PureScript by Example

This repository contains a community fork of PureScript by Example by Phil Freeman, also known as "the PureScript book". This version differs from the original in that it has been updated so that the code and exercises work with up-to-date versions of the compiler, libraries, and tools. Some chapters have also been rewritten to showcase the latest features of the PureScript ecosystem.

If you enjoyed the book or found it useful, please consider buying a copy of the original on Leanpub.

Status

This book is being continuously updated as the language evolves, so please report any issues you discover with the material. We appreciate any feedback you have to share, even if it's as simple as pointing out a confusing section that we could make more beginner-friendly.

Unit tests are also being added to each chapter so you can check if your answers to the exercises are correct. See #79 for the latest status on tests.

PureScript is a small strongly, statically typed programming language with expressive types, written in and inspired by Haskell, and compiling to Javascript.

Functional programming in JavaScript has seen quite a lot of popularity recently, but large-scale application development is hindered by the lack of a disciplined environment in which to write code. PureScript aims to solve that problem by bringing the power of strongly-typed functional programming to the world of JavaScript development.

This book will show you how to get started with the PureScript programming language, from the basics (setting up a development environment) to the advanced.

Each chapter will be motivated by a particular problem, and in the course of solving that problem, new functional programming tools and techniques will be introduced. Here are some examples of problems that will be solved in this book:

• Transforming data structures with maps and folds
• Form field validation using applicative functors
• Testing code with QuickCheck
• Using the canvas
• Domain specific language implementation
• Working with the DOM
• JavaScript interoperability
• Parallel asynchronous execution

Some text is derived from the PureScript Documentation Repo, which uses the same license, and is copyright various contributors.

Introduction

Functional JavaScript

Functional programming techniques have been making appearances in JavaScript for some time now:

• Libraries such as UnderscoreJS allow the developer to leverage tried-and-trusted functions such as map, filter and reduce to create larger programs from smaller programs by composition:

var sumOfPrimes =
_.chain(_.range(1000))
.filter(isPrime)
.reduce(function(x, y) {
return x + y;
})
.value();

• Asynchronous programming in NodeJS leans heavily on functions as first-class values to define callbacks.

require('fs').readFile(sourceFile, function (error, data) {
if (!error) {
require('fs').writeFile(destFile, data, function (error) {
if (!error) {
console.log("File copied");
}
});
}
});

• Libraries such as React and virtual-dom model views as pure functions of application state.

Functions enable a simple form of abstraction which can yield great productivity gains. However, functional programming in JavaScript has its own disadvantages: JavaScript is verbose, untyped, and lacks powerful forms of abstraction. Unrestricted JavaScript code also makes equational reasoning very difficult.

PureScript is a programming language which aims to address these issues. It features lightweight syntax, which allows us to write very expressive code which is still clear and readable. It uses a rich type system to support powerful abstractions. It also generates fast, understandable code, which is important when interoperating with JavaScript, or other languages which compile to JavaScript. All in all, I hope to convince you that PureScript strikes a very practical balance between the theoretical power of purely functional programming, and the fast-and-loose programming style of JavaScript.

Types and Type Inference

The debate over statically typed languages versus dynamically typed languages is well-documented. PureScript is a statically typed language, meaning that a correct program can be given a type by the compiler which indicates its behavior. Conversely, programs which cannot be given a type are incorrect programs, and will be rejected by the compiler. In PureScript, unlike in dynamically typed languages, types exist only at compile-time, and have no representation at runtime.

It is important to note that in many ways, the types in PureScript are unlike the types that you might have seen in other languages like Java or C#. While they serve the same purpose at a high level, the types in PureScript are inspired by languages like ML and Haskell. PureScript's types are expressive, allowing the developer to assert strong claims about their programs. Most importantly, PureScript's type system supports type inference - it requires far fewer explicit type annotations than other languages, making the type system a tool rather than a hindrance. As a simple example, the following code defines a number, but there is no mention of the Number type anywhere in the code:

iAmANumber =
let square x = x * x
in square 42.0


A more involved example shows that type-correctness can be confirmed without type annotations, even when there exist types which are unknown to the compiler:

iterate f 0 x = x
iterate f n x = iterate f (n - 1) (f x)


Here, the type of x is unknown, but the compiler can still verify that iterate obeys the rules of the type system, no matter what type x might have.

In this book, I will try to convince you (or reaffirm your belief) that static types are not only a means of gaining confidence in the correctness of your programs, but also an aid to development in their own right. Refactoring a large body of code in JavaScript can be difficult when using any but the simplest of abstractions, but an expressive type system together with a type checker can even make refactoring into an enjoyable, interactive experience.

In addition, the safety net provided by a type system enables more advanced forms of abstraction. In fact, PureScript provides a powerful form of abstraction which is fundamentally type-driven: type classes, made popular in the functional programming language Haskell.

Polyglot Web Programming

Functional programming has its success stories - applications where it has been particularly successful: data analysis, parsing, compiler implementation, generic programming, parallelism, to name a few.

It would be possible to practice end-to-end application development in a functional language like PureScript. PureScript provides the ability to import existing JavaScript code, by providing types for its values and functions, and then to use those functions in regular PureScript code. We'll see this approach later in the book.

However, one of PureScript's strengths is its interoperability with other languages which target JavaScript. Another approach would be to use PureScript for a subset of your application's development, and to use one or more other languages to write the rest of the JavaScript.

Here are some examples:

• Core logic written in PureScript, with the user interface written in JavaScript.
• Application written in JavaScript or another compile-to-JS language, with tests written in PureScript.
• PureScript used to automate user interface tests for an existing application.

In this book, we'll focus on solving small problems with PureScript. The solutions could be integrated into a larger application, but we will also look at how to call PureScript code from JavaScript, and vice versa.

Prerequisites

The software requirements for this book are minimal: the first chapter will guide you through setting up a development environment from scratch, and the tools we will use are available in the standard repositories of most modern operating systems.

The PureScript compiler itself can be downloaded as a binary distribution, or built from source on any system running an up-to-date installation of the GHC Haskell compiler, and we will walk through this process in the next chapter.

The code in this version of the book is compatible with versions 0.13.* of the PureScript compiler.

I will assume that you are familiar with the basics of JavaScript. Any prior familiarity with common tools from the JavaScript ecosystem, such as NPM and Gulp, will be beneficial if you wish to customize the standard setup to your own needs, but such knowledge is not necessary.

No prior knowledge of functional programming is required, but it certainly won't hurt. New ideas will be accompanied by practical examples, so you should be able to form an intuition for the concepts from functional programming that we will use.

Readers who are familiar with the Haskell programming language will recognize a lot of the ideas and syntax presented in this book, because PureScript is heavily influenced by Haskell. However, those readers should understand that there are a number of important differences between PureScript and Haskell. It is not necessarily always appropriate to try to apply ideas from one language in the other, although many of the concepts presented here will have some interpretation in Haskell.

The chapters in this book are largely self contained. A beginner with little functional programming experience would be well-advised, however, to work through the chapters in order. The first few chapters lay the groundwork required to understand the material later on in the book. A reader who is comfortable with the ideas of functional programming (especially one with experience in a strongly-typed language like ML or Haskell) will probably be able to gain a general understanding of the code in the later chapters of the book without reading the preceding chapters.

Each chapter will focus on a single practical example, providing the motivation for any new ideas introduced. Code for each chapter are available from the book's GitHub repository. Some chapters will include code snippets taken from the chapter's source code, but for a full understanding, you should read the source code from the repository alongside the material from the book. Longer sections will contain shorter snippets which you can execute in the interactive mode PSCi to test your understanding.

Code samples will appear in a monospaced font, as follows:

module Example where

import Effect.Console (log)

main = log "Hello, World!"


Commands which should be typed at the command line will be preceded by a dollar symbol:

$spago build  Usually, these commands will be tailored to Linux/Mac OS users, so Windows users may need to make small changes such as modifying the file separator, or replacing shell built-ins with their Windows equivalents. Commands which should be typed at the PSCi interactive mode prompt will be preceded by an angle bracket: > 1 + 2 3  Each chapter will contain exercises, labelled with their difficulty level. It is strongly recommended that you attempt the exercises in each chapter to fully understand the material. This book aims to provide an introduction to the PureScript language for beginners, but it is not the sort of book that provides a list of template solutions to problems. For beginners, this book should be a fun challenge, and you will get the most benefit if you read the material, attempt the exercises, and most importantly of all, try to write some code of your own. Getting Help If you get stuck at any point, there are a number of resources available online for learning PureScript: • The #purescript and #purescript-beginners FP Slack channels are a great place to chat about issues you may be having. Use this link to gain access the Slack chatrooms. • The Purescript Discourse Forum is another good place to search for solutions to common problems. Questions you ask here will be available to help future readers, whereas on Slack, message history is only kept for approximately 2 weeks. • PureScript: Jordan's Reference is an alternative learning resource that goes into great depth. If a concept in this book is difficult to understand, consider reading the corresponding section in that reference. • Pursuit is a searchable database of PureScript types and functions. Read Pursuit's help page to learn what kinds of searches you can do. • The unofficial PureScript Cookbook provides answers via code to "How do I do X?"-type questions. • The PureScript documentation repository collects articles and examples on a wide variety of topics, written by PureScript developers and users. • The PureScript website contains links to several learning resources, including code samples, videos and other resources for beginners. • Try PureScript! is a website which allows users to compile PureScript code in the web browser, and contains several simple examples of code. If you prefer to learn by reading examples, the purescript, purescript-node and purescript-contrib GitHub organizations contain plenty of examples of PureScript code. About the Author I am the original developer of the PureScript compiler. I'm based in Los Angeles, California, and started programming at an early age in BASIC on an 8-bit personal computer, the Amstrad CPC. Since then I have worked professionally in a variety of programming languages (including Java, Scala, C#, F#, Haskell and PureScript). Not long into my professional career, I began to appreciate functional programming and its connections with mathematics, and enjoyed learning functional concepts using the Haskell programming language. I started working on the PureScript compiler in response to my experience with JavaScript. I found myself using functional programming techniques that I had picked up in languages like Haskell, but wanted a more principled environment in which to apply them. Solutions at the time included various attempts to compile Haskell to JavaScript while preserving its semantics (Fay, Haste, GHCJS), but I was interested to see how successful I could be by approaching the problem from the other side - attempting to keep the semantics of JavaScript, while enjoying the syntax and type system of a language like Haskell. I maintain a blog, and can be reached on Twitter. Acknowledgements I would like to thank the many contributors who helped PureScript to reach its current state. Without the huge collective effort which has been made on the compiler, tools, libraries, documentation and tests, the project would certainly have failed. The PureScript logo which appears on the cover of this book was created by Gareth Hughes, and is gratefully reused here under the terms of the Creative Commons Attribution 4.0 license. Finally, I would like to thank everyone who has given me feedback and corrections on the contents of this book. Getting Started Chapter Goals In this chapter, we'll set up a working PureScript development environment, solve some exercises, and use the tests provided with this book to check our answers. You may also find a video walkthrough of this chapter helpful if that better suits your learning style. Environment Setup First, work through this Getting Started Guide in the Documentation Repo to setup your environment and learn a few basics about the language. Don't worry if the code in the example solution to the Project Euler problem is confusing or contains unfamiliar syntax. We'll cover all of this in great detail in the upcoming chapters. Solving Exercises Now that you've installed the necessary development tools, clone this book's repo. git clone https://github.com/purescript-contrib/purescript-book.git  The book repo contains PureScript example code and unit tests for the exercises that accompany each chapter. There's some initial setup required to reset the exercise solutions so they are ready to be solved by you. Use the resetSolutions.sh script to simplify this process. While you're at it, you should also strip out all the anchor comments with the removeAnchors.sh script (these anchors are used for copying code snippets into the book's rendered markdown, and you probably don't need this clutter in your local repo): cd purescript-book ./scripts/resetSolutions.sh ./scripts/removeAnchors.sh git add . git commit --all --message "Exercises ready to be solved"  Now run the tests for this chapter: cd exercises/chapter2 spago test  You should see the following successful test output: → Suite: Euler - Sum of Multiples ✓ Passed: below 10 ✓ Passed: below 1000 All 2 tests passed! 🎉  Note that the answer function (found in src/Euler.purs) has been modified to find the multiples of 3 and 5 below any integer. The test suite (found in test/Main.purs) for this answer function is more comprehensive than the test in the earlier getting-started guide. Don't worry about understanding how this test framework code works while reading these early chapters. The remainder of the book contains lots of exercises. If you write your solutions in the Test.MySolutions module (test/MySolutions.purs), you can check your work against the provided test suite. Let's work through this next exercise together in test-driven-development style. Exercise: 1. (Medium) Write a diagonal function to compute the length of the diagonal (or hypotenuse) of a right-angled triangle when given the lengths of the two other sides. Solution We'll start by enabling the tests for this exercise. Move the start of the block-comment down a few lines as shown below. Block comments start with {- and end with -}:  suite "diagonal" do test "3 4 5" do Assert.equal 5.0 (diagonal 3.0 4.0) test "5 12 13" do Assert.equal 13.0 (diagonal 5.0 12.0) {- Move this block comment starting point to enable more tests  If we attempt to run the test now, we'll encounter a compilation error because we have not yet implemented our diagonal function. $ spago test

Error found:
in module Test.Main
at test/Main.purs:21:27 - 21:35 (line 21, column 27 - line 21, column 35)

Unknown value diagonal


Let's first take a look at what happens with a faulty version of this function. Add the following code to test/MySolutions.purs:

import Math (sqrt)

diagonal w h = sqrt (w * w + h)


And check our work by running spago test:

→ Suite: diagonal
☠ Failed: 3 4 5 because expected 5.0, got 3.605551275463989
☠ Failed: 5 12 13 because expected 13.0, got 6.082762530298219

2 tests failed:


Uh-oh, that's not quite right. Let's fix this with the correct application of the Pythagorean formula by changing the function to:

diagonal w h = sqrt (w * w + h * h)


Trying spago test again now shows all tests are passing:

→ Suite: Euler - Sum of Multiples
✓ Passed: below 10
✓ Passed: below 1000
→ Suite: diagonal
✓ Passed: 3 4 5
✓ Passed: 5 12 13

All 4 tests passed! 🎉


Success! Now you're ready to try these next exercises on your own.

Exercises

1. (Easy) Write a function circleArea which computes the area of a circle with a given radius. Use the pi constant, which is defined in the Math module. Hint: don't forget to import pi by modifying the import Math statement.
2. (Medium) Write a function leftoverCents which takes an Integer and returns what's leftover after dividing by 100. Use the rem function. Search Pursuit for this function to learn about usage and which module to import it from. Note: Your IDE may support auto-importing of this function if you accept the auto-completion suggestion.

Conclusion

In this chapter, we installed the PureScript compiler and the Spago tool. We also learned how to write solutions to exercises and check these for correctness.

There will be many more exercises in the chapters ahead, and working through those really helps with learning the material. If you're stumped by any of the exercises, please reach out to any of the community resources listed in the Getting Help section of this book, or even file an issue in this book's repo. This reader feedback on which exercises could be made more approachable helps us improve the book.

Once you solve all the exercises in a chapter, you may compare your answers against those in the no-peeking/Solutions.purs. No peeking please without putting in an honest effort to solve these yourself though. And even if you are stuck, try asking a community member for help first, as we would prefer to give you a small hint rather than spoil the exercise. If you found a more elegant solution (that still only requires knowledge of covered content), please send us a PR.

The repo is continuously being revised, so be sure to check for updates before starting each new chapter.

Functions and Records

Chapter Goals

This chapter will introduce two building blocks of PureScript programs: functions and records. In addition, we'll see how to structure PureScript programs, and how to use types as an aid to program development.

We will build a simple address book application to manage a list of contacts. This code will introduce some new ideas from the syntax of PureScript.

The front-end of our application will be the interactive mode PSCi, but it would be possible to build on this code to write a front-end in JavaScript. In fact, we will do exactly that in later chapters, adding form validation and save/restore functionality.

Project Setup

The source code for this chapter is contained in the file src/Data/AddressBook.purs. This file starts with a module declaration and its import list:

module Data.AddressBook where

import Prelude

import Control.Plus (empty)
import Data.Maybe (Maybe)


Here, we import several modules:

• The Control.Plus module, which defines the empty value.
• The Data.List module, which is provided by the lists package which can be installed using Spago. It contains a few functions which we will need for working with linked lists.
• The Data.Maybe module, which defines data types and functions for working with optional values.

Notice that the imports for these modules are listed explicitly in parentheses. This is generally a good practice, as it helps to avoid conflicting imports.

Assuming you have cloned the book's source code repository, the project for this chapter can be built using Spago, with the following commands:

$cd chapter3$ spago build


Simple Types

PureScript defines three built-in types which correspond to JavaScript's primitive types: numbers, strings and booleans. These are defined in the Prim module, which is implicitly imported by every module. They are called Number, String, and Boolean, respectively, and you can see them in PSCi by using the :type command to print the types of some simple values:

