre Sevinc 72 CHAPTER 5 Strictness and laziness EXERCISE 5.7 Implement map, filter, append, and flatMap using foldRight. The append method should be non-strict in its argument. Note that these implementations are incremental—they don’t fully generate their answers. It’s not until some other computation looks at the elements of the resulting Stream that the computation to generate that Stream actually takes place—and then it will do just enough work to generate the requested elements. Because of this incre- mental nature, we can call these functions one after another without fully instantiat- ing the intermediate results. Let’s look at a simplified program trace for (a fragment of) the motivating exam- ple we started this chapter with, Stream(1,2,3,4).map(_ + 10).filter(_ % 2 == 0). We’ll convert this expression to a List to force evaluation. Take a minute to work through this trace to understand what’s happening. It’s a bit more challenging than the trace we looked at earlier in this chapter. Remember, a trace like this is just the same expression over and over again, evaluated by one more step each time. Stream(1,2,3,4).map(_ + 10).filter(_%2==0).toList cons(11, Stream(2,3,4).map(_ + 10)).filter(_%2==0).toList Stream(2,3,4).map(_ + 10).filter(_%2==0).toList cons(12, Stream(3,4).map(_ + 10)).filter(_%2==0).toList 12 :: Stream(3,4).map(_ + 10).filter(_%2==0).toList 12 :: cons(13, Stream(4).map(_ + 10)).filter(_%2==0).toList 12 :: Stream(4).map(_ + 10).filter(_%2==0).toList 12 :: cons(14, Stream().map(_ + 10)).filter(_%2==0).toList 12 :: 14 :: Stream().map(_ + 10).filter(_%2==0).toList 12 :: 14 :: List() The thing to notice in this trace is how the filter and map transformations are inter- leaved—the computation alternates between generating a single element of the out- put of map, and testing with filter to see if that element is divisible by 2 (adding it to Listing 5.3 Program trace for Stream Apply map to the first element.Apply filter to the first element. Apply map to the second element. Apply filter to the second element. Produce the first element of the result. Apply filter to the fourth element and produce the final element of the result. map and filter have no more work to do, and the empty stream becomes the empty list. Licensed to Emre Sevinc 73Infinite streams and corecursion the output list if it is). Note that we don’t fully instantiate the intermediate stream that results from the map. It’s exactly as if we had interleaved the logic using a special- purpose loop. For this reason, people sometimes describe streams as “first-class loops” whose logic can be combined using higher-order functions like map and filter. Since intermediate streams aren’t instantiated, it’s easy to reuse existing combina- tors in novel ways without having to worry that we’re doing more processing of the stream than necessary. For example, we can reuse filter to define find, a method to return just the first element that matches if it exists. Even though filter transforms the whole stream, that transformation is done lazily, so find terminates as soon as a match is found: def find(p: A => Boolean): Option[A] = filter(p).headOption The incremental nature of stream transformations also has important consequences for memory usage. Because intermediate streams aren’t generated, a transformation of the stream requires only enough working memory to store and transform the current element. For instance, in the transformation Stream(1,2,3,4).map(_ + 10).filter (_ % 2 == 0), the garbage collector can reclaim the space allocated for the values 11 and 13 emitted by map as soon as filter determines they aren’t needed. Of course, this is a simple example; in other situations we might be dealing with larger numbers of ele- ments, and the stream elements themselves could be large objects that retain signifi- cant amounts of memory. Being able to reclaim this memory as quickly as possible can cut down on the amount of memory required by your program as a whole. We’ll have a lot more to say about defining memory-efficient streaming calcula- tions, in particular calculations that require I/O, in part 4 of this book. 5.4 Infinite streams and corecursion Because they’re incremental, the functions we’ve written also work for infinite streams. Here’s an example of an infinite Stream of 1s: val ones: Stream[Int] = Stream.cons(1, ones) Although ones is infinite, the functions we’ve written so far only inspect the portion of the stream needed to generate the demanded output. For example: scala> ones.take(5).toList res0: List[Int] = List(1, 1, 1, 1, 1) scala> ones.exists(_%2!=0) res1: Boolean = true Try playing with a few other examples: ones.map(_ + 1).exists(_ % 2 == 0) ones.takeWhile(_ == 1) ones.forAll(_ != 1) Licensed to Emre Sevinc 74 CHAPTER 5 Strictness and laziness In each case, we get back a result immediately. Be careful though, since it’s easy to write expressions that never terminate or aren’t stack-safe. For example, ones.forAll(_ == 1) will forever need to inspect more of the series since it’ll never encounter an element that allows it to terminate with a definite answer (this will man- ifest as a stack overflow rather than an infinite loop).6 Let’s see what other functions we can discover for generating streams. EXERCISE 5.8 Generalize ones slightly to the function constant, which returns an infinite Stream of a given value. def constant[A](a: A): Stream[A] EXERCISE 5.9 Write a function that generates an infinite stream of integers, starting from n, then n + 1, n + 2, and so on.7 def from(n: Int): Stream[Int] 6 It’s possible to define a stack-safe version of forAll using an ordinary recursive loop. 7 In Scala, the Int type is a 32-bit signed integer, so this stream will switch from positive to negative values at some point, and will repeat itself after about four billion elements. 1 ones.take(3) => [1, 1, 1]“Ones” is a stream that refers to itself; it generates an inﬁnite series of 1‘s. Expressions that only evaluate a finite number of strean elements can accept an infinite stream as input. Many functions can be evaluated using finite resources even if their inputs generate infinite sequences. Ones 1111. . . Ones = An infinite stream Licensed to Emre Sevinc 75Infinite streams and corecursion EXERCISE 5.10 Write a function fibs that generates the infinite stream of Fibonacci numbers: 0, 1, 1, 2, 3, 5, 8, and so on. EXERCISE 5.11 Write a more general stream-building function called unfold. It takes an initial state, and a function for producing both the next state and the next value in the generated stream. def unfold[A, S](z: S)(f: S => Option[(A, S)]): Stream[A] Option is used to indicate when the Stream should be terminated, if at all. The func- tion unfold is a very general Stream-building function. The unfold function is an example of what’s sometimes called a corecursive func- tion. Whereas a recursive function consumes data, a corecursive function produces data. And whereas recursive functions terminate by recursing on smaller inputs, core- cursive functions need not terminate so long as they remain productive, which just means that we can always evaluate more of the result in a finite amount of time. The unfold function is productive as long as f terminates, since we just need to run the function f one more time to generate the next element of the Stream. Corecursion is also sometimes called guarded recursion, and productivity is also sometimes called cotermination. These terms aren’t that important to our discussion, but you’ll hear them used sometimes in the context of functional programming. If you’re curious to learn where they come from and understand some of the deeper connections, follow the references in the chapter notes. EXERCISE 5.12 Write fibs, from, constant, and ones in terms of unfold.8 8 Using unfold to define constant and ones means that we don’t get sharing as in the recursive definition val ones: Stream[Int] = cons(1, ones). The recursive definition consumes constant memory even if we keep a reference to it around while traversing it, while the unfold-based implementation does not. Preserv- ing sharing isn’t something we usually rely on when programming with streams, since it’s extremely delicate and not tracked by the types. For instance, sharing is destroyed when calling even xs.map(x => x). Licensed to Emre Sevinc 76 CHAPTER 5 Strictness and laziness EXERCISE 5.13 Use unfold to implement map, take, takeWhile, zipWith (as in chapter 3), and zipAll. The zipAll function should continue the traversal as long as either stream has more elements—it uses Option to indicate whether each stream has been exhausted. def zipAll[B](s2: Stream[B]): Stream[(Option[A],Option[B])] Now that we have some practice writing stream functions, let’s return to the exercise we covered at the end of chapter 3—a function, hasSubsequence, to check whether a list contains a given subsequence. With strict lists and list-processing functions, we were forced to write a rather tricky monolithic loop to implement this function without doing extra work. Using lazy lists, can you see how you could implement hasSubsequence by combining some other functions we’ve already written?9 Try to think about it on your own before continuing. EXERCISE 5.14 Hard: Implement startsWith using functions you’ve written. It should check if one Stream is a prefix of another. For instance, Stream(1,2,3) startsWith Stream(1,2) would be true. def startsWith[A](s: Stream[A]): Boolean EXERCISE 5.15 Implement tails using unfold. For a given Stream, tails returns the Stream of suf- fixes of the input sequence, starting with the original Stream. For example, given Stream(1,2,3), it would return Stream(Stream(1,2,3), Stream(2,3), Stream(3), Stream()). def tails: Stream[Stream[A]] We can now implement hasSubsequence using functions we’ve written: def hasSubsequence[A](s: Stream[A]): Boolean = tails exists (_ startsWith s) 9 This small example, of assembling hasSubsequence from simpler functions using laziness, is from Cale Gibbard. See this post: http://lambda-the-ultimate.org/node/1277#comment-14313. Licensed to Emre Sevinc 77Summary This implementation performs the same number of steps as a more monolithic imple- mentation using nested loops with logic for breaking out of each loop early. By using laziness, we can compose this function from simpler components and still retain the efficiency of the more specialized (and verbose) implementation. EXERCISE 5.16 Hard: Generalize tails to the function scanRight, which is like a foldRight that returns a stream of the intermediate results. For example: scala> Stream(1,2,3).scanRight(0)(_ + _).toList res0: List[Int] = List(6,5,3,0) This example should be equivalent to the expression List(1+2+3+0, 2+3+0, 3+0, 0). Your function should reuse intermediate results so that traversing a Stream with n elements always takes time linear in n. Can it be implemented using unfold? How, or why not? Could it be implemented using another function we’ve written? 5.5 Summary In this chapter, we introduced non-strictness as a fundamental technique for imple- menting efficient and modular functional programs. Non-strictness can be thought of as a technique for recovering some efficiency when writing functional code, but it’s also a much bigger idea—non-strictness can improve modularity by separating the description of an expression from the how-and-when of its evaluation. Keeping these concerns separate lets us reuse a description in multiple contexts, evaluating different portions of our expression to obtain different results. We weren’t able to do that when description and evaluation were intertwined as they are in strict code. We saw a num- ber of examples of this principle in action over the course of the chapter, and we’ll see many more in the remainder of the book. We’ll switch gears in the next chapter and talk about purely functional approaches to state. This is the last building block needed before we begin exploring the process of functional design. Licensed to Emre Sevinc 78 Purely functional state In this chapter, we’ll see how to write purely functional programs that manipulate state, using the simple domain of random number generation as the example. Although by itself it’s not the most compelling use case for the techniques in this chapter, the simplicity of random number generation makes it a good first exam- ple. We’ll see more compelling use cases in parts 3 and 4 of the book, especially part 4, where we’ll say a lot more about dealing with state and effects. The goal here is to give you the basic pattern for how to make any stateful API purely func- tional. As you start writing your own functional APIs, you’ll likely run into many of the same questions that we’ll explore here. 6.1 Generating random numbers using side effects If you need to generate random1 numbers in Scala, there’s a class in the standard library, scala.util.Random,2 with a pretty typical imperative API that relies on side effects. Here’s an example of its use. scala> val rng = new scala.util.Random scala> rng.nextDouble res1: Double = 0.9867076608154569 scala> rng.nextDouble res2: Double = 0.8455696498024141 scala> rng.nextInt 1 Actually, pseudo-random, but we’ll ignore this distinction. 2 Scala API link: http://mng.bz/3DP7. Listing 6.1 Using scala.util.Random to generate random numbers Creates a new random number generator seeded with the current system time Licensed to Emre Sevinc 79Generating random numbers using side effects res3: Int = -623297295 scala> rng.nextInt(10) res4: Int = 4 Even if we didn’t know anything about what happens inside scala.util.Random, we can assume that the object rng has some internal state that gets updated after each invocation, since we’d otherwise get the same value each time we called nextInt or nextDouble. Because the state updates are performed as a side effect, these methods aren’t referentially transparent. And as we know, this implies that they aren’t as test- able, composable, modular, and easily parallelized as they could be. Let’s just take testability as an example. If we want to write a method that makes use of randomness, we need tests to be reproducible. Let’s say we had the following side-effecting method, intended to simulate the rolling of a single six-sided die, which should return a value between 1 and 6, inclusive: def rollDie: Int = { val rng = new scala.util.Random rng.nextInt(6) } This method has an off-by-one error. Whereas it’s supposed to return a value between 1 and 6, it actually returns a value from 0 to 5. But even though it doesn’t work prop- erly, five out of six times a test of this method will meet the specification! And if a test did fail, it would be ideal if we could reliably reproduce the failure. Note that what’s important here is not this specific example, but the general idea. In this case, the bug is obvious and easy to reproduce. But we can easily imagine a situ- ation where the method is much more complicated and the bug far more subtle. The more complex the program and the subtler the bug, the more important it is to be able to reproduce bugs in a reliable way. One suggestion might be to pass in the random number generator. That way, when we want to reproduce a failed test, we can pass the same generator that caused the test to fail: def rollDie(rng: scala.util.Random): Int = rng.nextInt(6) But there’s a problem with this solution. The “same” generator has to be both created with the same seed, and also be in the same state, which means that its methods have been called a certain number of times since it was created. That will be really difficult to guarantee, because every time we call nextInt, for example, the previous state of the random number generator is destroyed. Do we now need a separate mechanism to keep track of how many times we’ve called the methods on Random? No! The answer to all of this, of course, is that we should eschew side effects on principle! Gets a random integer between 0 and 9 Returns a random number from 0 to 5 Licensed to Emre Sevinc 80 CHAPTER 6 Purely functional state 6.2 Purely functional random number generation The key to recovering referential transparency is to make the state updates explicit. Don’t update the state as a side effect, but simply return the new state along with the value that we’re generating. Here’s one possible interface to a random number generator: trait RNG { def nextInt: (Int, RNG) } This method should generate a random Int. We’ll later define other functions in terms of nextInt. Rather than returning only the generated random number (as is done in scala.util.Random) and updating some internal state by mutating it in place, we return the random number and the new state, leaving the old state unmodified.3 In effect, we separate the concern of computing what the next state is from the concern of communicating the new state to the rest of the program. No global mutable memory is being used—we simply return the next state back to the caller. This leaves the caller of nextInt in complete control of what to do with the new state. Note that we’re still encapsulating the state, in the sense that users of this API don’t know anything about the implementation of the random number generator itself. 3 Recall that (A,B) is the type of two-element tuples, and given p: (A,B), you can use p._1 to extract the A and p._2 to extract the B. . . . Each call to SimpleRNG.nextInt returns the next random number in the sequence and the SimpleRNG object needed to continue the sequence. SimpleRNG SimpleRNG (1) SimpleRNG with new seed A random number 384748 .nextInt .nextInt SimpleRNG (25214903928) –1151252339 .nextInt SimpleRNG (206026503493683) –549383847 .nextInt SimpleRNG (245470556921330) 1612966641 .nextInt seed A functional RNG Licensed to Emre Sevinc 81Making stateful APIs pure But we do need to have an implementation, so let’s pick a simple one. The following is a simple random number generator that uses the same algorithm as scala.util .Random, which happens to be what’s called a linear congruential generator (http:// mng.bz/r046). The details of this implementation aren’t really important, but notice that nextInt returns both the generated value and a new RNG to use for generating the next value. case class SimpleRNG(seed: Long) extends RNG { def nextInt: (Int, RNG) = { val newSeed = (seed * 0x5DEECE66DL + 0xBL) & 0xFFFFFFFFFFFFL val nextRNG = SimpleRNG(newSeed) val n = (newSeed >>> 16).toInt (n, nextRNG) } } Here’s an example of using this API from the interpreter: scala> val rng = SimpleRNG(42) rng: SimpleRNG = SimpleRNG(42) scala> val (n1, rng2) = rng.nextInt n1: Int = 16159453 rng2: RNG = SimpleRNG(1059025964525) scala> val (n2, rng3) = rng2.nextInt n2: Int = -1281479697 rng3: RNG = SimpleRNG(197491923327988) We can run this sequence of statements as many times as we want and we’ll always get the same values. When we call rng.nextInt, it will always return 16159453 and a new RNG, whose nextInt will always return -1281479697. In other words, this API is pure. 6.3 Making stateful APIs pure This problem of making seemingly stateful APIs pure and its solution (having the API compute the next state rather than actually mutate anything) aren’t unique to random number generation. It comes up frequently, and we can always deal with it in this same way.4 Listing 6.2 A purely functional random number generator 4 An efficiency loss comes with computing next states using pure functions, because it means we can’t actually mutate the data in place. (Here, it’s not really a problem since the state is just a single Long that must be cop- ied.) This loss of efficiency can be mitigated by using efficient purely functional data structures. It’s also pos- sible in some cases to mutate the data in place without breaking referential transparency, which we’ll talk about in part 4. & is bitwise AND. We use the current seed to generate a new seed.The next state, which is an RNG instance created from the new seed. >>> is right binary shift with zero fill. The value n is the new pseudo- random integer.The return value is a tuple containing both a pseudo-random integer and the next RNG state. Let’s choose an arbitrary seed value, 42. Syntax for declaring two values by deconstructing the pair returned by rng.nextInt. Licensed to Emre Sevinc 82 CHAPTER 6 Purely functional state For instance, suppose you have a class like this: class Foo { private var s: FooState = ... def bar: Bar def baz: Int } Suppose bar and baz each mutate s in some way. We can mechanically translate this to the purely functional API by making explicit the transition from one state to the next: trait Foo { def bar: (Bar, Foo) def baz: (Int, Foo) } Whenever we use this pattern, we make the caller responsible for passing the com- puted next state through the rest of the program. For the pure RNG interface just shown, if we reuse a previous RNG, it will always generate the same value it generated before. For instance: def randomPair(rng: RNG): (Int,Int) = { val (i1,_) = rng.nextInt val (i2,_) = rng.nextInt (i1,i2) } Here i1 and i2 will be the same! If we want to generate two distinct numbers, we need to use the RNG returned by the first call to nextInt to generate the second Int: def randomPair(rng: RNG): ((Int,Int), RNG) = { val (i1,rng2) = rng.nextInt val (i2,rng3) = rng2.nextInt ((i1,i2), rng3) } You can see the general pattern, and perhaps you can also see how it might get tedious to use this API directly. Let’s write a few functions to generate random values and see if we notice any repetition that we can factor out. EXERCISE 6.1 Write a function that uses RNG.nextInt to generate a random integer between 0 and Int.maxValue (inclusive). Make sure to handle the corner case when nextInt returns Int.MinValue, which doesn’t have a non-negative counterpart. def nonNegativeInt(rng: RNG): (Int, RNG) Note use of rng2 here. We return the final state after generating the two random numbers. This lets the caller generate more random values using the new state. Licensed to Emre Sevinc 83Making stateful APIs pure EXERCISE 6.2 Write a function to generate a Double between 0 and 1, not including 1. Note: You can use Int.MaxValue to obtain the maximum positive integer value, and you can use x.toDouble to convert an x: Int to a Double. def double(rng: RNG): (Double, RNG) EXERCISE 6.3 Write functions to generate an (Int, Double) pair, a (Double, Int) pair, and a (Double, Double, Double) 3-tuple. You should be able to reuse the functions you’ve already written. def intDouble(rng: RNG): ((Int,Double), RNG) def doubleInt(rng: RNG): ((Double,Int), RNG) def double3(rng: RNG): ((Double,Double,Double), RNG) EXERCISE 6.4 Write a function to generate a list of random integers. def ints(count: Int)(rng: RNG): (List[Int], RNG) Dealing with awkwardness in functional programming As you write more functional programs, you’ll sometimes encounter situations like this where the functional way of expressing a program feels awkward or tedious. Does this imply that purity is the equivalent of trying to write an entire novel without using the letter E? Of course not. Awkwardness like this is almost always a sign of some missing abstraction waiting to be discovered. When you encounter these situations, we encourage you to plow ahead and look for common patterns that you can factor out. Most likely, this is a problem that others have encountered, and you may even rediscover the “standard” solution yourself. Even if you get stuck, struggling to puzzle out a clean solution yourself will help you to better understand what solutions others have discovered to deal with similar problems. With practice, experience, and more familiarity with the idioms contained in this book, expressing a program functionally will become effortless and natural. Of course, good design is still hard, but programming using pure functions greatly simplifies the design space. Licensed to Emre Sevinc 84 CHAPTER 6 Purely functional state 6.4 A better API for state actions Looking back at our implementations, we’ll notice a common pattern: each of our functions has a type of the form RNG => (A, RNG) for some type A. Functions of this type are called state actions or state transitions because they transform RNG states from one to the next. These state actions can be combined using combinators, which are higher-order functions that we’ll define in this section. Since it’s pretty tedious and repetitive to pass the state along ourselves, we want our combinators to pass the state from one action to the next automatically. To make the type of actions convenient to talk about, and to simplify our thinking about them, let’s make a type alias for the RNG state action data type: type Rand[+A] = RNG => (A, RNG) We can think of a value of type Rand[A] as “a randomly generated A,” although that’s not really precise. It’s really a state action—a program that depends on some RNG, uses it to generate an A, and also transitions the RNG to a new state that can be used by another action later. We can now turn methods such as RNG’s nextInt into values of this new type: val int: Rand[Int] = _.nextInt We want to write combinators that let us combine Rand actions while avoiding explic- itly passing along the RNG state. We’ll end up with a kind of domain-specific language that does all of the passing for us. For example, a simple RNG state transition is the unit action, which passes the RNG state through without using it, always returning a constant value rather than a random value: def unit[A](a: A): Rand[A] = rng => (a, rng) There’s also map for transforming the output of a state action without modifying the state itself. Remember, Rand[A] is just a type alias for a function type RNG => (A, RNG), so this is just a kind of function composition: def map[A,B](s: Rand[A])(f: A => B): Rand[B] = rng => { val (a, rng2) = s(rng) (f(a), rng2) } As an example of how map is used, here’s nonNegativeEven, which reuses nonNegative- Int to generate an Int that’s greater than or equal to zero and divisible by two: def nonNegativeEven: Rand[Int] = map(nonNegativeInt)(i =>i-i%2) Licensed to Emre Sevinc 85A better API for state actions EXERCISE 6.5 Use map to reimplement double in a more elegant way. See exercise 6.2. 6.4.1 Combining state actions Unfortunately, map isn’t powerful enough to implement intDouble and doubleInt from exercise 6.3. What we need is a new combinator map2 that can combine two RNG actions into one using a binary rather than unary function. EXERCISE 6.6 Write the implementation of map2 based on the following signature. This function takes two actions, ra and rb, and a function f for combining their results, and returns a new action that combines them: def map2[A,B,C](ra: Rand[A], rb: Rand[B])(f: (A, B) => C): Rand[C] We only have to write the map2 combinator once, and then we can use it to combine arbitrary RNG state actions. For example, if we have an action that generates values of type A and an action to generate values of type B, then we can combine them into one action that generates pairs of both A and B: def both[A,B](ra: Rand[A], rb: Rand[B]): Rand[(A,B)] = map2(ra, rb)((_, _)) We can use this to reimplement intDouble and doubleInt from exercise 6.3 more succinctly: val randIntDouble: Rand[(Int, Double)] = both(int, double) val randDoubleInt: Rand[(Double, Int)] = both(double, int) EXERCISE 6.7 Hard: If you can combine two RNG transitions, you should be able to combine a whole list of them. Implement sequence for combining a List of transitions into a single transition. Use it to reimplement the ints function you wrote before. For the latter, Licensed to Emre Sevinc 86 CHAPTER 6 Purely functional state you can use the standard library function List.fill(n)(x) to make a list with x repeated n times. def sequence[A](fs: List[Rand[A]]): Rand[List[A]] 6.4.2 Nesting state actions A pattern is beginning to emerge: we’re progressing toward implementations that don’t explicitly mention or pass along the RNG value. The map and map2 combinators allowed us to implement, in a rather succinct and elegant way, functions that were oth- erwise tedious and error-prone to write. But there are some functions that we can’t very well write in terms of map and map2. One such function is nonNegativeLessThan, which generates an integer between 0 (inclusive) and n (exclusive): def nonNegativeLessThan(n: Int): Rand[Int] A first stab at an implementation might be to generate a non-negative integer mod- ulo n: def nonNegativeLessThan(n: Int): Rand[Int] = map(nonNegativeInt){_%n} This will certainly generate a number in the range, but it’ll be skewed because Int.MaxValue may not be exactly divisible by n. So numbers that are less than the remainder of that division will come up more frequently. When nonNegativeInt gen- erates numbers higher than the largest multiple of n that fits in a 32-bit integer, we should retry the generator and hope to get a smaller number. We might attempt this: def nonNegativeLessThan(n: Int): Rand[Int] = map(nonNegativeInt){i=> val mod=i%n if (i + (n-1) - mod >= 0) mod else nonNegativeLessThan(n)(???) } This is moving in the right direction, but nonNegativeLessThan(n) has the wrong type to be used right there. Remember, it should return a Rand[Int] which is a func- tion that expects an RNG! But we don’t have one right there. What we would like is to chain things together so that the RNG that’s returned by nonNegativeInt is passed along to the recursive call to nonNegativeLessThan. We could pass it along explicitly instead of using map, like this: def nonNegativeLessThan(n: Int): Rand[Int] = { rng => val (i, rng2) = nonNegativeInt(rng) val mod=i%n if (i + (n-1) - mod >= 0) (mod, rng2) else nonNegativeLessThan(n)(rng) } Retry recursively if the Int we got is higher than the largest multiple of n that fits in a 32-bit Int. Licensed to Emre Sevinc 87A general state action data type But it would be better to have a combinator that does this passing along for us. Nei- ther map nor map2 will cut it. We need a more powerful combinator, flatMap. EXERCISE 6.8 Implement flatMap, and then use it to implement nonNegativeLessThan. def flatMap[A,B](f: Rand[A])(g: A => Rand[B]): Rand[B] flatMap allows us to generate a random A with Rand[A], and then take that A and choose a Rand[B] based on its value. In nonNegativeLessThan, we use it to choose whether to retry or not, based on the value generated by nonNegativeInt. EXERCISE 6.9 Reimplement map and map2 in terms of flatMap. The fact that this is possible is what we’re referring to when we say that flatMap is more powerful than map and map2. We can now revisit our example from the beginning of this chapter. Can we make a more testable die roll using our purely functional API? Here’s an implementation of rollDie using nonNegativeLessThan, including the off-by-one error we had before: def rollDie: Rand[Int] = nonNegativeLessThan(6) If we test this function with various RNG states, we’ll pretty soon find an RNG that causes this function to return 0: scala> val zero = rollDie(SimpleRNG(5))._1 zero: Int = 0 And we can re-create this reliably by using the same SimpleRNG(5) random generator, without having to worry that its state is destroyed after it’s been used. Fixing the bug is trivial: def rollDie: Rand[Int] = map(nonNegativeLessThan(6))(_ + 1) 6.5 A general state action data type The functions we’ve just written—unit, map, map2, flatMap, and sequence—aren’t really specific to random number generation at all. They’re general-purpose functions for working with state actions, and don’t care about the type of the state. Note that, for instance, map doesn’t care that it’s dealing with RNG state actions, and we can give it a more general signature: Licensed to Emre Sevinc 88 CHAPTER 6 Purely functional state def map[S,A,B](a: S => (A,S))(f: A => B): S => (B,S) Changing this signature doesn’t require modifying the implementation of map! The more general signature was there all along; we just didn’t see it. We should then come up with a more general type than Rand, for handling any type of state: type State[S,+A]=S=>(A,S) Here State is short for computation that carries some state along, or state action, state transi- tion, or even statement (see the next section). We might want to write it as its own class, wrapping the underlying function like this: case class State[S,+A](run: S => (A,S)) The representation doesn’t matter too much. What’s important is that we have a sin- gle, general-purpose type, and using this type we can write general-purpose functions for capturing common patterns of stateful programs. We can now just make Rand a type alias for State: type Rand[A] = State[RNG, A] EXERCISE 6.10 Generalize the functions unit, map, map2, flatMap, and sequence. Add them as meth- ods on the State case class where possible. Otherwise you should put them in a State companion object. The functions we’ve written here capture only a few of the most common patterns. As you write more functional code, you’ll likely encounter other patterns and discover other functions to capture them. 6.6 Purely functional imperative programming In the preceding sections, we were writing functions that followed a definite pattern. We’d run a state action, assign its result to a val, then run another state action that used that val, assign its result to another val, and so on. It looks a lot like imperative programming. In the imperative programming paradigm, a program is a sequence of statements where each statement may modify the program state. That’s exactly what we’ve been doing, except that our “statements” are really State actions, which are really func- tions. As functions, they read the current program state simply by receiving it in their argument, and they write to the program state simply by returning a value. Licensed to Emre Sevinc 89Purely functional imperative programming We implemented some combinators like map, map2, and ultimately flatMap to handle the propagation of the state from one statement to the next. But in doing so, we seem to have lost a bit of the imperative mood. Consider as an example the following (which assumes that we’ve made Rand[A] a type alias for State[RNG, A]): val ns: Rand[List[Int]] = int.flatMap(x => int.flatMap(y => ints(x).map(xs => xs.map(_ % y)))) It’s not clear what’s going on here. But since we have map and flatMap defined, we can use a for-comprehension to recover the imperative style: val ns: Rand[List[Int]] = for { x<-int y<-int xs <- ints(x) } yield xs.map(_ % y) This code is much easier to read (and write), and it looks like what it is—an impera- tive program that maintains some state. But it’s the same code. We get the next Int and assign it to x, get the next Int after that and assign it to y, then generate a list of length x, and finally return the list with all of its elements modulo y. To facilitate this kind of imperative programming with for-comprehensions (or flatMaps), we really only need two primitive State combinators—one for reading the state and one for writing the state. If we imagine that we have a combinator get for Aren’t imperative and functional programming opposites? Absolutely not. Remember, functional programming is simply programming without side effects. Imperative programming is about programming with statements that modify some program state, and as we’ve seen, it’s entirely reasonable to maintain state without side effects. Functional programming has excellent support for writing imperative programs, with the added benefit that such programs can be reasoned about equationally because they’re referentially transparent. We’ll have much more to say about equational rea- soning about programs in part 2, and imperative programs in particular in parts 3 and 4. int is a value of type Rand[Int] that generates a single random integer. ints(x) generates a list of length x. Replaces every element in the list with its remainder when divided by y. Generates an integer x. Generates another integer y. Generates a list xs of length x. Returns the list xs with each element replaced with its remainder when divided by y. Licensed to Emre Sevinc 90 CHAPTER 6 Purely functional state getting the current state, and a combinator set for setting a new state, we could implement a combinator that can modify the state in arbitrary ways: def modify[S](f: S => S): State[S, Unit] = for { s<-get _ <- set(f(s)) } yield () This method returns a State action that modifies the incoming state by the function f. It yields Unit to indicate that it doesn’t have a return value other than the state. What would the get and set actions look like? They’re exceedingly simple. The get action simply passes the incoming state along and returns it as the value: def get[S]: State[S, S] = State(s => (s, s)) The set action is constructed with a new state s. The resulting action ignores the incoming state, replaces it with the new state, and returns () instead of a meaningful value: def set[S](s: S): State[S, Unit] = State(_ => ((), s)) These two simple actions, together with the State combinators that we wrote—unit, map, map2, and flatMap—are all the tools we need to implement any kind of state machine or stateful program in a purely functional way. EXERCISE 6.11 Hard: To gain experience with the use of State, implement a finite state automaton that models a simple candy dispenser. The machine has two types of input: you can insert a coin, or you can turn the knob to dispense candy. It can be in one of two states: locked or unlocked. It also tracks how many candies are left and how many coins it contains. sealed trait Input case object Coin extends Input case object Turn extends Input case class Machine(locked: Boolean, candies: Int, coins: Int) The rules of the machine are as follows: Inserting a coin into a locked machine will cause it to unlock if there’s any candy left. Turning the knob on an unlocked machine will cause it to dispense candy and become locked. Turning the knob on a locked machine or inserting a coin into an unlocked machine does nothing. A machine that’s out of candy ignores all inputs. Gets the current state and assigns it to s Sets the new state to f applied to s Licensed to Emre Sevinc 91Summary The method simulateMachine should operate the machine based on the list of inputs and return the number of coins and candies left in the machine at the end. For exam- ple, if the input Machine has 10 coins and 5 candies, and a total of 4 candies are suc- cessfully bought, the output should be (14, 1). def simulateMachine(inputs: List[Input]): State[Machine, (Int, Int)] 6.7 Summary In this chapter, we touched on the subject of how to write purely functional programs that have state. We used random number generation as the motivating example, but the overall pattern comes up in many different domains. The idea is simple: use a pure function that accepts a state as its argument, and it returns the new state along- side its result. Next time you encounter an imperative API that relies on side effects, see if you can provide a purely functional version of it, and use some of the functions we wrote here to make working with it more convenient. Licensed to Emre Sevinc Licensed to Emre Sevinc Part 2 Functional design and combinator libraries We said in chapter 1 that functional programming is a radical premise that affects how we write and organize programs at every level. In part 1, we cov- ered the fundamentals of FP and saw how the commitment to using only pure functions affects the basic building blocks of programs: loops, data structures, exceptions, and so on. In part 2, we’ll see how the assumptions of functional pro- gramming affect library design. We’ll create three useful libraries in part 2—one for parallel and asynchro- nous computation, another for testing programs, and a third for parsing text. There won’t be much in the way of new syntax or language features, but we’ll make heavy use of the material already covered. Our primary goal isn’t to teach you about parallelism, testing, and parsing. The primary goal is to help you develop skill in designing functional libraries, even for domains that look nothing like the ones here. This part of the book will be a somewhat meandering journey. Functional design can be a messy, iterative process. We hope to show at least a stylized view of how functional design proceeds in the real world. Don’t worry if you don’t fol- low every bit of discussion. These chapters should be like peering over the shoul- der of someone as they think through possible designs. And because no two people approach this process the same way, the particular path we walk in each case might not strike you as the most natural one—perhaps it considers issues in what seems like an odd order, skips too fast, or goes too slow. Keep in mind that when you design your own functional libraries, you get to do it at your own pace, Licensed to Emre Sevinc 94 Functional design and combinator libraries take whatever path you want, and, whenever questions come up about design choices, you get to think through the consequences in whatever way makes sense for you, which could include running little experiments, creating prototypes, and so on. There are no right answers in functional library design. Instead, we have a collec- tion of design choices, each