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Hal Daum´ III e
Copyright (c) Hal Daume III, 2002-2006. The preprint version of this tutorial is intended to be free to the entire Haskell community, so we grant permission to copy and distribute it for any purpose, provided that it is reproduced in its entirety, including this notice. Modified versions may not be distributed without prior concent of the author, and must still maintain a copy of this notice. The author retains the right to change or modify this copyright at any time, as well as to make the book no longer free of charge.
About This Report
The goal of the Yet Another Haskell Tutorial is to provide a complete intoduction to the Haskell programming language. It assumes no knowledge of the Haskell language or familiarity with functional programming in general. However, general familiarity with programming concepts (such as algorithms) will be helpful. This is not intended to be an introduction to programming in general; rather, to programming in Haskell. Sufficient familiarity with your operating system and a text editor is also necessary (this report only discusses installation on configuration on Windows and *Nix system; other operating systems may be supported – consult the documentation of your chosen compiler for more information on installing on other platforms).
What is Haskell?
Haskell is called a lazy, pure functional programming language. It is called lazy because expressions which are not needed to determine the answer to a problem are not evaluated. The opposize of lazy is strict, which is the evaluation strategry of most common programming languages (C, C++, Java, even ML). A strict language is one in which every expression is evaluated, whether the result of its computation is important or not. (This is probably not entirely true as optimizing compilers for strict languages often do what's called "dead code elmination" – this removes unused expressions from the program.) It is called pure because it does not allow side effects (A side effect is something that affects the "state" of the world. For instance, a function that prints something to the screen is said to be side-effecting, as is a function which affects the value of a global variable.) – of course, a programming language without side effects would be horribly useless; Haskell uses a system of monads to isolate all impure computations from the rest of the program and perform them in the safe way (see Chapter 9 for a discussion of monads proper or Chapter 5 for how to do input/output in a pure language). Haskell is called a functional language because the evaluation of a program is equivalent to evaluating a function in the pure mathematical sense. This also differs from standard languages (like C and Java) which evaluate a sequence of statements, one after the other (this is termed an imperative langauge). i
The History of Haskell
The history of Haskell is best described using the words of the authors. The following text is quoted from the published version of the Haskell 98 Report: In September of 1987 a meeting was held at the conference on Functional Programming Languages and Computer Architecture (FPCA '87) in Portland, Oregon, to discuss an unfortunate situation in the functional programming community: there had come into being more than a dozen nonstrict, purely functional programming languages, all similar in expressive power and semantic underpinnings. There was a strong consensus at this meeting that more widespread use of this class of functional languages was being hampered by the lack of a common language. It was decided that a committee should be formed to design such a language, providing faster communication of new ideas, a stable foundation for real applications development, and a vehicle through which others would be encouraged to use functional languages. This document describes the result of that committee's efforts: a purely functional programming language called Haskell, named after the logician Haskell B. Curry whose work provides the logical basis for much of ours. The committee's primary goal was to design a language that satisfied these constraints: 1. It should be suitable for teaching, research, and applications, including building large systems. 2. It should be completely described via the publication of a formal syntax and semantics. 3. It should be freely available. Anyone should be permitted to implement the language and distribute it to whomever they please. 4. It should be based on ideas that enjoy a wide consensus. 5. It should reduce unnecessary diversity in functional programming languages. The committee intended that Haskell would serve as a basis for future research in language design, and hoped that extensions or variants of the language would appear, incorporating experimental features. Haskell has indeed evolved continuously since its original publication. By the middle of 1997, there had been four iterations of the language design (the latest at that point being Haskell 1.4). At the 1997 Haskell Workshop in Amsterdam, it was decided that a stable variant of Haskell was needed; this stable language is the subject of this Report, and is called "Haskell 98". Haskell 98 was conceived as a relatively minor tidy-up of Haskell 1.4, making some simplifications, and removing some pitfalls for the unwary.
however. I have consistently found the Haskell community to be incredibly helpful. A separate effort was therefore begun by a distinct (but overlapping) committee to fix the Haskell 98 Libraries. My personal reason for using Haskell is that I have found that I write more bug-free code in less time using Haskell than any other language. which often outperforms even C++. The language is constantly evolving (that's not to say it's instable. . a set of libraries would have to be standardised too. The first issue certainly has come up a few times in my experience. Some people may find that the speed hit taken for using Haskell is unbearable. The second problem tends not be to a very big issue: most of the code I've written is not buggy. Of course. If you don't find this sufficient. and (2) it tends to be difficult to debug. I would suggest taking a look at the language O'Caml.iii It is intended to be a "stable" language in sense the implementors are committed to supporting Haskell 98 exactly as specified. However. together with a standard library called the Prelude. Perhaps most importantly. rather that there are numerous extensions that have been added to some compilers which I find very useful) and user suggestions are often heeded when new extensions are to be implemented. yet also has many of the benefits of Haskell. Haskell has a standardized foreign-function interface which allow you to link in code written in other languages. and the complaints of most Haskellers I know. are: (1) the generated code tends to be slower than equivalent programs written in a language like C. By the time Haskell 98 was stabilised. as most of the common sources of bugs in other languages simply don't exist in Haskell. however. Why Not Use Haskell? My two biggest complaints. There are many motivations for using Haskell. Why Use Haskell? Clearly you're interested in Haskell since you're reading this tutorial. I also find it very readable and extensible. The original Haskell Report covered only the language. for the foreseeable future. for when you need to get the most speed out of your code. CPU time is almost always cheaper than programmer time and if I have to wait a little longer for my results after having saved a few days programming and debugging. it had become clear that many programs need access to a larger set of library functions (notably concerning input/output and simple interaction with the operating system). this isn't the case of all applications. If these program were to be portable.
none of the known tutorials introduce input/output and iteractivity soon enough (Paul Hudak's book is an exception in that it does introduce IO by page 35. • Online Haskell Course is a short course (in German) for beginning with Haskell. though the focus and aim of that book and this tutorial are very different). It became clear that there is a strong need for a tutorial which is introductory in the sense that it does not assume knowledge of functional programming. A brief survey of the tutorials available yields: • A Gentle Introduction to Haskell is an introduction to Haskell. The majority of this book will cover the Haskell 98 standard. many compilers offer certain extensions to the standard which are useful. • Haskell Tutorial is based on a course given at the 3rd International Summer School on Advanced Functional Programming.iv Target Audience There have been many books and tutorials written about Haskell. given that the reader is familiar with functional programming en large. • Haskell Companion is a short reference of common concepts and definitions. The goals of this tutorial are: • to be practical above all else • to provide a comprehensive. Moreover. • Two Dozen Short Lessons in Haskell is the draft of an excellent textbook that emphasizes user involvement. Though all of these tutorials is excellent. The Haskell language underwent a standardization process and the result is called Haskell 98. Any deviations from the standard will be noted (for instance. This tutorial is not for beginning programmers. • PLEAC-Haskell is a tutorial in the style of the Perl Cookbook. • Haskell for Miranda Programmers assumes knowledge of the language Miranda. free introduction to the Haskell language • to point out common pitfalls and their solutions • to provide a good sense of how Haskell can be used in the real world . some of these may be discussed). as can be witnessed by reading through the archives of the Haskell mailing list. or not cover everything. Haskell is full of pitfalls for new programmers and experienced non-functional programmers alike. they are on their own incomplete: The "Gentle Introduction" is far too advanced for beginning Haskellers and the others tend to end too early. some experience and knowledge of programming and computers is assumed (though the appendix does contain some background information). visit the http://haskell. for a (nearly) complete list. but which is advanced in the sense that it does assume some background in programming.org/bookshelf (Haskell Bookshelf) at the Haskell homepage.
Dick Kieburtz. Dave Barton. Jon Fairbairn.org/classes/ Acknowledgements It would be inappropriate not to give credit also to the original designers of Haskell.cs. Kevin Hammond. Rishiyur Nikhil. Alastair Reid.org/aboutHaskell. .geoinfo. John Hughes. Sigbjorn Finne and Sven Panne. John Hughes. Thomas Johnsson. but these are those who come to mind. Mike Reeve. Lennart Augustsson. Ralf Hinze. Manuel Chakravarty. Finally. Warren Burton. Andy Gordon. I would like to specifically thank Simon Peyton Jones. Also thanks to the many people who have reported "bugs" in the first edition. Simon Marlow.tuwien. Philip Wadler. Maria Guzman. Brian Boutel.html Haskell Wiki : http://haskell. David Wise.nl/˜afie/haskell/tourofprelude. Paul Hudak. Joseph Fasel. John Launchbury. Colin Runciman. John Peterson. pdf Tour of the Haskell Prelude : http://www. There were doubtless others who helped and are not listed. Koen Classen. Erik Meijer. Those are: Arvind. Mark Jones.uu.Hal Daum´ III e .at/navratil/HaskellTutorial.v Additional Online Sources of Information A Short Introduction to Haskell : http://haskell. Alastair Reid. all of whom have made my life learning Haskell all the more enjoyable by always being supportive.ac. Simon Peyton Jones. Jonathan Young.org/hawiki/ Haskell-Tutorial : ftp://ftp.html Courses in Haskell : http://haskell.
vi .
. 2. 2. . . . 2. . . . . . . . . 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Program options . . . . . . . . . .1 Strings . . .5 Functions . . . . .3. . 3. . . . . . . . . . 2. . . . . . . . . . 2. . Language Basics 3. . . . . . . 2. . . . . . . . . . .3 Lists . . . . . . . . . . .5.3. . . . . . . . . . . . . . . . . . 3. . . . . . . . .2. . . . . . . . . . . . 2. . .6 How to get help . . 2. . . . . . . . . . .2.4 Program options . . . . . . . . . 2.1. . . .3. . . . 2. .3 How to run the compiler . .Contents 1 2 Introduction Getting Started 2. . . .2. . . . .4 Editors .4 Source Code Files . . . . . . . . . . . . . .1.1 Where to get it . . . . . . . .2 Simple List Functions 3. . . . . . . . . . . . . . . . . . . . . . . . . . 3. . .1 Where to get it . . . .1 Let Bindings . . . . . . . . . . . . . . . . . . . . . . . . .3. . 3. . . . . . . . . . 2. . . . . . . . .1. . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Installation procedures . . . . . . . . . . . . . 3. . . . . . . . . . . .5. . . . . . . . . . .1 Arithmetic . . .1. . . . . . . . .3 How to run it . . . . . . . . . . . . . . .3 How to run it . . . . . . 3 5 5 6 6 6 7 7 7 7 8 8 8 9 9 9 9 9 9 9 9 9 11 13 14 15 17 18 20 22 27 28 . .2 Pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 How to get help . . . .1 Where to get it . 2. . . . . . . . . . . . . . . . . . .5 How to get help . . . .2. . . . . . .2.2 Infix . . . . . . . . .2 Installation procedures . . . . 2. . . . . . . . . .3. . . . . . . . . . . . .2. . . . . . . 2. . . . . . . 3. . . . . . vii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Triples and More . . . . . . . . . . . . . . .3 NHC . . . . . . . . . . . . . . . . . . . . . . . .5 Program options . . . 3 . . . . . . . . . . . . 2. . . . . . . . . . . . . .3. . . . . . . . . . . . . . . . . . .1. . .2 Glasgow Haskell Compiler . . . . . . . . . . 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Hugs . . . . . . . . . . . .2 Installation procedures .3. . . . . . . . .4 How to run the interpreter 2. . . . . . . . 2. . 2. . . . .
. . . . .3 Partial Application . . . . . 4. . . . . .3. . . . . . . . . .viii 3. . . . . . . . . . . . . .4. . .6 The Unit type . . . . . . 4. .3. . . . 5 . . . . . . . . .3 Hierarchical Imports . . . . . . . . . . .1 Bird-scripts . . . . . . . . . . . . .4.5. . . .4 The Show Class . . . . . . . . . . . . . .3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Advanced Features 7. . . . . . . . . . . . . . . . . . . . . . . . .3 The Num Class . .4. . . . . . . . . . . . . . . . . . . . . . . . 6 Modules 6. . . . . . . . 4.3 That Pesky IO Type . . . . . . Interactivity . . .2 Local Declarations . . . . . . . . . . . . . . . . . . . . . . . . .1 Sections and Infix Operators 7. . . . . . .1 Simple Types . .5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. . . . . . . . . . . . . . . . . . . . . . . . . . 5. . . . . . . . . . .2 Polymorphic Types . . . . . . . . .1 Motivation . . . . . . . . . . . . . . . . . . . . . . . .2 Multiple Constructors . . . 4. . . . 4. . . . . . . . . . . . . .1 Lambda Calculus .2 LaTeX-scripts . . . . . . . . . . . . . . . . . . . . . .2 Equality Testing . . . . . . . . . . . . . . . . . . .2 Higher-Order Types . . .8 4 CONTENTS Comments . . . . . . . . . . . . . . 4. . . . . 4. 4. 6. . . . . . . .4 Literate Versus Non-Literate 6. . .5 Enumerated Sets . . . . . . . . . . . . . . . . . . . . . . . 7. . . . . . . . . . . .5 Data Types . . . . . .6 3. . . . . . 28 29 31 37 37 39 40 40 41 41 41 42 42 42 44 45 46 47 47 49 51 51 52 53 53 57 57 58 62 64 67 67 69 70 71 71 72 73 73 74 76 80 Type Basics 4. Basic Input/Output 5. . . . . . . . . . . . . . . . . . .4 Binary Trees . . . . . . .4. . . . . . 4. . . . . . . . . . . . . . . . 5. 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Pairs . . . . . .4 Function Types . . . . 4. . . .3 Recursive Datatypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. . . . . . . . . .4. . . . . . . . . . . 4. . . . . . . .2 Imports . . . . . . . . . . . . . . . 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. . . . .5. . . . . 4. . . . . 6. . . . . . . .3 The IO Library . . . . . . . . . . . . . . . .4 Explicit Type Declarations 4. . . . . 4. . . . . . . . . . . . . . . . . . . . . . 4. . . . . . .4. . . . . . . . . . . . . . . . . . . . . . . .4 Pattern Matching . . .4. . . . . . . .3 Type Classes .6 Continuation Passing Style . . . .2 Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. . . . . . . . . . . . .1 Exports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5. .7 3. 6. Recursion . . . . . .5. . . . . . . . .4 A File Reading Program . . . . . . . . . . . .5 Functional Arguments . . . . . . . . . . . . . .5. . . . . .3.1 The RealWorld Solution 5. . . . . . . . . . . 7 . . . . . .
. . . . . . . . . . . . .7. . . . . . . . . . . . .1 Pong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3. . . . . . . . . . . . . . . .1 Strict Fields . . . . .7 Datatypes Revisited . . . . . . . . . . . . . . . . . . . . .CONTENTS Guards . 7. . . . . . 7. . . . . . . . . 9. . . . . . . . . . . . . . . . . . . .6. . . 8. . . . . 7. . . . .11 Layout . . . . . . . . . . . . . . . . . . . .1 Named Fields . . . . . . 9. . . . 9. . . . . . . . ix 83 85 85 86 87 89 89 90 90 92 92 94 96 97 99 99 103 103 104 105 105 108 108 109 113 115 117 117 119 120 122 124 130 134 137 139 144 144 151 . . . . 8. . . . . . .8 Parsing Monads . . . . . 9 Monads 9. . . . . . . . . . .4. . . .4 Classes . . . . . . . . . . . . . . . . . . . . . . . . Instance Declarations . . . . . . .5 7. . . . . . . . . . . . . . . . . . . . . . . 8. . .2 Definition . . . . . .8. . . . 9.4 Class Contexts . . . . . . . . . . . . . . . . .3 Datatypes . . . . . . . . . . . . . . . . .10 Finite Maps . . 7. . . . . . . .6. . . . . . 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 A Simple State Monad . . . . . . . . .6. . . . . . . . . . . . . . . 8. . . . . . . . 8. . . . . . . . . . . . . . . . . . . .8 Default . . . . . . . . . . . 8. 7. . . . . . . . . . . . . . . . . . . . . . . . . . .5 Deriving Classes . . . . . . . . . . . . . . . . . .2 List Comprehensions . . . . . . . . . . . . . . . . . . . 9. . . . . . . . . . . . . .1 A Simple Parsing Monad 9. . . . . . . .8. . 9. . . . . . .8. . . . . . . . . . .4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 Kinds . . . . . 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 MonadPlus . . . . . . . .3 Other Important Classes 7. . . . . . . 7. . . . . . . . . . . . . . . . .1 The Eq Class . . . . . . . . . . . 7. . . . . . . . . . . . . . . . . .1 Do Notation . . . .6. . . . . . . . . . . . .2 Parsec . . . . . . . . . . . . . . . . . . . .8 More Lists . . . . . . . . . . . . 9. . . . 7. . . . . . . .2 The Show Class .5 Monadic Combinators . . . . . . . . . . . . . . .6 . . . . . . . . . . . . 8. . . 8 Advanced Types 8. . . . . . . . . . . . . . . .7 Monad Transformers . . . . . . . . . . . . .7 Class Hierarchies . . .1 Type Synonyms . . . . . . . . . . . . . . . .5 Instances . . . . . . . . . . . . 8. . . . . . . . . . . .2 Newtypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Standard List Functions 7. . .8. . .9 Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Computations 8. . . . . . . . . . . . . . . . . . . . . . .4 Common Monads . . . . . . . . . . . . . . . 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6. . . . . 7. . . . . .12 The Final Word on Lists . . . . . . . . . . . . . . . . . . . . . . . . . . 8. . . . . . . . . . . . . . . . . . . 9. .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .CONTENTS 10 Advanced Techniques 10. . . . . 10. . . . . . 10. . . . . . . . . . . .1 Exceptions . . .7 Dynamic Types . . . . . . . . . . . . . . . . . . . . . . . . . .4 The ST Monad . . . . . . . . . . . .5 Concurrency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. . . . . . . . . .3 Mutable References . . . . . . . . . . .6 Regular Expressions 10. . . . . . . . . . A Brief Complexity Theory B Recursion and Induction C Solutions To Exercises 1 157 157 157 157 157 157 157 157 159 161 163 . . . . . . . . . . . . . . . . . . . . .2 Mutable Arrays . . . . . . . . . . . . . . . . . . . 10. . . . 10. . .
These are often for making comparisons to other programming languages or adding helpful information. NOTE Notes will appear like this. all of which should have been included in its distribution. we will refer to interaction betwen you and the operating system and/or the interactive shell (more on this in Section 2). WARNING Warnings will appear like this. Occasionally. If we're covering a difficult or confusing topic and there is something you should watch out for.Chapter 1 Introduction This tutorial contains a whole host of example code. Haskell keywords will appear like this: where. we will often make additional notes to something written. 3 . This will look something like this: map :: (a− > b)− > [a]− > [b] Within the body text. Finally. we will sometimes make reference to built-in functions (so-called Preludefunctions). identifiers as map. If not. Interaction will look like this. we will place a warning. types as String and classes as Eq. This book is formatted to make example code stand out from the rest of the text. Strewn throughout the tutorial. please refer to the links off of the Haskell web site (haskell.org) to get it. Code will look like this.
1 Hugs Hugs supports almost all of the Haskell 98 standard (it lacks some of the libraries). cannot create stand-alones. written in C. NHC . It may have changed – see http://haskell. so if you had to go with just one. GHC is both an interpreter (like Hugs) and a compiler which will produce stand-alone programs.less used and no interactive environment. I've tried to make a list of some of the differences in the following list but of course this is far from exhaustive: Hugs . As such. Which you use is entirely up to you. but can test and debug programs in an interactive environment. works on almost every platform. since it is much faster. Following is a descrition of how to download and install each of this as of the time this tutorial was written. NHC is exclusively a compiler.Chapter 2 Getting Started There are three well known Haskell system: Hugs. I'd get GHC. GHC and NHC. have all of them installed and use them for different purposes.very fast. Hugs is exclusively an interpreter. built-in support for module browsing. including: multi-parameter 5 . 2. in a sense the "de facto" standard. but allows function definitions in the environment (in Hugs you have to put them in a file). GHC . implements almost all of Haskell 98 (the standard) and most extensions. but produces smaller and often faster executables than does GHC. this is what I would suggest. However. since it contains both a compiler and interactive environment. personally. I. build in graphics library. I tend to use GHC to compile (primarily because I'm most familiar with it) and the Hugs interactive environment. implements all of Haskell 98 and extensions.org (the Haskell website) for up-to-date information. meaning that you cannot compile stand-alone programs with it.interactive environment is slower than Hugs. that is a fair amount to download an install. supports Haskell 98 and some extensions. good support for interfacing with other languages. as well as a number of advanced/experimental extensions.
then this must be specified using options on the command line. extensible records. scoped type variables..1. simply choose "Run This Program" and the installation will begin automatically. 2. On Windows 95/NT. • Internal data structures are initialized. GETTING STARTED type classes. From there. the Hugs interpreter is usually started with a command line of the form: hugs [option — file] . (This manual assumes that Hugs has already been successfully installed on your system. installation for Hugs is more of less identical to installation for any program on your platform. • Command line options are processed. The variable HUGSFLAGS holds these options. the heap is initialized. and its size is fixed at this point. the interpreter performs the following tasks: • Options in the environment are processed. .2 Installation procedures Once you've downloaded Hugs.1 Where to get it The official Hugs web page is at: http://haskell.hs or . there is a link titled "downloading" which will send you to the download page. Presumably if you're using a system which isn't otherwise supported. you know enough about your system to be able to run configure scripts and make things by hand. Hugs may be started by selecting it from the start menu or by double clicking on a file with the . just follow the on-screen instructions.1.org/hugs (http://haskell. installation differs depending on your platform.org/hugs) If you go there. in the environment or in the registry.6 CHAPTER 2.1. For Windows when you click on the "msi" file to download. For source first gunzip the file.and are used to customize the behaviour of the interpreter. you can download the appropriate version of Hugs for your computer. the registry is also queried for Hugs option settings. When Hugs starts. 2. rank-2 polymorphism. On Windows .lhs extension. then untar it. and restricted type synonyms. existentials. In particular..) Hugs uses options to set system parameters. if you want to run the interpreter with a heap size other than the default. From that page. 2. These options are distinguished by a leading + or . For RPMs use whatever RPM installation program you know best.3 How to run it On Unix machines. however.
haskell. If the prelude.. The current version as of the writing of this tutorial is 5. The most important option at this point is "+98" or "-98". 2. 2. cannot be found in one of the path directories or in the current directory. and a space and time profiler. Further information on the Hugs options is in the manual: http://cvs. In particular. which turns off all extensions.4 Program options To list all of the options would take too much space. multi-parameter type classes. the interpreter will not terminate if a problem occurs while it is trying to load one of the specified files.html (). If you've downloaded someone elses code and you're having trouble loading it. When you start in "-98". first make sure you have the "98" flag set properly. 2. The effect of a command hugs f1 . existential and universal quantification.2. then Hugs will terminate.1. optimising compiler and interactive environment for Haskell 98. go to the Hugs web page. but it will abort the attempted load command.org/ghc (GHC) to download the latest release. unboxed types.hs. The environment variables and command line options used by Hugs are described in the following sections.. Hugs will not run without the prelude file.. located in the file Prelude. • Program files specified on the command line are loaded.2 Glasgow Haskell Compiler The Glasgow Haskell Compiler (GHC) is a robust. When you start hugs with "+98" it is in Haskell 98 mode.1. exceptions. go to the Haskell web page. scoped type variables. 2. you are in Hugs mode and all extensions are turned on. From that page. fn is the same as starting up Hugs with the hugs command and then typing :load f1 ..1 Where to get it Go to the official GHC web page http://haskell. To get general Haskell help. for example: concurrency. It implements numerous experimental language extensions to Haskell 98.2.2 and can be downloaded off of the GHC download page (follow the "Download" link).org/Hugs/pages/hugsman/started. . The interpreter will look for the prelude file on the path specified by the -P option.5 How to get help To get Hugs specific help. a foreign language interface. GLASGOW HASKELL COMPILER 7 • The prelude file is loaded.2. GHC comes with a generational garbage collector.04. and so on. fully-featured. you can download the appropriate version of GHC for your computer. weak pointers. GHC compiles Haskell to either native code or C. fn.
then untar it.4 How to run the interpreter GHCi is invoked with the command "ghci" or "ghc –interactive". Presumably if you're using a system which isn't otherwise supported.2.2 Installation procedures Once you've downloaded GHC.hs" stipulates the name of the file to compile. From there. 2. just as if you had said :load modules at the GHCi prompt. "Main. you should say "-o main. look at the GHC users manual under "Installing GHC". For Windows when you click on the "msi" file to download. and the "-o main" means that you want to put the output in a file called "main". just follow the on-screen instructions.2. You can then run the program by simply typing "main" at the prompt. For source first gunzip the file. simply choose "Run This Program" and the installation will begin automatically. 2. Assuming that you have a program written with a main function in a file called Main. you know enough about your system to be able to run configure scripts and make things by hand. For RPMs use whatever RPM installation program you know best.8 CHAPTER 2. however. you can compile it simply by writing: % ghc --make Main.exe" to tell Windows that this is an executable file. GETTING STARTED 2.hs -o main The "–make" option tells GHC that this is a program and not just a library and you want to build it and all modules it depends on. installation differs depending on your platform. installation for GHC is more of less identical to installation for any program on your platform. this instructs GHCi to load the specified modules or filenames (and all the modules they depend on). For a more detailed description of the installation procedure. .hs. NOTE In Windows. One or more modules or filenames can also be specified on the command line.2.3 How to run the compiler Running the compiler is fairly easy.
2.3. When you start with "-fglasgowexts".3.2 2. you can get along with simplistic editor capable of just cut-n-paste. first make sure you have this flag set properly. Of course. .3 NHC About NHC. go to the Haskell web page. With respect to programming in Haskell. 2. go to the GHC web page. but good editor is capable of doing most of the chores for you.4 Editors With good text editor. NHC 9 2.4 2.5 Program options To list all of the options would take too much space.2. good text editor should have as much as possible of the following features: • Syntax highlighting for source files • Indentation of source files • Interaction with Haskell interpreter (be it Hugs or GHCi) • Computer-aided code navigation • Code completion . Further information on the GHC and GHCi options are in the manual off of the GHC web page. letting you concentrate on what you are writing. all extensions are turned on.3.5 Where to get it Installation procedures How to run it Program options How to get help 2.6 How to get help To get GHC(i) specific help. which turns off all extensions. The most important option at this point is "-fglasgow-exts".2. If you've downloaded someone elses code and you're having trouble loading it.3 2.3. 2.3. 2. When you start GHCi without "-fglasgow-exts" it is in Haskell 98 mode.1 2. programming is fun. . To get general Haskell help.3.
. . . You can quickly jump to the definition of chosen function with the help of "Definitions" menu. very smart and have to be seen in action to be believed. and . Indentation is aware about Haskell's 2-dimensional layout rules (see Section 7. GETTING STARTED At the time of writing. . having all the features listed above.haskell. . What's else available?.org/haskel ()). several options were available: Emacs/XEmacs support Haskell via haskell-mode and accompanying Elist code (available from http://www.10 CHAPTER 2. (X)Emacs seem to do the best job. . and name of the currently edited function is always displayed in the modeline.11.
However. that you can define infinitely large data structures. } By looking at this code. current. using imperative-esque psuedo-code. current." In the above psuedo-code. which will probably be quite alien if you are used to languages like C and Java. when we "run" makeList on the fifth line. because makeList would keep calling itself ad infinitum. Most importantly. the program would never terminate. The equivalent code in Haskell is: 11 lazy strict . we can see what it's trying to do: it creates a new list. for instance. we introduce the basic syntax of Haskell. the opposite of lazy. which means that no computation takes place unless it is forced to take place when the result of that computation is used. In addition to familiarizing you with the interactive environments and showing you how to compile a basic program. This means." while lazy languages are referred to as "call by name. we need to establish some general properties of Haskell.next = makeList().Chapter 3 Language Basics In this chapter we present the basic concepts of Haskell. This is because we assume this imperative-esque language is strict. This leads to an infinite loop. provided that you never use the entire structure. Strict languages are often referred to as "call by value. we could create an infinite list containing the number 1 in each position by doing something like: List makeList() { List current = new List(). before we talk about specifics of the language. if you actually wrote this code and called it. Haskell is a lazy language. For instance. return current. sets its value to 1 and then recursively calls itself to make the rest of the list.value = 1. Of course. we attempt to get a value out of it.
. Haskell is case-sensitive. LANGUAGE BASICS makeList = 1 : makeList This program reads: we're defining something called makeList (this is what goes on the left-hand side of the equals sign). we do not actually attempt to evaluate what makeList is at this point: we simply remember that if ever in the future we need the second element of makeList. print it to the screen. only the first 10 elements are ever calculated. 3. However. you are able to modify "global" variables from within a function. or calculate the sum of its elements. This is laziness. For instance. Functions that do not have side effects are called pure. we give the definition of makeList. are all side effecting operations. in a language like C or Java. a memory location or anything else of that nature. On the right-hand side. This is a side effect because the modification of this global variable is not related to the output produced by the function. . Now. you must not think of x as a register. . Furthermore. just as "Hal" is my name. "hello". Haskell eschews side effects. . or the square-root function). characters: 'a'. you're going to have to start thinking differently. since Haskell is lazy (or "call by name"). case-sensitive values types side effects pure . reading a file. A side effect is essentially something that happens in the course of executing a function that is not related to the output produced by that function. . this is not unusual. strings: "abc". x is simply a name. If you only use the first 10 elements. if you have a value x. the operation won't terminate because it would have to evaluate an infinitely long list. . What is unusual is that Haskell requires that the names given to functions and values begin with a lower-case letter and that the names given to types begin with an upper-case letter. Most languages have some system of types. be sure you haven't named your function Foo. By itself. An easy test for whether or not a function is pure is to ask yourself a simple question: "Given the same arguments. Being a functional language. numbers: 1. but Haskell actually uses case to give meaning. In Haskell. the colon operator is used to create lists (we'll talk more about this soon). . Second. You cannot arbitrarily decide to store a different person in my name any more than you can arbitrarily decide to store a different value in x. will this function always produce the same result?".. . This means that code that might look like the following C code is invalid (and has no counterpart) in Haskell: int x = 5. 'b'. Many languages are. ' '. if you attempt to write makeList to a file. However. The moral is: if your otherwise correct program won't compile. . the fact that the list is infinitely long doesn't matter. This right-hand side says that the value of makeList is the element 1 stuck on to the beginning of the value of makeList. . and types (the categories to which values belong). modifying the state of the real world is considered a side effect: printing something to the screen. Haskell distinguishes between values (for instance.12 CHAPTER 3. ). if you simply use a finite portion of the list (say the first 10 elements). . the function squares a value. All of this means that if you're used to writing code in an imperative language (like C or Java). 2. etc. we need to just look at makeList. Most importantly. or something else beginning with a capital letter. even functions: for instance.
and then. including plus (+). Try some of the following: Prelude> 5*4+3 23 Prelude> 5ˆ5-2 3123 Prelude> sqrt 2 1. In this way. see Chapter 2 for installation instructions). exponentiation (ˆ) and square-root (sqrt). They all carry the same interpretation. By not allowing destructive updates (or any other such side effecting operations). This property is called referential transparency and basically states that if two functions f and g produce the same values for the same arguments. minus (-). Start up your favorite interactive shell (Hugs or GHCi. when we define a function f. the number 5 is an expression (its value is 5). a Haskell shell can be used as a powerful calculator. That is. The shell will output to the screen a few lines talking about itself and what it's doing and then should finish with the cursor on a line reading: Prelude> From here. you can begin to evaluate expressions. and call that function with a particular argument a in the beginning of a program. NOTE There is no agreed-upon exact definition of referential transparency. most simple arithmetic operations are supported by Haskell. then we may replace f with g (and vice-versa). Destructive update does not exist in Haskell. we know we will get out the same result. For instance. it didn't increment a global counter). again call f with the same argument a. ARITHMETIC x = x + 1. the differences lie in how they are formalized.3. Values can be built up from other values. for instance. times (*).1. The definition given above is the one I like best. at the end of the program. Haskell code is very easy to comprehend.4142135623730951 expressions . An expression is basically something that has a value. destructive update referential transparency 3. You can experiment with these by asking the interactive shell to evaluate expressions and to give you their value.1 Arithmetic Let's begin our foray into Haskell with simple arithmetic. 5 + 6 is an expression (its value is 11). divided-by (/). In fact. 13 A call like x = x + 1 is called destructive update because we are destroying whatever was in x before and replacing it with a new value. This is because we know that a cannot have changed and because we know that f only depends on a (for instance.
we simply wrote sqrt 2. but in Haskell. The reason for this is that the "understood" grouping of the first expression is (5*4)+3.e.3).3) . Exercises Exercise 3. parentheses aren't required. Haskell also allows grouping by parentheses. since function application is so common. You could write it with the parentheses. and if extra parentheses make the intent of the code clearer. you may find it odd that sqrt 2 comes back with a decimal point (i. we may want to refer to a position by its x/y coordinate. Also note that parentheses aren't required around function arguments. in addition to the standard arithmetic operations..14 Prelude> 5*(4+3) 35 CHAPTER 3. as would be required in most other languages. Try the following: Prelude> (5. For instance. For instance. due to operator precedence. WARNING Even though parentheses are not always needed. LANGUAGE BASICS operator precedence We can see that. which would be a pair of integers. not sqrt(2). To make a pair of integers is simple: you enclose the pair in parenthesis and separate them with a comma. sometimes it is better to leave them in anyway. This interchangability of numeric types is due to Haskell's system of type classes and will be discussed in detail in Section 4. is a floating point number) even though the argument to the function seems to be an integer. Can you think of a way to determine whether function application binds more or less tightly than multiplication? 3.3) (5. we should also address multiple values. hence the difference between the values of 5*4+3 and 5*(4+3). Triples and More In addition to single values.2 Pairs. use them. Does it work? NOTE If you're familiar with programming in other languages. other people will probably have to read your code.1 We've seen that multiplication binds more tightly than division. Now try entering 2ˆ5000.