$spago repl > :type 1.0 Number > :type "test" String > :type true Boolean  PureScript defines some other built-in types: integers, characters, arrays, records, and functions. Integers are differentiated from floating point values of type Number by the lack of a decimal point: > :type 1 Int  Character literals are wrapped in single quotes, unlike string literals which use double quotes: > :type 'a' Char  Arrays correspond to JavaScript arrays, but unlike in JavaScript, all elements of a PureScript array must have the same type: > :type [1, 2, 3] Array Int > :type [true, false] Array Boolean > :type [1, false] Could not match type Int with type Boolean.  The error in the last example is an error from the type checker, which unsuccessfully attempted to unify (i.e. make equal) the types of the two elements. Records correspond to JavaScript's objects, and record literals have the same syntax as JavaScript's object literals: > author = { name: "Phil", interests: ["Functional Programming", "JavaScript"] } > :type author { name :: String , interests :: Array String }  This type indicates that the specified object has two fields, a name field which has type String, and an interests field, which has type Array String, i.e. an array of Strings. Fields of records can be accessed using a dot, followed by the label of the field to access: > author.name "Phil" > author.interests ["Functional Programming","JavaScript"]  PureScript's functions correspond to JavaScript's functions. The PureScript standard libraries provide plenty of examples of functions, and we will see more in this chapter: > import Prelude > :type flip forall a b c. (a -> b -> c) -> b -> a -> c > :type const forall a b. a -> b -> a  Functions can be defined at the top-level of a file by specifying arguments before the equals sign: add :: Int -> Int -> Int add x y = x + y  Alternatively, functions can be defined inline, by using a backslash character followed by a space-delimited list of argument names. To enter a multi-line declaration in PSCi, we can enter "paste mode" by using the :paste command. In this mode, declarations are terminated using the Control-D key sequence: > :paste … add :: Int -> Int -> Int … add = \x y -> x + y … ^D  Having defined this function in PSCi, we can apply it to its arguments by separating the two arguments from the function name by whitespace: > add 10 20 30  Quantified Types In the previous section, we saw the types of some functions defined in the Prelude. For example, the flip function had the following type: > :type flip forall a b c. (a -> b -> c) -> b -> a -> c  The keyword forall here indicates that flip has a universally quantified type. It means that we can substitute any types for a, b and c, and flip will work with those types. For example, we might choose the type a to be Int, b to be String and c to be String. In that case we could specialize the type of flip to (Int -> String -> String) -> String -> Int -> String  We don't have to indicate in code that we want to specialize a quantified type - it happens automatically. For example, we can just use flip as if it had this type already: > flip (\n s -> show n <> s) "Ten" 10 "10Ten"  While we can choose any types for a, b and c, we have to be consistent. The type of the function we passed to flip had to be consistent with the types of the other arguments. That is why we passed the string "Ten" as the second argument, and the number 10 as the third. It would not work if the arguments were reversed: > flip (\n s -> show n <> s) 10 "Ten" Could not match type Int with type String  Notes On Indentation PureScript code is indentation-sensitive, just like Haskell, but unlike JavaScript. This means that the whitespace in your code is not meaningless, but rather is used to group regions of code, just like curly braces in C-like languages. If a declaration spans multiple lines, then any lines except the first must be indented past the indentation level of the first line. Therefore, the following is valid PureScript code: add x y z = x + y + z  But this is not valid code: add x y z = x + y + z  In the second case, the PureScript compiler will try to parse two declarations, one for each line. Generally, any declarations defined in the same block should be indented at the same level. For example, in PSCi, declarations in a let statement must be indented equally. This is valid: > :paste … x = 1 … y = 2 … ^D  but this is not: > :paste … x = 1 … y = 2 … ^D  Certain PureScript keywords (such as where, of and let) introduce a new block of code, in which declarations must be further-indented: example x y z = foo + bar where foo = x * y bar = y * z  Note how the declarations for foo and bar are indented past the declaration of example. The only exception to this rule is the where keyword in the initial module declaration at the top of a source file. Defining Our Types A good first step when tackling a new problem in PureScript is to write out type definitions for any values you will be working with. First, let's define a type for records in our address book: type Entry = { firstName :: String , lastName :: String , address :: Address }  This defines a type synonym called Entry - the type Entry is equivalent to the type on the right of the equals symbol: a record type with three fields - firstName, lastName and address. The two name fields will have type String, and the address field will have type Address, defined as follows: type Address = { street :: String , city :: String , state :: String }  Note that records can contain other records. Now let's define a third type synonym, for our address book data structure, which will be represented simply as a linked list of entries: type AddressBook = List Entry  Note that List Entry is not the same as Array Entry, which represents an array of entries. Type Constructors and Kinds List is an example of a type constructor. Values do not have the type List directly, but rather List a for some type a. That is, List takes a type argument a and constructs a new type List a. Note that just like function application, type constructors are applied to other types simply by juxtaposition: the type List Entry is in fact the type constructor List applied to the type Entry - it represents a list of entries. If we try to incorrectly define a value of type List (by using the type annotation operator ::), we will see a new type of error: > import Data.List > Nil :: List In a type-annotated expression x :: t, the type t must have kind Type  This is a kind error. Just like values are distinguished by their types, types are distinguished by their kinds, and just like ill-typed values result in type errors, ill-kinded types result in kind errors. There is a special kind called Type which represents the kind of all types which have values, like Number and String. There are also kinds for type constructors. For example, the kind Type -> Type represents a function from types to types, just like List. So the error here occurred because values are expected to have types with kind Type, but List has kind Type -> Type. To find out the kind of a type, use the :kind command in PSCi. For example: > :kind Number Type > import Data.List > :kind List Type -> Type > :kind List String Type  PureScript's kind system supports other interesting kinds, which we will see later in the book. Displaying Address Book Entries Let's write our first function, which will render an address book entry as a string. We start by giving the function a type. This is optional, but good practice, since it acts as a form of documentation. In fact, the PureScript compiler will give a warning if a top-level declaration does not contain a type annotation. A type declaration separates the name of a function from its type with the :: symbol: showEntry :: Entry -> String  This type signature says that showEntry is a function, which takes an Entry as an argument and returns a String. Here is the code for showEntry: showEntry entry = entry.lastName <> ", " <> entry.firstName <> ": " <> showAddress entry.address  This function concatenates the three fields of the Entry record into a single string, using the showAddress function to turn the record inside the address field into a String. showAddress is defined similarly: showAddress :: Address -> String showAddress addr = addr.street <> ", " <> addr.city <> ", " <> addr.state  A function definition begins with the name of the function, followed by a list of argument names. The result of the function is specified after the equals sign. Fields are accessed with a dot, followed by the field name. In PureScript, string concatenation uses the diamond operator (<>), instead of the plus operator like in JavaScript. Test Early, Test Often The PSCi interactive mode allows for rapid prototyping with immediate feedback, so let's use it to verify that our first few functions behave as expected. First, build the code you've written: $ spago build


Next, load PSCi, and use the import command to import your new module:

$spago repl > import Data.AddressBook  We can create an entry by using a record literal, which looks just like an anonymous object in JavaScript. > address = { street: "123 Fake St.", city: "Faketown", state: "CA" }  Now, try applying our function to the example: > showAddress address "123 Fake St., Faketown, CA"  Let's also test showEntry by creating an address book entry record containing our example address: > entry = { firstName: "John", lastName: "Smith", address: address } > showEntry entry "Smith, John: 123 Fake St., Faketown, CA"  Creating Address Books Now let's write some utility functions for working with address books. We will need a value which represents an empty address book: an empty list. emptyBook :: AddressBook emptyBook = empty  We will also need a function for inserting a value into an existing address book. We will call this function insertEntry. Start by giving its type: insertEntry :: Entry -> AddressBook -> AddressBook  This type signature says that insertEntry takes an Entry as its first argument, and an AddressBook as a second argument, and returns a new AddressBook. We don't modify the existing AddressBook directly. Instead, we return a new AddressBook which contains the same data. As such, AddressBook is an example of an immutable data structure. This is an important idea in PureScript - mutation is a side-effect of code, and inhibits our ability to reason effectively about its behavior, so we prefer pure functions and immutable data where possible. To implement insertEntry, we can use the Cons function from Data.List. To see its type, open PSCi and use the :type command: $ spago repl

> import Data.List
> :type Cons

forall a. a -> List a -> List a


This type signature says that Cons takes a value of some type a, and a list of elements of type a, and returns a new list with entries of the same type. Let's specialize this with a as our Entry type:

Entry -> List Entry -> List Entry


But List Entry is the same as AddressBook, so this is equivalent to

Entry -> AddressBook -> AddressBook


In our case, we already have the appropriate inputs: an Entry, and a AddressBook, so can apply Cons and get a new AddressBook, which is exactly what we wanted!

Here is our implementation of insertEntry:

insertEntry entry book = Cons entry book


This brings the two arguments entry and book into scope, on the left hand side of the equals symbol, and then applies the Cons function to create the result.

Curried Functions

Functions in PureScript take exactly one argument. While it looks like the insertEntry function takes two arguments, it is in fact an example of a curried function.

The -> operator in the type of insertEntry associates to the right, which means that the compiler parses the type as

Entry -> (AddressBook -> AddressBook)


That is, insertEntry is a function which returns a function! It takes a single argument, an Entry, and returns a new function, which in turn takes a single AddressBook argument and returns a new AddressBook.

This means that we can partially apply insertEntry by specifying only its first argument, for example. In PSCi, we can see the result type:

> :type insertEntry entry



As expected, the return type was a function. We can apply the resulting function to a second argument:

> :type (insertEntry entry) emptyBook


Note though that the parentheses here are unnecessary - the following is equivalent:

> :type insertEntry entry emptyBook


This is because function application associates to the left, and this explains why we can just specify function arguments one after the other, separated by whitespace.

Note that in the rest of the book, I will talk about things like "functions of two arguments". However, it is to be understood that this means a curried function, taking a first argument and returning another function.

Now consider the definition of insertEntry:

insertEntry :: Entry -> AddressBook -> AddressBook
insertEntry entry book = Cons entry book


If we explicitly parenthesize the right-hand side, we get (Cons entry) book. That is, insertEntry entry is a function whose argument is just passed along to the (Cons entry) function. But if two functions have the same result for every input, then they are the same function! So we can remove the argument book from both sides:

insertEntry :: Entry -> AddressBook -> AddressBook
insertEntry entry = Cons entry


But now, by the same argument, we can remove entry from both sides:

insertEntry :: Entry -> AddressBook -> AddressBook
insertEntry = Cons


This process is called eta conversion, and can be used (along with some other techniques) to rewrite functions in point-free form, which means functions defined without reference to their arguments.

In the case of insertEntry, eta conversion has resulted in a very clear definition of our function - "insertEntry is just cons on lists". However, it is arguable whether point-free form is better in general.

Property Accessors

One common pattern is to use a function to access individual fields (or "properties") of a record. An inline function to extract an Address from an Entry could be written as:

\entry -> entry.address


PureScript also allows property accessor shorthand, where an underscore acts as the anonymous function argument, so the inline function above is equivalent to:

_.address


This works with any number of levels or properties, so a function to extract the city associated with an Entry could be written as:

_.address.city


For example:

> address = { street: "123 Fake St.", city: "Faketown", state: "CA" }
> _.lastName entry
"Smith"

"Faketown"


The last function we need to implement for our minimal address book application will look up a person by name and return the correct Entry. This will be a nice application of building programs by composing small functions - a key idea from functional programming.

We can first filter the address book, keeping only those entries with the correct first and last names. Then we can simply return the head (i.e. first) element of the resulting list.

With this high-level specification of our approach, we can calculate the type of our function. First open PSCi, and find the types of the filter and head functions:

$spago repl > import Data.List > :type filter forall a. (a -> Boolean) -> List a -> List a > :type head forall a. List a -> Maybe a  Let's pick apart these two types to understand their meaning. filter is a curried function of two arguments. Its first argument is a function, which takes an element of the list and returns a Boolean value as a result. Its second argument is a list of elements, and the return value is another list. head takes a list as its argument, and returns a type we haven't seen before: Maybe a. Maybe a represents an optional value of type a, and provides a type-safe alternative to using null to indicate a missing value in languages like JavaScript. We will see it again in more detail in later chapters. The universally quantified types of filter and head can be specialized by the PureScript compiler, to the following types: filter :: (Entry -> Boolean) -> AddressBook -> AddressBook head :: AddressBook -> Maybe Entry  We know that we will need to pass the first and last names that we want to search for, as arguments to our function. We also know that we will need a function to pass to filter. Let's call this function filterEntry. filterEntry will have type Entry -> Boolean. The application filter filterEntry will then have type AddressBook -> AddressBook. If we pass the result of this function to the head function, we get our result of type Maybe Entry. Putting these facts together, a reasonable type signature for our function, which we will call findEntry, is: findEntry :: String -> String -> AddressBook -> Maybe Entry  This type signature says that findEntry takes two strings, the first and last names, and a AddressBook, and returns an optional Entry. The optional result will contain a value only if the name is found in the address book. And here is the definition of findEntry: findEntry firstName lastName book = head (filter filterEntry book) where filterEntry :: Entry -> Boolean filterEntry entry = entry.firstName == firstName && entry.lastName == lastName  Let's go over this code step by step. findEntry brings three names into scope: firstName, and lastName, both representing strings, and book, an AddressBook. The right hand side of the definition combines the filter and head functions: first, the list of entries is filtered, and the head function is applied to the result. The predicate function filterEntry is defined as an auxiliary declaration inside a where clause. This way, the filterEntry function is available inside the definition of our function, but not outside it. Also, it can depend on the arguments to the enclosing function, which is essential here because filterEntry uses the firstName and lastName arguments to filter the specified Entry. Note that, just like for top-level declarations, it was not necessary to specify a type signature for filterEntry. However, doing so is recommended as a form of documentation. Infix Function Application Most of the functions discussed so far used prefix function application, where the function name was put before the arguments. For example, when using the insertEntry function to add an Entry (john) to an empty AddressBook, we might write: > book1 = insertEntry john emptyBook  However, this chapter has also included examples of infix binary operators, such as the == operator in the definition of filterEntry, where the operator is put between the two arguments. These infix operators are actually defined in the PureScript source as infix aliases for their underlying prefix implementations. For example, == is defined as an infix alias for the prefix eq function with the line: infix 4 eq as ==  and therefore entry.firstName == firstName in filterEntry could be replaced with the eq entry.firstName firstName. We'll cover a few more examples of defining infix operators later in this section. There are situations where putting a prefix function in an infix position as an operator leads to more readable code. One example is the mod function: > mod 8 3 2  This is fine, but doesn't line up with common usage (in conversation, one might say "eight mod three"). Wrapping a prefix function in backticks () lets you use that it in infix position as an operator, e.g., > 8 mod 3 2  In the same way, wrapping insertEntry in backticks turns it into an infix operator, such that book1 and book2 below are equivalent: book1 = insertEntry john emptyBook book2 = john insertEntry emptyBook  We can make an AddressBook with multiple entries by using multiple applications of insertEntry as a prefix function (book3) or as an infix operator (book4) as shown below: book3 = insertEntry john (insertEntry peggy (insertEntry ned emptyBook)) book4 = john insertEntry (peggy insertEntry (ned insertEntry emptyBook))  We can also define an infix operator alias (or synonym) for insertEntry. We'll arbitrarily choose ++ for this operator, give it a precedence of 5, and make it right associative using infixr: infixr 5 insertEntry as ++  This new operator lets us rewrite the above book4 example as: book5 = john ++ (peggy ++ (ned ++ emptyBook))  and the right associativity of our new ++ operator lets us get rid of the parentheses without changing the meaning: book6 = john ++ peggy ++ ned ++ emptyBook  Another common technique for eliminating parens is to use apply's infix operator $, along with your standard prefix functions.

For example, the earlier book3 example could be rewritten as:

book7 = insertEntry john $insertEntry peggy$ insertEntry ned emptyBook


Substituting $ for parens is usually easier to type and (arguably) easier to read. A mnemonic to remember the meaning of this symbol is to think of the dollar sign as being drawn from two parens that are also being crossed-out, suggesting the parens are now unnecessary. Note that $ isn't special syntax that's hardcoded into the language. It's simply the infix operator for a regular function called apply, which is defined in the Prelude as follows:

apply :: forall a b. (a -> b) -> a -> b
apply f x = f x

infixr 0 apply as $ The apply function takes another function (of type (a -> b)) as its first argument and a value (of type a) as its second argument, then calls that function with that value. If it seems like this function doesn't contribute anything meaningful, you are absolutely correct! Your program is logically identical without it (see referential transparency). The syntactic utility of this function comes from the special properties assigned to its infix operator. $ is a right-associative (infixr), low precedence (0) operator, which lets us remove sets of parentheses for deeply-nested applications.

Another parens-busting opportunity for the $ operator is in our earlier findEntry function: findEntry firstName lastName book = head$ filter filterEntry book


We'll see an even more elegant way to rewrite this line with "function composition" in the next section.

If you'd like to use a concise infix operator alias as a prefix function, you can surround it in parentheses:

> 8 + 3
11

> (+) 8 3
11


Alternatively, operators can be partially applied by surrounding the expression with parentheses and using _ as an operand in an operator section. You can think of this as a more convenient way to create simple anonymous functions (although in the below example, we're then binding that anonymous function to a name, so it's not so anonymous anymore):

> add3 = (3 + _)
5


To summarize, the following are equivalent definitions of a function that adds 5 to its argument:

add5 x = 5 + x
add5 x = (+) 5 x
add5 x = 5 add x
add5   = \x -> 5 + x
add5 x = 5 (+) x  -- Yo Dawg, I herd you like infix, so we put infix in your infix!


Function Composition

Just like we were able to simplify the insertEntry function by using eta conversion, we can simplify the definition of findEntry by reasoning about its arguments.

Note that the book argument is passed to the filter filterEntry function, and the result of this application is passed to head. In other words, book is passed to the composition of the functions filter filterEntry and head.

In PureScript, the function composition operators are <<< and >>>. The first is "backwards composition", and the second is "forwards composition".

We can rewrite the right-hand side of findEntry using either operator. Using backwards-composition, the right-hand side would be

(head <<< filter filterEntry) book


In this form, we can apply the eta conversion trick from earlier, to arrive at the final form of findEntry:

findEntry firstName lastName = head <<< filter filterEntry
where
...


An equally valid right-hand side would be:

filter filterEntry >>> head


Either way, this gives a clear definition of the findEntry function: "findEntry is the composition of a filtering function and the head function".

I will let you make your own decision which definition is easier to understand, but it is often useful to think of functions as building blocks in this way - each function executing a single task, and solutions assembled using function composition.

Exercises

1. (Easy) Test your understanding of the findEntry function by writing down the types of each of its major subexpressions. For example, the type of the head function as used is specialized to AddressBook -> Maybe Entry. Note: There is no test for this exercise.
2. (Medium) Write a function findEntryByStreet :: String -> AddressBook -> Maybe Entry which looks up an Entry given a street address. Hint reusing the existing code in findEntry. Test your function in PSCi and by running spago test.
3. (Medium) Rewrite findEntryByStreet to replace filterEntry with the composition (using <<< or >>>) of: a property accessor (using the _. notation); and a function that tests whether its given string argument is equal to the given street address.
4. (Medium) Write a function isInBook which tests whether a name appears in a AddressBook, returning a Boolean value. Hint: Use PSCi to find the type of the Data.List.null function, which tests whether a list is empty or not.
5. (Difficult) Write a function removeDuplicates which removes "duplicate" address book entries. We'll consider entries duplicated if they share the same first and last names, while ignoring address fields. Hint: Use PSCi to find the type of the Data.List.nubByEq function, which removes duplicate elements from a list based on an equality predicate. Note that the first element in each set of duplicates (closest to list head) is the one that is kept.

Conclusion

In this chapter, we covered several new functional programming concepts:

• How to use the interactive mode PSCi to experiment with functions and test ideas.
• The role of types as both a correctness tool, and an implementation tool.
• The use of curried functions to represent functions of multiple arguments.
• Creating programs from smaller components by composition.
• Structuring code neatly using where expressions.
• How to avoid null values by using the Maybe type.
• Using techniques like eta conversion and function composition to refactor code into a clear specification.

In the following chapters, we'll build on these ideas.

Recursion, Maps And Folds

Chapter Goals

In this chapter, we will look at how recursive functions can be used to structure algorithms. Recursion is a basic technique used in functional programming, which we will use throughout this book.

We will also cover some standard functions from PureScript's standard libraries. We will see the map and fold functions, as well as some useful special cases, like filter and concatMap.

The motivating example for this chapter is a library of functions for working with a virtual filesystem. We will apply the techniques learned in this chapter to write functions which compute properties of the files represented by a model of a filesystem.

Project Setup

The source code for this chapter is contained in src/Data/Path.purs and test/Examples.purs. The Data.Path module contains a model of a virtual filesystem. You do not need to modify the contents of this module. Implement your solutions to the exercises in the Test.MySolutions module. Enable accompanying tests in the Test.Main module as you complete each exercise and check your work by running spago test.

The project has the following dependencies:

• maybe, which defines the Maybe type constructor
• arrays, which defines functions for working with arrays
• strings, which defines functions for working with JavaScript strings
• foldable-traversable, which defines functions for folding arrays and other data structures
• console, which defines functions for printing to the console

Introduction

Recursion is an important technique in programming in general, but particularly common in pure functional programming, because, as we will see in this chapter, recursion helps to reduce the mutable state in our programs.

Recursion is closely linked to the divide and conquer strategy: to solve a problem on certain inputs, we can break down the inputs into smaller parts, solve the problem on those parts, and then assemble a solution from the partial solutions.

Let's see some simple examples of recursion in PureScript.

Here is the usual factorial function example:

fact :: Int -> Int
fact n =
if n == 0 then
1
else
n * fact (n - 1)


Here, we can see how the factorial function is computed by reducing the problem to a subproblem - that of computing the factorial of a smaller integer. When we reach zero, the answer is immediate.

Here is another common example, which computes the Fibonacci function:

fib :: Int -> Int
fib n =
if n == 0 || n == 1 then
1
else
fib (n - 1) + fib (n - 2)


Again, this problem is solved by considering the solutions to subproblems. In this case, there are two subproblems, corresponding to the expressions fib (n - 1) and fib (n - 2). When these two subproblems are solved, we assemble the result by adding the partial results.