with different trade-offs. Our goal is that you start to under- stand these trade-offs and what different choices mean. Sometimes, when designing a library, we’ll come to a fork in the road. In this text we may, for pedagogical purposes, deliberately make a choice with undesirable consequences that we’ll uncover later. We want you to see this process first-hand, because it’s part of what actually occurs when designing functional programs. We’re less interested in the particular libraries cov- ered here in part 2, and more interested in giving you insight into how functional design proceeds and how to navigate situations that you will likely encounter. Library design is not something that only a select few people get to do; it’s part of the day-to- day work of ordinary functional programming. In these chapters and beyond, you should absolutely feel free to experiment, play with different design choices, and develop your own aesthetic. One final note: as you work through part 2, you may notice repeated patterns of similar-looking code. Keep this in the back of your mind. When we get to part 3, we’ll discuss how to remove this duplication, and we’ll discover an entire world of funda- mental abstractions that are common to all libraries. Licensed to Emre Sevinc 95 Purely functional parallelism Because modern computers have multiple cores per CPU, and often multiple CPUs, it’s more important than ever to design programs in such a way that they can take advantage of this parallel processing power. But the interaction of programs that run with parallelism is complex, and the traditional mechanism for communica- tion among execution threads—shared mutable memory—is notoriously difficult to reason about. This can all too easily result in programs that have race conditions and deadlocks, aren’t readily testable, and don’t scale well. In this chapter, we’ll build a purely functional library for creating parallel and asynchronous computations. We’ll rein in the complexity inherent in parallel pro- grams by describing them using only pure functions. This will let us use the substi- tution model to simplify our reasoning and hopefully make working with concurrent computations both easy and enjoyable. What you should take away from this chapter is not only how to write a library for purely functional parallelism, but how to approach the problem of designing a purely functional library. Our main concern will be to make our library highly composable and modular. To this end, we’ll keep with our theme of separating the concern of describing a computation from actually running it. We want to allow users of our library to write programs at a very high level, insulating them from the nitty-gritty of how their programs will be executed. For example, towards the end of the chap- ter we’ll develop a combinator, parMap, that will let us easily apply a function f to every element in a collection simultaneously: val outputList = parMap(inputList)(f) Licensed to Emre Sevinc 96 CHAPTER 7 Purely functional parallelism To get there, we’ll work iteratively. We’ll begin with a simple use case that we’d like our library to handle, and then develop an interface that facilitates this use case. Only then will we consider what our implementation of this interface should be. As we keep refining our design, we’ll oscillate between the interface and implementation as we gain a better understanding of the domain and the design space through progres- sively more complex use cases. We’ll emphasize algebraic reasoning and introduce the idea that an API can be described by an algebra that obeys specific laws. Why design our own library? Why not just use the concurrency primitives that come with Scala’s standard library in the scala.concurrent package? This is partially for pedagogical purposes—we want to show you how easy it is to design your own prac- tical libraries. But there’s another reason: we want to encourage the view that no exist- ing library is authoritative or beyond reexamination, even if designed by experts and labeled “standard.” There’s a certain safety in doing what everybody else does, but what’s conventional isn’t necessarily the most practical. Most libraries contain a lot of arbitrary design choices, many made unintentionally. When you start from scratch, you get to revisit all the fundamental assumptions that went into designing the library, take a different path, and discover things about the problem space that others may not have even considered. As a result, you might arrive at your own design that suits your purposes better. In this particular case, our fundamental assumption will be that our library admits absolutely no side effects. We’ll write a lot of code in this chapter, largely posed as exercises for you, the reader. As always, you can find the answers in the downloadable content that goes along with the book. 7.1 Choosing data types and functions When you begin designing a functional library, you usually have some general ideas about what you want to be able to do, and the difficulty in the design process is in refining these ideas and finding a data type that enables the functionality you want. In our case, we’d like to be able to “create parallel computations,” but what does that mean exactly? Let’s try to refine this into something we can implement by examining a simple, parallelizable computation—summing a list of integers. The usual left fold for this would be as follows: def sum(ints: Seq[Int]): Int = ints.foldLeft(0)((a,b) =>a+b) Here Seq is a superclass of lists and other sequences in the standard library. Impor- tantly, it has a foldLeft method. Instead of folding sequentially, we could use a divide-and-conquer algorithm; see the following listing. Licensed to Emre Sevinc 97Choosing data types and functions def sum(ints: IndexedSeq[Int]): Int = if (ints.size <= 1) ints.headOption getOrElse 0 else { val (l,r) = ints.splitAt(ints.length/2) sum(l) + sum(r) } We divide the sequence in half using the splitAt function, recursively sum both halves, and then combine their results. And unlike the foldLeft-based implementation, this implementation can be parallelized—the two halves can be summed in parallel. As we think about what sort of data types and functions could enable parallelizing this computation, we can shift our perspective. Rather than focusing on how this parallelism will ultimately be implemented and forcing ourselves to work with the implementation APIs directly (likely related to java.lang.Thread and the java.util.concurrent library), we’ll instead design our own ideal API as illuminated by our examples and work backward from there to an implementation. 7.1.1 A data type for parallel computations Look at the line sum(l) + sum(r), which invokes sum on the two halves recursively. Just from looking at this single line, we can see that any data type we might choose to rep- resent our parallel computations needs to be able to contain a result. That result will have some meaningful type (in this case Int), and we require some way of extracting this result. Let’s apply this newfound knowledge to our design. For now, we can just Listing 7.1 Summing a list using a divide-and-conquer algorithm IndexedSeq is a superclass of random-access sequences like Vector in the standard library. Unlike lists, these sequences provide an efficient splitAt method for dividing them into two parts at a particular index. headOption is a method defined on all collections in Scala. We saw this function in chapter 4. Divides the sequence in half using the splitAt function. Recursively sums both halves and adds the results together. The importance of simple examples Summing integers is in practice probably so fast that parallelization imposes more overhead than it saves. But simple examples like this are exactly the kind that are most helpful to consider when designing a functional library. Complicated examples include all sorts of incidental structure and extraneous detail that can confuse the initial design process. We’re trying to explain the essence of the problem domain, and a good way to do this is to start with trivial examples, factor out common con- cerns across these examples, and gradually add complexity. In functional design, our goal is to achieve expressiveness not with mountains of special cases, but by build- ing a simple and composable set of core data types and functions. Licensed to Emre Sevinc 98 CHAPTER 7 Purely functional parallelism invent a container type for our result, Par[A] (for parallel), and legislate the existence of the functions we need: def unit[A](a: => A): Par[A], for taking an unevaluated A and returning a computation that might evaluate it in a separate thread. We call it unit because in a sense it creates a unit of parallelism that just wraps a single value. def get[A](a: Par[A]): A, for extracting the resulting value from a parallel computation. Can we really do this? Yes, of course! For now, we don’t need to worry about what other functions we require, what the internal representation of Par might be, or how these functions are implemented. We’re simply reading off the needed data types and functions by inspecting our simple example. Let’s update this example now. def sum(ints: IndexedSeq[Int]): Int = if (ints.size <= 1) ints headOption getOrElse 0 else { val (l,r) = ints.splitAt(ints.length/2) val sumL: Par[Int] = Par.unit(sum(l)) val sumR: Par[Int] = Par.unit(sum(r)) Par.get(sumL) + Par.get(sumR) } We’ve wrapped the two recursive calls to sum in calls to unit, and we’re calling get to extract the two results from the two subcomputations. Listing 7.2 Updating sum with our custom data type Computes the left half in parallel. Computes the right half in parallel. Extracts both results and sums them. The problem with using concurrency primitives directly What of java.lang.Thread and Runnable? Let’s take a look at these classes. Here’s a partial excerpt of their API, transcribed into Scala: trait Runnable { def run: Unit } class Thread(r: Runnable) { def start: Unit def join: Unit } Already, we can see a problem with both of these types—none of the methods return a meaningful value. Therefore, if we want to get any information out of a Runnable, it has to have some side effect, like mutating some state that we can inspect. This is bad for compositionality—we can’t manipulate Runnable objects generically since we always need to know something about their internal behavior. Thread also has the disadvantage that it maps directly onto operating system threads, which are a scarce resource. It would be preferable to create as many “logical threads” as is nat- ural for our problem, and later deal with mapping these onto actual OS threads. Begins running r in a separate thread. Blocks the calling thread until r finishes running. Licensed to Emre Sevinc 99Choosing data types and functions We now have a choice about the meaning of unit and get—unit could begin evaluat- ing its argument immediately in a separate (logical) thread,1 or it could simply hold onto its argument until get is called and begin evaluation then. But note that in this example, if we want to obtain any degree of parallelism, we require that unit begin evaluating its argument concurrently and return immediately. Can you see why?2 But if unit begins evaluating its argument concurrently, then calling get arguably breaks referential transparency. We can see this by replacing sumL and sumR with their definitions—if we do so, we still get the same result, but our program is no longer parallel: Par.get(Par.unit(sum(l))) + Par.get(Par.unit(sum(r))) If unit starts evaluating its argument right away, the next thing to happen is that get will wait for that evaluation to complete. So the two sides of the + sign won’t run in parallel if we simply inline the sumL and sumR variables. We can see that unit has a def- inite side effect, but only with regard to get. That is, unit simply returns a Par[Int] in this case, representing an asynchronous computation. But as soon as we pass that Par to get, we explicitly wait for it, exposing the side effect. So it seems that we want to avoid calling get, or at least delay calling it until the very end. We want to be able to combine asynchronous computations without waiting for them to finish. 1 We’ll use the term logical thread somewhat informally throughout this chapter to mean a computation that runs concurrently with the main execution thread of our program. There need not be a one-to-one corre- spondence between logical threads and OS threads. We may have a large number of logical threads mapped onto a smaller number of OS threads via thread pooling, for instance. 2 Function arguments in Scala are strictly evaluated from left to right, so if unit delays execution until get is called, we will both spawn the parallel computation and wait for it to finish before spawning the second par- allel computation. This means the computation is effectively sequential! This kind of thing can be handled by something like java.util.concurrent .Future, ExecutorService, and friends. Why don’t we use them directly? Here’s a portion of their API: class ExecutorService { def submit[A](a: Callable[A]): Future[A] } trait Future[A] { def get: A } Though these are a tremendous help in abstracting over physical threads, these prim- itives are still at a much lower level of abstraction than the library we want to create in this chapter. A call to Future.get, for example, blocks the calling thread until the ExecutorService has finished executing it, and its API provides no means of com- posing futures. Of course, we can build the implementation of our library on top of these tools (and this is in fact what we end up doing later in the chapter), but they don’t present a modular and compositional API that we’d want to use directly from functional programs. Licensed to Emre Sevinc 100 CHAPTER 7 Purely functional parallelism Before we continue, note what we’ve done. First, we conjured up a simple, almost trivial example. We next explored this example a bit to uncover a design choice. Then, via some experimentation, we discovered an interesting consequence of one option and in the process learned something fundamental about the nature of our problem domain! The overall design process is a series of these little adventures. You don’t need any special license to do this sort of exploration, and you don’t need to be an expert in functional programming either. Just dive in and see what you find. 7.1.2 Combining parallel computations Let’s see if we can avoid the aforementioned pitfall of combining unit and get. If we don’t call get, that implies that our sum function must return a Par[Int]. What conse- quences does this change reveal? Again, let’s just invent functions with the required signatures: def sum(ints: IndexedSeq[Int]): Par[Int] = if (ints.size <= 1) Par.unit(ints.headOption getOrElse 0) else { val (l,r) = ints.splitAt(ints.length/2) Par.map2(sum(l), sum(r))(_ + _) } EXERCISE 7.1 Par.map2 is a new higher-order function for combining the result of two parallel com- putations. What is its signature? Give the most general signature possible (don’t assume it works only for Int). Observe that we’re no longer calling unit in the recursive case, and it isn’t clear whether unit should accept its argument lazily anymore. In this example, accepting the argument lazily doesn’t seem to provide any benefit, but perhaps this isn’t always the case. Let’s come back to this question later. What about map2—should it take its arguments lazily? It would make sense for map2 to run both sides of the computation in parallel, giving each side equal opportunity to run (it would seem arbitrary for the order of the map2 arguments to matter—we sim- ply want map2 to indicate that the two computations being combined are indepen- dent, and can be run in parallel). What choice lets us implement this meaning? As a simple test case, consider what happens if map2 is strict in both arguments, and we’re evaluating sum(IndexedSeq(1,2,3,4)). Take a minute to work through and under- stand the following (somewhat stylized) program trace. Licensed to Emre Sevinc 101Choosing data types and functions sum(IndexedSeq(1,2,3,4)) map2( sum(IndexedSeq(1,2)), sum(IndexedSeq(3,4)))(_ + _) map2( map2( sum(IndexedSeq(1)), sum(IndexedSeq(2)))(_ + _), sum(IndexedSeq(3,4)))(_ + _) map2( map2( unit(1), unit(2))(_ + _), sum(IndexedSeq(3,4)))(_ + _) map2( map2( unit(1), unit(2))(_ + _), map2( sum(IndexedSeq(3)), sum(IndexedSeq(4)))(_ + _))(_ + _) ... In this trace, to evaluate sum(x), we substitute x into the definition of sum, as we’ve done in previous chapters. Because map2 is strict, and Scala evaluates arguments left to right, whenever we encounter map2(sum(x),sum(y))(_ + _), we have to then evalu- ate sum(x) and so on recursively. This has the rather unfortunate consequence that we’ll strictly construct the entire left half of the tree of summations first before moving on to (strictly) constructing the right half. Here sum(IndexedSeq(1,2)) gets fully expanded before we consider sum(IndexedSeq(3,4)). And if map2 evaluates its argu- ments in parallel (using whatever resource is being used to implement the parallel- ism, like a thread pool), that implies the left half of our computation will start executing before we even begin constructing the right half of our computation. What if we keep map2 strict, but don’t have it begin execution immediately? Does this help? If map2 doesn’t begin evaluation immediately, this implies a Par value is merely constructing a description of what needs to be computed in parallel. Nothing actually occurs until we evaluate this description, perhaps using a get-like function. The problem is that if we construct our descriptions strictly, they’ll be rather heavy- weight objects. Looking back at our trace, our description will have to contain the full tree of operations to be performed: map2( map2( unit(1), unit(2))(_ + _), map2( unit(3), unit(4))(_ + _))(_ + _) Listing 7.3 Program trace for sum Licensed to Emre Sevinc 102 CHAPTER 7 Purely functional parallelism Whatever data structure we use to store this description, it’ll likely occupy more space than the original list itself! It would be nice if our descriptions were more lightweight. It seems we should make map2 lazy and have it begin immediate execution of both sides in parallel. This also addresses the problem of giving neither side priority over the other. 7.1.3 Explicit forking Something still doesn’t feel right about our latest choice. Is it always the case that we want to evaluate the two arguments to map2 in parallel? Probably not. Consider this simple hypothetical example: Par.map2(Par.unit(1), Par.unit(1))(_ + _) In this case, we happen to know that the two computations we’re combining will exe- cute so quickly that there isn’t much point in spawning off a separate logical thread to evaluate them. But our API doesn’t give us any way of providing this sort of informa- tion. That is, our current API is very inexplicit about when computations get forked off the main thread—the programmer doesn’t get to specify where this forking should occur. What if we make the forking more explicit? We can do that by inventing another function, def fork[A](a: => Par[A]): Par[A], which we can take to mean that the given Par should be run in a separate logical thread: def sum(ints: IndexedSeq[Int]): Par[Int] = if (ints.length <= 1) Par.unit(ints.headOption getOrElse 0) else { val (l,r) = ints.splitAt(ints.length/2) Par.map2(Par.fork(sum(l)), Par.fork(sum(r)))(_ + _) } With fork, we can now make map2 strict, leaving it up to the programmer to wrap arguments if they wish. A function like fork solves the problem of instantiating our parallel computations too strictly, but more fundamentally it puts the parallelism explicitly under programmer control. We’re addressing two concerns here. The first is that we need some way to indicate that the results of the two parallel tasks should be combined. Separate from this, we have the choice of whether a particular task should be performed asynchronously. By keeping these concerns separate, we avoid having any sort of global policy for parallelism attached to map2 and other combinators we write, which would mean making tough (and ultimately arbitrary) choices about what global policy is best. Let’s now return to the question of whether unit should be strict or lazy. With fork, we can now make unit strict without any loss of expressiveness. A non-strict ver- sion of it, let’s call it lazyUnit, can be implemented using unit and fork: def unit[A](a: A): Par[A] def lazyUnit[A](a: => A): Par[A] = fork(unit(a)) Licensed to Emre Sevinc 103Choosing data types and functions The function lazyUnit is a simple example of a derived combinator, as opposed to a primitive combinator like unit. We were able to define lazyUnit just in terms of other operations. Later, when we pick a representation for Par, lazyUnit won’t need to know anything about this representation—its only knowledge of Par is through the operations fork and unit that are defined on Par.3 We know we want fork to signal that its argument gets evaluated in a separate logi- cal thread. But we still have the question of whether it should begin doing so immedi- ately upon being called, or hold on to its argument, to be evaluated in a logical thread later, when the computation is forced using something like get. In other words, should evaluation be the responsibility of fork or of get? Should evaluation be eager or lazy? When you’re unsure about a meaning to assign to some function in your API, you can always continue with the design process—at some point later the trade-offs of differ- ent choices of meaning may become clear. Here we make use of a helpful trick—we’ll think about what sort of information is required to implement fork and get with various meanings. If fork begins evaluating its argument immediately in parallel, the implementa- tion must clearly know something, either directly or indirectly, about how to create threads or submit tasks to some sort of thread pool. Moreover, this implies that the thread pool (or whatever resource we use to implement the parallelism) must be (globally) accessible and properly initialized wherever we want to call fork.4 This means we lose the ability to control the parallelism strategy used for different parts of our program. And though there’s nothing inherently wrong with having a global resource for executing parallel tasks, we can imagine how it would be useful to have more fine-grained control over what implementations are used where (we might like for each subsystem of a large application to get its own thread pool with different parameters, for example). It seems much more appropriate to give get the responsi- bility of creating threads and submitting execution tasks. Note that coming to these conclusions didn’t require knowing exactly how fork and get will be implemented, or even what the representation of Par will be. We just rea- soned informally about the sort of information required to actually spawn a parallel task, and examined the consequences of having Par values know about this information. In contrast, if fork simply holds on to its unevaluated argument until later, it requires no access to the mechanism for implementing parallelism. It just takes an unevaluated Par and “marks” it for concurrent evaluation. Let’s now assume this meaning for fork. With this model, Par itself doesn’t need to know how to actually implement the parallelism. It’s more a description of a parallel computation that gets interpreted at a later time by something like the get function. This is a shift from before, where we were considering Par to be a container of a value that we could simply 3 This sort of indifference to representation is a hint that the operations are actually more general, and can be abstracted to work for types other than just Par. We’ll explore this topic in detail in part 3. 4 Much like the credit card processing system was accessible to the buyCoffee method in our Cafe example in chapter 1. Licensed to Emre Sevinc 104 CHAPTER 7 Purely functional parallelism get when it becomes available. Now it’s more of a first-class program that we can run. So let’s rename our get function to run, and dictate that this is where the parallelism actually gets implemented: def run[A](a: Par[A]): A Because Par is now just a pure data structure, run has to have some means of imple- menting the parallelism, whether it spawns new threads, delegates tasks to a thread pool, or uses some other mechanism. 7.2 Picking a representation Just by exploring this simple example and thinking through the consequences of dif- ferent choices, we’ve sketched out the following API. def unit[A](a: A): Par[A] def map2[A,B,C](a: Par[A], b: Par[B])(f: (A,B) => C): Par[C] def fork[A](a: => Par[A]): Par[A] def lazyUnit[A](a: => A): Par[A] = fork(unit(a)) def run[A](a: Par[A]): A We’ve also loosely assigned meaning to these various functions: unit promotes a constant value to a parallel computation. map2 combines the results of two parallel computations with a binary function. fork marks a computation for concurrent evaluation. The evaluation won’t actually occur until forced by run. lazyUnit wraps its unevaluated argument in a Par and marks it for concurrent evaluation. run extracts a value from a Par by actually performing the computation. At any point while sketching out an API, you can start thinking about possible represen- tations for the abstract types that appear. EXERCISE 7.2 Before continuing, try to come up with representations for Par that make it possible to implement the functions of our API. Listing 7.4 Basic sketch for an API for Par Creates a computation that immediately results in the value a. Combines the results of two parallel computations with a binary function. Marks a computation for concurrent evaluation by run. Wraps the expression a for concurrent evaluation by run. Fully evaluates a given Par, spawning parallel computations as requested by fork and extracting the resulting value. Licensed to Emre Sevinc 105Refining the API Let’s see if we can come up with a representation. We know run needs to execute asynchronous tasks somehow. We could write our own low-level API, but there’s already a class that we can use in the Java Standard Library, java.util.concurrent .ExecutorService. Here is its API, excerpted and transcribed to Scala: class ExecutorService { def submit[A](a: Callable[A]): Future[A] } trait Callable[A] { def call: A } trait Future[A] { def get: A def get(timeout: Long, unit: TimeUnit): A def cancel(evenIfRunning: Boolean): Boolean def isDone: Boolean def isCancelled: Boolean } So ExecutorService lets us submit a Callable value (in Scala we’d probably just use a lazy argument to submit) and get back a corresponding Future that’s a handle to a computation that’s potentially running in a separate thread. We can obtain a value from a Future with its get method (which blocks the current thread until the value is available), and it has some extra features for cancellation (throwing an exception after blocking for a certain amount of time, and so on). Let’s try assuming that our run function has access to an ExecutorService and see if that suggests anything about the representation for Par: def run[A](s: ExecutorService)(a: Par[A]): A The simplest possible model for Par[A] might be ExecutorService => A. This would obviously make run trivial to implement. But it might be nice to defer the decision of how long to wait for a computation, or whether to cancel it, to the caller of run. So Par[A] becomes ExecutorService => Future[A], and run simply returns the Future: type Par[A] = ExecutorService => Future[A] def run[A](s: ExecutorService)(a: Par[A]): Future[A] = a(s) Note that since Par is represented by a function that needs an ExecutorService, the creation of the Future doesn’t actually happen until this ExectorService is provided. Is it really that simple? Let’s assume it is for now, and revise our model if we find it doesn’t allow some functionality we’d like. 7.3 Refining the API The way we’ve worked so far is a bit artificial. In practice, there aren’t such clear boundaries between designing the API and choosing a representation, and one doesn’t necessarily precede the other. Ideas for a representation can inform the API, the API can inform the choice of representation, and it’s natural to shift fluidly between these two perspectives, run experiments as questions arise, build prototypes, and so on. Essentially just a lazy A Licensed to Emre Sevinc 106 CHAPTER 7 Purely functional parallelism We’ll devote this section to exploring our API. Though we got a lot of mileage out of considering a simple example, before we add any new primitive operations, let’s try to learn more about what’s expressible using those we already have. With our primi- tives and choices of meaning for them, we’ve carved out a little universe for ourselves. We now get to discover what ideas are expressible in this universe. This can and should be a fluid process—we can change the rules of our universe at any time, make a fundamental change to our representation or introduce a new primitive, and explore how our creation then behaves. Let’s begin by implementing the functions of the API that we’ve developed so far. Now that we have a representation for Par, a first crack at it should be straightfor- ward. What follows is a simplistic implementation using the representation of Par that we’ve chosen. object Par { def unit[A](a: A): Par[A] = (es: ExecutorService) => UnitFuture(a) private case class UnitFuture[A](get: A) extends Future[A] { def isDone = true def get(timeout: Long, units: TimeUnit) = get def isCancelled = false def cancel(evenIfRunning: Boolean): Boolean = false } def map2[A,B,C](a: Par[A], b: Par[B])(f: (A,B) => C): Par[C] = (es: ExecutorService) => { val af = a(es) val bf = b(es) UnitFuture(f(af.get, bf.get)) } Listing 7.5 Basic implementation for Par unit is represented as a function that returns a UnitFuture, which is a simple implementation of Future that just wraps a constant value. It doesn’t use the ExecutorService at all. It’s always done and can’t be cancelled. Its get method simply returns the value that we gave it. map2 doesn’t evaluate the call to f in a separate logical thread, in accord with our design choice of having fork be the sole function in the API for controlling parallelism. We can always do fork(map2(a,b)(f)) if we want the evaluation of f to occur in a separate thread. This implementation of map2 does not respect timeouts. It simply passes the ExecutorService on to both Par values, waits for the results of the Futures af and bf, applies f to them, and wraps them in a UnitFuture. In order to respect timeouts, we’d need a new Future implementation that records the amount of time spent evaluating af, and then subtracts that time from the available time allocated for evaluating bf. Licensed to Emre Sevinc 107Refining the API def fork[A](a: => Par[A]): Par[A] = es => es.submit(new Callable[A] { def call = a(es).get }) } We should note that Future doesn’t have a purely functional interface. This is part of the reason why we don’t want users of our library to deal with Future directly. But importantly, even though methods on Future rely on side effects, our entire Par API remains pure. It’s only after the user calls run and the implementation receives an ExecutorService that we expose the Future machinery. Our users therefore program to a pure interface whose implementation nevertheless relies on effects at the end of the day. But since our API remains pure, these effects aren’t side effects. In part 4 we’ll discuss this distinction in detail. EXERCISE 7.3 Hard: Fix the implementation of map2 so that it respects the contract of timeouts on Future. EXERCISE 7.4 This API already enables a rich set of operations. Here’s a simple example: using lazyUnit, write a function to convert any function A => B to one that evaluates its result asynchronously. def asyncF[A,B](f: A => B): A => Par[B] This is the simplest and most natural implementation of fork,but there are some problems with it—for one, the outer Callable will block waiting for the “inner” task to complete. Since this blocking occupies a thread in our thread pool, or whatever resource backs the ExecutorService, this implies that we’re losing out on some potential parallelism. Essentially, we’re using two threads when one should suffice. This is a symptom of a more serious problem with the implementation that we’ll discuss later in the chapter. Adding infix syntax using implicit conversions If Par were an actual data type, functions like map2 could be placed in the class body and then called with infix syntax like x.map2(y)(f) (much like we did for Stream and Option). But since Par is just a type alias, we can’t do this directly. There’s a trick to add infix syntax to any type using implicit conversions. We won’t discuss that here since it isn’t that relevant to what we’re trying to cover, but if you’re interested, check out the answer code associated with this chapter. Licensed to Emre Sevinc 108 CHAPTER 7 Purely functional parallelism What else can we express with our existing combinators? Let’s look at a more concrete example. Suppose we have a Par[List[Int]] representing a parallel computation that pro- duces a List[Int], and we’d like to convert this to a Par[List[Int]] whose result is sorted: def sortPar(parList: Par[List[Int]]): Par[List[Int]] We could of course run the Par, sort the resulting list, and repackage it in a Par with unit. But we want to avoid calling run. The only other combinator we have that allows us to manipulate the value of a Par in any way is map2. So if we passed parList to one side of map2, we’d be able to gain access to the List inside and sort it. And we can pass whatever we want to the other side of map2, so let’s just pass a no-op: def sortPar(parList: Par[List[Int]]): Par[List[Int]] = map2(parList, unit(()))((a, _) => a.sorted) That was easy. We can now tell a Par[List[Int]] that we’d like that list sorted. But we might as well generalize this further. We can “lift” any function of type A => B to become a function that takes Par[A] and returns Par[B]; we can map any function over a Par: def map[A,B](pa: Par[A])(f: A => B): Par[B] = map2(pa, unit(()))((a,_) => f(a)) For instance, sortPar is now simply this: def sortPar(parList: Par[List[Int]]) = map(parList)(_.sorted) That’s terse and clear. We just combined the operations to make the types line up. And yet, if you look at the implementations of map2 and unit, it should be clear this implementation of map means something sensible. Was it cheating to pass a bogus value, unit(()), as an argument to map2, only to ignore its value? Not at all! The fact that we can implement map in terms of map2, but not the other way around, just shows that map2 is strictly more powerful than map. This sort of thing happens a lot when we’re designing libraries—often, a function that seems to be primitive will turn out to be expressible using some more powerful primitive. What else can we implement using our API? Could we map over a list in parallel? Unlike map2, which combines two parallel computations, parMap (let’s call it) needs to combine N parallel computations. It seems like this should somehow be expressible: def parMap[A,B](ps: List[A])(f: A => B): Par[List[B]] We could always just write parMap as a new primitive. Remember that Par[A] is simply an alias for ExecutorService => Future[A]. There’s nothing wrong with implementing operations as new primitives. In some cases, we can even implement the operations more efficiently by assuming something about the underlying representation of the data types we’re working with. But right now we’re interested in exploring what operations are expressible using our existing Licensed to Emre Sevinc 109Refining the API API, and grasping the relationships between the various operations we’ve defined. Understanding what combinators are truly primitive will become more important in part 3, when we show how to abstract over common patterns across libraries.5 Let’s see how far we can get implementing parMap in terms of existing combinators: def parMap[A,B](ps: List[A])(f: A => B): Par[List[B]] = { val fbs: List[Par[B]] = ps.map(asyncF(f)) ... } Remember, asyncF converts an A => B to an A => Par[B] by forking a parallel com- putation to produce the result. So we can fork off our N parallel computations pretty easily, but we need some way of collecting their results. Are we stuck? Well, just from inspecting the types, we can see that we need some way of converting our List[Par[B]] to the Par[List[B]] required by the return type of parMap. EXERCISE 7.5 Hard: Write this function, called sequence. No additional primitives are required. Do not call run. def sequence[A](ps: List[Par[A]]): Par[List[A]] Once we have sequence, we can complete our implementation of parMap: def parMap[A,B](ps: List[A])(f: A => B): Par[List[B]] = fork { val fbs: List[Par[B]] = ps.map(asyncF(f)) sequence(fbs) } Note that we’ve wrapped our implementation in a call to fork. With this implementa- tion, parMap will return immediately, even for a huge input list. When we later call run, it will fork a single asynchronous computation which itself spawns N parallel com- putations, and then waits for these computations to finish, collecting their results into a list. EXERCISE 7.6 Implement parFilter, which filters elements of a list in parallel. def parFilter[A](as: List[A])(f: A => Boolean): Par[List[A]] 5 In this case, there’s another good reason not to implement parMap as a new primitive—it’s challenging to do correctly, particularly if we want to properly respect timeouts. It’s frequently the case that primitive combina- tors encapsulate some rather tricky logic, and reusing them means we don’t have to duplicate this logic. Licensed to Emre Sevinc 110 CHAPTER 7 Purely functional parallelism Can you think of any other useful functions to write? Experiment with writing a few parallel computations of your own to see which ones can be expressed without addi- tional primitives. Here are some ideas to try: Is there a more general version of the parallel summation function we wrote at the beginning of this chapter? Try using it to find the maximum value of an IndexedSeq in parallel. Write a function that takes a list of paragraphs (a List[String]) and returns the total number of words across all paragraphs, in parallel. Generalize this function as much as possible. Implement map3, map4, and map5, in terms of map2. 7.4 The algebra of an API As the previous section demonstrates, we often get far just by writing down the type signature for an operation we want, and then “following the types” to an implementa- tion. When working this way, we can almost forget the concrete domain (for instance, when we implemented map in terms of map2 and unit) and just focus on lining up types. This isn’t cheating; it’s a natural style of reasoning, analogous to the reasoning one does when simplifying an algebraic equation. We’re treating the API as an algebra,6 or an abstract set of operations along with a set of laws or properties we assume to be true, and simply doing formal symbol manipulation following the rules of the game specified by this algebra. Up until now, we’ve been reasoning somewhat informally about our API. There’s nothing wrong with this, but it can be helpful to take a step back and formalize what laws you expect to hold (or would like to hold) for your API.7 Without realizing it, you’ve probably mentally built up a model of what properties or laws you expect. Actu- ally writing these down and making them precise can highlight design choices that wouldn’t be otherwise apparent when reasoning informally. 7.4.1 The law of mapping Like any design choice, choosing laws has consequences—it places constraints on what the operations can mean, determines what implementation choices are possible, and affects what other properties can be true. Let’s look at an example. We’ll just make up a possible law that seems reasonable. This might be used as a test case if we were writ- ing tests for our library: map(unit(1))(_ + 1) == unit(2) 6 We do mean algebra in the mathematical sense of one or more sets, together with a collection of functions operating on objects of these sets, and a set of axioms. Axioms are statements assumed true from which we can derive other theorems that must also be true. In our case, the sets are particular types like Par[A] and List[Par[A]], and the functions are operations like map2, unit, and sequence. 