LISTS 15 ogeneous Here. and so on are called tuples and can store fixed amounts of heterogeneous data. 3. We can define a list like: . if you try to use them on a larger tuple. They are.3. In Haskell. A data structure that can hold an arbitrary number of elements is a list. In general. we write: Prelude> (1. "hello") 5 Prelude> snd (5."foo"). There are two predefined functions that allow you to extract the first and second elements of a pair. and so on.3) Prelude> (1. triples hold three. pairs are allowed to be heterogeneous. fst and snd.3). 5 and 3. respectively. the first element of a pair need not have the same type as the second element: that is.3. respectively.4) (1. To define a triple and a quadruple. which must be made up of elements of all the same type (we will discuss lists further in Section 3. The meaning of this error message will be explained in Chapter 4. you will get a message stating that there was a type error. we have a pair of integers. Lists are assembled in a very similar fashion to tuples.2.'a').2 Use a combination of fst and snd to extract the character out of the tuple ((1. tuples Exercises Exercise 3. NOTE The functions fst and snd won't work on anything longer than a pair.3. You can see how they work below: Prelude> fst (5. "hello") "hello" In addition to pairs. you can define triples.4) And so on. except that they use square brackets instead of parentheses.2.2. quadruples etc. This contrasts with lists.3.2. triples. pairs. you can have a pair of an integer with a string.3 Lists The primary limitation of tuples is that they hold only a fixed number of elements: pairs hold two. For instance.3) (1.
4).2. Try some of the following: Prelude> [(184.108.40.206].[5.2.4] [5.2.9). Unlike tuples. The colon is called the "cons" operator.3. Of course. "syntactic sugar" is a strictly unnecessary language feature.(4.1.1). NOTE In general.(2. similarly.3. they can also contain tuples or even other lists.2. lists must be homogenous.(4.2] [0. a type error will be reported.4).3. lists don't have to just contain integers or strings.9)." The etymology of this is that we are constructing a new list from an element and an old list. we can very easily add an element on to the beginning of the list using the colon operator.4] We can actually build any list by using the cons operator (the colon) and the empty list: Prelude> 5:1:2:3:4:[] [5.3] Lists don't need to have any elements. If we write something using the [5.4] notation.4] syntactic sugar cons operator In fact.(2. If you try to.2] Prelude> [1.3. We can see the cons operator in action in the following examples: Prelude> 0:[1.1.[5.7]) homogenous .3] [1.16 CHAPTER 3.2.4] syntax is "syntactic sugar" for the expression using the explicit cons operators and empty list. while tuples are heterogeneous.6.3. the process of adding an element is called "consing. This means that you cannot have a list that holds both integers and strings.16)] [(1. LANGUAGE BASICS Prelude> [1. which is added to make the syntax nicer. Tuples. can contain lists and other tuples. The empty list is simply [].2] [1.1).1. One further difference between lists and tuples is that.7]) ([1.2.1.(3.3.2] Prelude> 5:[1.4]. the [5. the compiler simply translates it to the expression using (:) and [].3.(18.104.22.168)] Prelude> ([1.1.
However.2. and the tail function returns all but the first element of a (non-empty) list. the interpreter has inferred that you're trying to add three to something. LISTS 17 There are two basic list functions: head and tail.10] 1 Prelude> length (tail [1.4. This means that when we execute read "Hello".3.3. so an error is reported.4. we can create the string "Hello" as: Prelude> 'H':'e':'l':'l':'o':[] "Hello" Lists (and. the exact error message is implementation dependent. However. you use the length function: Prelude> length [1. non-string values can be converted to strings using the show function.2. strings) can be concatenated using the ++ operator: Prelude> "Hello " ++ "World" "Hello World" Additionally. and strings can be converted to non-string values using the read function. of course. an error will be reported (note that this is a run-time error.10] 5 Prelude> head [1. not a compile-time error): Prelude> "Five squared is " ++ show (5*5) "Five squared is 25" Prelude> read "5" + 3 8 Prelude> read "Hello" + 3 Program error: Prelude. To get the length of a list. we expect to be returned a number.10]) 4 3. The head function returns the first element of a (non-empty) list. if you try to read a value that's malformed. So.read: no parse Above.3.3.1 Strings In Haskell. . "Hello" cannot be parsed as a number.3. Of course.3.4.2. a String is simply a list of Chars.
an initial value and a list. There are three primary list-processing functions: map.isLower "Hello World" "elloorld" The function foldr takes a little more getting used to. we get: 3 + 8 + 12 + 5 + 0 which sums the list. When you map across a list.toUpper that takes as input a Char and produces a Char that is the upper-case version of the original argument. there is a built-in function Char. In Hugs. the function Char. filter and foldr (also foldl).toUpper "Hello World" "HELLO WORLD" WARNING Hugs users: Hugs doesn't like qualified names like Char.5] 160 . the length of the list never changes – only the individual values in the list change. if we have a list: 3 : 8 : 12 : 5 : [] and we apply foldr (+) 0 to it.12. We can test this: Prelude> foldr (+) 0 [3. For instance.8.3. Thus. To remove elements from the list. So. we can map the toUpper function across the entire list: Prelude> map Char.2 Simple List Functions Much of the computation in Haskell programs is done by processing lists. foldr takes three arguments: a function. simply use toUpper. The best way to think about foldr is that it replaces occurences of the list cons operator (:) with the function parameter and replaces the empty list constructor ([]) with the initial value. you can use the filter function. The map function takes as arguments a list of values and a function that should be applied to each of the values. LANGUAGE BASICS 3. This function allows you to remove certain elements from a list depending on their value.18 CHAPTER 3.toUpper. We can filter out all non-lowercase characters using this: Prelude> filter Char. to covert an entire string (which is simply a list of characters) to upper case. For instance.5] 28 We can perform the same sort of operation to calculate the product of all the elements on a list: Prelude> foldr (*) 1 [4. but not on their context.isLower tells you whether a given character is lower case.8.
taking the last element and combining it with the initial value via the provided function. This raises a question as to what happens when the function isn't associative (a function (·) is associative if a · (b · c) = (a · b) · c). The way it accomplishes this is essentially by going all the way down the list. The derivation here proceeds in the opposite fashion: foldl foldl foldl foldl (-) (-) (-) (-) 1 [4.4) .5]) 4 . Thus.1)) 4 .(5 . do we mean ((4 · 8) · 5) · 1 or 4 · (8 · ((5 · 1))? foldr assumes the function is right-associative (i.5] ((1 .4) [8.5] 4 .foldr (-) 1 [5]) 4 .8.(8 .4) .4 0 ==> ==> ==> ==> ==> ==> ==> The foldl function goes the other way and effectively produces the opposite bracketing.8.8. It does so until there is no more list left. we need to specify where to put the parentheses.12. the correct bracketing is the latter). when we use it on a non-associtive function (like minus).5] 28 However.8.5] (1 .8.3.8) . foldl looks the same when applied.5) [] ==> ==> ==> .(8 .3.(8 ..e. LISTS 19 We said earlier that folding is like replacing (:) with a particular function and ([]) with an initial element.5] -16 This is because foldl uses the opposite bracketing. we get different results when using the non-associative function minus: Prelude> foldl (-) 1 [4.8) [5] (((1 .(5 . Namely.(8 .foldr (-) 1 [])) 4 . so we could have done summing just as well with foldl: Prelude> foldl (+) 0 [3. When we write 4 · 8 · 5 · 1.4) 4 .5] 0 The exact derivation of this looks something like: foldr (-) 1 [4. we can see the effect: Prelude> foldr (-) 1 [4. It then takes the second-to-last element in the list and combines it to this new value.(foldr (-) 1 [8.
save it and then use it.False. Here. However. So. Exercises Exercise 3. LANGUAGE BASICS Note that once the foldl goes away. on "aBCde" it should return 3. That is. write a function using a fold that will return the maximum value in a list (and zero if the list is empty). For instance.8) .5 (-11) .5] simply returns the same list. each element in the new list representing whether or not the original element was a lower-case character. Exercise 3.3 Use map to convert a string into a list of booleans.2 and 2.3 how to write a Hello World program and how to compile it. when provided with [(18.104.22.168 ((-3) .5 -16 CHAPTER 3. create a file called Test.20 ==> ==> ==> ==> ((1 .8.hs and enter the following code: .5 We've seen how to calculate sums and products using folding functions. Assume that the values in the list are always ≥ 0. This is because before foldl does anything. Exercise 3.8) . it should take the string "aBCde" and return [True.'a')]. NOTE foldl is often more efficient than foldr for reasons that we will discuss in Section 7. A similar function using foldl would fail to produce any output. Exercise 3.(6. while foldl cannot.4 Source Code Files As programmers.10. foldr starts producing output immediately. On the other hand. that's okay.4. foldr can work on infinite lists. Given that the function max returns the maximum of two numbers. Explain to yourself why it works. We'll discuss them further in Section 7.2.False. If this discussion of the folding functions is still somewhat unclear. it would produce output.True.'c').'b').4 Use the functions mentioned in this section (you will need two of them) to compute the number of lower-case letters in a string. when applied to [5.True]. Even if the list were infinite.2. We already saw in Sections 2.1] it will return 10. write code in our editor of choice.4) . we show how to use functions defined in a source-code file in the interactive environment. it has to go to the end of the list. For instance.8. To do this. So. the parenthesization is exactly the opposite of the foldr. we don't want to simply evaluate small expressions like these – we want to sit down.6 Write a function that takes a list of pairs of length at least 2 and returns the first component of the second element in the list. it will return 1.(1. 3. foldr (:) [] [1.
the interpreter will print various data to explain what it is doing."Hello") Test> z 30 Perfect. The Prelude module (usually simply referred to as "the Prelude") is always loaded and contains the standard definitions (for instance. For example: Test> x 5 Test> y (6. if you already have one of them loaded. we can use things that were defined in it.hs % ghci Test. there are three definitions: x. It defines a module called "Test" (in general module names should match file names.hs This will start Hugs or GHCi. Now that we've loaded Test.hs .. Once you've written and saved this file. Exactly correct. by executing either of the following: % hugs Test. fst. load this in your favorite interpreter. in the directory in which you saved it. You'll notice that where it used to say "Prelude" it now says "Test. If any errors appear.3.4. double check and then try again. In order for a program to be an executable. Alternatively. as: Prelude> :l Test. see Section 6 for more on this). you probably mistyped something in the file. it must have the module name the Prelude . You're probably thinking that "Prelude" must also be a module. just as we expected! One final issue regarding how to compile programs to stand-alone executables remains. the (:) operator for lists. "Hello") z = x * fst y This is a very simple "program" written in Haskell.. SOURCE CODE FILES 21 module Test where x = 5 y = (6. Test> Between the first and last line. and load the file. respectively." That means that Test is the current module. or (+) or (*). In this module. y and z. snd and so on). you can use the ":load" command (or just ":l") to load a module.
exe" This will create a file called "test" (or on Windows.exe") that you can then run. % . For example. as it is a functional language. 0 if its argument is 0. functions are central to Haskell.hs and rename it to "Main" (change the line that reads module Test to module Main). we can start writing functions.hs -o test NOTE For Windows.22 CHAPTER 3.hs file. As you might have expected. We then say that the value of square x is equal to x * x. Try this: main = putStrLn "Hello World" Now. and 1 if its argument is greater than 0 (this is called the signum function): . which we call x. in GHC. on Windows: C:\> test. you would say: % ghc --make Test. This means that the evaluation of a program is simply the evaluation of a function. it would be "-o test. we simply need to add a main function. "test./test Hello World NOTE Or. and compile it (refer back to Section 2 for information on how to do this for your compiler). we could define a function that returns −1 if its argument is less than 0. LANGUAGE BASICS "Main" and must contain a function called main. save the file.exe Hello World 3.5 Functions Now that we've seen how to write code in a file. For instance. So. Haskell also supports standard conditional expressions. We can write a simple function to square a number and enter it into our Test. we say that we're defining a function square that takes one argument (aka parameter). if you go in to Test. We might define this as follows: square x = x * x In this function definition.
instead of typing :l Test. It evaluates the condition (in this case x < 0 and. it evaluates the else condition).5. however.hs again. You can test this program by editing the file and loading it back into your interpreter. Writing this function using if statements would be long and very unreadable. you must have both a then and an else clause.4 for all of then details). a value of 2 if its argument were 2. also supports case constructions. if this evaluates to True.3. if the condition evaluated to False. a value of 5 if its argument were 1. you can simply type :reload or just :r to reload the current file. The if/then/else construct in Haskell is very similar to that of most other programming languages. it evaluates the then condition. the system will think you are trying to subtract the value "1" from the value "signum. If Test is already the current module." which is illtyped. Haskell. These are used when there are multiple values that you want to check against (case expressions are actually quite a bit more powerful than this – see Section 7. so we write it using a case statement as follows (we call this function f): f x = case x 0 -> 1 -> 2 -> _ -> if/then/else case/of of 1 5 2 -1 . and a value of −1 in all other instances. if missing. This is usually much faster. like many other languages. FUNCTIONS 23 signum x = if x < 0 then -1 else if x > 0 then 1 else 0 You can experiment with this as: Test> 1 Test> 0 Test> -1 Test> -1 signum 5 signum 0 signum (5-10) signum (-1) Note that the parenthesis around "-1" in the last example are required. Suppose we wanted to define a function that had a value of 1 if its argument were 0.
or you're likely to run in to problems. then the value of f is 2. a semicolon is inserted before every new line that is indented the same amount. WARNING Because whitespace matters in Haskell. From then on. that's probably better. This is perfectly acceptable.24 CHAPTER 3.11 for a more complete discussion of layout). you're free to structure the code as you wish. a close-brace is inserted. but if you follow the general rule of indenting after each of those keywords. Functions can also be defined piece-wise. the value of f is −1 (the underscore can be thought of as a "wildcard" – it will match anything) . Haskell uses a system called "layout" to structure its code (the programming language Python uses a similar system). If it maches 2. meaning that you can write one version of your function for certain parameters and then another version for other parameters. If it matches 0. If it matches 1. In this style. The following is also equally valid: f x = case x of { 0 -> 1 . we're defining f to take an argument x and then inspect the value of x. 1 -> 5 . you'll never have to remember it (see Section 7. If a following line is indented less. If you can configure your editor to never use tabs. and the column position at which the next command appears is remembered. This may sound complicated. For instance. 2 -> 2 . 2 -> 2 . do and of. The indentation here is important. _ -> 1 } However. and if it hasn't matched anything by that point. the above function f could also be written as: f f f f 0 1 2 _ = = = = 1 5 2 -1 . Some people prefer not to use layout and write the braces and semicolons explicitly. _ -> 1 } Of course. the above function might look like: f x = case x of { 0 -> 1 . make sure your tabs are always 8 spaces long. let. you need to be careful about whether you are using tabs or spaces. The layout system allows you to write code without the explicit semicolons and braces that other languages like C and Java require. The general rule for layout is that an open-brace is inserted after the keywords where. if you write the braces and semicolons explicitly. the value of f is 1. 1 -> 5 . the value of f is 5. structuring your code like this only serves to make it unreadable (in this case). LANGUAGE BASICS wildcard In this program. If not.
square) means that it creates a new function that takes an argument. regardless of its argument (most compilers will warn you about this. and then applying that to f . This style of piece-wise definition is very popular and will be used quite frequently throughout this tutorial. applies f to that argument and then applies square to the result. Conversely. more mathematically oriented. That is. when we wrote 5*4+3. we were evaluating 5 ∗ 4 and then applying +3 to the result. We can do the same thing with our square and f functions: Test> 25 Test> 4 Test> 5 Test> -1 square (f 1) square (f 2) f (square 1) f (square 2) The result of each of these function applications is fairly straightforward. NOTE In mathematics we write f ◦ g to mean "f following g.) function (called the function composition function). too. if we write (square . 1 or 2 were applied to it (most compilers will warn you about this. g also to mean "f following g. There is another. takes two functions and makes them in to one. this means that it creates a new function that takes an argument. More complicated functions can be built from simpler functions using function composition. using the (. saying something about overlapping patterns). in the first line. and f would return -1. way to express function composition.1). Function composition is simply taking the result of the application of one function and using that as an argument for another. taking the result. applies square to that argument and then applies f to the result.3.5. These two definitions of f are actually equivalent – this piece-wise version is translated into the case expression." in Haskell we write f . The parentheses around the inner function are necessary." which has no meaning. though. The (. applying the value x to the function f ◦ g is the same as applying it to g. the order is important. We can see this by testing it as before: Test> (square . Function application like this is fairly standard in most programming languages. If we had put the last line first. (f . it would have matched every argument. f). This (. f) 1 25 function composition . We've already seen this way back in arithmetic (Section 3.) (just a single period) function. the interpreter would think that you were trying to get the value of "square f. saying something about incomplete patterns). In this. f would produce an error if anything other than 0. If we had not included this last line. For instance." The meaning of f ◦ g is simply that (f ◦ g)(x) = f (g(x)). FUNCTIONS 25 Here.) function is supposed to look like the (◦) operator in mathematics. otherwise.
the Haskell compiler will think we're trying to compose square with the value f 1 in the first line.2.4] head [1.2) snd (5. square) 2 -1 CHAPTER 3.3.4] tail [1.41421 Prelude> "hello" Prelude> 5 Prelude> 5 Prelude> 2 Prelude> True Prelude> False Prelude> 1 Prelude> [2.26 Test> (square .3.4] [1. at some point. f) 2 4 Test> (f . Here are some simple functions. which makes no sense since f 1 isn't even a function.3.3. Undoubtedly.5.6] . otherwise.2. but if we can keep this to a minimum.2) null [] null [1. some of which we've already seen: sqrt the square root function id the identity function: id x = x fst extracts the first element from a pair snd extracts the second element from a pair null tells you whether or not a list is empty head returns the first element on a non-empty list tail returns everything but the first element of a non-empty list ++ concatenates two lists == checks to see if two elements are equal /= checks to see if two elements are unequal Here. LANGUAGE BASICS Here.4] Prelude> sqrt 2 id "hello" id 5 fst (5. that would save a lot of time. square) 1 5 Test> (f . you will accidentally rewrite some already-existing function (I've done it more times than I can count).2.2. we show example usages of each of these functions: Prelude> 1.3] ++ [4. we must enclose the function composition in parentheses. It would probably be wise to take a little time-out to look at some of the functions that are defined in the Prelude.
2.4*a*c) twice_a = 2*a in ((-b + det) / twice_a.3. Just make sure they're indented the same amount. you can provide multiple declarations inside a let. or you will have layout problems: roots a b c = let det = sqrt (b*b . For instance.4*a*c) occurred.4. We can do this using a let/in declaration: roots a b c = let det = sqrt (b*b . That is. We could write the following function to compute the two values of x: roots a b c = ((-b + sqrt(b*b .2.6] Prelude> [1. 3.5. For instance.5.1 Let Bindings Often we wish to provide local declarations for use in our functions. det and then use that in both places where sqrt(b*b . (-b . Haskell allows for local bindings. FUNCTIONS [1.sqrt(b*b . we can create values inside of a function that only that function can see. the following equation is used √ find the roots (zeros) of a polynomial of the form ax2 + bx + c = 0: x = to (−b ± b2 − 4ac)/2a. say. if you remember back to your grade school mathematics courses.4*a*c)) / (2*a).2.3] == [1. we could create a local binding for sqrt(b*b-4*a*c) and call it.det) / (2*a)) In fact.4*a*c)) / (2*a)) To remedy this problem.det) / twice_a) . (-b .3] True Prelude> 'a' /= 'b' True Prelude> head [] Program error: {head []} 27 We can see that applying head to an empty list gives an error (the exact error message depends on whether you're using GHCi or Hugs – the shown error message is from Hugs).4*a*c) in ((-b + det) / (2*a). (-b .3.5.
In Hugs.f is a function which takes an integer and produces integer.28 CHAPTER 3. You can use them in non-infix mode by enclosing them in parentheses.this is an embedded .toUpper. Block comments begin with {. Block comments can be nested. and {and -} correspond to /* and */. 3. NOTE The -. (*). {. (++) are all infix functions. Hence.and extend until the end of the line. rather than letters. For example: module Test2 where main = putStrLn "Hello World" -.in Haskell corresponds to // in C++ or Java. For instance.and extend to a corresponding -}. Comments are used to explain your program in English and are completely ignored by compilers and interpreters.2 Infix Infix functions are ones that are composed of symbols.6 Comments There are two types of comments in Haskell: line comments and block comments. non-infix functions (like map) can be made infix by enclosing them in backquotes (the ticks on the tilde key on American keyboards): Prelude> map Char.toUpper "Hello World" "HELLO WORLD" Prelude> Char.toUpper 'map' "Hello World" "HELLO WORLD" WARNING Hugs users: Hugs doesn't like qualified names like Char.write a string -. (+). Line comments begin with the token -. simply use toUpper.to the screen {.5. the two following expressions are the same: Prelude> 5 + 10 15 Prelude> (+) 5 10 15 Similarly. LANGUAGE BASICS 3.
1 maps to 5 2 -> 2 -. for loops don't make much sense in Haskell because they require destructive update (the index variable is constantly being updated).7 Recursion In imperative languages like C and Java. return fact. If you think of ! as a function.3. 3. then it is calling itself. .2 maps to 2 _ -> -1 -. for (int i=2. We can translate this definition almost verbatim into Haskell code: factorial 1 = 1 factorial n = n * factorial (n-1) This is likely the simplest recursive function you'll ever see. The prototypical recursive function is the factorial function. you might write this as something like: int factorial(int n) { int fact = 1. usually given as: n! = 1 n ∗ (n − 1)! n=1 otherwise This definition itself is exactly a recursive definition: namely the value of n! depends on the value of (n − 1)!.everything else maps to -1 29 This example program shows the use of both line comments and (embedded) block comments.7. } While this code fragment will successfully compute factorials for positive integers. the most basic control structure is a loop (like a for loop). However. A function is recursive if it calls itself (see Appendix B for more). i <= n. In an imperative language. but it is correct. Haskell uses recursion. it somehow ignores the basic definition of factorial. i++) fact = fact * i. RECURSION comment -} the original comment extends to the matching end-comment token: -} f x = case x of 0 -> 1 -. Instead.0 maps to 1 1 -> 5 -. Recursive functions exist also in C and Java but are used less than they are in functional languages.
this translates directly into Haskell code: exponent a 1 = a exponent a b = a * exponent a (b-1) Just as we can define recursive functions on integers. usually the base case is the empty list []. it is often useful to try to differentiate these two cases. This problem has a single base case: namely when b is 1. Consider the task of calculating the length of a list.e. } but this is likely to be much slower than the loop version in C. if we have a cons list. In this case.. However. We can again break this down into two cases: either we have an empty list or we have a non-empty list. we can define a length function as: my_length [] = 0 my_length (x:xs) = 1 + my_length xs NOTE Whenever we provide alternative definitions for standard Haskell functions. We can write a general form as: ab = a a ∗ ab−1 b=1 otherwise Again. so can we define recursive functions on lists. there is one base case (when n = 1) and one recursive case (when n > 1). In the case of factorial. and that we want to calculate ab . Recursion can be a difficult thing to master. For designing your own recusive algorithms.30 CHAPTER 3. Clearly the length of an empty list is zero. else return n * factorial(n-1). Turning now to the task of exponentiation. . It is completely analogous to the concept of induction in mathematics (see Chapter B for a more formal treatment of this). we prefix them with my so the compiler doesn't become confused. LANGUAGE BASICS NOTE Of course. then the length of this list is just the length of its tail plus one. suppose that we have two positive integers a and b. usually a problem can be thought of as having one or more base cases and one or more recursive-cases. an imperative recursive version could be written: int factorial(int n) { if (n == 1) return 1. Furthermore. and the recursive case is a cons list (i. Thus. The recursive case is when b > 1. a value consed on to another list).
3. but only uses addition (i.7 The fibonacci sequence is defined by: Fn = 1 Fn−2 + Fn−1 n = 1 or n = 2 otherwise Write a recursive function fib that takes a positive integer n as a parameter and calculates Fn . no fair just using multiplication). when presented with an empty list.9 Define a recursive function my map that behaves identically to the standard function map. the base case is the empty list. When presented with a list of the form (x:xs).. Begin by making a mathematical definition in the style of the previous exercise and the rest of this section. we can consider the filter function. this time. then we exclude x and return the result of filtering the tail of the list.8. we need to decide whether or not to keep the value x. we simply return an empty list. we're choosing whether to keep an element. However. . and the recursive case is a cons list. then we return a list that begins with x followed by the result of filtering the tail of the list. INTERACTIVITY 31 Similarly. depending on whether or not a particular predicate holds. Exercise 3.8 Define a recursive function mult that takes two positive integers a and b and returns a*b. To do this. it is not entirely clear how one should handle operations like user input.8 Interactivity If you are familiar with books on other (imperative) languages. Again. Exercise 3. See the exercises for the definition of map and Chapter 7 for the folds. This is because filter cannot add elements. it can only remove them. The reason for this is simple: Being a pure functional language.e. We can also define map and both fold functions using explicit recursion. you might be wondering why you haven't seen many of the standard programs written in tutorials of other languages (like ones that ask the user for his name and then says "Hi" to him by name). We can define the filter function as: my_filter p [] = [] my_filter p (x:xs) = if p x then x : my_filter p xs else my_filter p xs In this code. If p x is false. If p x is true. we use an if statement and the predicate p. 3. Exercises Exercise 3.
the second would be interpreted as (putStrLn "Hello. so without the parenthesis. we import the IO library.getLine putStrLn ("Hello. The way to do this is to use the do keyword. This allows us to specify the order of operations (remember that normally. to write a simple program that asks a user for his name and then address him directly. so that we can access the IO . LANGUAGE BASICS monads do notation After all. then you no longer have a function. how are you?") NOTE The parentheses are required on the second instance of putStrLn but not the first. but for now.. since Haskell is a lazy language. and the user types something the first time and something else the second time. though. a branch of formal mathematics: monads. think of them simply as a convenient way to express operations like input/output. So.." so that we can compile it. The solution to this was found in the depths of category theory.." so that the compile knows that this is the function to run when the program is run. We'll discuss them in this context much more in Chapter 5 and then discuss monads for monads' sake in Chapter 9. On the fourth line. If you call this function twice. We name the primary function "main. enter the following code into "Name. Suppose we want to write a function that's interactive.32 CHAPTER 3. Let's go back and look at the code a little. You can then either load this code in your interpreter and execute main by simply typing "main. We're not yet ready to talk about monads formally. This is because ++ binds more tightly than function application.hs": module Main where import IO main = do hSetBuffering stdin LineBuffering putStrLn "Please enter your name: " name <. " ++ name ++ ". I'll show the results of the interactive approach: Main> main Please enter your name: Hal Hello. the order in which operations are evaluated in it is unspecified). since it would return two different values." or you can compile it and run it from the command line. ") ++ name ++ . suppose you have a function that reads a string from the keyboard. Hal. how are you? Main> And there's interactivity. We name the module "Main.
it waits until it's gotten a whole block. it expects to read it in rather large blocks." This would normally be written "name = getLine." The last line constructs a string using what we read in the previous line and then prints it to the screen.getLine let guessNum = read guess if guessNum < num then do putStrLn "Too low!" doGuessing num else if read guess > num then do putStrLn "Too high!" doGuessing num else do putStrLn "You Win!" Let's examine this code. We want to get rid of this. Using this. On the fifth line we write "import Random" to tell the . Enter the following code into "Guess.3.randomRIO (1::Int. telling Haskell that we're executing a sequence of commands. A typical person's name is nowhere near large enough to fill this block. In this context." but using the arrow instead of the equal sign shows that getLine isn't a real function and can return different values. so we tell it to use LineBuffering instead of block buffering. when GHC reads input. INTERACTIVITY 33 functions. which prints a string to the screen. Another example of a function that isn't really a function would be one that returns a random value.getLine. Thus.hs": module Main where import IO import Random main = do hSetBuffering stdin LineBuffering num <. when we try to read from stdin. this is only required by GHC – in Hugs you can get by without it). On the seventh line. we say "name <. and store the results in name. a function that does this is called randomRIO. The necessity for this is because. we start with do.8. On the ninth line. The first command is hSetBuffering stdin LineBuffering. This command means "run the action getLine. The next command is putStrLn. we can write a "guess the number" program. 100) putStrLn "I'm thinking of a number between 1 and 100" doGuessing num doGuessing num = do putStrLn "Enter your guess:" guess <. which you should probably ignore for now (incidentally.
we ask for a random number in the range (1. we check to see if they guessed too high. However. In the first line of main. The if statement checks first to see if their guess is too low. We need to write ::Int to tell the compiler that we're using integers here – not floating point numbers or other numbers. on the last line of main. so they must have gotten it correct. we tell the user what's going on. You can either compile this code or load it into your interpreter. and num is an integer. Note that while we're in do notation. the "obvious" way to write the command is actually . we don't need ins for lets. On the next line. We'll talk more about this in Section 4. they didn't guess too low and they didn't guess too high. The fact that we exit is implicit in the fact that there are no commands following this. pure function (and not an IO action). First. If they did. If they guessed too low. we cannot). Otherwise. we tell them and start doGuessing again.34 CHAPTER 3. since guess is a string. it asks the user to guess and then accepts their guess (which is a String) from the keyboard. 100). we tell the compiler to execute the command doGuessing. we simply bind the value to guessNum. Since "read guess" is a plain. We don't need an explicit return () statement. We tell them that they won and exit. The doGuessing function takes the number the user is trying to guess as an argument. and then. we don't need to use the <. In the case when it does. we inform them and then start doGuessing over again. If they didn't guess too low. and you will get something like: Main> main I'm thinking of a number between 1 and 100 Enter your guess: 50 Too low! Enter your guess: 75 Too low! Enter your guess: 85 Too high! Enter your guess: 80 Too high! Enter your guess: 78 Too low! Enter your guess: 79 You Win! The recursive action that we just saw doesn't actually return a value that we use in any way. LANGUAGE BASICS compiler that we're going to be using some random functions (these aren't built into the Prelude). we first need to convert guess to an integer by reading it.notation (in fact.
its factorial. we will give the incorrect version. then give the correct version. specifically with the term word : askForWords.getLine if word == "" then return [] else do rest <. the product of all the numbers. Let's say we're writing a simple program that repeatedly asks the user to type in a few words. The primary function (actually. a session might look like: .askForWords return (word : rest) Here. askForWords). Here. when run. That means that before we can attach anything to the front. you should have a good understanding of how to write simple functions. he just hits enter without typing anything).10 Write a program that will repeatedly ask the user for numbers until she types in zero. In this case. for each number. we need to run the action and take the result. Remeber that when using (:).e. However. INTERACTIVITY 35 incorrect. we are making a list out of an element (in this case word) and another list (in this case. By now. take the result and store it in the variable rest. The incorrect formulation of this might look something like: askForWords = do putStrLn "Please enter a word:" word <. and then either continues or ends. it is an action that. explain why it is wrong.8. checks to see if it's empty. If at any point the user enters the empty word (i. will produce a list. The error is on the last line.getLine if word == "" then return [] else return (word : askForWords) Before reading ahead. see if you can figure out what is wrong with the above code. we return the list created from word and rest.3. the program prints out everything he's typed up until that point and then exits. we first run askForWords. an action) in this program is one that asks the user for a word. Exercises Exercise 3. askForWords is not a list.. and manipulate lists. compile them. and. at which point it will tell her the sum of all the numbers. we want to do something like: askForWords = do putStrLn "Please enter a word:" word <. test functions and programs in the interactive environment. For instance. Then.
. it recurses itself and then makes a list out of the number it just read and the result of the recursive call. if it's zero.36 CHAPTER 3. returns the empty list. If it's not zero. LANGUAGE BASICS Give me a number (or 5 Give me a number (or 8 Give me a number (or 2 Give me a number (or 0 The sum is 15 The product is 80 5 factorial is 120 8 factorial is 40320 2 factorial is 2 0 to stop): 0 to stop): 0 to stop): 0 to stop): Hint: write an IO action that reads a number and.
when you define a variable. Haskell uses a system of type inference. it is sometimes considered good style to explicitly specify the types of outermost functions. int. Both Hugs and GHCi allow you to apply type inference to an expression to find its type. for "character. In Haskell. if you have a function which expects an argument of a certain type and you give it the wrong type. This means that you don't even need to specify the type of expressions. a compile-time error will be generated (that is. String (for 37 .1 Simple Types There are a slew of built-in types. etc. you need to specify its type (for instance. you needn't do this – the type will be inferred from context.). including Int (for integers. NOTE If you want. For comparison." Then. 4. char. you will not be able to compile the program). This vastly reduces the number of bugs that can creep into your program. In fact. For instance. For instance 'a' would have type Char. Furthermore. this often helps debugging. both positive and negative). Char (for single characters).Chapter 4 Type Basics Haskell uses a system of static type checking. Double (for floating point numbers). you certainly are allowed to explicitely specify the type of an expression. This is done by using the :t command. This means that every expression in Haskell is assigned a type. in C. start up your favorite shell and try the following: Prelude> :t 'c' 'c' :: Char This tells us that the expression 'c' has type Char (the double colon :: is used throughout Haskell to specify types).
for instance. The third line tells us the inferred type of this term and the fourth line tells us what it needs to have matched. instead of "a" in the previous example. a test of equality: Prelude> :t 'a' == 'b' 'a' == 'b' :: Bool You should note than even though this expression is false. NOTE Bool is short for Boolean (usually pronounced "boo-lee-uhn". and others. For instance. it still has a type. namely the type Bool. this has no effect since there's only one possible interpretation of "a". the equality operator requires that the type of both of its arguments are of the same type. consider the case of numbers. In this case. it says that type type Char doesn't match the type [Char] (a list a characters – a string in Haskell is represented as a list of characters). We can see that Char and String are of different types by trying to compare a character to a string: Prelude> :t 'a' == ERROR . You can try: Prelude> :t 5 :: Int 5 :: Int Prelude> :t 5 :: Double 5 :: Double .Type error *** Expression *** Term *** Type *** Does not match "a" in application : 'a' == "a" : 'a' : Char : [Char] The first line of the error (the line containing "Expression") tells us the expression in which the type error occured. we could have written ("a"::String). As mentioned before. let's examime one of type String: Prelude> :t "Hello" "Hello" :: String You can also enter more complicated expressions. We have already seen an expression of type Char. In this case.38 CHAPTER 4. though I've heard "boo-leen" once or twice) and has two possible values: True and False. TYPE BASICS strings). However. You can observe the process of type checking and type inference by trying to get the shell to give you the type of an ill-typed expression. The second line tells us which part of this expression is ill-typed. For instance. you can explicitely specify the type of an expression using the :: operator.