Note that, while the above examples of fact and fib work as intended, a more idiomatic implementation would use pattern matching instead of if/then/else. Pattern matching techniques are discussed in a later chapter.

Recursion on Arrays

We are not limited to defining recursive functions over the Int type! We will see recursive functions defined over a wide array of data types when we cover pattern matching later in the book, but for now, we will restrict ourselves to numbers and arrays.

Just as we branch based on whether the input is non-zero, in the array case, we will branch based on whether the input is non-empty. Consider this function, which computes the length of an array using recursion:

import Prelude

import Data.Array (null, tail)
import Data.Maybe (fromMaybe)

length :: forall a. Array a -> Int
length arr =
if null arr
then 0
else 1 + (length $fromMaybe []$ tail arr)


In this function, we use an if .. then .. else expression to branch based on the emptiness of the array. The null function returns true on an empty array. Empty arrays have length zero, and a non-empty array has a length that is one more than the length of its tail.

The tail function returns a Maybe wrapping the given array without its first element. If the array is empty (i.e. it doesn't have a tail) Nothing is returned. The fromMaybe function takes a default value and a Maybe value. If the latter is Nothing it returns the default, in the other case it returns the value wrapped by Just.

This example is obviously a very impractical way to find the length of an array in JavaScript, but should provide enough help to allow you to complete the following exercises:

Exercises

1. (Easy) Write a recursive function isEven which returns true if and only if its input is an even integer.
2. (Medium) Write a recursive function countEven which counts the number of even integers in an array. Hint: the function head (also available in Data.Array) can be used to find the first element in a non-empty array.

Maps

The map function is an example of a recursive function on arrays. It is used to transform the elements of an array by applying a function to each element in turn. Therefore, it changes the contents of the array, but preserves its shape (i.e. its length).

When we cover type classes later in the book we will see that the map function is an example of a more general pattern of shape-preserving functions which transform a class of type constructors called functors.

Let's try out the map function in PSCi:

$spago repl > import Prelude > map (\n -> n + 1) [1, 2, 3, 4, 5] [2, 3, 4, 5, 6]  Notice how map is used - we provide a function which should be "mapped over" the array in the first argument, and the array itself in its second. Infix Operators The map function can also be written between the mapping function and the array, by wrapping the function name in backticks: > (\n -> n + 1) map [1, 2, 3, 4, 5] [2, 3, 4, 5, 6]  This syntax is called infix function application, and any function can be made infix in this way. It is usually most appropriate for functions with two arguments. There is an operator which is equivalent to the map function when used with arrays, called <$>. This operator can be used infix like any other binary operator:

> (\n -> n + 1) <$> [1, 2, 3, 4, 5] [2, 3, 4, 5, 6]  Let's look at the type of map: > :type map forall a b f. Functor f => (a -> b) -> f a -> f b  The type of map is actually more general than we need in this chapter. For our purposes, we can treat map as if it had the following less general type: forall a b. (a -> b) -> Array a -> Array b  This type says that we can choose any two types, a and b, with which to apply the map function. a is the type of elements in the source array, and b is the type of elements in the target array. In particular, there is no reason why map has to preserve the type of the array elements. We can use map or <$> to transform integers to strings, for example:

> show <$> [1, 2, 3, 4, 5] ["1","2","3","4","5"]  Even though the infix operator <$> looks like special syntax, it is in fact just an alias for a regular PureScript function. The function is simply applied using infix syntax. In fact, the function can be used like a regular function by enclosing its name in parentheses. This means that we can used the parenthesized name (<$>) in place of map on arrays: > (<$>) show [1, 2, 3, 4, 5]
["1","2","3","4","5"]


Infix function names are defined as aliases for existing function names. For example, the Data.Array module defines an infix operator (..) as a synonym for the range function, as follows:

infix 8 range as ..


We can use this operator as follows:

> import Data.Array

> 1 .. 5
[1, 2, 3, 4, 5]

> show <$> (1 .. 5) ["1","2","3","4","5"]  Note: Infix operators can be a great tool for defining domain-specific languages with a natural syntax. However, used excessively, they can render code unreadable to beginners, so it is wise to exercise caution when defining any new operators. In the example above, we parenthesized the expression 1 .. 5, but this was actually not necessary, because the Data.Array module assigns a higher precedence level to the .. operator than that assigned to the <$> operator. In the example above, the precedence of the .. operator was defined as 8, the number after the infix keyword. This is higher than the precedence level of <$>, meaning that we do not need to add parentheses: > show <$> 1 .. 5
["1","2","3","4","5"]


If we wanted to assign an associativity (left or right) to an infix operator, we could do so with the infixl and infixr keywords instead.

Filtering Arrays

The Data.Array module provides another function filter, which is commonly used together with map. It provides the ability to create a new array from an existing array, keeping only those elements which match a predicate function.

For example, suppose we wanted to compute an array of all numbers between 1 and 10 which were even. We could do so as follows:

> import Data.Array

> filter (\n -> n mod 2 == 0) (1 .. 10)
[2,4,6,8,10]


Exercises

1. (Easy) Write a function squared which calculates the squares of an array of numbers. Hint: Use the map or <$> function. 2. (Easy) Write a function keepNonNegative which removes the negative numbers from an array of numbers. Hint: Use the filter function. 3. (Medium) • Define an infix synonym <$?> for filter. Note: Infix synonyms may not be defined in the REPL, but you can define it in a file.
• Write a keepNonNegativeRewrite function, which is the same as keepNonNegative, but replaces filter with your new infix operator <$?>. • Experiment with the precedence level and associativity of your operator in PSCi. Note: There are no unit tests for this step. Flattening Arrays Another standard function on arrays is the concat function, defined in Data.Array. concat flattens an array of arrays into a single array: > import Data.Array > :type concat forall a. Array (Array a) -> Array a > concat [[1, 2, 3], [4, 5], [6]] [1, 2, 3, 4, 5, 6]  There is a related function called concatMap which is like a combination of the concat and map functions. Where map takes a function from values to values (possibly of a different type), concatMap takes a function from values to arrays of values. Let's see it in action: > import Data.Array > :type concatMap forall a b. (a -> Array b) -> Array a -> Array b > concatMap (\n -> [n, n * n]) (1 .. 5) [1,1,2,4,3,9,4,16,5,25]  Here, we call concatMap with the function \n -> [n, n * n] which sends an integer to the array of two elements consisting of that integer and its square. The result is an array of ten integers: the integers from 1 to 5 along with their squares. Note how concatMap concatenates its results. It calls the provided function once for each element of the original array, generating an array for each. Finally, it collapses all of those arrays into a single array, which is its result. map, filter and concatMap form the basis for a whole range of functions over arrays called "array comprehensions". Array Comprehensions Suppose we wanted to find the factors of a number n. One simple way to do this would be by brute force: we could generate all pairs of numbers between 1 and n, and try multiplying them together. If the product was n, we would have found a pair of factors of n. We can perform this computation using an array comprehension. We will do so in steps, using PSCi as our interactive development environment. The first step is to generate an array of pairs of numbers below n, which we can do using concatMap. Let's start by mapping each number to the array 1 .. n: > pairs n = concatMap (\i -> 1 .. n) (1 .. n)  We can test our function > pairs 3 [1,2,3,1,2,3,1,2,3]  This is not quite what we want. Instead of just returning the second element of each pair, we need to map a function over the inner copy of 1 .. n which will allow us to keep the entire pair: > :paste … pairs' n = … concatMap (\i -> … map (\j -> [i, j]) (1 .. n) … ) (1 .. n) … ^D > pairs' 3 [[1,1],[1,2],[1,3],[2,1],[2,2],[2,3],[3,1],[3,2],[3,3]]  This is looking better. However, we are generating too many pairs: we keep both [1, 2] and [2, 1] for example. We can exclude the second case by making sure that j only ranges from i to n: > :paste … pairs'' n = … concatMap (\i -> … map (\j -> [i, j]) (i .. n) … ) (1 .. n) … ^D > pairs'' 3 [[1,1],[1,2],[1,3],[2,2],[2,3],[3,3]]  Great! Now that we have all of the pairs of potential factors, we can use filter to choose the pairs which multiply to give n: > import Data.Foldable > factors n = filter (\pair -> product pair == n) (pairs'' n) > factors 10 [[1,10],[2,5]]  This code uses the product function from the Data.Foldable module in the foldable-traversable library. Excellent! We've managed to find the correct set of factor pairs without duplicates. Do Notation However, we can improve the readability of our code considerably. map and concatMap are so fundamental, that they (or rather, their generalizations map and bind) form the basis of a special syntax called do notation. Note: Just like map and concatMap allowed us to write array comprehensions, the more general operators map and bind allow us to write so-called monad comprehensions. We'll see plenty more examples of monads later in the book, but in this chapter, we will only consider arrays. We can rewrite our factors function using do notation as follows: factors :: Int -> Array (Array Int) factors n = filter (\xs -> product xs == n)$ do
i <- 1 .. n
j <- i .. n
pure [i, j]


The keyword do introduces a block of code which uses do notation. The block consists of expressions of a few types:

• Expressions which bind elements of an array to a name. These are indicated with the backwards-facing arrow <-, with a name on the left, and an expression on the right whose type is an array.
• Expressions which do not bind elements of the array to names. The do result is an example of this kind of expression and is illustrated in the last line, pure [i, j].
• Expressions which give names to expressions, using the let keyword.

This new notation hopefully makes the structure of the algorithm clearer. If you mentally replace the arrow <- with the word "choose", you might read it as follows: "choose an element i between 1 and n, then choose an element j between i and n, and return [i, j]".

In the last line, we use the pure function. This function can be evaluated in PSCi, but we have to provide a type:

> pure [1, 2] :: Array (Array Int)
[[1, 2]]


In the case of arrays, pure simply constructs a singleton array. In fact, we could modify our factors function to use this form, instead of using pure:

factorsV2 :: Int -> Array (Array Int)
factorsV2 n = filter (\xs -> product xs == n) $do i <- 1 .. n j <- i .. n [[i, j]]  and the result would be the same. Guards One further change we can make to the factors function is to move the filter inside the array comprehension. This is possible using the guard function from the Control.MonadZero module (from the control package): import Control.MonadZero (guard) factorsV3 :: Int -> Array (Array Int) factorsV3 n = do i <- 1 .. n j <- i .. n guard$ i * j == n
pure [i, j]


Just like pure, we can apply the guard function in PSCi to understand how it works. The type of the guard function is more general than we need here:

> import Control.MonadZero

> :type guard
forall m. MonadZero m => Boolean -> m Unit


In our case, we can assume that PSCi reported the following type:

Boolean -> Array Unit


For our purposes, the following calculations tell us everything we need to know about the guard function on arrays:

> import Data.Array

> length $guard true 1 > length$ guard false
0


That is, if guard is passed an expression which evaluates to true, then it returns an array with a single element. If the expression evaluates to false, then its result is empty.

This means that if the guard fails, then the current branch of the array comprehension will terminate early with no results. This means that a call to guard is equivalent to using filter on the intermediate array. Depending on the application, you might prefer to use guard instead of a filter. Try the two definitions of factors to verify that they give the same results.

Exercises

1. (Easy) Write a function isPrime which tests if its integer argument is prime or not. Hint: Use the factors function.
2. (Medium) Write a function cartesianProduct which uses do notation to find the cartesian product of two arrays, i.e. the set of all pairs of elements a, b, where a is an element of the first array, and b is an element of the second.
3. (Medium) Write a function triples :: Int -> Array (Array Int) which takes a number n and returns all Pythagorean triples whose components (the a, b and c values) are each less than or equal to n. A Pythagorean triple is an array of numbers [a, b, c] such that a² + b² = c². Hint: Use the guard function in an array comprehension.
4. (Difficult) Write a function factorize which produces the prime factorization of n, i.e. the array of prime integers whose product is n. Hint: for an integer greater than 1, break the problem down into two subproblems: finding the first factor, and finding the remaining factors.

Folds

Left and right folds over arrays provide another class of interesting functions which can be implemented using recursion.

Start by importing the Data.Foldable module, and inspecting the types of the foldl and foldr functions using PSCi:

> import Data.Foldable

> :type foldl
forall a b f. Foldable f => (b -> a -> b) -> b -> f a -> b

> :type foldr
forall a b f. Foldable f => (a -> b -> b) -> b -> f a -> b


These types are actually more general than we are interested in right now. For the purposes of this chapter, we can assume that PSCi had given the following (more specific) answer:

> :type foldl
forall a b. (b -> a -> b) -> b -> Array a -> b

> :type foldr
forall a b. (a -> b -> b) -> b -> Array a -> b


In both of these cases, the type a corresponds to the type of elements of our array. The type b can be thought of as the type of an "accumulator", which will accumulate a result as we traverse the array.

The difference between the foldl and foldr functions is the direction of the traversal. foldl folds the array "from the left", whereas foldr folds the array "from the right".

Let's see these functions in action. Let's use foldl to sum an array of integers. The type a will be Int, and we can also choose the result type b to be Int. We need to provide three arguments: a function Int -> Int -> Int, which will add the next element to the accumulator, an initial value for the accumulator of type Int, and an array of Ints to add. For the first argument, we can just use the addition operator, and the initial value of the accumulator will be zero:

> foldl (+) 0 (1 .. 5)
15


In this case, it didn't matter whether we used foldl or foldr, because the result is the same, no matter what order the additions happen in:

> foldr (+) 0 (1 .. 5)
15


Let's write an example where the choice of folding function does matter, in order to illustrate the difference. Instead of the addition function, let's use string concatenation to build a string:

> foldl (\acc n -> acc <> show n) "" [1,2,3,4,5]
"12345"

> foldr (\n acc -> acc <> show n) "" [1,2,3,4,5]
"54321"


This illustrates the difference between the two functions. The left fold expression is equivalent to the following application:

((((("" <> show 1) <> show 2) <> show 3) <> show 4) <> show 5)


whereas the right fold is equivalent to this:

((((("" <> show 5) <> show 4) <> show 3) <> show 2) <> show 1)


Tail Recursion

Recursion is a powerful technique for specifying algorithms, but comes with a problem: evaluating recursive functions in JavaScript can lead to stack overflow errors if our inputs are too large.

It is easy to verify this problem, with the following code in PSCi:

> f 0 = 0
> f n = 1 + f (n - 1)

> f 10
10

> f 100000
RangeError: Maximum call stack size exceeded


This is a problem. If we are going to adopt recursion as a standard technique from functional programming, then we need a way to deal with possibly unbounded recursion.

PureScript provides a partial solution to this problem in the form of tail recursion optimization.

Note: more complete solutions to the problem can be implemented in libraries using so-called trampolining, but that is beyond the scope of this chapter. The interested reader can consult the documentation for the free and tailrec packages.

The key observation which enables tail recursion optimization is the following: a recursive call in tail position to a function can be replaced with a jump, which does not allocate a stack frame. A call is in tail position when it is the last call made before a function returns. This is the reason why we observed a stack overflow in the example - the recursive call to f was not in tail position.

In practice, the PureScript compiler does not replace the recursive call with a jump, but rather replaces the entire recursive function with a while loop.

Here is an example of a recursive function with all recursive calls in tail position:

factTailRec :: Int -> Int -> Int
factTailRec 0 acc = acc
factTailRec n acc = factTailRec (n - 1) (acc * n)


Notice that the recursive call to factTailRec is the last thing that happens in this function - it is in tail position.

Accumulators

One common way to turn a function which is not tail recursive into a tail recursive function is to use an accumulator parameter. An accumulator parameter is an additional parameter which is added to a function which accumulates a return value, as opposed to using the return value to accumulate the result.

For example, consider again the length function presented in the beginning of the chapter:

length :: forall a. Array a -> Int
length arr =
if null arr
then 0
else 1 + (length $fromMaybe []$ tail arr)


This implementation is not tail recursive, so the generated JavaScript will cause a stack overflow when executed on a large input array. However, we can make it tail recursive, by introducing a second function argument to accumulate the result instead:

lengthTailRec :: forall a. Array a -> Int
lengthTailRec arr = length' arr 0
where
length' :: Array a -> Int -> Int
length' arr' acc =
if null arr'
then acc
else length' (fromMaybe [] $tail arr') (acc + 1)  In this case, we delegate to the helper function length', which is tail recursive - its only recursive call is in the last case, and is in tail position. This means that the generated code will be a while loop, and will not blow the stack for large inputs. To understand the implementation of lengthTailRec, note that the helper function length' essentially uses the accumulator parameter to maintain an additional piece of state - the partial result. It starts out at 0, and grows by adding 1 for every element in the input array. Note also that while we might think of the accumulator as "state", there is no direct mutation going on. Prefer Folds to Explicit Recursion If we can write our recursive functions using tail recursion, then we can benefit from tail recursion optimization, so it becomes tempting to try to write all of our functions in this form. However, it is often easy to forget that many functions can be written directly as a fold over an array or similar data structure. Writing algorithms directly in terms of combinators such as map and fold has the added advantage of code simplicity - these combinators are well-understood, and as such, communicate the intent of the algorithm much better than explicit recursion. For example, we can reverse an array using foldr: > import Data.Foldable > :paste … reverse :: forall a. Array a -> Array a … reverse = foldr (\x xs -> xs <> [x]) [] … ^D > reverse [1, 2, 3] [3,2,1]  Writing reverse in terms of foldl will be left as an exercise for the reader. Exercises 1. (Easy) Write a function allTrue which uses foldl to test whether an array of boolean values are all true. 2. (Medium - No Test) Characterize those arrays xs for which the function foldl (==) false xs returns true. In other words, complete the sentence: "The function returns true when xs contains ..." 3. (Medium) Write a function fibTailRec which is the same as fib but in tail recursive form. Hint: Use an accumulator parameter. 4. (Medium) Write reverse in terms of foldl. A Virtual Filesystem In this section, we're going to apply what we've learned, writing functions which will work with a model of a filesystem. We will use maps, folds and filters to work with a predefined API. The Data.Path module defines an API for a virtual filesystem, as follows: • There is a type Path which represents a path in the filesystem. • There is a path root which represents the root directory. • The ls function enumerates the files in a directory. • The filename function returns the file name for a Path. • The size function returns the file size for a Path which represents a file. • The isDirectory function tests whether a Path is a file or a directory. In terms of types, we have the following type definitions: root :: Path ls :: Path -> Array Path filename :: Path -> String size :: Path -> Maybe Int isDirectory :: Path -> Boolean  We can try out the API in PSCi: $ spago repl

> import Data.Path

> root
/

> isDirectory root
true

> ls root
[/bin/,/etc/,/home/]


The Test.Solutions module defines functions which use the Data.Path API. You do not need to modify the Data.Path module, or understand its implementation. We will work entirely in the Test.Solutions module.

Listing All Files

Let's write a function which performs a deep enumeration of all files inside a directory. This function will have the following type:

allFiles :: Path -> Array Path


We can define this function by recursion. First, we can use ls to enumerate the immediate children of the directory. For each child, we can recursively apply allFiles, which will return an array of paths. concatMap will allow us to apply allFiles and flatten the results at the same time.

Finally, we use the cons operator : to include the current file:

allFiles file = file : concatMap allFiles (ls file)


Note: the cons operator : actually has poor performance on immutable arrays, so it is not recommended in general. Performance can be improved by using other data structures, such as linked lists and sequences.

Let's try this function in PSCi:

> import Test.Solutions
> import Data.Path

> allFiles root

[/,/bin/,/bin/cp,/bin/ls,/bin/mv,/etc/,/etc/hosts, ...]


Great! Now let's see if we can write this function using an array comprehension using do notation.