7 We’ll have much more to say about this throughout the rest of this book. In the next chapter, we’ll design a declarative testing library that lets us define properties we expect functions to satisfy, and automatically gen- erates test cases to check these properties. And in part 3 we’ll introduce abstract interfaces specified only by sets of laws. Licensed to Emre Sevinc 111The algebra of an API We’re saying that mapping over unit(1) with the _ + 1 function is in some sense equiv- alent to unit(2). (Laws often start out this way, as concrete examples of identities 8 we expect to hold.) In what sense are they equivalent? This is an interesting question. For now, let’s say two Par objects are equivalent if for any valid ExecutorService argument, their Future results have the same value. We can check that this holds for a particular ExecutorService with a function like this: def equal[A](e: ExecutorService)(p: Par[A], p2: Par[A]): Boolean = p(e).get == p2(e).get Laws and functions share much in common. Just as we can generalize functions, we can generalize laws. For instance, the preceding could be generalized this way: map(unit(x))(f) == unit(f(x)) Here we’re saying this should hold for any choice of x and f, not just 1 and the _ + 1 function. This places some constraints on our implementation. Our implementation of unit can’t, say, inspect the value it receives and decide to return a parallel computa- tion with a result of 42 when the input is 1—it can only pass along whatever it receives. Similarly for our ExecutorService—when we submit Callable objects to it for execu- tion, it can’t make any assumptions or change behavior based on the values it receives. More concretely, this law disallows downcasting or isInstanceOf checks (often grouped under the term typecasing) in the implementations of map and unit. Much like we strive to define functions in terms of simpler functions, each of which do just one thing, we can define laws in terms of simpler laws that each say just one thing. Let’s see if we can simplify this law further. We said we wanted this law to hold for any choice of x and f. Something interesting happens if we substitute the identity function for f.9 We can simplify both sides of the equation and get a new law that’s considerably simpler:10 map(unit(x))(f) == unit(f(x)) map(unit(x))(id) == unit(id(x)) map(unit(x))(id) == unit(x) map(y)(id) == y Fascinating! Our new, simpler law talks only about map—apparently the mention of unit was an extraneous detail. To get some insight into what this new law is saying, let’s think about what map can’t do. It can’t, say, throw an exception and crash the computa- tion before applying the function to the result (can you see why this violates the law?). All it can do is apply the function f to the result of y, which of course leaves y 8 Here we mean identity in the mathematical sense of a statement that two expressions are identical or equivalent. 9 The identity function is defined as def id[A](a: A): A = a. 10 This is the same sort of substitution and simplification one might do when solving an algebraic equation. Initial law. Substitute identity function for f. Simplify. Substitute y for unit(x) on both sides. Licensed to Emre Sevinc 112 CHAPTER 7 Purely functional parallelism unaffected when that function is id.11 Even more interestingly, given map(y)(id) == y, we can perform the substitutions in the other direction to get back our original, more complex law. (Try it!) Logically, we have the freedom to do so because map can’t possi- bly behave differently for different function types it receives. Thus, given map(y)(id) == y, it must be true that map(unit(x))(f) == unit(f(x)). Since we get this second law or theorem for free, simply because of the parametricity of map, it’s sometimes called a free theorem.12 EXERCISE 7.7 Hard: Given map(y)(id) == y, it’s a free theorem that map(map(y)(g))(f) == map(y)(f compose g). (This is sometimes called map fusion, and it can be used as an optimization—rather than spawning a separate parallel computation to compute the second mapping, we can fold it into the first mapping.)13 Can you prove it? You may want to read the paper “Theorems for Free!” (http://mng.bz/Z9f1) to better under- stand the “trick” of free theorems. 7.4.2 The law of forking As interesting as all this is, this particular law doesn’t do much to constrain our imple- mentation. You’ve probably been assuming these properties without even realizing it (it would be strange to have any special cases in the implementations of map, unit, or ExecutorService.submit, or have map randomly throwing exceptions). Let’s consider a stronger property—that fork should not affect the result of a parallel computation: fork(x) == x This seems like it should be obviously true of our implementation, and it is clearly a desirable property, consistent with our expectation of how fork should work. fork(x) should do the same thing as x, but asynchronously, in a logical thread separate from the main thread. If this law didn’t always hold, we’d have to somehow know when it was safe to call without changing meaning, without any help from the type system. Surprisingly, this simple property places strong constraints on our implementation of fork. After you’ve written down a law like this, take off your implementer hat, put on your debugger hat, and try to break your law. Think through any possible corner cases, try to come up with counterexamples, and even construct an informal proof that the law holds—at least enough to convince a skeptical fellow programmer. 11 We say that map is required to be structure-preserving in that it doesn’t alter the structure of the parallel com- putation, only the value “inside” the computation. 12 The idea of free theorems was introduced by Philip Wadler in the classic paper “Theorems for Free!” (http://mng.bz/Z9f1). 13 Our representation of Par doesn’t give us the ability to implement this optimization, since it’s an opaque function. If it were reified as a data type, we could pattern match and discover opportunities to apply this rule. You may want to try experimenting with this idea on your own. Licensed to Emre Sevinc 113The algebra of an API 7.4.3 Breaking the law: a subtle bug Let’s try this mode of thinking. We’re expecting that fork(x) == x for all choices of x, and any choice of ExecutorService. We have a good sense of what x could be—it’s some expression making use of fork, unit, and map2 (and other combinators derived from these). What about ExecutorService? What are some possible implementations of it? There’s a good listing of different implementations in the class java.util .concurrent.Executors (API link: http://mng.bz/urQd). EXERCISE 7.8 Hard: Take a look through the various static methods in Executors to get a feel for the different implementations of ExecutorService that exist. Then, before continuing, go back and revisit your implementation of fork and try to find a counterexample or convince yourself that the law holds for your implementation. There’s actually a rather subtle problem that will occur in most implementations of fork. When using an ExecutorService backed by a thread pool of bounded size (see Executors.newFixedThreadPool), it’s very easy to run into a deadlock.14 Suppose we have an ExecutorService backed by a thread pool where the maximum number of threads is 1. Try running the following example using our current implementation: 14 In the next chapter, we’ll write a combinator library for testing that can help discover problems like these automatically. Why laws about code and proofs are important It may seem unusual to state and prove properties about an API. This certainly isn’t something typically done in ordinary programming. Why is it important in FP? In functional programming it’s easy, and expected, to factor out common functionality into generic, reusable components that can be composed. Side effects hurt compo- sitionality, but more generally, any hidden or out-of-band assumption or behavior that prevents us from treating our components (be they functions or anything else) as black boxes makes composition difficult or impossible. In our example of the law for fork, we can see that if the law we posited didn’t hold, many of our general-purpose combinators, like parMap, would no longer be sound (and their usage might be dangerous, since they could, depending on the broader par- allel computation they were used in, result in deadlocks). Giving our APIs an algebra, with laws that are meaningful and aid reasoning, makes the APIs more usable for clients, but also means we can treat the objects of our APIs as black boxes. As we’ll see in part 3, this is crucial for our ability to factor out com- mon patterns across the different libraries we’ve written. Licensed to Emre Sevinc 114 CHAPTER 7 Purely functional parallelism val a = lazyUnit(42 + 1) val S = Executors.newFixedThreadPool(1) println(Par.equal(S)(a, fork(a))) Most implementations of fork will result in this code deadlocking. Can you see why? Let’s look again at our implementation of fork: def fork[A](a: => Par[A]): Par[A] = es => es.submit(new Callable[A] { def call = a(es).get }) Note that we’re submitting the Callable first, and within that Callable, we’re submit- ting another Callable to the ExecutorService and blocking on its result (recall that a(es) will submit a Callable to the ExecutorService and get back a Future). This is a problem if our thread pool has size 1. The outer Callable gets submitted and picked up by the sole thread. Within that thread, before it will complete, we submit and block waiting for the result of another Callable. But there are no threads avail- able to run this Callable. They’re waiting on each other and therefore our code deadlocks. EXERCISE 7.9 Hard: Show that any fixed-size thread pool can be made to deadlock given this imple- mentation of fork. When you find counterexamples like this, you have two choices—you can try to fix your implementation such that the law holds, or you can refine your law a bit, to state more explicitly the conditions under which it holds (you could simply stipulate that you require thread pools that can grow unbounded). Even this is a good exercise—it forces you to document invariants or assumptions that were previously implicit. Can we fix fork to work on fixed-size thread pools? Let’s look at a different imple- mentation: def fork[A](fa: => Par[A]): Par[A] = es => fa(es) This certainly avoids deadlock. The only problem is that we aren’t actually forking a separate logical thread to evaluate fa. So fork(hugeComputation)(es) for some ExecutorService es, would run hugeComputation in the main thread, which is exactly what we wanted to avoid by calling fork. This is still a useful combinator, though, since it lets us delay instantiation of a computation until it’s actually needed. Let’s give it a name, delay: def delay[A](fa: => Par[A]): Par[A] = es => fa(es) Waits for the result of one Callable inside another Callable. Licensed to Emre Sevinc 115The algebra of an API But we’d really like to be able to run arbitrary computations over fixed-size thread pools. In order to do that, we’ll need to pick a different representation of Par. 7.4.4 A fully non-blocking Par implementation using actors In this section, we’ll develop a fully non-blocking implementation of Par that works for fixed-size thread pools. Since this isn’t essential to our overall goals of discussing various aspects of functional design, you may skip to the next section if you prefer. Otherwise, read on. The essential problem with the current representation is that we can’t get a value out of a Future without the current thread blocking on its get method. A representa- tion of Par that doesn’t leak resources this way has to be non-blocking in the sense that the implementations of fork and map2 must never call a method that blocks the cur- rent thread like Future.get. Writing such an implementation correctly can be chal- lenging. Fortunately we have our laws with which to test our implementation, and we only have to get it right once. After that, the users of our library can enjoy a compos- able and abstract API that does the right thing every time. In the code that follows, you don’t need to understand exactly what’s going on with every part of it. We just want to show you, using real code, what a correct repre- sentation of Par that respects the laws might look like. THE BASIC IDEA How can we implement a non-blocking representation of Par? The idea is simple. Instead of turning a Par into a java.util.concurrent.Future that we can get a value out of (which requires blocking), we’ll introduce our own version of Future with which we can register a callback that will be invoked when the result is ready. This is a slight shift in perspective: sealed trait Future[A] { private[parallelism] def apply(k: A => Unit): Unit } type Par[+A] = ExecutorService => Future[A] Our Par type looks identical, except we’re now using our new version of Future, which has a different API than the one in java.util.concurrent. Rather than calling get to obtain the result from our Future, our Future instead has an apply method that receives a function k that expects the result of type A and uses it to perform some effect. This kind of function is sometimes called a continuation or a callback. The apply method is marked private[parallelism] so that we don’t expose it to users of our library. Marking it private[parallelism] ensures that it can only be accessed from code within the fpinscala.parallelism package. This is so that our API remains pure and we can guarantee that our laws hold. The apply method is declared private to the fpinscala.parallelism package, which means that it can only be accessed by code within that package. Par looks the same, but we’re using our new non-blocking Future instead of the one in java.util.concurrent. Licensed to Emre Sevinc 116 CHAPTER 7 Purely functional parallelism With this representation of Par, let’s look at how we might implement the run func- tion first, which we’ll change to just return an A. Since it goes from Par[A] to A, it will have to construct a continuation and pass it to the Future value’s apply method. def run[A](es: ExecutorService)(p: Par[A]):A={ val ref = new AtomicReference[A] val latch = new CountDownLatch(1) p(es) { a => ref.set(a); latch.countDown } latch.await ref.get } It should be noted that run blocks the calling thread while waiting for the latch. It’s not possible to write an implementation of run that doesn’t block. Since it needs to return a value of type A, it has to wait for that value to become available before it can return. For this reason, we want users of our API to avoid calling run until they defi- nitely want to wait for a result. We could even go so far as to remove run from our API altogether and expose the apply method on Par instead so that users can register asynchronous callbacks. That would certainly be a valid design choice, but we’ll leave our API as it is for now. Listing 7.6 Implementing run for Par Using local side effects for a pure API The Future type we defined here is rather imperative. An A => Unit? Such a function can only be useful for executing some side effect using the given A, as we certainly aren’t using the returned result. Are we still doing functional programming in using a type like Future? Yes, but we’re making use of a common technique of using side effects as an implementation detail for a purely functional API. We can get away with this because the side effects we use are not observable to code that uses Par. Note that Future.apply is protected and can’t even be called by outside code. As we go through the rest of our implementation of the non-blocking Par, you may want to convince yourself that the side effects employed can’t be observed by outside code. The notion of local effects, observability, and subtleties of our definitions of purity and referential transparency are discussed in much more detail in chapter 14, but for now an informal understanding is fine. A mutable, thread-safe reference to use for storing the result. See the java.util.concurrent .atomic package for more information about these classes. A java.util.concurrent .CountDownLatch allows threads to wait until its countDown method is called a certain number of times. Here the countDown method will be called once when we’ve received the value of type A from p, and we want the run implementation to block until that happens. When we receive the value, sets the result and releases the latch. Waits until the result becomes available and the latch is released. Once we’ve passed the latch, we know ref has been set, and we return its value. Licensed to Emre Sevinc 117The algebra of an API Let’s look at an example of actually creating a Par. The simplest one is unit: def unit[A](a: A): Par[A] = es => new Future[A] { def apply(cb: A => Unit): Unit = cb(a) } Since unit already has a value of type A available, all it needs to do is call the continu- ation cb, passing it this value. If that continuation is the one from our run implemen- tation, for example, this will release the latch and make the result available immediately. What about fork? This is where we introduce the actual parallelism: def fork[A](a: => Par[A]): Par[A] = es => new Future[A] { def apply(cb: A => Unit): Unit = eval(es)(a(es)(cb)) } def eval(es: ExecutorService)(r: => Unit): Unit = es.submit(new Callable[Unit] { def call=r}) When the Future returned by fork receives its continuation cb, it will fork off a task to evaluate the by-name argument a. Once the argument has been evaluated and called to produce a Future[A], we register cb to be invoked when that Future has its result- ing A. What about map2? Recall the signature: def map2[A,B,C](a: Par[A], b: Par[B])(f: (A,B) => C): Par[C] Here, a non-blocking implementation is considerably trickier. Conceptually, we’d like map2 to run both Par arguments in parallel. When both results have arrived, we want to invoke f and then pass the resulting C to the continuation. But there are several race conditions to worry about here, and a correct non-blocking implementation is difficult using only the low-level primitives of java.util.concurrent. A BRIEF INTRODUCTION TO ACTORS To implement map2, we’ll use a non-blocking concurrency primitive called actors. An Actor is essentially a concurrent process that doesn’t constantly occupy a thread. Instead, it only occupies a thread when it receives a message. Importantly, although multiple threads may be concurrently sending messages to an actor, the actor pro- cesses only one message at a time, queueing other messages for subsequent process- ing. This makes them useful as a concurrency primitive when writing tricky code that must be accessed by multiple threads, and which would otherwise be prone to race conditions or deadlocks. It’s best to illustrate this with an example. Many implementations of actors would suit our purposes just fine, including one in the Scala standard library (see Simply passes the value to the continuation. Note that the ExecutorService isn’t needed. eval forks off evaluation of a and returns immediately. The callback will be invoked asynchronously on another thread. A helper function to evaluate an action asynchronously using some ExecutorService. Licensed to Emre Sevinc 118 CHAPTER 7 Purely functional parallelism scala.actors.Actor), but in the interest of simplicity we’ll use our own minimal actor implementation included with the chapter code in the file Actor.scala: scala> import fpinscala.parallelism._ scala> val S = Executors.newFixedThreadPool(4) S: java.util.concurrent.ExecutorService = ... scala> val echoer = Actor[String](S) { | msg => println (s"Got message: '$msg'") | } echoer: fpinscala.parallelism.Actor[String] = ... Let’s try out this Actor: scala> echoer ! "hello" Got message: 'hello' scala> scala> echoer ! "goodbye" Got message: 'goodbye' scala> echoer ! "You're just repeating everything I say, aren't you?" Got message: 'You're just repeating everything I say, aren't you?' It’s not at all essential to understand the Actor implementation. A correct, efficient implementation is rather subtle, but if you’re curious, see the Actor.scala file in the chapter code. The implementation is just under 100 lines of ordinary Scala code.15 IMPLEMENTING MAP2 VIA ACTORS We can now implement map2 using an Actor to collect the result from both argu- ments. The code is straightforward, and there are no race conditions to worry about, since we know that the Actor will only process one message at a time. def map2[A,B,C](p: Par[A], p2: Par[B])(f: (A,B) => C): Par[C] = es => new Future[C] { def apply(cb: C => Unit): Unit = { var ar: Option[A] = None var br: Option[B] = None 15 The main trickiness in an actor implementation has to do with the fact that multiple threads may be messag- ing the actor simultaneously. The implementation needs to ensure that messages are processed only one at a time, and also that all messages sent to the actor will be processed eventually rather than queued indefinitely. Even so, the code ends up being short. Listing 7.7 Implementing map2 with Actor An actor uses an ExecutorService to process messages when they arrive, so we create one here. This is a very simple actor that just echoes the String messages it receives. Note we supply S, an ExecutorService to use for processing messages. Sends the "hello" message to the actor. Note that echoer doesn’t occupy a thread at this point, since it has no further messages to process. Sends the "goodbye" message to the actor. The actor reacts by submitting a task to its ExecutorService to process that message. Two mutable vars are used to store the two results. Licensed to Emre Sevinc 119The algebra of an API val combiner = Actor[Either[A,B]](es) { case Left(a) => br match { case None => ar = Some(a) case Some(b) => eval(es)(cb(f(a, b))) } case Right(b) => ar match { case None => br = Some(b) case Some(a) => eval(es)(cb(f(a, b))) } } p(es)(a => combiner ! Left(a)) p2(es)(b => combiner ! Right(b)) } } Given these implementations, we should now be able to run Par values of arbitrary complexity without having to worry about running out of threads, even if the actors only have access to a single JVM thread. Let’s try this out in the REPL: scala> import java.util.concurrent.Executors scala> val p = parMap(List.range(1, 100000))(math.sqrt(_)) p: ExecutorService => Future[List[Double]] = < function > scala> val x = run(Executors.newFixedThreadPool(2))(p) x: List[Double] = List(1.0, 1.4142135623730951, 1.7320508075688772, 2.0, 2.23606797749979, 2.449489742783178, 2.6457513110645907, 2.828 4271247461903, 3.0, 3.1622776601683795, 3.3166247903554, 3.46410... That will call fork about 100,000 times, starting that many actors to combine these val- ues two at a time. Thanks to our non-blocking Actor implementation, we don’t need 100,000 JVM threads to do that in. Fantastic. Our law of forking now holds for fixed-size thread pools. EXERCISE 7.10 Hard: Our non-blocking representation doesn’t currently handle errors at all. If at any point our computation throws an exception, the run implementation’s latch never counts down and the exception is simply swallowed. Can you fix that? Taking a step back, the purpose of this section hasn’t necessarily been to figure out the best non-blocking implementation of fork, but more to show that laws are impor- tant. They give us another angle to consider when thinking about the design of a An actor that awaits both results, combines them with f, and passes the result to cb. If the A result came in first, stores it in ar and waits for the B. If the A result came last and we already have our B, calls f with both results and passes the resulting C to the callback, cb. Analogously, if the B result came in first, stores it in br and waits for the A. If the B result came last and we already have our A, calls f with both results and passes the resulting C to the callback, cb. Passes the actor as a continuation to both sides. On the A side, we wrap the result in Left, and on the B side, we wrap it in Right. These are the constructors of the Either data type, and they serve to indicate to the actor where the result came from. Licensed to Emre Sevinc 120 CHAPTER 7 Purely functional parallelism library. If we hadn’t tried writing out some of the laws of our API, we may not have dis- covered the thread resource leak in our first implementation until much later. In general, there are multiple approaches you can consider when choosing laws for your API. You can think about your conceptual model, and reason from there to postulate laws that should hold. You can also just invent laws you think might be useful or instructive (like we did with our fork law), and see if it’s possible and sensible to ensure that they hold for your model. And lastly, you can look at your implementation and come up with laws you expect to hold based on that.16 7.5 Refining combinators to their most general form Functional design is an iterative process. After you write down your API and have at least a prototype implementation, try using it for progressively more complex or real- istic scenarios. Sometimes you’ll find that these scenarios require new combinators. But before jumping right to implementation, it’s a good idea to see if you can refine the combinator you need to its most general form. It may be that what you need is just a specific case of some more general combinator. Let’s look at an example of this. Suppose we want a function to choose between two forking computations based on the result of an initial computation: def choice[A](cond: Par[Boolean])(t: Par[A], f: Par[A]): Par[A] This constructs a computation that proceeds with t if cond results in true, or f if cond results in false. We can certainly implement this by blocking on the result of the cond, and then using this result to determine whether to run t or f. Here’s a simple blocking implementation:17 def choice[A](cond: Par[Boolean])(t: Par[A], f: Par[A]): Par[A] = es => if (run(es)(cond).get) t(es) else f(es) But before we call ourselves good and move on, let’s think about this combinator a bit. What is it doing? It’s running cond and then, when the result is available, it runs either t or f. This seems reasonable, but let’s see if we can think of some variations to get at 16 This last way of generating laws is probably the weakest, since it can be too easy to just have the laws reflect the implementation, even if the implementation is buggy or requires all sorts of unusual side conditions that make composition difficult. 17 See Nonblocking.scala in the chapter code for the non-blocking implementation. About the exercises in this section The exercises and answers in this section use our original simpler (blocking) repre- sentation of Par[A]. If you’d like to work through the exercises and answers using the non-blocking implementation we developed in the previous section instead, see the file Nonblocking.scala in both the exercises and answers projects. Notice we are blocking on the result of cond. Licensed to Emre Sevinc 121Refining combinators to their most general form the essence of this combinator. There’s something rather arbitrary about the use of Boolean here, and the fact that we’re only selecting among two possible parallel com- putations, t and f. Why just two? If it’s useful to be able to choose between two paral- lel computations based on the results of a first, it should be certainly be useful to choose between N computations: def choiceN[A](n: Par[Int])(choices: List[Par[A]]): Par[A] Let’s say that choiceN runs n, and then uses that to select a parallel computation from choices. This is a bit more general than choice. EXERCISE 7.11 Implement choiceN and then choice in terms of choiceN. Note what we’ve done so far. We’ve refined our original combinator, choice, to choiceN, which turns out to be more general, capable of expressing choice as well as other use cases not supported by choice. But let’s keep going to see if we can refine choice to an even more general combinator. EXERCISE 7.12 There’s still something rather arbitrary about choiceN. The choice of List seems overly specific. Why does it matter what sort of container we have? For instance, what if, instead of a list of computations, we have a Map of them:18 def choiceMap[K,V](key: Par[K])(choices: Map[K,Par[V]]): Par[V] If you want, stop reading here and see if you can come up with a new and more general combinator in terms of which you can implement choice, choiceN, and choiceMap. The Map encoding of the set of possible choices feels overly specific, just like List. If we look at our implementation of choiceMap, we can see we aren’t really using much of the API of Map. Really, the Map[A,Par[B]] is used to provide a function, A => Par[B]. And now that we’ve spotted that, looking back at choice and choiceN, we can see that for choice, the pair of arguments was just being used as a function of type Boolean => Par[A] (where the Boolean selects one of the two Par[A] arguments), and for choiceN the list was just being used as a function of type Int => Par[A]! 18 Map[K,V] (API link: http://mng.bz/eZ4l) is a purely functional data structure in the Scala standard library. It associates keys of type K with values of type V in a one-to-one relationship, and allows us to look up the value by the associated key. Licensed to Emre Sevinc 122 CHAPTER 7 Purely functional parallelism Let’s make a more general signature that unifies them all: def chooser[A,B](pa: Par[A])(choices: A => Par[B]): Par[B] EXERCISE 7.13 Implement this new primitive chooser, and then use it to implement choice and choiceN. Whenever you generalize functions like this, take a critical look at your generalized function when you’re finished. Although the function may have been motivated by some specific use case, the signature and implementation may have a more general meaning. In this case, chooser is perhaps no longer the most appropriate name for this operation, which is actually quite general—it’s a parallel computation that, when run, will run an initial computation whose result is used to determine a second com- putation. Nothing says that this second computation needs to even exist before the first computation’s result is available. It doesn’t need to be stored in a container like List or Map. Perhaps it’s being generated from whole cloth using the result of the first computation. This function, which comes up often in functional libraries, is usually called bind or flatMap: def flatMap[A,B](a: Par[A])(f: A => Par[B]): Par[B] Is flatMap really the most primitive possible function, or can we generalize further? Let’s play around with it a bit more. The name flatMap is suggestive of the fact that this operation could be decomposed into two steps: mapping f: A => Par[B] over our Par[A], which generates a Par[Par[B]], and then flattening this nested Par[Par[B]] to a Par[B]. But this is interesting—it suggests that all we needed to do was add an even simpler combinator, let’s call it join, for converting a Par[Par[X]] to Par[X] for any choice of X: def join[A](a: Par[Par[A]]): Par[A] Again we’re just following the types. We have an example that demands a function with the given signature, and so we just bring it into existence. Now that it exists, we can think about what the signature means. We call it join since conceptually it’s a par- allel computation that, when run, will execute the inner computation, wait for it to finish (much like Thread.join), and then return its result. EXERCISE 7.14 Implement join. Can you see how to implement flatMap using join? And can you implement join using flatMap? Licensed to Emre Sevinc 123Summary We’ll stop here, but you’re encouraged to explore this algebra further. Try more com- plicated examples, discover new combinators, and see what you find! Here are some questions to consider: Can you implement a function with the same signature as map2, but using flatMap and unit? How is its meaning different than that of map2? Can you think of laws relating join to the other primitives of the algebra? Are there parallel computations that can’t be expressed using this algebra? Can you think of any computations that can’t even be expressed by adding new primitives to the algebra? 7.6 Summary We’ve now completed the design of a library for defining parallel and asynchronous computations in a purely functional way. Although this domain is interesting, the pri- mary goal of this chapter was to give you a window into the process of functional design, a sense of the sorts of issues you’re likely to encounter, and ideas for how you can handle those issues. Chapters 4 through 6 had a strong theme of separation of concerns: specifically, the idea of separating the description of a computation from the interpreter that then runs it. In this chapter, we saw that principle in action in the design of a library that describes parallel computations as values of a data type Par, with a separate inter- preter run to actually spawn the threads to execute them. In the next chapter, we’ll look at a completely different domain, take another meandering journey toward an API for that domain, and draw further lessons about functional design. Recognizing the expressiveness and limitations of an algebra As you practice more functional programming, one of the skills you’ll develop is the ability to recognize what functions are expressible from an algebra, and what the limitations of that algebra are. For instance, in the preceding example, it may not have been obvious at first that a function like choice couldn’t be expressed purely in terms of map, map2, and unit, and it may not have been obvious that choice was just a special case of flatMap. Over time, observations like this will come quickly, and you’ll also get better at spotting how to modify your algebra to make some needed combinator expressible. These skills will be helpful for all of your API design work. As a practical consideration, being able to reduce an API to a minimal set of primitive functions is extremely useful. As we noted earlier when we implemented parMap in terms of existing combinators, it’s frequently the case that primitive combinators encapsulate some rather tricky logic, and reusing them means we don’t have to dupli- cate this logic. Licensed to Emre Sevinc 124 Property-based testing In chapter 7 we worked through the design of a functional library for expressing parallel computations. There we introduced the idea that an API should form an algebra—that is, a collection of data types, functions over these data types, and importantly, laws or properties that express relationships between these functions. We also hinted at the idea that it might be possible to somehow check these laws automatically. This chapter will take us toward a simple but powerful library for property-based testing. The general idea of such a library is to decouple the specification of pro- gram behavior from the creation of test cases. The programmer focuses on specify- ing the behavior of programs and giving high-level constraints on the test cases; the framework then automatically generates test cases that satisfy these constraints, and runs tests to ensure that programs behave as specified. Although a library for testing has a very different purpose than a library for par- allel computations, we’ll discover that these libraries have a lot of surprisingly simi- lar combinators. This similarity is something we’ll return to in part 3. 8.1 A brief tour of property-based testing As an example, in ScalaCheck (http://mng.bz/n2j9), a property-based testing library for Scala, a property looks something like this. val intList = Gen.listOf(Gen.choose(0,100)) val prop = forAll(intList)(ns => ns.reverse.reverse == ns) && Listing 8.1 ScalaCheck properties A generator of lists of integers between 0 and 100. A property that specifies the behavior of the List.reverse method. Check that reversing a list twice gives back the original list. Licensed to Emre Sevinc 125A brief tour of property-based testing forAll(intList)(ns => ns.headOption == ns.reverse.lastOption) val failingProp = forAll(intList)(ns => ns.reverse == ns) And we can check properties like so: scala> prop.check + OK, passed 100 tests. scala> failingProp.check ! Falsified after 6 passed tests. > ARG_0: List(0, 1) Here, intList is not a List[Int], but a Gen[List[Int]], which is something that knows how to generate test data of type List[Int]. We can sample from this generator, and it will produce lists of different lengths, filled with random numbers between 0 and 100. Generators in a property-based testing library have a rich API. We can com- bine and compose generators in different ways, reuse them, and so on. The function forAll creates a property by combining a generator of type Gen[A] with some predicate of type A => Boolean. The property asserts that all values pro- duced by the generator should satisfy the predicate. Like generators, properties can also have a rich API. In this simple example we’ve used && to combine two properties. The resulting property will hold only if neither property can be falsified by any of the generated test cases. Together, the two properties form a partial specification of the correct behavior of the reverse method.1 1 The goal of this sort of testing is not necessarily to fully specify program behavior, but to give greater confi- dence in the code. Like testing in general, we can always make our properties more complete, but we should do the usual cost-benefit analysis to determine if the additional work is worth doing. Check that the first element becomes the last element after reversal. A property which is obviously false. A Gen object generates a variety of different objects to pass to a Boolean expression, searching for one that will make it false. forAll(intList)(ns => ns.reverse.reverse == ns) ns => ns.reverse.reverse == ns intList List(54, 24, 18, …, 99) List(2, 61, 14, 84, 12) List(5, 5, 5) List(99, 98, 97, …, 3, 2, 1) List() List(1) Gen.listOf(Gen.choose(0, 100)) Generators and properties Licensed to Emre Sevinc 126 CHAPTER 8 Property-based testing When we invoke prop.check, ScalaCheck will randomly generate List[Int] values to try to find a case that falsifies the predicates that we’ve supplied. The output indicates that ScalaCheck has generated 100 test cases (of type List[Int]) and that they all sat- isfied the predicates. Properties can of course fail—the output of failingProp.check indicates that the predicate tested false for some input, which is helpfully printed out to facilitate further testing or debugging. EXERCISE 8.1 To get used to thinking about testing in this way, come up with properties that specify the implementation of a sum: List[Int] => Int function. You don’t have to write your properties down as executable ScalaCheck code—an informal description is fine. Here are some ideas to get you started: Reversing a list and summing it should give the same result as summing the original, nonreversed list. What should the sum be if all elements of the list are the same value? Can you think of other properties? EXERCISE 8.2 What properties specify a function that finds the maximum of a List[Int]? Property-based testing libraries often come equipped with other useful features. We’ll talk more about some of these features later, but just to give an idea of what’s possible: Test case minimization—In the event of a failing test, the framework tries smaller sizes until it finds the smallest test case that also fails, which is more illuminating for debugging purposes. For instance, if a property fails for a list of size 10, the framework tries smaller lists and reports the smallest list that fails the test. Exhaustive test case generation—We call the set of values that could be produced by some Gen[A] the domain.2 When the domain is small enough (for instance, if it’s all even integers less than 100), we may exhaustively test all its values, rather than generate sample values. If the property holds for all values in a domain, we have an actual proof, rather than just the absence of evidence to the contrary. 2 This is the same usage of “domain” as the domain of a function (http://mng.bz/ZP8q)—generators describe possible inputs to functions we’d like to test. Note that we’ll also still sometimes use “domain” in the more colloquial sense, to refer to a subject or area of interest, for example, “the domain of functional parallelism” or “the error-handling domain.” Licensed to Emre Sevinc 127Choosing data types and functions ScalaCheck is just one property-based testing library. And while there’s nothing wrong with it, we’ll derive our own library in this chapter, starting from scratch. Like in chap- ter 7, this is mostly for pedagogical purposes, but also partly because we should con- sider no library to be the final word on any subject. There’s certainly nothing wrong with using an existing library like ScalaCheck, and existing libraries can be a good source of ideas. But even if you decide you like the existing library’s solution, spend- ing an hour or two playing with designs and writing down some type signatures is a great way to learn more about the domain and understand the design trade-offs. 8.2 Choosing data types and functions This section will be another messy and iterative process of discovering data types and functions for our library. This time around, we’re designing a library for property- based testing. As before, this is a chance to peer over the shoulder of someone working through possible designs. The particular path we take and the library we arrive at isn’t necessarily the same as what you would come up with on your own. If property-based testing is unfamiliar to you, even better; this is a chance to explore a new domain and its design space, and make your own discoveries about it. If at any point you’re feeling inspired or have ideas of your own about how to design a library like this, don’t wait for an exercise to prompt you—put the book down and go off to play with your ideas. You can always come back to the chapter if you run out of ideas or get stuck. 