(5::Int) + (10::Double) 4. This essentially means that you can have type variables.9] [6. (5. 'h':'e':'l':'l':'o':[] 2.9] Prelude> tail "hello" "ello" Prelude> tail ["the".6.8. briefly."happy"] This is possible because tail has a polymorphic type: [α] → [α].7. POLYMORPHIC TYPES 39 Here.1 Figure out for yourself. [5."is". and then verify the types of the following expressions. The way to read this. For instance. What if we don't specify the type? Prelude> :t 5 5 :: Num a => a Not quite what you expected? What this means. (a."is".2 Polymorphic Types Haskell employs a polymorhpic type system. then type type of the expression 5 can be of type a. is that if some type a is an instance of the Num class." Exercises Exercise 4.'a'] 3.b) -> a . If that made no sense.3 we talk extensively about type classes (which is what this is).7. that's okay for now. In Section 4.8. if they have a type.4.'a') 4. we can see that the number 5 can be instantiated as either an Int our a Double. note that a function like tail doesn't care what the elements in the list are: Prelude> tail [5."man". which we have alluded to before."happy"] ["man". That means it can take as an argument any list and return a value which is a list of the same type. though. Also note if the expression is a type error: 1. is to say "a being an instance of Num implies a. (5::Int) + 10 5. The same analysis can explain the type of fst: Prelude> :t fst forall a b .2.
b) to a. Java.40 CHAPTER 4.). Before we delve too deeply into the subject of type classes. That is. however.3 Type Classes We saw last section some strange typing having to do with the number five. type variables will be written using the first part of the Greek alphabet: α. . so you should learn to love it. . That is. the cannonical example is the equality function. The one chosen in Haskell is the system of type classes. Also note if the expression is a type error: 1. etc. equality testing) will be defined for. if they have a type. we want to use an α-compare. we should use a character comparison. Unfortunately. null 4.3. tail head 4. . . this presents some problems for static type checking. snd 2. we should use a floating point comparison. if we want to compare two things which have type α.1 Motivation In many languages (C++. let's take a step back and see some of the motivation. a function can be written that takes parameters of differing types. head . we should use an integer comparison. β. head . There are as many solutions to this problem as there are statically typed languages (perhaps a slight exageration. and then verify the types of the following expressions. 4. if we want to compare two characters. if we want to compare two floating point numbers. since the type checker doesn't know which types a certain operation (for instance. NOTE In general. γ.2 Figure out for yourself. TYPE BASICS Here. the one we have. We call α a type variable since it is a variable whose value is a type. Exercises Exercise 4. but not so much so). Whether this is the "correct" solution or the "best" solution of course depends on your application domain. GHCi has made explicit the universal quantification of the type values. δ. fst is a function from (a. . If we want to compare two integers. In general. It is. there exists a system of overloading. For instance. head 3. it is saying that for all types a and b. 5.
show applied to the character 'a' is the three-character string "'a"' (the first and last characters are apostrophes). not part of the interpreter printing the value. we need to talk a little more about functions. Before we do that. We have only skimmed the surface of the power (and complexity) of type classes here. the compiler or interpreter will give you some cryptic error mes- . The basic numeric types (Int.4 The Show Class Another of the standard classes in Haskell is the Show class. just for some.2 Equality Testing Returning to the issue of equality testing. 1. 2.4. It defines the Num class to contain all of these numbers and certain minimal operations over them (addition. we say that α is an instance of that class. If a specific type α belongs to a certain type class (that is. for instance). Thus. we associate this function == with a type class. each of the same type (call it α). 4. For instance show applied to the integer 5 is the string "5". etc. what we want to be able to do is define a function == (the equality operator) which takes two parameters. Types which are members of the Show class have functions which convert values of that type to a string. 3. Int is an instance of Eq since equality is defined over integers.3..4. show applied to a string simply puts quotes around it. and returns a boolean. There will be much more discussion of them in Section 8. but we need some more background before we can get there.e.3 The Num Class In addition to overloading operators like ==. The actual string doesn't contain the backslashes. Some types are note instances of Show.3. which we call Eq. 4.3. Haskell has overloaded numeric constants (i. meaning that they are part of the string. But this function may not be defined for every type. all functions associated with that class are implemented for α). For instance. functions for example. Double) are defined to be instances of Num. You can test this in the interpreter: Prelude> "5" Prelude> "'a'" Prelude> "\"Hello show 5 show 'a' show "Hello World" World\"" NOTE The reason the backslashes appear in the last line is because the interior quotes are "escaped". TYPE CLASSES 41 4. the compiler is free to say 5 is an integer or floating point number as it sees fit.). This function is called show. This was done so that when you type in a number like 5.3. If you try to show a function (like sqrt).
if we evaluate (λx. we would write: λxλy. The way we would write a squaring function in lambda calculus is: λx. TYPE BASICS sage. When we apply a value to a lambda expression. which means that we take a value. To get the type of this. yielding (5 ∗ 5) which is 25. The λ is called "lambda abstraction. with an arrow. of course. one corresponding to x and the other corresponding to y. meaning that just as 1 or 'c' are values which have a type. If we want to write a function that takes two numbers. in Haskell we have to give them names if we're defining functions): square = \x -> x*x f = \x y -> 2*x + y You can also evaluate lambda expressions in your interactive shell: Prelude> (\x -> x*x) 5 25 Prelude> (\x y -> 2*x + y) 5 4 14 We can see in the second example that we need to give the lambda abstraction two arguments. The type given to functions mimicks the lambda calculus representation of the functions. it won't be too painful) and talk about the lambda calculus.2∗x+y. functions are first class values. 4. we need to make a short diversion into very theoretical computer science (don't worry." In general.42 CHAPTER 4. doubles the first and adds it to the second. also we don't need to repeat the lambdas. complaining about a missing instance declaration or an illegal class constraint. For instance.1 Lambda Calculus The name "Lambda Calculus".x ∗ x. 4. we first ask ourselves what the type of x . means) and then multiply it by itself. we remove the lambda and replace every occurrence of x with 5. Haskell is largely based on an extension of the lambda calculus. and these two expressions can be written directly in Haskell (we simply replace the λ with a backslash and the .x ∗ x)5.x∗x. the definition of square gives λx. Before we talk too much about functions.2 Higher-Order Types "Higher-Order Types" is the name given to functions. lambdas can only have one parameter. so are functions like square or ++. while perhaps daunting. which we will call x (that's what "λx. In fact. describes a fairly simple system for representing functions. we remove the outermost λ and replace every occurrence of the lambda variable with the value.4 Function Types In Haskell.4. For instance. and.4. 4.
b) -> a Prelude> :t snd snd :: (a. of which we're not sure. produces a new value. This isn't entirely accurate. 5. Then. We know that when we multiply two Ints together. For instance. We can easily find the type of Prelude functions using ":t" as before: Prelude> :t head head :: [a] -> a Prelude> :t tail tail :: [a] -> [a] Prelude> :t null null :: [a] -> Bool Prelude> :t fst fst :: (a.4.2(5) + y.b)" and gives back something of type "a".b) -> b We read this as: "head" is a function that takes a list containing values of type "a" and gives back a value of type "a". But we know the type of this value is the type of λy.4. they are of type Num a ⇒ a.2x + y)5 which becomes our new value λy. if you have α → β → γ it is assume that β → γ is grouped. "null" takes a list of "a"s and gives back a boolean. "tail" takes a list of "a"s and gives back another list of "a"s. if we take f and apply only one number to it. Say we decide x is an Int. So the type of f is Int → (Int → Int). We apply the above analysis and find out that this expression has type Int → Int. f takes an Int and produces something which has type Int → Int. functions are values) is something which takes a value x and given that value. The value of this function (remember. with α → β grouped. we notice that the function square takes an Int and produces a value x*x. and so on. numbers like 5 aren't really of type Int. So we know that f takes an Int and produces a value of some type. . If you want the other way. we get (λxλy. you need to put parentheses around them. Thus. in function types. it only means that it gives back something with the same type as the first element. so the type of the results of square is also an Int. We can apply a similar analysis to the function f above.2(5) + y. As we saw before. which takes a value y and produces 2*x+y. we say the type of square is Int → Int. where all occurances of x have been replaced with the applied value. NOTE The parentheses are not necessary. NOTE Saying that the type of fst is (a. Thus. FUNCTION TYPES 43 is. we get another Int. "fst" takes a pair of type "(a. b) → a does not necessarily mean that it simply gives back the first element.
we might say that + takes two values of type a and produces a value of type a. the do notation. so how do I get rid of the IO. in order to do this we need to put them in parentheses. any function which is used infix (meaning in the middle of two arguments rather than before them) must be put in parentheses when getting its type. but this is less precise. for some type a which is an instance of Num. However. The only way to use things with an IO type is to combine them with other functions using (for example). Suppose you have a function f which takes a String and produces an Int. for a given type a.4. and mean that + is a function which. : takes a value of type a and another value of type [a] (list of as) and produces another value of type [a].44 CHAPTER 4. In short hand.3 That Pesky IO Type You might be tempted to try getting the type of a function like putStrLn: Prelude> putStrLn Prelude> readFile :t :: :t :: putStrLn String -> IO () readFile FilePath -> IO String What in the world is that IO thing? It's basically Haskell's way of representing that these functions aren't really functions. you can combine these as: . that. in shorthand. you cannot write a function with type IO String → String. ++ takes two lists of as and produces a new list of as. if you're reading a file using readFile. In general. you can't directly remove it. 4. Similarly. Prelude> :t (+) (+) :: Num a => a -> a -> a Prelude> :t (*) (*) :: Num a => a -> a -> a Prelude> :t (++) (++) :: [a] -> [a] -> [a] Prelude> :t (:) (:) :: a -> [a] -> [a] The types of + and * are the same. You can't directly apply f to the result of readFile since the input to f is String and the output of readFile is IOString and these don't match. takes a value of type a and produces another function which takes a value of type a and produces a value of type a. presumably you want to do something with the string it returns (otherwise. The type of ++ means. They're called "IO Actions" (hence the IO). why would you read the file in the first place). That is. TYPE BASICS We can also get the type of operators like + and * and ++ and :. The immediate question which arises is: okay. however. In brief. For example.
We then. Also note that we don't write i <. if you knew apriori that square were only going to be applied to value of type Int.4. This is because we are in a do block. for one (or more) of the following reasons: • Clarity • Speed • Debugging Some people consider it good software engineering to specify the types of all toplevel functions.4 Explicit Type Declarations It is sometimes desirable to explicitly specify the types of some elements or functions. However. not an IO action. If nothing else. However.4.readFile "somefile" let i = f s putStrLn (show i) Here.4. For instance.f s because f is just a normal function. Double. it may be easier to figure out where the error is. FUNCTION TYPES 45 main = do s <. In this definition. you could apply square to anything which is an instance of Num: Int. Type declarations are written separatly from the function definition. we could explicitly type the function square as in the following code (an explicitly declared type is called a type signature): square :: Num a => a -> a square x = x*x These two lines do not even have to be next to eachother. Note that the let here doesn't have a corresponding in. etc. you could only apply square to values of type Int. 4. you could refine its type as: square :: Int -> Int square x = x*x Now. if you're trying to compile a program and you get type errors that you cannot understand. the type that you specify must match the inferred type of the function definition (or be more specific). if you declare the types of some of your functions explicitly. print i to the screen. with this definition. for example. Moreover. the compiler doesn't have to generate the general code specified in the original . we use the arrow convention to "get the string out of the IO action" and then apply f to the string (called s).
If you have extensions turned on ("-98" in Hugs or "-fglasgow-exts" in GHC(i)). an initial value of type a and a list of bs. a list of as to produce a final value of type a. For instance. For instance. So map will take a value of type [a] and produce a value of type [b]. so it may be able to generate faster code. These two lists don't necessarily have to have the same types of elements. 4. the type of map is (a → b) → [a] → [b]. foldl has a more general type: (a → b → a) → a → [b] → a. an initial value of type a. however. You can of course you filter and length to do this.46 CHAPTER 4. Thus.4. In fact. TYPE BASICS function definition since it knows you will only apply square to Ints. To see this. this function must have type a → b. these are well-typed. it leaves the compiler alone to infer the type of the rest of the expression. It's job is to take a list of elements and produce another list of elements. map took a function to apply to each element in a list. you can also add a type signature to expressions and not just functions. and foldl took a function which told it how to combine list elements together.5 Functional Arguments In Section 3. What is the type of square in this example? Make your guess then you can check it either by entering this code into a file and loading it into your interpreter or by asking for the type of the expression: Prelude> :t (\(x :: Int) -> x*x) since this lambda abstraction is equivalent to the above function declaration.3 we saw examples of functions taking other functions as arguments. So it takes a function which turn an a and a b into an a. you could write: square (x :: Int) = x*x which tells the compiler that x is an Int. which you can verify in your interpreter with ":t". It produces an a. filter took a function that told it which elements of a list to keep. As with every other function in Haskell. Let's first think about the map function. meaning that you take a function which combines two as into another one. In order to convert an a to a b. but we will also do it using foldr: module Count where import Char . We can apply the same sort of analysis to filter and discern that it has type (a → Bool) → [a] → [a]. you might be tempted to give it type (a → a → a) → a → [a] → a. How does it do this? It uses the user-supplied function to convert. As we presented the foldl function. we can write a function count which counts how many members of a list satisfy a given constraint.
\x y z -> (x. \x -> "hello.1 Pairs For instance. It filters the list l according to the predicate p. However. it is often desirable to be able to define our own data structures and functions over them. it applies the lambda expression shown. 4. It checks to see if p holds about x. build-in pair type) could be: data Pair a b = Pair a b . it just returns c.5 Data Types Tuples and lists are nice. On the other hand. a definition of a pair of elements (much like the standard. it returns the new value c+1. If it does. \x -> x + 5 4. Exercises Exercise 4. then takes the length of the resulting list. \x -> x x 7.5.y:z:[]) 3. the old count. DATA TYPES 47 count1 p l = length (filter p l) count2 p l = foldr (\x c -> if p x then c+1 else c) 0 l The functioning of count1 is simple. If it doesn't. common ways to define structured values.5.3 Figure out for yourself. \x -> x 'a' 6. increasing the count of elements for which the predicate holds. count2 uses the intial value (which is an integer) to hold the current count.4. For each element in the list l. Also note if the expression is a type error: 1. This takes two arguments. if they have a type. and then verify the types of the following expressions. c which holds the current count and x which is the current element in the list that we're looking at. \x -> [x] 2. world" 5. \x -> x + x 4. So-called "datatypes" are defined using the data keyword.
Let's walk through this code one word at a time. First we say "data" meaning that we're defining a datatype. We then give the name of the datatype, in this case, "Pair." The "a" and "b" that follow "Pair" are type parameters, just like the "a" is the type of the function map. So up until this point, we've said that we're going to define a data structure called "Pair" which is parameterized over two types, a and b. After the equals sign, we specify the constructors of this data type. In this case, there is a single constructor, "Pair" (this doesn't necessarily have to have the same name as the type, but in this case it seems to make more sense). After this pair, we again write "a b", which means that in order to construct a Pair we need two values, one of type a and one of type b. This definition introduces a function, Pair :: a -> b -> Pair a b that you can use to construct Pairs. If you enter this code into a file and load it, you can see how these are constructed: Datatypes> :t Pair Pair :: a -> b -> Pair a b Datatypes> :t Pair 'a' Pair 'a' :: a -> Pair Char a Datatypes> :t Pair 'a' "Hello" :t Pair 'a' "Hello" Pair 'a' "Hello" :: Pair Char [Char] So, by giving Pair two values, we have completely constructed a value of type Pair. We can write functions involving pairs as: pairFst (Pair x y) = x pairSnd (Pair x y) = y In this, we've used the pattern matching capabilities of Haskell to look at a pair an extract values from it. In the definition of pairFst we take an entire Pair and extract the first element; similarly for pairSnd. We'll discuss pattern matching in much more detail in Section 7.4.
Exercise 4.4 Write a data type declaration for Triple, a type which contains three elements, all of different types. Write functions tripleFst, tripleSnd and tripleThr to extract respectively the first, second and third elements. Exercise 4.5 Write a datatype Quadruple which holds four elements. However, the first two elements must be the same type and the last two elements must be the same type. Write a function firstTwo which returns a list containing the first two elements and a function lastTwo which returns a list containing the last two elements. Write type signatures for these functions
Multiple Constructors
We have seen an example of the data type with one constructor: Pair. It is also possible (and extremely useful) to have multiple constructors. Let us consider a simple function which searches through a list for an element satisfying a given predicate and then returns the first element satisfying that predicate. What should we do if none of the elements in the list satisfy the predicate? A few options are listed below: • Raise an error • Loop indefinitely • Write a check function • Return the first element • ... Raising an error is certainly an option (see Section 10.1 to see how to do this). The problem is that it is difficult/impossible to recover from such errors. Looping indefinitely is possible, but not terribly useful. We could write a sister function which checks to see if the list contains an element satisfying a predicate and leave it up to the user to always use this function first. We could return the first element, but this is very ad-hoc and difficult to remember. The fact that there is no basic option to solve this problem simply means we have to think about it a little more. What are we trying to do? We're trying to write a function which might succeed and might not. Furthermore, if it does succeed, it returns some sort of value. Let's write a datatype: data Maybe a = Nothing | Just a This is one of the most common datatypes in Haskell and is defined in the Prelude. Here, we're saying that there are two possible ways to create something of type Maybe a. The first is to use the nullary constructor Nothing, which takes no arguments (this is what "nullary" means). The second is to use the constructor Just, together with a value of type a. The Maybe type is useful in all sorts of circumstances. For instance, suppose we want to write a function (like head) which returns the first element of a given list. However, we don't want the program to die if the given list is empty. We can accomplish this with a function like: firstElement :: [a] -> Maybe a firstElement [] = Nothing firstElement (x:xs) = Just x
The type signature here says that firstElement takes a list of as and produces something with type Maybe a. In the first line of code, we match against the empty list []. If this match succeeds (i.e., the list is, in fact, empty), we return Nothing. If the first match fails, then we try to match against x:xs which must succeed. In this case, we return Just x. For our findElement function, we represent failure by the value Nothing and success with value a by Just a. Our function might look something like this: findElement :: (a -> Bool) -> [a] -> Maybe a findElement p [] = Nothing findElement p (x:xs) = if p x then Just x else findElement p xs
The first line here gives the type of the function. In this case, our first argument is the predicate (and takes an element of type a and returns True if and only if the element satisfies the predicate); the second argument is a list of as. Our return value is maybe an a. That is, if the function succeeds, we will return Just a and if not, Nothing. Another useful datatype is the Either type, defined as: data Either a b = Left a | Right b
This is a way of expressing alternation. That is, something of type Either a b is either a value of type a (using the Left constructor) or a value of type b (using the Right constructor).
Exercise 4.6 Write a datatype Tuple which can hold one, two, three or four elements, depending on the constructor (that is, there should be four constructors, one for each number of arguments). Also provide functions tuple1 through tuple4 which take a tuple and return Just the value in that position, or Nothing if the number is invalid (i.e., you ask for the tuple4 on a tuple holding only two elements). Exercise 4.7 Based on our definition of Tuple from the previous exercise, write a function which takes a Tuple and returns either the value (if it's a one-tuple), a Haskell-pair (i.e., ('a',5)) if it's a two-tuple, a Haskell-triple if it's a three-tuple or a Haskell-quadruple if it's a four-tuple. You will need to use the Either type to represent this.
DATA TYPES 51 4. we apply the listLength function to xs and add one to the result. We can write our own length function for our lists as: listLength Nil = 0 listLength (Cons x xs) = 1 + listLength xs This function is slightly more complicated and uses recursion to calculate the length of a List. For instance. except that uses special syntax where [] corresponds to Nil and : corresponds to Cons. This is almost identical to the actual definition of the list datatype in Haskell. This much is obvious. we have defined what it means to be of type List a. The first line says that the length of an empty list (a Nil) is 0. Each of these children is another node. Thus. it's the length of its tail (the value of xs) plus one (to account for x). A binary tree is a structure that has a single root node. if it's a branch. each node in the tree is either a "leaf" or a "branch. listTail.5. We say that a list is either empty (Nil) or it's the Cons of a value of type a and another value of type List a. A non-empty list must be of the form Cons x xs for some values of x and xs. 4." If it's a leaf.4. but function on our List datatype.5. or it's a branch with a left child (which is a BinaryTree of as). a . Don't worry about exceptional conditions on the first two.4 Binary Trees We can define datatypes that are more complicated that lists. We know that xs is another list and we know that whatever the length of the current list is. listFoldl and listFoldr which are equivalent to their Prelude twins. We can define such a data type as: data BinaryTree a = Leaf a | Branch (BinaryTree a) a (BinaryTree a) In this datatype declaration we say that a BinaryTree of as is either a Leaf which holds an a. This gives us the length of the entire list. we could define a list datatype as: data List a = Nil | Cons a (List a) In this definition.5. These are datatypes whose definitions are based on themselves. Suppose we want to define a structure that looks like a binary tree.3 Recursive Datatypes We can also define recursive datatypes. it holds a value. Exercises Exercise 4. it holds a value and a left child and a right child. The second line tells us how to calculate the length of a nonempty list.8 Write functions listHead.
Exercise 4. for instance.52 CHAPTER 4. a type which can only have a constrained number of values. Suppose we were using this to write a drawing program.255.e.9 Write a function elements which returns the elements in a BinaryTree in a bottom-up. followed by the other child's value. plus one. We could define a color type: data Color = Red | Orange | Yellow | Green | Blue | Purple | White | Black This would be sufficient to deal with simple colors. we say that the size of a leaf is 1 and the size of a branch is the size of its left child.0) (0. it calculates the number of nodes in a BinaryTree. TYPE BASICS node value (which is an a).5. as: colorToRGB colorToRGB colorToRGB colorToRGB colorToRGB Red Orange Yellow Green Blue = = = = = (255.0) (255. It is simple to modify the listLength function so that instead of calculating the length of lists.0. Exercises Exercise 4. we could then write a function to convert between a Color and a RGB triple. left-to-right manner (i.0) (255. The solution is given below: treeSize (Leaf x) = 1 treeSize (Branch left x right) = 1 + treeSize left + treeSize right Here. The result type should be a normal Haskell list.0. 4. Can you figure out how? We can call this function treeSize..255. followed by its parent's value. and so on).255) .5 Enumerated Sets You can also use datatypes to define things like enumerated sets. We can write a colorToRGB function.0) (0.10 Write a fold function for BinaryTrees and rewrite elements in terms of it (call the new one elements2). plus the size of its right child. and a right child (which is also a BinaryTree of as).128. the first element returned in the left-most leaf.
we could change the Color datatype to something like: data Color = Red | Orange | Yellow | Green | Blue | Purple | White | Black | Custom Int Int Int -. The task of converting a string of Java tokens to a Java abstract syntax tree is an example of a .0. I will handwave through an example which is too complex to write out at this point and then give a real example.3. though one with less motivation. The idea behind CPS is to pass around as a function argument what to do next. Consider the problem of parsing.4 and 8.b) 4. CONTINUATION PASSING STYLE colorToRGB Purple = (255. The idea here is that we have a sequence of tokens (words.0) 53 If we wanted also to allow the user to define his own custom colors. 4. We'll dwell much more on data types in Sections 7.255) colorToRGB Black = (0. letters.255.0. It's definition is: data () = () The only true value of this type is ().5. This is essentially the same as a void type in a langauge like C or Java and will be useful when we talk about IO in Chapter 5.4.6 The Unit type A final useful datatype defined in Haskell (from the Prelude) is the unit type.255) colorToRGB White = (255.R G B components And add a final definition for colorToRGB: colorToRGB (Custom r g b) = (r. whatever) and we want to ascribe structure to them.6 Continuation Passing Style There is a style of functional programming called "Continuation Passing Style" (also simply "CPS").g.6.
What the parseFunction function would do is to parse an identifier. [Token]) parseIdentifier :: [Token] -> Maybe (String. it repeatedly calls parseIdentifier until that fails. [String]) -> [Token] -> a) -> ([Token] -> a) -> a parseIdentifier :: [Token] -> (String -> [Token] -> a) -> ([Token] -> a) -> a Let's consider parseIdentifier. TYPE BASICS parsing problem. then that something is the argument paired with the rest of the tokens. it fails. parseIdentifier will parse one of the arguments. however. we write functions: parseFunction :: [Token] -> ((String. But for simplicity. it calls the first continuation with that identifier and the remaining tokens as arguments. then it's not an argument. If it returns Nothing. another way to think about this problem. then. If reading the identifier fails. it continues and tries to parse a open parenthesis. It then tries to parse a close parenthesis. then for a close parenthesis. it calls the second continuation with all the tokens. [String]). The second continuation is what to do if you fail. The general approach to solving this would be to write a function which parses function calls like this one. If this fails. [Token]) The idea would be that if we call parseFunction. assume they are not separated by commas. then it's done. if it doesn't return Nothing. together with whatever is left after parsing the function. Similarly. Otherwise. then for zero or more identifiers. a function call looks like myFunction(x y z). then it returns the pair described earlier. "y" and "z". Instead of the above. We want to convert this into something like a pair containing first the string "myFunction" and then a list with three string elements: "x". . If that succeeds. If that succeeds. The first continuation is what to do when you succeed. That is. is try to read an identifier. then for an open parenthesis. This takes three arguments: a list of tokens and two continuations. The advantage to this solution is that functions no longer need to return the remaining tokens (which tends to get ugly). If this succeeds. if it returns Just something. What parseIdentifier does. So is the task of taking an English sentence and creating a parse tree (though the latter is quite a bit harder). it fails itself. Suppose we're parsing something like C or Java where functions take arguments in parentheses. One way to do this would be to have two functions: parseFunction :: [Token] -> Maybe ((String.54 CHAPTER 4. There is. First it would look for an identifier ("myFunction"). Otherwise.
2.4.6. but it is perhaps too complex at the moment. we simply need to define this function which looks for an open parenthesis. This enables us to change.9.. The first argument it gives is the list of tokens.. The first continuation (which is what parseIdentifier should do if it succeeds) is in turn a function which will look for an open parenthesis. This is easy.7. We can write a CPS fold as: cfold' f z [] = z cfold' f z (x:xs) = f x z (\y -> cfold' f y xs) In this code. Now.10] CPS> cfold' (\x t g -> g (x : t)) [] [1. the first thing it does is call parseIdentifier. slightly different from the standard folds. zero or more arguments and a close parethesis. Thus. and the third is the continuation: basicially. the second is the accumulated element.2.2.3] [1. We write a function which looks for the open parenthesis and then calls parseIdentifier with a success continuation that looks for more identifiers. zero or more identifiers and a close parenthesis.10] [10.5.3. what to do next.3. zero or more arguments and a close parethesis.6.5.9. an open parenthesis. Instead. We can write a wrapper function for cfold' that will make it behave more like a normal fold: cfold f z l = cfold' (\x t g -> f x (g t)) z l We can test that this function behaves as we desire: CPS> cfold (+) 0 [1.4. consider the problem of folding across a list.3] One thing that's nice about formulating cfold in terms of the helper function cfold' is that we can use the helper function directly.2. cfold' take a function f which takes three arguments. and a "failure" continuation which looks for the close parenthesis (note that this failure doesn't really mean failure – it just means there are no more arguments left).1] .8. CONTINUATION PASSING STYLE 55 Now consider parseFunction. I realize this discussion has been quite abstract. z.6.10] [1. the evaluation order of the fold very easily: CPS> cfold' (\x t g -> (x : g t)) [] [1.4] 10 CPS> cfold (:) [] [1.2. Recall that it wants to read an identifier. The first is the current list element.4.8.7. x.3. The second argument (the failure argument) is just going to be the failure function given to parseFunction. for instance. I would willingly give code for all this parsing.
3] cfold' f [] [1.3]) (\g -> g [1]) (\y -> cfold' f y [2.3] cfold' (\x t g -> g (x:t)) [] [1. If not.1] ==> ==> ==> ==> ==> ==> ==> ==> ==> ==> ==> ==> ==> ==> ==> ==> ==> ==> ==> ==> ==> ==> In general.11 Test whether the CPS-style fold mimicks either of foldr and foldl.3]) (\y -> cfold' f y [2.3]) 1 : (f 2 [] (\y -> cfold' f y [3])) 1 : (2 : ((\y -> cfold' f y [3]) [])) 1 : (2 : (cfold' f [] [3])) 1 : (2 : (f 3 [] (\y -> cfold' f y []))) 1 : (2 : (3 : (cfold' f [] []))) 1 : (2 : (3 : [])) [1.56 CHAPTER 4.2.3]) []) 1 : (cfold' f [] [2.2. though it can be difficult to master. We will revisit the topic more thoroughly later in the book. Exercises Exercise 4.2.1]) [] [3.3] f 1 [] (\y -> cfold' f y [2. this slight difference changes the behavior for being like foldr to being like foldl.12 Write map and filter using continuation passing style. where is the difference? Exercise 4.3]) [1] cfold' f [1] [2. continuation passing style is a very powerful abstraction.2. .3] (\x t g -> g (x:t)) 2 [1] (\y -> cfold' f y [3]) cfold' f (2:[1]) [3] cfold' f [2.3] cfold' f [] [1.1] [3] (\x t g -> g (x:t)) 3 [2. As it turns out. We can evaluate both of these calls as follows (let f be the folding function): cfold' (\x t g -> (x : g t)) [] [1. TYPE BASICS The only difference between these calls to cfold' is whether we call the continuation before or after constructing the list.3] (\x t g -> g (x:t)) 1 [] (\y -> cfold' f y [2.2.1] (\y -> cfold' f y []) cfold' f (3:[2.2.3]) 1 : ((\y -> cfold' f y [2.
Let's first consider the initial two examples and think about what their types should be. but it doesn't seem to require an argument.Chapter 5 Basic Input/Output As we mentioned earlier. due to referential transparency. similarly. Before we give the solution. The second operation. 5. Certainly the first operation (I hesitate to call it a "function") should take a String argument and produce something. We want both of these operations to be functions. let's take a step back and think about the difficulties inherent in such a task." Their values depend directly on the real world. we might write these functions as having type: 57 . But clearly this does not have the desired effect.1 The RealWorld Solution In a sense. should return a String. but what should it produce? It could produce a unit (). as it will not return the same String every time. Any IO library should provide a host of functions. it is difficult to think of a good. Supposing we had a type RealWorld. there should be no problem with replacing it with a function f = (). but they are by definition not functions. since there is essentially no return value from printing a string. the reason that these items are not functions is that they interact with the "real world. The item that reads a string from the keyboard cannot be a function. clean way to integrate operations like input/output into a pure functional language. And if the first function simply returns () every time. containing (at a minimum) operations like: • print a string to the screen • read a string from a keyboard • write data to a file • read data from a file There are two issues here.
5. IO. the reference to rW'' on the last line has been changed to a reference to rW'. But that means that the RealWorld has been updated. It also doesn't model the fact that the program below makes no sense: main rW = let rW' = printAString rW "Please enter your name: " (rW''. In this style (assuming an initial RealWorld state were an argument to main).name) = readAString rW' in printAString rW'' ("Hello.hs" program from Section 3. " ++ name ++ ". though it is more than somewhat unweildy. we can use them to express a variety of constructions like concurrence. exceptions. Similarly. " ++ name ++ ". However. Moreover. In fact. it then modifies the state of the world in such a way that the string is now printed and returns this new value. Clearly. . it must read a string in order to have a value for name to be printed. our "Name. non-determinism and much more. This would be a possible way to do IO. printAString takes a current state of the world and a string to print. BASIC INPUT/OUTPUT printAString :: RealWorld -> String -> RealWorld readAString :: RealWorld -> (RealWorld.8 would look something like: main rW = let rW' = printAString rW "Please enter your name: " (rW''. readAString takes a current state of the world and returns a new state of the world. they can be defined within Haskell with no special handling from the compiler (though compilers often choose to optimize monadic operations). how are you?") This is not only hard to read. how are you?") In this program.name) = readAString rW' in printAString rW' -. if you accidentally use the wrong version of the RealWorld. It is completely unclear what this program should do.2 Actions The breakthrough for solving this problem came when Phil Wadler realized that monads would be a good way to think about IO computations. there is nothing special about them. monads are able to express much more than just the simple operations described above.OOPS! ("Hello. String) That is. but prone to error.58 CHAPTER 5. paired with the String that was typed. then we try to ignore this update by using an "old version" of the RealWorld. There is clearly something wrong happening here. Suffice it to say that doing IO operations in a pure lazy functional language is not trivial.