Recall that a backwards arrow corresponds to choosing an element from an array. The first step is to choose an element from the immediate children of the argument. Then we simply call the function recursively for that file. Since we are using do notation, there is an implicit call to concatMap which concatenates all of the recursive results.

Here is the new version:

allFiles' :: Path -> Array Path
allFiles' file = file : do
child <- ls file
allFiles' child


Try out the new version in PSCi - you should get the same result. I'll let you decide which version you find clearer.

Exercises

1. (Easy) Write a function onlyFiles which returns all files (not directories) in all subdirectories of a directory.

2. (Medium) Write a function whereIs to search for a file by name. The function should return a value of type Maybe Path, indicating the directory containing the file, if it exists. It should behave as follows:

> whereIs root "ls"
Just (/bin/)

> whereIs root "cat"
Nothing


Hint: Try to write this function as an array comprehension using do notation.

3. (Difficult) Write a function largestSmallest which takes a Path and returns an array containing the single largest and single smallest files in the Path. Note: consider the cases where there are zero or one files in the Path by returning an empty array or a one-element array respectively.

Conclusion

In this chapter, we covered the basics of recursion in PureScript, as a means of expressing algorithms concisely. We also introduced user-defined infix operators, standard functions on arrays such as maps, filters and folds, and array comprehensions which combine these ideas. Finally, we showed the importance of using tail recursion in order to avoid stack overflow errors, and how to use accumulator parameters to convert functions to tail recursive form.

Pattern Matching

Chapter Goals

This chapter will introduce two new concepts: algebraic data types, and pattern matching. We will also briefly cover an interesting feature of the PureScript type system: row polymorphism.

Pattern matching is a common technique in functional programming and allows the developer to write compact functions which express potentially complex ideas, by breaking their implementation down into multiple cases.

Algebraic data types are a feature of the PureScript type system which enable a similar level of expressiveness in the language of types - they are closely related to pattern matching.

The goal of the chapter will be to write a library to describe and manipulate simple vector graphics using algebraic types and pattern matching.

Project Setup

The source code for this chapter is defined in the file src/Data/Picture.purs

The project uses some packages which we have already seen, and adds the following new dependency:

• math, which provides access to the JavaScript Math module.

The Data.Picture module defines a data type Shape for simple shapes, and a type Picture for collections of shapes, along with functions for working with those types.

The module imports the Data.Foldable module, which provides functions for folding data structures:

module Data.Picture where

import Prelude
import Data.Foldable (foldl)
import Data.Number (infinity)


The Data.Picture module also imports the Math module, but this time using the as keyword:

import Math as Math


This makes the types and functions in that module available for use, but only by using the qualified name, like Math.max. This can be useful to avoid overlapping imports, or just to make it clearer which modules certain things are imported from.

Note: it is not necessary to use the same module name as the original module for a qualified import. Shorter qualified names like import Math as M are possible, and quite common.

Simple Pattern Matching

Let's begin by looking at an example. Here is a function which computes the greatest common divisor of two integers using pattern matching:

gcd :: Int -> Int -> Int
gcd n 0 = n
gcd 0 m = m
gcd n m = if n > m
then gcd (n - m) m
else gcd n (m - n)


This algorithm is called the Euclidean Algorithm. If you search for its definition online, you will likely find a set of mathematical equations which look a lot like the code above. This is one benefit of pattern matching: it allows you to define code by cases, writing simple, declarative code which looks like a specification of a mathematical function.

A function written using pattern matching works by pairing sets of conditions with their results. Each line is called an alternative or a case. The expressions on the left of the equals sign are called patterns, and each case consists of one or more patterns, separated by spaces. Cases describe which conditions the arguments must satisfy before the expression on the right of the equals sign should be evaluated and returned. Each case is tried in order, and the first case whose patterns match their inputs determines the return value.

For example, the gcd function is evaluated using the following steps:

• The first case is tried: if the second argument is zero, the function returns n (the first argument).
• If not, the second case is tried: if the first argument is zero, the function returns m (the second argument).
• Otherwise, the function evaluates and returns the expression in the last line.

Note that patterns can bind values to names - each line in the example binds one or both of the names n and m to the input values. As we learn about different kinds of patterns, we will see that different types of patterns correspond to different ways to choose names from the input arguments.

Simple Patterns

The example code above demonstrates two types of patterns:

• Integer literals patterns, which match something of type Int, only if the value matches exactly.
• Variable patterns, which bind their argument to a name

There are other types of simple patterns:

• Number, String, Char and Boolean literals
• Wildcard patterns, indicated with an underscore (_), which match any argument, and which do not bind any names.

Here are two more examples which demonstrate using these simple patterns:

fromString :: String -> Boolean
fromString "true" = true
fromString _      = false

toString :: Boolean -> String
toString true  = "true"
toString false = "false"


Try these functions in PSCi.

Guards

In the Euclidean algorithm example, we used an if .. then .. else expression to switch between the two alternatives when m > n and m <= n. Another option in this case would be to use a guard.

A guard is a boolean-valued expression which must be satisfied in addition to the constraints imposed by the patterns. Here is the Euclidean algorithm rewritten to use a guard:

gcdV2 :: Int -> Int -> Int
gcdV2 n 0 = n
gcdV2 0 n = n
gcdV2 n m | n > m     = gcdV2 (n - m) m
| otherwise = gcdV2 n (m - n)


In this case, the third line uses a guard to impose the extra condition that the first argument is strictly larger than the second.

As this example demonstrates, guards appear on the left of the equals symbol, separated from the list of patterns by a pipe character (|).

Exercises

1. (Easy) Write the factorial function using pattern matching. Hint: Consider the two corner cases of zero and non-zero inputs. Note: This is a repeat of an example from the previous chapter, but see if you can rewrite it here on your own.
2. (Medium) Write a function binomial which finds the coefficient of the x^kth term in the polynomial expansion of (1 + x)^n. This is the same as the number of ways to choose a subset of k elements from a set of n elements. Use the formula n! / k! (n - k)!, where ! is the factorial function written earlier. Hint: Use pattern matching to handle corner cases.
3. (Medium) Write a function pascal which uses Pascals Rule for computing the same binomial coefficients as the previous exercise.

Array Patterns

Array literal patterns provide a way to match arrays of a fixed length. For example, suppose we want to write a function isEmpty which identifies empty arrays. We could do this by using an empty array pattern ([]) in the first alternative:

isEmpty :: forall a. Array a -> Boolean
isEmpty [] = true
isEmpty _ = false


Here is another function which matches arrays of length five, binding each of its five elements in a different way:

takeFive :: Array Int -> Int
takeFive [0, 1, a, b, _] = a * b
takeFive _ = 0


The first pattern only matches arrays with five elements, whose first and second elements are 0 and 1 respectively. In that case, the function returns the product of the third and fourth elements. In every other case, the function returns zero. For example, in PSCi:

> :paste
… takeFive [0, 1, a, b, _] = a * b
… takeFive _ = 0
… ^D

> takeFive [0, 1, 2, 3, 4]
6

> takeFive [1, 2, 3, 4, 5]
0

> takeFive []
0


Array literal patterns allow us to match arrays of a fixed length, but PureScript does not provide any means of matching arrays of an unspecified length, since destructuring immutable arrays in these sorts of ways can lead to poor performance. If you need a data structure which supports this sort of matching, the recommended approach is to use Data.List. Other data structures exist which provide improved asymptotic performance for different operations.

Record Patterns and Row Polymorphism

Record patterns are used to match - you guessed it - records.

Record patterns look just like record literals, but instead of values on the right of the colon, we specify a binder for each field.

For example: this pattern matches any record which contains fields called first and last, and binds their values to the names x and y respectively:

showPerson :: { first :: String, last :: String } -> String
showPerson { first: x, last: y } = y <> ", " <> x


Record patterns provide a good example of an interesting feature of the PureScript type system: row polymorphism. Suppose we had defined showPerson without a type signature above. What would its inferred type have been? Interestingly, it is not the same as the type we gave:

> showPerson { first: x, last: y } = y <> ", " <> x

> :type showPerson
forall r. { first :: String, last :: String | r } -> String


What is the type variable r here? Well, if we try showPerson in PSCi, we see something interesting:

> showPerson { first: "Phil", last: "Freeman" }
"Freeman, Phil"

> showPerson { first: "Phil", last: "Freeman", location: "Los Angeles" }
"Freeman, Phil"


We are able to append additional fields to the record, and the showPerson function will still work. As long as the record contains the first and last fields of type String, the function application is well-typed. However, it is not valid to call showPerson with too few fields:

> showPerson { first: "Phil" }

Type of expression lacks required label "last"


We can read the new type signature of showPerson as "takes any record with first and last fields which are Strings and any other fields, and returns a String".

This function is polymorphic in the row r of record fields, hence the name row polymorphism.

Note that we could have also written

> showPerson p = p.last <> ", " <> p.first


and PSCi would have inferred the same type.

Record Puns

Recall that the showPerson function matches a record inside its argument, binding the first and last fields to values named x and y. We could alternatively just reuse the field names themselves, and simplify this sort of pattern match as follows:

showPersonV2 :: { first :: String, last :: String } -> String
showPersonV2 { first, last } = last <> ", " <> first


Here, we only specify the names of the fields, and we do not need to specify the names of the values we want to introduce. This is called a record pun.

It is also possible to use record puns to construct records. For example, if we have values named first and last in scope, we can construct a person record using { first, last }:

unknownPerson :: { first :: String, last :: String }
unknownPerson = { first, last }
where
first = "Jane"
last  = "Doe"


This may improve readability of code in some circumstances.

Nested Patterns

Array patterns and record patterns both combine smaller patterns to build larger patterns. For the most part, the examples above have only used simple patterns inside array patterns and record patterns, but it is important to note that patterns can be arbitrarily nested, which allows functions to be defined using conditions on potentially complex data types.

For example, this code combines two record patterns:

type Address = { street :: String, city :: String }

livesInLA :: Person -> Boolean
livesInLA { address: { city: "Los Angeles" } } = true
livesInLA _ = false


Named Patterns

Patterns can be named to bring additional names into scope when using nested patterns. Any pattern can be named by using the @ symbol.

For example, this function sorts two-element arrays, naming the two elements, but also naming the array itself:

sortPair :: Array Int -> Array Int
sortPair arr@[x, y]
| x <= y = arr
| otherwise = [y, x]
sortPair arr = arr


This way, we save ourselves from allocating a new array if the pair is already sorted. Note that if the input array does not contain exactly two elements, then this function simply returns it unchanged, even if it's unsorted.

Exercises

1. (Easy) Write a function sameCity which uses record patterns to test whether two Person records belong to the same city.
2. (Medium) What is the most general type of the sameCity function, taking into account row polymorphism? What about the livesInLA function defined above? Note: There is no test for this exercise.
3. (Medium) Write a function fromSingleton which uses an array literal pattern to extract the sole member of a singleton array. If the array is not a singleton, your function should return a provided default value. Your function should have type forall a. a -> Array a -> a

Case Expressions

Patterns do not only appear in top-level function declarations. It is possible to use patterns to match on an intermediate value in a computation, using a case expression. Case expressions provide a similar type of utility to anonymous functions: it is not always desirable to give a name to a function, and a case expression allows us to avoid naming a function just because we want to use a pattern.

Here is an example. This function computes "longest zero suffix" of an array (the longest suffix which sums to zero):

import Data.Array (tail)
import Data.Foldable (sum)
import Data.Maybe (fromMaybe)

lzs :: Array Int -> Array Int
lzs [] = []
lzs xs = case sum xs of
0 -> xs
$spago repl > import Data.Picture > showPicture [ Line { x: 0.0, y: 0.0 } { x: 1.0, y: 1.0 } ] ["Line [start: (0.0, 0.0), end: (1.0, 1.0)]"]  Computing Bounding Rectangles The example code for this module contains a function bounds which computes the smallest bounding rectangle for a Picture. The Bounds type defines a bounding rectangle. type Bounds = { top :: Number , left :: Number , bottom :: Number , right :: Number }  bounds uses the foldl function from Data.Foldable to traverse the array of Shapes in a Picture, and accumulate the smallest bounding rectangle: bounds :: Picture -> Bounds bounds = foldl combine emptyBounds where combine :: Bounds -> Shape -> Bounds combine b shape = union (shapeBounds shape) b  In the base case, we need to find the smallest bounding rectangle of an empty Picture, and the empty bounding rectangle defined by emptyBounds suffices. The accumulating function combine is defined in a where block. combine takes a bounding rectangle computed from foldl's recursive call, and the next Shape in the array, and uses the union function to compute the union of the two bounding rectangles. The shapeBounds function computes the bounds of a single shape using pattern matching. Exercises 1. (Medium) Extend the vector graphics library with a new operation area which computes the area of a Shape. For the purpose of this exercise, the area of a line or a piece of text is assumed to be zero. 2. (Difficult) Extend the Shape type with a new data constructor Clipped, which clips another Picture to a rectangle. Extend the shapeBounds function to compute the bounds of a clipped picture. Note that this makes Shape into a recursive data type. Conclusion In this chapter, we covered pattern matching, a basic but powerful technique from functional programming. We saw how to use simple patterns as well as array and record patterns to match parts of deep data structures. This chapter also introduced algebraic data types, which are closely related to pattern matching. We saw how algebraic data types allow concise descriptions of data structures, and provide a modular way to extend data types with new operations. Finally, we covered row polymorphism, a powerful type of abstraction which allows many idiomatic JavaScript functions to be given a type. In the rest of the book, we will use ADTs and pattern matching extensively, so it will pay dividends to become familiar with them now. Try creating your own algebraic data types and writing functions to consume them using pattern matching. Type Classes Chapter Goals This chapter will introduce a powerful form of abstraction which is enabled by PureScript's type system - type classes. This motivating example for this chapter will be a library for hashing data structures. We will see how the machinery of type classes allow us to hash complex data structures without having to think directly about the structure of the data itself. We will also see a collection of standard type classes from PureScript's Prelude and standard libraries. PureScript code leans heavily on the power of type classes to express ideas concisely, so it will be beneficial to familiarize yourself with these classes. Project Setup The source code for this chapter is defined in the file src/Data/Hashable.purs. The project has the following dependencies: • maybe, which defines the Maybe data type, which represents optional values. • tuples, which defines the Tuple data type, which represents pairs of values. • either, which defines the Either data type, which represents disjoint unions. • strings, which defines functions which operate on strings. • functions, which defines some helper functions for defining PureScript functions. The module Data.Hashable imports several modules provided by these packages. Show Me! Our first simple example of a type class is provided by a function we've seen several times already: the show function, which takes a value and displays it as a string. show is defined by a type class in the Prelude module called Show, which is defined as follows: class Show a where show :: a -> String  This code declares a new type class called Show, which is parameterized by the type variable a. A type class instance contains implementations of the functions defined in a type class, specialized to a particular type. For example, here is the definition of the Show type class instance for Boolean values, taken from the Prelude: instance showBoolean :: Show Boolean where show true = "true" show false = "false"  This code declares a type class instance called showBoolean - in PureScript, type class instances are named to aid the readability of the generated JavaScript. We say that the Boolean type belongs to the Show type class. We can try out the Show type class in PSCi, by showing a few values with different types: > import Prelude > show true "true" > show 1.0 "1.0" > show "Hello World" "\"Hello World\""  These examples demonstrate how to show values of various primitive types, but we can also show values with more complicated types: > import Data.Tuple > show (Tuple 1 true) "(Tuple 1 true)" > import Data.Maybe > show (Just "testing") "(Just \"testing\")"  The output of show should be a string that you can paste back into the repl (or .purs file) to recreate the item being shown. Here we'll use logShow, which just calls show then log, to render the string without quotes. Ignore the unit print - that will covered in Chapter 8 when we examine Effects, like log. > import Effect.Console > logShow (Tuple 1 true) (Tuple 1 true) unit > logShow (Just "testing") (Just "testing") unit  If we try to show a value of type Data.Either, we get an interesting error message: > import Data.Either > show (Left 10) The inferred type forall a. Show a => String has type variables which are not mentioned in the body of the type. Consider adding a type annotation.  The problem here is not that there is no Show instance for the type we intended to show, but rather that PSCi was unable to infer the type. This is indicated by the unknown type a in the inferred type. We can annotate the expression with a type, using the :: operator, so that PSCi can choose the correct type class instance: > show (Left 10 :: Either Int String) "(Left 10)"  Some types do not have a Show instance defined at all. One example of this is the function type ->. If we try to show a function from Int to Int, we get an appropriate error message from the type checker: > import Prelude > show$ \n -> n + 1

No type class instance was found for

Data.Show.Show (Int -> Int)


Exercises

1. (Easy) Define a Show instance for Point. Match the same output as the showPoint function from the previous chapter. Note: Point is now a newtype (instead of a type synonym), which allows us to customize how to show it. Otherwise, we'd be stuck with the default Show instance for records.

newtype Point
= Point
{ x :: Number
, y :: Number
}


Common Type Classes

In this section, we'll look at some standard type classes defined in the Prelude and standard libraries. These type classes form the basis of many common patterns of abstraction in idiomatic PureScript code, so a basic understanding of their functions is highly recommended.

Eq

The Eq type class defines the eq function, which tests two values for equality. The == operator is actually just an alias for eq.

class Eq a where
eq :: a -> a -> Boolean


Note that in either case, the two arguments must have the same type: it does not make sense to compare two values of different types for equality.

Try out the Eq type class in PSCi:

> 1 == 2
false

> "Test" == "Test"
true


Ord

The Ord type class defines the compare function, which can be used to compare two values, for types which support ordering. The comparison operators < and > along with their non-strict companions <= and >=, can be defined in terms of compare.

data Ordering = LT | EQ | GT

class Eq a <= Ord a where
compare :: a -> a -> Ordering


The compare function compares two values, and returns an Ordering, which has three alternatives:

• LT - if the first argument is less than the second.
• EQ - if the first argument is equal to the second.
• GT - if the first argument is greater than the second.

Again, we can try out the compare function in PSCi:

> compare 1 2
LT

> compare "A" "Z"
LT


Field

The Field type class identifies those types which support numeric operators such as addition, subtraction, multiplication and division. It is provided to abstract over those operators, so that they can be reused where appropriate.

Note: Just like the Eq and Ord type classes, the Field type class has special support in the PureScript compiler, so that simple expressions such as 1 + 2 * 3 get translated into simple JavaScript, as opposed to function calls which dispatch based on a type class implementation.

class EuclideanRing a <= Field a


The Field type class is composed from several more general superclasses. This allows us to talk abstractly about types which support some but not all of the Field operations. For example, a type of natural numbers would be closed under addition and multiplication, but not necessarily under subtraction, so that type might have an instance of the Semiring class (which is a superclass of Num), but not an instance of Ring or Field.

Superclasses will be explained later in this chapter, but the full numeric type class hierarchy (cheatsheet) is beyond the scope of this chapter. The interested reader is encouraged to read the documentation for the superclasses of Field in prelude.

Semigroups and Monoids

The Semigroup type class identifies those types which support an append operation to combine two values:

class Semigroup a where
append :: a -> a -> a


Strings form a semigroup under regular string concatenation, and so do arrays. Several other standard instances are provided by the prelude package.

The <> concatenation operator, which we have already seen, is provided as an alias for append.

The Monoid type class (provided by the prelude package) extends the Semigroup type class with the concept of an empty value, called mempty:

class Semigroup m <= Monoid m where
mempty :: m


Again, strings and arrays are simple examples of monoids.