8.2.1 Initial snippets of an API With that said, let’s get started. What data types should we use for our testing library? What primitives should we define, and what might they mean? What laws should our functions satisfy? As before, we can look at a simple example and “read off” the needed data types and functions, and see what we find. For inspiration, let’s look at the ScalaCheck example we showed earlier: val intList = Gen.listOf(Gen.choose(0,100)) val prop = forAll(intList)(ns => ns.reverse.reverse == ns) && forAll(intList)(ns => ns.headOption == ns.reverse.lastOption) Without knowing anything about the implementation of Gen.choose or Gen.listOf, we can guess that whatever data type they return (let’s call it Gen, short for generator) must be parametric in some type. That is, Gen.choose(0,100) probably returns a Gen[Int], and Gen.listOf is then a function with the signature Gen[Int] => Gen[List[Int]]. But since it doesn’t seem like Gen.listOf should care about the type of the Gen it receives as input (it would be odd to require separate combinators for cre- ating lists of Int, Double, String, and so on), let’s go ahead and make it polymorphic: def listOf[A](a: Gen[A]): Gen[List[A]] We can learn many things by looking at this signature. Notice what we’re not specify- ing—the size of the list to generate. For this to be implementable, our generator must therefore either assume or be told the size. Assuming a size seems a bit inflexible—any Licensed to Emre Sevinc 128 CHAPTER 8 Property-based testing assumption is unlikely to be appropriate in all contexts. So it seems that generators must be told the size of test cases to generate. We can imagine an API where this is made explicit: def listOfN[A](n: Int, a: Gen[A]): Gen[List[A]] This would certainly be a useful combinator, but not having to explicitly specify sizes is powerful as well. It means that whatever function runs the tests has the freedom to choose test case sizes, which opens up the possibility of doing the test case minimiza- tion we mentioned earlier. If the sizes are always fixed and specified by the program- mer, the test runner won’t have this flexibility. Keep this concern in mind as we get further along in our design. What about the rest of this example? The forAll function looks interesting. We can see that it accepts a Gen[List[Int]] and what looks to be a corresponding predi- cate, List[Int] => Boolean. But again, it doesn’t seem like forAll should care about the types of the generator and the predicate, as long as they match up. We can express this with the type: def forAll[A](a: Gen[A])(f: A => Boolean): Prop Here, we’ve simply invented a new type, Prop (short for property, following the ScalaCheck naming), for the result of binding a Gen to a predicate. We might not know the internal representation of Prop or what other functions it supports, but based on this example we can see that it has an && operator, so let’s introduce that: trait Prop { def &&(p: Prop): Prop } 8.2.2 The meaning and API of properties Now that we have a few fragments of an API, let’s discuss what we want our types and functions to mean. First, consider Prop. We know there exist functions forAll (for cre- ating a property), && (for composing properties), and check (for running a property). In ScalaCheck, this check method has a side effect of printing to the console. It’s fine to expose this as a convenience function, but it’s not a basis for composition. For instance, we couldn’t implement && for Prop if its representation were just the check method:3 trait Prop { def check: Unit def &&(p: Prop): Prop = ??? } Since check has a side effect, the only option for implementing && in this case would be to run both check methods. So if check prints out a test report, then we would get two of them, and they would print failures and successes independently of each other. 3 This might remind you of similar problems that we discussed in chapter 7, when we looked at using Thread and Runnable for parallelism. Licensed to Emre Sevinc 129Choosing data types and functions That’s likely not a correct implementation. The problem is not so much that check has a side effect, but more generally that it throws away information. In order to combine Prop values using combinators like &&, we need check (or whatever function “runs” properties) to return some meaningful value. What type should that value have? Well, let’s consider what sort of information we’d like to get out of checking our properties. At a minimum, we need to know whether the property succeeded or failed. This lets us implement &&. EXERCISE 8.3 Assuming the following representation of Prop, implement && as a method of Prop. trait Prop { def check: Boolean } In this representation, Prop is nothing more than a non-strict Boolean, and any of the usual Boolean functions (AND, OR, NOT, XOR, and so on) can be defined for Prop. But a Boolean alone is probably insufficient. If a property fails, we might want to know how many tests succeeded first, and what arguments produced the failure. And if a property succeeds, it would be useful to know how many tests it ran. Let’s try return- ing an Either to indicate success or failure: object Prop { type SuccessCount = Int ... } trait Prop { def check: Either[???,SuccessCount] } What type shall we return in the failure case? We don’t know anything about the type of the test cases being generated. Should we add a type parameter to Prop and make it Prop[A]? Then check could return Either[A,Int]. Before going too far down this path, let’s ask ourselves whether we really care about the type of the value that caused the property to fail. We don’t really. We would only care about the type if we were going to do further computation with the failure. Most likely we’re just going to end up printing it to the screen for inspection by the person running the tests. After all, the goal here is to find bugs, and to indicate to someone what test cases trigger those bugs so they can go and fix them. As a general rule, we shouldn’t use String to repre- sent data that we want to compute with. But for values that we’re just going to show to human beings, a String is absolutely appropriate. This suggests that we can get away with the following representation for Prop: object Prop { type FailedCase = String type SuccessCount = Int } Type aliases like this can help the readability of an API. Licensed to Emre Sevinc 130 CHAPTER 8 Property-based testing trait Prop { def check: Either[(FailedCase, SuccessCount), SuccessCount] } In the case of failure, check returns a Left((s,n)), where s is some String that rep- resents the value that caused the property to fail, and n is the number of cases that suc- ceeded before the failure occurred. That takes care of the return value of check, at least for now, but what about the arguments to check? Right now, the check method takes no arguments. Is this suffi- cient? We can think about what information Prop will have access to just by inspecting the way Prop values are created. In particular, let’s look at forAll: def forAll[A](a: Gen[A])(f: A => Boolean): Prop Without knowing more about the representation of Gen, it’s hard to say whether there’s enough information here to be able to generate values of type A (which is what we need to implement check). So for now let’s turn our attention to Gen, to get a bet- ter idea of what it means and what its dependencies might be. 8.2.3 The meaning and API of generators We determined earlier that a Gen[A] was something that knows how to generate values of type A. What are some ways it could do that? Well, it could randomly generate these values. Look back at the example from chapter 6—there, we gave an interface for a purely functional random number generator RNG and showed how to make it conve- nient to combine computations that made use of it. We could just make Gen a type that wraps a State transition over a random number generator:4 case class Gen[A](sample: State[RNG,A]) EXERCISE 8.4 Implement Gen.choose using this representation of Gen. It should generate integers in the range start to stopExclusive. Feel free to use functions you’ve already written. def choose(start: Int, stopExclusive: Int): Gen[Int] EXERCISE 8.5 Let’s see what else we can implement using this representation of Gen. Try implement- ing unit, boolean, and listOfN. def unit[A](a: => A): Gen[A] 4 Recall the definition: case class State[S,A](run: S => (A,S)). Always generates the value a Licensed to Emre Sevinc 131Choosing data types and functions def boolean: Gen[Boolean] def listOfN[A](n: Int, g: Gen[A]): Gen[List[A]] As we discussed in chapter 7, we’re interested in understanding what operations are primitive and what operations are derived, and in finding a small yet expressive set of primitives. A good way to explore what is expressible with a given set of primitives is to pick some concrete examples you’d like to express, and see if you can assemble the functionality you want. As you do so, look for patterns, try factoring out these patterns into combinators, and refine your set of primitives. We encourage you to stop reading here and simply play with the primitives and combinators we’ve written so far. If you want some concrete examples to inspire you, here are some ideas: If we can generate a single Int in some range, do we need a new primitive to generate an (Int,Int) pair in some range? Can we produce a Gen[Option[A]] from a Gen[A]? What about a Gen[A] from a Gen[Option[A]]? Can we generate strings somehow using our existing primitives? 8.2.4 Generators that depend on generated values Suppose we’d like a Gen[(String,String)] that generates pairs where the second string contains only characters from the first. Or that we had a Gen[Int] that chooses an inte- ger between 0 and 11, and we’d like to make a Gen[List[Double]] that then generates lists of whatever length is chosen. In both of these cases there’s a dependency—we Generates lists of length n using the generator g The importance of play You don’t have to wait around for a concrete example to force exploration of the design space. In fact, if you rely exclusively on concrete, obviously useful or important examples to design your API, you’ll often miss out on aspects of the design space and generate APIs with ad hoc, overly specific features. We don’t want to overfit our design to the particular examples we happen to think of right now. We want to reduce the problem to its essence, and sometimes the best way to do this is play. Don’t try to solve important problems or produce useful functionality. Not right away. Just experiment with different representations, primitives, and operations, let questions naturally arise, and explore whatever piques your curiosity. (“These two functions seem similar. I wonder if there’s some more general operation hiding inside,” or “Would it make sense to make this data type polymorphic?” or “What would it mean to change this aspect of the representation from a single value to a List of values?”) There’s no right or wrong way to do this, but there are so many different design choices that it’s impossible not to run headlong into fascinating questions to play with. It doesn’t matter where you begin—if you keep playing, the domain will inexora- bly guide you to make all the design choices that are required. Licensed to Emre Sevinc 132 CHAPTER 8 Property-based testing generate a value, and then use that value to determine what generator to use next. For this we need flatMap, which lets one generator depend on another. EXERCISE 8.6 Implement flatMap, and then use it to implement this more dynamic version of listOfN. Put flatMap and listOfN in the Gen class. def flatMap[B](f: A => Gen[B]): Gen[B] def listOfN(size: Gen[Int]): Gen[List[A]] EXERCISE 8.7 Implement union, for combining two generators of the same type into one, by pulling values from each generator with equal likelihood. def union[A](g1: Gen[A], g2: Gen[A]): Gen[A] EXERCISE 8.8 Implement weighted, a version of union that accepts a weight for each Gen and gener- ates values from each Gen with probability proportional to its weight. def weighted[A](g1: (Gen[A],Double), g2: (Gen[A],Double)): Gen[A] 8.2.5 Refining the Prop data type Now that we know more about our representation of generators, let’s return to our definition of Prop. Our Gen representation has revealed information about the requirements for Prop. Our current definition of Prop looks like this: trait Prop { def check: Either[(FailedCase, SuccessCount), SuccessCount] } Prop is nothing more than a non-strict Either. But it’s missing some information. We have the number of successful test cases in SuccessCount, but we haven’t specified how many test cases to examine before we consider the property to have passed the test. We could certainly hardcode something, but it would be better to abstract over this dependency: type TestCases = Int type Result = Either[(FailedCase, SuccessCount), SuccessCount] case class Prop(run: TestCases => Result) Licensed to Emre Sevinc 133Choosing data types and functions Also, we’re recording the number of successful tests on both sides of that Either. But when a property passes, it’s implied that the number of passed tests will be equal to the argument to run. So the caller of run learns nothing new by being told the success count. Since we don’t currently need any information in the Right case of that Either, we can turn it into an Option: type Result = Option[(FailedCase, SuccessCount)] case class Prop(run: TestCases => Result) This seems a little weird, since None will mean that all tests succeeded and the prop- erty passed and Some will indicate a failure. Until now, we’ve only used the None case of Option to indicate failure. But in this case we’re using it to represent the absence of a failure. That’s a perfectly legitimate use for Option, but its intent isn’t very clear. So let’s make a new data type, equivalent to Option[(FailedCase, SuccessCount)], that shows our intent very clearly. sealed trait Result { def isFalsified: Boolean } case object Passed extends Result { def isFalsified = false } case class Falsified(failure: FailedCase, successes: SuccessCount) extends Result { def isFalsified = true } Is this now a sufficient representation of Prop? Let’s take another look at forAll. Can forAll be implemented? Why not? def forAll[A](a: Gen[A])(f: A => Boolean): Prop We can see that forAll doesn’t have enough information to return a Prop. Besides the number of test cases to try, Prop.run must have all the information needed to gener- ate test cases. If it needs to generate random test cases using our current representa- tion of Gen, it’s going to need an RNG. Let’s go ahead and propagate that dependency to Prop: case class Prop(run: (TestCases,RNG) => Result) If we think of other dependencies that it might need, besides the number of test cases and the source of randomness, we can just add these as extra parameters to Prop.run later. We now have enough information to actually implement forAll. Here’s a simple implementation. Listing 8.2 Creating a Result data type Indicates that all tests passed Indicates that one of the test cases falsified the property Licensed to Emre Sevinc 134 CHAPTER 8 Property-based testing def forAll[A](as: Gen[A])(f: A => Boolean): Prop = Prop { (n,rng) => randomStream(as)(rng).zip(Stream.from(0)).take(n).map { case (a, i) => try { if (f(a)) Passed else Falsified(a.toString, i) } catch { case e: Exception => Falsified(buildMsg(a, e), i) } }.find(_.isFalsified).getOrElse(Passed) } def randomStream[A](g: Gen[A])(rng: RNG): Stream[A] = Stream.unfold(rng)(rng => Some(g.sample.run(rng))) def buildMsg[A](s: A, e: Exception): String = s"test case: $s\n" + s"generated an exception: ${e.getMessage}\n" + s"stack trace:\n ${e.getStackTrace.mkString("\n")}" Notice that we’re catching exceptions and reporting them as test failures, rather than letting the run throw the error (which would lose information about what argument triggered the failure). EXERCISE 8.9 Now that we have a representation of Prop, implement && and || for composing Prop values. Notice that in the case of failure we don’t know which property was responsi- ble, the left or the right. Can you devise a way of handling this, perhaps by allowing Prop values to be assigned a tag or label which gets displayed in the event of a failure? def &&(p: Prop): Prop def ||(p: Prop): Prop 8.3 Test case minimization Earlier, we mentioned the idea of test case minimization. That is, ideally we’d like our framework to find the smallest or simplest failing test case, to better illustrate the problem and facilitate debugging. Let’s see if we can tweak our representations to sup- port this outcome. There are two general approaches we could take: Shrinking—After we’ve found a failing test case, we can run a separate proce- dure to minimize the test case by successively decreasing its “size” until it no longer fails. This is called shrinking, and it usually requires us to write separate code for each data type to implement this minimization process. Listing 8.3 Implementing forAll A stream of pairs (a, i) where a is a random value and i is its index in the stream. When a test fails, record the failed case and its index so we know how many tests succeeded before it. If a test case generates an exception, record it in the result. Generates an infinite stream of A values by repeatedly sampling a generator. String interpolation syntax. A string starting with s" can refer to a Scala value v as $v or ${v} in the string. The Scala compiler will expand this to v.toString. Licensed to Emre Sevinc 135Test case minimization Sized generation—Rather than shrinking test cases after the fact, we simply gener- ate our test cases in order of increasing size and complexity. So we start small and increase the size until we find a failure. This idea can be extended in vari- ous ways to allow the test runner to make larger jumps in the space of possible sizes while still making it possible to find the smallest failing test. ScalaCheck, incidentally, takes the first approach: shrinking. There’s nothing wrong with this approach (it’s also used by the Haskell library QuickCheck that ScalaCheck is based on: http://mng.bz/E24n), but we’ll see what we can do with sized generation. It’s a bit simpler and in some ways more modular, because our generators only need to know how to generate a test case of a given size. They don’t need to be aware of the “schedule” used to search the space of test cases, and the function that runs the tests therefore has the freedom to choose this schedule. We’ll see how this plays out shortly. Instead of modifying our Gen data type, for which we’ve already written a number of useful combinators, let’s introduce sized generation as a separate layer in our library. A simple representation of a sized generator is just a function that takes a size and produces a generator: case class SGen[+A](forSize: Int => Gen[A]) EXERCISE 8.10 Implement helper functions for converting Gen to SGen. You can add this as a method on Gen. def unsized: SGen[A] EXERCISE 8.11 Not surprisingly, SGen at a minimum supports many of the same operations as Gen, and the implementations are rather mechanical. Define some convenience functions on SGen that simply delegate to the corresponding functions on Gen.5 EXERCISE 8.12 Implement a listOf combinator that doesn’t accept an explicit size. It should return an SGen instead of a Gen. The implementation should generate lists of the requested size. def listOf[A](g: Gen[A]): SGen[List[A]] 5 In part 3 we’ll discuss ways of factoring out this sort of duplication. Licensed to Emre Sevinc 136 CHAPTER 8 Property-based testing Let’s see how SGen affects the definition of Prop and Prop.forAll. The SGen version of forAll looks like this: def forAll[A](g: SGen[A])(f: A => Boolean): Prop Can you see why it’s not possible to implement this function? SGen is expecting to be told a size, but Prop doesn’t receive any size information. Much like we did with the source of randomness and number of test cases, we simply need to add this as a dependency to Prop. But since we want to put Prop in charge of invoking the underly- ing generators with various sizes, we’ll have Prop accept a maximum size. Prop will then generate test cases up to and including the maximum specified size. This will also allow it to search for the smallest failing test case. Let’s see how this works out.6 type MaxSize = Int case class Prop(run: (MaxSize,TestCases,RNG) => Result) def forAll[A](g: SGen[A])(f: A => Boolean): Prop = forAll(g(_))(f) def forAll[A](g: Int => Gen[A])(f: A => Boolean): Prop = Prop { (max,n,rng) => val casesPerSize = (n + (max - 1)) / max val props: Stream[Prop] = Stream.from(0).take((n min max) + 1).map(i => forAll(g(i))(f)) val prop: Prop = props.map(p => Prop { (max, _, rng) => p.run(max, casesPerSize, rng) }).toList.reduce(_ && _) prop.run(max,n,rng) } 8.4 Using the library and improving its usability We’ve converged on what seems like a reasonable API. We could keep tinkering with it, but at this point let’s try using the library to construct tests and see if we notice any deficiencies, either in what it can express or in its general usability. Usability is some- what subjective, but we generally like to have convenient syntax and appropriate helper functions for common usage patterns. We aren’t necessarily aiming to make the library more expressive, but we want to make it pleasant to use. Listing 8.4 Generating test cases up to a given maximum size 6 This rather simplistic implementation gives an equal number of test cases to each size being generated, and increases the size by 1 starting from 0. We could imagine a more sophisticated implementation that does something more like a binary search for a failing test case size—starting with sizes 0,1,2,4,8,16..., and then narrowing the search space in the event of a failure. For each size, generate this many random cases. Make one property per size, but no more than n properties. Combine them all into one property. Licensed to Emre Sevinc 137Using the library and improving its usability 8.4.1 Some simple examples Let’s revisit an example that we mentioned at the start of this chapter—specifying the behavior of the function max, available as a method on List (API docs link: http:// mng.bz/Pz86). The maximum of a list should be greater than or equal to every other element in the list. Let’s specify this: val smallInt = Gen.choose(-10,10) val maxProp = forAll(listOf(smallInt)) { ns => val max = ns.max !ns.exists(_ > max) } At this point, calling run directly on a Prop is rather cumbersome. We can introduce a helper function for running our Prop values and printing their result to the console in a useful format. Let’s make this a method on the Prop companion object. def run(p: Prop, maxSize: Int = 100, testCases: Int = 100, rng: RNG = RNG.Simple(System.currentTimeMillis)): Unit = p.run(maxSize, testCases, rng) match { case Falsified(msg, n) => println(s"! Falsified after $n passed tests:\n $msg") case Passed => println(s"+ OK, passed $testCases tests.") } We’re taking advantage of default arguments here. This makes the method more con- venient to call. We want the default number of tests to be enough to get good cover- age, but not too many or they’ll take too long to run. If we try running run(maxProp), we notice that the property fails! Property-based testing has a way of revealing hidden assumptions that we have about our code, and forcing us to be more explicit about these assumptions. The standard library’s imple- mentation of max crashes when given the empty list. We need to fix our property to take this into account. EXERCISE 8.13 Define listOf1 for generating nonempty lists, and then update your specification of max to use this generator. Let’s try a few more examples. Listing 8.5 A run helper function for Prop No value greater than max should exist in ns. A default argument of 100 Licensed to Emre Sevinc 138 CHAPTER 8 Property-based testing EXERCISE 8.14 Write a property to verify the behavior of List.sorted (API docs link: http://mng.bz/ Pz86), which you can use to sort (among other things) a List[Int].7 For instance, List(2,1,3).sorted is equal to List(1,2,3). 8.4.2 Writing a test suite for parallel computations Recall that in chapter 7 we discovered laws that should hold for our parallel computa- tions. Can we express these laws with our library? The first “law” we looked at was actu- ally a particular test case: map(unit(1))(_ + 1) == unit(2) We certainly can express this, but the result is somewhat ugly.8 val ES: ExecutorService = Executors.newCachedThreadPool val p1 = Prop.forAll(Gen.unit(Par.unit(1)))(i => Par.map(i)(_ + 1)(ES).get == Par.unit(2)(ES).get) We’ve expressed the test, but it’s verbose, cluttered, and the idea of the test is obscured by details that aren’t really relevant here. Notice that this isn’t a question of the API being expressive enough—yes, we can express what we want, but a combination of missing helper functions and poor syntax obscures the intent. PROVING PROPERTIES Let’s improve on this. Our first observation is that forAll is a bit too general for this test case. We aren’t varying the input to this test, we just have a hardcoded example. Hardcoded examples should be just as convenient to write as in a traditional unit test- ing library. Let’s introduce a combinator for it (on the Prop companion object): def check(p: => Boolean): Prop How would we implement this? One possible way is to use forAll: def check(p: => Boolean): Prop = { lazy val result = p forAll(unit(()))(_ => result) } But this doesn’t seem quite right. We’re providing a unit generator that only gener- ates a single value, and then we’re proceeding to ignore that value just to drive the evaluation of the given Boolean. Even though we memoize the result so that it’s not evaluated more than once, the test runner will still generate multiple test cases and test the Boolean multiple times. 7 sorted takes an implicit Ordering for the elements of the list, to control the sorting strategy. 8 This is assuming our representation of Par[A] that’s just an alias for the function type ExecutorService => Future[A]. Note that we are non-strict here. Result is memoized to avoid recomputation. Licensed to Emre Sevinc 139Using the library and improving its usability For example, if we say run(check(true)), this will test the property 100 times and print “OK, passed 100 tests.” But checking a property that is always true 100 times is a terrible waste of effort. What we need is a new primitive. Remember, the representation of Prop that we have so far is just a function of type (MaxSize, TestCases, RNG) => Result, where Result is either Passed or Falsified. A simple implementation of a check primitive is to construct a Prop that ignores the number of test cases: def check(p: => Boolean): Prop = Prop { (_, _, _) => if (p) Passed else Falsified("()", 0) } This is certainly better than using forAll, but run(check(true)) will still print “passed 100 tests” even though it only tests the property once. It’s not really true that such a property has “passed” in the sense that it remains unfalsified after a number of tests. It is proved after just one test. It seems that we want a new kind of Result: case object Proved extends Result Then we can just return Proved instead of Passed in a property created by check. We’ll need to modify the test runner to take this case into account. def run(p: Prop, maxSize: Int = 100, testCases: Int = 100, rng: RNG = RNG.Simple(System.currentTimeMillis)): Unit = p.run(maxSize, testCases, rng) match { case Falsified((msg, n)) => println(s"! Falsified after $n passed tests:\n $msg") case Passed => println(s"+ OK, passed $testCases tests.") case Proved => println(s"+ OK, proved property.") } We also have to modify our implementations of Prop combinators like &&. These changes are quite trivial, since such combinators don’t need to distinguish between Passed and Proved results. EXERCISE 8.15 Hard: A check property is easy to prove conclusively because the test just involves eval- uating the Boolean argument. But some forAll properties can be proved as well. For instance, if the domain of the property is Boolean, then there are really only two cases to test. If a property forAll(p) passes for both p(true) and p(false), then it is proved. Some domains (like Boolean and Byte) are so small that they can be exhaus- tively checked. And with sized generators, even infinite domains can be exhaustively Listing 8.6 Using run to return a Proved object Licensed to Emre Sevinc 140 CHAPTER 8 Property-based testing checked up to the maximum size. Automated testing is very useful, but it’s even better if we can automatically prove our code correct. Modify our library to incorporate this kind of exhaustive checking of finite domains and sized generators. This is less of an exer- cise and more of an extensive, open-ended design project. TESTING PAR Getting back to proving the property that Par.map(Par.unit(1))(_ + 1) is equal to Par.unit(2), we can use our new Prop.check primitive to express this in a way that doesn’t obscure the intent: val p2 = Prop.check { val p = Par.map(Par.unit(1))(_ + 1) val p2 = Par.unit(2) p(ES).get == p2(ES).get } This is now pretty clear. But can we do something about the p(ES).get and p2(ES).get noise? There’s something rather unsatisfying about it. For one, we’re forcing this code to be aware of the internal implementation details of Par simply to compare two Par val- ues for equality. One improvement is to lift the equality comparison into Par using map2, which means we only have to run a single Par at the end to get our result: def equal[A](p: Par[A], p2: Par[A]): Par[Boolean] = Par.map2(p,p2)(_ == _) val p3 = check { equal( Par.map(Par.unit(1))(_ + 1), Par.unit(2) )(ES).get } This is a bit nicer than having to run each side separately. But while we’re at it, why don’t we move the running of Par out into a separate function, forAllPar. This also gives us a good place to insert variation across different parallel strategies, without it cluttering up the property we’re specifying: val S = weighted( choose(1,4).map(Executors.newFixedThreadPool) -> .75, unit(Executors.newCachedThreadPool) -> .25) def forAllPar[A](g: Gen[A])(f: A => Par[Boolean]): Prop = forAll(S.map2(g)((_,_))) { case (s,a) => f(a)(s).get } This generator creates a fixed thread pool executor 75% of the time and an unbounded one 25% of the time. a -> b is syntactic sugar for (a,b). Licensed to Emre Sevinc 141Using the library and improving its usability S.map2(g)((_,_)) is a rather noisy way of combining two generators to produce a pair of their outputs. Let’s quickly introduce a combinator to clean that up:9 def **[B](g: Gen[B]): Gen[(A,B)] = (this map2 g)((_,_)) Much nicer: def forAllPar[A](g: Gen[A])(f: A => Par[Boolean]): Prop = forAll(S ** g) { case (s,a) => f(a)(s).get } We can even introduce ** as a pattern using custom extractors (http://mng.bz/ 4pUc), which lets us write this: def forAllPar[A](g: Gen[A])(f: A => Par[Boolean]): Prop = forAll(S ** g) { case s ** a => f(a)(s).get } This syntax works nicely when tupling up multiple generators—when pattern match- ing, we don’t have to nest parentheses like using the tuple pattern directly would require. To enable ** as a pattern, we define an object called ** with an unapply function: object ** { def unapply[A,B](p: (A,B)) = Some(p) } See the custom extractors documentation for more details on this technique. So S is a Gen[ExecutorService] that will vary over fixed-size thread pools from 1–4 threads, and also consider an unbounded thread pool. And now our property looks a lot cleaner:10 val p2 = checkPar { equal ( Par.map(Par.unit(1))(_ + 1), Par.unit(2) ) } These might seem like minor changes, but this sort of factoring and cleanup can greatly improve the usability of our library, and the helper functions we’ve written make the properties easier to read and more pleasant to write. You may want to add a forAllPar version for sized generators as well. Let’s look at some other properties from chapter 7. Recall that we generalized our test case: map(unit(x))(f) == unit(f(x)) 9 Calling this ** is actually appropriate, since this function is taking the product of two generators, in the sense we discussed in chapter 3. 10 We can’t use the standard Java/Scala equals method, or the == method in Scala (which delegates to the equals method), since that method returns a Boolean directly, and we need to return a Par[Boolean]. Some infix syntax for equal might be nice. See the answer file for chapter 7 for an example of how to do this. Licensed to Emre Sevinc 142 CHAPTER 8 Property-based testing We then simplified it to the law that mapping the identity function over a computa- tion should have no effect: map(y)(x => x) == y Can we express this? Not exactly. This property implicitly states that the equality holds for all choices of y, for all types. We’re forced to pick particular values for y: val pint = Gen.choose(0,10) map (Par.unit(_)) val p4 = forAllPar(pint)(n => equal(Par.map(n)(y => y), n)) We can certainly range over more choices of y, but what we have here is probably good enough. The implementation of map can’t care about the values of our parallel com- putation, so there isn’t much point in constructing the same test for Double, String, and so on. What can affect map is the structure of the parallel computation. If we wanted greater assurance that our property held, we could provide richer generators for the structure. Here, we’re only supplying Par expressions with one level of nesting. EXERCISE 8.16 Hard: Write a richer generator for Par[Int], which builds more deeply nested parallel computations than the simple ones we gave previously. EXERCISE 8.17 Express the property about fork from chapter 7, that fork(x) == x. 8.5 Testing higher-order functions and future directions So far, our library seems quite expressive, but there’s one area where it’s lacking: we don’t currently have a good way to test higher-order functions. While we have lots of ways of generating data using our generators, we don’t really have a good way of gen- erating functions. For instance, let’s consider the takeWhile function defined for List and Stream. Recall that this function returns the longest prefix of its input whose elements all sat- isfy a predicate. For instance, List(1,2,3).takeWhile(_ < 3) results in List(1,2). A simple property we’d like to check is that for any list, s: List[A], and any f: A => Boolean, the expression s.takeWhile(f).forall(f) evaluates to true. That is, every element in the returned list satisfies the predicate.11 11 In the Scala standard library, forall is a method on List and Stream with the signature def forall[A] (f: A => Boolean): Boolean. Licensed to Emre Sevinc 143Testing higher-order functions and future directions EXERCISE 8.18 Come up with some other properties that takeWhile should satisfy. Can you think of a good property expressing the relationship between takeWhile and dropWhile? We could certainly take the approach of only examining particular arguments when testing higher-order functions. For instance, here’s a more specific property for takeWhile: val isEven = (i: Int) => i%2 == 0 val takeWhileProp = Prop.forAll(Gen.listOf(int))(ns => ns.takeWhile(isEven).forall(isEven)) This works, but is there a way we could let the testing framework handle generating functions to use with takeWhile?12 Let’s consider our options. To make this concrete, let’s suppose we have a Gen[Int] and would like to produce a Gen[String => Int]. What are some ways we could do that? Well, we could produce String => Int func- tions that simply ignore their input string and delegate to the underlying Gen[Int]: def genStringIntFn(g: Gen[Int]): Gen[String => Int] = g map (i => (s => i)) This approach isn’t sufficient though. We’re simply generating constant functions that ignore their input. In the case of takeWhile, where we need a function that returns a Boolean, this will be a function that always returns true or always returns false— clearly not very interesting for testing the behavior of our function. EXERCISE 8.19 Hard: We want to generate a function that uses its argument in some way to select which Int to return. Can you think of a good way of expressing this? This is a very open- ended and challenging design exercise. See what you can discover about this problem and if there’s a nice general solution that you can incorporate into the library we’ve developed so far. EXERCISE 8.20 You’re strongly encouraged to venture out and try using the library we’ve developed! See what else you can test with it, and see if you discover any new idioms for its use or 12 Recall that in chapter 7 we introduced the idea of free theorems and discussed how parametricity frees us from having to inspect the behavior of a function for every type of argument. Still, there are many situations where being able to generate functions for testing is useful. Licensed to Emre Sevinc 144 CHAPTER 8 Property-based testing perhaps ways it could be extended further or made more convenient. Here are a few ideas to get you started: Write properties to specify the behavior of some of the other functions we wrote for List and Stream, for instance, take, drop, filter, and unfold. Write a sized generator for producing the Tree data type defined in chapter 3, and then use this to specify the behavior of the fold function we defined for Tree. Can you think of ways to improve the API to make this easier? Write properties to specify the behavior of the sequence function we defined for Option and Either. 8.6 The laws of generators Isn’t it interesting that many of the functions we’ve implemented for our Gen type look quite similar to other functions we defined on Par, List, Stream, and Option? As an example, for Par we defined this: def map[A,B](a: Par[A])(f: A => B): Par[B] And in this chapter we defined map for Gen (as a method on Gen[A]): def map[B](f: A => B): Gen[B] We’ve also defined similar-looking functions for Option, List, Stream, and State. We have to wonder, is it merely that our functions share similar-looking signatures, or do they satisfy the same laws as well? Let’s look at a law we introduced for Par in chapter 7: map(x)(id) == x Does this law hold for our implementation of Gen.map? What about for Stream, List, Option, and State? Yes, it does! Try it and see. This indicates that not only do these functions share similar-looking signatures, they also in some sense have analogous meanings in their respective domains. It appears there are deeper forces at work! We’re uncovering some fundamental patterns that cut across domains. In part 3, we’ll learn the names for these patterns, discover the laws that govern them, and under- stand what it all means. 8.7 Summary In this chapter, we worked through another extended exercise in functional library design, using the domain of property-based testing as inspiration. We reiterate that our goal was not necessarily to learn about property-based testing as such, but to highlight particular aspects of functional design. First, we saw that oscil- lating between the abstract algebra and the concrete representation lets the two inform each other. This avoids overfitting the library to a particular representation, and also avoids ending up with a floating abstraction disconnected from the end goal. Licensed to Emre Sevinc 145Summary Second, we noticed that this domain led us to discover many of the same combina- tors we’ve now seen a few times before: map, flatMap, and so on. Not only are the sig- natures of these functions analogous, the laws satisfied by the implementations are analogous too. There are a great many seemingly distinct problems being solved in the world of software, yet the space of functional solutions is much smaller. Many libraries are just simple combinations of certain fundamental structures that appear over and over again across a variety of different domains. This is an opportunity for code reuse that we’ll exploit in part 3, when we learn both the names of some of these structures and how to spot more general abstractions. In the next and final chapter of part 2, we’ll look at another domain, parsing, with its own unique challenges. We’ll take a slightly different approach in that chapter, but once again familiar patterns will emerge. Licensed to Emre Sevinc 146 Parser combinators In this chapter, we’ll work through the design of a combinator library for creating parsers. We’ll use JSON parsing (http://mng.bz/DpNA) as a motivating use case. Like chapters 7 and 8, this chapter is not so much about parsing as it is about pro- viding further insight into the process of functional design. This chapter will introduce a design approach that we’ll call algebraic design. This is just a natural evolution of what we’ve already been doing to different degrees in past chapters—designing our interface first, along with associated laws, and letting this guide our choice of data type representations. At a few key points during this chapter, we’ll give more open-ended exercises, intended to mimic the scenarios you might encounter when writing your own What is a parser? A parser is a specialized program that takes unstructured data (such as text, or any kind of stream of symbols, numbers, or tokens) as input, and outputs a struc- tured representation of that data. For example, we can write a parser to turn a comma-separated file into a list of lists, where the elements of the outer list rep- resent the records, and the elements of each inner list represent the comma-sep- arated fields of each record. Another example is a parser that takes an XML or JSON document and turns it into a tree-like data structure. In a parser combinator library, like the one we’ll build in this chapter, a parser doesn’t have to be anything quite that complicated, and it doesn’t have to parse entire documents. It can do something as elementary as recognizing a single char- acter in the input. We then use combinators to assemble composite parsers from elementary ones, and still more complex parsers from those. Licensed to Emre Sevinc 147Designing an algebra, first libraries from scratch. You’ll get the most out of this chapter if you use these opportu- nities to put the book down and spend some time investigating possible approaches. When you design your own libraries, you won’t be handed a nicely chosen sequence of type signatures to fill in with implementations. You’ll have to make the decisions about what types and combinators you need, and a goal in part 2 of this book has been to prepare you for doing this on your own. As always, if you get stuck on one of the exercises or want some more ideas, you can keep reading or consult the answers. It may also be a good idea to do these exercises with another person, or compare notes with other readers online. 