5. The compiled code then executes this action. ACTIONS 59 As pointed out before. Furthermore. which means that it is an IO action that returns nothing. we're sequencing four actions: setting buffering. since they are not (in the pure mathematical sense). You cannot actually run an action yourself. a program is. a single action that is run when the compiled program is executed. in order to get the value out of the action. However. The getLine action has type IO String. Thus. we write name <. in this program. Let's consider the original name program: main = do hSetBuffering stdin LineBuffering putStrLn "Please enter your name: " name <.getLine. we cannot think of things like "print a string to the screen" or "read data from a file" as functions.2. In fact. This action has type: putStrLn :: String -> IO () As expected. " ++ name ++ ". Therefore. the compiler requires that the main function have type IO (). and put the results in the variable called name. while you are not allowed to run actions yourself. so the fully applied action has type IO (). What it returns is of type IO (). instead. which prints a string to the screen. will have type String. you are allowed to combine actions. but we will gloss over this for now. Moreover. a getLine and another putStrLn. the result will have type (). You can probably already guess the type of getLine: getLine :: IO String This means that getLine is an IO action that. so it is okay to execute it directly. so we provide it a String. The putStrLn action has type String → IO (). One particularly useful action is putStrLn. This is something that is left up to the compiler. itself. we give them a special type. which basically means "run getLine.notation is a way to get the value out of an action. this type means that putStrLn is an action within the IO monad. we give them another name: actions. So. However. a putStrLn. The question immediately arises: "how do you 'run' an action?". the <. This is something that we are allowed to execute. This means that this function is actually an action (that is what the IO means)." . NOTE Actually. Not only do we give them a special name. putStrLn takes a string argument. when this action is evaluated (or "run") . we have already seen one way to do this using the do notation (how to really do this will be revealed in Chapter 9).getLine putStrLn ("Hello. how are you?") We can consider the do notation as a way to combine a sequence of actions. when run.
The code here is: do putStrLn "Too low!" doGuessing num Here. NOTE In this code. A similar argument shows that the type of the "else" branch is also IO (). The type of the entire if/then/else construction is then the type of the two branches. "else putStrLn "You Win!"" would have been sufficient. I already started a do block. I don't need another one. and the "else" branch. the type of the "then" branch is also IO ().. but you need to be somewhat careful. It is incorrect to think to yourself "Well. we have: do . The type result of the entire computation is precisely the type of the final computation. This is somewhat overly verbose. the "then" branch.60 CHAPTER 5.. BASIC INPUT/OUTPUT Normal Haskell constructions like if/then/else and case/of can be used within the do notation. Since we have only one action here. it essentially takes three arguments: the condition.. since do is only necessary to sequence actions. and the two branches can have any type. it is superfluous. The condition needs to have type Bool." and hence write something like: do if (read guess) < num then putStrLn "Too low!" doGuessing num else . which is fine. For instance. The first has type IO (). This clearly has the correct type. if (read guess) < num then do putStrLn "Too low!" doGuessing num else if read guess > num then do putStrLn "Too high!" doGuessing num else do putStrLn "You Win!" If we think about how the if/then/else construction works. which is just what we want. we are sequencing two actions: putStrLn and doGuessing. . In fact. The second also has type IO (). In the outermost comparison.. which is fine. in our "guess the number" program. Thus. we have (read guess) < num as the condition. Let's just consider the "then" branch. the last line is "else do putStrLn "You Win!"". This means the type of the entire if/then/else construction is IO (). provided that they have the same type.
the dos after the ->s are necessary on the first two options. int guess = atoi(readLine()). EQ. the compiler doesn't know that the putStrLn and doGuessing calls are supposed to be sequenced. ACTIONS 61 Here. since we didn't repeat the do. In particular. LT. if (guess == num) { print "You win!". To do this. depending on whether the first is greater than. the value of type Int). It will certainly complain (though the error may be somewhat difficult to comprehend at this point). } } . one of type IO Int) and makes it into an action that returns the given value (for instance. because we are sequencing actions. again. in an imperative language. } // we won't get here if guess == num if (guess < num) { print "Too low!". less than or equal to the second. If you're used to programming in an imperative language like C or Java. you might think that return will exit you from the current function. doGuessing(num). } else { print "Too high!". We can write the same doGuessing function using a case statement. the function doGuessing and the integer num. and the compiler will think you're trying to call putStrLn with three arguments: the string. we first introduce the Prelude function compare.getLine case compare (read guess) num of LT -> do putStrLn "Too low!" doGuessing num GT -> do putStrLn "Too high!" doGuessing num EQ -> putStrLn "You Win!" Here. doGuessing(num). In Haskell. return ().2. doGuessing num = do putStrLn "Enter your guess:" guess <.5. you might write this function as: void doGuessing(int num) { print "Enter your guess:". which takes two values of the same type (in the Ord class) and returns one of GT. This is not so in Haskell. return simply takes a normal value (for instance.
First of all. If the name is Koen. it will first print "You win!. which might look something like: doGuessing num = do putStrLn "Enter your guess:" guess <. If the name is one of Simon.62 CHAPTER 5. so the else branch is taken.3 The IO Library The IO Library (available by importing the IO module) contains many definitions.getLine case compare (read guess) num of EQ -> do putStrLn "You win!" return () -. tell the user that you think Haskell is a great programming language. Exercises Exercise 5. the most common of which are listed below: data IOMode = ReadMode | WriteMode | AppendMode | ReadWriteMode :: FilePath -> IOMode -> IO Handle openFile . otherwise." but it won't exit. Of course it is not. it won't have a pattern that matches. 5. However. and it will check whether guess is less than num. In either case. tell them that you think debugging Haskell is fun (Koen Classen is one of the people who works on Haskell debugging).we don't expect to get here unless guess == num if (read guess < num) then do print "Too low!". if you guess incorrectly. and the program will fail immediately with an exception.1 Write a program that asks the user for his or her name. John or Phil. we expect the code to exit there (and in mode imperative languages. it will try to evaluate the case statement and get either LT or GT as the result of the compare. because we have the return () in the first if match. doGuessing will not behave as you expect. one using if statements. and it will print "Too high!" and then ask you to guess again. if you guess correctly. the equivalent code in Haskell. the other using a case statement. it does). tell the user that you don't know who he or she is. BASIC INPUT/OUTPUT Here. doGuessing else do print "Too high!". Write two different versions of this program. On the other hand.
The bracket function is used to perform actions safely. using the IOMode argument as the mode for opening the file. See Section 8. produces the contents of that file. The getChar.3. hPutChar prints a character to a file. writes a character to it. respectively. hGetContents reads the entire file. for instance. if there were an error at some point. THE IO LIBRARY hClose hIsEOF :: Handle -> IO () :: Handle -> IO Bool 63 hGetChar :: Handle -> IO Char hGetLine :: Handle -> IO String hGetContents :: Handle -> IO String getChar getLine getContents hPutChar hPutStr hPutStrLn putChar putStr putStrLn readFile writeFile :: IO Char :: IO String :: IO String :: Handle -> Char -> IO () :: Handle -> String -> IO () :: Handle -> String -> IO () :: Char -> IO () :: String -> IO () :: String -> IO () :: FilePath -> IO String :: FilePath -> String -> IO () bracket :: IO a -> (a -> IO b) -> (a -> IO c) -> IO c NOTE The type FilePath is a type synonym for String. The openFile and hClose functions open and close a file. hGetChar and hGetLine read a character or line (respectively) from a file. and then closes the file. The bracket function makes this easy. hPutStr prints a string. When writing such a function. the readFile function takes a String (the file to read) and returns an action that.5. Most of these functions are self-explanatory. and hPutStrLn prints a string with a newline character at the end. the file is still successfully closed. getLine and getContents variants read from standard input. The readFile and writeFile functions read an entire file without having to open it first. hIsEOF tests for end-of file. The variants without the h prefix work on standard output. there is no difference between FilePath and String. when run. one needs to be careful to ensure that.1 for more about type synonyms. That is. So. It takes . Consider a function that opens a file.
That way.64 CHAPTER 5. The second is the action to perform at the end. BASIC INPUT/OUTPUT three arguments: The first is the action to perform at the beginning. if writing the character fails. The interface is admittedly poor. regardless of whether there's an error or not. and it does not catch all errors (try reading a non-existant file).hs.4 A File Reading Program We can write a simple program that allows a user to read and write files. Enter the following code into "FileRead. it should give a fairly complete example of how to use IO. Nevertheless. our character-writing function might look like: writeChar :: FilePath -> Char -> IO () writeChar fp c = bracket (openFile fp ReadMode) hClose (\h -> hPutChar h c) This will open the file. you don't need to worry too much about catching the exceptions and about closing all of your handles.getLine case command of 'q':_ -> return () 'r':filename -> do putStrLn ("Reading " ++ filename) doRead filename doLoop 'w':filename -> do putStrLn ("Writing " ++ filename) doWrite filename doLoop _ -> doLoop . However. and the exception will be reraised afterwards. hClose will still be executed. For instance. which might result in an error. 5." and compile/run: module Main where import IO main = do hSetBuffering stdin LineBuffering doLoop doLoop = do putStrLn "Enter a command rFN wFN or q to quit:" command <. The third is the action to perform in the middle. write the character and then close the file.
NOTE The return function is a function that takes a value of type a and returns an action of type IO a. Thus. A FILE READING PROGRAM 65 doRead filename = bracket (openFile filename ReadMode) hClose (\h -> do contents <. It then performs a case switch on the command and checks first to see if the first character is a 'q. It opens a file in ReadMode. If the first character of the command wasn't a 'q. The check for 'w' is nearly identical.5. We would need to catch exceptions raised by openFile.getLine bracket (openFile filename WriteMode) hClose (\h -> hPutStrLn h contents) What does this program do? First. but they were written in the extended fashion to show how the more complex functions are used. the wildcard character. They only catch exceptions within the main body. and then writes it to the file specified. reads its contents and prints the first 100 characters (the take function takes an integer n and a list and returns the first n elements of the list).' If it is. in order to make this complete.1. it issues a short string of instructions and reads a command. NOTE Both doRead and doWrite could have been made simpler by using readFile and writeFile.hGetContents h putStrLn "The first 100 chars:" putStrLn (take 100 contents)) doWrite filename = do putStrLn "Enter text to go into the file:" contents <.' the program checks to see if it was an 'r' followed by some string that is bound to the variable filename.4. We will do this when we talk about exceptions in more detail in Section 10. in these cases). not within the startup or shutdown functions (openFile and hClose. Otherwise. It then tells you that it's reading the file. it matches . reads it from the keyboard. . You may think that the calls to bracket in doRead and doWrite should take care of this. The doRead function uses the bracket function to make sure there are no problems reading the file. The doWrite function asks for some text. the type of return () is IO (). it returns a value of unit type. does the read and runs doLoop again. but they don't. The only major problem with this program is that it will die if you try to read a file that doesn't already exists or if you specify some bad filename like *\ˆ# @. and loops to doLoop.
the program should ask him for a file name and print that file to the screen (if the file doesn't exist. [write] a file write Enter a file name to write: foo Enter text (dot on a line by itself to end): this is some text for foo . with ".66 CHAPTER 5. [write] a file read Enter a file name to read: foo .. If the user responds quit.2 Write a program that first asks whether the user wants to read from a file..." signaling completion. Do you want to [read] a file.contents of foo. All but the ". the program should exit. [write] a file blech I don't understand the command blech.. write to a file or quit. that file does not exist. BASIC INPUT/OUTPUT Exercises Exercise 5. If he responds read. Do you want to [read] a file." should be written to the file. Do you want to [read] a file. it should ask him for a file name and then ask him for text to write to the file. the program may crash). running this program might produce: Do you want to [read] a file. Do you want to [read] a file. [write] a file read Enter a file name to read: foo this is some text for foo Do you want to [read] a file. If he responds write. [write] a file quit Goodbye! or [quit]? or [quit]? or [quit]? or [quit]? or [quit]? or [quit]? . [write] a file read Enter a file name to read: foof Sorry. For example.
'deal deck n' deals 'n' cards from 'deck' deal :: Deck -> Int -> [Card] deal deck n = dealHelper deck n [] 67 . I may wish to have a separate module called "Cards" to handle the generation of cards. 6. shuffle :: . -> Deck -> Deck shuffle = .. suppose I am writing a game of poker.. For instance. program subcomponents are divided into modules.. I have left out the implementation details.hs" extension.... and then use this "Cards" module in my "Poker" modules... but suppose the skeleton of our module looks something like this: module Cards where data Card = .. data Deck = . the shuffling and the dealing functions.Chapter 6 Modules In Haskell... -> Deck newDeck = .1 Exports Suppose as suggested we are writing a cards module.. if I ever go back and want to write a blackjack program. That way. of course). -. I don't have to rewrite all the code for the cards. I can simply import the old "Cards" module. Each module sits in its own file and the name of the module should match the name of the file (without the ". if you wish to ever use that module in a larger program. newDeck :: .
so people who use this module won't be able to access our dealHelper function.68 dealHelper = . CHAPTER 6. ) .Queen.Spades. For instance if our definition of Card were: data Card = data Suit = | | | data Face = | | | | Card Suit Face Hearts Spades Diamonds Clubs Jack Queen King Ace Number Int Then users of our module would be able to use things of type Card. shuffle.King. we have specified exactly what functions the module exports. Here.Number). The () after Card and Deck specify that we are exporting the type but none of the constructors. we would have to specify it in the export list: module Cards ( Card(Card). Suit(Hearts. In order to do this. deal ) where . newDeck..Diamonds. The implementation of this helper function is very dependent on the exact data structures you used for Card and Deck so we don't want other people to be able to call this function. we create an export list.. which we insert just after the module name declaration: module Cards ( Card().... Deck().Ace. the function deal calls a helper function dealHelper. but wouldn't be able to construct their own Cards and wouldn't be able to extract any of the suit/face information stored in them..Clubs). MODULES In this code. . Face(Jack. If we wanted users of our module to be able to access all of this information.
as before. Suppose. You are now writing your poker module and you want to import all the definitions from the "Cards" module. IMPORTS where . Cards.. ) where . you wrote a module called "Cards" which you saved in the file "Cards. you can simply write (. so if you want to export them all. And this will automatically export all the constructors. 69 This can get frustrating if you're exporting datatypes with many constructors. but as long as you stay away from the corner cases. Face(. Suit(.6.. as in: module Cards ( Card(. You may refer to them simply by their name in the "Cards" module (as. To do this.. you can import one of the modules qualified which means that you would no longer be able to simply use the newDeck format but must use the longer Cards.). Suppose we knew the only function from "Cards" that we wanted was newDeck.2 Imports There are a few idiosyncracies in the module import system. or you may refer to them explicitely as imported from "Cards" (as. for instance.2. for instance.). all you need to do is write: module Poker where import Cards This will enable to you use any of the functions.newDeck format.newDeck)..). you should be fine. 6. ... newDeck)...hs". In these cases.. we could import only this function by writing: ..). It may be the case that two module export functions or types of the same name. you would write: import qualified Cards Another way to avoid problems with overlapping function definitions is to import only certain functions from modules. types and constructors exported by the module "Cards". to remove ambiguity. If you wanted to import "Cards" in this qualified form.
as in: module Cards. for Windows haskell\Cards\Cards. for instance. but that we didn't need the "Cards" version of that function. in Hugs it's "-P"). as: import Cards. suppose we want to import "Cards" as a qualified module. Hierarchical imports allow you to specify (to a certain degree) where in the directory structure a module exists. MODULES import Cards (newDeck) On the other hand...) where . For instance. This was designed to get rid of clutter in the directories in which modules are stored. but don't want to have to type Cards.70 CHAPTER 6.3 Hierarchical Imports Though technically not part of the Haskell 98 standard..hs (or.. You could them import it in any module. C. so you can write: . for instance.Cards(. out all the time and would rather just type. you created a directory specifically for it called "Cards". Suppose instead of saving the "Cards" module in your general haskell directory.Cards". I highly recommend using the as keyword to shorten the names. regardless of this module's directory.Cards If you start importing these module qualified. 6. We could hide the definition of deal and import everything else by writing: import Cards hiding (deal) Finally. most Haskell compilers support hierarchical imports. if you have a "haskell" directory on your computer and this directory is in your compiler's path (see your compiler notes for how to set this. in GHC it's "-i". – we could do this using the as keyword: import qualified Cards as C These options can be mixed and matched – you can give explicit import lists on qualified/as imports. If you then change the name of the Cards module to "Cards.hs). suppose we knew that that the deal function overlapped with another module. The full path of the Cards. then you can specify module locations in subdirectories to that directory.hs file is then haskell/Cards/Cards.
.newDeck . Literate programming swaps these preconceptions.. literate scripts must have the extension lhs instead of hs to tell the compiler that the program is written in a literate style. 6...4. the first uses so-called BirdA scripts and the second uses LTEX-style markup.4.6. Everything else remains the same.Cards. we write code without any special annotation.Cards ... comments are default and code is introduced with a leading greater-than sign (">"). instead of: import qualified Cards..Cards as Cards . Each will be discussed individually..newDeck . No matter which you use. That is. this program will have the exact same properties as the nonliterate version from Section 3. .4 Literate Versus Non-Literate The idea of literate programming is a relatively simple one.4. When we think about programming. For example.. but comments are annotated with either -. 6.. Cards. which tends to get ugly.or {. LITERATE VERSUS NON-LITERATE 71 import qualified Cards. we think about the code being the default mode of entry and comments being secondary.1 Bird-scripts In a Bird-style literate program. our Hello World program would be written in Bird-style as: This is a simple (literate!) Hello World program. > module Main > where All our main function does is print a string: > main = putStrLn "Hello World" Note that the spaces between the lines of code and the "comments" are necessary (your compiler will probably complain if you are missing them). Cards. -}. When compiled or loaded in an interpreter. but took quite a while to become popularized. There are two types of literate programs in Haskell..
.2 LaTeX-scripts A LTEX is a text-markup language very popular in the academic community for publishA ing. If you are unfamiliar with LTEX. the blank lines are not necessary.72 CHAPTER 6. \begin{code} module Main where \end{code} All our main function does is print a string: \begin{code} main = putStrLn "Hello World" \end{code} A In LTEX-style scripts. you may not find this section terribly useful. a literate Hello World program written in LTEX-style would look like: This is another simple (literate!) Hello World program. A Again. MODULES 6.4.
126.96.36.199. This won't work. though.x − 2. For binary infix operators (like +). these are called sections.3. because while the + in +2 is parsed as the standard plus operator (as there is no ambiguity).2. In general.6. there is a more concise way to write this: Prelude> map (*2) [1.in -2 is interpreted as the unary minus. not the binary minus.0] Prelude> map (2/) [1.4] [0.5.5] You might be tempted to try to subtract values from elements in a list by mapping -2 across a list.8.2.3. For example: 73 .9] Prelude> map (/2) [1. Thus -2 here is the number −22.214.171.124.4.2. not the function λx.4] [2.4] [126.96.36.199.3.4.8] This type of thing can be done for any infix function: Prelude> map (+5) [1.2.666667.0.Chapter 7 Advanced Features Discussion 7.1 Sections and Infix Operators We've already seen how to double the values of elements in a list using map: Prelude> map (\x -> x*2) [1.4] [22.214.171.124] [6. we can cause the function to become prefix by enclosing it in paretheses.8] However.2. the .6.
(-b ..4.0 Prelude> (3/) 6 0. For instance.74 CHAPTER 7.5. (-b .sqrt(b*b .12] 188.8.131.52). we can do this using a where clause.5. ADVANCED FEATURES Prelude> (+) 5 3 8 Prelude> (-) 5 3 2 Additionally. where clauses come immediately after function definitions and introduce a new level of layout (see Section 7. example: Prelude> (+2) 'map' [1. we can provide either of its argument to make a section.5 Non-infix functions can be made infix by enclosing them in backquotes ("` For '').9.8.4*a*c)) / (2*a). We write this as: roots a b c = ((-b + det) / (2*a).4*a*c)) / (2*a)) In addition to the let bindings introduced there. There.2 Local Declarations Recall back from Section 3.10. For example: Prelude> (+5) 3 8 Prelude> (/3) 6 2. we considered the function for computing the roots of a quadratic polynomial: roots a b c = ((-b + sqrt(b*b . if we had the following code block: .11.10] [3.det) / (2*a)) where det = sqrt(b*b-4*a*c) Any values defined in a where clause shadow any other values with the same name. there are many computations which require using the result of the same computation in multiple places in a function.
However. (-b . Where clauses can contain any number of subexpressions. but they must be aligned for layout. we could also pull out the 2*a computation and get the following code: roots a b c = ((-b + det) / (a2).4*a*c) a2 = 2*a These two types of clauses can be mixed (i. which is the string "Hello World. Furthermore. it looks like: roots a b c = ((-b + det) / a2. Sometimes it is more convenient to put the local definitions before the actual expression of the function. f is a function which takes any argument to that string.. since it is shadowed by the local definition (the fact that the types don't match doesn't matter either).det) / (2*a)) where det = sqrt(b*b-4*a*c) f _ = det The value of roots doesn't notice the top-level declaration of det. LOCAL DECLARATIONS 75 det = "Hello World" roots a b c = ((-b + det) / (2*a). (-b ." Thus.det) / (a2)) where det = sqrt(b*b-4*a*c) a2 = 2*a Sub-expressions in where clauses must come after function definitions. This is strongly advised against.4*a*c) a2 = 2*a in ((-b + det) / a2.7. (-b . since f cannot "see inside" of roots. values in the let clause shadow those in the where clause. where clauses are virtually identical to their let clause cousins except for their placement. This can be done by using let/in clauses. you can write a function which has both a let cause and a where clause). if you choose to do it. The same roots function can be written using let as: roots a b c = let det = sqrt (b*b . We have already seen let clauses. as it tends to make code difficult to read. (-b . So if you define the function: .2. the only thing it knows about det is what is available at the top level. For instance.det) / a2) where det = sqrt (b*b .det) / a2) Using a where clause.e.
76 CHAPTER 7. Now. suppose we are writting a function lcaseString which converts a whole string into lower case. Thus. This all is related to type type of map. but we've only given it one. where clauses are probably more desirable because they allow the reader to see immediately what a function does. We could write this as: lcaseString s = map toLower s Here. consider the task of converting a string to lowercase and remove all non letter characters. the section (+1) is a partial application of +. whether you should use let clauses or where clauses is largely a matter of personal preference. In our case. Usually. we can remove it by performing eta reduction. No one should have to remember this rule and by shadowing where-defined values in a let clause only makes your code difficult to understand. there is no partial application (though you could argue that applying no arguments to toLower could be considered partial application). we notice that the application of s occurs at the end of both lcaseString and of map toLower. However. Of course. the names you give to the subexpressions should be sufficiently expressive that without reading their definitions any reader of your code should be able to figure out what they do. toLower is of type Char → Char. in real life. In this case. we have a partial application of map: it expects a function and a list. though I think where clauses are more common. For instance. I plead with you to never ever write code that looks like this. In which case let clauses may be better. which is (a → b) → ([a] → [b]). We might write this as: eta reduction . For instance. we saw a form of "partial application" in which functions like + were partially applied. but we've only given it the function.2. in the expression map (+1) [1. When discussing sections in Section 7. as desired. we get a function of type [Char] → [Char]. when parentheses are all included. if we supply this function to map. Partial application is very common in function definitions and sometimes goes by the name "eta reduction". ADVANCED FEATURES f x = let y = x+1 in y where y = x+2 The value of f 5 is 6. In general. In fact. values are often given cryptic names. Either is probably okay.3 Partial Application Partial application is when you take a function which takes n arguments and you supply it with < n of them. not 7. 7. This is because + really takes two arguments.1.3]. to get: lcaseString = map toLower Now. However.
The ($) function is also useful when combined with other infix functions.) is function composition. For instance. which makes no sense. filter isAlpha) s And again. By removing the point from the function definition. we have a point-free function. The definition of ($) from the Prelude is very simple: f $ x = f x However. For instance. PARTIAL APPLICATION 77 lcaseLetters s = map toLower (filter isAlpha s) But note that we can actually write this in terms of function composition: lcaseLetters s = (map toLower . which means that it can be used to replace parentheses. A function similar to (. filter isAlpha Writing functions in this style is very common among advanced Haskell users. One way to write this would be: point-free programming $ function application . Whereas (. It is call point free because in the original definition of lcaseLetters. However. Consider now the task of extracting from a list of tuples all the ones whose first component is greater than zero. using the function application function. we can think of the value s as a point on which the function is operating. we cannot write: Prelude> putStrLn "5+3=" ++ show (5+3) because this is interpreted as (putStrLn "5+3=") ++ (show (5+3)). we can rewrite this as: foo x y = bar y $ baz $ fluff $ ork x This moderately resembles the function composition syntax. we're left with an eta reducible function: lcaseLetters = map toLower .7. we can fix this by writing instead: Prelude> putStrLn $ "5+3=" ++ show (5+3) Which works fine. this function is given very low fixity. ($) is function application.3. we might write a function: foo x y = bar y (baz (fluff (ork x))) However.) is ($). In fact it has a name: point-free programming (not to be confused with pointless programming).
For instance. b) → c. For instance. (3+) .(5.b) -> a>0) Now.fst) This definition is simultaneously shorter and easier to understand than the original. yielding: fstGt0 = filter (\ (a. While converting to point free style often results in clearer code. b) → c and produces a function of type a → b → c. This is useful. (4*) .6)] [2.12. for example. converting the following map to point free style yields something nearly uninterpretable: foo = map (\x -> sqrt (3+4*(xˆ2))) foo = map (sqrt . We can clearly see exactly what it is doing: we're filtering a list by checking whether something is greater than zero. when mapping across a list of pairs: Prelude> map (uncurry (*)) [(1.(3. we can sort a list in reverse order by using flip compare: .2).b) -> a>0) l We can first apply eta reduction to the whole function. it takes a function of type a → b → c and produces a function of type b → a → c. (ˆ2)) There are a handful of combinators defined in the Prelude which are useful for point free programming: • uncurry takes a function of type a → b → c and converts it into a function of type (a. we can rewrite the lambda function to use the fst function instead of the pattern matching: fstGt0 = filter (\x -> fst x > 0) Now. this is of course not always the case. What are we checking? The fst element.4). • flip reverse the order of arguments to a function. fst) x) And finally we can eta reduce: fstGt0 = filter ((>0). ADVANCED FEATURES fstGt0 l = filter (\ (a.78 CHAPTER 7. we can use function composition between fst and > to get: fstGt0 = filter (\x -> ((>0) .30] • curry is the opposite of uncurry and takes a function of type (a. That is.
3.8. func1 x l = map (\y -> y*x) l func2 f g l = filter f (map g l) func3 f l = l ++ map f l func4 l = map (\y -> y+2) (filter (\z -> z 'elem' [1.8.sortBy (flip compare) [5. Exercises Exercise 7. if we can define other functions.8] Prelude> List.1.1.1] only shorter.3. Of course. For instance.7.10]) (5:l)) func5 f l = foldr (\x y -> f (y. not all functions can be written in point free style. For instance: square x = x*x Cannot be written in point free style. if possible.3..sortBy compare [5.1] This is the same as saying: Prelude> List.sortBy (\a b -> compare b a) [5.3] [8. pair But in this case.1 Convert the following functions into point-free style.5.5.5. we can write: pair x = (x.3] [1.3.3] [8.x) square = uncurry (*) . PARTIAL APPLICATION 79 Prelude> List.x)) 0 l .8.1. this is not terribly useful. without some other combinators.
Eq) -. and if we see Yellow.255) Black = (0.255.0.5. Finally. It is most commonly used in conjunction with case expressions.0) Green = (0. ADVANCED FEATURES 7. So we write that (remember that piecewise function definitions are just case statements): colorToRGB Red = (255. in our interpreter.b) Then. Let's return to our Color example from Section 4. Specifically.0. if we type: Color> colorToRGB Yellow (255.255.255. so we write: colorToRGB colorToRGB colorToRGB colorToRGB colorToRGB colorToRGB colorToRGB colorToRGB Orange = (255. since this is the RGB value for red.255. which we have already seen in Section 3.80 CHAPTER 7.255) Purple = (255.0).128.R G B components We then want to write a function that will convert between something of type Color and a triple of Ints.0.0) Yellow = (255. which is comprised of three components.255.255) White = (255.0) (Custom r g b) = (r.0) . we want to return (255. which correspond to the RGB values.0) If we see a Color which is Orange. respectively. and so on.5. if we see a custom color.0).128. we want to return (255. if we see a Color which is Red.0).0.0) Blue = (0. we want to make a triple out of these. I'll repeat the definition we already had for the datatype: data Color = Red | Orange | Yellow | Green | Blue | Purple | White | Black | Custom Int Int Int deriving (Show. we want to return (255.g.0.4 Pattern Matching Pattern matching is one of the most powerful features of Haskell (and most functional programming languages).
_. g and b. Suppose instead. g and b are essentially wild cards. which tells us if a given RGB value is bright or not. As a "side-effect" of this matching.4. as expected. but will not do the binding that would happen if you put a variable name there. if this match fails. so then we go see if the subelements match. the variables r. We could define this function as: isBright = isBright' . r gets the value 50. The isBright function is the composition of our previously defined function colorToRGB and a helper function isBright'. it goes on to the next line. PATTERN MATCHING 81 What is happening is this: we create a value. If this match succeeds.200. This match will succeed if the value is Custom x y z for any x.255) = True isBright' _ = False Let's dwell on this definition for a second. so there is no trouble matching r with 50. We then get to try to match Custom 50 200 100 against Custom r g b. Red is not equal to Yellow. call it x.0). We continue down the definitions of colorToRGB and try to match Yellow against Orange. however. This fails. but it's simply an example). We can also write a function to check to see if a Color is a custom color or not: isCustomColor (Custom _ _ _) = True isCustomColor _ = False When we apply a value to isCustomColor it tries to match that value against . so we use this function definition. failing on all values from Red to Black._) = True isBright' (_.255. We check to see if we can "match" x against Red. which succeeds. We can see that the Custom part matches. where my definition of "bright" is that one of its RGB components is equal to 255 (admittedly and arbitrary definition._. Custom y and z. colorToRGB where isBright' (255. g with 200 and b with 100._) = True isBright' (_.255. too.7. So the entire match succeeded and we look at the definition of this part of the function and bundle up the triple using the matched values of r. We then apply this to colorToRGB. We could replace the first line here . The (underscore) character is a "wildcard" and will match anything. which will match anything and then return False. This match fails because according to the definition of Eq Color. In the matching. We the try to match Yellow against Yellow. which has value Red. we used a custom color: Color> colorToRGB (Custom 50 200 100) (50. g gets the value 200 and b gets the value 100. which simply returns the value (255. For some reason we might want to define a function which tells us whether a given color is "bright" or not.100) We apply the same matching process. the function returns True.
respectively. Again. which corresponds to a failure. the . the RGB value is valid.invalid RGB value The first eight lines match the RGB arguments against the predefined values and. if they match. The fourth line. isBright' returns True and so does isBright. we don't want to allow the user to construct custom colors like (600. so I will try to use it frequently in this tutorial. on the other hand. However. ) which succeeds if the value has 255 in its first position. rgbToColor returns Just the appropriate color. the last definition of rgbToColor matches the first argument against r.-40. If. The isBright' helper function takes the RGB triple produced by colorToRGB.99) since these are invalid RGB values. this function composition style of programming takes some getting used to.82 CHAPTER 7. The code to do this is: rgbToColor 255 0 0 = Just Red rgbToColor 255 128 0 = Just Orange rgbToColor 255 255 0 = Just Yellow rgbToColor 0 255 0 = Just Green rgbToColor 0 0 255 = Just Blue rgbToColor 255 0 255 = Just Purple rgbToColor 255 255 255 = Just White rgbToColor 0 0 0 = Just Black rgbToColor r g b = if 0 <= r && r <= 255 && 0 <= g && g <= 255 && 0 <= b && b <= 255 then Just (Custom r g b) else Nothing -. . we want to use the Custom slot only for values which don't match the predefined colors. If this match succeeds. We could simple stick everything in a Custom constructor. we return Nothing. If none of these matches. If the RGB value passed to our function is invalid. matches everything else and reports it as not bright. It first tries to match it against (255. ADVANCED FEATURES with isBright c = isBright' (colorToRGB c) but there is no need to explicitly write the parameter here. so we don't. but this can be difficult to deal with. we use the Maybe datatype. we create the appropriate Color value and return Just that. The second and third line of definition check for 255 in the second and third position in the triple. This is defined (in the Prelude) as: data Maybe a = Nothing | Just a The way we use this is as follows: our rgbToColor function returns a value of type Maybe Color. We might want to also write a function to convert between RGB triples and Colors. but this would defeat the purpose. the fallthrough. We could throw an error if such a value is given. Instead.
it matches anything and returns False. Guards appear after all arguments to a function but before the equals sign." So we say that the value of comparison x y "such that" x is less than y is "The first is less. if not. it returns Just (Custom r g b). g x = x ..1). You can only match against datatypes. we can write a function that checks to see if a right RGB value is valid: rgbIsValid r g b = rgbIsValid' (rgbToColor r g b) where rgbIsValid' (Just _) = True rgbIsValid' _ = False Here. We could use guards to write a simple function which returns a string telling you the result of comparing two elements: comparison x y | x < y = "The first is less" | x > y = "The second is less" | otherwise = "They are equal" You can read the vertical bar as "such that. it returns True. Pattern matching isn't magic.7. we can see that this works: . If not. Using this.5. so it can't perform matches like this. though. it returns Nothing corresponding to an invalid color. you cannot match against functions.e. They enable you to allow piecewise function definitions to be taken according to arbitrary boolean expressions. If so. the following is invalid: f x = x + 1 g (f x) = x Even though the intended meaning of g is clear (i. we compose the helper function rgbIsValid' with our function rgbToColor. It then checks to see if these values are valid (each is greater than or equal to zero and less than or equal to 255). the compiler doesn't know in general that f has an inverse function. The helper function checks to see if the value returned by rgbToColor is Just anything (the wildcard). and are begun with a vertical bar. The keyword otherwise is simply defined to be equal to True and thus matches anything that falls through that far. So. GUARDS 83 second against g and the third against b (which causes the side-effect of binding these values).5 Guards Guards can be thought of as an extension to the pattern matching facility." The value such that x is greater than y is "The second is less" and the value otherwise is "They are equal". If so. For instance. 7.