A Monoid type class instance for a type describes how to accumulate a result with that type, by starting with an "empty" value, and combining new results. For example, we can write a function which concatenates an array of values in some monoid by using a fold. In PSCi:

> import Prelude
> import Data.Monoid
> import Data.Foldable

> foldl append mempty ["Hello", " ", "World"]
"Hello World"

> foldl append mempty [[1, 2, 3], [4, 5], [6]]
[1,2,3,4,5,6]


The prelude package provides many examples of monoids and semigroups, which we will use in the rest of the book.

Foldable

If the Monoid type class identifies those types which act as the result of a fold, then the Foldable type class identifies those type constructors which can be used as the source of a fold.

The Foldable type class is provided in the foldable-traversable package, which also contains instances for some standard containers such as arrays and Maybe.

The type signatures for the functions belonging to the Foldable class are a little more complicated than the ones we've seen so far:

class Foldable f where
foldr :: forall a b. (a -> b -> b) -> b -> f a -> b
foldl :: forall a b. (b -> a -> b) -> b -> f a -> b
foldMap :: forall a m. Monoid m => (a -> m) -> f a -> m


It is instructive to specialize to the case where f is the array type constructor. In this case, we can replace f a with Array a for any a, and we notice that the types of foldl and foldr become the types that we saw when we first encountered folds over arrays.

What about foldMap? Well, that becomes forall a m. Monoid m => (a -> m) -> Array a -> m. This type signature says that we can choose any type m for our result type, as long as that type is an instance of the Monoid type class. If we can provide a function which turns our array elements into values in that monoid, then we can accumulate over our array using the structure of the monoid, and return a single value.

Let's try out foldMap in PSCi:

> import Data.Foldable

> foldMap show [1, 2, 3, 4, 5]
"12345"


Here, we choose the monoid for strings, which concatenates strings together, and the show function which renders an Int as a String. Then, passing in an array of integers, we see that the results of showing each integer have been concatenated into a single String.

But arrays are not the only types which are foldable. foldable-traversable also defines Foldable instances for types like Maybe and Tuple, and other libraries like lists define Foldable instances for their own data types. Foldable captures the notion of an ordered container.

Functor, and Type Class Laws

The Prelude also defines a collection of type classes which enable a functional style of programming with side-effects in PureScript: Functor, Applicative and Monad. We will cover these abstractions later in the book, but for now, let's look at the definition of the Functor type class, which we have seen already in the form of the map function:

class Functor f where
map :: forall a b. (a -> b) -> f a -> f b


The map function (and its alias <$>) allows a function to be "lifted" over a data structure. The precise definition of the word "lifted" here depends on the data structure in question, but we have already seen its behavior for some simple types: > import Prelude > map (\n -> n < 3) [1, 2, 3, 4, 5] [true, true, false, false, false] > import Data.Maybe > import Data.String (length) > map length (Just "testing") (Just 7)  How can we understand the meaning of the map function, when it acts on many different structures, each in a different way? Well, we can build an intuition that the map function applies the function it is given to each element of a container, and builds a new container from the results, with the same shape as the original. But how do we make this concept precise? Type class instances for Functor are expected to adhere to a set of laws, called the functor laws: • map id xs = xs • map g (map f xs) = map (g <<< f) xs The first law is the identity law. It states that lifting the identity function (the function which returns its argument unchanged) over a structure just returns the original structure. This makes sense since the identity function does not modify its input. The second law is the composition law. It states that mapping one function over a structure, and then mapping a second, is the same thing as mapping the composition of the two functions over the structure. Whatever "lifting" means in the general sense, it should be true that any reasonable definition of lifting a function over a data structure should obey these rules. Many standard type classes come with their own set of similar laws. The laws given to a type class give structure to the functions of that type class and allow us to study its instances in generality. The interested reader can research the laws ascribed to the standard type classes that we have seen already. Deriving Instances Rather than writing instances manually, you can let the compiler do most of the work for you. Take a look at this Type Class Deriving guide. That information will help you solve the following exercises. Exercises The following newtype represents a complex number: newtype Complex = Complex { real :: Number , imaginary :: Number }  1. (Easy) Define a Show instance for Complex. Match the output format expected by the tests (e.g. 1.2+3.4i, 5.6-7.8i, etc.). 2. (Easy) Derive an Eq instance for Complex. Note: You may instead write this instance manually, but why do more work if you don't have to? 3. (Medium) Define a Semiring instance for Complex. Note: You can use wrap and over2 from Data.Newtype to create a more concise solution. If you do so, you will also need to import class Newtype from Data.Newtype and derive a Newtype instance for Complex. 4. (Easy) Derive (via newtype) a Ring instance for Complex. Note: You may instead write this instance manually, but that's not as convenient. Here's the Shape ADT from the previous chapter: data Shape = Circle Point Number | Rectangle Point Number Number | Line Point Point | Text Point String  1. (Medium) Derive (via Generic) a Show instance for Shape. How does the amount of code written and String output compare to showShape from the previous chapter? Note: You may instead write this instance manually, but you'll need to pay close attention to the output format expected by the tests. Type Class Constraints Types of functions can be constrained by using type classes. Here is an example: suppose we want to write a function which tests if three values are equal, by using equality defined using an Eq type class instance. threeAreEqual :: forall a. Eq a => a -> a -> a -> Boolean threeAreEqual a1 a2 a3 = a1 == a2 && a2 == a3  The type declaration looks like an ordinary polymorphic type defined using forall. However, there is a type class constraint Eq a, separated from the rest of the type by a double arrow =>. This type says that we can call threeAreEqual with any choice of type a, as long as there is an Eq instance available for a in one of the imported modules. Constrained types can contain several type class instances, and the types of the instances are not restricted to simple type variables. Here is another example which uses Ord and Show instances to compare two values: showCompare :: forall a. Ord a => Show a => a -> a -> String showCompare a1 a2 | a1 < a2 = show a1 <> " is less than " <> show a2 showCompare a1 a2 | a1 > a2 = show a1 <> " is greater than " <> show a2 showCompare a1 a2 = show a1 <> " is equal to " <> show a2  Note that multiple constraints can be specified by using the => symbol multiple times, just like we specify curried functions of multiple arguments. But remember not to confuse the two symbols: • a -> b denotes the type of functions from type a to type b, whereas • a => b applies the constraint a to the type b. The PureScript compiler will try to infer constrained types when a type annotation is not provided. This can be useful if we want to use the most general type possible for a function. To see this, try using one of the standard type classes like Semiring in PSCi: > import Prelude > :type \x -> x + x forall a. Semiring a => a -> a  Here, we might have annotated this function as Int -> Int, or Number -> Number, but PSCi shows us that the most general type works for any Semiring, allowing us to use our function with both Ints and Numbers. Instance Dependencies Just as the implementation of functions can depend on type class instances using constrained types, so can the implementation of type class instances depend on other type class instances. This provides a powerful form of program inference, in which the implementation of a program can be inferred using its types. For example, consider the Show type class. We can write a type class instance to show arrays of elements, as long as we have a way to show the elements themselves: instance showArray :: Show a => Show (Array a) where ...  If a type class instance depends on multiple other instances, those instances should be grouped in parentheses and separated by commas on the left hand side of the => symbol: instance showEither :: (Show a, Show b) => Show (Either a b) where ...  These two type class instances are provided in the prelude library. When the program is compiled, the correct type class instance for Show is chosen based on the inferred type of the argument to show. The selected instance might depend on many such instance relationships, but this complexity is not exposed to the developer. Exercises 1. (Easy) The following declaration defines a type of non-empty arrays of elements of type a: data NonEmpty a = NonEmpty a (Array a)  Write an Eq instance for the type NonEmpty a which reuses the instances for Eq a and Eq (Array a). Note: you may instead derive the Eq instance. 2. (Medium) Write a Semigroup instance for NonEmpty a by reusing the Semigroup instance for Array. 3. (Medium) Write a Functor instance for NonEmpty. 4. (Medium) Given any type a with an instance of Ord, we can add a new "infinite" value which is greater than any other value: data Extended a = Infinite | Finite a  Write an Ord instance for Extended a which reuses the Ord instance for a. 5. (Difficult) Write a Foldable instance for NonEmpty. Hint: reuse the Foldable instance for arrays. 6. (Difficult) Given a type constructor f which defines an ordered container (and so has a Foldable instance), we can create a new container type which includes an extra element at the front: data OneMore f a = OneMore a (f a)  The container OneMore f also has an ordering, where the new element comes before any element of f. Write a Foldable instance for OneMore f: instance foldableOneMore :: Foldable f => Foldable (OneMore f) where ...  7. (Medium) Write a dedupShapes :: Array Shape -> Array Shape function which removes duplicate Shapes from an array using the nubEq function. 8. (Medium) Write a dedupShapesFast function which is the same as dedupShapes, but uses the more efficient nub function. Multi Parameter Type Classes It's not the case that a type class can only take a single type as an argument. This is the most common case, but in fact, a type class can be parameterized by zero or more type arguments. Let's see an example of a type class with two type arguments. module Stream where import Data.Array as Array import Data.Maybe (Maybe) import Data.String.CodeUnits as String class Stream stream element where uncons :: stream -> Maybe { head :: element, tail :: stream } instance streamArray :: Stream (Array a) a where uncons = Array.uncons instance streamString :: Stream String Char where uncons = String.uncons  The Stream module defines a class Stream which identifies types which look like streams of elements, where elements can be pulled from the front of the stream using the uncons function. Note that the Stream type class is parameterized not only by the type of the stream itself, but also by its elements. This allows us to define type class instances for the same stream type but different element types. The module defines two type class instances: an instance for arrays, where uncons removes the head element of the array using pattern matching, and an instance for String, which removes the first character from a String. We can write functions which work over arbitrary streams. For example, here is a function which accumulates a result in some Monoid based on the elements of a stream: import Prelude import Data.Monoid (class Monoid, mempty) foldStream :: forall l e m. Stream l e => Monoid m => (e -> m) -> l -> m foldStream f list = case uncons list of Nothing -> mempty Just cons -> f cons.head <> foldStream f cons.tail  Try using foldStream in PSCi for different types of Stream and different types of Monoid. Functional Dependencies Multi-parameter type classes can be very useful, but can easily lead to confusing types and even issues with type inference. As a simple example, consider writing a generic tail function on streams using the Stream class given above: genericTail xs = map _.tail (uncons xs)  This gives a somewhat confusing error message: The inferred type forall stream a. Stream stream a => stream -> Maybe stream has type variables which are not mentioned in the body of the type. Consider adding a type annotation.  The problem is that the genericTail function does not use the element type mentioned in the definition of the Stream type class, so that type is left unsolved. Worse still, we cannot even use genericTail by applying it to a specific type of stream: > map _.tail (uncons "testing") The inferred type forall a. Stream String a => Maybe String has type variables which are not mentioned in the body of the type. Consider adding a type annotation.  Here, we might expect the compiler to choose the streamString instance. After all, a String is a stream of Chars, and cannot be a stream of any other type of elements. The compiler is unable to make that deduction automatically, and cannot commit to the streamString instance. However, we can help the compiler by adding a hint to the type class definition: class Stream stream element | stream -> element where uncons :: stream -> Maybe { head :: element, tail :: stream }  Here, stream -> element is called a functional dependency. A functional dependency asserts a functional relationship between the type arguments of a multi-parameter type class. This functional dependency tells the compiler that there is a function from stream types to (unique) element types, so if the compiler knows the stream type, then it can commit to the element type. This hint is enough for the compiler to infer the correct type for our generic tail function above: > :type genericTail forall stream element. Stream stream element => stream -> Maybe stream > genericTail "testing" (Just "esting")  Functional dependencies can be quite useful when using multi-parameter type classes to design certain APIs. Nullary Type Classes We can even define type classes with zero type arguments! These correspond to compile-time assertions about our functions, allowing us to track global properties of our code in the type system. An important example is the Partial class which we saw earlier when discussing partial functions. Take for example the functions head and tail defined in Data.Array.Partial that allow us to get the head or tail of an array without wrapping them in a Maybe, so they can fail if the array is empty: head :: forall a. Partial => Array a -> a tail :: forall a. Partial => Array a -> Array a  Note that there is no instance defined for the Partial type class! Doing so would defeat its purpose: attempting to use the head function directly will result in a type error: > head [1, 2, 3] No type class instance was found for Prim.Partial  Instead, we can republish the Partial constraint for any functions making use of partial functions: secondElement :: forall a. Partial => Array a -> a secondElement xs = head (tail xs)  We've already seen the unsafePartial function, which allows us to treat a partial function as a regular function (unsafely). This function is defined in the Partial.Unsafe module: unsafePartial :: forall a. (Partial => a) -> a  Note that the Partial constraint appears inside the parentheses on the left of the function arrow, but not in the outer forall. That is, unsafePartial is a function from partial values to regular values: > unsafePartial head [1, 2, 3] 1 > unsafePartial secondElement [1, 2, 3] 2  Superclasses Just as we can express relationships between type class instances by making an instance dependent on another instance, we can express relationships between type classes themselves using so-called superclasses. We say that one type class is a superclass of another if every instance of the second class is required to be an instance of the first, and we indicate a superclass relationship in the class definition by using a backwards facing double arrow. We've already seen some examples of superclass relationships: the Eq class is a superclass of Ord, and the Semigroup class is a superclass of Monoid. For every type class instance of the Ord class, there must be a corresponding Eq instance for the same type. This makes sense, since in many cases, when the compare function reports that two values are incomparable, we often want to use the Eq class to determine if they are in fact equal. In general, it makes sense to define a superclass relationship when the laws for the subclass mention the members of the superclass. For example, it is reasonable to assume, for any pair of Ord and Eq instances, that if two values are equal under the Eq instance, then the compare function should return EQ. In other words, a == b should be true exactly when compare a b evaluates to EQ. This relationship on the level of laws justifies the superclass relationship between Eq and Ord. Another reason to define a superclass relationship is in the case where there is a clear "is-a" relationship between the two classes. That is, every member of the subclass is a member of the superclass as well. Exercises 1. (Medium) Define a partial function unsafeMaximum :: Partial => Array Int -> Int which finds the maximum of a non-empty array of integers. Test out your function in PSCi using unsafePartial. Hint: Use the maximum function from Data.Foldable. 2. (Medium) The Action class is a multi-parameter type class which defines an action of one type on another: class Monoid m <= Action m a where act :: m -> a -> a  An action is a function which describes how monoidal values can be used to modify a value of another type. There are two laws for the Action type class: • act mempty a = a • act (m1 <> m2) a = act m1 (act m2 a) That is, the action respects the operations defined by the Monoid class. For example, the natural numbers form a monoid under multiplication: newtype Multiply = Multiply Int instance semigroupMultiply :: Semigroup Multiply where append (Multiply n) (Multiply m) = Multiply (n * m) instance monoidMultiply :: Monoid Multiply where mempty = Multiply 1  Write an instance which implements this action: instance actionMultiplyInt :: Action Multiply Int  Does this instance satisfy the laws listed above? 3. (Medium) Write an Action instance which repeats an input string some number of times: instance actionMultiplyString :: Action Multiply String  Hint: Search Pursuit for a helper-function with the signature String -> Int -> String. Note that String might appear as a more generic type (such as Monoid). Does this instance satisfy the laws listed above? 4. (Medium) Write an instance Action m a => Action m (Array a), where the action on arrays is defined by acting on each array element independently. 5. (Difficult) Given the following newtype, write an instance for Action m (Self m), where the monoid m acts on itself using append: newtype Self m = Self m  Note: The testing framework requires Show and Eq instances for the Self and Multiply types. You may either write these instances manually, or let the compiler handle this for you with derive newtype instance shorthand. 6. (Difficult) Should the arguments of the multi-parameter type class Action be related by some functional dependency? Why or why not? Note: There is no test for this exercise. A Type Class for Hashes In the last section of this chapter, we will use the lessons from the rest of the chapter to create a library for hashing data structures. Note that this library is for demonstration purposes only, and is not intended to provide a robust hashing mechanism. What properties might we expect of a hash function? • A hash function should be deterministic, and map equal values to equal hash codes. • A hash function should distribute its results approximately uniformly over some set of hash codes. The first property looks a lot like a law for a type class, whereas the second property is more along the lines of an informal contract, and certainly would not be enforceable by PureScript's type system. However, this should provide the intuition for the following type class: newtype HashCode = HashCode Int instance hashCodeEq :: Eq HashCode where eq (HashCode a) (HashCode b) = a == b hashCode :: Int -> HashCode hashCode h = HashCode (h mod 65535) class Eq a <= Hashable a where hash :: a -> HashCode  with the associated law that a == b implies hash a == hash b. We'll spend the rest of this section building a library of instances and functions associated with the Hashable type class. We will need a way to combine hash codes in a deterministic way: combineHashes :: HashCode -> HashCode -> HashCode combineHashes (HashCode h1) (HashCode h2) = hashCode (73 * h1 + 51 * h2)  The combineHashes function will mix two hash codes and redistribute the result over the interval 0-65535. Let's write a function which uses the Hashable constraint to restrict the types of its inputs. One common task which requires a hashing function is to determine if two values hash to the same hash code. The hashEqual relation provides such a capability: hashEqual :: forall a. Hashable a => a -> a -> Boolean hashEqual = eq on hash  This function uses the on function from Data.Function to define hash-equality in terms of equality of hash codes, and should read like a declarative definition of hash-equality: two values are "hash-equal" if they are equal after each value has been passed through the hash function. Let's write some Hashable instances for some primitive types. Let's start with an instance for integers. Since a HashCode is really just a wrapped integer, this is simple - we can use the hashCode helper function: instance hashInt :: Hashable Int where hash = hashCode  We can also define a simple instance for Boolean values using pattern matching: instance hashBoolean :: Hashable Boolean where hash false = hashCode 0 hash true = hashCode 1  With an instance for hashing integers, we can create an instance for hashing Chars by using the toCharCode function from Data.Char: instance hashChar :: Hashable Char where hash = hash <<< toCharCode  To define an instance for arrays, we can map the hash function over the elements of the array (if the element type is also an instance of Hashable) and then perform a left fold over the resulting hashes using the combineHashes function: instance hashArray :: Hashable a => Hashable (Array a) where hash = foldl combineHashes (hashCode 0) <<< map hash  Notice how we build up instances using the simpler instances we have already written. Let's use our new Array instance to define an instance for Strings, by turning a String into an array of Chars: instance hashString :: Hashable String where hash = hash <<< toCharArray  How can we prove that these Hashable instances satisfy the type class law that we stated above? We need to make sure that equal values have equal hash codes. In cases like Int, Char, String and Boolean, this is simple because there are no values of those types which are equal in the sense of Eq but not equal identically. What about some more interesting types? To prove the type class law for the Array instance, we can use induction on the length of the array. The only array with length zero is []. Any two non-empty arrays are equal only if they have equal head elements and equal tails, by the definition of Eq on arrays. By the inductive hypothesis, the tails have equal hashes, and we know that the head elements have equal hashes if the Hashable a instance must satisfy the law. Therefore, the two arrays have equal hashes, and so the Hashable (Array a) obeys the type class law as well. The source code for this chapter includes several other examples of Hashable instances, such as instances for the Maybe and Tuple type. Exercises 1. (Easy) Use PSCi to test the hash functions for each of the defined instances. Note: There is no provided unit test for this exercise. 2. (Medium) Write a function arrayHasDuplicates which tests if an array has any duplicate elements based on both hash and value equality. First check for hash equality with the hashEqual function, then check for value equality with == if a duplicate pair of hashes is found. Hint: the nubByEq function in Data.Array should make this task much simpler. 3. (Medium) Write a Hashable instance for the following newtype which satisfies the type class law: newtype Hour = Hour Int instance eqHour :: Eq Hour where eq (Hour n) (Hour m) = mod n 12 == mod m 12  The newtype Hour and its Eq instance represent the type of integers modulo 12, so that 1 and 13 are identified as equal, for example. Prove that the type class law holds for your instance. 4. (Difficult) Prove the type class laws for the Hashable instances for Maybe, Either and Tuple. Note: There is no test for this exercise. Conclusion In this chapter, we've been introduced to type classes, a type-oriented form of abstraction which enables powerful forms of code reuse. We've seen a collection of standard type classes from the PureScript standard libraries, and defined our own library based on a type class for computing hash codes. This chapter also gave an introduction to the notion of type class laws, a technique for proving properties about code which uses type classes for abstraction. Type class laws are part of a larger subject called equational reasoning, in which the properties of a programming language and its type system are used to enable logical reasoning about its programs. This is an important idea, and will be a theme which we will return to throughout the rest of the book. Applicative Validation Chapter Goals In this chapter, we will meet an important new abstraction - the applicative functor, described by the Applicative type class. Don't worry if the name sounds confusing - we will motivate the concept with a practical example - validating form data. This technique allows us to convert code which usually involves a lot of boilerplate checking into a simple, declarative description of our form. We will also meet another type class, Traversable, which describes traversable functors, and see how this concept also arises very naturally from solutions to real-world problems. The example code for this chapter will be a continuation of the address book example from chapter 3. This time, we will extend our address book data types, and write functions to validate values for those types. The understanding is that these functions could be used, for example in a web user interface, to display errors to the user as part of a data entry form. Project Setup The source code for this chapter is defined in the files src/Data/AddressBook.purs and src/Data/AddressBook/Validation.purs. The project has a number of dependencies, many of which we have seen before. There are two new dependencies: • control, which defines functions for abstracting control flow using type classes like Applicative. • validation, which defines a functor for applicative validation, the subject of this chapter. The Data.AddressBook module defines data types and Show instances for the types in our project, and the Data.AddressBook.Validation module contains validation rules for those types. Generalizing Function Application To explain the concept of an applicative functor, let's consider the type constructor Maybe that we met earlier. The source code for this module defines a function address which has the following type: address :: String -> String -> String -> Address  This function is used to construct a value of type Address from three strings: a street name, a city, and a state. We can apply this function easily and see the result in PSCi: > import Data.AddressBook > address "123 Fake St." "Faketown" "CA" { street: "123 Fake St.", city: "Faketown", state: "CA" }  However, suppose we did not necessarily have a street, city, or state, and wanted to use the Maybe type to indicate a missing value in each of the three cases. In one case, we might have a missing city. If we try to apply our function directly, we will receive an error from the type checker: > import Data.Maybe > address (Just "123 Fake St.") Nothing (Just "CA") Could not match type Maybe String with type String  Of course, this is an expected type error - address takes strings as arguments, not values of type Maybe String. However, it is reasonable to expect that we should be able to "lift" the address function to work with optional values described by the Maybe type. In fact, we can, and the Control.Apply provides the function lift3 function which does exactly what we need: > import Control.Apply > lift3 address (Just "123 Fake St.") Nothing (Just "CA") Nothing  In this case, the result is Nothing, because one of the arguments (the city) was missing. If we provide all three arguments using the Just constructor, then the result will contain a value as well: > lift3 address (Just "123 Fake St.") (Just "Faketown") (Just "CA") Just ({ street: "123 Fake St.", city: "Faketown", state: "CA" })  The name of the function lift3 indicates that it can be used to lift functions of 3 arguments. There are similar functions defined in Control.Apply for functions of other numbers of arguments. Lifting Arbitrary Functions So, we can lift functions with small numbers of arguments by using lift2, lift3, etc. But how can we generalize this to arbitrary functions? It is instructive to look at the type of lift3: > :type lift3 forall a b c d f. Apply f => (a -> b -> c -> d) -> f a -> f b -> f c -> f d  In the Maybe example above, the type constructor f is Maybe, so that lift3 is specialized to the following type: forall a b c d. (a -> b -> c -> d) -> Maybe a -> Maybe b -> Maybe c -> Maybe d  This type says that we can take any function with three arguments, and lift it to give a new function whose argument and result types are wrapped with Maybe. Certainly, this is not possible for every type constructor f, so what is it about the Maybe type which allowed us to do this? Well, in specializing the type above, we removed a type class constraint on f from the Apply type class. Apply is defined in the Prelude as follows: class Functor f where map :: forall a b. (a -> b) -> f a -> f b class Functor f <= Apply f where apply :: forall a b. f (a -> b) -> f a -> f b  The Apply type class is a subclass of Functor, and defines an additional function apply. As <$> was defined as an alias for map, the Prelude module defines <*> as an alias for apply. As we'll see, these two operators are often used together.