9.1 Designing an algebra, first Recall that we defined algebra to mean a collection of functions operating over some data type(s), along with a set of laws specifying relationships between these functions. In past chapters, we moved rather fluidly between inventing functions in our algebra, refining the set of functions, and tweaking our data type representations. Laws were somewhat of an afterthought—we worked out the laws only after we had a representa- tion and an API fleshed out. There’s nothing wrong with this style of design,1 but here we’ll take a different approach. We’ll start with the algebra (including its laws) and decide on a representation later. This approach—let’s call it algebraic design—can be used for any design problem but works particularly well for parsing, because it’s easy to imagine what combinators are required for parsing different kinds of inputs.2 This lets us keep an eye on the concrete goal even as we defer deciding on a representation. 1 For more about different functional design approaches, see the chapter notes for this chapter. 2 As we’ll see, there’s a connection between algebras for parsers and the classes of languages (regular, context- free, context-sensitive) studied by computer science. Parser combinators versus parser generators You might be familiar with parser generator libraries like Yacc (http://mng.bz/w3zZ) or similar libraries in other languages (for instance, ANTLR in Java: http://mng.bz/ aj8K). These libraries generate code for a parser based on a specification of the grammar. This approach works fine and can be quite efficient, but comes with all the usual problems of code generation—the libraries produce as their output a mono- lithic chunk of code that’s difficult to debug. It’s also difficult to reuse fragments of logic, since we can’t introduce new combinators or helper functions to abstract over common patterns in our parsers. In a parser combinator library, parsers are just ordinary first-class values. Reusing parsing logic is trivial, and we don’t need any sort of external tool separate from our programming language. Licensed to Emre Sevinc 148 CHAPTER 9 Parser combinators There are many different kinds of parsing libraries.3 Ours will be designed for expressiveness (we’d like to be able to parse arbitrary grammars), speed, and good error reporting. This last point is important. Whenever we run a parser on input that it doesn’t expect—which can happen if the input is malformed—it should generate a parse error. If there are parse errors, we want to be able to point out exactly where the error is in the input and accurately indicate its cause. Error reporting is often an after- thought in parsing libraries, but we’ll make sure we give careful attention to it. OK, let’s begin. For simplicity and for speed, our library will create parsers that operate on strings as input.4 We need to pick some parsing tasks to help us discover a good algebra for our parsers. What should we look at first? Something practical like parsing an email address, JSON, or HTML? No! These tasks can come later. A good and simple domain to start with is parsing various combinations of repeated letters and gibberish words like "abracadabra" and "abba". As silly as this sounds, we’ve seen before how simple examples like this help us ignore extraneous details and focus on the essence of the problem. So let’s start with the simplest of parsers, one that recognizes the single character input 'a'. As in past chapters, we can just invent a combinator for the task, char: def char(c: Char): Parser[Char] What have we done here? We’ve conjured up a type, Parser, which is parameterized on a single parameter indicating the result type of the Parser. That is, running a parser shouldn’t simply yield a yes/no response—if it succeeds, we want to get a result that has some useful type, and if it fails, we expect information about the failure. The char('a') parser will succeed only if the input is exactly the character 'a' and it will return that same character 'a' as its result. This talk of “running a parser” makes it clear our algebra needs to be extended somehow to support that. Let’s invent another function for it: def run[A](p: Parser[A])(input: String): Either[ParseError,A] Wait a minute; what is ParseError? It’s another type we just conjured into existence! At this point, we don’t care about the representation of ParseError, or Parser for that matter. We’re in the process of specifying an interface that happens to make use of two types whose representation or implementation details we choose to remain igno- rant of for now. Let’s make this explicit with a trait: 3 There’s even a parser combinator library in Scala’s standard libraries. As in the previous chapter, we’re deriv- ing our own library from first principles partially for pedagogical purposes, and to further encourage the idea that no library is authoritative. The standard library’s parser combinators don’t really satisfy our goals of pro- viding speed and good error reporting (see the chapter notes for some additional discussion). 4 This is certainly a simplifying design choice. We can make the parsing library more generic, at some cost. See the chapter notes for more discussion. Licensed to Emre Sevinc 149Designing an algebra, first trait Parsers[ParseError, Parser[+_]] { def run[A](p: Parser[A])(input: String): Either[ParseError,A] def char(c: Char): Parser[Char] } What’s with the funny Parser[+_] type argument? It’s not too important for right now, but that’s Scala’s syntax for a type parameter that is itself a type constructor.5 Making ParseError a type argument lets the Parsers interface work for any representation of ParseError, and making Parser[+_] a type parameter means that the interface works for any representation of Parser. The underscore just means that whatever Parser is, it expects one type argument to represent the type of the result, as in Parser[Char]. This code will compile as it is. We don’t need to pick a representation for ParseError or Parser, and we can continue placing additional combinators in the body of this trait. Our char function should satisfy an obvious law—for any Char, c, run(char(c))(c.toString) == Right(c) Let’s continue. We can recognize the single character 'a', but what if we want to rec- ognize the string "abracadabra"? We don’t have a way of recognizing entire strings right now, so let’s add that: def string(s: String): Parser[String] Likewise, this should satisfy an obvious law—for any String, s, run(string(s))(s) == Right(s) What if we want to recognize either the string "abra"or the string "cadabra"? We could add a very specialized combinator for it: def orString(s1: String, s2: String): Parser[String] But choosing between two parsers seems like something that would be more generally useful, regardless of their result type, so let’s go ahead and make this polymorphic: def or[A](s1: Parser[A], s2: Parser[A]): Parser[A] We expect that or(string("abra"),string("cadabra")) will succeed whenever either string parser succeeds: run(or(string("abra"),string("cadabra")))("abra") == Right("abra") run(or(string("abra"),string("cadabra")))("cadabra") == Right("cadabra") Incidentally, we can give this or combinator nice infix syntax like s1 | s2 or alternately s1 or s2, using implicits like we did in chapter 7. 5 We’ll say much more about this in the next few chapters. Parser is a type parameter that itself is a covariant type constructorHere the Parser type constructor is applied to Char. Licensed to Emre Sevinc 150 CHAPTER 9 Parser combinators trait Parsers[ParseError, Parser[+_]] { self => ... def or[A](s1: Parser[A], s2: Parser[A]): Parser[A] implicit def string(s: String): Parser[String] implicit def operators[A](p: Parser[A]) = ParserOps[A](p) implicit def asStringParser[A](a: A)(implicit f: A => Parser[String]): ParserOps[String] = ParserOps(f(a)) case class ParserOps[A](p: Parser[A]) { def |[B>:A](p2: Parser[B]): Parser[B] = self.or(p,p2) def or[B>:A](p2: => Parser[B]): Parser[B] = self.or(p,p2) } } We’ve also made string an implicit conversion and added another implicit asStringParser. With these two functions, Scala will automatically promote a String to a Parser, and we get infix operators for any type that can be converted to a Parser[String]. So given val P: Parsers, we can then import P._ to let us write expressions like "abra" | "cadabra" to create parsers. This will work for all imple- mentations of Parsers. Other binary operators or methods can be added to the body of ParserOps. We’ll follow the discipline of keeping the primary definition directly in Parsers and delegating in ParserOps to this primary definition. See the code for this chapter for more examples. We’ll use the a | b syntax liberally throughout the rest of this chapter to mean or(a,b). We can now recognize various strings, but we don’t have a way of talking about rep- etition. For instance, how would we recognize three repetitions of our "abra" | "cadabra" parser? Once again, let’s add a combinator for it:6 def listOfN[A](n: Int, p: Parser[A]): Parser[List[A]] We made listOfN parametric in the choice of A, since it doesn’t seem like it should care whether we have a Parser[String], a Parser[Char], or some other type of parser. Here are some examples of what we expect from listOfN: run(listOfN(3, "ab" | "cad"))("ababcad") == Right("ababcad") run(listOfN(3, "ab" | "cad"))("cadabab") == Right("cadabab") run(listOfN(3, "ab" | "cad"))("ababab") == Right("ababab") At this point, we’ve just been collecting up required combinators, but we haven’t tried to refine our algebra into a minimal set of primitives, and we haven’t talked much about more general laws. We’ll start doing this next, but rather than give the game away, we’ll ask you to examine a few more simple use cases yourself and try to design a Listing 9.1 Adding infix syntax to parsers 6 This should remind you of a similar function we wrote in the previous chapter. This introduces the name self to refer to this Parsers instance; it’s used later in ParserOps. Use self to explicitly disambiguate reference to the or method on the trait. Licensed to Emre Sevinc 151Designing an algebra, first minimal algebra with associated laws. This should be a challenging exercise, but enjoy struggling with it and see what you can come up with. Here are additional parsing tasks to consider, along with some guiding questions: A Parser[Int] that recognizes zero or more 'a' characters, and whose result value is the number of 'a' characters it has seen. For instance, given "aa", the parser results in 2; given "" or "b123" (a string not starting with 'a'), it results in 0; and so on. A Parser[Int] that recognizes one or more 'a' characters, and whose result value is the number of 'a' characters it has seen. (Is this defined somehow in terms of the same combinators as the parser for 'a' repeated zero or more times?) The parser should fail when given a string without a starting 'a'. How would you like to handle error reporting in this case? Could the API support giv- ing an explicit message like "Expected one or more 'a'" in the case of failure? A parser that recognizes zero or more 'a', followed by one or more 'b', and which results in the pair of counts of characters seen. For instance, given "bbb", we get (0,3), given "aaaab", we get (4,1), and so on. And additional considerations: If we’re trying to parse a sequence of zero or more "a" and are only interested in the number of characters seen, it seems inefficient to have to build up, say, a List[Char] only to throw it away and extract the length. Could something be done about this? Are the various forms of repetition primitive in our algebra, or could they be defined in terms of something simpler? We introduced a type ParseError earlier, but so far we haven’t chosen any func- tions for the API of ParseError and our algebra doesn’t have any way of letting the programmer control what errors are reported. This seems like a limitation, given that we’d like meaningful error messages from our parsers. Can you do something about it? Does a | b mean the same thing as b | a? This is a choice you get to make. What are the consequences if the answer is yes? What about if the answer is no? Does a | (b | c) mean the same thing as (a | b) | c? If yes, is this a primitive law for your algebra, or is it implied by something simpler? Try to come up with a set of laws to specify your algebra. You don’t necessarily need the laws to be complete; just write down some laws that you expect should hold for any Parsers implementation. Spend some time coming up with combinators and possible laws based on this guid- ance. When you feel stuck or at a good stopping point, then continue by reading the next section, which walks through one possible design. Licensed to Emre Sevinc 152 CHAPTER 9 Parser combinators 7 9.2 A possible algebra We’ll walk through the discovery of a set of combinators for the parsing tasks men- tioned earlier. If you worked through this design task yourself, you likely took a differ- ent path and may have ended up with a different set of combinators, which is fine. First, let’s consider the parser that recognizes zero or more repetitions of the char- acter 'a' and returns the number of characters it has seen. We can start by adding a primitive combinator for it; let’s call it many: def many[A](p: Parser[A]): Parser[List[A]] This isn’t exactly what we’re after—we need a Parser[Int] that counts the number of elements. We could change the many combinator to return a Parser[Int], but that feels too specific—undoubtedly there will be occasions where we care about more than just the list length. Better to introduce another combinator that should be famil- iar by now, map: def map[A,B](a: Parser[A])(f: A => B): Parser[B] We can now define our parser like this: map(many(char('a')))(_.size) Let’s add map and many as methods in ParserOps, so we can write the same thing with nicer syntax: val numA: Parser[Int] = char('a').many.map(_.size) We expect that, for instance, run(numA)("aaa") gives Right(3), and run(numA)("b") gives Right(0). We have a strong expectation for the behavior of map—it should merely transform the result value if the Parser was successful. No additional input characters should be examined by map, and a failing parser can’t become a successful one via map or vice 7 This sort of viewpoint might also be associated with object-oriented design, although OO hasn’t traditionally placed much emphasis on algebraic laws. Furthermore, a big reason for encapsulation in OO is that objects often have some mutable state, and making this public would allow client code to violate invariants. That con- cern isn’t relevant in FP. The advantages of algebraic design When you design the algebra of a library first, representations for the data types of the algebra don’t matter as much. As long as they support the required laws and func- tions, you don’t even need to make your representations public. There’s an idea here that a type is given meaning based on its relationship to other types (which are specified by the set of functions and their laws), rather than its inter- nal representation.7 This viewpoint is often associated with category theory, a branch of mathematics we’ve mentioned before. See the chapter notes for more on this con- nection if you’re interested. Licensed to Emre Sevinc 153A possible algebra versa. In general, we expect map to be structure preserving much like we required for Par and Gen. Let’s formalize this by stipulating the now-familiar law: map(p)(a => a) == p How should we document this law? We could put it in a documentation comment, but in the preceding chapter we developed a way to make our laws executable. Let’s use that library here. import fpinscala.testing._ trait Parsers[ParseError, Parser[+_]] ... object Laws { def equal[A](p1: Parser[A], p2: Parser[A])(in: Gen[String]): Prop = forAll(in)(s => run(p1)(s) == run(p2)(s)) def mapLaw[A](p: Parser[A])(in: Gen[String]): Prop = equal(p, p.map(a => a))(in) } } This will come in handy later when we test that our implementation of Parsers behaves as we expect. When we discover more laws later on, you’re encouraged to write them out as actual properties inside the Laws object.8 Incidentally, now that we have map, we can actually implement char in terms of string: def char(c: Char): Parser[Char] = string(c.toString) map (_.charAt(0)) And similarly another combinator, succeed, can be defined in terms of string and map: def succeed[A](a: A): Parser[A] = string("") map (_ => a) This parser always succeeds with the value a, regardless of the input string (since string("") will always succeed, even if the input is empty). Does this combinator seem familiar to you? We can specify its behavior with a law: run(succeed(a))(s) == Right(a) Listing 9.2 Combining Parser with map 8 Again, see the chapter code for more examples. In the interest of keeping this chapter shorter, we won’t give Prop implementations of all the laws, but that doesn’t mean you shouldn’t write them yourself! Licensed to Emre Sevinc 154 CHAPTER 9 Parser combinators 9.2.1 Slicing and nonempty repetition The combination of many and map certainly lets us express the parsing task of counting the number of 'a' characters, but it seems inefficient to construct a List[Char] only to discard its values and extract its length. It would be nice if we could run a Parser purely to see what portion of the input string it examines. Let’s conjure up a combina- tor for that purpose: def slice[A](p: Parser[A]): Parser[String] We call this combinator slice since we intend for it to return the portion of the input string examined by the parser if successful. As an example, run(slice(('a'|'b') .many))("aaba") results in Right("aaba")—we ignore the list accumulated by many and simply return the portion of the input string matched by the parser. With slice, our parser that counts 'a' characters can now be written as char('a') .many.slice.map(_.size) (assuming we add an alias for slice to ParserOps). The _.size function here is now referencing the size method on String, which takes constant time, rather than the size method on List, which takes time proportional to the length of the list (and requires us to actually construct the list). Note that there’s no implementation here yet. We’re still just coming up with our desired interface. But slice does put a constraint on the implementation, namely, that even if the parser p.many.map(_.size) will generate an intermediate list when run, slice(p.many).map(_.size) will not. This is a strong hint that slice is primi- tive, since it will have to have access to the internal representation of the parser. Let’s consider the next use case. What if we want to recognize one or more 'a' characters? First, we introduce a new combinator for it, many1: def many1[A](p: Parser[A]): Parser[List[A]] It feels like many1 shouldn’t have to be primitive, but should be defined somehow in terms of many. Really, many1(p) is just p followed by many(p). So it seems we need some way of running one parser, followed by another, assuming the first is successful. Let’s add that: def product[A,B](p: Parser[A], p2: Parser[B]): Parser[(A,B)] We can add ** and product as methods on ParserOps, where a ** b and a product b both delegate to product(a,b). EXERCISE 9.1 Using product, implement the now-familiar combinator map2 and then use this to implement many1 in terms of many. Note that we could have chosen to make map2 primitive and defined product in terms of map2 as we’ve done in previous chapters. The choice is up to you. def map2[A,B,C](p: Parser[A], p2: Parser[B])(f: (A,B) => C): Parser[C] Licensed to Emre Sevinc 155A possible algebra With many1, we can now implement the parser for zero or more 'a' followed by one or more 'b' as follows: char('a').many.slice.map(_.size) ** char('b').many1.slice.map(_.size) EXERCISE 9.2 Hard: Try coming up with laws to specify the behavior of product. Now that we have map2, is many really primitive? Let’s think about what many(p) will do. It tries running p, followed by many(p) again, and again, and so on until the attempt to parse p fails. It’ll accumulate the results of all successful runs of p into a list. As soon as p fails, the parser returns the empty List. EXERCISE 9.3 Hard: Before continuing, see if you can define many in terms of or, map2, and succeed. EXERCISE 9.4 Hard: Using map2 and succeed, implement the listOfN combinator from earlier. def listOfN[A](n: Int, p: Parser[A]): Parser[List[A]] Now let’s try to implement many. Here’s an implementation in terms of or, map2, and succeed: def many[A](p: Parser[A]): Parser[List[A]] = map2(p, many(p))(_ :: _) or succeed(List()) This code looks nice and tidy. We’re using map2 to say that we want p followed by many(p) again, and that we want to combine their results with :: to construct a list of results. Or, if that fails, we want to succeed with the empty list. But there’s a problem with this implementation. Can you spot what it is? We’re calling many recursively in the second argument to map2, which is strict in evaluating its second argument. Consider a simplified program trace of the evaluation of many(p) for some parser p. We’re only showing the expansion of the left side of the or here: many(p) map2(p, many(p))(_ :: _) map2(p, map2(p, many(p))(_ :: _))(_ :: _) map2(p, map2(p, map2(p, many(p))(_ :: _))(_ :: _))(_ :: _) ... Licensed to Emre Sevinc 156 CHAPTER 9 Parser combinators Because a call to map2 always evaluates its second argument, our many function will never terminate! That’s no good. This indicates that we need to make product and map2 non-strict in their second argument: def product[A,B](p: Parser[A], p2: => Parser[B]): Parser[(A,B)] def map2[A,B,C](p: Parser[A], p2: => Parser[B])( f: (A,B) => C): Parser[C] = product(p, p2) map (f.tupled) EXERCISE 9.5 We could also deal with non-strictness with a separate combinator like we did in chap- ter 7. Try this here and make the necessary changes to your existing combinators. What do you think of that approach in this instance? Now our implementation of many should work fine. Conceptually, product should have been non-strict in its second argument anyway, since if the first Parser fails, the second won’t even be consulted. We now have good combinators for parsing one thing followed by another, or mul- tiple things of the same kind in succession. But since we’re considering whether com- binators should be non-strict, let’s revisit the or combinator from earlier: def or[A](p1: Parser[A], p2: Parser[A]): Parser[A] We’ll assume that or is left-biased, meaning it tries p1 on the input, and then tries p2 only if p1 fails.9 In this case, we ought to make it non-strict in its second argument, which may never even be consulted: def or[A](p1: Parser[A], p2: => Parser[A]): Parser[A] 9.3 Handling context sensitivity Let’s take a step back and look at the primitives we have so far: string(s)—Recognizes and returns a single String slice(p)—Returns the portion of input inspected by p if successful succeed(a)—Always succeeds with the value a map(p)(f)—Applies the function f to the result of p, if successful product(p1,p2)—Sequences two parsers, running p1 and then p2, and returns the pair of their results if both succeed or(p1,p2)—Chooses between two parsers, first attempting p1, and then p2 if p1 fails 9 This is a design choice. You may wish to think about the consequences of having a version of or that always runs both p1 and p2. Licensed to Emre Sevinc 157Handling context sensitivity Using these primitives, we can express repetition and nonempty repetition (many, listOfN, and many1) as well as combinators like char and map2. Would it surprise you if these primitives were sufficient for parsing any context-free grammar, including JSON? Well, they are! We’ll get to writing that JSON parser soon, but what can’t we express yet? Suppose we want to parse a single digit, like '4', followed by that many 'a' charac- ters (this sort of problem should feel familiar from previous chapters). Examples of valid input are "0", "1a", "2aa", "4aaaa", and so on. This is an example of a context- sensitive grammar. It can’t be expressed with product because our choice of the second parser depends on the result of the first (the second parser depends on its con- text). We want to run the first parser, and then do a listOfN using the number extracted from the first parser’s result. Can you see why product can’t express this? This progression might feel familiar to you. In past chapters, we encountered simi- lar expressiveness limitations and dealt with it by introducing a new primitive, flat- Map. Let’s introduce that here (and we’ll add an alias to ParserOps so we can write parsers using for-comprehensions): def flatMap[A,B](p: Parser[A])(f: A => Parser[B]): Parser[B] Can you see how this signature implies an ability to sequence parsers where each parser in the chain depends on the output of the previous one? EXERCISE 9.6 Using flatMap and any other combinators, write the context-sensitive parser we couldn’t express earlier. To parse the digits, you can make use of a new primitive, regex, which promotes a regular expression to a Parser.10 In Scala, a string s can be promoted to a Regex object (which has methods for matching) using s.r, for instance, "[a-zA-Z_][a-zA-Z0-9_]*".r. implicit def regex(r: Regex): Parser[String] EXERCISE 9.7 Implement product and map2 in terms of flatMap. EXERCISE 9.8 map is no longer primitive. Express it in terms of flatMap and/or other combinators. 10 In theory this isn’t necessary; we could write out "0" | "1" | ... "9" to recognize a single digit, but this isn’t likely to be very efficient. Licensed to Emre Sevinc 158 CHAPTER 9 Parser combinators So it appears we have a new primitive, flatMap, which enables context-sensitive pars- ing and lets us implement map and map2. This is not the first time flatMap has made an appearance. We now have an even smaller set of just six primitives: string, regex, slice, succeed, or, and flatMap. But we also have more power than before. With flatMap, instead of the less-general map and product, we can parse not just arbitrary context- free grammars like JSON, but context-sensitive grammars as well, including extremely complicated ones like C++ and PERL! 9.4 Writing a JSON parser Let’s write that JSON parser now, shall we? We don’t have an implementation of our algebra yet, and we’ve yet to add any combinators for good error reporting, but we can deal with these things later. Our JSON parser doesn’t need to know the internal details of how parsers are represented. We can simply write a function that produces a JSON parser using only the set of primitives we’ve defined and any derived combinators. That is, for some JSON parse result type (we’ll explain the JSON format and the parse result type shortly), we’ll write a function like this: def jsonParser[Err,Parser[+_]](P: Parsers[Err,Parser]): Parser[JSON] = { import P._ val spaces = char(' ').many.slice ... } This might seem like a peculiar thing to do, since we won’t actually be able to run our parser until we have a concrete implementation of the Parsers interface. But we’ll proceed, because in FP, it’s common to define an algebra and explore its expressive- ness without having a concrete implementation. A concrete implementation can tie us down and makes changes to the API more difficult. Especially during the design phase of a library, it can be much easier to refine an algebra without having to commit to any particular implementation, and part of our goal here is to get you comfortable with this style of working. After this section, we’ll return to the question of adding better error reporting to our parsing API. We can do this without disturbing the overall structure of the API or changing our JSON parser very much. And we’ll also come up with a concrete, runna- ble representation of our Parser type. Importantly, the JSON parser we’ll implement in this next section will be completely independent of that representation. 9.4.1 The JSON format If you aren’t already familiar with the JSON format, you may want to read Wikipedia’s description (http://mng.bz/DpNA) and the grammar specification (http://json.org). Here’s an example JSON document: { "Company name" : "Microsoft Corporation", "Ticker" : "MSFT", Gives access to all the combinators Licensed to Emre Sevinc 159Writing a JSON parser "Active" : true, "Price" : 30.66, "Shares outstanding" : 8.38e9, "Related companies" : [ "HPQ", "IBM", "YHOO", "DELL", "GOOG" ] } A value in JSON can be one of several types. An object in JSON is a comma-separated sequence of key-value pairs, wrapped in curly braces ({}). The keys must be strings like "Ticker" or "Price", and the values can be either another object, an array like ["HPQ", "IBM" ... ] that contains further values, or a literal like "MSFT", true, null, or 30.66. We’ll write a rather dumb parser that simply parses a syntax tree from the docu- ment without doing any further processing.11 We’ll need a representation for a parsed JSON document. Let’s introduce a data type for this: trait JSON object JSON { case object JNull extends JSON case class JNumber(get: Double) extends JSON case class JString(get: String) extends JSON case class JBool(get: Boolean) extends JSON case class JArray(get: IndexedSeq[JSON]) extends JSON case class JObject(get: Map[String, JSON]) extends JSON } 9.4.2 A JSON parser Recall that we’ve built up the following set of primitives: string(s): Recognizes and returns a single String regex(s): Recognizes a regular expression s slice(p): Returns the portion of input inspected by p if successful succeed(a): Always succeeds with the value a flatMap(p)(f): Runs a parser, then uses its result to select a second parser to run in sequence or(p1,p2): Chooses between two parsers, first attempting p1, and then p2 if p1 fails We used these primitives to define a number of combinators like map, map2, many, and many1. EXERCISE 9.9 Hard: At this point, you are going to take over the process. You’ll be creating a Parser[JSON] from scratch using the primitives we’ve defined. You don’t need to 11 See the chapter notes for discussion of alternate approaches. Licensed to Emre Sevinc 160 CHAPTER 9 Parser combinators worry (yet) about the representation of Parser. As you go, you’ll undoubtedly dis- cover additional combinators and idioms, notice and factor out common patterns, and so on. Use the skills you’ve been developing throughout this book, and have fun! If you get stuck, you can always consult the answers. Here’s some minimal guidance: Any general-purpose combinators you discover can be added to the Parsers trait directly. You’ll probably want to introduce combinators that make it easier to parse the tokens of the JSON format (like string literals and numbers). For this you could use the regex primitive we introduced earlier. You could also add a few primi- tives like letter, digit, whitespace, and so on, for building up your token parsers. Consult the hints if you’d like more guidance. A full JSON parser is given in the file JSON.scala in the answers. 9.5 Error reporting So far we haven’t discussed error reporting at all. We’ve focused exclusively on discov- ering a set of primitives that let us express parsers for different grammars. But besides just parsing a grammar, we want to be able to determine how the parser should respond when given unexpected input. Even without knowing what an implementation of Parsers will look like, we can reason abstractly about what information is being specified by a set of combinators. None of the combinators we’ve introduced so far say anything about what error message should be reported in the event of failure or what other information a ParseError should contain. Our existing combinators only specify what the grammar is and what to do with the result if successful. If we were to declare ourselves done and move to implementation at this point, we’d have to make some arbitrary decisions about error reporting and error messages that are unlikely to be universally appropriate. EXERCISE 9.10 Hard: If you haven’t already done so, spend some time discovering a nice set of combi- nators for expressing what errors get reported by a Parser. For each combinator, try to come up with laws specifying what its behavior should be. This is a very open-ended design task. Here are some guiding questions: Given the parser "abra".**(" ".many).**("cadabra"), what sort of error would you like to report given the input "abra cAdabra" (note the capital 'A')? Only something like Expected 'a'? Or Expected "cadabra"? What if you wanted to choose a different error message, like "Magic word incorrect, try again!"? Licensed to Emre Sevinc 161Error reporting Given a or b, if a fails on the input, do we always want to run b, or are there cases where we might not want to? If there are such cases, can you think of addi- tional combinators that would allow the programmer to specify when or should consider the second parser? How do you want to handle reporting the location of errors? Given a or b, if a and b both fail on the input, might we want to support report- ing both errors? And do we always want to report both errors, or do we want to give the programmer a way to specify which of the two errors is reported? We suggest you continue reading once you’re satisfied with your design. The next sec- tion works through a possible design in detail. 9.5.1 A possible design Now that you’ve spent some time coming up with some good error-reporting combi- nators, we’ll work through one possible design. Again, you may have arrived at a dif- ferent design and that’s totally fine. This is just another opportunity to see a worked design process. We’ll progressively introduce our error-reporting combinators. To start, let’s intro- duce an obvious one. None of the primitives so far let us assign an error message to a parser. We can introduce a primitive combinator for this, label: def label[A](msg: String)(p: Parser[A]): Parser[A] The intended meaning of label is that if p fails, its ParseError will somehow incorpo- rate msg. What does this mean exactly? Well, we could just assume type ParseError = String and that the returned ParseError will equal the label. But we’d like our parse error to also tell us where the problem occurred. Let’s tentatively add this to our algebra: case class Location(input: String, offset: Int = 0) { lazy val line = input.slice(0,offset+1).count(_ == '\n') + 1 lazy val col = input.slice(0,offset+1).lastIndexOf('\n') match { case -1 => offset + 1 case lineStart => offset - lineStart } } def errorLocation(e: ParseError): Location def errorMessage(e: ParseError): String Combinators specify information In a typical library design scenario, where we have at least some idea of a concrete representation, we often think of functions in terms of how they will affect this repre- sentation. By starting with the algebra first, we’re forced to think differently—we must think of functions in terms of what information they specify to a possible implemen- tation. The signatures determine what information is given to the implementation, and the implementation is free to use this information however it wants as long as it respects any specified laws. Licensed to Emre Sevinc 162 CHAPTER 9 Parser combinators We’ve picked a concrete representation for Location here that includes the full input, an offset into this input, and the line and column numbers, which can be computed lazily from the full input and offset. We can now say more precisely what we expect from label. In the event of failure with Left(e), errorMessage(e) will equal the mes- sage set by label. This can be specified with a Prop: def labelLaw[A](p: Parser[A], inputs: SGen[String]): Prop = forAll(inputs ** Gen.string) { case (input, msg) => run(label(msg)(p))(input) match { case Left(e) => errorMessage(e) == msg case _=>true } } What about the Location? We’d like for this to be filled in by the Parsers implemen- tation with the location where the error occurred. This notion is still a bit fuzzy—if we have a or b and both parsers fail on the input, which location is reported, and which label(s)? We’ll discuss this in the next section. 9.5.2 Error nesting Is the label combinator sufficient for all our error-reporting needs? Not quite. Let’s look at an example: val p = label("first magic word")("abra") ** " ".many ** label("second magic word")("cadabra") What sort of ParseError would we like to get back from run(p)("abra cAdabra")? (Note the capital A in cAdabra.) The immediate cause is that capital 'A' instead of the expected lowercase 'a'. That error will have a location, and it might be nice to report it somehow. But reporting only that low-level error wouldn’t be very informative, espe- cially if this were part of a large grammar and we were running the parser on a larger input. We have some more context that would be useful to know—the immediate error occurred in the Parser labeled "second magic word". This is certainly helpful information. Ideally, the error message should tell us that while parsing "second magic word", there was an unexpected capital 'A'. That pinpoints the error and gives us the context needed to understand it. Perhaps the top-level parser (p in this case) might be able to provide an even higher-level description of what the parser was doing when it failed ("parsing magic spell", say), which could also be informative. So it seems wrong to assume that one level of error reporting will always be suffi- cient. Let’s therefore provide a way to nest labels: def scope[A](msg: String)(p: Parser[A]): Parser[A] Unlike label, scope doesn’t throw away the label(s) attached to p—it merely adds additional information in the event that p fails. Let’s specify what this means exactly. First, we modify the functions that pull information out of a ParseError. Skip whitespace Licensed to Emre Sevinc 163Error reporting Rather than containing just a single Location and String message, we should get a List[(Location,String)]: case class ParseError(stack: List[(Location,String)]) This is a stack of error messages indicating what the Parser was doing when it failed. We can now specify what scope does—if run(p)(s) is Left(e1), then run(scope(msg) (p)) is Left(e2), where e2.stack.head will be msg and e2.stack.tail will be e1. We can write helper functions later to make constructing and manipulating ParseError values more convenient, and to format them nicely for human consump- tion. For now, we just want to make sure it contains all the relevant information for error reporting, and it seems like ParseError will be sufficient for most purposes. Let’s pick this as our concrete representation and remove the abstract type parameter from Parsers: trait Parsers[Parser[+_]] { def run[A](p: Parser[A])(input: String): Either[ParseError,A] ... } Now we’re giving the Parsers implementation all the information it needs to construct nice, hierarchical errors if it chooses. As users of the Parsers library, we’ll judiciously sprinkle our grammar with label and scope calls that the Parsers implementation can use when constructing parse errors. Note that it would be perfectly reasonable for implementations of Parsers to not use the full power of ParseError and retain only basic information about the cause and location of errors. 