This means that once a pattern matches. Pattern matching then stops completely. not in conjunction with pattern matching. if none of the guards succeed. yielding an RGB triple which is matched (using pattern matching!) against (r. . they are matched against x and y. Neither of the guards succeeds. ADVANCED FEATURES Guards> comparison 5 10 "The first is less" Guards> comparison 10 5 "The second is less" Guards> comparison 7 7 "They are equal" One thing to note about guards is that they are tested after pattern matching.g.g. The guards are then activated and x and y are compared. The second and third guard check g and b against 255. and applies colorToRGB to it. so an error is raised. which succeeds and the values are bound. One nicety about guards is that where clauses are common to all guards. g and b are bound to their respective values. further pattern matches will not be attempted. returns true. though.hs:8: Non-exhaustive patterns in function comparison2 If we think about what is happening in the compiler this makes sense. It takes a color. if we had instead defined: comparison2 x y | x < y = "The first is less" | x > y = "The second is less" comparison2 _ _ = "They are equal" The intention would be that if both of the guards failed. So. Guards> comparison2 7 7 *** Exception: Guards. The last guard fires as a last resort and returns False.b). When we apply two sevens to comparison2. The first guard checks to see if r is 255 and. but performs its calculation slightly differently. This is not what happens. it would "fall through" to the final match and say that they were equal.b) = colorToRGB c The function is equivalent to the previous version. c. This match succeeds and the values r. if so. So another possible definition for our isBright function from the previous section would be: isBright2 c | r == 255 = True | g == 255 = True | b == 255 = True | otherwise = False where (r. respectively and return true if they match.84 CHAPTER 7. since it has succeeded.
1 The Eq Class The Eq class has two members (i. and vice versa).6 Instance Declarations In order to declare a type to be an instance of a class.. These declarations must be provided inside the instance declaration. The type signature of /= (not equal) is identical. Most classes provide what's called a "minimal complete definition. Suppose we have our color example. you need to provide an instance declaration for it. you can declare it an instance of the class. Once you've written these functions for your type. two functions): (==) :: Eq a => a -> a -> Bool (/=) :: Eq a => a -> a -> Bool The first of these type signatures reads that the function == is a function which takes two as which are members of Eq and produces a Bool.7. This is best demonstrated by example." This means the functions which must be implemented for this class in order for its definition to be satisfied. A minimal complete definition for the Eq class requires that either one of these functions be defined (if you define ==.6. 7.6. then /= is defined automatically by negating the result of ==.e.R G B components We can define Color to be an instance of Eq by the following declaration: instance Eq Color where Red == Red = True Orange == Orange = True Yellow == Yellow = True Green == Green = True Blue == Blue = True Purple == Purple = True White == White = True . repeded here for convenience: data Color = Red | Orange | Yellow | Green | Blue | Purple | White | Black | Custom Int Int Int -. INSTANCE DECLARATIONS 85 7.
Finally there's the method declaration. Eq. The first eight lines of the method declaration are basically identical. g == g' and b == b' are all equal. On the left hand side. We then say that these two custom colors are equal precisely when r == r'. 7. respectively. The first one. for instance. . Again. Color which is going to be an instance of this class.86 CHAPTER 7. there's the where keyword. says that the value of the expression Red == Red is equal to True. ADVANCED FEATURES Black == Black = True (Custom r g b) == (Custom r' g' b') = r == r' && g == g' && b == b' _ == _ = False The first line here begins with the keyword instance telling the compiler that we're making an instance declaration. g' and b' to the components. On the right hand side. Lines two through eight are identical. and the type.2 The Show Class The Show class is used to display arbitrary values as strings. the first eight lines are identical and simply take a Color and produce a string. we bind r. we bind r'. g and b to the components.6. The fallthrough says that any pair we haven't previously declared as equal are unequal. It then specifies the class. The declaration for custom colors is a bit different. We can define our Color datatype to be an instance of Show with the following instance declaration: instance show show show show show show show show show Show Color where Red = "Red" Orange = "Orange" Yellow = "Yellow" Green = "Green" Blue = "Blue" Purple = "Purple" White = "White" Black = "Black" (Custom r g b) = "Custom " ++ show r ++ " " ++ show g ++ " " ++ show b This declaration specifies exactly how to convert values of type Color to Strings. Following that. We pattern match Custom on both sides of ==. This class has three methods: show :: Show a => a -> String showsPrec :: Show a => Int -> a -> String -> String showList :: Show a => [a] -> String -> String A minimal complete definition is either show or showsPrec (we will talk about showsPrec later – it's in there for efficiency reasons).
how you can do this should be clear by now. INSTANCE DECLARATIONS 87 The last line for handling custom colors matches out the RGB components and creates a string by concattenating the result of showing the components individually (with spaces in between and "Custom" at the beginning). that is. I won't provide example instance declarations.4). The Ord Class The ordering class. we get: Prelude> compare 5 7 LT Prelude> compare 6 6 EQ Prelude> compare 7 5 GT In order to declare a type to be an instance of Ord you must already have declared it an instance of Eq (in other words. it is recommended that you implement compare if you implement only one. though. This function returns a value of type Ordering which is defined as: data Ordering = LT | EQ | GT So. the functions are: compare :: Ord a => a -> a -> Ordering (<=) :: Ord a => a -> a -> Bool (>) :: Ord a => a -> a -> Bool (>=) :: Ord a => a -> a -> Bool (<) :: Ord a => a -> a -> Bool min :: Ord a => a -> a -> a max :: Ord a => a -> a -> a The almost any of the functions alone is a minimal complete definition. It's methods are: . for types where each element has a successor and a predecessor.3 Other Important Classes There are a few other important classes which I will mention briefly because either they are commonly used or because we will be using them shortly. Ord is a subclass of Eq – more about this in Section 8.6.6. 7. for instance. The Enum Class The Enum class is for enumerated types.7.
The methods for Read are: readsPrec :: Read a => Int -> String -> [(a. negate x means −x. That is. The enum functions enumerate lists of elements. which uses readsPrec as: read s = fst (head (readsPrec 0 s)) . The most important function related to this is read. For instance. The Num Class The Num class provides the standard arithmetic operations: (-) :: Num a => a -> (*) :: Num a => a -> (+) :: Num a => a -> negate :: Num a => a signum :: Num a => a abs :: Num a => a -> fromInteger :: Num a a -> a a -> a a -> a -> a -> a a => Integer -> a All of these are obvious except for perhaps negate which is the unary minus. which converts from and to Ints. enumFrom x lists all elements after x. String)] readList :: String -> [([a]. String)] The minimal complete definition is readsPrec. respectively. It is a way to take a string and read in from it a value of arbitrary type. ADVANCED FEATURES pred :: Enum a => a -> a succ :: Enum a => a -> a toEnum :: Enum a => Int -> a fromEnum :: Enum a => a -> Int enumFrom :: Enum a => a -> [a] enumFromThen :: Enum a => a -> a -> [a] enumFromTo :: Enum a => a -> a -> [a] enumFromThenTo :: Enum a => a -> a -> a -> [a] The minimal complete definition contains both toEnum and fromEnum. The Read Class The Read class is the opposite of the Show class. The To functions end the enumeration at the given element. The pred and succ functions give the predecessor and successor.88 CHAPTER 7. enumFromThen x step lists all elements starting at x in steps of size step.
_)] -> Just a _ -> Nothing How to write and use readsPrec directly will be discussed further in the examples. we need to know that a is an instance of Eq otherwise we can't write a declaration.. Luckily for us. the compiler can automatically derive some of the most basic classes. Similarly. The definition would be something like: data Maybe a = Nothing | Just a Now. Eq. To do this.7. If you write a datatype that's "simple enough" (almost any datatype you'll write unless you start writing fixed point types). Ord.6. as in: data Color = Red | . the declaration: .5 Deriving Classes Writing obvious Eq. 7. You could define a maybeRead function as: maybeRead s = case readsPrec 0 s of [(a. for.R G B components deriving (Eq. Read) This will automatically create instances of the Color datatype of the named classes. it is." 7. We express this as: instance Eq a => Eq (Maybe a) where Nothing == Nothing = True (Just x) == (Just x') == x == x' This first line can be read "That a is an instance of Eq implies (=>) that Maybe a is an instance of Eq.4 Class Contexts Suppose we are definition the Maybe datatype from scratch.6. Ord. you simply add a deriving clause to after the datatype declaration.. | Custom Int Int Int -. INSTANCE DECLARATIONS 89 This will fail if parsing the string fails. say.6. Read and Show classes like these is tedious and should be automated. Show. when we go to write the instance declarations.
would allow for instance declarations for any class to be derived. I refer you to the literature. instead. 7. you might often find yourself wondering "wait. Consider the following made-up configuration type for a terminal program: data Configuration = Configuration String String String Bool Bool String String Integer deriving (Eq. a datatype that holds many many values. 7. Show) --------- user name local host remote host is guest? is super user? current directory home directory time connected You could then write accessor functions.90 CHAPTER 7. among other things. They are. you really only care about one or possibly two of the many settings. Enum. Datatypes offer a sort of notational convenience if you have. ADVANCED FEATURES data Maybe a = Nothing | Just a deriving (Eq. Bounded. if many of the settings have the same type. Ord. Show Read) derives these classes just when a is appropriate. Moreover. This is much beyond the scope of this tutorial. Show and Read. All in all. incredibly important. like (I've only listed a few): getUserName (Configuration un _ _ _ _ _ _ _) = un getLocalHost (Configuration _ lh _ _ _ _ _ _) = lh getRemoteHost (Configuration _ _ rh _ _ _ _ _) = rh . for instance. otherwise I wouldn't devote so much time to them. Ord. you are allowed to derive instances of Eq. was this the fourth or fifth element?" One thing you could do would be to write accessor functions. however. There is considerable work in the area of "polytypic programming" or "generic programming" which.1 Named Fields Consider a datatype whose purpose is to hold configuration settings.7 Datatypes Revisited I know by this point you're probably terribly tired of hearing about datatypes.7. Usually when you extract members from this type. These are called named fields.
it gives us very convenient update methods. We simply give names to the fields in the datatype declaration. or remove one.make sure the directory exists if directoryExists newDir then -. as follows: data Configuration = Configuration { username localhost remotehost isguest issuperuser currentdir homedir timeconnected } :: :: :: :: :: :: :: :: String. 91 You could also write update functions to update a single element. However.change our current directory cfg{currentdir = newDir} else error "directory does not exist" postWorkingDir :: Configuration -> String -.. Bool. This is highly annoying and is an easy place for bugs to slip in. each should be separated by commas. you write y{x=z}. to update the field x in a datatype y to z.retrieve our current directory postWorkingDir cfg = currentdir cfg So..7. now if you add an element to the configuration. c=d}. for instance. . in general. Moreover. Of course. y{x=z. all of these functions now have to take a different number of arguments. a=b. You can change more than one. String.. Here is a short example for a "post working directory" and "change directory" like functions that work on Configurations: changeDir :: Configuration -> String -> Configuration changeDir cfg newDir = -..7. Bool. DATATYPES REVISITED getIsGuest (Configuration _ _ _ ig _ _ _ _) = ig . String. Integer This will automatically generate the following accessor functions for us: username :: Configuration -> String localhost :: Configuration -> String . there's a solution. String. String.
You can of course continue to pattern match against Configurations as you did before. The named fields are simply syntactic sugar; you can still write something like: getUserName (Configuration un _ _ _ _ _ _ _) = un But there is little reason to. Finally, you can pattern match against named fields as in: getHostData (Configuration {localhost=lh,remotehost=rh}) = (lh,rh) This matches the variable lh against the localhost field on the Configuration and the variable rh against the remotehost field on the Configuration. These matches of course succeed. You could also constrain the matches by putting values instead of variable names in these positions, as you would for standard datatypes. You can create values of Configuration in the old way as shown in the first definition below, or in the named-field's type, as shown in the second definition below: initCFG = Configuration "nobody" "nowhere" "nowhere" False False "/" "/" 0 initCFG' = Configuration { username="nobody", localhost="nowhere", remotehost="nowhere", isguest=False, issuperuser=False, currentdir="/", homedir="/", timeconnected=0 } Though the second is probably much more understandable unless you litter your code with comments.
7.8 More Lists
todo: put something here
Standard List Functions
Recall that the definition of the built-in Haskell list datatype is equivalent to: data List a = Nil | Cons a (List a)
With the exception that Nil is called [] and Cons x xs is called x:xs. This is simply to make pattern matching easier and code smaller. Let's investigate how some of the standard list functions may be written. Consider map. A definition is given below: map _ [] = [] map f (x:xs) = f x : map f xs Here, the first line says that when you map across an empty list, no matter what the function is, you get an empty list back. The second line says that when you map across a list with x as the head and xs as the tail, the result is f applied to x consed onto the result of mapping f on xs. The filter can be defined similarly: filter _ [] = [] filter p (x:xs) | p x = x : filter p xs | otherwise = filter p xs How this works should be clear. For an empty list, we return an empty list. For a non empty list, we return the filter of the tail, perhaps with the head on the front, depending on whether it satisfies the predicate p or not. We can define foldr as: foldr _ z [] = z foldr f z (x:xs) = f x (foldr f z xs) Here, the best interpretation is that we are replacing the empty list ([]) with a particular value and the list constructor (:) with some function. On the first line, we can see the replacement of [] for z. Using backquotes to make f infix, we can write the second line as: foldr f z (x:xs) = x 'f' (foldr f z xs) From this, we can directly see how : is being replaced by f. Finally, foldl: foldl _ z [] = z foldl f z (x:xs) = foldl f (f z x) xs This is slightly more complicated. Remember, z can be thought of as the current state. So if we're folding across a list which is empty, we simply return the current state. On the other hand, if the list is not empty, it's of the form x:xs. In this case, we get a new state by appling f to the current state z and the current list element x and then recursively call foldl on xs with this new state. There is another class of functions: the zip and unzip functions, which respectively take multiple lists and make one or take one lists and split them apart. For instance, zip does the following:
Prelude> zip "hello" [1,2,3,4,5] [('h',1),('e',2),('l',3),('l',4),('o',5)] Basically, it pairs the first elements of both lists and makes that the first element of the new list. It then pairs the second elements of both lists and makes that the second element, etc. What if the lists have unequal length? It simply stops when the shorter one stops. A reasonable definition for zip is: zip [] _ = [] zip _ [] = [] zip (x:xs) (y:ys) = (x,y) : zip xs ys The unzip function does the opposite. It takes a zipped list and returns the two "original" lists: Prelude> unzip [('f',1),('o',2),('o',3)] ("foo",[1,2,3]) There are a whole slew of zip and unzip functions, named zip3, unzip3, zip4, unzip4 and so on; the ...3 functions use triples instead of pairs; the ...4 functions use 4-tuples, etc. Finally, the function take takes an integer n and a list and returns the first n elements off the list. Correspondingly, drop takes an integer n and a list and returns the result of throwing away the first n elements off the list. Neither of these functions produces an error; if n is too large, they both will just return shorter lists.
List Comprehensions
There is some syntactic sugar for dealing with lists whose elements are members of the Enum class (see Section 7.6), such as Int or Char. If we want to create a list of all the elements from 1 to 10, we can simply write: Prelude> [1..10] [1,2,3,4,5,6,7,8,9,10] We can also introduce an amount to step by: Prelude> [1,3..10] [1,3,5,7,9] Prelude> [1,4..10] [1,4,7,10] These expressions are short hand for enumFromTo and enumFromThenTo, respectively. Of course, you don't need to specify an upper bound. Try the following (but be ready to hit Control+C to stop the computation!):
Of course.7. However. we would write something like {f (x)|x ∈ s ∧ p(x)} to mean the set of all values of f when applied to elements of s which satisfy p.('o'.5)] Finally. remove all the lower-case letters and convert the rest of the letters to upper case."Hello World".2.8.s.6. We can see that this works: Prelude> assignID "hello" [('h'. based on standard set-notation in mathematics.9.('l'.8. Haskell is lazy.5. But if we only evaluate an initial segment of this list. we want to assign an ID to each element in a list. As we said before.1). We could not have put p x before x <.] [1. there is some useful syntactic sugar for map and filter.. That means that a list of all numbers from 1 on is perfectly well formed and that's exactly what this list is. MORE LISTS 95 Prelude> [1.11. This is equivalent to the Haskell statement map f (filter p s).. because of laziness.2.7. if you attempt to print the list (which we're implicitly doing by typing it in the interpreter).] [1. p x].3. In math.s otherwise the compiler wouldn't know yet what x was. We could do this in either of the following two equivalent ways: Prelude> map toLower (filter isUpper "Hello World") "hw" Prelude> [toLower x | x <.] And we'll get exactly what we want.4).. Suppose we want to take a string.('l'. we're fine: Prelude> take 3 [1. isUpper x] "hw" .('e'. we can also use more math-like notation and write [f x | x <. While in math the ordering of the statements on the side after the pipe is free.3] Prelude> take 3 (drop 5 [1. we can simply write: assignID l = zip l [1. However.. say. We can use this to do simple string processing.Int)] assignID l = zip l [1.8] This comes in useful if.length l] Which means that the list will be traversed twice.10. it won't halt. it is not so in Haskell.3). Without laziness we'd have to write something like this: assignID :: [a] -> [(a.4.7.]) [6.2).12{Interrupted!} Probably yours printed a few thousand more elements than this..
(1.8).(2.3). and so on.(2.7).4).4). having average complexity O(n) to access an arbitrary element (if you don't know what O(.10)] Arrays> listArray (1.(5. the accumArray function takes an accumulation function. one might be more readable than the other.5]] array (1.6).y) | x <.5) [(i. and.(1.(4.7.96 CHAPTER 7. the listArray function. they are bad for random access.5). There are a few methods for creating arrays. ADVANCED FEATURES These two are equivalent.10] array (1. ) means.5) [3. Doing this manually with lists and maps would be cumbersome and possibly difficult to read.7)] If you reverse the order of the x <. For instance.clauses.5) [(1..2*i) | i <.(4. y could no longer depend on x and you would need to make x depend on y but this is trivial). in the first example.(5.0) and (5.(1.5).3).(3. as in: .(4.5) [(i.6).(1.[1. Finally. they are printed with an association list. It is easy to add elements to the beginning of them and to manipulate them in various ways that change the length of the list. in that case.(2.. So.7).(1.7]] [(1.(3. and an association list which specifies the initial values of the array.5].6).i) | i <.(1.(4. an initial value and an association list and accumulates pairs from the list into the array.(4. y <.(5.6). .5.6).3).5).(4.1.(5.5) [(1. You can extract an element of an array using the ! function. the array function. which takes an array and an index.1).5). and the accumArray function.5) [(1.[1.10)] Arrays> accumArray (+) 2 (1.(5.9 Arrays Lists are nice for many things.1).(2. The array function takes a pair which is the bounds of the array.(3.(2..4).2). 7.6). There's more you can do here.(2. the association list says that the value of the array at 1 is 2.3). The listArray function takes bounds and then simply a list of values.7)] When arrays are printed out (via the show function).3).2).[x.6).5).(3. you should use arrays instead of lists. the value of the array at 2 is 4. though. In order to use arrays you must import the Array module.[1.(2. one for each point between (0.(5.4). if you're willing to give up fast insertion and deletion because you need random access. depending on the exact functions you're using. .5). a two-page introduction to complexity theory).7).7).7). Suppose you want to create a list of pairs. (4.(3. Here are some examples of arrays being created: Arrays> array (1.(3.5]] array (1.and y <.4).5).(3.4). However. It couldn't be easier than with list comprehensions: Prelude> [(x. the order in which the space is traversed will be reversed (of course.7) below the diagonal. you can either ignore it or take a quick detour and read Appendix A.2).(2.(3. (2..
4.(4.5) [3.1). functional arrays are pretty much only useful when you're filling them up once and then only reading.1. They are in fact O(n) update.5. This takes an array and an association list and updates the positions specified in the list: Arrays> (listArray (1.(3.2.1). Thus. 7.10 Finite Maps The FiniteMap datatype (which is available in the FiniteMap module.7. which are discussed in Section 7. A brief comparison is: .(3.5. they are not O(1) update.99). the result of these functions applied to arr are: Arrays> bounds arr (1.5. FINITE MAPS 97 Arrays> (listArray (1. or Data.-99).10]) ! 3 5 Moreover. n.5) [3. Finite maps can be compared to lists are arrays in terms of the time it takes to perform various operations on those datatypes of a fixed size.(3.10]) // [(2. If you need fast access and update.5) [(1.(2. the array must be copied in order to make an update.5] Arrays> elems arr [3.7).5) [3.10].188.8.131.52.1.7.(5.FiniteMap module in the hierarchical libraries) is a purely functional implementation of balanced trees.5.-99)] array (1. since in order to maintain purity.10 and have O(log n) access and update.99).3).3.3).(2. you can update elements in the array using the // function.(5.(4. you should probably use FiniteMaps.5).7.10)] There are a few other functions which are of interest: bounds returns the bounds of an array indices returns a list of all indices of the array elems returns a list of all the values in the array in order assocs returns an association list for the array If we define arr to be listArray (1.10] Arrays> assocs arr [(1.1.5) Arrays> indices arr [1.10)] Note that while arrays are O(1) access.
('d'.('d'. The type of a finite map is for the form FiniteMapkeyelt where key is the type of the keys and elt is the type of the elements.98 CHAPTER 7.1).('b'.1). lists provide fast insertion (but slow everything else). finite maps are lookup tables from type key to type elt.10). There are also function listToFM and fmToList to convert lists to and from finite maps. Try the following: Prelude> :m FiniteMap FiniteMap> let fm = listToFM [('a'.6)] FiniteMap> lookupFM myFM 'e' Just 6 FiniteMap> lookupFM fm 'e' Nothing You can also experiment with the other commands.2)] FiniteMap> fmToList myFM [('a'.('b'.10).2).5).('c'. That is.('b'. as they are not instances of Show: FiniteMap> show myFM .('d'.5).('c'. which is a bit slower than lists or arrays).2)] FiniteMap> let myFM = addToFM fm 'e' 6 FiniteMap> fmToList fm [('a'. The basic finite map functions are: emptyFM addToFM :: FiniteMap key elt :: FiniteMap key elt -> FiniteMap key elt delFromFM :: FiniteMap key elt -> FiniteMap key elt elemFM :: key -> FiniteMap key lookupFM :: FiniteMap key elt -> key -> elt -> key -> elt -> Bool key -> Maybe elt In all these cases. Note that you cannot show a finite map. the type key must be an instance of Ord (and hence also an instance of Eq).5). ADVANCED FEATURES List Array FiniteMap insert O(1) O(n) O(log n) update O(n) O(n) O(log n) delete O(n) O(n) O(log n) find O(n) O(1) O(log n) map O(n) O(n) O(n log n) As we can see. arrays provide fast lookup (but slow everything else) and finite maps provide moderately fast everything (except mapping.1).10).('e'.('c'.
you first need to use fmToList.7.12 The Final Word on Lists You are likely tired of hearing about lists at this point. so it is the identity function. In fact. f) a) [] ((:) . In fact. 99 7. we're keeping both the same. you can convert any function of the following style into a foldr: myfunc [] = z myfunc (x:xs) = f x (myfunc xs) By writing the last line with f in infix form. we can simplify this definition through a sequence of steps: foldr foldr foldr foldr foldr (\a b -> f a : b) [] (\a b -> (:) (f a) b) [] (\a -> (:) (f a)) [] (\a -> ((:) . A primitive recursive function is essentially one which can be calculated using only "for" loops. b is the accumulator (i. the result list) and a is the element being currently considered.11. In fact. foldr f z can be thought of as replacing the [] in lists by z and the : by f. but they are so fundamental to Haskell (and really all of functional programming) that it would be terrible not to talk about them some more. This is because.e. LAYOUT <interactive>:1: No instance for (Show (FiniteMap Char Integer)) arising from use of 'show' at <interactive>:1 In the definition of 'it': show myFM In order to inspect the elements.11 Layout 7. but not "while" loops. It turns out that foldr is actually quite a powerful function: it can compute an primitive recursive function. we can fairly easily define map in terms of foldr: map2 f = foldr (\a b -> f a : b) [] Here.. In this case. this should be obvious: myfunc [] = z myfunc (x:xs) = x 'f' (myfunc xs) . as mentioned before. f) [] primitive recursive ==> ==> ==> ==> This is directly related to the fact that foldr (:) [] is the identity function on lists.
First. we can figure out that z is supposed to be []. suppose that we call the result of calling filter p xs simply b. ADVANCED FEATURES Clearly. First. Based on the first line. then we can rewrite this as: filter p [] = [] filter p (x:xs) = if p x then x : b else b Given this. the question is whether we can write this in fold notation. we can apply eta reduction to the first line to give: (++) [] = id Through a sequence of steps. we are just replacing [] with z and : with f. just like in the map case. (++) xs Thus.100 CHAPTER 7. The definition for ++ is: (++) [] ys = ys (++) (x:xs) ys = x : (xs ++ ys) Now. (++) xs Now. we notice that the base case converts [] into id. we get that an eta-reduced defintion of ++ is: (++) [] = id (++) (x:xs) = (x:) . we can also eta-reduce the second line: ==> ==> ==> (++) (++) (++) (++) (x:xs) (x:xs) (x:xs) (x:xs) ys = x : ((++) xs ys) ys = (x:) ((++) xs ys) ys = ((x:) . we can get the following definition in terms of foldr: . Consider the filter function: filter p filter p if p x then else [] = [] (x:xs) = x : filter p xs filter p xs This function also follows the form above. Now. we can transform filter into a fold: filter p = foldr (\a b -> if p a then a:b else b) [] Let's consider a slightly more complicated function: ++. (++) xs) ys = (x:) . Now. we can try to put this into fold notation. if we assume (++) xs is called b and x is called a.
(++) = foldr (\a b -> (a:) . b) id This actually makes sense intuitively. If we only think about applying ++ to one argument, we can think of it as a function which takes a list and creates a function which, when applied, will prepend this list to another list. In the lambda function, we assume we have a function b which will do this for the rest of the list and we need to create a function which will do this for b as well as the single element a. In order to do this, we first apply b and then further add a to the front. We can further reduce this expression to a point-free style through the following sequence: ==> ==> ==> ==> ==> ==> (++) (++) (++) (++) (++) (++) = = = = = = foldr foldr foldr foldr foldr foldr (\a b -> (a:) . b) id (\a b -> (.) (a:) b) id (\a -> (.) (a:)) id (\a -> (.) ((:) a)) id (\a -> ((.) . (:)) a) id ((.) . (:)) id
This final version is point free, though not necessarily understandable. Presumbably the original version is clearer. As a final example, consider concat. We can write this as: concat [] = [] concat (x:xs) = x ++ concat xs It should be immediately clear that the z element for the fold is [] and that the recursive function is ++, yielding: concat = foldr (++) []
Exercise 7.2 The function and takes a list of booleans and returns True if and only if all of them are True. It also returns True on the empty list. Write this function in terms of foldr. Exercise 7.3 The function concatMap behaves such that concatMap f is the same as concat . map f. Write this function in terms of foldr.
Chapter 8
Advanced Types
As you've probably ascertained by this point, the type system is integral to Haskell. While this chapter is called "Advanced Types", you will probably find it to be more general than that and it must not be skipped simply because you're not interested in the type system.
8.1 Type Synonyms
Type synonyms exist in Haskell simply for convenience: their removal would not make Haskell any less powerful. Consider the case when you are constantly dealing with lists of three-dimensional points. For instance, you might have a function with type [(Double, Double, Double)] → Double → [(Double, Double, Double)]. Since you are a good software engineer, you want to place type signatures on all your top-level functions. However, typing [(Double, Double, Double)] all the time gets very tedious. To get around this, you can define a type synonym: type List3D = [(Double,Double,Double)] Now, the type signature for your functions may be written List3D → Double → List3D. We should note that type synonyms cannot be self-referential. That is, you cannot have: type BadType = Int -> BadType This is because this is an "infinite type." Since Haskell removes type synonyms very early on, any instance of BadType will be replaced by Int → BadType, which will result in an infinite loop. Type synonyms can also be parameterized. For instance, you might want to be able to change the types of the points in the list of 3D points. For this, you could define: type List3D a = [(a,a,a)] 103
What you want is to define a type which is isomorphic to Int.) Disregarding that subtlety. NOTE "Isomorphic" is a common term in mathematics which basically means "structurally identical. instead of simply storing an integer. we can define: newtype MyInt = MyInt Int But we cannot define any of: newtype Bad1 = Bad1a Int | Bad1b Double newtype Bad2 = Bad2 Int Double . Unfortunately. we have to store a pointer to an integer and have to follow that pointer whenever we need the value of a MyInt. However. For instance. Double. Haskell has a newtype construction.2 Newtypes Consider the problem in which you need to have a type which is very much like Int. but it really has one more value: | (pronounced "bottom"). To get around these problems. if you have two graphs which are identical except they have different labels on the nodes. Perhaps you wish to order Ints first by even numbers then by odd numbers (that is. (See Section ?? for more information on bottom. ADVANCED TYPES Then your references to [(Double.104 CHAPTER 8. you cannot define a new instance of Ord for Int because then Haskell won't know which one to use. since datatypes can themselves be undefined. they are isomorphic. 8. but its ordering is defined differently. we usually think that it takes all values of integers. A newtype is a cross between a datatype and a type synonym: it has a constructor like a datatype. When we think of the type Int. but it can have only one constructor and this constructor can have only one argument." For instance. there may be efficiency issues with this representation: now. One way to do this would be to define a new datatype: data MyInt = MyInt Int We could then write appropriate code for this datatype. two types are isomorphic if they have the same underlying structure. MyInt has not only values MyInt 0. but also MyInt | . ordering is standard). MyInt 1 and so on. Double)] would become List3D Double. which is used to represent erroneous or undefined computations. all odd numbers are greater than any even number and within the odd/even subsets. Thus. it has an additional value: | which differs from MyInt | and this makes the types non-isomorphic. In our context. The problem (and this is very subtle) is that this type is not truly isomorphic to Int: it has one more value. in graph theory.
on newtypes.8. Before we talk about the solution. we could derive Num on MyInt to provide arithmetic functions over it. This enables use to write our desired instance of Ord as: instance Ord MyInt where MyInt i < MyInt j | odd i && odd j = i < j | even i && even j = i < j | even i = True | otherwise = False where odd x = (x 'mod' 2) == 0 even = not . Suppose we've defined the unit datatype (this one of the simplest datatypes you can define): . DATATYPES 105 Of course. Pattern matching over newtypes is exactly as in datatypes. Moreover.3 Datatypes We've already seen datatypes used in a variety of contexts. sometimes this leads to inefficiencies. We can write constructor and destructor functions for MyInt as follows: mkMyInt i = MyInt i unMyInt (MyInt i) = i 8. suppose we've defined MyInt as a newtype. 8. let's spend some time to get a bit more comfortable with how bottom works in to the picture (for more theory. Int) is an instance.1 Strict Fields One of the great things about Haskell is that computation is performed lazily.3. This section concludes some of the discussion and introduces some of the common datatypes in Haskell. For example. see Section ??).3. in recent versions of GHC (see Section 2. It also provides a more theoretical underpinning to what datatypes actually are. you are allowed to derive any class of which the base type (in this case.Double) Now. However. the fact that we cannot define Bad2 as above is not a big issue: we can simply define the following by pairing the types: newtype Good2 = Good2 (Int.2). I'm implicitly assuming we have derived Eq over MyInt – where is my assumption in the above code?). we can still derive classes like Show and Eq over newtypes (in fact. odd Like datatype. One way around this problem is to use datatypes with strict fields.
since foo has type a and thus can have any type. The other is bottom (written | ). which takes no arguments. it could be the fact that you. know that you will never come across the third of these. In fact.106 CHAPTER 8. Unit. Nothing. there are two values of type Unit. all you need to do is evaluate foo. it can also have type Unit." What is the type of foo? Simply a. we have found a second value with type Unit. We cannot say anything more about it than that. We can define a strict version of Maybe as: data SMaybe a = SNothing | SJust !a There are now only three values of SMaybe. However. Just | and Just Unit. there would be exactly one value of type Unit: namely. You use an exclamation point to specify a constructor as strict. They are: | . as a programmer. suppose we define the value: foo = foo This is perfectly valid Haskell code and simply says that when you want to evaluate foo. One of them is Unit. ADVANCED TYPES data Unit = Unit This datatype has exactly one constructor. Clearly this is an "infinite loop. In a strict language like ML. The fact that foo has type a in fact tells us that it must be an infinite loop (or some other such strange value). that there are actually four values with type Maybe Unit. This means that if the argument to Just is bottom. However. for instance: foo :: Unit foo = foo Thus. for instance. In fact. We could write. For instance. We can see the difference by writing the follwing program: module Main where import System data SMaybe a = SNothing | SJust !a deriving Show . This mean. This is not quite so in Haskell. then the entire structure becomes bottom. Any other non-terminating function or error-producing function will have exactly the same effect as foo (though Haskell provides some more utility with the function error). you want the argument to Just to be strict. we have found all values of type Unit. Unit. You can think of bottom as representing a computation which won't halt. Namely.