The type of apply looks a lot like the type of map. The difference between map and apply is that map takes a function as an argument, whereas the first argument to apply is wrapped in the type constructor f. We'll see how this is used soon, but first, let's see how to implement the Apply type class for the Maybe type:

instance functorMaybe :: Functor Maybe where
map f (Just a) = Just (f a)
map f Nothing  = Nothing

instance applyMaybe :: Apply Maybe where
apply (Just f) (Just x) = Just (f x)
apply _        _        = Nothing


This type class instance says that we can apply an optional function to an optional value, and the result is defined only if both are defined.

Now we'll see how map and apply can be used together to lift functions of arbitrary number of arguments.

For functions of one argument, we can just use map directly.

For functions of two arguments, we have a curried function g with type a -> b -> c, say. This is equivalent to the type a -> (b -> c), so we can apply map to g to get a new function of type f a -> f (b -> c) for any type constructor f with a Functor instance. Partially applying this function to the first lifted argument (of type f a), we get a new wrapped function of type f (b -> c). If we also have an Apply instance for f, then we can then use apply to apply the second lifted argument (of type f b) to get our final value of type f c.

Putting this all together, we see that if we have values x :: f a and y :: f b, then the expression (g <$> x) <*> y has type f c (remember, this expression is equivalent to apply (map g x) y). The precedence rules defined in the Prelude allow us to remove the parentheses: g <$> x <*> y.

In general, we can use <$> on the first argument, and <*> for the remaining arguments, as illustrated here for lift3: lift3 :: forall a b c d f . Apply f => (a -> b -> c -> d) -> f a -> f b -> f c -> f d lift3 f x y z = f <$> x <*> y <*> z


It is left as an exercise for the reader to verify the types involved in this expression.

As an example, we can try lifting the address function over Maybe, directly using the <$> and <*> functions: > address <$> Just "123 Fake St." <*> Just "Faketown" <*> Just "CA"
Just ({ street: "123 Fake St.", city: "Faketown", state: "CA" })

> address <$> Just "123 Fake St." <*> Nothing <*> Just "CA" Nothing  Try lifting some other functions of various numbers of arguments over Maybe in this way. Alternatively applicative do notation can be used for the same purpose in a way that looks similar to the familiar do notation. Here is lift3 using applicative do notation. Note ado is used instead of do, and in is used on the final line to denote the yielded value: lift3 :: forall a b c d f . Apply f => (a -> b -> c -> d) -> f a -> f b -> f c -> f d lift3 f x y z = ado a <- x b <- y c <- z in f a b c  The Applicative Type Class There is a related type class called Applicative, defined as follows: class Apply f <= Applicative f where pure :: forall a. a -> f a  Applicative is a subclass of Apply and defines the pure function. pure takes a value and returns a value whose type has been wrapped with the type constructor f. Here is the Applicative instance for Maybe: instance applicativeMaybe :: Applicative Maybe where pure x = Just x  If we think of applicative functors as functors which allow lifting of functions, then pure can be thought of as lifting functions of zero arguments. Intuition for Applicative Functions in PureScript are pure and do not support side-effects. Applicative functors allow us to work in larger "programming languages" which support some sort of side-effect encoded by the functor f. As an example, the functor Maybe represents the side effect of possibly-missing values. Some other examples include Either err, which represents the side effect of possible errors of type err, and the arrow functor r -> which represents the side-effect of reading from a global configuration. For now, we'll only consider the Maybe functor. If the functor f represents this larger programming language with effects, then the Apply and Applicative instances allow us to lift values and function applications from our smaller programming language (PureScript) into the new language. pure lifts pure (side-effect free) values into the larger language, and for functions, we can use map and apply as described above. This raises a question: if we can use Applicative to embed PureScript functions and values into this new language, then how is the new language any larger? The answer depends on the functor f. If we can find expressions of type f a which cannot be expressed as pure x for some x, then that expression represents a term which only exists in the larger language. When f is Maybe, an example is the expression Nothing: we cannot write Nothing as pure x for any x. Therefore, we can think of PureScript as having been enlarged to include the new term Nothing, which represents a missing value. More Effects Let's see some more examples of lifting functions over different Applicative functors. Here is a simple example function defined in PSCi, which joins three names to form a full name: > import Prelude > fullName first middle last = last <> ", " <> first <> " " <> middle > fullName "Phillip" "A" "Freeman" Freeman, Phillip A  Now suppose that this function forms the implementation of a (very simple!) web service with the three arguments provided as query parameters. We want to make sure that the user provided each of the three parameters, so we might use the Maybe type to indicate the presence or otherwise absence of a parameter. We can lift fullName over Maybe to create an implementation of the web service which checks for missing parameters: > import Data.Maybe > fullName <$> Just "Phillip" <*> Just "A" <*> Just "Freeman"
Just ("Freeman, Phillip A")

> fullName <$> Just "Phillip" <*> Nothing <*> Just "Freeman" Nothing  or with applicative do > import Data.Maybe > :paste… … ado … f <- Just "Phillip" … m <- Just "A" … l <- Just "Freeman" … in fullName f m l … ^D (Just "Freeman, Phillip A") … ado … f <- Just "Phillip" … m <- Nothing … l <- Just "Freeman" … in fullName f m l … ^D Nothing  Note that the lifted function returns Nothing if any of the arguments was Nothing. This is good, because now we can send an error response back from our web service if the parameters are invalid. However, it would be better if we could indicate which field was incorrect in the response. Instead of lifting over Maybe, we can lift over Either String, which allows us to return an error message. First, let's write an operator to convert optional inputs into computations which can signal an error using Either String: > import Data.Either > :paste … withError Nothing err = Left err … withError (Just a) _ = Right a … ^D  Note: In the Either err applicative functor, the Left constructor indicates an error, and the Right constructor indicates success. Now we can lift over Either String, providing an appropriate error message for each parameter: > :paste … fullNameEither first middle last = … fullName <$> (first  withError "First name was missing")
…            <*> (middle withError "Middle name was missing")
…            <*> (last   withError "Last name was missing")
… ^D


or with applicative do

> :paste
… fullNameEither first middle last = ado
…  f <- first  withError "First name was missing"
…  m <- middle withError "Middle name was missing"
…  l <- last   withError "Last name was missing"
…  in fullName f m l
… ^D

> :type fullNameEither
Maybe String -> Maybe String -> Maybe String -> Either String String


Now our function takes three optional arguments using Maybe, and returns either a String error message or a String result.

We can try out the function with different inputs:

> fullNameEither (Just "Phillip") (Just "A") (Just "Freeman")
(Right "Freeman, Phillip A")

> fullNameEither (Just "Phillip") Nothing (Just "Freeman")
(Left "Middle name was missing")

> fullNameEither (Just "Phillip") (Just "A") Nothing
(Left "Last name was missing")


In this case, we see the error message corresponding to the first missing field, or a successful result if every field was provided. However, if we are missing multiple inputs, we still only see the first error:

> fullNameEither Nothing Nothing Nothing
(Left "First name was missing")


This might be good enough, but if we want to see a list of all missing fields in the error, then we need something more powerful than Either String. We will see a solution later in this chapter.

Combining Effects

As an example of working with applicative functors abstractly, this section will show how to write a function which will generically combine side-effects encoded by an applicative functor f.

What does this mean? Well, suppose we have a list of wrapped arguments of type f a for some a. That is, suppose we have a list of type List (f a). Intuitively, this represents a list of computations with side-effects tracked by f, each with return type a. If we could run all of these computations in order, we would obtain a list of results of type List a. However, we would still have side-effects tracked by f. That is, we expect to be able to turn something of type List (f a) into something of type f (List a) by "combining" the effects inside the original list.

For any fixed list size n, there is a function of n arguments which builds a list of size n out of those arguments. For example, if n is 3, the function is \x y z -> x : y : z : Nil. This function has type a -> a -> a -> List a. We can use the Applicative instance for List to lift this function over f, to get a function of type f a -> f a -> f a -> f (List a). But, since we can do this for any n, it makes sense that we should be able to perform the same lifting for any list of arguments.

That means that we should be able to write a function

combineList :: forall f a. Applicative f => List (f a) -> f (List a)


This function will take a list of arguments, which possibly have side-effects, and return a single wrapped list, applying the side-effects of each.

To write this function, we'll consider the length of the list of arguments. If the list is empty, then we do not need to perform any effects, and we can use pure to simply return an empty list:

combineList Nil = pure Nil


In fact, this is the only thing we can do!

If the list is non-empty, then we have a head element, which is a wrapped argument of type f a, and a tail of type List (f a). We can recursively combine the effects in the tail, giving a result of type f (List a). We can then use <$> and <*> to lift the Cons constructor over the head and new tail: combineList (Cons x xs) = Cons <$> x <*> combineList xs


Again, this was the only sensible implementation, based on the types we were given.

We can test this function in PSCi, using the Maybe type constructor as an example:

> import Data.List
> import Data.Maybe

> combineList (fromFoldable [Just 1, Just 2, Just 3])
(Just (Cons 1 (Cons 2 (Cons 3 Nil))))

> combineList (fromFoldable [Just 1, Nothing, Just 2])
Nothing


When specialized to Maybe, our function returns a Just only if every list element was Just, otherwise it returns Nothing. This is consistent with our intuition of working in a larger language supporting optional values - a list of computations which return optional results only has a result itself if every computation contained a result.

But the combineList function works for any Applicative! We can use it to combine computations which possibly signal an error using Either err, or which read from a global configuration using r ->.

We will see the combineList function again later, when we consider Traversable functors.

Exercises

1. (Medium) Write versions of the numeric operators +, -, * and / which work with optional arguments (i.e. arguments wrapped in Maybe) and return a value wrapped in Maybe. Name these functions addMaybe, subMaybe, mulMaybe, and divMaybe. Hint: Use lift2.
2. (Medium) Extend the above exercise to work with all Apply types (not just Maybe). Name these new functions addApply, subApply, mulApply, and divApply.
3. (Difficult) Write a function combineMaybe which has type forall a f. Applicative f => Maybe (f a) -> f (Maybe a). This function takes an optional computation with side-effects, and returns a side-effecting computation which has an optional result.

Applicative Validation

The source code for this chapter defines several data types which might be used in an address book application. The details are omitted here, but the key functions which are exported by the Data.AddressBook module have the following types:

address :: String -> String -> String -> Address

phoneNumber :: PhoneType -> String -> PhoneNumber

person :: String -> String -> Address -> Array PhoneNumber -> Person


where PhoneType is defined as an algebraic data type:

data PhoneType
= HomePhone
| WorkPhone
| CellPhone
| OtherPhone


These functions can be used to construct a Person representing an address book entry. For example, the following value is defined in Data.AddressBook:

examplePerson :: Person
examplePerson =
person "John" "Smith"
(address "123 Fake St." "FakeTown" "CA")
[ phoneNumber HomePhone "555-555-5555"
, phoneNumber CellPhone "555-555-0000"
]


Test this value in PSCi (this result has been formatted):

> import Data.AddressBook

> examplePerson
{ firstName: "John"
, lastName: "Smith"
{ street: "123 Fake St."
, city: "FakeTown"
, state: "CA"
}
, phones:
[ { type: HomePhone
, number: "555-555-5555"
}
, { type: CellPhone
, number: "555-555-0000"
}
]
}


We saw in a previous section how we could use the Either String functor to validate a data structure of type Person. For example, provided functions to validate the two names in the structure, we might validate the entire data structure as follows:

nonEmpty1 :: String -> Either String String
nonEmpty1 ""     = Left "Field cannot be empty"
nonEmpty1 value  = Right value

validatePerson1 :: Person -> Either String Person
validatePerson1 p =
person <$> nonEmpty1 p.firstName <*> nonEmpty1 p.lastName <*> pure p.homeAddress <*> pure p.phones  or with applicative do validatePerson1Ado :: Person -> Either String Person validatePerson1Ado p = ado f <- nonEmpty1 p.firstName l <- nonEmpty1 p.lastName in person f l p.homeAddress p.phones  In the first two lines, we use the nonEmpty1 function to validate a non-empty string. nonEmpty1 returns an error indicated with the Left constructor if its input is empty, otherwise it returns the value wrapped with the Right constructor. The final lines do not perform any validation but simply provide the address and phones fields to the person function as the remaining arguments. This function can be seen to work in PSCi, but has a limitation which we have seen before: > validatePerson$ person "" "" (address "" "" "") []
(Left "Field cannot be empty")


The Either String applicative functor only provides the first error encountered. Given the input here, we would prefer to see two errors - one for the missing first name, and a second for the missing last name.

There is another applicative functor which is provided by the validation library. This functor is called V, and it provides the ability to return errors in any semigroup. For example, we can use V (Array String) to return an array of Strings as errors, concatenating new errors onto the end of the array.

The Data.AddressBook.Validation module uses the V (Array String) applicative functor to validate the data structures in the Data.AddressBook module.

Here is an example of a validator taken from the Data.AddressBook.Validation module:

type Errors
= Array String

nonEmpty :: String -> String -> V Errors String
nonEmpty field ""     = invalid [ "Field '" <> field <> "' cannot be empty" ]
nonEmpty _     value  = pure value

lengthIs :: String -> Int -> String -> V Errors String
lengthIs field len value | length value /= len =
invalid [ "Field '" <> field <> "' must have length " <> show len ]
lengthIs _     _   value = pure value

address <$> nonEmpty "Street" a.street <*> nonEmpty "City" a.city <*> lengthIs "State" 2 a.state  or with applicative do validateAddressAdo :: Address -> V Errors Address validateAddressAdo a = ado street <- nonEmpty "Street" a.street city <- nonEmpty "City" a.city state <- lengthIs "State" 2 a.state in address street city state  validateAddress validates an Address structure. It checks that the street and city fields are non-empty, and checks that the string in the state field has length 2. Notice how the nonEmpty and lengthIs validator functions both use the invalid function provided by the Data.Validation module to indicate an error. Since we are working in the Array String semigroup, invalid takes an array of strings as its argument. We can try this function in PSCi: > import Data.AddressBook > import Data.AddressBook.Validation > validateAddress$ address "" "" ""
(invalid [ "Field 'Street' cannot be empty"
, "Field 'City' cannot be empty"
, "Field 'State' must have length 2"
])

> validateAddress $address "" "" "CA" (invalid [ "Field 'Street' cannot be empty" , "Field 'City' cannot be empty" ])  This time, we receive an array of all validation errors. Regular Expression Validators The validatePhoneNumber function uses a regular expression to validate the form of its argument. The key is a matches validation function, which uses a Regex from the Data.String.Regex module to validate its input: matches :: String -> Regex -> String -> V Errors String matches _ regex value | test regex value = pure value matches field _ _ = invalid [ "Field '" <> field <> "' did not match the required format" ]  Again, notice how pure is used to indicate successful validation, and invalid is used to signal an array of errors. validatePhoneNumber is built from the matches function in the same way as before: validatePhoneNumber :: PhoneNumber -> V Errors PhoneNumber validatePhoneNumber pn = phoneNumber <$> pure pn."type"
<*> matches "Number" phoneNumberRegex pn.number


or with applicative do

validatePhoneNumberAdo :: PhoneNumber -> V Errors PhoneNumber
tpe    <- pure pn."type"
number <- matches "Number" phoneNumberRegex pn.number
in phoneNumber tpe number


Again, try running this validator against some valid and invalid inputs in PSCi:

> validatePhoneNumber $phoneNumber HomePhone "555-555-5555" pure ({ type: HomePhone, number: "555-555-5555" }) > validatePhoneNumber$ phoneNumber HomePhone "555.555.5555"
invalid (["Field 'Number' did not match the required format"])


Exercises

1. (Easy) Write a regular expression stateRegex :: Regex to check that a string only contains two alphabetic characters. Hint: see the source code for phoneNumberRegex.
2. (Medium) Write a regular expression nonEmptyRegex :: Regex to check that a string is not entirely whitespace. Hint: If you need help developing this regex expression, check out RegExr which has a great cheatsheet and interactive test environment.
3. (Medium) Write a function validateAddressImproved that is similar to validateAddress, but uses the above stateRegex to validate the state field and nonEmptyRegex to validate the street and city fields. Hint: see the source for validatePhoneNumber for an example of how to use matches.