9.5.3 Controlling branching and backtracking There’s one last concern regarding error reporting that we need to address. As we just discussed, when we have an error that occurs inside an or combinator, we need some way of determining which error(s) to report. We don’t want to only have a global con- vention for this; we sometimes want to allow the programmer to control this choice. Let’s look at a more concrete motivating example: val spaces = " ".many val p1 = scope("magic spell") { "abra" ** spaces ** "cadabra" } val p2 = scope("gibberish") { "abba" ** spaces ** "babba" } val p = p1 or p2 What ParseError would we like to get back from run(p)("abra cAdabra")? (Again, note the capital A in cAdabra.) Both branches of the or will produce errors on the input. The "gibberish"-labeled parser will report an error due to expecting the first word to be "abba", and the "magic spell" parser will report an error due to the Licensed to Emre Sevinc 164 CHAPTER 9 Parser combinators accidental capitalization in "cAdabra". Which of these errors do we want to report back to the user? In this instance, we happen to want the "magic spell" parse error—after success- fully parsing the "abra" word, we’re committed to the "magic spell" branch of the or, which means if we encounter a parse error, we don’t examine the next branch of the or. In other instances, we may want to allow the parser to consider the next branch of the or. So it appears we need a primitive for letting the programmer indicate when to commit to a particular parsing branch. Recall that we loosely assigned p1 or p2 to mean try running p1 on the input, and then try running p2 on the same input if p1 fails. We can change its meaning to try running p1 on the input, and if it fails in an uncommitted state, try running p2 on the same input; otherwise, report the failure. This is useful for more than just providing good error messages—it also improves efficiency by letting the implementation avoid examining lots of possible parsing branches. One common solution to this problem is to have all parsers commit by default if they examine at least one character to produce a result.12 We then introduce a combinator, attempt, which delays committing to a parse: def attempt[A](p: Parser[A]): Parser[A] It should satisfy something like this:13 attempt(p flatMap (_ => fail)) or p2 == p2 Here fail is a parser that always fails (we could introduce this as a primitive combina- tor if we like). That is, even if p fails midway through examining the input, attempt reverts the commit to that parse and allows p2 to be run. The attempt combinator can be used whenever there’s ambiguity in the grammar and multiple tokens may have to be examined before the ambiguity can be resolved and parsing can commit to a single branch. As an example, we might write this: (attempt("abra" ** spaces ** "abra") ** "cadabra") or ( "abra" ** spaces "cadabra!") Suppose this parser is run on "abra cadabra!"—after parsing the first "abra", we don’t know whether to expect another "abra" (the first branch) or "cadabra!" (the second branch). By wrapping an attempt around "abra" ** spaces ** "abra", we allow the second branch to be considered up until we’ve finished parsing the second "abra", at which point we commit to that branch. 12 See the chapter notes for more discussion of this. 13 This is not quite an equality. Even though we want to run p2 if the attempted parser fails, we may want p2 to somehow incorporate the errors from both branches if it fails. Licensed to Emre Sevinc 165Implementing the algebra EXERCISE 9.11 Can you think of any other primitives that might be useful for letting the programmer specify what error(s) in an or chain get reported? Note that we still haven’t written an implementation of our algebra! But this exercise has been more about making sure our combinators provide a way for users of our library to convey the right information to the implementation. It’s up to the imple- mentation to figure out how to use this information in a way that satisfies the laws we’ve stipulated. 9.6 Implementing the algebra By this point, we’ve fleshed out our algebra and defined a Parser[JSON] in terms of it.14 Aren’t you curious to try running it? Let’s again recall our set of primitives: string(s)—Recognizes and returns a single String regex(s)—Recognizes a regular expression s slice(p)—Returns the portion of input inspected by p if successful label(e)(p)—In the event of failure, replaces the assigned message with e scope(e)(p)—In the event of failure, adds e to the error stack returned by p flatMap(p)(f)—Runs a parser, and then uses its result to select a second parser to run in sequence attempt(p)—Delays committing to p until after it succeeds or(p1,p2)—Chooses between two parsers, first attempting p1, and then p2 if p1 fails in an uncommitted state on the input EXERCISE 9.12 Hard: In the next section, we’ll work through a representation for Parser and imple- ment the Parsers interface using this representation. But before we do that, try to come up with some ideas on your own. This is a very open-ended design task, but the algebra we’ve designed places strong constraints on possible representations. You should be able to come up with a simple, purely functional representation of Parser that can be used to implement the Parsers interface.15 14 You may want to revisit your parser to make use of some of the error-reporting combinators we just discussed in the previous section. 15 Note that if you try running your JSON parser once you have an implementation of Parsers, you may get a stack overflow error. See the end of the next section for a discussion of this. Licensed to Emre Sevinc 166 CHAPTER 9 Parser combinators Your code will likely look something like this: class MyParser[+A](...) { ... } object MyParsers extends Parsers[MyParser] { // implementations of primitives go here } Replace MyParser with whatever data type you use for representing your parsers. When you have something you’re satisfied with, get stuck, or want some more ideas, keep reading. 9.6.1 One possible implementation We’re now going to discuss an implementation of Parsers. Our parsing algebra sup- ports a lot of features. Rather than jumping right to the final representation of Parser, we’ll build it up gradually by inspecting the primitives of the algebra and rea- soning about the information that will be required to support each one. Let’s begin with the string combinator: def string(s: String): Parser[A] We know we need to support the function run: def run[A](p: Parser[A])(input: String): Either[ParseError,A] As a first guess, we can assume that our Parser is simply the implementation of the run function: type Parser[+A] = String => Either[ParseError,A] We could use this to implement the string primitive: def string(s: String): Parser[A] = (input: String) => if (input.startsWith(s)) Right(s) else Left(Location(input).toError("Expected:"+s)) The else branch has to build up a ParseError. These are a little inconvenient to con- struct right now, so we’ve introduced a helper function, toError, on Location: def toError(msg: String): ParseError = ParseError(List((this, msg))) 9.6.2 Sequencing parsers So far, so good. We have a representation for Parser that at least supports string. Let’s move on to sequencing of parsers. Unfortunately, to represent a parser like "abra" ** "cadabra", our existing representation isn’t going to suffice. If the parse of "abra" is successful, then we want to consider those characters consumed and run the Uses toError, defined later, to construct a ParseError Licensed to Emre Sevinc 167Implementing the algebra "cadabra" parser on the remaining characters. So in order to support sequencing, we require a way of letting a Parser indicate how many characters it consumed. Captur- ing this is pretty easy:16 type Parser[+A] = Location => Result[A] trait Result[+A] case class Success[+A](get: A, charsConsumed: Int) extends Result[A] case class Failure(get: ParseError) extends Result[Nothing] We introduced a new type here, Result, rather than just using Either. In the event of success, we return a value of type A as well as the number of characters of input con- sumed, which the caller can use to update the Location state.17 This type is starting to get at the essence of what a Parser is—it’s a kind of state action that can fail, similar to what we built in chapter 6. It receives an input state, and if successful, returns a value as well as enough information to control how the state should be updated. This understanding—that a Parser is just a state action—gives us a way of framing a representation that supports all the fancy combinators and laws we’ve stipulated. We simply consider what each primitive requires our state type to track (just a Location may not be sufficient), and work through the details of how each combinator trans- forms this state. EXERCISE 9.13 Implement string, regex, succeed, and slice for this initial representation of Parser. Note that slice is less efficient than it could be, since it must still construct a value only to discard it. We’ll return to this later. 9.6.3 Labeling parsers Moving down our list of primitives, let’s look at scope next. In the event of failure, we want to push a new message onto the ParseError stack. Let’s introduce a helper func- tion for this on ParseError. We’ll call it push:18 def push(loc: Location, msg: String): ParseError = copy(stack = (loc,msg) :: stack) 16 Recall that Location contains the full input string and an offset into this string. 17 Note that returning an (A,Location) would give Parser the ability to change the input stored in the Location. That’s granting it too much power! 18 The copy method comes for free with any case class. It returns a copy of the object, but with one or more attributes modified. If no new value is specified for a field, it will have the same value as in the original object. Behind the scenes, this just uses the ordinary mechanism for default arguments in Scala. A parser now returns a Result that’s either a success or a failure. In the success case, we return the number of characters consumed by the parser. Licensed to Emre Sevinc 168 CHAPTER 9 Parser combinators With this we can implement scope: def scope[A](msg: String)(p: Parser[A]): Parser[A] = s => p(s).mapError(_.push(s.loc,msg)) The function mapError is defined on Result—it just applies a function to the failing case: def mapError(f: ParseError => ParseError): Result[A] = this match { case Failure(e) => Failure(f(e)) case _=>this } Because we push onto the stack after the inner parser has returned, the bottom of the stack will have more detailed messages that occurred later in parsing. For example, if scope(msg1)(a ** scope(msg2)(b)) fails while parsing b, the first error on the stack will be msg1, followed by whatever errors were generated by a, then msg2, and finally errors generated by b. We can implement label similarly, but instead of pushing onto the error stack, it replaces what’s already there. We can write this again using mapError: def label[A](msg: String)(p: Parser[A]): Parser[A] = s => p(s).mapError(_.label(msg)) We added a helper function to ParseError, also named label. We’ll make a design decision that label trims the error stack, cutting off more detailed messages from inner scopes, using only the most recent location from the bottom of the stack: def label[A](s: String): ParseError = ParseError(latestLoc.map((_,s)).toList) def latestLoc: Option[Location] = latest map (_._1) def latest: Option[(Location,String)] = stack.lastOption EXERCISE 9.14 Revise your implementation of string to use scope and/or label to provide a mean- ingful error message in the event of an error. 9.6.4 Failover and backtracking Let’s now consider or and attempt. Recall what we specified for the expected behav- ior of or: it should run the first parser, and if that fails in an uncommitted state, it should In the event of failure, push msg onto the error stack. Calls a helper method on ParseError, which is also named label. Gets the last element of the stack or None if the stack is empty. Licensed to Emre Sevinc 169Implementing the algebra run the second parser on the same input. We said that consuming at least one charac- ter should result in a committed parse, and that attempt(p) converts committed fail- ures of p to uncommitted failures. We can support the behavior we want by adding one more piece of information to the Failure case of Result—a Boolean value indicating whether the parser failed in a committed state: case class Failure(get: ParseError, isCommitted: Boolean) extends Result[Nothing] The implementation of attempt just cancels the commitment of any failures that occur. It uses a helper function, uncommit, which we can define on Result: def attempt[A](p: Parser[A]): Parser[A] = s => p(s).uncommit def uncommit: Result[A] = this match { case Failure(e,true) => Failure(e,false) case _=>this } Now the implementation of or can simply check the isCommitted flag before running the second parser. In the parser x or y, if x succeeds, then the whole thing succeeds. If x fails in a committed state, we fail early and skip running y. Otherwise, if x fails in an uncommitted state, we run y and ignore the result of x: def or[A](x: Parser[A], y: => Parser[A]): Parser[A] = s=>x(s)match { case Failure(e,false) => y(s) case r=>r } 9.6.5 Context-sensitive parsing Now for the final primitive in our list, flatMap. Recall that flatMap enables context- sensitive parsers by allowing the selection of a second parser to depend on the result of the first parser. The implementation is simple—we advance the location before call- ing the second parser. Again we use a helper function, advanceBy, on Location. There is one subtlety—if the first parser consumes any characters, we ensure that the second parser is committed, using a helper function, addCommit, on ParseError. def flatMap[A,B](f: Parser[A])(g: A => Parser[B]): Parser[B] = s=>f(s)match { case Success(a,n) => g(a)(s.advanceBy(n)) .addCommit(n != 0) .advanceSuccess(n) case e@Failure(_,_) => e } Listing 9.3 Using addCommit to make sure our parser is committed Committed failure or success skips running y. Advance the source location before calling the second parser. Commit if the first parser has consumed any characters. If successful, we increment the number of characters consumed by n, to account for characters already consumed by f. Licensed to Emre Sevinc 170 CHAPTER 9 Parser combinators advanceBy has the obvious implementation. We simply increment the offset: def advanceBy(n: Int): Location = copy(offset = offset+n) Likewise, addCommit, defined on ParseError, is straightforward: def addCommit(isCommitted: Boolean): Result[A] = this match { case Failure(e,c) => Failure(e, c || isCommitted) case _=>this } And finally, advanceSuccess increments the number of consumed characters of a suc- cessful result. We want the total number of characters consumed by flatMap to be the sum of the consumed characters of the parser f and the parser produced by g. We use advanceSuccess on the result of g to ensure this: def advanceSuccess(n: Int): Result[A] = this match { case Success(a,m) => Success(a,n+m) case _=>this } EXERCISE 9.15 Implement the rest of the primitives, including run, using this representation of Parser, and try running your JSON parser on various inputs.19 EXERCISE 9.16 Come up with a nice way of formatting a ParseError for human consumption. There are a lot of choices to make, but a key insight is that we typically want to combine or group labels attached to the same location when presenting the error as a String for display. EXERCISE 9.17 Hard: The slice combinator is still less efficient than it could be. For instance, many(char('a')).slice will still build up a List[Char], only to discard it. Can you think of a way of modifying the Parser representation to make slicing more efficient? 19 You’ll find, unfortunately, that it causes stack overflow for large inputs (for instance, [1,2,3,...10000]). One simple solution to this is to provide a specialized implementation of many that avoids using a stack frame for each element of the list being built up. So long as any combinators that do repetition are defined in terms of many (which they all can be), this solves the problem. See the answers for discussion of more general approaches. If unsuccessful, leave the result alone. Licensed to Emre Sevinc 171Summary EXERCISE 9.18 Some information is lost when we combine parsers with the or combinator. If both parsers fail, we’re only keeping the errors from the second parser. But we might want to show both error messages, or choose the error from whichever branch got furthest without failing. Change the representation of ParseError to keep track of errors that occurred in other branches of the parser. 9.7 Summary In this chapter, we introduced algebraic design, an approach to writing combinator libraries, and we used it to design a parser library and to implement a JSON parser. Along the way, we discovered a number of combinators similar to what we saw in previ- ous chapters, and these were again related by familiar laws. In part 3, we’ll finally understand the nature of the connection between these libraries and learn how to abstract over their common structure. This is the final chapter in part 2. We hope you’ve come away from these chapters with a basic sense of how functional design can proceed, and more importantly, we hope these chapters have motivated you to try your hand at designing your own func- tional libraries, for whatever domains interest you. Functional design isn’t something reserved only for experts—it should be part of the day-to-day work done by functional programmers at all levels of experience. Before you start on part 3, we encourage you to venture beyond this book, write some more functional code, and design some of your own libraries. Have fun, enjoy struggling with design problems that come up, and see what you discover. When you come back, a universe of patterns and abstrac- tions awaits in part 3. Licensed to Emre Sevinc Licensed to Emre Sevinc Part 3 Common structures in functional design We’ve now written a number of libraries using the principles of functional design. In part 2, we saw these principles applied to a few concrete problem domains. By now you should have a good grasp of how to approach a program- ming problem in your own work while striving for compositionality and alge- braic reasoning. Part 3 takes a much wider perspective. We’ll look at the common patterns that arise in functional programming. In part 2, we experimented with various libraries that provided concrete solutions to real-world problems, and now we want to integrate what we’ve learned from our experiments into abstract theo- ries that describe the common structure among those libraries. This kind of abstraction has a direct practical benefit: the elimination of duplicate code. We can capture abstractions as classes, interfaces, and functions that we can refer to in our actual programs. But the primary benefit is conceptual integration. When we recognize common structure among different solutions in different contexts, we unite all of those instances of the structure under a single definition and give it a name. As you gain experience with this, you can look at the general shape of a problem and say, for example: “That looks like a monad!” You’re then already far along in finding the shape of the solution. A secondary benefit is that if other people have developed the same kind of vocabulary, you can communicate your designs to them with extraordinary efficiency. Licensed to Emre Sevinc 174 Common structures in functional design Part 3 won’t be a sequence of meandering journeys in the style of part 2. Instead, we’ll begin each chapter by introducing an abstract concept, give its definition, and then tie it back to what we’ve seen already. The primary goal will be to train you in rec- ognizing patterns when designing your own libraries, and to write code that takes advantage of such patterns. Licensed to Emre Sevinc 175 Monoids By the end of part 2, we were getting comfortable with considering data types in terms of their algebras—that is, the operations they support and the laws that gov- ern those operations. Hopefully you will have noticed that the algebras of very dif- ferent data types tend to share certain patterns in common. In this chapter, we’ll begin identifying these patterns and taking advantage of them. This chapter will be our first introduction to purely algebraic structures. We’ll consider a simple structure, the monoid,1 which is defined only by its algebra. Other than satisfying the same laws, instances of the monoid interface may have little or nothing to do with one another. Nonetheless, we’ll see how this algebraic structure is often all we need to write useful, polymorphic functions. We choose to start with monoids because they’re simple, ubiquitous, and useful. Monoids come up all the time in everyday programming, whether we’re aware of them or not. Working with lists, concatenating strings, or accumulating the results of a loop can often be phrased in terms of monoids. We’ll see how monoids are use- ful in two ways: they facilitate parallel computation by giving us the freedom to break our problem into chunks that can be computed in parallel; and they can be composed to assemble complex calculations from simpler pieces. 10.1 What is a monoid? Let’s consider the algebra of string concatenation. We can add "foo" + "bar" to get "foobar", and the empty string is an identity element for that operation. That is, if we say (s + "") or ("" + s), the result is always s. Furthermore, if we combine three 1 The name monoid comes from mathematics. In category theory, it means a category with one object. This mathematical connection isn’t important for our purposes in this chapter, but see the chapter notes for more information. Licensed to Emre Sevinc 176 CHAPTER 10 Monoids strings by saying (r + s + t), the operation is associative—it doesn’t matter whether we parenthesize it: ((r + s) + t) or (r + (s + t)). The exact same rules govern integer addition. It’s associative, since (x + y) + z is always equal to x + (y + z), and it has an identity element, 0, which “does nothing” when added to another integer. Ditto for multiplication, whose identity element is 1. The Boolean operators && and || are likewise associative, and they have identity elements true and false, respectively. These are just a few simple examples, but algebras like this are virtually every- where. The term for this kind of algebra is monoid. The laws of associativity and iden- tity are collectively called the monoid laws. A monoid consists of the following: Some type A An associative binary operation, op, that takes two values of type A and com- bines them into one: op(op(x,y), z) == op(x, op(y,z)) for any choice of x: A, y: A, z: A A value, zero: A, that is an identity for that operation: op(x, zero) == x and op(zero, x) == x for any x: A We can express this with a Scala trait: trait Monoid[A] { def op(a1: A, a2: A): A def zero: A } An example instance of this trait is the String monoid: val stringMonoid = new Monoid[String] { def op(a1: String, a2: String) = a1 + a2 val zero = "" } List concatenation also forms a monoid: def listMonoid[A] = new Monoid[List[A]] { def op(a1: List[A], a2: List[A]) = a1 ++ a2 val zero = Nil } Satisfies op(op(x,y), z) == op(x, op(y,z)) Satisfies op(x, zero) == x and op(zero, x) == x The purely abstract nature of an algebraic structure Notice that other than satisfying the monoid laws, the various Monoid instances don’t have much to do with each other. The answer to the question “What is a monoid?” is simply that a monoid is a type, together with the monoid operations and a set of laws. A monoid is the algebra, and nothing more. Of course, you may build some other intuition by considering the various concrete instances, but this intuition is necessarily imprecise and nothing guarantees that all monoids you encounter will match your intuition! Licensed to Emre Sevinc 177What is a monoid? EXERCISE 10.1 Give Monoid instances for integer addition and multiplication as well as the Boolean operators. val intAddition: Monoid[Int] val intMultiplication: Monoid[Int] val booleanOr: Monoid[Boolean] val booleanAnd: Monoid[Boolean] EXERCISE 10.2 Give a Monoid instance for combining Option values. def optionMonoid[A]: Monoid[Option[A]] EXERCISE 10.3 A function having the same argument and return type is sometimes called an endofunc- tion.2 Write a monoid for endofunctions. def endoMonoid[A]: Monoid[A => A] EXERCISE 10.4 Use the property-based testing framework we developed in part 2 to implement a property for the monoid laws. Use your property to test the monoids we’ve written. def monoidLaws[A](m: Monoid[A], gen: Gen[A]): Prop 2 The Greek prefix endo- means within, in the sense that an endofunction’s codomain is within its domain. Having versus being a monoid There is a slight terminology mismatch between programmers and mathematicians when they talk about a type being a monoid versus having a monoid instance. As a programmer, it’s tempting to think of the instance of type Monoid[A] as being a monoid. But that’s not accurate terminology. The monoid is actually both things—the type together with the instance satisfying the laws. It’s more accurate to say that the type Licensed to Emre Sevinc 178 CHAPTER 10 Monoids Just what is a monoid, then? It’s simply a type A and an implementation of Monoid[A] that satisfies the laws. Stated tersely, a monoid is a type together with a binary operation (op) over that type, satisfying associativity and having an identity element (zero). What does this buy us? Just like any abstraction, a monoid is useful to the extent that we can write useful generic code assuming only the capabilities provided by the abstraction. Can we write any interesting programs, knowing nothing about a type other than that it forms a monoid? Absolutely! Let’s look at some examples. 10.2 Folding lists with monoids Monoids have an intimate connection with lists. If you look at the signatures of fold- Left and foldRight on List, you might notice something about the argument types: def foldRight[B](z: B)(f: (A, B) => B): B def foldLeft[B](z: B)(f: (B, A) => B): B What happens when A and B are the same type? def foldRight(z: A)(f: (A, A) => A): A def foldLeft(z: A)(f: (A, A) => A): A The components of a monoid fit these argument types like a glove. So if we had a list of Strings, we could simply pass the op and zero of the stringMonoid in order to reduce the list with the monoid and concatenate all the strings: scala> val words = List("Hic", "Est", "Index") words: List[String] = List(Hic, Est, Index) scala> val s = words.foldRight(stringMonoid.zero)(stringMonoid.op) s: String = "HicEstIndex" scala> val t = words.foldLeft(stringMonoid.zero)(stringMonoid.op) t: String = "HicEstIndex" Note that it doesn’t matter if we choose foldLeft or foldRight when folding with a monoid;3 we should get the same result. This is precisely because the laws of associativity and identity hold. A left fold associates operations to the left, whereas a right fold asso- ciates to the right, with the identity element on the left and right respectively: 3 Given that both foldLeft and foldRight have tail-recursive implementations. (continued) type A forms a monoid under the operations defined by the Monoid[A] instance. Less precisely, we might say that “type A is a monoid,” or even that “type A is monoidal.” In any case, the Monoid[A] instance is simply evidence of this fact. This is much the same as saying that the page or screen you’re reading “forms a rect- angle” or “is rectangular.” It’s less accurate to say that it “is a rectangle” (although that still makes sense), but to say that it “has a rectangle” would be strange. Licensed to Emre Sevinc 179Associativity and parallelism words.foldLeft("")(_ + _) == (("" + "Hic") + "Est") + "Index" words.foldRight("")(_ + _) == "Hic" + ("Est" + ("Index" + "")) We can write a general function concatenate that folds a list with a monoid: def concatenate[A](as: List[A], m: Monoid[A]): A = as.foldLeft(m.zero)(m.op) But what if our list has an element type that doesn’t have a Monoid instance? Well, we can always map over the list to turn it into a type that does: def foldMap[A,B](as: List[A], m: Monoid[B])(f: A => B): B EXERCISE 10.5 Implement foldMap. EXERCISE 10.6 Hard: The foldMap function can be implemented using either foldLeft or fold- Right. But you can also write foldLeft and foldRight using foldMap! Try it. 10.3 Associativity and parallelism The fact that a monoid’s operation is associative means we can choose how we fold a data structure like a list. We’ve already seen that operations can be associated to the left or right to reduce a list sequentially with foldLeft or foldRight. But if we have a monoid, we can reduce a list using a balanced fold, which can be more efficient for some operations and also allows for parallelism. As an example, suppose we have a sequence a, b, c, d that we’d like to reduce using some monoid. Folding to the right, the combination of a, b, c, and d would look like this: op(a, op(b, op(c, d))) Folding to the left would look like this: op(op(op(a, b), c), d) But a balanced fold looks like this: op(op(a, b), op(c, d)) Note that the balanced fold allows for parallelism, because the two inner op calls are independent and can be run simultaneously. But beyond that, the more balanced tree structure can be more efficient in cases where the cost of each op is proportional to Licensed to Emre Sevinc 180 CHAPTER 10 Monoids the size of its arguments. For instance, consider the runtime performance of this expression: List("lorem", "ipsum", "dolor", "sit").foldLeft("")(_ + _) At every step of the fold, we’re allocating the full intermediate String only to discard it and allocate a larger string in the next step. Recall that String values are immuta- ble, and that evaluating a + b for strings a and b requires allocating a fresh character array and copying both a and b into this new array. It takes time proportional to a.length + b.length. Here’s a trace of the preceding expression being evaluated: List("lorem", ipsum", "dolor", "sit").foldLeft("")(_ + _) List("ipsum", "dolor", "sit").foldLeft("lorem")(_ + _) List("dolor", "sit").foldLeft("loremipsum")(_ + _) List("sit").foldLeft("loremipsumdolor")(_ + _) List().foldLeft("loremipsumdolorsit")(_ + _) "loremipsumdolorsit" Note the intermediate strings being created and then immediately discarded. A more efficient strategy would be to combine the sequence by halves, which we call a balanced fold—we first construct "loremipsum" and "dolorsit", and then add those together. EXERCISE 10.7 Implement a foldMap for IndexedSeq.4 Your implementation should use the strategy of splitting the sequence in two, recursively processing each half, and then adding the answers together with the monoid. def foldMapV[A,B](v: IndexedSeq[A], m: Monoid[B])(f: A => B): B EXERCISE 10.8 Hard: Also implement a parallel version of foldMap using the library we developed in chapter 7. Hint: Implement par, a combinator to promote Monoid[A] to a Monoid [Par[A]],5 and then use this to implement parFoldMap. import fpinscala.parallelism.Nonblocking._ def par[A](m: Monoid[A]): Monoid[Par[A]] def parFoldMap[A,B](v: IndexedSeq[A], m: Monoid[B])(f: A => B): Par[B] 4 Recall that IndexedSeq is the interface for immutable data structures supporting efficient random access. It also has efficient splitAt and length methods. 5 The ability to “lift” a Monoid into the Par context is something we’ll discuss more generally in chapters 11 and 12. Licensed to Emre Sevinc 181Example: Parallel parsing EXERCISE 10.9 Hard: Use foldMap to detect whether a given IndexedSeq[Int] is ordered. You’ll need to come up with a creative Monoid. 10.4 Example: Parallel parsing As a nontrivial use case, let’s say that we wanted to count the number of words in a String. This is a fairly simple parsing problem. We could scan the string character by character, looking for whitespace and counting up the number of runs of consecutive nonwhitespace characters. Parsing sequentially like that, the parser state could be as simple as tracking whether the last character seen was a whitespace. But imagine doing this not for just a short string, but an enormous text file, possi- bly too big to fit in memory on a single machine. It would be nice if we could work with chunks of the file in parallel. The strategy would be to split the file into manage- able chunks, process several chunks in parallel, and then combine the results. In that case, the parser state needs to be slightly more complicated, and we need to be able to combine intermediate results regardless of whether the section we’re looking at is at the beginning, middle, or end of the file. In other words, we want the combining operation to be associative. To keep things simple and concrete, let’s consider a short string and pretend it’s a large file: "lorem ipsum dolor sit amet, " If we split this string roughly in half, we might split it in the middle of a word. In the case of our string, that would yield "lorem ipsum do" and "lor sit amet, ". When we add up the results of counting the words in these strings, we want to avoid double- counting the word dolor. Clearly, just counting the words as an Int isn’t sufficient. We need to find a data structure that can handle partial results like the half words do and lor, and can track the complete words seen so far, like ipsum, sit, and amet. The partial result of the word count could be represented by an algebraic data type: sealed trait WC case class Stub(chars: String) extends WC case class Part(lStub: String, words: Int, rStub: String) extends WC A Stub is the simplest case, where we haven’t seen any complete words yet. But a Part keeps the number of complete words we’ve seen so far, in words. The value lStub holds any partial word we’ve seen to the left of those words, and rStub holds any par- tial word on the right. Licensed to Emre Sevinc 182 CHAPTER 10 Monoids For example, counting over the string "lorem ipsum do" would result in Part ("lorem", 1, "do") since there’s one certainly complete word, "ipsum". And since there’s no whitespace to the left of lorem or right of do, we can’t be sure if they’re complete words, so we don’t count them yet. Counting over "lor sit amet, " would result in Part("lor", 2, ""). EXERCISE 10.10 Write a monoid instance for WC and make sure that it meets the monoid laws. val wcMonoid: Monoid[WC] EXERCISE 10.11 Use the WC monoid to implement a function that counts words in a String by recur- sively splitting it into substrings and counting the words in those substrings. 6 6 Homomorphism comes from Greek, homo meaning “same” and morphe meaning “shape.” Monoid homomorphisms If you have your law-discovering cap on while reading this chapter, you may notice that there’s a law that holds for some functions between monoids. Take the String con- catenation monoid and the integer addition monoid. If you take the lengths of two strings and add them up, it’s the same as taking the length of the concatenation of those two strings: "foo".length + "bar".length == ("foo" + "bar").length Here, length is a function from String to Int that preserves the monoid structure. Such a function is called a monoid homomorphism.6 A monoid homomorphism f between monoids M and N obeys the following general law for all values x and y: M.op(f(x), f(y)) == f(N.op(x, y)) The same law should hold for the homomorphism from String to WC in the present exercise. This property can be useful when designing your own libraries. If two types that your library uses are monoids, and there exist functions between them, it’s a good idea to think about whether those functions are expected to preserve the monoid structure and to check the monoid homomorphism law with automated tests. Licensed to Emre Sevinc 183Foldable data structures 10.5 Foldable data structures In chapter 3, we implemented the data structures List and Tree, both of which could be folded. In chapter 5, we wrote Stream, a lazy structure that also can be folded much like a List can, and now we’ve just written a fold for IndexedSeq. When we’re writing code that needs to process data contained in one of these structures, we often don’t care about the shape of the structure (whether it’s a tree or a list), or whether it’s lazy or not, or provides efficient random access, and so forth. For example, if we have a structure full of integers and want to calculate their sum, we can use foldRight: ints.foldRight(0)(_ + _) Looking at just this code snippet, we shouldn’t have to care about the type of ints. It could be a Vector, a Stream, or a List, or anything at all with a foldRight method. We can capture this commonality in a trait: trait Foldable[F[_]] { def foldRight[A,B](as: F[A])(z: B)(f: (A,B) => B): B def foldLeft[A,B](as: F[A])(z: B)(f: (B,A) => B): B def foldMap[A,B](as: F[A])(f: A => B)(mb: Monoid[B]): B def concatenate[A](as: F[A])(m: Monoid[A]): A = foldLeft(as)(m.zero)(m.op) } Here we’re abstracting over a type constructor F, much like we did with the Parser type in the previous chapter. We write it as F[_], where the underscore indicates that F is not a type but a type constructor that takes one type argument. Just like functions that take other functions as arguments are called higher-order functions, something like Foldable is a higher-order type constructor or a higher-kinded type.7 7 Just like values and functions have types, types and type constructors have kinds. Scala uses kinds to track how many type arguments a type constructor takes, whether it’s co- or contravariant in those arguments, and what the kinds of those arguments are. Sometimes there will be a homomorphism in both directions between two monoids. If they satisfy a monoid isomorphism (iso- meaning equal), we say that the two monoids are isomorphic. A monoid isomorphism between M and N has two homomor- phisms f and g, where both f andThen g and g andThen f are an identity function. For example, the String and List[Char] monoids with concatenation are isomor- phic. The two Boolean monoids (false, ||) and (true, &&) are also isomorphic, via the ! (negation) function. Licensed to Emre Sevinc 184 CHAPTER 10 Monoids EXERCISE 10.12 Implement Foldable[List], Foldable[IndexedSeq], and Foldable[Stream]. Remember that foldRight, foldLeft, and foldMap can all be implemented in terms of each other, but that might not be the most efficient implementation. EXERCISE 10.13 Recall the binary Tree data type from chapter 3. Implement a Foldable instance for it. sealed trait Tree[+A] case object Leaf[A](value: A) extends Tree[A] case class Branch[A](left: Tree[A], right: Tree[A]) extends Tree[A] EXERCISE 10.14 Write a Foldable[Option] instance. EXERCISE 10.15 Any Foldable structure can be turned into a List. Write this conversion in a generic way: def toList[A](fa: F[A]): List[A] 10.6 Composing monoids The Monoid abstraction in itself is not all that compelling, and with the generalized foldMap it’s only slightly more interesting. The real power of monoids comes from the fact that they compose. This means, for example, that if types A and B are monoids, then the tuple type (A, B) is also a monoid (called their product). EXERCISE 10.16 Prove it. Notice that your implementation of op is obviously associative so long as A.op and B.op are both associative. def productMonoid[A,B](A: Monoid[A], B: Monoid[B]): Monoid[(A,B)] Licensed to Emre Sevinc 185Composing monoids 10.6.1 Assembling more complex monoids Some data structures form interesting monoids as long as the types of the elements they contain also form monoids. For instance, there’s a monoid for merging key-value Maps, as long as the value type is a monoid. def mapMergeMonoid[K,V](V: Monoid[V]): Monoid[Map[K, V]] = new Monoid[Map[K, V]] { def zero = Map[K,V]() def op(a: Map[K, V], b: Map[K, V]) = (a.keySet ++ b.keySet).foldLeft(zero) { (acc,k) => acc.updated(k, V.op(a.getOrElse(k, V.zero), b.getOrElse(k, V.zero))) } } Using this simple combinator, we can assemble more complex monoids fairly easily: scala> val M: Monoid[Map[String, Map[String, Int]]] = | mapMergeMonoid(mapMergeMonoid(intAddition)) M: Monoid[Map[String, Map[String, Int]]] = $anon$1@21dfac82 This allows us to combine nested expressions using the monoid, with no additional programming: scala> val m1 = Map("o1" -> Map("i1" -> 1, "i2" -> 2)) m1: Map[String,Map[String,Int]] = Map(o1 -> Map(i1 -> 1, i2 -> 2)) scala> val m2 = Map("o1" -> Map("i2" -> 3)) m2: Map[String,Map[String,Int]] = Map(o1 -> Map(i2 -> 3)) scala> val m3 = M.op(m1, m2) m3: Map[String,Map[String,Int]] = Map(o1 -> Map(i1 -> 1, i2 -> 5)) EXERCISE 10.17 Write a monoid instance for functions whose results are monoids. def functionMonoid[A,B](B: Monoid[B]): Monoid[A => B] EXERCISE 10.18 A bag is like a set, except that it’s represented by a map that contains one entry per element with that element as the key, and the value under that key is the number of times the element appears in the bag. For example: scala> bag(Vector("a", "rose", "is", "a", "rose")) res0: Map[String,Int] = Map(a -> 2, rose -> 2, is -> 1) Listing 10.1 Merging key-value Maps Licensed to Emre Sevinc 186 CHAPTER 10 Monoids Use monoids to compute a “bag” from an IndexedSeq. def bag[A](as: IndexedSeq[A]): Map[A, Int] 10.6.2 Using composed monoids to fuse traversals The fact that multiple monoids can be composed into one means that we can perform multiple calculations simultaneously when folding a data structure. For example, we can take the length and sum of a list at the same time in order to calculate the mean: scala> val m = productMonoid(intAddition, intAddition) m: Monoid[(Int, Int)] = $anon$1@8ff557a scala> val p = listFoldable.foldMap(List(1,2,3,4))(a => (1, a))(m) p: (Int, Int) = (4, 10) scala> val mean = p._1 / p._2.toDouble mean: Double = 2.5 It can be tedious to assemble monoids by hand using productMonoid and foldMap. Part of the problem is that we’re building up the Monoid separately from the mapping function of foldMap, and we must manually keep these “aligned” as we did here. But we can create a combinator library that makes it much more convenient to assemble these composed monoids and define complex computations that may be parallelized and run in a single pass. Such a library is beyond the scope of this chapter, but see the chapter notes for a brief discussion and links to further material. 