depending on what command line argument is passed./strict f Nothing \% .undefined \% . we will do something different.undefined \% .3./strict a Fail: Prelude./strict c Just Fail: Prelude./strict d Just () \% ./strict g Fail: Prelude.undefined . print x Here.8.undefined \% ./strict b Nothing \% . print x printJust :: SMaybe () -> IO () printSJust SNothing = putStrLn "Nothing" printSJust (SJust x) = do putStr "Just "./strict e Fail: Prelude. DATATYPES main = do [cmd] <case cmd "a" -> "b" -> "c" -> "d" -> "e" "f" "g" "h" -> -> -> -> 107 getArgs of printJust printJust printJust printJust undefined Nothing (Just undefined) (Just ()) printSJust undefined printSJust SNothing printSJust (SJust undefined) printSJust (SJust ()) printJust :: Maybe () -> IO () printJust Nothing = putStrLn "Nothing" printJust (Just x) = do putStr "Just ". The outputs for the various options are: \% .
there are three things drawn on the screen: the two paddles and the ball. We will begin the discussion by talking about Pong and then move on to a useful generalization of computations.Int) getVelocity :: a -> (Int. they share many commonalities. In this case. We can express these commonalities by defining a class for Pong entities.Int) getColor :: a -> Color getShape :: a -> Shape This code defines a typeclass Entity. However. getColor and getShape with the corresponding types.108 \% . but only in the context of previously existing type classes. since the constructor is strict. 8.Int) getAcceleration :: a -> (Int. . ADVANCED TYPES The thing worth noting here is the difference between cases "c" and "g". paddleVelX. To see how we can write an instance of this class.1 Pong The discussion here will be motivated by the construction of the game Pong (see Appendix ?? for the full code). :: Color.4 Classes We have already encountered type classes a few times. ". the Just is printed. paddleAccY :: Int. This section is about how to define your own. velocity. as soon as you match the SJust. The first line here uses the keyword class to introduce a new typeclass. . in the "g" case. We can read this typeclass definition as "There is a typeclass 'Entity'. 8./strict h Just () CHAPTER 8. shape. While the paddles and the ball are different in a few respects. so the whole thing fails before it gets a chance to do anything. because this is printed before the undefined value is evaluated. such as position. getAcceleration. acceleration. In Pong. This class has five methods: getPosition. . and so on. paddleAccX. We make such a definition as follows: class Entity a where getPosition :: a -> (Int. the value is undefined. a type 'a' is an instance of Entity if it provides the following five functions: . paddleVelY. getVelocity. let us define a player (paddle) datatype: data Paddle = Paddle { paddlePosX. color.4. which we call Entity. you also match the value. paddleColor paddlePosY. In the "c" case.
we can define Paddle to be an instance of Entity: instance Entity Paddle where getPosition p = (paddlePosX p. Suppose we want to search for a path between two vertices. Allow us to take a small aside to set up a small graph library: data Graph v e = Graph [(Int.4. the second is the list (set) of edges. If it succeeds. it will turn out that many of our routines will need entities to also be instances of Eq. playerNumber :: Int } Given this data declaration. To represent this. paddleHeight 109 The actual Haskell types of the class functions all have included the context Entity a =>. We can therefore choose to make Entity a subclass of Eq: namely. paddleVelY p) getAcceleration p = (paddleAccX p. getPosition has type Entity a ⇒ a → (Int.e)] The Graph datatype takes two type arguments which correspond to vertex and edge labels. For example. we will use the Maybe datatype.2 Computations Let's think back to our original motivation for defining the Maybe datatype from Section ??. To do this. we change the first line of the class declaration to: class Eq a => Entity a where Now. Our search function could be written (naively) as follows: search :: Graph v e -> Int -> Int -> Maybe [Int] search g@(Graph vl el) src dst | src == dst = Just [src] . paddlePosY p) getVelocity p = (paddleVelX p. We wanted to be able to express that functions (i. paddleAccY p) getColor = paddleColor getShape = Rectangle 5 . The first argument to the Graph constructor is a list (set) of vertices. Let us consider the case of performing search on a graph. Int). Perhaps there is no path between those vertices. 8.. However.e. in order to define Paddles to be instances of Entity we will first need them to be instances of Eq – we can do this by deriving the class.4. computations) can fail. it will return the list of vertices traversed. CLASSES paddleHeight :: Int. We will assume these lists are always sorted and that each vertex has a unique id and that there is at most one edge between any two vertices. you can only be an instance of Entity if you are already an instance of Eq.v)] [(Int.8.Int.
v. This algorithm is terrible: namely. If we're traversing the edge-list and it is empty. we're looking at an edge from u to v. we've failed. we put our current position before the path found and return. it is sufficent for now. we have found our way and just return the trivial solution.110 CHAPTER 8. so we return Nothing. it can loop indefinitely. we try the rest of the edges. If they are. If u is not our source. if the graph contains cycles. there are cases where the Maybe datatype is not sufficient: perhaps we wish to include an error message together with the failure. Nevertheless. first we check to see if these are equal. Otherwise. We could define a datatype to express this as: data Failable a = Success a | Fail String Now._):es) | src == u = case search2 g v dst of Success p -> Success (u:p) _ -> search' es | otherwise = search' es This code is a straightforward translation of the above. but all possible paths. Now. Be sure you understand it well: things only get more complicated. The basic idea is the same: . If this fails. There is another option for this computation: perhaps we want not just one path. Otherwise.v. If u is our source. We can express this as a function which returns a list of lists of vertices. failures come with a failure string to express what went wrong. ADVANCED TYPES | otherwise = search' el where search' [] = Nothing search' ((u. then we consider this step and recursively search the graph from v to dst. we want to traverse the edge-list. this edge is useless and we continue traversing the edge-list._):es) | src == u = case search g v dst of Just p -> Just (u:p) Nothing -> search' es | otherwise = search' es This algorithm works as follows (try to read along): to search in a graph g from src to dst. We can rewrite our search function to use this datatype: search2 :: Graph v e -> Int -> Int -> Failable [Int] search2 g@(Graph vl el) src dst | src == dst = Success [src] | otherwise = search' el where search' [] = Fail "No path" search' ((u. if this succeeds.
we're saying that c is an instance of the class Computation if it provides four functions: success. the first success is chosen. they are strung together). The augment function takes some previously given computation (namely. failure. we need to be able to augment a previous success (if there was one) with some new value. How augment works is probably best shown by example. CLASSES 111 search3 :: Graph v e -> Int -> Int -> [[Int]] search3 g@(Graph vl el) src dst | src == dst = [[src]] | otherwise = search' el where search' [] = [] search' ((u. The success function takes a value of type a and returns it wrapped up in c. We may ask ourselves what all of these have in common and try to gobble up those commonalities in a class. The augment function is a bit more complex. The combine function takes two previous computation and produces a new one which is the combination of both. we need a way to combine two successes (in the first two cases. Finally. representing a successful computation. giving augment the type c a → (a → a) → c a would have been sufficient. In essense. since a is always [Int]. We can fit this all into a class as follows: class Computation c where success :: a -> c a failure :: String -> c a augment :: c a -> (a -> c b) -> c b combine :: c a -> c a -> c a In this class declaration.8. in the third.v. Failable and [] to be instances of Computation as: instance Computation Maybe where success = Just failure = const Nothing . Furthermore. augment and combine.4. c a) and a function which takes the value of that computation (the a) and returns a b and produces a b inside of that computation._):es) | src == u = map (u:) (search3 g v dst) ++ search' es | otherwise = search' es The code here has gotten a little shorter. Note that in our current situation. but we make it this more general time just for generality. we need some way of representing success and some way of representing failure. The failure function takes a String and returns a computation representing a failure. We can define Maybe. though it is essentially the same. thanks to the standard prelude map function.
_):es) | src == u = (searchAll g v dst 'augment' (success . we ignore the function and return Nothing. applies f to it. If we failed before (and thus had a Nothing). we apply the function to each element in the list and then concatenate the results. success is represented with Just and failure ignores its argument and returns Nothing. Finally.v. Using these computations. (u:))) 'combine' search' es | otherwise = search' es . if we did. Finally: instance Computation [] where success a = [a] failure = const [] augment l f = concat (map f l) combine = (++) Here. The combine function takes the first success we found and ignores the rest. The function augment checks to see if we succeeded before (and thus had a Just something) and. we can express all of the above versions of search as: searchAll g@(Graph vl el) src dst | src == dst = success [src] | otherwise = search' el where search' [] = failure "no path" search' ((u. ADVANCED TYPES Here. instance Computation Failable where success = Success failure = Fail augment (Success x) f = f x augment (Fail s) _ = Fail s combine (Fail _) y = y combine x _ = x These definitions are obvious.112 augment augment combine combine (Just x) f = f x Nothing _ = Nothing Nothing y = y x _ = x CHAPTER 8. Failure is represented with the empty list and to combine previous successes we simply catentate them. the value of a successful computation is a singleton list containing that value. augmenting a computation amounts to mapping the function across the list of previous computations and concatentate them.
We can write the instance: instance Functor BinTree where fmap f (Leaf a) = Leaf (f a) fmap f (Branch left right) = Branch (fmap f left) (fmap f right) . fmap is essentially a generalization of map to arbitrary structures (and. There. In fact. a functor is like a function. we can also define other structures to be instances of functors. 8. There is a naturally associated functor which goes along with this raising. lists are already instances of Functor). However. The combine function isn't actually required by monads. of course. The definition of the functor class is: class Functor f where fmap :: (a -> b) -> f a -> f b The type definition for fmap (not to mention its name) is very similar to the function map over lists. you are in a very good position as you have understood the concept of monads. probably the most difficult concept in Haskell.5. allow us to consider some more advanced classes here. like "monad" comes from category theory. INSTANCES 113 In this. read through it again and then wait for the proper discussion of monads in Chapter 9. There is a Functor class defined in the Functor module. but instead of mapping elements to elements. we see the uses of all the functions from the class Computation. In fact.5 Instances We have already seen how to declare instances of some simple classes. the Computation class is almost exactly the Monad class. If you didn't understand everything here. NOTE The name "functor". failure is called fail and augment is called >>= (read "bind"). If you've understood this discussion of computations. Consider the following datatype for binary trees: data BinTree a = Leaf a | Branch (BinTree a) (BinTree a) We can immediately identify that the BinTree type essentially "raises" a type a into trees of that type. except that success is called return.8. it maps structures to structures. but is found in the MonadPlus class for reasons which will become obvious later.
which we could in turn reduce to trying to find out if that type is an instance of Eq. we've seen how to make something like BinTree an instance of Eq by using the deriving keyword. and so on. The "Eq a =>" part of the definition is called the "context. but here we will do it by hand." We should note that there are some restrictions on what can appear in the context and what can appear in the declaration. However. we're assuming. We want to make BinTree as instances of Eq but obviously we cannot do this unless a is itself an instance of Eq. We can specify this dependence in the instance declaration: instance Eq a => Eq (BinTree a) where Leaf a == Leaf b = a == b Branch l r == Branch l' r' = l == l' && r == r' _ == _ = False The first line of this can be read "if a is an instance of Eq. that there won't be any declarations like the second. ADVANCED TYPES Now. For instance. then BinTree a is also an instance of Eq". if elsewhere in a program we had the definition: instance MyEq a => Eq a where (==) = myeq In this case. To see why.114 CHAPTER 8. The closed world assumption can also bite you in cases like: class OnlyInts a where foo :: a -> a -> Bool . That is. when we write a definition like the first one. we're not allowed to have instance declarations that don't contain type constructors on the right hand side. if we're trying to establish if some type is an instance of Eq. consider the following declarations: class MyEq a where myeq :: a -> a -> Bool instance Eq a => MyEq a where myeq = (==) As it stands. We then provide the definitions. If we did not include the "Eq a =>" part. The compiler protects itself against this by refusing the first instance declaration. the compiler would complain because we're trying to use the == function on as in the second line. However. we could reduce it to trying to find out if that type is an instance of MyEq. This is commonly known as the closed-world assumption. there doesn't seem to be anything wrong with this definition. this assumption is invalid because there's nothing to prevent the second declaration (or some equally evil declaration).
instance OnlyInts Int where foo == (==) bar :: OnlyInts a => a -> Bool bar = foo 5
We've again made the closed-world assumption: we've assumed that the only instance of OnlyInts is Int, but there's no reason another instance couldn't be defined elsewhere, ruining our defintion of bar.
8.6 Kinds
Let us take a moment and think about what types are available in Haskell. We have simple types, like Int, Char, Double and so on. We then have type constructors like Maybe which take a type (like Char) and produce a new type, Maybe Char. Similarly, the type constructor [] (lists) takes a type (like Int) and produces [Int]. We have more complex things like → (function arrow) which takes two types (say Int and Bool) and produces a new type Int → Bool. In a sense, these types themselves have type. Types like Int have some sort of basic type. Types like Maybe have a type which takes something of basic type and returns something of basic type. And so forth. Talking about the types of types becomes unwieldy and highly ambiguous, so we call the types of types "kinds." What we have been calling "basic types" have kind "*". Something of kind * is something which can have an actual value. There is also a single kind constructor, → with which we can build more complex kinds. Consider Maybe. This takes something of kind * and produces something of kind *. Thus, the kind of Maybe is * -> *. Recall the definition of Pair from Section 4.5.1: data Pair a b = Pair a b Here, Pair is a type constructor which takes two arguments, each of kind * and produces a type of kind *. Thus, the kind of Pair is * -> (* -> *). However, we again assume associativity so we just write * -> * -> *. Let us make a slightly strange datatype definition: data Strange c a b = MkStrange (c a) (c b) Before we analyze the kind of Strange, let's think about what it does. It is essentially a pairing constructor, though it doesn't pair actual elements, but elements within another constructor. For instance, think of c as Maybe. Then MkStrange pairs Maybes of the two types a and b. However, c need not be Maybe but could instead by [], or many other things.
What do we know about c, though? We know that it must have kind * -> *. This is because we have c a on the right hand side. The type variables a and b each have kind * as before. Thus, the kind of Strange is (* -> *) -> * -> * -> *. That is, it takes a constructor (c) of kind * -> * together with two types of kind * and produces something of kind *. A question may arise regarding how we know a has kind * and not some other kind k. In fact, the inferred kind for Strange is (k -> *) -> k -> k -> *. However, this requires polymorphism on the kind level, which is too complex, so we make a default assumption that k = *. NOTE There are extensions to GHC which allow you to specify the kind of constructors directly. For instance, if you wanted a different kind, you could write this explicitly: data Strange (c :: (* -> *) -> *) a b = MkStrange (c a) (c b) to give a different kind to Strange. The notation of kinds suggests that we can perform partial application, as we can for functions. And, in fact, we can. For instance, we could have: type MaybePair = Strange Maybe The kind of MaybePair is, not surprisingly, * -> * -> *. We should note here that all of the following definitions are acceptable: type MaybePair1 = Strange Maybe type MaybePair2 a = Strange Maybe a type MaybePair3 a b = Strange Maybe a b These all appear to be the same, but they are in fact not identical as far as Haskell's type system is concerned. The following are all valid type definitions using the above: type MaybePair1a = MaybePair1 type MaybePair1b = MaybePair1 Int type MaybePair1c = MaybePair1 Int Double type MaybePair2b = MaybePair2 Int type MaybePair2c = MaybePair2 Int Double type MaybePair3c = MaybePair3 Int Double But the following are not valid:
type MaybePair2a = MaybePair2 type MaybePair3a = MaybePair3 type MaybePair3b = MaybePair3 Int This is because while it is possible to partially apply type constructors on datatypes, it is not possible on type synonyms. For instance, the reason MaybePair2a is invalid is because MaybePair2 is defined as a type synonym with one argument and we have given it none. The same applies for the invalid MaybePair3 definitions.
8.7 Class Hierarchies 8.8 Default
what is it?
rephrased slightly): main = do s <. In both cases. while learning Haskell.Chapter 9 Monads The most difficult concept to master. We can distinguish two subcomponents here: (1) learning how to use existing monads and (2) learning how to write new ones." Still.4. if you can grasp writing your own monads. we can abstract get away from the problems plaguing the RealWorld solution to IO presented in Chapter 5. by using monads. you must learn to use existing monads.2. So far we've seen two uses of monads. you will only need to learn to write your own monads if you want to become a "super Haskell guru. Essentially. The first use was IO actions: We've seen that. programming in Haskell will be much more pleasant. is that of understanding and using monads. On the other hand. we can write the above code as: 119 computations . The second use was representing different types of computations in Section 8.4. we needed a way to sequence operations and saw that a sufficient definition (at least for computations) was: class Computation c where success :: a -> c a failure :: String -> c a augment :: c a -> (a -> c b) -> c b combine :: c a -> c a -> c a Let's see if this definition will enable us to also perform IO. we need a way to represent taking a value out of an action and performing some new operation on it (as in the example from Section 4.3.readFile "somefile" putStrLn (show (f s)) But this is exactly what augment does. Using augment. If you want to use Haskell.
MONADS main = -.1 Do Notation We have hinted that there is a connection between monads and the do notation. as mentioned before. Here. The >> (read: "then" ) method is simply a version of >>= that ignores the a. we make that relationship concrete. fail is equivalent to our failure. And. using our Computation class.120 CHAPTER 9. in fact. The Monad class has four methods (but the fourth method can be defined in terms of the third): class Monad return fail (>>=) (>>) m where :: a -> m :: String :: m a -> :: m a -> a -> m a (a -> m b) -> m b m b -> m b bind then In this definition. return is equivalent to our success. this program really reads: main = readFile "somefile" >>= \s -> putStrLn (show (f s)) syntactic sugar .note the lack of a "do" readFile "somefile" 'augment' \s -> putStrLn (show (f s)) This certainly seems to be sufficient. we could define our above program as: main = readFile "somefile" 'augment' \s -> putStrLn (show (f s)) But we now know that augment is called >>= in the monadic world. As we mentioned earlier. although. This will turn out to be useful. and >>= (read: "bind" ) is equivalent to our augment. There is actually nothing magic about the do notation – it is simply "syntactic sugar" for monadic operations. it can be defined in terms of >>=: a >> x = a >>= \_ -> x 9. it turns out to be more than sufficient. The definition of a monad is a slightly trimmed-down version of our Computation class. Thus.
let .1. print the string) and then do es. Translation Rule 3 This states that do {let decls. Talking about do becomes easier if we do not use implicit layout (see Section ?? for how to do this). es} is to perform e (i. we will elaborate on these one at a time: = fail ". es} → e >> do {es}. es} → let ok p = do {es} . There are four translation rules: 1..f g x into f >>= \x -> g x. es} → let decls in do {es} 4. we make use of the >> function. defined earlier.. es}. For instance. es} → e >> do {es} 3. es} → let decls in do {es}.9. do {e. DO NOTATION 121 And this is completely valid Haskell at this point: if you defined a function f :: Show a => String -> a. This rule simple states that to do {e. do {let decls. This is clearly what we want. and then do es." Translation Rule 1 The first translation rule.e. the other three translation rules handle the cases where there is more than one action. We lift the declarations within the let out and do whatever comes after the declarations.. having a do or not is irrelevant. Here.e. This tells us what to do if we have an action (e) followed by a list of actions (es). ok in e >>= ok Again. This is exactly what the compiler does. This is essentially the base case for an inductive definition of do. throw away the result. then the translation of do {e. do {p <. you could compile and run this program) This suggests that we can translate: x <. states (as we have stated before) that when performing a single action. This rule tells us how to deal with lets inside of a do statement. The base case has one action (namely e here). do {e} → e 2. if e is putStrLn s for some string s. we first perform the action e. Translation Rule 2 This states that do {e. do {e} → e.
Apart from this. The compiler could simply throw an error. We run the action e." This rule states that binding a computation whose result is "Hello World" to putStrLn is the same as simply printing it to the screen.. However. this law states that the following two programs are equivalent: . The reason for the complex definition is that p doesn't need to simply be a variable. like usual.getLine putStrLn (x:xs) In this. the two definitions are equivalent. 9. allows the compiler to fill in the .e. Here. it is clear what is happening. The question becomes what should happen if this pattern match fails.2 Definition monad laws There are three rules that all monads must obey called the "Monad Laws" (and it is up to you to ensure that your monads obey these rules) : 1. suppose f is the function putStrLn and a is the string "Hello World. MONADS Translation Rule 4 This states that do {p <. with an appropriate error message about the pattern matching having failed. f >>= (\x -> g x >>= h) ≡ (f >>= g) >>= h Let's look at each of these individually: Law 1 This states that return a >>= f ≡ f a. the translation.. and then send the results into es. ok = fail ".122 CHAPTER 9. Thus. which is roughly equivalent. return a >>= f ≡ f a 2." in e >>= ok. since we're within a monad. then this is equivalent to simply performing the computation f on a directly. we're assuming that the results of the action getLine will begin with the string "abc" and will have at least one more character. it could be some complex pattern. This seems to make sense. for failed pattern matches. as defined. and we'd prefer to fail using that function. For instance. f >>= return ≡ f 3. However. but first give the result the name p. and then bind it together with some other computation f. an alternate formulation of this rule. This means that if we create a trivial computation that simply returns the value a regardless of anything else (this is the return a part). In do notation. is: do {p <e.. it is not exactly obvious what is going on here. For example.. Again. Suppose we think about monads as computations. es} → let ok p = do {es} . es} → e >>= \p -> es. the following is valid code: foo = do ('a':'b':'c':x:xs) <. rather than the "catch all" error function. we have access to a special fail function.
+ and * are associative. DEFINITION 123 law1a = do x <. and then returns that newly read string. this law is not as easy to grasp as the other two. a function · is associative if (f · g) · h = f · (g · h). then all we have done is to perform the computation. In this instance. reads another string via getLine. . If we throw away the messiness with the lambdas. the law states that if we perform the computation f and then pass the result on to the trivial return function. For instance. take f to be getLine. 5 − (3 − 1) = (5 − 3) − 1.2. Take h to be putStrLn. In do notation. this means that it's going to associative .return a f x law1b = do f a Law 2 The second monad law states that f >>= return ≡ f. for example. That this law must hold should be obvious. f is some computation.and / are not associative since. At first glance. since bracketing on these functions doesn't make a difference. Take g to be an action which takes a value as input. To see this. prints it to the screen. it doesn't matter how we group them. It takes a value called x. we see that this law states: f >>= (g >>= h) ≡ (f >>= g) >>= h. feeding the results into h. For a concrete example. On the other hand. and runs g on it. the law states that the following two programs are equivalent: law2a = do x <.f return x law2b = do f Law 3 This states that f >>= (\x -> g x >>= h) ≡ (f >>= g) >>= h. The intuition behind this law is that when we string together actions. Let's consider what (\x -> g x >>= h) does. NOTE Outside the world of monads. think of f as getLine (reads a string from the keyboard). This law states that reading a string and then returning the value read is exactly the same as just reading the string. That is.9. It is essentially an associativity law for monads.
Clearly. these two actions are the same.f do g <. This action reads a string from the keyboard. returning that newly read string as a result.do x <. In do notation. we get an action that does the action described by (f >>= g).f g x h y 9. and the text of the law is also quite complicated. all state information usually must be passed to functions explicitly as arguments. In general. read another value and then print that. prints it. Using monads. the new version of f takes the original parameter of type a and a new state parameter. and we compose them in the same order. we can encode it by changing the type of f to a → state → (state. The rest is just notation. we can write a simple map function to apply some function to each value in the leaves: mapTree :: (a -> b) -> Tree a -> Tree b mapTree f (Leaf a) = Leaf (f a) .124 CHAPTER 9. and we need to add state to this function. it also returns an updated state. That is. Suppose we have a function f of type a → b. print it. it doesn't matter where we put the parentheses. Thus. and then reads another string. b). When we bind this with h as on the right hand side of the law. consider (f >>= g). we can effectively hide some state information. and then prints the results. in addition to returning the value of type b. While this explanation is quite complicated. the entire left hand side of the law first reads a string and then does what we've just described. encoded in a tuple. And. MONADS take a value. the actual meaning is simple: if we have three actions. For instance. says that the following two programs are equivalent: law3a = do x <. In Haskell.3 A Simple State Monad One of the simplest monads that we can craft is a state-passing monad.x h y law3b = do y <. suppose we have a binary tree defined as: data Tree a = Leaf a | Branch (Tree a) (Tree a) Now. if state is of type state. On the other hand.
we need to add state. Leaf b) mapTreeState f (Branch lhs rhs) state = let (state' . the differences between mapTree and mapTreeState are: (1) the augmented f type. A SIMPLE STATE MONAD mapTree f (Branch lhs rhs) = Branch (mapTree f lhs) (mapTree f rhs) 125 This works fine until we need to write a function that numbers the leaves left to right. If we think of the st as the state. That is. The bindState function looks distinctly like the interior let declarations in mapTreeState. a) bindState :: State st a -> (a -> State st b) -> State st b bindState m k = \st -> let (st'. Tree b) mapTreeState f (Leaf a) state = let (state'. It takes two arguments. a) To go along with this type. a) = m st m' = k a in m' st' Let's examine each of these in turn. then this is a function that doesn't change the state and returns the value a. In a sense. b) = f a state in (state'. Notice that both types changed in exactly the same way. We can augment the function to something like: mapTreeState :: (a -> state -> (state.3. The first function. (2) we replaced the type -> Tree b with -> state -> (state. and the value of type a as the value. and the type signature is getting harder and harder to understand. What we want to do is abstract away the state passing part. rhs') = mapTreeState f rhs state' in (state''. Tree b). to the mapTree function. we write two functions: returnState :: a -> State st a returnState a = \st -> (st. lhs') = mapTreeState f lhs state (state''. The first argument is an action that returns something of type . which keeps track of how many leaves we've numbered so far. b)) -> Tree a -> state -> (state. returnState. takes a value of type a and creates something of type State st a. Branch lhs' rhs') This is beginning to get a bit unweildy. We can abstract this away with a type synonym declaration: type State st a = st -> (st.9.
a)) st in (f a) st' ==> \st -> let (st'. binding the result to a function that recurses on the right-hand-side. The definition of bindState takes an initial state. MONADS a with state st. st. Let's calculate on the left hand side. In fact. we recurse on the left-hand-side. we can verify that State st a obeys the monad laws: Law 1 states: return a >>= f ≡ f a. binding that to a simple function that returns the newly created Branch. a) = (\st -> (st. a) = (st. It then lets the function k act on a. producing something of type State st b. mapTreeStateM and give it the type: mapTreeStateM :: (a -> State st b) -> Tree a -> State st (Tree b) Using these "plumbing" functions (returnState and bindState) we can write this function without ever having to explicitly talk about the state: mapTreeStateM f (Leaf a) = f a 'bindState' \b -> returnState (Leaf b) mapTreeStateM f (Branch lhs rhs) = mapTreeStateM f lhs 'bindState' \lhs' -> mapTreeStateM f rhs 'bindState' \rhs' -> returnState (Branch lhs' rhs') In the Leaf case. This gives back a new state st' and a value a. a) = (returnState a) st m' = f a in m' st' ==> \st -> let (st'. and bindState is analogous to the overloaded >>= method. It first applies this to the State st a argument called m. We finally run m' with the new state st'. We write a new function.126 CHAPTER 9. substituting our names: returnState a 'bindState' f ==> \st -> let (st'. State st is a monad. As you have probably guessed by this point. The result of bindState is essentially the result of transforming the a into a b. called m'. we apply f to a and then bind the result to a function that takes the result and returns a Leaf with the new value. returnState is analogous to the overloaded return method. In the Branch case. The second is a function that takes this a and produces something of type b also with the same state. a) in (f a) st' .
In the fourth step. In the last step. we simplify the last two lines and substitute the definition of returnState. we simply substitute the definition of bindState. a) = f st (st''. b) = g a st' (st'''. Moving on to Law 2. This is much more involved to show. we apply st to the lambda function. so we will only sketch the proof here.3. This is shown as follows: f 'bindState' returnState ==> \st -> let (st'. we need to show that f >>= return ≡ f. a) ==> \st -> f st ==> f Finally. a) = f st in (\x -> g x 'bindState' h) a st' ==> \st -> let (st'.c) . b) = g a in h b st'') st' ==> \st -> let (st' .9. a) = f st in (st'. a) = f st in (returnState a) st' ==> \st -> let (st'. a) = f st in (g a 'bindState' h) st' ==> \st -> let (st'. a)) st' ==> \st -> let (st'. we rename st' to st and remove the let. Notice that we can write the left-hand-side as: \st -> let (st'. In the second step. we need to show that State obeys the third law: f >>= (\x -> g x >>= h) ≡ (f >>= g) >>= h. a) = f st in (\st -> (st. a) = f st in (\st' -> let (st''.c) = h b st'' in (st'''. A SIMPLE STATE MONAD ==> \st -> (f a) st ==> f a 127 In the first step. In the third step. we eta reduce.
.128 CHAPTER 9. Now that we know that State st is actually a monad.mapTreeM f lhs rhs' <.f a return (Leaf b) mapTreeM f (Branch lhs rhs) = do lhs' <. } This is because you cannot make instances out of non-fully-applied type synonyms. if we remove the type signature. Of course. a)) State run >>= action = State run' where run' st = let (st'. We can't write: instance Monad (State st) where { .. this is an informal. we'd like to make it an instance of the Monad class. we get the more general type: mapTreeM :: Monad m => (a -> m b) -> Tree a -> m (Tree b) . "hand waving" argument and it would take us a few more derivations to actually prove. what we need to do instead is convert the type synonym into a newtype.mapTreeM f rhs return (Branch lhs' rhs') which is significantly cleaner than before. In fact. this means that we can put whichever bracketing we prefer and the results will not change. Unfortunately. a)) Unfortunately. Instead. a) = run st State run'' = action a in run'' st' mapTreeM Now. but this gives the general idea. MONADS The interesting thing to note here is that we have both action applications on the same let level. but it's not terribly difficult: instance Monad (State state) where return a = State (\state -> (state. we can write our mapTreeM function as: mapTreeM :: (a -> State state b) -> Tree a -> State state (Tree b) mapTreeM f (Leaf a) = do b <. Since let is associative. as: newtype State st a = State (st -> (st. the straightforward way of doing this doesn't work. this means that we need to do some packing and unpacking of the State constructor in the Monad instance declaration.
3.5))) getState putState . These look like: getState :: State state state getState = State (\state -> (state.2)) (Leaf ('c'. The putState function takes a new state and produces an action that ignores the current state and inserts the new one. Int)) numberTree tree = mapTreeM number tree where number v = do cur <. passes it through unchanged.3)))) (Branch (Leaf ('d'. ())) Here. the nice thing about encapsulating the stateful aspect of the computation like this is that we can provide functions to get and change the current state. we can provide an example Tree: testTree = Branch (Branch (Leaf 'a') (Branch (Leaf 'b') (Leaf 'c'))) (Branch (Leaf 'd') (Leaf 'e')) and number it: State> runStateM (numberTree testTree) 1 Branch (Branch (Leaf ('a'.1)) (Branch (Leaf ('b'. we need to be able to run the action by providing an initial state: runStateM :: State state a -> state -> a runStateM (State f) st = snd (f st) Now. Now.9. getState is a monadic operation that takes the current state. mapTreeM can be run in any monad. state)) putState :: state -> State state () putState new = State (\_ -> (new.getState putState (cur+1) return (v. A SIMPLE STATE MONAD 129 That is.cur) Finally. we can write our numberTree function as: numberTree :: Tree a -> State Int (Tree (a. and then returns it as the value. not just our State monad. Now.4)) (Leaf ('e'.
and our putState is called put. for instance.4 Common Monads lists It turns out that many of our favorite datatypes are actually monads themselves. All this is built in to the Control.getState putState (v:cur) return (v:cur) and can see it in action: State> runStateM (fluffLeaves testTree) [] Branch (Branch (Leaf "a") (Branch (Leaf "ba") (Leaf "cba"))) (Branch (Leaf "dcba") (Leaf "edcba")) In fact. 9. We can also print out the leaves of the tree in a left-to-right fashion as: State> mapTreeM print testTree 'a' 'b' 'c' 'd' 'e' This crucially relies on the fact that mapTreeM has the more general type involving arbitrary monads – not just the state monad. which we will discuss in Section 9. our getState is called get. we can write an action that will make each leaf value equal to its old value as well as all the values preceeding: fluffLeaves tree = mapTreeM fluff tree where fluff v = do cur <. MONADS This may seem like a large amount of work to do something simple. Consider. our runStateM is called evalState. This module also contains a state transformer monad.State module. lists.Monad. However.7. There. you don't even need to write your own monad instance and datatype. note the new power of mapTreeM. They have a monad definition that looks something like: instance Monad return x = l >>= f = fail _ = [] where [x] concatMap f l [] . Furthermore.130 CHAPTER 9.
'e').('a'.('a'.'e'). However. given the definition: cross l1 l2 = do x <. with failure being represented as Nothing and with success as Just.'f')] List comprehension form is simply an abbreviated form of a monadic statement using lists."def"] [('a'.('b'.'f').'d').('a'.('b'. COMMON MONADS This enables us to use lists in do notation. the list comprehension form could be used for any monad – not just lists.'b') Monads> cross (Nothing :: Maybe Char) (Just 'b') Nothing Monads> cross (Just 'a') (Nothing :: Maybe Char) Nothing Monads> cross (Nothing :: Maybe Char) (Nothing :: Maybe Char) Nothing list comprehensions Maybe .'d').'d'). in older versions of Haskell."ab".l2 return (x. ('b'.'d'). The Maybe type is also a monad.'e').'f')] 131 It is not a coincidence that this looks very much like the list comprehension form: Prelude> [(x.4. in the current version of Haskell. this is no longer allowed.('b'. This is because the do notation works for any monad.('a'. For instance. and there's nothing specific to lists about the cross function.'f'). Monads> cross (Just 'a') (Just 'b') Just ('a'.y) we get a cross-product function: Monads> cross "ab" "cde" [('a'. We get the following instance declaration: instance Monad Maybe where return a = Just a Nothing >>= f = Nothing Just x >>= f = f x fail _ = Nothing We can use the same cross product function that we did for lists on Maybes. ('b'.('b'.9.l1 y <.y) | x <. y <.'e'). In fact.