Traversable Functors

The remaining validator is validatePerson, which combines the validators we have seen so far to validate an entire Person structure, including the following new validatePhoneNumbers function:

validatePhoneNumbers :: String -> Array PhoneNumber -> V Errors (Array PhoneNumber)
validatePhoneNumbers field []      =
invalid [ "Field '" <> field <> "' must contain at least one value" ]
validatePhoneNumbers _     phones  =
traverse validatePhoneNumber phones

validatePerson :: Person -> V Errors Person
validatePerson p =
person <$> nonEmpty "First Name" p.firstName <*> nonEmpty "Last Name" p.lastName <*> validateAddress p.homeAddress <*> validatePhoneNumbers "Phone Numbers" p.phones  or with applicative do validatePersonAdo :: Person -> V Errors Person validatePersonAdo p = ado firstName <- nonEmpty "First Name" p.firstName lastName <- nonEmpty "Last Name" p.lastName address <- validateAddress p.homeAddress numbers <- validatePhoneNumbers "Phone Numbers" p.phones in person firstName lastName address numbers  validatePhoneNumbers uses a new function we haven't seen before - traverse. traverse is defined in the Data.Traversable module, in the Traversable type class: class (Functor t, Foldable t) <= Traversable t where traverse :: forall a b m. Applicative m => (a -> m b) -> t a -> m (t b) sequence :: forall a m. Applicative m => t (m a) -> m (t a)  Traversable defines the class of traversable functors. The types of its functions might look a little intimidating, but validatePerson provides a good motivating example. Every traversable functor is both a Functor and Foldable (recall that a foldable functor was a type constructor which supported a fold operation, reducing a structure to a single value). In addition, a traversable functor provides the ability to combine a collection of side-effects which depend on its structure. This may sound complicated, but let's simplify things by specializing to the case of arrays. The array type constructor is traversable, which means that there is a function: traverse :: forall a b m. Applicative m => (a -> m b) -> Array a -> m (Array b)  Intuitively, given any applicative functor m, and a function which takes a value of type a and returns a value of type b (with side-effects tracked by m), we can apply the function to each element of an array of type Array a to obtain a result of type Array b (with side-effects tracked by m). Still not clear? Let's specialize further to the case where m is the V Errors applicative functor above. Now, we have a function of type traverse :: forall a b. (a -> V Errors b) -> Array a -> V Errors (Array b)  This type signature says that if we have a validation function m for a type a, then traverse m is a validation function for arrays of type Array a. But that's exactly what we need to be able to validate the phones field of the Person data structure! We pass validatePhoneNumber to traverse to create a validation function which validates each element successively. In general, traverse walks over the elements of a data structure, performing computations with side-effects and accumulating a result. The type signature for Traversable's other function sequence might look more familiar: sequence :: forall a m. Applicative m => t (m a) -> m (t a)  In fact, the combineList function that we wrote earlier is just a special case of the sequence function from the Traversable type class. Setting t to be the type constructor List, we recover the type of the combineList function: combineList :: forall f a. Applicative f => List (f a) -> f (List a)  Traversable functors capture the idea of traversing a data structure, collecting a set of effectful computations, and combining their effects. In fact, sequence and traverse are equally important to the definition of Traversable - each can be implemented in terms of each other. This is left as an exercise for the interested reader. The Traversable instance for lists given in the Data.List module is: instance traversableList :: Traversable List where -- traverse :: forall a b m. Applicative m => (a -> m b) -> List a -> m (List b) traverse _ Nil = pure Nil traverse f (Cons x xs) = Cons <$> f x <*> traverse f xs


(The actual definition was later modified to improve stack safety. You can read more about that change here.)

In the case of an empty list, we can simply return an empty list using pure. If the list is non-empty, we can use the function f to create a computation of type f b from the head element. We can also call traverse recursively on the tail. Finally, we can lift the Cons constructor over the applicative functor m to combine the two results.

But there are more examples of traversable functors than just arrays and lists. The Maybe type constructor we saw earlier also has an instance for Traversable. We can try it in PSCi:

> import Data.Maybe
> import Data.Traversable

> traverse (nonEmpty "Example") Nothing
pure (Nothing)

> traverse (nonEmpty "Example") (Just "")
invalid (["Field 'Example' cannot be empty"])

> traverse (nonEmpty "Example") (Just "Testing")
pure ((Just unit))


These examples show that traversing the Nothing value returns Nothing with no validation, and traversing Just x uses the validation function to validate x. That is, traverse takes a validation function for type a and returns a validation function for Maybe a, i.e. a validation function for optional values of type a.

Other traversable functors include Array, and Tuple a and Either a for any type a. Generally, most "container" data type constructors have Traversable instances. As an example, the exercises will include writing a Traversable instance for a type of binary trees.

Exercises

1. (Easy) Write Eq and Show instances for the following binary tree data structure:

data Tree a = Leaf | Branch (Tree a) a (Tree a)


Recall from the previous chapter that you may either write these instances manually or let the compiler derive them for you.

There are many "correct" formatting options for Show output. The test for this exercise expects the following whitespace style. This happens to match the default formatting of generic show, so you only need to make note of this if you're planning on writing this instance manually.

(Branch (Branch Leaf 8 Leaf) 42 Leaf)

2. (Medium) Write a Traversable instance for Tree a, which combines side-effects from left-to-right. Hint: There are some additional instance dependencies that need to be defined for Traversable.

3. (Medium) Write a function traversePreOrder :: forall a m b. Applicative m => (a -> m b) -> Tree a -> m (Tree b) that performs a pre-order traversal of the tree. This means the order of effect execution is root-left-right, instead of left-root-right as was done for the previous in-order traverse exercise. Hint: No additional instances need to be defined, and you don't need to call any of the the functions defined earlier. Applicative do notation (ado) is the easiest way to write this function.

4. (Medium) Write a function traversePostOrder that performs a post-order traversal of the tree where effects are executed left-right-root.

5. (Medium) Create a new version of the Person type where the homeAddress field is optional (using Maybe). Then write a new version of validatePerson (renamed as validatePersonOptionalAddress) to validate this new Person. Hint: Use traverse to validate a field of type Maybe a.

6. (Difficult) Write a function sequenceUsingTraverse which behaves like sequence, but is written in terms of traverse.

7. (Difficult) Write a function traverseUsingSequence which behaves like traverse, but is written in terms of sequence.

Applicative Functors for Parallelism

In the discussion above, I chose the word "combine" to describe how applicative functors "combine side-effects". However, in all the examples given, it would be equally valid to say that applicative functors allow us to "sequence" effects. This would be consistent with the intuition that traversable functors provide a sequence function to combine effects in sequence based on a data structure.

However, in general, applicative functors are more general than this. The applicative functor laws do not impose any ordering on the side-effects that their computations perform. In fact, it would be valid for an applicative functor to perform its side-effects in parallel.

For example, the V validation functor returned an array of errors, but it would work just as well if we picked the Set semigroup, in which case it would not matter what order we ran the various validators. We could even run them in parallel over the data structure!

As a second example, the parallel package provides a type class Parallel which supports parallel computations. Parallel provides a function parallel which uses some Applicative functor to compute the result of its input computation in parallel:

f <$> parallel computation1 <*> parallel computation2  This computation would start computing values asynchronously using computation1 and computation2. When both results have been computed, they would be combined into a single result using the function f. We will see this idea in more detail when we apply applicative functors to the problem of callback hell later in the book. Applicative functors are a natural way to capture side-effects which can be combined in parallel. Conclusion In this chapter, we covered a lot of new ideas: • We introduced the concept of an applicative functor which generalizes the idea of function application to type constructors which capture some notion of side-effect. • We saw how applicative functors gave a solution to the problem of validating data structures, and how by switching the applicative functor we could change from reporting a single error to reporting all errors across a data structure. • We met the Traversable type class, which encapsulates the idea of a traversable functor, or a container whose elements can be used to combine values with side-effects. Applicative functors are an interesting abstraction which provide neat solutions to a number of problems. We will see them a few more times throughout the book. In this case, the validation applicative functor provided a way to write validators in a declarative style, allowing us to define what our validators should validate and not how they should perform that validation. In general, we will see that applicative functors are a useful tool for the design of domain specific languages. In the next chapter, we will see a related idea, the class of monads, and extend our address book example to run in the browser! The Effect Monad Chapter Goals In the last chapter, we introduced applicative functors, an abstraction which we used to deal with side-effects: optional values, error messages and validation. This chapter will introduce another abstraction for dealing with side-effects in a more expressive way: monads. The goal of this chapter is to explain why monads are a useful abstraction, and their connection with do notation. Project Setup The project adds the following dependencies: • effect - defines the Effect monad, the subject of the second half of the chapter. This dependency is often listed in every starter project (it's been a dependency of every chapter so far), so you'll rarely have to explicitly install it. • react-basic-hooks - a web framework that we will use for our Address Book app. Monads and Do Notation Do notation was first introduced when we covered array comprehensions. Array comprehensions provide syntactic sugar for the concatMap function from the Data.Array module. Consider the following example. Suppose we throw two dice and want to count the number of ways in which we can score a total of n. We could do this using the following non-deterministic algorithm: • Choose the value x of the first throw. • Choose the value y of the second throw. • If the sum of x and y is n then return the pair [x, y], else fail. Array comprehensions allow us to write this non-deterministic algorithm in a natural way: import Prelude import Control.Plus (empty) import Data.Array ((..)) countThrows :: Int -> Array (Array Int) countThrows n = do x <- 1 .. 6 y <- 1 .. 6 if x + y == n then pure [ x, y ] else empty  We can see that this function works in PSCi: > import Test.Examples > countThrows 10 [[4,6],[5,5],[6,4]] > countThrows 12 [[6,6]]  In the last chapter, we formed an intuition for the Maybe applicative functor, embedding PureScript functions into a larger programming language supporting optional values. In the same way, we can form an intuition for the array monad, embedding PureScript functions into a larger programming language supporting non-deterministic choice. In general, a monad for some type constructor m provides a way to use do notation with values of type m a. Note that in the array comprehension above, every line contains a computation of type Array a for some type a. In general, every line of a do notation block will contain a computation of type m a for some type a and our monad m. The monad m must be the same on every line (i.e. we fix the side-effect), but the types a can differ (i.e. individual computations can have different result types). Here is another example of do notation, this type applied to the type constructor Maybe. Suppose we have some type XML representing XML nodes, and a function child :: XML -> String -> Maybe XML  which looks for a child element of a node, and returns Nothing if no such element exists. In this case, we can look for a deeply-nested element by using do notation. Suppose we wanted to read a user's city from a user profile which had been encoded as an XML document: userCity :: XML -> Maybe XML userCity root = do prof <- child root "profile" addr <- child prof "address" city <- child addr "city" pure city  The userCity function looks for a child element profile, an element address inside the profile element, and finally an element city inside the address element. If any of these elements are missing, the return value will be Nothing. Otherwise, the return value is constructed using Just from the city node. Remember, the pure function in the last line is defined for every Applicative functor. Since pure is defined as Just for the Maybe applicative functor, it would be equally valid to change the last line to Just city. The Monad Type Class The Monad type class is defined as follows: class Apply m <= Bind m where bind :: forall a b. m a -> (a -> m b) -> m b class (Applicative m, Bind m) <= Monad m  The key function here is bind, defined in the Bind type class. Just like for the <$> and <*> operators in the Functor and Apply type classes, the Prelude defines an infix alias >>= for the bind function.

The Monad type class extends Bind with the operations of the Applicative type class that we have already seen.

It will be useful to see some examples of the Bind type class. A sensible definition for Bind on arrays can be given as follows:

instance bindArray :: Bind Array where
bind xs f = concatMap f xs


This explains the connection between array comprehensions and the concatMap function that has been alluded to before.

Here is an implementation of Bind for the Maybe type constructor:

instance bindMaybe :: Bind Maybe where
bind Nothing  _ = Nothing
bind (Just a) f = f a


This definition confirms the intuition that missing values are propagated through a do notation block.

Let's see how the Bind type class is related to do notation. Consider a simple do notation block which starts by binding a value from the result of some computation:

do value <- someComputation
whatToDoNext


Every time the PureScript compiler sees this pattern, it replaces the code with this:

bind someComputation \value -> whatToDoNext


or, written infix:

someComputation >>= \value -> whatToDoNext


The computation whatToDoNext is allowed to depend on value.

If there are multiple binds involved, this rule is applied multiple times, starting from the top. For example, the userCity example that we saw earlier gets desugared as follows:

userCity :: XML -> Maybe XML
userCity root =
child root "profile" >>= \prof ->
child addr "city" >>= \city ->
pure city


It is worth noting that code expressed using do notation is often much clearer than the equivalent code using the >>= operator. However, writing binds explicitly using >>= can often lead to opportunities to write code in point-free form - but the usual warnings about readability apply.

The Monad type class comes equipped with three laws, called the monad laws. These tell us what we can expect from sensible implementations of the Monad type class.

It is simplest to explain these laws using do notation.

Identity Laws

The right-identity law is the simplest of the three laws. It tells us that we can eliminate a call to pure if it is the last expression in a do notation block:

do
x <- expr
pure x


The right-identity law says that this is equivalent to just expr.

The left-identity law states that we can eliminate a call to pure if it is the first expression in a do notation block:

do
x <- pure y
next


This code is equivalent to next, after the name x has been replaced with the expression y.

The last law is the associativity law. It tells us how to deal with nested do notation blocks. It states that the following piece of code:

c1 = do
y <- do
x <- m1
m2
m3


is equivalent to this code:

c2 = do
x <- m1
y <- m2
m3


Each of these computations involves three monadic expression m1, m2 and m3. In each case, the result of m1 is eventually bound to the name x, and the result of m2 is bound to the name y.

In c1, the two expressions m1 and m2 are grouped into their own do notation block.

In c2, all three expressions m1, m2 and m3 appear in the same do notation block.

The associativity law tells us that it is safe to simplify nested do notation blocks in this way.

Note that by the definition of how do notation gets desugared into calls to bind, both of c1 and c2 are also equivalent to this code:

c3 = do
x <- m1
do
y <- m2
m3


As an example of working with monads abstractly, this section will present a function which works with any type constructor in the Monad type class. This should serve to solidify the intuition that monadic code corresponds to programming "in a larger language" with side-effects, and also illustrate the generality which programming with monads brings.

The function we will write is called foldM. It generalizes the foldl function that we met earlier to a monadic context. Here is its type signature:

foldM :: forall m a b. Monad m => (a -> b -> m a) -> a -> List b -> m a
foldl :: forall   a b.            (a -> b ->   a) -> a -> List b ->   a


Notice that this is the same as the type of foldl, except for the appearance of the monad m.

Intuitively, foldM performs a fold over a list in some context supporting some set of side-effects.

For example, if we picked m to be Maybe, then our fold would be allowed to fail by returning Nothing at any stage - every step returns an optional result, and the result of the fold is therefore also optional.

If we picked m to be the Array type constructor, then every step of the fold would be allowed to return zero or more results, and the fold would proceed to the next step independently for each result. At the end, the set of results would consist of all folds over all possible paths. This corresponds to a traversal of a graph!

To write foldM, we can simply break the input list into cases.

If the list is empty, then to produce the result of type a, we only have one option: we have to return the second argument:

foldM _ a Nil = pure a


Note that we have to use pure to lift a into the monad m.

What if the list is non-empty? In that case, we have a value of type a, a value of type b, and a function of type a -> b -> m a. If we apply the function, we obtain a monadic result of type m a. We can bind the result of this computation with a backwards arrow <-.

It only remains to recurse on the tail of the list. The implementation is simple:

foldM f a (b : bs) = do
a' <- f a b
foldM f a' bs


Note that this implementation is almost identical to that of foldl on lists, with the exception of do notation.

We can define and test this function in PSCi. Here is an example - suppose we defined a "safe division" function on integers, which tested for division by zero and used the Maybe type constructor to indicate failure:

safeDivide :: Int -> Int -> Maybe Int
safeDivide _ 0 = Nothing
safeDivide a b = Just (a / b)


Then we can use foldM to express iterated safe division:

> import Test.Examples
> import Data.List (fromFoldable)

> foldM safeDivide 100 (fromFoldable [5, 2, 2])
(Just 5)

> foldM safeDivide 100 (fromFoldable [2, 0, 4])
Nothing


The foldM safeDivide function returns Nothing if a division by zero was attempted at any point. Otherwise it returns the result of repeatedly dividing the accumulator, wrapped in the Just constructor.

Every instance of the Monad type class is also an instance of the Apply type class, by virtue of the superclass relationship between the two classes.

However, there is also an implementation of the Apply type class which comes "for free" for any instance of Monad, given by the ap function:

ap :: forall m a b. Monad m => m (a -> b) -> m a -> m b
ap mf ma = do
f <- mf
a <- ma
pure (f a)


If m is a law-abiding member of the Monad type class, then there is a valid Apply instance for m given by ap.

The interested reader can check that ap agrees with apply for the monads we have already encountered: Array, Maybe and Either e.

If every monad is also an applicative functor, then we should be able to apply our intuition for applicative functors to every monad. In particular, we can reasonably expect a monad to correspond, in some sense, to programming "in a larger language" augmented with some set of additional side-effects. We should be able to lift functions of arbitrary arities, using map and apply, into this new language.

But monads allow us to do more than we could do with just applicative functors, and the key difference is highlighted by the syntax of do notation. Consider the userCity example again, in which we looked for a user's city in an XML document which encoded their user profile:

userCity :: XML -> Maybe XML
userCity root = do
prof <- child root "profile"
pure city


Do notation allows the second computation to depend on the result prof of the first, and the third computation to depend on the result addr of the second, and so on. This dependence on previous values is not possible using only the interface of the Applicative type class.

Try writing userCity using only pure and apply: you will see that it is impossible. Applicative functors only allow us to lift function arguments which are independent of each other, but monads allow us to write computations which involve more interesting data dependencies.

In the last chapter, we saw that the Applicative type class can be used to express parallelism. This was precisely because the function arguments being lifted were independent of one another. Since the Monad type class allows computations to depend on the results of previous computations, the same does not apply - a monad has to combine its side-effects in sequence.

Exercises

1. (Easy) Write a function third which returns the third element of an array with three or more elements. Your function should return an appropriate Maybe type. Hint: Look up the types of the head and tail functions from the Data.Array module in the arrays package. Use do notation with the Maybe monad to combine these functions.