10.7 Summary Our goal in part 3 is to get you accustomed to working with more abstract structures, and to develop the ability to recognize them. In this chapter, we introduced one of the simplest purely algebraic abstractions, the monoid. When you start looking for it, you’ll find ample opportunity to exploit the monoidal structure of your own libraries. The associative property enables folding any Foldable data type and gives the flexibil- ity of doing so in parallel. Monoids are also compositional, and you can use them to assemble folds in a declarative and reusable way. Monoid has been our first purely abstract algebra, defined only in terms of its abstract operations and the laws that govern them. We saw how we can still write useful functions that know nothing about their arguments except that their type forms a monoid. This more abstract mode of thinking is something we’ll develop further in the rest of part 3. We’ll consider other purely algebraic interfaces and show how they encapsulate common patterns that we’ve repeated throughout this book. Licensed to Emre Sevinc 187 Monads In the previous chapter, we introduced a simple algebraic structure, the monoid. This was our first instance of a completely abstract, purely algebraic interface, and it led us to think about interfaces in a new way. A useful interface may be defined only by a collection of operations related by laws. In this chapter, we’ll continue this mode of thinking and apply it to the problem of factoring out code duplication across some of the libraries we wrote in parts 1 and 2. We’ll discover two new abstract interfaces, Functor and Monad, and get more general experience with spotting these sorts of abstract structures in our code.1 11.1 Functors: generalizing the map function In parts 1 and 2, we implemented several different combinator libraries. In each case, we proceeded by writing a small set of primitives and then a number of combi- nators defined purely in terms of those primitives. We noted some similarities between derived combinators across the libraries we wrote. For instance, we imple- mented a map function for each data type, to lift a function taking one argument “into the context of” some data type. For Gen, Parser, and Option, the type signa- tures were as follows: def map[A,B](ga: Gen[A])(f: A => B): Gen[B] def map[A,B](pa: Parser[A])(f: A => B): Parser[B] def map[A,B](oa: Option[A])(f: A => B): Option[A] 1 The names functor and monad come from the branch of mathematics called category theory, but it isn’t nec- essary to have any category theory background to follow the content in this chapter. You may be interested in following some of the references in the chapter notes for more information. Licensed to Emre Sevinc 188 CHAPTER 11 Monads These type signatures differ only in the concrete data type (Gen, Parser, or Option). We can capture as a Scala trait the idea of “a data type that implements map”: trait Functor[F[_]] { def map[A,B](fa: F[A])(f: A => B): F[B] } Here we’ve parameterized map on the type constructor, F[_], much like we did with Foldable in the previous chapter.2 Instead of picking a particular F[_], like Gen or Parser, the Functor trait is parametric in the choice of F. Here’s an instance for List: val listFunctor = new Functor[List] { def map[A,B](as: List[A])(f: A => B): List[B] = as map f } We say that a type constructor like List (or Option, or F) is a functor, and the Functor[F] instance constitutes proof that F is in fact a functor. What can we do with this abstraction? As we did in several places throughout this book, we can discover useful functions just by playing with the operations of the inter- face, in a purely algebraic way. You may want to pause here to see what (if any) useful operations you can define only in terms of map. Let’s look at one example. If we have F[(A, B)] where F is a functor, we can “dis- tribute” the F over the pair to get (F[A], F[B]): trait Functor[F[_]] { ... def distribute[A,B](fab: F[(A, B)]): (F[A], F[B]) = (map(fab)(_._1), map(fab)(_._2)) } We wrote this just by following the types, but let’s think about what it means for con- crete data types like List, Gen, Option, and so on. For example, if we distribute a List[(A, B)], we get two lists of the same length, one with all the As and the other with all the Bs. That operation is sometimes called unzip. So we just wrote a generic unzip function that works not just for lists, but for any functor! And when we have an operation on a product like this, we should see if we can con- struct the opposite operation over a sum or coproduct: def codistribute[A,B](e: Either[F[A], F[B]]): F[Either[A, B]] = e match { case Left(fa) => map(fa)(Left(_)) case Right(fb) => map(fb)(Right(_)) } What does codistribute mean for Gen? If we have either a generator for A or a gener- ator for B, we can construct a generator that produces either A or B depending on which generator we actually have. 2 Recall that a type constructor is applied to a type to produce a type. For example, List is a type constructor, not a type. There are no values of type List, but we can apply it to the type Int to produce the type List[Int]. Likewise, Parser can be applied to String to yield Parser[String]. Licensed to Emre Sevinc 189Functors: generalizing the map function We just came up with two really general and potentially useful combinators based purely on the abstract interface of Functor, and we can reuse them for any type that allows an implementation of map. 11.1.1 Functor laws Whenever we create an abstraction like Functor, we should consider not only what abstract methods it should have, but which laws we expect to hold for the implementa- tions. The laws you stipulate for an abstraction are entirely up to you,3 and of course Scala won’t enforce any of these laws. But laws are important for two reasons: Laws help an interface form a new semantic level whose algebra may be rea- soned about independently of the instances. For example, when we take the prod- uct of a Monoid[A] and a Monoid[B] to form a Monoid[(A,B)], the monoid laws let us conclude that the “fused” monoid operation is also associative. We don’t need to know anything about A and B to conclude this. More concretely, we often rely on laws when writing various combinators derived from the functions of some abstract interface like Functor. We’ll see examples of this later. For Functor, we’ll stipulate the familiar law we first introduced in chapter 7 for our Par data type:4 map(x)(a => a) == x In other words, mapping over a structure x with the identity function should itself be an identity. This law is quite natural, and we noticed later in part 2 that this law was sat- isfied by the map functions of other types besides Par. This law (and its corollaries given by parametricity) capture the requirement that map(x) “preserves the structure” of x. Implementations satisfying this law are restricted from doing strange things like throwing exceptions, removing the first element of a List, converting a Some to None, and so on. Only the elements of the structure are modified by map; the shape or struc- ture itself is left intact. Note that this law holds for List, Option, Par, Gen, and most other data types that define map! To give a concrete example of this preservation of structure, we can consider distribute and codistribute, defined earlier. Here are their signatures again: def distribute[A,B](fab: F[(A, B)]): (F[A], F[B]) def codistribute[A,B](e: Either[F[A], F[B]]): F[Either[A, B]] Since we know nothing about F other than that it’s a functor, the law assures us that the returned values will have the same shape as the arguments. If the input to distribute is a list of pairs, the returned pair of lists will be of the same length as the 3 Though if you’re going to borrow the name of some existing mathematical abstraction like functor or monoid, we recommend using the laws already specified by mathematics. 4 This law also comes from the mathematical definition of functor. Licensed to Emre Sevinc 190 CHAPTER 11 Monads input, and corresponding elements will appear in the same order. This kind of alge- braic reasoning can potentially save us a lot of work, since we don’t have to write sepa- rate tests for these properties. 11.2 Monads: generalizing the flatMap and unit functions Functor is just one of many abstractions that we can factor out of our libraries. But Functor isn’t too compelling, as there aren’t many useful operations that can be defined purely in terms of map. Next, we’ll look at a more interesting interface, Monad. Using this interface, we can implement a number of useful operations, once and for all, factoring out what would otherwise be duplicated code. And it comes with laws with which we can reason that our libraries work the way we expect. Recall that for several of the data types in this book so far, we implemented map2 to “lift” a function taking two arguments. For Gen, Parser, and Option, the map2 function could be implemented as follows. def map2[A,B,C]( fa: Gen[A], fb: Gen[B])(f: (A,B) => C): Gen[C] = fa flatMap (a => fb map (b => f(a,b))) def map2[A,B,C]( fa: Parser[A], fb: Parser[B])(f: (A,B) => C): Parser[C] = fa flatMap (a => fb map (b => f(a,b))) def map2[A,B,C]( fa: Option[A], fb: Option[B])(f: (A,B) => C): Option[C] = fa flatMap (a => fb map (b => f(a,b))) These functions have more in common than just the name. In spite of operating on data types that seemingly have nothing to do with one another, the implementations are identical! The only thing that differs is the particular data type being operated on. This confirms what we’ve suspected all along—that these are particular instances of some more general pattern. We should be able to exploit that fact to avoid repeating ourselves. For example, we should be able to write map2 once and for all in such a way that it can be reused for all of these data types. We’ve made the code duplication particularly obvious here by choosing uniform names for our functions, taking the arguments in the same order, and so on. It may be more difficult to spot in your everyday work. But the more libraries you write, the bet- ter you’ll get at identifying patterns that you can factor out into common abstractions. Listing 11.1 Implementing map2 for Gen, Parser, and Option Makes a generator of a random C that runs random generators fa and fb, combining their results with the function f. Makes a parser that produces C by combining the results of parsers fa and fb with the function f. Combines two Options with the function f, if both have a value; otherwise returns None. Licensed to Emre Sevinc 191Monads: generalizing the flatMap and unit functions 11.2.1 The Monad trait What unites Parser, Gen, Par, Option, and many of the other data types we’ve looked at is that they’re monads. Much like we did with Functor and Foldable, we can come up with a Scala trait for Monad that defines map2 and numerous other functions once and for all, rather than having to duplicate their definitions for every concrete data type. In part 2 of this book, we concerned ourselves with individual data types, finding a minimal set of primitive operations from which we could derive a large number of useful combinators. We’ll do the same kind of thing here to refine an abstract interface to a small set of primitives. Let’s start by introducing a new trait, called Mon for now. Since we know we want to eventually define map2, let’s go ahead and add that. trait Mon[F[_]] { def map2[A,B,C](fa: F[A], fb: F[B])(f: (A,B) => C): F[C] = fa flatMap (a => fb map (b => f(a,b))) } Here we’ve just taken the implementation of map2 and changed Parser, Gen, and Option to the polymorphic F of the Mon[F] interface in the signature.5 But in this polymorphic context, this won’t compile! We don’t know anything about F here, so we certainly don’t know how to flatMap or map over an F[A]. What we can do is simply add map and flatMap to the Mon interface and keep them abstract. The syntax for calling these functions changes a bit (we can’t use infix syntax anymore), but the structure is otherwise the same. trait Mon[F[_]] { def map[A,B](fa: F[A])(f: A => B): F[B] def flatMap[A,B](fa: F[A])(f: A => F[B]): F[B] def map2[A,B,C]( fa: F[A], fb: F[B])(f: (A,B) => C): F[C] = flatMap(fa)(a => map(fb)(b => f(a,b))) } This translation was rather mechanical. We just inspected the implementation of map2, and added all the functions it called, map and flatMap, as suitably abstract meth- ods on our interface. This trait will now compile, but before we declare victory and move on to defining instances of Mon[List], Mon[Parser], Mon[Option], and so on, Listing 11.2 Creating a Mon trait for map2 5 Our decision to call the type constructor argument F here was arbitrary. We could have called this argument Foo, w00t, or Blah2, though by convention, we usually give type constructor arguments one-letter uppercase names, such as F, G, and H, or sometimes M and N, or P and Q. Listing 11.3 Adding map and flatMap to our trait This won’t compile, since map and flatMap are undefined in this context. We’re calling the (abstract) functions map and flatMap in the Mon interface. Licensed to Emre Sevinc 192 CHAPTER 11 Monads let’s see if we can refine our set of primitives. Our current set of primitives is map and flatMap, from which we can derive map2. Is flatMap and map a minimal set of primi- tives? Well, the data types that implemented map2 all had a unit, and we know that map can be implemented in terms of flatMap and unit. For example, on Gen: def map[A,B](f: A => B): Gen[B] = flatMap(a => unit(f(a))) So let’s pick unit and flatMap as our minimal set. We’ll unify under a single concept all data types that have these functions defined. The trait is called Monad, it has flat- Map and unit abstract, and it provides default implementations for map and map2. trait Monad[F[_]] extends Functor[F] { def unit[A](a: => A): F[A] def flatMap[A,B](ma: F[A])(f: A => F[B]): F[B] def map[A,B](ma: F[A])(f: A => B): F[B] = flatMap(ma)(a => unit(f(a))) def map2[A,B,C](ma: F[A], mb: F[B])(f: (A, B) => C): F[C] = flatMap(ma)(a => map(mb)(b => f(a, b))) } To tie this back to a concrete data type, we can implement the Monad instance for Gen. object Monad { val genMonad = new Monad[Gen] { def unit[A](a: => A): Gen[A] = Gen.unit(a) def flatMap[A,B](ma: Gen[A])(f: A => Gen[B]): Gen[B] = ma flatMap f } } We only need to implement unit and flatMap, and we get map and map2 at no addi- tional cost. We’ve implemented them once and for all, for any data type for which it’s possible to supply an instance of Monad! But we’re just getting started. There are many more functions that we can implement once and for all in this manner. Listing 11.4 Creating our Monad trait Listing 11.5 Implementing Monad for Gen Since Monad provides a default implementation of map, it can extend Functor. All monads are functors, but not all functors are monads. The name monad We could have called Monad anything at all, like FlatMappable, Unicorn, or Bicycle. But monad is already a perfectly good name in common use. The name comes from category theory, a branch of mathematics that has inspired a lot of functional pro- gramming concepts. The name monad is intentionally similar to monoid, and the two concepts are related in a deep way. See the chapter notes for more information. Licensed to Emre Sevinc 193Monadic combinators EXERCISE 11.1 Write monad instances for Par, Parser, Option, Stream, and List. EXERCISE 11.2 Hard: State looks like it would be a monad too, but it takes two type arguments and you need a type constructor of one argument to implement Monad. Try to implement a State monad, see what issues you run into, and think about possible solutions. We’ll discuss the solution later in this chapter. 11.3 Monadic combinators Now that we have our primitives for monads, we can look back at previous chapters and see if there were some other functions that we implemented for each of our monadic data types. Many of them can be implemented once for all monads, so let’s do that now. EXERCISE 11.3 The sequence and traverse combinators should be pretty familiar to you by now, and your implementations of them from various prior chapters are probably all very simi- lar. Implement them once and for all on Monad[F]. def sequence[A](lma: List[F[A]]): F[List[A]] def traverse[A,B](la: List[A])(f: A => F[B]): F[List[B]] One combinator we saw for Gen and Parser was listOfN, which allowed us to repli- cate a parser or generator n times to get a parser or generator of lists of that length. We can implement this combinator for all monads F by adding it to our Monad trait. We should also give it a more generic name such as replicateM (meaning “replicate in a monad”). EXERCISE 11.4 Implement replicateM. def replicateM[A](n: Int, ma: F[A]): F[List[A]] Licensed to Emre Sevinc 194 CHAPTER 11 Monads EXERCISE 11.5 Think about how replicateM will behave for various choices of F. For example, how does it behave in the List monad? What about Option? Describe in your own words the general meaning of replicateM. There was also a combinator for our Gen data type, product, to take two generators and turn them into a generator of pairs, and we did the same thing for Par computa- tions. In both cases, we implemented product in terms of map2. So we can definitely write it generically for any monad F: def product[A,B](ma: F[A], mb: F[B]): F[(A, B)] = map2(ma, mb)((_, _)) We don’t have to restrict ourselves to combinators that we’ve seen already. It’s impor- tant to play around and see what we find. EXERCISE 11.6 Hard: Here’s an example of a function we haven’t seen before. Implement the function filterM. It’s a bit like filter, except that instead of a function from A => Boolean, we have an A => F[Boolean]. (Replacing various ordinary functions like this with the monadic equivalent often yields interesting results.) Implement this function, and then think about what it means for various data types. def filterM[A](ms: List[A])(f: A => F[Boolean]): F[List[A]] The combinators we’ve seen here are only a small sample of the full library that Monad lets us implement once and for all. We’ll see some more examples in chapter 13. 11.4 Monad laws In this section, we’ll introduce laws to govern our Monad interface.6 Certainly we’d expect the functor laws to also hold for Monad, since a Monad[F]is a Functor[F], but what else do we expect? What laws should constrain flatMap and unit? 11.4.1 The associative law For example, if we wanted to combine three monadic values into one, which two should we combine first? Should it matter? To answer this question, let’s for a moment 6 These laws, once again, come from the concept of monads from category theory, but a background in cate- gory theory isn’t necessary to understand this section. Licensed to Emre Sevinc 195Monad laws take a step down from the abstract level and look at a simple concrete example using the Gen monad. Say we’re testing a product order system and we need to mock up some orders. We might have an Order case class and a generator for that class. case class Order(item: Item, quantity: Int) case class Item(name: String, price: Double) val genOrder: Gen[Order] = for { name <- Gen.stringN(3) price <- Gen.uniform.map(_ * 10) quantity <- Gen.choose(1,100) } yield Order(Item(name, price), quantity) Here we’re generating the Item inline (from name and price), but there might be places where we want to generate an Item separately. So we could pull that into its own generator: val genItem: Gen[Item] = for { name <- Gen.stringN(3) price <- Gen.uniform.map(_ * 10) } yield Item(name, price) Then we can use that in genOrder: val genOrder: Gen[Order] = for { item <- genItem quantity <- Gen.choose(1,100) } yield Order(item, quantity) And that should do exactly the same thing, right? It seems safe to assume that. But not so fast. How can we be sure? It’s not exactly the same code. Let’s expand both implementations of genOrder into calls to map and flatMap to better see what’s going on. In the former case, the translation is straightforward: Gen.nextString.flatMap(name => Gen.nextDouble.flatMap(price => Gen.nextInt.map(quantity => Order(Item(name, price), quantity)))) But the second case looks like this (inlining the call to genItem): Gen.nextString.flatMap(name => Gen.nextInt.map(price => Item(name, price))).flatMap(item => Gen.nextInt.map(quantity => Order(item, quantity) Once we expand them, it’s clear that those two implementations aren’t identical. And yet when we look at the for-comprehension, it seems perfectly reasonable to assume that the two implementations do exactly the same thing. In fact, it would be Listing 11.6 Defining our Order class A random string of length 3 A uniform random Double between 0 and 10 A random Int between 0 and 100 Licensed to Emre Sevinc 196 CHAPTER 11 Monads surprising and weird if they didn’t. It’s because we’re assuming that flatMap obeys an associative law: x.flatMap(f).flatMap(g) == x.flatMap(a => f(a).flatMap(g)) And this law should hold for all values x, f, and g of the appropriate types—not just for Gen but for Parser, Option, and any other monad. 11.4.2 Proving the associative law for a specific monad Let’s prove that this law holds for Option. All we have to do is substitute None or Some(v) for x in the preceding equation and expand both sides of it. We start with the case where x is None, and then both sides of the equal sign are None: None.flatMap(f).flatMap(g) == None.flatMap(a => f(a).flatMap(g)) Since None.flatMap(f) is None for all f, this simplifies to None == None Thus, the law holds if x is None. What about if x is Some(v) for an arbitrary choice of v? In that case, we have x.flatMap(f).flatMap(g) == x.flatMap(a => f(a).flatMap(g)) Some(v).flatMap(f).flatMap(g) == Some(v).flatMap(a => f(a).flatMap(g)) f(v).flatMap(g) == (a => f(a).flatMap(g))(v) f(v).flatMap(g) == f(v).flatMap(g) Thus, the law also holds when x is Some(v) for any v. We’re now done, as we’ve shown that the law holds when x is None or when x is Some, and these are the only two possi- bilities for Option. KLEISLI COMPOSITION: A CLEARER VIEW ON THE ASSOCIATIVE LAW It’s not so easy to see that the law we just discussed is an associative law. Remember the associative law for monoids? That was clear: op(op(x,y), z) == op(x, op(y,z)) But our associative law for monads doesn’t look anything like that! Fortunately, there’s a way we can make the law clearer if we consider not the monadic values of types like F[A], but monadic functions of types like A => F[B]. Functions like that are called Kleisli arrows,7 and they can be composed with one another: def compose[A,B,C](f: A => F[B], g: B => F[C]): A => F[C] 7 Kleisli arrow comes from category theory and is named after the Swiss mathematician Heinrich Kleisli. Original law Substitute Some(v) for x on both sides Apply definition of Some(v).flatMap(...) Simplify function application (a => ..)(v) Licensed to Emre Sevinc 197Monad laws EXERCISE 11.7 Implement the Kleisli composition function compose. We can now state the associative law for monads in a much more symmetric way: compose(compose(f, g), h) == compose(f, compose(g, h)) EXERCISE 11.8 Hard: Implement flatMap in terms of compose. It seems that we’ve found another minimal set of monad combinators: compose and unit. EXERCISE 11.9 Show that the two formulations of the associative law, the one in terms of flatMap and the one in terms of compose, are equivalent. 11.4.3 The identity laws The other monad law is now pretty easy to see. Just like zero was an identity element for append in a monoid, there’s an identity element for compose in a monad. Indeed, that’s exactly what unit is, and that’s why we chose this name for this operation:8 def unit[A](a: => A): F[A] This function has the right type to be passed as an argument to compose.9 The effect should be that anything composed with unit is that same thing. This usually takes the form of two laws, left identity and right identity: compose(f, unit) == f compose(unit, f) == f We can also state these laws in terms of flatMap, but they’re less clear that way: flatMap(x)(unit) == x flatMap(unit(y))(f) == f(y) 8 The name unit is often used in mathematics to mean an identity for some operation. 9 Not quite, since it takes a non-strict A to F[A] (it’s an (=> A) => F[A]), and in Scala this type is different from an ordinary A => F[A]. We’ll ignore this distinction for now, though. Licensed to Emre Sevinc 198 CHAPTER 11 Monads EXERCISE 11.10 Prove that these two statements of the identity laws are equivalent. EXERCISE 11.11 Prove that the identity laws hold for a monad of your choice. EXERCISE 11.12 There’s a third minimal set of monadic combinators: map, unit, and join. Implement join in terms of flatMap. def join[A](mma: F[F[A]]): F[A] EXERCISE 11.13 Implement either flatMap or compose in terms of join and map. EXERCISE 11.14 Restate the monad laws to mention only join, map, and unit. EXERCISE 11.15 Write down an explanation, in your own words, of what the associative law means for Par and Parser. EXERCISE 11.16 Explain in your own words what the identity laws are stating in concrete terms for Gen and List. 11.5 Just what is a monad? Let’s now take a wider perspective. There’s something unusual about the Monad inter- face. The data types for which we’ve given monad instances don’t seem to have much Licensed to Emre Sevinc 199Just what is a monad? to do with each other. Yes, Monad factors out code duplication among them, but what is a monad exactly? What does “monad” mean? You may be used to thinking of interfaces as providing a relatively complete API for an abstract data type, merely abstracting over the specific representation. After all, a singly linked list and an array-based list may be implemented differently behind the scenes, but they’ll share a common interface in terms of which a lot of useful and con- crete application code can be written. Monad, like Monoid, is a more abstract, purely algebraic interface. The Monad combinators are often just a small fragment of the full API for a given data type that happens to be a monad. So Monad doesn’t generalize one type or another; rather, many vastly different data types can satisfy the Monad interface and laws. We’ve seen three minimal sets of primitive Monad combinators, and instances of Monad will have to provide implementations of one of these sets: unit and flatMap unit and compose unit, map, and join And we know that there are two monad laws to be satisfied, associativity and identity, that can be formulated in various ways. So we can state plainly what a monad is: A monad is an implementation of one of the minimal sets of monadic combinators, satisfying the laws of associativity and identity. That’s a perfectly respectable, precise, and terse definition. And if we’re being pre- cise, this is the only correct definition. A monad is precisely defined by its operations and laws; no more, no less. But it’s a little unsatisfying. It doesn’t say much about what it implies—what a monad means. The problem is that it’s a self-contained definition. Even if you’re a beginning programmer, you have by now obtained a vast amount of knowledge related to programming, and this definition integrates with none of that. In order to really understand what’s going on with monads, try to think about monads in terms of things you already know and connect them to a wider context. To develop some intuition for what monads mean, let’s look at another couple of monads and compare their behavior. 11.5.1 The identity monad To distill monads to their essentials, let’s look at the simplest interesting specimen, the identity monad, given by the following type: case class Id[A](value: A) EXERCISE 11.17 Implement map and flatMap as methods on this class, and give an implementation for Monad[Id]. Licensed to Emre Sevinc 200 CHAPTER 11 Monads Now, Id is just a simple wrapper. It doesn’t really add anything. Applying Id to A is an identity since the wrapped type and the unwrapped type are totally isomorphic (we can go from one to the other and back again without any loss of information). But what is the meaning of the identity monad? Let’s try using it in the REPL: scala> Id("Hello, ") flatMap (a => | Id("monad!") flatMap (b => | Id(a + b))) res0: Id[java.lang.String] = Id(Hello, monad!) When we write the exact same thing with a for-comprehension, it might be clearer: scala> for { | a <- Id("Hello, ") | b <- Id("monad!") | } yielda+b res1: Id[java.lang.String] = Id(Hello, monad!) So what is the action of flatMap for the identity monad? It’s simply variable substitu- tion. The variables a and b get bound to "Hello, " and "monad!", respectively, and then substituted into the expression a + b. We could have written the same thing without the Id wrapper, using just Scala’s own variables: scala> val a = "Hello, " a: java.lang.String = "Hello, " scala> val b = "monad!" b: java.lang.String = monad! scala> a + b res2: java.lang.String = Hello, monad! Besides the Id wrapper, there’s no difference. So now we have at least a partial answer to the question of what monads mean. We could say that monads provide a context for introducing and binding variables, and performing variable substitution. Let’s see if we can get the rest of the answer. 11.5.2 The State monad and partial type application Look back at the discussion of the State data type in chapter 6. Recall that we imple- mented some combinators for State, including map and flatMap. case class State[S, A](run: S => (A, S)) { def map[B](f: A => B): State[S, B] = State(s => { val (a, s1) = run(s) (f(a), s1) }) def flatMap[B](f: A => State[S, B]): State[S, B] = State(s => { Listing 11.7 Revisiting our State data type Licensed to Emre Sevinc 201Just what is a monad? val (a, s1) = run(s) f(a).run(s1) }) } It looks like State definitely fits the profile for being a monad. But its type constructor takes two type arguments, and Monad requires a type constructor of one argument, so we can’t just say Monad[State]. But if we choose some particular S, then we have something like State[S, _], which is the kind of thing expected by Monad. So State doesn’t just have one monad instance but a whole family of them, one for each choice of S. We’d like to be able to partially apply State to where the S type argument is fixed to be some concrete type. This is much like how we might partially apply a function, except at the type level. For example, we can create an IntState type constructor, which is an alias for State with its first type argument fixed to be Int: type IntState[A] = State[Int, A] And IntState is exactly the kind of thing that we can build a Monad for: object IntStateMonad extends Monad[IntState] { def unit[A](a: => A): IntState[A] = State(s => (a, s)) def flatMap[A,B](st: IntState[A])(f: A => IntState[B]): IntState[B] = st flatMap f } Of course, it would be really repetitive if we had to manually write a separate Monad instance for each specific state type. Unfortunately, Scala doesn’t allow us to use underscore syntax to simply say State[Int, _] to create an anonymous type construc- tor like we create anonymous functions. But instead we can use something similar to lambda syntax at the type level. For example, we could have declared IntState directly inline like this: object IntStateMonad extends Monad[({type IntState[A] = State[Int, A]})#IntState] { ... } This syntax can be jarring when you first see it. But all we’re doing is declaring an anonymous type within parentheses. This anonymous type has, as one of its members, the type alias IntState, which looks just like before. Outside the parentheses we’re then accessing its IntState member with the # syntax. Just like we can use a dot (.) to access a member of an object at the value level, we can use the # symbol to access a type member (see the “Type Projection” section of the Scala Language Specification: http://mng.bz/u70U). A type constructor declared inline like this is often called a type lambda in Scala. We can use this trick to partially apply the State type constructor and declare a State- Monad trait. An instance of StateMonad[S] is then a monad instance for the given state type S: Licensed to Emre Sevinc 202 CHAPTER 11 Monads def stateMonad[S] = new Monad[({type f[x] = State[S,x]})#f] { def unit[A](a: => A): State[S,A] = State(s => (a, s)) def flatMap[A,B](st: State[S,A])(f: A => State[S,B]): State[S,B] = st flatMap f } Again, just by giving implementations of unit and flatMap, we get implementations of all the other monadic combinators for free. EXERCISE 11.18 Now that we have a State monad, you should try it out to see how it behaves. What is the meaning of replicateM in the State monad? How does map2 behave? What about sequence? Let’s now look at the difference between the Id monad and the State monad. Remember that the primitive operations on State (besides the monadic operations unit and flatMap) are that we can read the current state with getState and we can set a new state with setState: def getState[S]: State[S, S] def setState[S](s: => S): State[S, Unit] Remember that we also discovered that these combinators constitute a minimal set of primitive operations for State. So together with the monadic primitives (unit and flatMap) they completely specify everything that we can do with the State data type. This is true in general for monads—they all have unit and flatMap, and each monad brings its own set of additional primitive operations that are specific to it. EXERCISE 11.19 What laws do you expect to mutually hold for getState, setState, unit, and flatMap? What does this tell us about the meaning of the State monad? Let’s study a simple example. The details of this code aren’t too important, but notice the use of getState and setState in the for block. val F = stateMonad[Int] def zipWithIndex[A](as: List[A]): List[(Int,A)] = as.foldLeft(F.unit(List[(Int, A)]()))((acc,a) => for { Listing 11.8 Getting and setting state with a for-comprehension The choice of the name f here is arbitrary. Licensed to Emre Sevinc 203Just what is a monad? xs <- acc n <- getState _ <- setState(n + 1) } yield (n, a) :: xs).run(0)._1.reverse This function numbers all the elements in a list using a State action. It keeps a state that’s an Int, which is incremented at each step. We run the whole composite state action starting from 0. We then reverse the result since we constructed it in reverse order.10 Note what’s going on with getState and setState in the for-comprehension. We’re obviously getting variable binding just like in the Id monad—we’re binding the value of each successive state action (getState, acc, and then setState) to vari- ables. But there’s more going on, literally between the lines. At each line in the for- comprehension, the implementation of flatMap is making sure that the current state is available to getState, and that the new state gets propagated to all actions that fol- low a setState. What does the difference between the action of Id and the action of State tell us about monads in general? We can see that a chain of flatMap calls (or an equivalent for-comprehension) is like an imperative program with statements that assign to vari- ables, and the monad specifies what occurs at statement boundaries. For example, with Id, nothing at all occurs except unwrapping and rewrapping in the Id constructor. With State, the most current state gets passed from one statement to the next. With the Option monad, a statement may return None and terminate the program. With the List monad, a statement may return many results, which causes statements that follow it to potentially run multiple times, once for each result. The Monad contract doesn’t specify what is happening between the lines, only that whatever is happening satisfies the laws of associativity and identity. EXERCISE 11.20 Hard: To cement your understanding of monads, give a monad instance for the follow- ing type, and explain what it means. What are its primitive operations? What is the action of flatMap? What meaning does it give to monadic functions like sequence, join, and replicateM? What meaning does it give to the monad laws?11 case class Reader[R, A](run: R => A) object Reader { def readerMonad[R] = new Monad[({type f[x] = Reader[R,x]})#f] { def unit[A](a: => A): Reader[R,A] def flatMap[A,B](st: Reader[R,A])(f: A => Reader[R,B]): Reader[R,B] } } 10 This is asymptotically faster than appending to the list in the loop. 11 See the chapter notes for further discussion of this data type. Licensed to Emre Sevinc 204 CHAPTER 11 Monads 11.6 Summary In this chapter, we took a pattern that we’ve seen repeated throughout the book and we unified it under a single concept: monad. This allowed us to write a number of combinators once and for all, for many different data types that at first glance don’t seem to have anything in common. We discussed laws that they all satisfy, the monad laws, from various perspectives, and we developed some insight into what it all means. An abstract topic like this can’t be fully understood all at once. It requires an itera- tive approach where you keep revisiting the topic from different perspectives. When you discover new monads or new applications of them, or see them appear in a new context, you’ll inevitably gain new insight. And each time it happens, you might think to yourself, “OK, I thought I understood monads before, but now I really get it.” Licensed to Emre Sevinc 205 Applicative and traversable functors In the previous chapter on monads, we saw how a lot of the functions we’ve been writing for different combinator libraries can be expressed in terms of a single interface, Monad. Monads provide a powerful interface, as evidenced by the fact that we can use flatMap to essentially write imperative programs in a purely func- tional way. In this chapter, we’ll learn about a related abstraction, applicative functors, which are less powerful than monads, but more general (and hence more common). The process of arriving at applicative functors will also provide some insight into how to discover such abstractions, and we’ll use some of these ideas to uncover another useful abstraction, traversable functors. It may take some time for the full significance and usefulness of these abstractions to sink in, but you’ll see them popping up again and again in your daily work with FP if you pay attention. 12.1 Generalizing monads By now we’ve seen various operations, like sequence and traverse, implemented many times for different monads, and in the last chapter we generalized the imple- mentations to work for any monad F: def sequence[A](lfa: List[F[A]]): F[List[A]] traverse(lfa)(fa => fa) def traverse[A,B](as: List[A])(f: A => F[B]): F[List[B]] as.foldRight(unit(List[B]()))((a, mbs) => map2(f(a), mbs)(_ :: _)) Licensed to Emre Sevinc 206 CHAPTER 12 Applicative and traversable functors Here, the implementation of traverse is using map2 and unit, and we’ve seen that map2 can be implemented in terms of flatMap: def map2[A,B,C](ma: F[A], mb: F[B])(f: (A,B) => C): F[C] = flatMap(ma)(a => map(mb)(b => f(a,b))) What you may not have noticed is that a large number of the useful combinators on Monad can be defined using only unit and map2. The traverse combinator is one example—it doesn’t call flatMap directly and is therefore agnostic to whether map2 is primitive or derived. Furthermore, for many data types, map2 can be implemented directly, without using flatMap. All this suggests a variation on Monad—the Monad interface has flatMap and unit as primitives, and derives map2, but we can obtain a different abstraction by letting unit and map2 be the primitives. We’ll see that this new abstraction, called an applicative functor, is less powerful than a monad, but we’ll also see that limitations come with benefits. 