This means that no matter what monad we're using at the moment. MONADS What this means is that if we write a function (like searchAll from Section 8.1. so that the interpreter knows what monad we're using. 'd')] [(0. (0. the searchAll function looks something like: searchAll g@(Graph vl el) src dst | src == dst = return [src] | otherwise = search' el where search' [] = fail "no path" search' ((u. there is no path. we can use it with any monad.3] We provide the type signature.4) only in terms of monadic operators. 'c').c and d.'m').b.3. we can compute the path from a to d: Monads> searchAll gr 0 3 :: Maybe [Int] Just [0.'n'). 'b').v. (2.1. this function will perform the calculation. Using real monadic functions (not do notation). (2.132 CHAPTER 9. There is an edge from a to both b and c.3. labelled a._):es) | src == u = searchAll g v dst >>= \path -> return (u:path) | otherwise = search' es The type of this function is Monad m => Graph v e -> Int -> Int -> m [Int]. (3. There is also an edge from both b and c to d.2.'l'). Using the Maybe monad. we get the empty list. depending on what we mean. (1. (1. If we try to search in the opposite direction. 'a'). The inability to find a path is represented as Nothing in the Maybe monad: Monads> searchAll gr 3 0 :: Maybe [Int] Nothing Note that the string "no path" has disappeared since there's no way for the Maybe monad to record this. If we perform the same impossible search in the list monad. indicating no path: Monads> searchAll gr 3 0 :: [[Int]] [] sea .'m')] This represents a graph with four nodes. Suppose we have the following graph: gr = Graph [(0.
6. We define: gr2 = Graph [(0. See Section 9. 'd')] [(0. There's no way for this implementation to recover. for instance. it does not.3. 'b').1. If we use the IO monad. Exercise 9.1 Verify that Maybe obeys the three monad laws.4. . Write an instance for it. and then try doing the search from this chapter using this monad. what happens if searchAll g v dst doesn't find a path.3]] 133 You may have expected this function call to return all paths. as coded. but. since IO knows how to keep track of error messages: Monads> searchAll gr 0 3 :: IO [Int] Monads> it [0. This has to do with the recursive call to searchAll inside of search'.'l').9.'m'). Hint: Your instance declaration should begin: instance Monad (Either String) where. nondeterminism Exercises Exercise 9. if we remove the edge from node b to node d. (2.2 The type Either String is a monad that can keep track of errors. we get back a list containing the first path: Monads> searchAll gr 0 3 :: [[Int]] [[0. 'a'). (1. There is one problem with this implementation of searchAll: if it finds an edge that does not lead to a solution. For instance. we can actually get at the error message. (3. but this algorithm can't find it.1.6 for more about using lists to represent nondeterminism. (2. We will see how to do this in Section 9. we need a function like combine from our Computation class.3] Monads> searchAll gr 3 0 :: IO [Int] *** Exception: user error Reason: no path In the first case.1. (0.2. we should still be able to find a path from a to d. COMMON MONADS If we perform the possible search. we needed to type it to get GHCi to actually evaluate the search. it won't be able to backtrack. Consider. 'c').'m')] and then try to search: Monads> searchAll gr2 0 3 *** Exception: user error Reason: no path To fix this.
These functions are incredibly useful (particularly foldM) when working with monads. except it takes its arguments in the opposite order. in the IO monad.5] 1 2 3 4 5 . In general. The ones we will discuss in this section.2. filterM and foldM are our old friends map. we can write either of the following: Monads> writeFile "foo" "hello world!" >> (readFile "foo" >>= putStrLn) hello world! Monads> writeFile "foo" "hello world!" >> (putStrLn =<< readFile "foo") hello world! mapM filterM foldM The mapM. filter and foldr wrapped up inside of monads. for instance. m is always assumed to be an instance of Monad. which haven't yet been thoroughly discussed.3. The =<< function is exactly the same as >>=.Monad library contains a few very useful monadic combinators.4.5 Monadic Combinators The Monad/Control.134 CHAPTER 9. except that they do not return any value. functions with an underscore at the end are equivalent to the ones without. to print a list of things to the screen: Monads> mapM_ print [1. are: • (=<<) :: • mapM :: • mapM :: • foldM :: (a -> m b) -> m a -> m b (a -> m b) -> [a] -> m [b] (a -> m b) -> [a] -> m () (a -> m Bool) -> [a] -> m [a] (a -> b -> m a) -> a -> [b] -> m a [m a] -> m [a] [m a] -> m () • filterM :: • sequence :: • sequence :: • liftM :: • when :: • join :: (a -> b) -> m a -> m b Bool -> m () -> m () m (m a) -> m a In the above. together with their types. We can use mapM . MONADS 9. For instance.
5. while the nonunderscored version returns the list of the return values.5] 0+1=1 1+2=3 3+3=6 6+4=10 10+5=15 Monads> it 15 135 The sequence and sequence functions simply "execute" a list of actions.7. we might want to write a function that prepends each line in a file with its line number. print 2. We can do this with: numberFile :: FilePath -> IO () numberFile fp = do text <.. print 2.9. MONADIC COMBINATORS We can use foldM to sum a list and print the intermediate sum at each step: Monads> foldM (\a b -> putStrLn (show a ++ "+" ++ show b ++ "=" ++ show (a+b)) >> return (a+b)) 0 [1.()] *Monads> sequence_ [print 1. we can shorten this using liftM: sequence liftM .) This is useful for shortening code (among other things).().readFile fp let l = lines text let n = zipWith (\n t -> show n ++ ' ' : t) [1. (Do not confuse this with the lift function used for monad transformers in Section 9. For instance. For instance: Monads> sequence [print 1. The liftM function "lifts" a non-monadic function to a monadic function.. print 'a'] 1 2 'a' *Monads> it () We can see that the underscored version doesn't return each value. print 'a'] 1 2 'a' *Monads> it [().] l mapM_ putStrLn n However.
lines 'liftM' readFile fp let n = zipWith (\n t -> show n ++ ' ' : t) [1.] l mapM_ putStrLn n In fact. MONADS numberFile :: FilePath -> IO () numberFile fp = do l <. perhaps we also want to split lines into words."ghi"] abc def ghi Of course. In other monads.136 CHAPTER 9."def". For instance: Monads> liftM (+1) (Just 5) Just 6 *Monads> liftM (+1) Nothing Nothing when The when function executes a monadic action only if a condition is met. For instance. Note that the parentheses are required. In fact.. the same could be accomplished with filter.."". join is exactly concat.. lines) 'liftM' readFile fp . it accomplishes a similar task: Monads> join (Just (Just 'a')) Just 'a' Monads> join (Just (Nothing :: Maybe Char)) Nothing Monads> join (Nothing :: Maybe (Maybe Char)) Nothing join . but sometimes when is more convenient.. Lifting pure functions into monads is also useful in other monads. you can apply any sort of (pure) processing to a file using liftM."abc". when m is the list monad. w <."". the join function is the monadic equivalent of concat on lists.. since the (. Finally.) function has the same fixity has 'liftM'. So. For instance liftM can be used to apply function inside of Just. if we only want to print non-empty lines: Monads> mapM_ (\l -> when (not $ null l) (putStrLn l)) ["".(map words . we can do this with: .
9. mplus is the same as ++.2.4. such a function is so generally useful that it exists in another class called MonadPlus.6.e. However. such that it will explore all possible paths. they are also both instances of MonadPlus. That is.. The definition is: class Monad m => MonadPlus m where mzero :: m a mplus :: m a -> m a -> m a In order to gain access to MonadPlus. the instance declarations look like: instance MonadPlus Maybe where mzero = Nothing mplus Nothing y = y mplus x _ = x instance MonadPlus [] where mzero = [] mplus x y = x ++ y We can use this class to reimplement the search function we've been exploring. if both elements are Nothing. For lists.3]. MONADPLUS Monads> join (return (putStrLn "hello")) hello Monads> return (putStrLn "hello") Monads> join [[1.2. In the case of Maybe.4.Monad in the hierarchical libraries). we showed that Maybe and list are both monads.6 MonadPlus Given only the >>= and return functions.5]] [1. instances of MonadPlus also have a "zero" element that is the identity under the "plus" (i.5] 137 These functions will turn out to be even more useful as we move on to more advanced topics in Chapter 10. In Section 9. otherwise. it is the first Just value. In fact. 9. The mplus operation on Maybe is Nothing. the zero element is Nothing. The new function looks like: searchAll2 g@(Graph vl el) src dst | src == dst = return [src] combine MonadPlus Maybe lists . In addition to having a combine function. it is impossible to write a function like combine with type c a → c a → c a.3. you need to import the Monad module (or Control.[4. combine) action. it is the empty list. in the case of lists.
and we come across a matching edge.3]. the matching case of search' looked like: search' es 'mplus' (searchAll2 g v dst >>= \path -> return (u:path)) . We can see that when using lists as the monad.2.v. The IO monad is not an instance of MonadPlus. we don't try to continue and execute search' es._):es) | src == u = do path <. we we're not able to execute the search with this monad. but we also continue to explore the out-edges of the current node in the recursive call to search'.e. not only do we explore this path. MPlus> searchAll2 gr 0 3 :: [[Int]] [[0.3 Suppose that we changed the order of arguments to mplus. MONADS | otherwise = search' el where search' [] = fail "no path" search' ((u.. Exercises Exercise 9. if the recursive call to searchAll2 fails.1. The call to mplus must be at the top level in order for it to work. Here.[0.3]] You might be tempted to implement this as: searchAll2 g@(Graph vl el) src dst | src == dst = return [src] | otherwise = search' el where search' [] = fail "no path" search' ((u._):es) | src == u = (searchAll2 g v dst >>= \path -> return (u:path)) 'mplus' search' es | otherwise = search' es Now.search' es return ((u:path) 'mplus' rest) | otherwise = search' es But note that this doesn't do what we want. when we're going through the edge list in search'.v.138 CHAPTER 9. I.3]] MPlus> searchAll2 gr2 0 3 :: [[Int]] [[0.2. we (a) get all possible paths in gr and (b) get a path in gr2.searchAll2 g v dst rest <.
7. In order to execute a command of type Monad m => m a.m return (s. provided that m is already a monad: MonadTrans lift state monad state transformer . This class has a simple method: lift. a)) Now. This gives rise to the following definition for a state transformer: newtype StateT state m a = StateT (state -> m (state. This is relatively straightforward. The class declaration for MonadTrans is: class MonadTrans t where lift :: Monad m => m a -> t m a The idea here is that t is the outer monad and that m lives inside of it. itself. we assume there's some other monad m and make the internal action into something of type state -> m (state. Before. a)) For instance. we defined a state monad as: newtype State state a = State (state -> (state. we introduce a MonadTrans class. we first lift it into the transformer. which is a state monad wrapped around an arbitrary monad. lifting a function from the realm of m to the realm of StateT state simply involves keeping the state (the s value) constant and executing the action. a). which essentially "lifts" the operations of one monad into another. a) as the monad. we also need to make StateT a monad. In this case.7 Monad Transformers Often we want to "piggyback" monads on top of each other. Of course. The simplest example of a transformer (and arguably the most useful) is the state transformer monad. instead of using a function of type state -> (state.a)) Here. we can think of m as IO. You can think of this as stacking monads on top of eachother. we make this an instance of MonadTrans: instance MonadTrans (StateT state) where lift m = StateT (\s -> do a <. For instance. MONAD TRANSFORMERS 139 How would you expect this to change the results when using the list monad on gr? Why? 9. First. In order to accomplish this. our state transformer monad is able to execute actions in the IO monad.9. there might be a case where you need access to both IO operations through the IO monad and state functions through some state monad.
putT and evalStateT . We can use state transformers to reimplement a version of our mapTreeM function from Section 9. Thus. in order to actually use this monad. This definition is nearly identical to the definition of bind for the standard (non-transformer) State monad described in Section 9. The fail function passes on the call to fail in the enclosed monad. First. Note. MONADS instance Monad m => Monad (StateT state m) where return a = StateT (\s -> return (s. ())) evalStateT :: Monad m => StateT s m a -> s -> m a evalStateT (StateT m) state = do (s'. a) <.a)) StateT m >>= k = StateT (\s -> do (s'.140 CHAPTER 9. This is typical of monad transformers: they don't know how to actually run things in their enclosed monad (they only know how to lift actions). we create a new StateT that takes a state s as an argument. we need to provide function getT . which you then need to run yourself. These are analogous to getState. IO). however. not the state transformer. a) <. It finally applies the new state to this transformer. Note that the use of return in the definition of return refers to the enclosed monad. we print out the value of the leaf. s)) putT :: Monad m => s -> StateT s m () putT s = StateT (\_ -> return (s. The only change here is that when we get to a leaf. In the definition of bind. since state transformers don't natively know how to deal with failure. It then runs the k action on this new state and gets a new transformer. putState and runStateM from Section 9. what you get out is a monadic action in the inside monad (in our case. we just print out "Branch. it applies this state to the first action (StateT m) and gets the new state and answer as a result.3: getT :: Monad m => StateT s m s getT = StateT (\s -> return (s. Of course.3.m s let StateT m' = k a m' s') fail s = StateT (\_ -> fail s) The idea behind the definition of return is that we keep the state constant and simply return the state/a pair in the enclosed monad. that the result of evalStateT is actually a monadic action in the enclosed monad.3." mapTreeM action (Leaf a) = do lift (putStrLn ("Leaf " ++ show a)) getT putT evalStateT . when we get to a branch.m state return a These functions should be straightforward.
The lift tells us that we're going to be executing a command in an enclosed monad. for a second. and the new versions of get and put. Monad (t IO). we simply change numberTree to use this version of mapTreeM.) as the first line. and we end up with: numberTree tree = mapTreeM number tree where number v = do cur <.getT putT (cur+1) return (v. This just as before.7. In this.. finally.mapTreeM action lhs rhs' <. Show a) => (a -> t IO a1) -> Tree a -> t IO (Tree a1) Ignoring.. we can run our monad: MTrans> evalStateT (numberTree testTree) 0 Branch Branch Leaf 'a' Branch Leaf 'b' Leaf 'c' Branch Leaf 'd' Leaf 'e' *MTrans> it . The type of this function is relatively complex: mapTreeM :: (MonadTrans t. we require that t IO is a monad. MONAD TRANSFORMERS b <. the class constraints. we require that t is a monad transformer (since we apply lift in it).9. the enclosed monad is IO. In this case.mapTreeM action rhs return (Branch lhs' rhs') 141 The only difference between this function and the one from Section 9. this says that mapTreeM takes an action and a tree and returns a tree. we require that a is an instance of show – this is simply because we use show to show the value of leaves. Now. since we use putStrLn we know that the enclosed monad is IO. since the command lifted is putStrLn.action a return (Leaf b) mapTreeM action (Branch lhs rhs) = do lift (putStrLn "Branch") lhs' <.3 is the calls to lift (putStrLn .cur) Using this.
'o').3. we need to keep track of which nodes we have visited. regardless of what monad we use.v. 'b').2)))) (Branch (Leaf ('d'. In order to get around this problem. (1. Consider: gr3 = Graph [(0. we can run the state transformer and get out only the correct paths.1.'l'). it will fail to terminate. if we move this erroneous edge to the end of the list (and call this gr4).2.4))) One problem not specified in our discussion of MonadPlus is that our search algorithm will fail to terminate on graphs with cycles. we need to introduce state.3. (1. there is a back edge from node b back to node a. 'c'). we add the current state to our set of visited states.3)) (Leaf ('e'. the result of searchAll2 gr4 0 3 will contain an infinite number of paths: presumably we only want paths that don't contain cycles. (2.getT putT (src:visited) if src 'elem' visited then mzero else search' el where search' [] = mzero search' ((u. MONADS Branch (Branch (Leaf ('a'.'n'). Namely.getT putT (src:visited) return [src] | otherwise = do visited <.'p')] In this graph. when we recurse. when we encounter a state we haven't yet visited. We only continue to recurse.0)) (Branch (Leaf ('b'. (0. 'd')] [(0. (1. We can do this as follows: searchAll5 g@(Graph vl el) src dst | src == dst = do visited <. so that we don't visit them again._):es) | src == u = (do path <.searchAll5 g v dst return (u:path)) 'mplus' search' es | otherwise = search' es Here. (2. (3. even on the cyclic graphs: cycles .142 CHAPTER 9. 'a').1)) (Leaf ('c'.'m').0. we implicitly use a state transformer (see the calls to getT and putT) to keep track of visited states. If we attempt to run searchAll2. Now. Futhermore. Moreover.
4 Write a function searchAll6. as it will return exactly the reverse of evalStateT (try it and find out!). Exercises Exercise 9.3]] In order to do this. the output generated for searchAll6 gr 0 3 should look like: Exploring 0 -> 3 Exploring 1 -> 3 Exploring 3 -> 3 Exploring 2 -> 3 Exploring 3 -> 3 MTrans> it [[0.2.9.3]..3].[0. for instance. This function looks like: execStateT :: Monad m => StateT s m a -> s -> m s execStateT (StateT m) state = do (s'.3]. you will have to define your own list monad transformer and make appropriate instances of it. that.7. it is empty. but can be useful in general (if. instead of returning a result.1. prints the search being conducted.1.[0. returns the final state.2. In our case. For instance.3]] Here. based on the code for searchAll2. We can also provide an execStateT method that. MONAD TRANSFORMERS 143 MTrans> evalStateT (searchAll5 gr3 0 3) [] :: [[Int]] [[0.1.5 Combine the searchAll5 function (from this section) with the searchAll6 function (from the previous exercise) into a single function called searchAll7.[0. the empty list provided as an argument to evalStateT is the initial state (i.e. This function should perform IO as in searchAll6 but should also keep track of state using a state transformer.3]] MTrans> evalStateT (searchAll5 gr4 0 3) [] :: [[Int]] [[0. the initial visited list). .2.m state return s' This is not so useful in our case. a) <. we need to know how many numbers are used in numberTree). Exercise 9. at every entry to the main function (not the recursion over the edge list).
we want to build up a library of paring "primitives.2.8.144 CHAPTER 9. In this chapter. c) . This function looks like: char :: Char -> Parser Char char c = Parser char' where char' [] = Left ("expecting " ++ show c ++ " got EOF") char' (x:xs) | x == c = Right (xs.1 A Simple Parsing Monad Consider the task of parsing. MONADS 9. A simple parsing monad is much like a state monad. where the state is the unparsed string. This yields standard instances of Monad and MonadPlus: instance Monad Parser where return a = Parser (\xl -> Right (xl. This makes the construction of parsing libraries in Haskell very clean.a)) fail s = Parser (\xl -> Left s) Parser m >>= k = Parser $ \xl -> case m xl of Left s -> Left s Right (xl'.8 Parsing Monads It turns out that a certain class of parsers are all monads. 9.8. a) -> let Parser n = k a in n xl' instance MonadPlus Parser where mzero = Parser (\xl -> Left "mzero") Parser p 'mplus' Parser q = Parser $ \xl -> case p xl of Right a -> Right a Left err -> case q xl of Right a -> Right a Left _ -> Left err primitives Now. a) } We again use Left err to be an error condition.8." The most basic primitive is a parser that will read a specific character.1 and then introduce the Parsec parsing library in Section 9. we begin by building our own (small) parsing library in Section 9. We can represent this exactly as: newtype Parser a = Parser { runParser :: String -> Either String (String.
we can write a case-insensitive "Hello" parser: runParser ."Hello") Parsing> runParser helloParser "hello World!" Left "expecting 'H' got 'h'" We can have a slightly more general function. PARSING MONADS | otherwise = Left 145 ("expecting " ++ show c ++ " got " ++ show x) Here. We don't need to worry about the underlying string – the monad takes care of that for us. the parser succeeds only if the first character of the input is the expected character. x) | otherwise = Left ("expecting char. All we need to do is combine these parser primatives.8. We can test this parser by using runParser and by supplying input: Parsing> runParser helloParser "Hello" Right ("".9."Hello") Parsing> runParser helloParser "Hello World!" Right (" World!". which will match any character fitting a description: matchChar :: (Char -> Bool) -> Parser Char matchChar c = Parser matchChar' where matchChar' [] = Left ("expecting char. We can use this parser to build up a parser for the string "Hello": helloParser :: Parser String helloParser = do char 'H' char 'e' char 'l' char 'l' char 'o' return "Hello" This shows how easy it is to combine these parsers. got EOF") matchChar' (x:xs) | c x = Right (xs. got " ++ show x) Using this.
c3."hELlO") Finally. for instance. but the above implementation works just as well.matchChar ('elem' return [c1. for instance. there would be many more functions of this sort. x) many toLower). On top of these primitives. using them. and they would be hidden away in a library. you could build up. However. p. In general. will take a parser that parses entities of type a and will make it into a parser that parses entities of type [a] (this is a Kleene-star operator): many :: Parser a -> Parser [a] many (Parser p) = Parser many' where many' xl = case p xl of Left err -> Right (xl.a) -> let Right (xl''.matchChar ('elem' c4 <. we usually build some combinators. The many combinator. we can have a function. a parser that parses (nonnegative) integers: . which will match any character: anyChar :: Parser Char anyChar = Parser anyChar' where anyChar' [] = Left ("expecting character.c2. We can test this function: Parsing> runParser ciHelloParser "hELlO world!" Right (" world!".matchChar ('elem' c2 <. we recurse and keep trying to apply p until it fails. If p succeeds. []) Right (xl'. we could have used something like matchChar ((=='h') . If this fails. we succeed but return the empty list.matchChar ('elem' c5 <. got EOF") anyChar' (x:xs) = Right (xs. rest) = many' xl' in Right (xl''.146 CHAPTER 9.c5] "Hh") "Ee") "Ll") "Ll") "Oo") Of course.c4.matchChar ('elem' c3 <. We then return the list of successes we've accumulated. a:rest) The idea here is that first we try to apply the given parser. MONADS ciHelloParser = do c1 <. so that users couldn't actually look inside the Parser type.
suppose we want to parse a Haskell-style list of Ints.int r <. or parsing a close brace and returning an empty list. using mplus. then we try the other. it doesn't know whether we're at the end yet. and so it first parses an int. but we don't know when this will happen. We can test this parser as before: Parsing> runParser int "54" Right ("".Char) and then match as many more digits as we can. at some point. on the other.(char '. The intList' function assumes that we're not yet at the end of the list. We then read the result and return it. it parses a close brace and returns the empty list.' >> intList') 'mplus' (char ']' >> return []) return (i:r) The first thing this code does is parse and open brace. This is where the fact that Parser is an instance of MonadPlus comes in handy: first we try one.9.54) *Parsing> runParser int "54abc" Right ("abc". it simply prepends the int it parsed itself to the beginning. so it again uses mplus. One thing that you should be careful of is the order in which you supply arguments to mplus.many (matchChar isDigit) return (read (t1:tr)) In this function. we're either going to parse a comma or a close brace. PARSING MONADS 147 int :: Parser Int int = do t1 <.54) *Parsing> runParser int "a54abc" Left "expecting char. Either way.matchChar isDigit tr <. It then parses the rest of the list. Consider the following code: intList :: Parser [Int] intList = do char '[' intList' 'mplus' (char ']' >> return []) where intList' = do i <. Then. got 'a'" Now. it tries one of two things: parsing using intList'. Consider the following parser: mplus . On the one hand. we first match a digit (the isDigit function comes from the module Char/Data.8. This becomes somewhat difficult because. it tries to parse a comma and then recurse. However.
as we can easily see: Parsing> runParser tricky2 "Hal" Right ("". This causes eof to produce an error message. end-of-string) as: eof :: Parser () eof = Parser eof' where eof' [] = Right ([]. got \"l\"" This is because." it parses just "Hal" and leaves the last "l" lying around to be parsed later. The correct way to implement this is: tricky3 = mplus (do s <.()) Parsing> runParser tricky2 "Hall" Left "Expecting EOF. You can see this with: Parsing> runParser tricky "Hal" Right ("". ()) eof' xl = Left ("Expecting EOF."Hal") Parsing> runParser tricky "Hall" Right ("l"." You can attempt to fix this by providing a parser primitive. the mplus doesn't know that it needs to parse the whole input." however. which detects end-of-file (really. So.mplus (string "Hal") (string "Hall") eof return s But this also doesn't work. it only parses the former. MONADS tricky = mplus (string "Hal") (string "Hall") You might expect this parser to parse both the words "Hal" and "Hall.148 CHAPTER 9.string "Hal" ." which succeeds. got " ++ show (take 10 xl)) You might then rewrite tricky using eof as: tricky2 = do s <. and then it doesn't bother trying to parse "Hall."Hal") This is because it tries to parse "Hal. when you provide it with "Hall. again.
Exercises Exercise 9. if we're parsing a large file. PARSING MONADS eof return s) (do s <..8.a)) fail s = Parser (\n xl -> Left (show n ++ ": " ++ s)) Parser m >>= k = Parser $ \n xl -> case m n xl of Left s -> Left s Right (n'."Hall") 149 This works precisely because each side of the mplus knows that it must read the end. xl'."Hal") Parsing> runParser tricky3 "Hall" Right ("".6 Write a parser intListSpace that will parse int lists but will allow arbitrary white space (spaces. if we cannot disambiguate immediately.g. Parsec. see Section 9. In this case. Given this monadic parser. it might be helpful to report the line number on which an error occurred.xl. tabs or newlines) between the commas and brackets. The solution most parser libraries (e. String.2) have adopted is to only recognize "LL(1)" grammars: that means that you must be able to disambiguate the input with a one token look-ahead. We could do this simply by extending the Parser type and by modifying the instances and the primitives: newtype Parser a = Parser { runParser :: Int -> String -> Either String (Int. For instance. due to the fact that we assumed we would be reading an end-of-file immediately afterwards. it is fairly easy to add information regarding source position. fixing the parser to accept both "Hal" and "Hall" was fairly simple.string "Hall" eof return s) We can see that this works: Parsing> runParser tricky3 "Hal" Right ("". and has little to do with monadic parsing. life becomes significantly more complicated. Unfortunately. a) -> let Parser m2 = k a line numbers .8. a) } instance Monad Parser where return a = Parser (\n xl -> Right (n. This is a general problem in parsing.9.
when we run a parser and there is an error."Hello") Parsing2> runParser int 1 "a54" Left "1: expecting char. Now. got 'a'" Parsing2> runParser intListSpace 1 .2 .3.2. it will tell us which line number contains the error: Parsing2> runParser helloParser 1 "Hello" Right (1.a\n]" Left "4: expecting char. 4 \n\n . 4 \n\n\n . x) | otherwise = Left ("expecting char. got 'a'" Parsing2> runParser intListSpace 1 "[1 . got EOF") matchChar' n (x:xs) | c x = Right (n+if x=='\n' then 1 else 0 . xs. MONADS instance MonadPlus Parser where mzero = Parser (\n xl -> Left "mzero") Parser p 'mplus' Parser q = Parser $ \n xl -> case p n xl of Right a -> Right a Left err -> case q n xl of Right a -> Right a Left _ -> Left err matchChar :: (Char -> Bool) -> Parser Char matchChar c = Parser matchChar' where matchChar' n [] = Left ("expecting char."".a]" Left "1: expecting ']' got '1'" We can use the intListSpace parser from the prior exercise to see that this does in fact work: Parsing2> runParser intListSpace 1 "[1 . The many function needs to be modified only to include the new state.a\n]" Left "3: expecting char. got 'a'" Parsing2> runParser intList 1 "[1. got " ++ show x) The definitions for char and anyChar are not given.150 in m2 n' xl' CHAPTER 9. since they can be written in terms of matchChar.2 .
Second. which takes a list of Chars and matches any of them. Graham Hutton and Daan Leijen have already done this for us in the Parsec library. 9. anyChar. In addition to the basic character parsing functions. Luckily. got 'a'" 151 We can see that the line number. which is the same as our char. PARSING MONADS "[1 . which is the same as our anyChar. string. in addition to a parser. the rest of the string isn't returned when we run parse. which parses a letter. you might want to add more and more features. which is the same as ours. 4 \n\n\n . which parses a digit. This is so it can give better error messages. These are: char. We can write our int and intList functions in Parsec as: char anyChar satisfy oneOf noneOf parse spaces space letter digit string . column 1): unexpected "b" expecting "a" ParsecI> parse (char 'H' >> char 'a' >> char 'l') "stdin" "Hal" Right 'l' ParsecI> parse (char 'H' >> char 'a' >> char 'l') "stdin" "Hap" Left "stdin" (line 1. the error messages produced are much better. Like our libarary. Parsec provides a few basic functions to build parsers from characters. on which the error occurs. satisfy. column 3): unexpected "p" expecting "l" Here. We can try parsing with the above functions: ParsecI> parse (char 'a') "stdin" "a" Right 'a' ParsecI> parse (char 'a') "stdin" "ab" Right 'a' ParsecI> parse (char 'a') "stdin" "b" Left "stdin" (line 1. The primary function Parsec uses to run a parser is parse. but it should be enough to get you started.a\n]" Left "5: expecting char. which is the same as ours. it by no means covers the whole library.8. letter. we can see a few differences between our parser and Parsec: first. digit. However.9. this function takes a string that represents the name of the file you're parsing. Parsec provides primitives for: spaces.2 Parsec As you continue developing your parser. This section is intended to be an introduction to the Parsec library. and noneOf. which is the same as our matchChar. which is the opposite of oneOf. space which parses a single space. increases as we add additional newlines before the erroneous "a". and a few others.8.\n2 . oneOf.
10]" Right [3.(char '.many digit return (read (i1:ir)) intList :: CharParser st [Int] intList = do char '[' intList' 'mplus' (char ']' >> return []) where intList' = do i <.int r <. we use the many function (built in to Parsec) together with the digit function (also built in to Parsec). that using mplus explicitly is not the preferred method of combining parsers: Parsec provides a <|> function that is a synonym of mplus.5. MONADS int :: CharParser st Int int = do i1 <.(char '.a. Parsec provides a few other useful ones: . column 6): unexpected "a" expecting digit In addition to these basic combinators.5. however.' >> intList') 'mplus' (char ']' >> return []) return (i:r) First. The intList function is actually identical to the one we wrote before. but that looks nicer: intList :: CharParser st [Int] intList = do char '[' intList' <|> (char ']' >> return []) where intList' = do i <.5.10] ParsecI> parse intList "stdin" "[3.10]" Left "stdin" (line 1. Note. note the type signatures.int r <.2.digit ir <.152 CHAPTER 9.2.' >> intList') <|> (char ']' >> return []) return (i:r) We can test this: ParsecI> parse intList "stdin" "[3. The st type variable is simply a state variable that we are not using. In the int function.
for simplicity. The second option uses between to parse text between parentheses. the parser alternates between two options (we could have used <|>.oneOf "+*" e2 <.parseExpr case op of '+' -> return (e1 :+: e2) '*' -> return (e1 :*: e2) ] Here. PARSING MONADS 153 • choice takes a list of parsers and performs an or operation (<|>) between all of them. between (char '(') (char ')') $ do e1 <. Show) And then write a parser for this language as: parseExpr :: Parser Expr parseExpr = choice [ do i <.parseExpr op <. • between takes three parsers: an open parser. We can give a datatype for this language as: data Expr = Value Int | Expr :+: Expr | Expr :*: Expr deriving (Eq. It runs them in order and returns the value of the between parser. • optional takes a parser that returns () and optionally runs it. The first simply parses an int and then wraps it up in the Value constructor. then another expression. What it parses is first an expression.9. return (Value i) . • notFollowedBy takes a parser and returns one that succeeds only if the given parser would have failed. Furthermore. it returns either e1 :+: e2 or e1 :*: e2. it simply computes the value: . but it uses the default value as the return. a close parser and a between parser.int. but I wanted to show the choice combinator in action). Suppose we want to parse a simple calculator language that includes only plus and times. • option takes a default value of type a and a parser that returns something of type a. for instance.8. assume each embedded expression must be enclosed in parentheses. Ord. if the parsing fails. then one of plus or times. Depending on what the operator is. This can be used. It then tries to parse with the parser. We can modify this parser. to take care of the brackets on our intList parser. so that instead of computing an Expr.
we want to also be able to say "let x = 5 in" inside of our expressions and then use the variables we've defined. MONADS parseValue :: Parser Int parseValue = choice [int .oneOf "+*" e2 <.parseValueLet op <. do c <. do string "let " c <.letter fm <.parseValueLet case op of '+' -> return (e1 + e2) .getState case lookupFM fm c of Nothing -> unexpected ("variable " ++ show c ++ " unbound") Just i -> return i .oneOf "+*" e2 <. parseValueLet :: CharParser (FiniteMap Char Int) Int parseValueLet = choice [ int .parseValueLet string " in " updateState (\fm -> addToFM fm c e) parseValueLet . In order to do this. That is.between (char '(') (char ')') $ do e1 <. we need to use the getState and setState (or updateState) functions built in to Parsec.parseValue case op of '+' -> return (e1 + e2) '*' -> return (e1 * e2) ] We can use this as: ParsecI> parse parseValue "stdin" "(3*(4+3))" Right 21 bindings getState setState updateState Now.parseValue op <. suppose we want to introduce bindings into our language. between (char '(') (char ')') $ do e1 <.letter char '=' e <.154 CHAPTER 9.
9. PARSING MONADS '*' -> return (e1 * e2) ] 155 The int and recursive cases remain the same. you will need to change the state to something like FiniteMap Char [Int]. which enables us to provide an initial state: ParsecI> runParser parseValueLet emptyFM "stdin" "let c=5 in ((5+4)*c)" Right 45 *ParsecI> runParser parseValueLet emptyFM "stdin" "let c=5 in ((5+4)*let x=2 in (c+x))" Right 63 *ParsecI> runParser parseValueLet emptyFM "stdin" "((let x=2 in 3+4)*x)" Right 14 Note that the bracketing does not affect the definitions of the variables. We add two more cases. we first parse a "let" string. runParser Exercises Exercise 9.8. one to deal with let-bindings. the use of "x" is. we continue and parse the rest. . Then. followed by it's value (a parseValueLet). our parser doesn't notice this. in the last example. we parse the " in " and update the state to include this binding. bindings would have to be removed (see the exercises). For instance. In the let-bindings case. Finally. However. In order to do this. However. we use the Parsec primitive unexpected to report an error. We can see this parser in action using the runParser command. followed by the character we're binding (the letter function is a Parsec primitive that parses alphabetic characters). outside the scope of the definition. since it operates in a strictly leftto-right fashion. In the usage case. In order to fix this omission. where the [Int] is a stack of definitions.7 Modify the parseValueLet parser. so that it obeys bracketing. the other to deal with usages. in some sense. we simply parse the character and then look it up in the state. if it doesn't exist.
3 Mutable References 10.2 Mutable Arrays 10.6 Regular Expressions 10.Chapter 10 Advanced Techniques 10.7 Dynamic Types 157 .4 The ST Monad 10.1 Exceptions 10.5 Concurrency 10.
The idea is to say how well a program scales with more data. for each element it would take a certain number of steps to do the addition and the recursive call (call this number x). and so on. Then. the total time this function will take is nx + y since it needs to do those additions n many times. The model we use is sort of an arbitrary "machine step. the function will take either 0 or 1 or 2 or some very small number of machine steps. it's not very useful (unless you know you'll only be working with small amounts of data. Consider the following Haskell function to return the sum of the elements in a list: sum [] = 0 sum (x:xs) = x + sum xs How long does it take this function to complete? That's a very difficult question. the length of the list. and actually dependent only on exactly how we define a machine step. it only depends on the length of the input list. it would take however much time the list of length 0 would take. since they are independent of n. If the input list is of length 0. it will take however many steps an empty list would take (call this value y) and then. This is far too much to deal with. it would depend on all sorts of things: your processor speed. how many other programs are running on your computer. If the input list is of length n. There are many good introductory books to complexity theory and the basics are explained in any good algorithms book. the exact way in which the addition is carried out.Appendix A Brief Complexity Theory Complexity Theory is the study of how long a program will take to run. so we really don't 159 . of course). your amount of memory. plus a few more steps for doing the first (and only element). What if the list is of length 1. so we need to invent a simpler model. If you have a program that runs quickly on very small amounts of data but chokes on huge amounts of data. depending exactly on how you count them (perhaps 1 step to do the pattern matching and 1 more to return the value 0). I'll keep the discussion here to a minimum." So the question is "how many machine steps will it take for this program to complete?" In this case. These x and y values are called constant values. depending on the size of its input. Well.
we have to insert x into this new list. you can access any element immediately. There's much more in complexity theory than this. we first have to sort the tail of the list first.160 APPENDIX A. this will take O(n − 1) = O(n) steps. What does this mean? Simply that for really long lists. or O(1). so we cannot throw it out. Here. Putting all of this together. the sum function won't take very long. which means that the entire complexity of this sorting algorithm is O(n2 ). Otherwise. the function takes nx+y machine steps to complete. the squared is not a constant value. However with arrays. but this should be enough to allow you to understand all the discussions in this tutorial. Basically saying something is O(n) means that for some constant factors x and y. in order to sort a list of n-many elements. Consider the following sorting algorithm for lists (commonly called "insertion sort"): sort [] = [] sort [x] = [x] sort (x:xs) = insert (sort xs) where insert [] = [x] insert (y:ys) | x <= y = x : y : ys | otherwise = y : insert ys The way this algorithm works is as follow: if we want to sort an empty list or a list of just one element. etc. Suppose it takes f (n) stepts to sort a list of length n. Let's analyze how long this function takes to complete. Then. It traverses the now-sorted tail and inserts x wherever it naturally fits. Consider the random access functions for lists and arrays. That's what the insert function does. BRIEF COMPLEXITY THEORY want to consider them all that important. which takes f (n − 1) time. In this case. Then. Therefore. which is basically as fast an any algorithm can go. Of course there are algorithms that run much more slowly that simply O(n2 ) and there are ones that run more quickly than O(n). If x has to go at the end. we return them as they are. we say that the complexity of this sum function is O(n) (read "order n"). we sort xs and then want to insert x in the appropriate location. Just keep in mind that O(1) is faster than O(n) is faster than O(n2 ). In the worst case. . accessing an arbitrary element in a list of length n will take O(n) time (think about accessing the last element). we see that we have to do O(n) amount of work O(n) many times. but that the sort function will take quite some time. we have a list of the form x:xs. as they are already sorted. which is said to be in constant time.
a function is recursive if its definition depends on itself. The base case is when n = 0 and the recursive case is when n > 0. We can actually implement this system of natural numbers in Haskell: data Nat = Zero | Succ Nat This is a recursive type definition. it has a successor. Clearly. The prototypical example is factorial. they'll say this is how it is). Recursive function definitions always contain a number of non-recursive base cases and a number of recursive cases. we can see that in order to calculate f act(5). all we need to do is add one to this result. One thing we might want to do is be able to convert back and forth beween Nats and Ints. and so on. . . One can actually think of the natural numbers themselves as recursive (in fact. we represent one as Succ Zero and three as Succ (Succ (Succ Zero)). . . In the case of factorial. That is. we realize that we're going to have something of the form Succ n. . 2 = succ(1). we need to calculate f act(3). This gives rise to our recursive case: natToInt (Succ n) = natToInt n + 1 161 . but in order to calculatef act(4). we need to calculate f act(4). there is a zero element and then for every element. Assuming we can do this. and so on forever. We can make the assumption that we'll be able to take n and produce an Int. . Here. whose definition is: f act(n) = 1 n ∗ f act(n − 1) n=0 n>0 Here. we can write a base case as: natToInt Zero = 0 In order to write the recursive case. That is 1 = succ(0). . we have one of each. 573 = succ(573). if you ask set theorists about this.Appendix B Recursion and Induction Informally.
Now. That is. In fact. thus in the base case the algorithm does the correct thing. . Now. obviously it does. in turn will knock over the third. by simply looking at the definition. That is. we know k! ≥ k. We now invoke the inductive hypothesis and claim that the statement holds for n = k. We know that 0 + m = m regardless of what m is. Now. Since n! = 1 > 0 = n. Now. suppose that n > 0. we wish to prove the statement when n = 0. it's going to knock over the second one. It's much like pushing down dominoes. so (k + 1) ∗ k! ≥ k + 1 if and only if k + 1 ≥ 1. Now. we assume that k! ≥ k. which means k + 1 ≥ 1. When n = 0. as the base case. Thus it is proven. First we prove the base case: does natToInt Zero evaluate to 0? Yes. suppose the first argument is Zero. We can continue this argument as long as we want and then see that it's true for all n. Let's consider a more complex example: addition of Nats. It may seem a bit counter-intuitive that we are assuming that the claim is true for k in our proof that it is true for n. we get that 0! ≥ 0 as desired.162 APPENDIX B. it is obvious that it does. The base case is like pushing down the first domino. Again. very analogous to our analysis of recursion. Now. But we know that k ≥ 0. and so on. our program is correct. You know that when you push down the first domino. we know it's true for n = 0 so using this we use our inductive argument to show that it's true for n = 1. We can write this concisely as: addNat Zero m = m addNat (Succ n) m = addNat n (Succ m) Now. Now. n! ≥ n if and only iff (k + 1)! ≥ (k + 1). First we formulate a base case: namely. We now apply the definition of factorial and get (k + 1)! = (k + 1) ∗ k!. First. suppose that addNat n m does the correct thing for all m and we want to show that addNat (Succ n) m does the correct thing. RECURSION AND INDUCTION There is a close connection between recursion and mathematical induction. You can think of it like this: we've proved the statement for the case when n = 0. we can assume that natToInt n evaluates to the correct value (this is the inductive hypothesis) and ask whether natToInt (Succ n) produces the correct value. we know that it is true for n = 1 so we reuse our inductive argument to show that it's true for n = 2. n! = 1 by definition. and the inductive case is like showing that pushing down domino k will cause the k + 1st domino to fall. Then n = k + 1 for some value k. we can use induction to prove that our natToInt function does the right thing. We know that (n + 1) + m = n + (m + 1) and thus since addNat n (Succ m) does the correct thing (by the inductive hypothesis). Induction is a proof technique which typically breaks problems down into base cases and "inductive" cases. This. we use k to formate the statement for our value of n. let's prove that this does the correct thing. Let's say we want to prove the statement n! ≥ n for all n ≥ 0.
yielding just 'a'. This is because the application of snd will leave you with fst "foo". You could also use foldl.1].8. the next-most inner application will return the maximum of whatever was the maximum before and the second-to-last element. the result would have been 3. However."foo")) you will have gotten a type error. 163 .19615 If multiplication bound more tightly. we can do something like: Prelude> sqrt 3 * 3 5. This is because first we want to take the first half the the tuple: (1.4 Solution: length (filter Char. This will continue on.5 foldr max 0 [5.Appendix C Solutions To Exercises Solution 3. The foldr case is easier to explain: we replace each cons with an application of max and the empty list with 0. so you cannot apply fst to it. Solution 3. Solution 3. the string "foo" isn't a tuple. actually.2 Solution: snd (fst ((1. function application binds more tightly than anything else."foo")).1 It binds more tightly.isLower "aBCde") Solution 3. To see this. If you tried fst (snd ((1. the inner-most application will take the maximum of 0 and the last element of the list (if it exists).2. Thus.isLower "aBCde" Solution 3. carrying to current maximum all the way back to the beginning of the list.10.3 Solution: map Char.'a'). Then.'a').'a') and then out of this we want to take the second half.
164 APPENDIX C. SOLUTIONS TO EXERCISES In the foldl case.'b').(1. This happens for each element and thus eventually returns the maximal element.'a')])) Solution 3. we replace our current state with that number and the continue. If it is. like: fib n = if n == 1 || n == 2 then 1 else fib (n-1) + fib (n-2) Either is acceptable. we can think of this as looking at each element in the list in order. Solution 3. Solution 3.9 We can define my map as: . but the first is perhaps more natural in Haskell. We could just as well have defined it as: mult 1 b = b mult a b = b + mult (a-1) b a a + a ∗ (b − 1) b=1 otherwise Solution 3.7 We can define a fibonacci function as: fib 1 = 1 fib 2 = 1 fib n = fib (n-1) + fib (n-2) We could also write it using explicit if statements.8 We can define: a∗b= And then type out code: mult a 1 = a mult a b = a + mult a (b-1) Note that it doesn't matter that of a and b we do the recursion on. We pull off the first element and check to see if it's bigger than our current state.(6.'c'). We start off our "state" with 0.6 fst (head (tail [(5.
it is an element x followed by a list xs. Assuming we've already properly applied my map to xs. the first of . module Main where import IO main = do nums <. The only tricky parts are the recursive calls in getNums and showFactorials.165 my_map f [] = [] my_map f (x:xs) = f x : my_map f xs Recall that the my map function is supposed to apply a function f to every element in the list. In the case that the list is non-empty.10 The code below appears in Numbers. In the case that the list is empty. Solution 3. so we just return the empty list. there are no elements to apply the function to.hs. For showFactorials. Suppose we have a list of numbers. we consider first the recursive call.getNums putStrLn ("The sum is " ++ show (sum nums)) putStrLn ("The product is " ++ show (product nums)) showFactorials nums getNums = do putStrLn "Give me a number (or 0 to stop):" num <. This is exactly what the second line does. then all we're left to do is apply f to x and then stick the results together.getNums return ((read num :: Int):rest) showFactorials [] = return () showFactorials (x:xs) = do putStrLn (show x ++ " factorial is " ++ show (factorial x)) showFactorials xs factorial 1 = 1 factorial n = n * factorial (n-1) The idea for getNums is just as spelled out in the hint.getLine if read num == 0 then return [] else do rest <.
Solution 4. The second and third argument must be of the same type. . Int 5. 5. so we don't need to do anything at all. hence the recursive call. We don't know anything about what it produces. [a]− > a 5.3 The types: 1.166 APPENDIX C. This ignores the first argument. [a]− > Bool 4. In this expression. 3. This function takes an element and returns the list containing only that element. (a. type error: lists are homogenous 3. 4. Solution 4. a− > String. type error: cannot add values of different types 1. though. Note that this must be return () instead of just () because if we simply wrote showFactorials [] = () then this wouldn't be an IO action.2 The types: 1. since they go into the same list. a− > b− > b− > (a. Num a => a− > a.1 String or [Char] 2. Then we print out the rest. so it can be any type. x must be a function which takes a Char as an argument. (Char− > a)− > a. 2. so we simply return (). a− > [a]. [b]). [[a]]− > a Solution 4. so we call it a. But what should we do in the case of the empty list? Clearly we are done. as it needs to be. First we print out the string showing the factorial. Since we apply (+) to a. Char) 4. Num a ⇒ (a. [a]− > a 3. you should probably just keep reading the tutorial. it must be an instance of Num. For more clarification on this. b)− > b 2. The first element can be of any type. SOLUTIONS TO EXERCISES which is x.
leading to an infinite type. this must be an instance of Num. Solution 4. Here. But then it must have type (b− > c)− > c. we assume x has type a.5 The code.y] lastTwo :: Quadruple a b -> [b] lastTwo (Quadruple x y z t) = [z.t] We note here that there are only two type variables.4 The definitions will be something like: data Triple a b c = Triple a b c tripleFst (Triple x y z) = x tripleSnd (Triple x y z) = y tripleThr (Triple x y z) = z Solution 4. is: data Quadruple a b = Quadruple a a b b firstTwo :: Quadruple a b -> [a] firstTwo (Quadruple x y z t) = [x.167 6. but then it must have type ((b− > c)− > c)− > c and so on. But x is applied to itself. Type error. since we apply (+). Num a => a− > a. Solution 4. so it must have type b− > c. 7.6 The code: data Tuple a b c d e = | | | tuple1 tuple1 tuple1 tuple1 (One (Two (Three (Four a ) a b ) a b c ) a b c d) One a Two a b Three a b c Four a b c d = = = = Just Just Just Just a a a a tuple2 (One a tuple2 (Two a b tuple2 (Three a b c ) = Nothing ) = Just b ) = Just b . Again. with type signatures. a and b associated with Quadruple.
168 tuple2 (Four tuple3 tuple3 tuple3 tuple3 tuple4 tuple4 tuple4 tuple4 (One (Two (Three (Four (One (Two (Three (Four APPENDIX C.b) ) (a.c. Solution 4.b.d)) Here.8 The code: listHead (Cons x xs) = x listTail (Cons x xs) = xs listFoldl f y Nil = y listFoldl f y (Cons x xs) = listFoldl f (f y x) xs listFoldr f y Nil = y listFoldr f y (Cons x xs) = f x (foldr f z xs) Solution 4.9 The code: elements (Leaf x) = [x] elements (Branch lhs x rhs) = elements lhs ++ [x] ++ elements rhs Solution 4.10 The code: . SOLUTIONS TO EXERCISES a b c d) = Just b a ) a b ) a b c ) a b c d) a ) a b ) a b c ) a b c d) = = = = = = = = Nothing Nothing Just c Just c Nothing Nothing Nothing Just d Solution 4.b.c) ) (a. we use embedded Eithers to represent the fact that there are four (instead of two) options.7 The code: fromTuple fromTuple fromTuple fromTuple (One (Two (Three (Four a ) a b ) a b c ) a b c d) = = = = Left Left Right Right (Left (Right (Left (Right a ) (a.
fold' (-) (\y -> 2 .3] fold' (-) (\y -> 0 .3] 0 . We can observe the difference in an interpreter: CPS> foldr (-) 0 [1..3] 2 CPS> foldl (-) 0 [1. By writing down the derivations of fold and foldr we can see exactly where they diverge: ==> ==> ==> ==> ==> foldr (-) 0 [1.foldr (-) 0 [])) 1 .y) [3]) 0 .2.(2 ..2.(2 . Solution 4.2.3)) -2 ==> ==> ==> ==> ==> Essentially.y) [1. . It's behavior most closely resembles foldr.foldr (-) 0 [2.2.(3 . 1 .0)) 2 fold (-) 0 [1.2. the primary difference is that in the foldr case.3] 1 .3] -6 CPS> fold (-) 0 [1.169 foldTree :: (a -> b -> b) -> b -> BinaryTree a -> b foldTree f z (Leaf x) = f x z foldTree f z (Branch lhs x rhs) = foldTree f (f x (foldTree f z rhs)) lhs elements2 = foldTree (:) [] or: elements2 tree = foldTree (\a b -> a:b) [] tree The first elements2 is simply a more compact version of the second.3] .fold' (-) (\y -> 1 . whereas in the CPS case. the initial value is used at the beginning.2.11 It mimicks neither exactly.(1 .3] 0 . but differs slightly in its treatment of the initial value.(3 .(1 .y) [2.(2 .3] -2 Clearly it behaves differently. the "initial value" is used at the end (replacing []).
getLine if name 'elem' ["Simon". since these are only one action commands.getLine putStrLn (if name 'elem' ["Simon"." Note that we don't need to repeat the dos inside the ifs. "John". We could also be a bit smarter and use the elem command which is built in to the Prelude: main = do putStrLn "Please enter your name:" name <.") Using case. "Phil"] then putStrLn "Haskell is great!" else if name == "Koen" then putStrLn "Debugging Haskell is fun!" else putStrLn "I don't know who you are. "John". We could instead write: main = do putStrLn "Please enter your name:" name <. we needn't put all the putStrLns inside the if statements.170 APPENDIX C. SOLUTIONS TO EXERCISES Solution 4. "Phil"] then "Haskell is great!" else if name == "Koen" then "Debugging Haskell is fun!" else "I don't know who you are.getLine if name == "Simon" || name == "John" || name == "Phil" then putStrLn "Haskell is great!" else if name == "Koen" then putStrLn "Debugging Haskell is fun!" else putStrLn "I don't know who you are. we get something like: .1 Using if. we get something like: main = do putStrLn "Please enter your name:" name <.12 Solution 5." Of course.
is actually not much cleaner.getLine bracket (openFile fn ReadMode) hClose (\h -> do txt <.") main doRead = do putStrLn "Enter a file name to read:" fn <.getLine case cmd of "quit" -> return () "read" -> do doRead.. main "write" -> do doWrite..171 main = do putStrLn "Please enter your name:" name <. Solution 5.." Which.):" .hGetContents h putStrLn txt) doWrite = do putStrLn "Enter a file name to write:" fn <.getLine case name of "Simon" -> putStrLn "Haskell is great!" "John" -> putStrLn "Haskell is great!" "Phil" -> putStrLn "Haskell is great!" "Koen" -> putStrLn "Debugging Haskell is fun!" _ -> putStrLn "I don't know who you are.getLine bracket (openFile fn WriteMode) hClose (\h -> do putStrLn "Enter text (. main _ -> do putStrLn ("I don't understand the command " ++ cmd ++ ".?" cmd <. . in this case..2 The code might look something like: module DoFile where import IO main = do putStrLn "Do you want to [read] a file.
The others look something like: func1 x = map (*x) func2 f g = filter f . we can clearly rewrite this as: and = foldr (&&) True Solution 7. which ensure the that the program lives on.. we're trying to use map as the second argument.getLine if l == ".172 APPENDIX C. In this case. Solution 7.2 We can start out with a recursive definition: and [] = True and (x:xs) = x && and xs From here. so it knows where to write the input to). In this case. this isn't possible. This is because the function composition operator (. map g func4 = map (+2) . map. But map takes two arguments.1 Function func3 cannot be converted into point-free style." then return () else do hPutStrLn h l writeLoop h The only interesting things here are the calls to bracket. and the writeLoop function. Solution 7. SOLUTIONS TO EXERCISES writeLoop h) writeLoop h = do l <. filter ('elem' [1.) expects a function which takes only one.10]) . trying to eta-reduce off the g. regardless of whether there's a failure or not. while (.3 We can write this recursively as: .) has type (b → c) → (a → b) → (a → c). (5:) func5 = foldr (uncurry $ flip f) 0 You might have been tempted to try to write func2 as filter f . Note that we need to pass the handle returned by openFile (through bracket to this function.
we get: ==> ==> f >>= return f >>= \x -> return x f >>= \x -> Just x At this point. there are two cases depending on whether f is Nothing or not. we get: ==> ==> ==> ==> Just a >>= \x -> Just x (\x -> Just x) a Just a f . we get: ==> ==> ==> return a >>= f Just a >>= \x -> f x (\x -> f x) a f a The second law is: f >>= return ≡ f. Here. Then. we can do point elimination to get: foldr foldr foldr foldr foldr (\a b -> f a ++ b) [] (\a b -> (++) (f a) b) [] (\a -> (++) (f a)) [] (\a -> ((++) . f) [] ==> ==> ==> ==> Solution 9. f is Just a. f) a) [] ((++) . we get: ==> ==> ==> Nothing >>= \x -> Just x Nothing f In the second case. In the case of Maybe.1 The first law is: return a >>= f ≡ f a.173 concatMap f [] = [] concatMap f (x:xs) = f x ++ concatMap f xs This hints that we can write this as: concatMap f = foldr (\a b -> f a ++ b) [] Now. In the first case.
we are doing depth-first search. When the recursive call to search' comes first. Solution 9. we are doing breadth-first search.3]. Thus.[0.2. Solution 9.1. Suppose f is Just a.1. Then the LHS reduces to g a >>= h and the RHS reduces to (Just a >>= \x -> g x) >>= h which in turn reduces to g a >>= h. we expect the solutions to come in the other order: MPlus> searchAll3 gr 0 3 :: [[Int]] [[0. Solution 9. This looks like: . SOLUTIONS TO EXERCISES And the second law is shown.4 This is a very difficult problem. If f is Nothing.174 APPENDIX C.3] Monads> searchAll gr 3 0 :: Either String [Int] Left "no path" which is exactly what we want.2 The idea is that we wish to use the Left constructor to represent errors on the Right constructor to represent successes.3]] Just as we expected. This leads to an instance declaration like: instance Monad (Either String) where return x = Right x Left s >>= _ = Left s Right x >>= f = f x fail s = Left s If we try to use this monad to do search. if you found that you were stuck immediately. so these two are the same. using the list monad. so they are the same.3 The order to mplus essentially determins the search order. we need to define a list transformer monad. The third law states: f >>= (\x -> g x >>= h) ≡ (f >>= g) >>= h. we get: Monads> searchAll gr 0 3 :: Either String [Int] Right [0. First. please just read as much of this solution as you need to try it yourself. then the left-hand-side clearly reduces to Nothing. The righthand-side reduces to Nothing >>= h which in turn reduces to Nothing. When the recursive call to searchAll2 comes first.
we need to make it an instance of MonadTrans: instance MonadTrans ListT where lift x = ListT (do a <.175 newtype ListT m e = ListT { unListT :: m [e] } The ListT constructor simply wraps a monadic action (in monad m) which returns a list.x. We now need to apply k to each of these elements (which will result in something of type ListT m [e2].m l' <. success is designated by a monadic action which returns a singleton list. Failure (like in the standard list monad) is represented by an empty list: of course. Binding happens essentially by running the action which will result in a list l. This has type [e]. We need to get rid of the ListTs around this (by using unListT) and then concatenate them to make a single list. a) out and then returning the singleton list. the zero element is a monadic action which returns an empty list.m2 return (l1 ++ l2) Here.m1 l2 <. We now need to make this a monad: instance Monad m => Monad (ListT m) where return x = ListT (return [x]) fail s = ListT (return [] ) ListT m >>= k = ListT $ do l <. return [a]) Lifting an action into ListT simply involves running it and getting the value (in this case. we need to make it an instance of MonadPlus instance Monad m => MonadPlus (ListT m) where mzero = ListT (return []) ListT m1 'mplus' ListT m2 = ListT $ do l1 <. k) l return (concat l') Here. it's actually an empty list returned from the enclosed monad. writing searchAll6 is fairly straightforward: searchAll6 g@(Graph vl el) src dst | src == dst = do lift $ putStrLn $ . Now. Finally. Addition is done by executing both actions and then concatenating the results. Once we have all this together.mapM (unListT .
we can run this by: MTrans> unListT (searchAll6 gr 0 3) Exploring 0 -> 3 Exploring 1 -> 3 Exploring 3 -> 3 Exploring 2 -> 3 Exploring 3 -> 3 MTrans> it [[0._):es) | src == u = (do path <. we could use this with any appropriate monad transformer.searchAll6 g v dst return (u:path)) 'mplus' search' es | otherwise = search' es The only change (besides changing the recursive call to call searchAll6 instead of searchAll2) here is that we call putStrLn with appropriate arguments.getT . This extra lift is required because now we're stacking two transformers on top of IO instead of just one.3]. in our case.v. Solution 9. we want to use ListT. lifted into the monad. after applying a graph and two ints) has type MonadTrans t. SOLUTIONS TO EXERCISES "Exploring " ++ show src ++ " -> " ++ show dst return [src] | otherwise = do lift $ putStrLn $ "Exploring " ++ show src ++ " -> " ++ show dst search' el where search' [] = mzero search' ((u. we see that the result (i.176 APPENDIX C. Thus. MonadPlus (t IO) => t IO [Int]). All we need to do is incorporate the calls to putT and getT into searchAll6 and add an extra lift to the IO calls.3]] This is precisely what we were looking for. If we look at the type of searchAll6. In theory.1.5 This exercise is actually simpler than the previous one.2.. searchAll7 g@(Graph vl el) src dst | src == dst = do lift $ lift $ putStrLn $ "Exploring " ++ show src ++ " -> " ++ show dst visited <.e.[0.
we run this beast by saying: MTrans> unListT (evalStateT (searchAll7 gr4 0 3) []) Exploring 0 -> 3 Exploring 1 -> 3 Exploring 3 -> 3 Exploring 0 -> 3 Exploring 2 -> 3 Exploring 3 -> 3 MTrans> it [[0.v. In our case. we simply sprinkle calls to spaces through intList to get intListSpace: .3]. MonadTrans t. Essentially this means that we've got something that's a state transformer wrapped on top of some other arbitrary transformer (t) which itself sits on top of IO. After applying the graph and two ints.getT putT (src:visited) if src 'elem' visited then mzero else search' el where search' [] = mzero search' ((u. t is going to be ListT. even on gr4. MonadPlus (StateT [Int] (t IO)) => StateT [Int] (t IO) [Int].6 First we write a function spaces which will parse out whitespaces: spaces :: Parser () spaces = many (matchChar isSpace) >> return () Now. Thus. using this.[0.1.searchAll7 g v dst return (u:path)) 'mplus' search' es | otherwise = search' es The type of this has grown significantly.2.177 putT (src:visited) return [src] | otherwise = do lift $ lift $ putStrLn $ "Exploring " ++ show src ++ " -> " ++ show dst visited <._):es) | src == u = (do path <. this has type Monad (t IO).3]] And it works. Solution 9.
5]) Parsing> runParser intListSpace "[1 .[1. 4 Right ("". got 'a'" \n\n .letter fm <.4. between (char '(') (char ')') $ do e1 <.2.7 We do this by replacing the state functions with push and pop functions as follows: parseValueLet2 :: CharParser (FiniteMap Char [Int]) Int parseValueLet2 = choice [ int .' >> spaces >> intList') 'mplus' (char ']' >> return []) return (i:r) We can test that this works: Parsing> runParser intListSpace "[1 .2 . do string "let " c <.(char '.getState case lookupFM fm c of Nothing -> unexpected ("variable " ++ show c ++ " unbound") Just (i:_) -> return i .letter char '=' e <.parseValueLet2 .int spaces r <.5\n]" \n\n . 4 Left "expecting char.parseValueLet2 string " in " pushBinding c e v <.a\n]" Solution 9.parseValueLet2 popBinding c return v .178 APPENDIX C. SOLUTIONS TO EXERCISES intListSpace :: Parser [Int] intListSpace = do char '[' spaces intList' 'mplus' (char ']' >> return []) where intList' = do i <.2 . do c <.
This enables us to modify only slightly the usage case.getState case lookupFM fm c of Just [_] -> setState (delFromFM fm c) Just (_:l) -> setState (addToFM fm c l) The primary difference here is that instead of calling updateState. we use two local functions.parseValueLet2 case op of '+' -> return (e1 + e2) '*' -> return (e1 * e2) ] where pushBinding c v = do fm <.179 op <. it completely removes the stack from the FiniteMap. the stack is never empty. we simply take the top element off the stack when we need to inspect the value of a variable. column 20): unexpected variable 'x' unbound . The pushBinding function takes a variable name and a value and adds the value onto the head of the list pointed to in the state FiniteMap. pushBinding and popBinding. We can test that this works: ParsecI> runParser parseValueLet2 emptyFM "stdin" "((let x=2 in 3+4)*x)" Left "stdin" (line 1. The popBinding function looks at the value and if there is only one element on the stack.getState case lookupFM fm c of Nothing -> setState (addToFM fm c [v]) Just l -> setState (addToFM fm c (v:l)) popBinding c = do fm <.oneOf "+*" e2 <. otherwise it just removes the first element. This means that if something is in the FiniteMap. this time.
117 Compilers. 31 do notation. 107–110 expressions. 72 λ.NHC. 140 /. 36 ::. 110 accumArray. 36 ˆ. 32 ++. 84 enumFromThenTo. 131–132 enumerated types. 95–100 arithmetic.. 87 ∗. 57 drop. 141 Eq.Interpreters concatenate. 52–53 common sub-expression. 143 arrays. 32 exceptions. 113 boolean. 101 180 brackets. 65 ==. see boolean . 134 do.Index (). 143 actions. 32 ΓE30F . 143–145 assocs. 27 elems. 123 false. 27–28 Emacs. 52 . 52 . 111 import. 32 +. 25 in Hugs. see GHC. 32 //. 65 equals.. 70 evaluation order. 141 enumFromTo. 38 −. see continuation passing style destructive update. 66 bounds. 144 Bird scripts. 87–90 CPS.. 65 []. 81 continuation passing style. 110 qualified. 57 hiding. 140 Editors. 32–34 array. 144 :. 144 bracket. 36 buffering. 127–129 equality. 38 cons. 32 extensions enabling in GHC. 122 as. 58 comments. 173–176 do notation. 36 constructors. 23 fallthrough. 50 . 72 {−−}. 32 −−. 144 Enum. 182 exports. 111 derive.
see tuples parentheses. 8 Haskell Bookshelf. 26–27 null. 101 head. 117 loop. 75. 79 let and do. 53 map. 19. 101 hGetContents. 137–143 comprehensions. 40 composition. 38. 144 induction. 54 infix. 140 cons. 138 FiniteMap. 72 LaTeX scripts. 101 hPutChar. 7 hClose. 29 length. 101 getLine. 50 type. 132 numeric types. 74 Num. 44–52 anonymous. 3 functions. 111–112 monads. 125–127 Haskell 98. 170–173 laws. 31 imports. 100–103 isLower. 71–80 getChar. 85–87 modules. 67 interactive programs. 94 and do. 101 getContents. 101 hPutStrLn. 36 empty. 145–147 foldl. 171 plus. 139 foldr. 35–42. see lambda as arguments. 101 hIsEOF. 19. 72 lambda calculus. 143 listArray. 19–23 ICFP. see GHC.Hugs IO. 181 state.INDEX files. see IO pairs. 38 lambda. see immutable named fields. 70 openFile. 101 operator precedence. 112–114 local bindings. 35 functional. 38. 107–114 hierarchical. 33 Ord. 36 literate style. 145–147 Maybe. 37. 37. 138 maps. 9 immutable. 42–44 filter. 135–137 NHC. 110–111 indices. 176–181 transformer. 38. 175 list comprehensions. 143 lists. 147 lazy. 138 fst. 101 Hugs. 181 mutable. 74 hGetChar. 130–131 output. see IO instance. 114 layout. 58. 101 hPutStr. 56–63 Interpreters. 78–80 associative. 115–116 input. 23–26 guards. 38. 101 hGetLin. 101 GHC. 91–106 library. 33 181 . 2. 173–176 definition of.
101 putStrLn. 87 unzip. 139 . see boolean tuples. 67–69 signatures. 101 putStr. 155–162 instances. 82. 101. 117 Show. 165–168 newtype. 149–152 polymorphic. 152– 155 strict. 74 take. 64 classes. 119–125 pure. 101 recursion. 73–75 inference. 37. 37–38 converting from/to. 152–155 default. 8 state. 168 higher-order. see interactive programs wildcard. 59 Read. 44. 59 randomRIO. 38 tail. 75–77 kinds. 122 writeFile. 31 robot. 101 zip. 9 sections. 2 putChar. 139 user input. 77 synonyms. 168 checking. 101 random numbers. 129–130 show. 38 readFile. 168 errors. 71 snd. 38 true. 80–87. 132–133 read. 152–155 strings. 69–71. 64 IO. 64. 115–116 shadowing. 32 standard. 38. 140 toUpper. 34–35 type. 148–149 INDEX Unit. 127–134. 53–56 referential tranparency. 2 strict.182 pattern matching. 77–78 hierarchy. 162–165 datatypes. 66 explicit declarations. 134–137. 2. 35 sqrt.
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