12.2 The Applicative trait Applicative functors can be captured by a new interface, Applicative, in which map2 and unit are primitives. trait Applicative[F[_]] extends Functor[F] { // primitive combinators def map2[A,B,C](fa: F[A], fb: F[B])(f: (A, B) => C): F[C] def unit[A](a: => A): F[A] // derived combinators def map[B](fa: F[A])(f: A => B): F[B] = map2(fa, unit(()))((a, _) => f(a)) def traverse[A,B](as: List[A])(f: A => F[B]): F[List[B]] as.foldRight(unit(List[B]()))((a, fbs) => map2(f(a), fbs)(_ :: _)) } This establishes that all applicatives are functors. We implement map in terms of map2 and unit, as we’ve done before for particular data types. The implementation is sug- gestive of laws for Applicative that we’ll examine later, since we expect this imple- mentation of map to preserve structure as dictated by the Functor laws. Note that the implementation of traverse is unchanged. We can similarly move other combinators into Applicative that don’t depend directly on flatMap or join. EXERCISE 12.1 Transplant the implementations of as many combinators as you can from Monad to Applicative, using only map2 and unit, or methods implemented in terms of them. Listing 12.1 Creating the Applicative interface We can implement map in terms of unit and map2. Recall () is the sole value of type Unit, so unit(()) is calling unit with the dummy value ().Definition of traverse is identical. Licensed to Emre Sevinc 207The Applicative trait def sequence[A](fas: List[F[A]]): F[List[A]] def replicateM[A](n: Int, fa: F[A]): F[List[A]] def product[A,B](fa: F[A], fb: F[A]): F[(A,B)] EXERCISE 12.2 Hard: The name applicative comes from the fact that we can formulate the Applicative interface using an alternate set of primitives, unit and the function apply, rather than unit and map2. Show that this formulation is equivalent in expressiveness by defining map2 and map in terms of unit and apply. Also establish that apply can be imple- mented in terms of map2 and unit. trait Applicative[F[_]] extends Functor[F] { def apply[A,B](fab: F[A => B])(fa: F[A]): F[B] def unit[A](a: => A): F[A] def map[A,B](fa: F[A])(f: A => B): F[B] def map2[A,B,C](fa: F[A], fb: F[B])(f: (A,B) => C): F[A] } EXERCISE 12.3 The apply method is useful for implementing map3, map4, and so on, and the pattern is straightforward. Implement map3 and map4 using only unit, apply, and the curried method available on functions.1 def map3[A,B,C,D](fa: F[A], fb: F[B], fc: F[C])(f: (A, B, C) => D): F[D] def map4[A,B,C,D,E](fa: F[A], fb: F[B], fc: F[C], fd: F[D])(f: (A, B, C, D) => E): F[E] Furthermore, we can now make Monad[F] a subtype of Applicative[F] by providing the default implementation of map2 in terms of flatMap. This tells us that all monads are applicative functors, and we don’t need to provide separate Applicative instances for all our data types that are already monads. 1 Recall that given f: (A,B) => C, f.curried has type A => B => C. A curried method exists for functions of any arity in Scala. Define in terms of map2 and unit. Define in terms of apply and unit. Licensed to Emre Sevinc 208 CHAPTER 12 Applicative and traversable functors trait Monad[F[_]] extends Applicative[F] { def flatMap[A,B](fa: F[A])(f: A => F[B]): F[B] = join(map(fa)(f)) def join[A](ffa: F[F[A]]): F[A] = flatMap(ffa)(fa => fa) def compose[A,B,C](f: A => F[B], g: B => F[C]): A => F[C] = a => flatMap(f(a))(g) def map[B](fa: F[A])(f: A => B): F[B] = flatMap(fa)((a: A) => unit(f(a))) def map2[A,B,C](fa: F[A], fb: F[B])(f: (A, B) => C): F[C] = flatMap(fa)(a => map(fb)(b => f(a,b))) } So far, we’ve been just rearranging the functions of our API and following the type sig- natures. Let’s take a step back to understand the difference in expressiveness between Monad and Applicative and what it all means. 12.3 The difference between monads and applicative functors In the last chapter, we noted there were several minimal sets of operations that defined a Monad: unit and flatMap unit and compose unit, map, and join Are the Applicative operations unit and map2 yet another minimal set of operations for monads? No. There are monadic combinators such as join and flatMap that can’t be implemented with just map2 and unit. To see convincing proof of this, take a look at join: def join[A](f: F[F[A]]): F[A] Just reasoning algebraically, we can see that unit and map2 have no hope of imple- menting this function. The join function “removes a layer” of F. But the unit func- tion only lets us add an F layer, and map2 lets us apply a function within F but does no flattening of layers. By the same argument, we can see that Applicative has no means of implementing flatMap either. So Monad is clearly adding some extra capabilities beyond Applicative. But what exactly? Let’s look at some concrete examples. Listing 12.2 Making Monad a subtype of Applicative A minimal implementation of Monad must implement unit and override either flatMap or join and map. Licensed to Emre Sevinc 209The difference between monads and applicative functors 12.3.1 The Option applicative versus the Option monad Suppose we’re using Option to work with the results of lookups in two Map objects. If we simply need to combine the results from two (independent) lookups, map2 is fine. val F: Applicative[Option] = ... val depts: Map[String,String] = ... val salaries: Map[String,Double] = ... val o: Option[String] = F.map2(depts.get("Alice"), salaries.get("Alice"))( (dept, salary) => s"Alice in $dept makes $salary per year" ) Here we’re doing two lookups, but they’re independent and we merely want to com- bine their results within the Option context. If we want the result of one lookup to affect what lookup we do next, then we need flatMap or join, as the following listing shows. val idsByName: Map[String,Int] val depts: Map[Int,String] = ... val salaries: Map[Int,Double] = ... val o: Option[String] = idsByName.get("Bob").flatMap { id => F.map2(depts.get(id), salaries.get(id))( (dept, salary) => s"Bob in $dept makes $salary per year" } Here depts is a Map[Int,String] indexed by employee ID, which is an Int. If we want to print out Bob’s department and salary, we need to first resolve Bob’s name to his ID, and then use this ID to do lookups in depts and salaries. We might say that with Applicative, the structure of our computation is fixed; with Monad, the results of pre- vious computations may influence what computations to run next. Listing 12.3 Combining results with the Option applicative Listing 12.4 Combining results with the Option monad Department, indexed by employee name.Salaries, indexed by employee name. String interpolation substitutes values for dept and salary. Employee ID, indexed by employee name. Department, indexed by employee ID. Salaries, indexed by employee ID. Look up Bob’s ID; then use result to do further lookups. “Effects” in FP Functional programmers often informally call type constructors like Par, Option, List, Parser, Gen, and so on effects. This usage is distinct from the term side effect, which implies some violation of referential transparency. These types are called effects because they augment ordinary values with “extra” capabilities. (Par adds the ability to define parallel computation, Option adds the possibility of failure, and so on.) Related to this usage of effects, we sometimes use the terms monadic effects or applicative effects to mean types with an associated Monad or Applicative instance. Licensed to Emre Sevinc 210 CHAPTER 12 Applicative and traversable functors 12.3.2 The Parser applicative versus the Parser monad Let’s look at one more example. Suppose we’re parsing a file of comma-separated val- ues with two columns: date and temperature. Here’s an example file: 1/1/2010, 25 2/1/2010, 28 3/1/2010, 42 4/1/2010, 53 ... If we know ahead of time the file will have the date and temperature columns in that order, we can just encode this order in the Parser we construct: case class Row(date: Date, temperature: Double) val F: Applicative[Parser] = ... val d: Parser[Date] = ... val temp: Parser[Double] = ... val row: Parser[Row] = F.map2(d, temp)(Row(_, _)) val rows: Parser[List[Row]] = row.sep("\n") If we don’t know the order of the columns and need to extract this information from the header, then we need flatMap. Here’s an example file where the columns happen to be in the opposite order: # Temperature, Date 25, 1/1/2010 28, 2/1/2010 42, 3/1/2010 53, 4/1/2010 ... To parse this format, where we must dynamically choose our Row parser based on first parsing the header (the first line starting with #), we need flatMap: case class Row(date: Date, temperature: Double) val F: Monad[Parser] = ... val d: Parser[Date] = ... val temp: Parser[Double] = ... val header: Parser[Parser[Row]] = ... val rows: Parser[List[Row]] = F.flatMap (header) { row => row.sep("\n") } Here we’re parsing the header, which gives us a Parser[Row] as its result. We then use this parser to parse the subsequent rows. Since we don’t know the order of the col- umns up front, we’re selecting our Row parser dynamically, based on the result of pars- ing the header. There are many ways to state the distinction between Applicative and Monad. Of course, the type signatures tell us all we really need to know and we can understand Licensed to Emre Sevinc 211The advantages of applicative functors the difference between the interfaces algebraically. But here are a few other common ways of stating the difference: Applicative computations have fixed structure and simply sequence effects, whereas monadic computations may choose structure dynamically, based on the result of previous effects. Applicative constructs context-free computations, while Monad allows for context sensitivity.2 Monad makes effects first class; they may be generated at “interpretation” time, rather than chosen ahead of time by the program. We saw this in our Parser example, where we generated our Parser[Row]as part of the act of parsing, and used this Parser[Row] for subsequent parsing. 12.4 The advantages of applicative functors The Applicative interface is important for a few reasons: In general, it’s preferable to implement combinators like traverse using as few assumptions as possible. It’s better to assume that a data type can provide map2 than flatMap. Otherwise we’d have to write a new traverse every time we encountered a type that’s Applicative but not a Monad! We’ll look at examples of such types next. Because Applicative is “weaker” than Monad, this gives the interpreter of applica- tive effects more flexibility. To take just one example, consider parsing. If we describe a parser without resorting to flatMap, this implies that the structure of our grammar is determined before we begin parsing. Therefore, our inter- preter or runner of parsers has more information about what it’ll be doing up front and is free to make additional assumptions and possibly use a more effi- cient implementation strategy for running the parser, based on this known structure. Adding flatMap is powerful, but it means we’re generating our pars- ers dynamically, so the interpreter may be more limited in what it can do. Power comes at a cost. See the chapter notes for more discussion of this issue. Applicative functors compose, whereas monads (in general) don’t. We’ll see how this works later. 12.4.1 Not all applicative functors are monads Let’s look at two examples of data types that are applicative functors but not monads. These are certainly not the only examples. If you do more functional programming, you’ll undoubtedly discover or create lots of data types that are applicative but not monadic.3 2 For example, a monadic parser allows for context-sensitive grammars while an applicative parser can only han- dle context-free grammars. 3 Monadic is the adjective form of monad. Licensed to Emre Sevinc 212 CHAPTER 12 Applicative and traversable functors THE APPLICATIVE FOR STREAMS The first example we’ll look at is (possibly infinite) streams. We can define map2 and unit for these streams, but not flatMap: val streamApplicative = new Applicative[Stream] { def unit[A](a: => A): Stream[A] = Stream.continually(a) def map2[A,B,C](a: Stream[A], b: Stream[B])( f: (A,B) => C): Stream[C] = a zip b map f.tupled } The idea behind this Applicative is to combine corresponding elements via zipping. EXERCISE 12.4 Hard: What is the meaning of streamApplicative.sequence? Specializing the signa- ture of sequence to Stream, we have this: def sequence[A](a: List[Stream[A]]): Stream[List[A]] VALIDATION: AN EITHER VARIANT THAT ACCUMULATES ERRORS In chapter 4, we looked at the Either data type and considered the question of how such a data type would have to be modified to allow us to report multiple errors. For a concrete example, think of validating a web form submission. Only reporting the first error means the user would have to repeatedly submit the form and fix one error at a time. This is the situation with Either if we use it monadically. First, let’s actually write the monad for the partially applied Either type. EXERCISE 12.5 Write a monad instance for Either. def eitherMonad[E]: Monad[({type f[x] = Either[E, x]})#f] Now consider what happens in a sequence of flatMap calls like the following, where each of the functions validName, validBirthdate, and validPhone has type Either[String, T] for a given type T: The infinite, constant stream. Combine elements pointwise. Licensed to Emre Sevinc 213The advantages of applicative functors validName(field1) flatMap (f1 => validBirthdate(field2) flatMap (f2 => validPhone(field3) map (f3 => WebForm(f1, f2, f3)) If validName fails with an error, then validBirthdate and validPhone won’t even run. The computation with flatMap inherently establishes a linear chain of depen- dencies. The variable f1 will never be bound to anything unless validName succeeds. Now think of doing the same thing with map3: map3( validName(field1), validBirthdate(field2), validPhone(field3))( WebForm(_,_,_)) Here, no dependency is implied between the three expressions passed to map3, and in principle we can imagine collecting any errors from each Either into a List. But if we use the Either monad, its implementation of map3 in terms of flatMap will halt after the first error. Let’s invent a new data type, Validation, that is much like Either except that it can explicitly handle more than one error: sealed trait Validation[+E, +A] case class Failure[E](head: E, tail: Vector[E] = Vector()) extends Validation[E, Nothing] case class Success[A](a: A) extends Validation[Nothing, A] EXERCISE 12.6 Write an Applicative instance for Validation that accumulates errors in Failure. Note that in the case of Failure there’s always at least one error, stored in head. The rest of the errors accumulate in the tail. To continue the example, consider a web form that requires a name, a birth date, and a phone number: case class WebForm(name: String, birthdate: Date, phoneNumber: String) This data will likely be collected from the user as strings, and we must make sure that the data meets a certain specification. If it doesn’t, we must give a list of errors to the user indicating how to fix the problem. The specification might say that name can’t be empty, that birthdate must be in the form "yyyy-MM-dd", and that phoneNumber must contain exactly 10 digits. Licensed to Emre Sevinc 214 CHAPTER 12 Applicative and traversable functors def validName(name: String): Validation[String, String] = if (name != "") Success(name) else Failure("Name cannot be empty") def validBirthdate(birthdate: String): Validation[String, Date] = try { import java.text._ Success((new SimpleDateFormat("yyyy-MM-dd")).parse(birthdate)) } catch { Failure("Birthdate must be in the form yyyy-MM-dd") } def validPhone(phoneNumber: String): Validation[String, String] = if (phoneNumber.matches("[0-9]{10}")) Success(phoneNumber) else Failure("Phone number must be 10 digits") And to validate an entire web form, we can simply lift the WebForm constructor with map3: def validWebForm(name: String, birthdate: String, phone: String): Validation[String, WebForm] = map3( validName(name), validBirthdate(birthdate), validPhone(phone))( WebForm(_,_,_)) If any or all of the functions produce Failure, the whole validWebForm method will return all of those failures combined. 12.5 The applicative laws This section walks through the laws for applicative functors.4 For each of these laws, you may want to verify that they’re satisfied by some of the data types we’ve been work- ing with so far (an easy one to verify is Option). 12.5.1 Left and right identity What sort of laws should we expect applicative functors to obey? Well, we should defi- nitely expect them to obey the functor laws: map(v)(id) == v map(map(v)(g))(f) == map(v)(f compose g) This implies some other laws for applicative functors because of how we’ve imple- mented map in terms of map2 and unit. Recall the definition of map: Listing 12.5 Validating user input in a web form 4 There are various other ways of presenting the laws for Applicative. See the chapter notes for more information. Licensed to Emre Sevinc 215The applicative laws def map[B](fa: F[A])(f: A => B): F[B] = map2(fa, unit(()))((a, _) => f(a)) Of course, there’s something rather arbitrary about this definition—we could have just as easily put the unit on the left side of the call to map2: def map[B](fa: F[A])(f: A => B): F[B] = map2(unit(()), fa)((_, a) => f(a)) The first two laws for Applicative might be summarized by saying that both these implementations of map respect the functor laws. In other words, map2 of some fa: F[A] with unit preserves the structure of fa. We’ll call these the left and right identity laws (shown here in the first and second lines of code, respectively): map2(unit(()), fa)((_,a) => a) == fa map2(fa, unit(()))((a,_) => a) == fa 12.5.2 Associativity To see the next law, associativity, let’s look at the signature of map3: def map3[A,B,C,D](fa: F[A], fb: F[B], fc: F[C])(f: (A, B, C) => D): F[D] We can implement map3 using apply and unit, but let’s think about how we might define it in terms of map2. We have to combine our effects two at a time, and we seem to have two choices—we can combine fa and fb, and then combine the result with fc. Or we could associate the operation the other way, grouping fb and fc together and combining the result with fa. The associativity law for applicative functors tells us that we should get the same result either way. This should remind you of the associativity laws for monoids and monads: op(a, op(b, c)) == op(op(a, b), c) compose(f, op(g, h)) == compose(compose(f, g), h) The associativity law for applicative functors is the same general idea. If we didn’t have this law, we’d need two versions of map3, perhaps map3L and map3R, depending on the grouping, and we’d get an explosion of other combinators based on having to distin- guish between different groupings. We can state the associativity law in terms of product.5 Recall that product just combines two effects into a pair, using map2: def product[A,B](fa: F[A], fb: F[B]): F[(A,B)] = map2(fa, fb)((_,_)) And if we have pairs nested on the right, we can always turn those into pairs nested on the left: 5 product, map, and unit are an alternate formulation of Applicative. Can you see how map2 can be imple- mented using product and map? Licensed to Emre Sevinc 216 CHAPTER 12 Applicative and traversable functors def assoc[A,B,C](p: (A,(B,C))): ((A,B), C) = p match { case (a, (b, c)) => ((a,b), c) } Using these combinators, product and assoc, the law of associativity for applicative functors is as follows: product(product(fa,fb),fc) == map(product(fa, product(fb,fc)))(assoc) Note that the calls to product are associated to the left on one side and to the right on the other side of the == sign. On the right side we’re then mapping the assoc func- tion to make the resulting tuples line up. 12.5.3 Naturality of product Our final law for applicative functors is naturality. To illustrate, let’s look at a simple example using Option. val F: Applicative[Option] = ... case class Employee(name: String, id: Int) case class Pay(rate: Double, hoursPerYear: Double) def format(e: Option[Employee], pay: Option[Pay]): Option[String] = F.map2(e, pay) { (e, pay) => s"${e.name} makes ${pay.rate * pay.hoursPerYear}" } val e: Option[Employee] = ... val pay: Option[Pay] = ... format(e, pay) Here we’re applying a transformation to the result of map2—from Employee we extract the name, and from Pay we extract the yearly wage. But we could just as easily apply these transformations separately, before calling format, giving format an Option [String] and Option[Double] rather than an Option[Employee] and Option[Pay]. This might be a reasonable refactoring, so that format doesn’t need to know the details of how the Employee and Pay data types are represented. val F: Applicative[Option] = ... def format(name: Option[String], pay: Option[Double]): Option[String] = F.map2(e, pay) { (e, pay) => s"$e makes $pay" } val e: Option[Employee] = ... val pay: Option[Pay] = ... format( F.map(e)(_.name), F.map(pay)(pay => pay.rate * pay.hoursPerYear)) Listing 12.6 Retrieving employee names and annual pay Listing 12.7 Refactoring format format now takes the employee name as a Option[String], rather than extracting the name from an Option[Employee]. Similarly for pay. Licensed to Emre Sevinc 217The applicative laws We’re applying the transformation to extract the name and pay fields before calling map2. We expect this program to have the same meaning as before, and this sort of pattern comes up frequently. When working with Applicative effects, we generally have the option of applying transformations before or after combining values with map2. The naturality law states that it doesn’t matter; we get the same result either way. Stated more formally, map2(a,b)(productF(f,g)) == product(map(a)(f), map(b)(g)) Where productF combines two functions into one function that takes both their argu- ments and returns the pair of their results: def productF[I,O,I2,O2](f: I => O, g: I2 => O2): (I,I2) => (O,O2) = (i,i2) => (f(i), g(i2)) The applicative laws are not surprising or profound. Just like the monad laws, these are simple sanity checks that the applicative functor works in the way that we’d expect. They ensure that unit, map, and map2 behave in a consistent and reasonable manner. EXERCISE 12.7 Hard: Prove that all monads are applicative functors by showing that if the monad laws hold, the Monad implementations of map2 and map satisfy the applicative laws. EXERCISE 12.8 Just like we can take the product of two monoids A and B to give the monoid (A, B), we can take the product of two applicative functors. Implement this function: def product[G[_]](G: Applicative[G]): Applicative[({type f[x] = (F[x], G[x])})#f] EXERCISE 12.9 Hard: Applicative functors also compose another way! If F[_] and G[_] are applicative functors, then so is F[G[_]]. Implement this function: def compose[G[_]](G: Applicative[G]): Applicative[({type f[x] = F[G[x]]})#f] EXERCISE 12.10 Hard: Prove that this composite applicative functor meets the applicative laws. This is an extremely challenging exercise. Licensed to Emre Sevinc 218 CHAPTER 12 Applicative and traversable functors EXERCISE 12.11 Try to write compose on Monad. It’s not possible, but it is instructive to attempt it and understand why this is the case. def compose[G[_]](G: Monad[G]): Monad[({type f[x] = F[G[x]]})#f] 12.6 Traversable functors We discovered applicative functors by noticing that our traverse and sequence func- tions (and several other operations) didn’t depend directly on flatMap. We can spot another abstraction by generalizing traverse and sequence once again. Look again at the signatures of traverse and sequence: def traverse[F[_],A,B](as: List[A])(f: A => F[B]): F[List[B]] def sequence[F[_],A](fas: List[F[A]]): F[List[A]] Any time you see a concrete type constructor like List showing up in an abstract interface like Applicative, you may want to ask the question, “What happens if I abstract over this type constructor?” Recall from chapter 10 that a number of data types other than List are Foldable. Are there data types other than List that are tra- versable? Of course! EXERCISE 12.12 On the Applicative trait, implement sequence over a Map rather than a List: def sequenceMap[K,V](ofa: Map[K,F[V]]): F[Map[K,V]] But traversable data types are too numerous for us to write specialized sequence and traverse methods for each of them. What we need is a new interface. We’ll call it Traverse:6 trait Traverse[F[_]] { def traverse[G[_]:Applicative,A,B](fa: F[A])(f: A => G[B]): G[F[B]] = sequence(map(fa)(f)) def sequence[G[_]:Applicative,A](fga: F[G[A]]): G[F[A]] = traverse(fga)(ga => ga) } 6 The name Traversable is already taken by an unrelated trait in the Scala standard library. Licensed to Emre Sevinc 219Uses of Traverse The interesting operation here is sequence. Look at its signature closely. It takes F[G[A]] and swaps the order of F and G, so long as G is an applicative functor. Now, this is a rather abstract, algebraic notion. We’ll get to what it all means in a minute, but first, let’s look at a few instances of Traverse. EXERCISE 12.13 Write Traverse instances for List, Option, and Tree. case class Tree[+A](head: A, tail: List[Tree[A]]) We now have instances for List, Option, Map, and Tree. What does this generalized traverse/sequence mean? Let’s just try plugging in some concrete type signatures for calls to sequence. We can speculate about what these functions do, just based on their signatures: List[Option[A]] => Option[List[A]] (a call to Traverse[List].sequence with Option as the Applicative) returns None if any of the input List is None; otherwise it returns the original List wrapped in Some. Tree[Option[A]] => Option[Tree[A]] (a call to Traverse[Tree].sequence with Option as the Applicative) returns None if any of the input Tree is None; otherwise it returns the original Tree wrapped in Some. Map[K, Par[A]] => Par[Map[K,A]] (a call to Traverse[Map[K,_]].sequence with Par as the Applicative) produces a parallel computation that evaluates all values of the map in parallel. There turns out to be a startling number of operations that can be defined in the most general possible way in terms of sequence and/or traverse. We’ll explore these in the next section. A traversal is similar to a fold in that both take some data structure and apply a func- tion to the data within in order to produce a result. The difference is that traverse preserves the original structure, whereas foldMap discards the structure and replaces it with the operations of a monoid. Look at the signature Tree[Option[A]] => Option[Tree[A]], for instance. We’re preserving the Tree structure, not merely col- lapsing the values using some monoid. 12.7 Uses of Traverse Let’s now explore the large set of operations that can be implemented quite gener- ally using Traverse. We’ll only scratch the surface here. If you’re interested, follow some of the references in the chapter notes to learn more, and do some exploring on your own. Licensed to Emre Sevinc 220 CHAPTER 12 Applicative and traversable functors EXERCISE 12.14 Hard: Implement map in terms of traverse as a method on Traverse[F]. This estab- lishes that Traverse is an extension of Functor and that the traverse function is a generalization of map (for this reason we sometimes call these traversable functors). Note that in implementing map, you can call traverse with your choice of Applicative[G]. trait Traverse[F[_]] extends Functor[F] { def traverse[G[_],A,B](fa: F[A])(f: A => G[B])( implicit G: Applicative[G]): G[F[B]] = sequence(map(fa)(f)) def sequence[G[_],A](fga: F[G[A]])( implicit G: Applicative[G]): G[F[A]] = traverse(fga)(ga => ga) def map[A,B](fa: F[A])(f: A => B): F[B] = ??? } But what is the relationship between Traverse and Foldable? The answer involves a connection between Applicative and Monoid. 12.7.1 From monoids to applicative functors We’ve just learned that traverse is more general than map. Next we’ll learn that traverse can also express foldMap and by extension foldLeft and foldRight! Take another look at the signature of traverse: def traverse[G[_]:Applicative,A,B](fa: F[A])(f: A => G[B]): G[F[B]] Suppose that our G were a type constructor ConstInt that takes any type to Int, so that ConstInt[A] throws away its type argument A and just gives us Int: type ConstInt[A] = Int Then in the type signature for traverse, if we instantiate G to be ConstInt, it becomes def traverse[A,B](fa: F[A])(f: A => Int): Int This looks a lot like foldMap from Foldable. Indeed, if F is something like List, then what we need to implement this signature is a way of combining the Int values returned by f for each element of the list, and a “starting” value for handling the empty list. In other words, we only need a Monoid[Int]. And that’s easy to come by. In fact, given a constant functor like we have here, we can turn any Monoid into an Applicative. Licensed to Emre Sevinc 221Uses of Traverse type Const[M, B] = M implicit def monoidApplicative[M](M: Monoid[M]) = new Applicative[({ type f[x] = Const[M, x] })#f] { def unit[A](a: => A): M = M.zero def map2[A,B,C](m1: M, m2: M)(f: (A,B) => C): M = M.op(m1,m2) } This means that Traverse can extend Foldable and we can give a default implemen- tation of foldMap in terms of traverse: trait Traverse[F[_]] extends Functor[F] with Foldable[F] { ... def foldMap[A,M](as: F[A])(f: A => M)(mb: Monoid[M]): M = traverse[({type f[x] = Const[M,x]})#f,A,Nothing]( as)(f)(monoidApplicative(mb)) } Note that Traverse now extends both Foldable and Functor! Importantly, Foldable itself can’t extend Functor. Even though it’s possible to write map in terms of a fold for most foldable data structures like List, it’s not possible in general. EXERCISE 12.15 Answer, to your own satisfaction, the question of why it’s not possible for Foldable to extend Functor. Can you think of a Foldable that isn’t a functor? So what is Traverse really for? We’ve already seen practical applications of particular instances, such as turning a list of parsers into a parser that produces a list. But in what kinds of cases do we want the generalization? What sort of generalized library does Tra- verse allow us to write? 12.7.2 Traversals with State The State applicative functor is a particularly powerful one. Using a State action to traverse a collection, we can implement complex traversals that keep some kind of internal state. An unfortunate amount of type annotation is necessary in order to partially apply State in the proper way, but traversing with State is common enough that we can cre- ate a special method for it and write those type annotations once and for all: def traverseS[S,A,B](fa: F[A])(f: A => State[S, B]): State[S, F[B]] = traverse[({type f[x] = State[S,x]})#f,A,B](fa)(f)(Monad.stateMonad) Listing 12.8 Turning a Monoid into an Applicative This is ConstInt generalized to any M, not just Int. A trait may list multiple supertraits, separated by the keyword with. Scala can’t infer the partially applied Const type alias here, so we have to provide an annotation. Licensed to Emre Sevinc 222 CHAPTER 12 Applicative and traversable functors To demonstrate this, here’s a State traversal that labels every element with its posi- tion. We keep an integer state, starting with 0, and add 1 at each step. def zipWithIndex[A](ta: F[A]): F[(A,Int)] = traverseS(ta)((a: A) => (for { i <- get[Int] _ <- set(i + 1) } yield (a, i))).run(0)._1 This definition works for List, Tree, or any other traversable. Continuing along these lines, we can keep a state of type List[A], to turn any tra- versable functor into a List. def toList[A](fa: F[A]): List[A] = traverseS(fa)((a: A) => (for { as <- get[List[A]] _ <- set(a :: as) } yield ()).run(Nil)._2.reverse We begin with the empty list Nil as the initial state, and at every element in the tra- versal, we add it to the front of the accumulated list. This will of course construct the list in the reverse order of the traversal, so we end by reversing the list that we get from running the completed state action. Note that we yield () because in this instance we don’t want to return any value other than the state. Of course, the code for toList and zipWithIndex is nearly identical. And in fact most traversals with State will follow this exact pattern: we get the current state, com- pute the next state, set it, and yield some value. We should capture that in a function. def mapAccum[S,A,B](fa: F[A], s: S)(f: (A, S) => (B, S)): (F[B], S) = traverseS(fa)((a: A) => (for { s1 <- get[S] (b, s2) = f(a, s1) _ <- set(s2) } yield b)).run(s) override def toList[A](fa: F[A]): List[A] = mapAccum(fa, List[A]())((a, s) => ((), a :: s))._2.reverse def zipWithIndex[A](fa: F[A]): F[(A, Int)] = mapAccum(fa, 0)((a, s) => ((a, s), s + 1))._1 Listing 12.9 Numbering the elements in a traversable Listing 12.10 Turning traversable functors into lists Listing 12.11 Factoring out our mapAccum function Get the current state, the accumulated list. Add the current element and set the new list as the new state. Licensed to Emre Sevinc 223Uses of Traverse EXERCISE 12.16 There’s an interesting consequence of being able to turn any traversable functor into a reversed list—we can write, once and for all, a function to reverse any traversable functor! Write this function, and think about what it means for List, Tree, and other traversable functors. def reverse[A](fa: F[A]): F[A] It should obey the following law, for all x and y of the appropriate types: toList(reverse(x)) ++ toList(reverse(y)) == reverse(toList(y) ++ toList(x)) EXERCISE 12.17 Use mapAccum to give a default implementation of foldLeft for the Traverse trait. 12.7.3 Combining traversable structures It’s the nature of a traversal that it must preserve the shape of its argument. This is both its strength and its weakness. This is well demonstrated when we try to combine two structures into one. Given Traverse[F], can we combine a value of some type F[A] and another of some type F[B] into an F[C]? We could try using mapAccum to write a generic version of zip. def zip[A,B](fa: F[A], fb: F[B]): F[(A, B)] = (mapAccum(fa, toList(fb)) { case (a, Nil) => sys.error("zip: Incompatible shapes.") case (a, b :: bs) => ((a, b), bs) })._1 Note that this version of zip is unable to handle arguments of different “shapes.” For example, if F is List, then it can’t handle lists of different lengths. In this implementa- tion, the list fb must be at least as long as fa. If F is Tree, then fb must have at least the same number of branches as fa at every level. We can change the generic zip slightly and provide two versions so that the shape of one side or the other is dominant. Listing 12.12 Combining two different structure types Licensed to Emre Sevinc 224 CHAPTER 12 Applicative and traversable functors def zipL[A,B](fa: F[A], fb: F[B]): F[(A, Option[B])] = (mapAccum(fa, toList(fb)) { case (a, Nil) => ((a, None), Nil) case (a, b :: bs) => ((a, Some(b)), bs) })._1 def zipR[A,B](fa: F[A], fb: F[B]): F[(Option[A], B)] = (mapAccum(fb, toList(fa)) { case (b, Nil) => ((None, b), Nil) case (b, a :: as) => ((Some(a), b), as) })._1 These implementations work out nicely for List and other sequence types. In the case of List, for example, the result of zipR will have the shape of the fb argument, and it will be padded with None on the left if fb is longer than fa. For types with more interesting structures, like Tree, these implementations may not be what we want. Note that in zipL, we’re simply flattening the right argument to a List[B] and discarding its structure. For Tree, this will amount to a preorder tra- versal of the labels at each node. We’re then “zipping” this sequence of labels with the values of our left Tree, fa; we aren’t skipping over nonmatching subtrees. For trees, zipL and zipR are most useful if we happen to know that both trees share the same shape. 12.7.4 Traversal fusion In chapter 5, we talked about how multiple passes over a structure can be fused into one. In chapter 10, we looked at how we can use monoid products to carry out multi- ple computations over a foldable structure in a single pass. Using products of applica- tive functors, we can likewise fuse multiple traversals of a traversable structure. EXERCISE 12.18 Use applicative functor products to write the fusion of two traversals. This function will, given two functions f and g, traverse fa a single time, collecting the results of both functions at once. def fuse[G[_],H[_],A,B](fa: F[A])(f: A => G[B], g: A => H[B]) (G: Applicative[G], H: Applicative[H]): (G[F[B]], H[F[B]]) 12.7.5 Nested traversals Not only can we use composed applicative functors to fuse traversals, traversable func- tors themselves compose. If we have a nested structure like Map[K,Option[List[V]]], Listing 12.13 A more flexible implementation of zip Licensed to Emre Sevinc 225Uses of Traverse then we can traverse the map, the option, and the list at the same time and easily get to the V value inside, because Map, Option, and List are all traversable. EXERCISE 12.19 Implement the composition of two Traverse instances. def compose[G[_]](implicit G: Traverse[G]): Traverse[({type f[x] = F[G[x]]})#f] 12.7.6 Monad composition Let’s now return to the issue of composing monads. As we saw earlier in this chapter, Applicative instances always compose, but Monad instances do not. If you tried before to implement general monad composition, then you would have found that in order to implement join for nested monads F and G, you’d have to write something of a type like F[G[F[G[A]]]] => F[G[A]]. And that can’t be written generally. But if G also happens to have a Traverse instance, we can sequence to turn G[F[_]] into F[G[_]], leading to F[F[G[G[A]]]]. Then we can join the adjacent F layers as well as the adjacent G layers using their respective Monad instances. EXERCISE 12.20 Hard: Implement the composition of two monads where one of them is traversable. def composeM[F[_],G[_]](F: Monad[F], G: Monad[G], T: Traverse[G]): Monad[({type f[x] = F[G[x]]})#f] Expressivity and power sometimes come at the price of compositionality and modular- ity. The issue of composing monads is often addressed with a custom-written version of each monad that’s specifically constructed for composition. This kind of thing is called a monad transformer. For example, the OptionT monad transformer composes Option with any other monad: case class OptionT[M[_],A](value: M[Option[A]])(implicit M: Monad[M]) { def flatMap[B](f: A => OptionT[M, B]): OptionT[M, B] = OptionT(value flatMap { case None => M.unit(None) case Some(a) => f(a).value }) } Option is added to the inside of the monad M. Licensed to Emre Sevinc 226 CHAPTER 12 Applicative and traversable functors The flatMap definition here maps over both M and Option, and flattens structures like M[Option[M[Option[A]]]] to just M[Option[A]]. But this particular implementation is specific to Option. And the general strategy of taking advantage of Traverse works only with traversable functors. To compose with State (which can’t be traversed), for example, a specialized StateT monad transformer has to be written. There’s no generic composition strategy that works for every monad. See the chapter notes for more information about monad transformers. 12.8 Summary In this chapter, we discovered two new useful abstractions, Applicative and Traverse, simply by playing with the signatures of our existing Monad interface. Appli- cative functors are a less expressive but more compositional generalization of monads. The functions unit and map allow us to lift values and functions, whereas map2 and apply give us the power to lift functions of higher arities. Traversable functors are the result of generalizing the sequence and traverse functions we’ve seen many times. Together, Applicative and Traverse let us construct complex nested and parallel tra- versals out of simple elements that need only be written once. As you write more func- tional code, you’ll learn to spot instances of these abstractions and how to make better use of them in your programs. This is the final chapter in part 3, but there are many abstractions beyond Monad, Applicative, and Traverse, and you can apply the techniques we’ve developed here to discover new structures yourself. Functional programmers have of course been dis- covering and cataloguing for a while, and there is by now a whole zoo of such abstrac- tions that captures various common patterns (arrows, categories, and comonads, just to name a few). Our hope is that these chapters have given you enough of an introduc- tion to start exploring this wide world on your own. The material linked in the chap- ter notes is a good place to start. In part 4 we’ll complete the functional programming story. So far we’ve been writ- ing libraries that might constitute the core of a practical application, but such applica- tions will ultimately need to interface with the outside world. In part 4 we’ll see that referential transparency can be made to apply even to programs that perform I/O operations or make use of mutable state. Even there, the principles and patterns we’ve learned so far allow us to write such programs in a compositional and reusable way. Licensed to Emre Sevinc