2. (Medium) Write a function possibleSums which uses foldM to determine all possible totals that could be made using a set of coins. The coins will be specified as an array which contains the value of each coin. Your function should have the following result:

> possibleSums []
[0]

> possibleSums [1, 2, 10]
[0,1,2,3,10,11,12,13]


Hint: This function can be written as a one-liner using foldM. You might want to use the nub and sort functions to remove duplicates and sort the result respectively.

3. (Medium) Confirm that the ap function and the apply operator agree for the Maybe monad. Note: There are no tests for this exercise.

4. (Medium) Verify that the monad laws hold for the Monad instance for the Maybe type, as defined in the maybe package. Note: There are no tests for this exercise.

5. (Medium) Write a function filterM which generalizes the filter function on lists. Your function should have the following type signature:

filterM :: forall m a. Monad m => (a -> m Boolean) -> List a -> m (List a)

6. (Difficult) Every monad has a default Functor instance given by:

map f a = do
x <- a
pure (f x)


Use the monad laws to prove that for any monad, the following holds:

lift2 f (pure a) (pure b) = pure (f a b)


where the Apply instance uses the ap function defined above. Recall that lift2 was defined as follows:

lift2 :: forall f a b c. Apply f => (a -> b -> c) -> f a -> f b -> f c
lift2 f a b = f <$> a <*> b  Note: There are no tests for this exercise. Native Effects We will now look at one particular monad which is of central importance in PureScript - the Effect monad. The Effect monad is defined in the Effect module. It is used to manage so-called native side-effects. If you are familiar with Haskell, it is the equivalent of the IO monad. What are native side-effects? They are the side-effects which distinguish JavaScript expressions from idiomatic PureScript expressions, which typically are free from side-effects. Some examples of native effects are: • Console IO • Random number generation • Exceptions • Reading/writing mutable state And in the browser: • DOM manipulation • XMLHttpRequest / AJAX calls • Interacting with a websocket • Writing/reading to/from local storage We have already seen plenty of examples of "non-native" side-effects: • Optional values, as represented by the Maybe data type • Errors, as represented by the Either data type • Multi-functions, as represented by arrays or lists Note that the distinction is subtle. It is true, for example, that an error message is a possible side-effect of a JavaScript expression, in the form of an exception. In that sense, exceptions do represent native side-effects, and it is possible to represent them using Effect. However, error messages implemented using Either are not a side-effect of the JavaScript runtime, and so it is not appropriate to implement error messages in that style using Effect. So it is not the effect itself which is native, but rather how it is implemented at runtime. Side-Effects and Purity In a pure language like PureScript, one question which presents itself is: without side-effects, how can one write useful real-world code? The answer is that PureScript does not aim to eliminate side-effects. It aims to represent side-effects in such a way that pure computations can be distinguished from computations with side-effects in the type system. In this sense, the language is still pure. Values with side-effects have different types from pure values. As such, it is not possible to pass a side-effecting argument to a function, for example, and have side-effects performed unexpectedly. The only way in which side-effects managed by the Effect monad will be presented is to run a computation of type Effect a from JavaScript. The Spago build tool (and other tools) provide a shortcut, by generating additional JavaScript to invoke the main computation when the application starts. main is required to be a computation in the Effect monad. The Effect Monad The Effect monad provides a well-typed API for computations with side-effects, while at the same time generating efficient JavaScript. Let's take a closer look at the return type of the familiar log function. Effect indicates that this function produces a native effect, console IO in this case. Unit indicates that no meaningful data is returned. You can think of Unit as being analogous to the void keyword in other languages, such as C, Java, etc. log :: String -> Effect Unit  Aside: You may encounter IDE suggestions for the more general (and more elaborately typed) log function from Effect.Class.Console. This is interchangeable with the one from Effect.Console when dealing with the basic Effect monad. Reasons for the more general version will become clearer after reading about "Monad Transformers" in the "Monadic Adventures" chapter. For the curious (and impatient), this works because there's a MonadEffect instance for Effect. log :: forall m. MonadEffect m => String -> m Unit  Now let's now consider an Effect that returns meaningful data. The random function from Effect.Random produces a random Number. random :: Effect Number  Here's a full example program (found in test/Random.purs of this chapter's exercises folder). module Test.Random where import Prelude import Effect (Effect) import Effect.Random (random) import Effect.Console (logShow) main :: Effect Unit main = do n <- random logShow n  Because Effect is a monad, we use do notation to unwrap the data it contains before passing this data on to the effectful logShow function. As a refresher, here's the equivalent code written using the bind operator: main :: Effect Unit main = random >>= logShow  Try running this yourself with: spago run --main Test.Random  You should see a randomly chosen number between 0.0 and 1.0 printed to the console. Aside: spago run defaults to searching in the Main module for a main function. You may also specify an alternate module as an entry point with the --main flag, as is done in the above example. Just be sure that this alternate module also contains a main function. Note that it's also possible to generate "random" (technically pseudorandom) data without resorting to impure effectful code. We'll cover these techniques in the "Generative Testing" chapter. As mentioned previously, the Effect monad is of central importance to PureScript. The reason why it's central is because it is the conventional way to interoperate with PureScript's Foreign Function Interface, which provides the mechanism to execute a program and perform side effects. While it's desireable to avoid using the Foreign Function Interface, it's fairly critical to understand how it works and how to use it, so I recommend reading that chapter before doing any serious PureScript work. That said, the Effect monad is fairly simple. It has a few helper functions, but aside from that it doesn't do much except encapsulate side effects. Exceptions Let's examine a function from the node-fs package that involves two native side effects: reading mutable state, and exceptions: readTextFile :: Encoding -> String -> Effect String  If we attempt to read a file that does not exist: import Node.Encoding (Encoding(..)) import Node.FS.Sync (readTextFile) main :: Effect Unit main = do lines <- readTextFile UTF8 "iDoNotExist.md" log lines  We encounter the following exception:  throw err; ^ Error: ENOENT: no such file or directory, open 'iDoNotExist.md' ... errno: -2, syscall: 'open', code: 'ENOENT', path: 'iDoNotExist.md'  To manage this exception gracefully, we can wrap the potentially problematic code in try to handle either outcome: main :: Effect Unit main = do result <- try$ readTextFile UTF8 "iDoNotExist.md"
case result of
Right lines -> log $"Contents: \n" <> lines Left error -> log$ "Couldn't open file. Error was: " <> message error


try runs an Effect and returns eventual exceptions as a Left value. If the computation succeeds, the result gets wrapped in a Right:

try :: forall a. Effect a -> Effect (Either Error a)


We can also generate our own exceptions. Here is an alternative implementation of Data.List.head which throws an exception if the list is empty, rather than returning a Maybe value of Nothing.

exceptionHead :: List Int -> Effect Int
exceptionHead l = case l of
x : _ -> pure x
Nil -> throwException $error "empty list"  Note that the exceptionHead function is a somewhat impractical example, as it is best to avoid generating exceptions in PureScript code and instead use non-native effects such as Either and Maybe to manage errors and missing values. Mutable State There is another effect defined in the core libraries: the ST effect. The ST effect is used to manipulate mutable state. As pure functional programmers, we know that shared mutable state can be problematic. However, the ST effect uses the type system to restrict sharing in such a way that only safe local mutation is allowed. The ST effect is defined in the Control.Monad.ST module. To see how it works, we need to look at the types of its actions: new :: forall a r. a -> ST r (STRef r a) read :: forall a r. STRef r a -> ST r a write :: forall a r. a -> STRef r a -> ST r a modify :: forall r a. (a -> a) -> STRef r a -> ST r a  new is used to create a new mutable reference cell of type STRef r a, which can be read using the read action, and modified using the write and modify actions. The type a is the type of the value stored in the cell, and the type r is used to indicate a memory region (or heap) in the type system. Here is an example. Suppose we want to simulate the movement of a particle falling under gravity by iterating a simple update function over a large number of small time steps. We can do this by creating a mutable reference cell to hold the position and velocity of the particle, and then using a for loop to update the value stored in that cell: import Prelude import Control.Monad.ST.Ref (modify, new, read) import Control.Monad.ST (ST, for, run) simulate :: forall r. Number -> Number -> Int -> ST r Number simulate x0 v0 time = do ref <- new { x: x0, v: v0 } for 0 (time * 1000) \_ -> modify ( \o -> { v: o.v - 9.81 * 0.001 , x: o.x + o.v * 0.001 } ) ref final <- read ref pure final.x  At the end of the computation, we read the final value of the reference cell, and return the position of the particle. Note that even though this function uses mutable state, it is still a pure function, so long as the reference cell ref is not allowed to be used by other parts of the program. We will see that this is exactly what the ST effect disallows. To run a computation with the ST effect, we have to use the run function: run :: forall a. (forall r. ST r a) -> a  The thing to notice here is that the region type r is quantified inside the parentheses on the left of the function arrow. That means that whatever action we pass to run has to work with any region r whatsoever. However, once a reference cell has been created by new, its region type is already fixed, so it would be a type error to try to use the reference cell outside the code delimited by run. This is what allows run to safely remove the ST effect, and turn simulate into a pure function! simulate' :: Number -> Number -> Int -> Number simulate' x0 v0 time = run (simulate x0 v0 time)  You can even try running this function in PSCi: > import Main > simulate' 100.0 0.0 0 100.00 > simulate' 100.0 0.0 1 95.10 > simulate' 100.0 0.0 2 80.39 > simulate' 100.0 0.0 3 55.87 > simulate' 100.0 0.0 4 21.54  In fact, if we inline the definition of simulate at the call to run, as follows: simulate :: Number -> Number -> Int -> Number simulate x0 v0 time = run do ref <- new { x: x0, v: v0 } for 0 (time * 1000) \_ -> modify ( \o -> { v: o.v - 9.81 * 0.001 , x: o.x + o.v * 0.001 } ) ref final <- read ref pure final.x  then the compiler will notice that the reference cell is not allowed to escape its scope, and can safely turn ref into a var. Here is the generated JavaScript for simulate inlined with run: var simulate = function (x0) { return function (v0) { return function (time) { return (function __do() { var ref = { value: { x: x0, v: v0 } }; Control_Monad_ST_Internal["for"](0)(time * 1000 | 0)(function (v) { return Control_Monad_ST_Internal.modify(function (o) { return { v: o.v - 9.81 * 1.0e-3, x: o.x + o.v * 1.0e-3 }; })(ref); })(); return ref.value.x; })(); }; }; };  Note that this resulting JavaScript is not as optimal as it could be. See this issue for more details. The above snippet should be updated once that issue is resolved. For comparison, this is the generated JavaScript of the non-inlined form: var simulate = function (x0) { return function (v0) { return function (time) { return function __do() { var ref = Control_Monad_ST_Internal["new"]({ x: x0, v: v0 })(); Control_Monad_ST_Internal["for"](0)(time * 1000 | 0)(function (v) { return Control_Monad_ST_Internal.modify(function (o) { return { v: o.v - 9.81 * 1.0e-3, x: o.x + o.v * 1.0e-3 }; })(ref); })(); var $$final = Control_Monad_ST_Internal.read(ref)(); return$$final.x; }; }; }; };  The ST effect is a good way to generate short JavaScript when working with locally-scoped mutable state, especially when used together with actions like for, foreach, and while which generate efficient loops. Exercises 1. (Medium) Rewrite the safeDivide function as exceptionDivide and throw an exception using throwException if the denominator is zero. Note: There is no unit test for this exercise because it's tricky to check for an expected exception within our unit test framework. Feel free to work on adding this test. 2. (Medium) Write a function estimatePi :: Int -> Number that uses n terms of the Gregory Series to calculate an approximation of pi. Hints: You can pattern your answer like the definition of simulate above. You might need to convert an Int into a Number using toNumber :: Int -> Number from Data.Int. 3. (Medium) Write a function fibonacci :: Int -> Int to compute the nth Fibonacci number, using ST to track the values of the previous two Fibonacci numbers. Using PSCi, compare the speed of your new ST-based implementation against the recursive implementation (fib) from Chapter 4. DOM Effects In the final sections of this chapter, we will apply what we have learned about effects in the Effect monad to the problem of working with the DOM. There are a number of PureScript packages for working directly with the DOM, or with open-source DOM libraries. For example: There are also PureScript libraries which build abstractions on top of these libraries, such as In this chapter, we will use the react-basic-hooks library to add a user interface to our address book application, but the interested reader is encouraged to explore alternative approaches. An Address Book User Interface Using the react-basic-hooks library, we will define our application as a React component. React components describe HTML elements in code as pure data structures, which are then efficiently rendered to the DOM. In addition, components can respond to events like button clicks. The react-basic-hooks library uses the Effect monad to describe how to handle these events. A full tutorial for the React library is well beyond the scope of this chapter, but the reader is encouraged to consult its documentation where needed. For our purposes, React will provide a practical example of the Effect monad. We are going to build a form which will allow a user to add a new entry into our address book. The form will contain text boxes for the various fields (first name, last name, city, state, etc.), and an area in which validation errors will be displayed. As the user types text into the text boxes, the validation errors will be updated. To keep things simple, the form will have a fixed shape: the different phone number types (home, cell, work, other) will be expanded into separate text boxes. You can launch the web app from the exercises/chapter8 directory with the following commands: $ npm install
$npx spago build$ npx parcel src/index.html --open


If development tools such as spago and parcel are installed globally, then the npx prefix may be omitted. You have likely already installed spago globally with npm i -g spago, and the same can be done for parcel.

parcel should launch a browser window with our "Address Book" app. If you keep the parcel terminal open, and rebuild with spago in another terminal, the page should automatically refresh with your latest edits. You can also configure automatic rebuilds (and therefore automatic page refresh) on file-save if you're using an editor that supports purs ide or are running pscid.

In this Address Book app, you should be able to enter some values into the form fields and see the validation errors printed onto the page.

Let's explore how it works.

The src/index.html file is minimal:

<!DOCTYPE html>
<html>
<meta charset="UTF-8">
<body>
<div id="container"></div>
<script src="./index.js"></script>
</body>
</html>


The <script line includes the JavaScript entry point, index.js, which contains this single line:

require("../output/Main/index.js").main();


It calls our generated JavaScript equivalent of the main function of module Main (src/main.purs). Recall that spago build puts all generated JavaScript in the output directory.

The main function uses the DOM and HTML APIs to render our address book component within the container element we defined in index.html:

main :: Effect Unit
main = do
-- Get window object
w <- window
-- Get window's HTML document
doc <- document w
-- Get "container" element in HTML
ctr <- getElementById "container" $toNonElementParentNode doc case ctr of Nothing -> throw "Container element not found." Just c -> do -- Create AddressBook react component addressBookApp <- mkAddressBookApp let -- Create JSX node from react component. Pass-in empty props app = element addressBookApp {} -- Render AddressBook JSX node in DOM "container" element D.render app c  Note that these three lines: w <- window doc <- document w ctr <- getElementById "container"$ toNonElementParentNode doc


Can be consolidated to:

doc <- document =<< window
ctr <- getElementById "container" $toNonElementParentNode doc  Or consolidated even further to: ctr <- getElementById "container" =<< (map toNonElementParentNode$ document =<< window)


It is a matter of personal preference whether the intermediate w and doc variables aid in readability.

Let's dig into our AddressBook reactComponent. We'll start with a simplified component, and then build up to the actual code in Main.purs.

Take a look at this minimal component. Feel free to substitute the full component with this one to see it run:

mkAddressBookApp :: Effect (ReactComponent {})
reactComponent
(\props -> pure $D.text "Hi! I'm an address book")  reactComponent has this intimidating signature: reactComponent :: forall hooks props. Lacks "children" props => Lacks "key" props => Lacks "ref" props => String -> ({ | props } -> Render Unit hooks JSX) -> Effect (ReactComponent { | props })  The important points to note are the arguments after all the type class constraints. It takes a String (an arbitrary component name), a function that describes how to convert props into rendered JSX, and returns our ReactComponent wrapped in an Effect. The props-to-JSX function is simply: \props -> pure$ D.text "Hi! I'm an address book"


props are ignored, D.text returns JSX, and pure lifts to rendered JSX. Now component has everything it needs to produce the ReactComponent.

Next we'll examine some of the additional complexities of the full Address Book component.

These are the first few lines of our full component:

mkAddressBookApp :: Effect (ReactComponent {})
Tuple person setPerson <- useState examplePerson


We track person as a piece of state with the useState hook.

Tuple person setPerson <- useState examplePerson


Note that you are free to break-up component state into multiple pieces of state with multiple calls to useState. For example, we could rewrite this app to use a separate piece of state for each record field of Person, but that happens to result in a slightly less convenient architecture in this case.

In other examples, you may encounter the /\ infix operator for Tuple. This is equivalent to the above line:

firstName /\ setFirstName <- useState p.firstName


useState takes a default initial value and returns the current value and a way to update the value. We can check the type of useState to gain more insight the types of person and setPerson:

useState ::
forall state.
state ->
Hook (UseState state) (Tuple state ((state -> state) -> Effect Unit))


We can strip the Hook (UseState state) wrapper off of the return value because useState is called within an R.do block. We'll elaborate on R.do later.

So now we can observe the following signatures:

person :: state
setPerson :: (state -> state) -> Effect Unit


The specific type of state is determined by our initial default value. Person Record in this case because that is the type of examplePerson.

person is how we access the current state at each rerender.

setPerson is how we update the state. We simply provide a function that describes how to transform the current state to the new state. The record update syntax is perfect for this when the type of state happens to be a Record, for example:

setPerson (\currentPerson -> currentPerson {firstName = "NewName"})



or as shorthand:

setPerson _ {firstName = "NewName"}


Non-Record states can also follow this update pattern. See this guide for more details on best practices.

Recall that useState is used within an R.do block. R.do is a special react hooks variant of do. The R. prefix "qualifies" this as coming from React.Basic.Hooks, and means we use their hooks-compatible version of bind in the R.do block. This is known as a "qualified do". It lets us ignore the Hook (UseState state) wrapping and bind the inner Tuple of values to variables.

Another possible state management strategy is with useReducer, but that is outside the scope of this chapter.

Rendering JSX occurs here:

pure
$D.div { className: "container" , children: renderValidationErrors errors <> [ D.div { className: "row" , children: [ D.form_$ [ D.h3_ [ D.text "Basic Information" ]
, formField "First Name" "First Name" person.firstName \s ->
setPerson _ { firstName = s }
, formField "Last Name" "Last Name" person.lastName \s ->
setPerson _ { lastName = s }
, D.h3_ [ D.text "Address" ]
, formField "Street" "Street" person.homeAddress.street \s ->
setPerson _ { homeAddress { street = s } }
, formField "City" "City" person.homeAddress.city \s ->
setPerson _ { homeAddress { city = s } }
, formField "State" "State" person.homeAddress.state \s ->
setPerson _ { homeAddress { state = s } }
, D.h3_ [ D.text "Contact Information" ]
]
<> renderPhoneNumbers
]
}
]
}


Here we produce JSX which represents the intended state of the DOM. This JSX is typically created by applying functions corresponding to HTML tags (e.g. div, form, h3, li, ul, label, input) which create single HTML elements. These HTML elements are actually React components themselves, converted to JSX. There are usually three variants of each of these functions:

• div_: Accepts an array of child elements. Uses default attributes.
• div: Accepts a Record of attributes. An array of child elements may be passed to the children field of this record.
• div': Same as div, but returns the ReactComponent before conversion to JSX.

To display validation errors (if any) at the top of our form, we create a renderValidationErrors helper function that turns the Errors structure into an array of JSX. This array is prepended to the rest of our form.

renderValidationErrors :: Errors -> Array R.JSX
renderValidationErrors [] = []
renderValidationErrors xs =
let
renderError :: String -> R.JSX
renderError err = D.li_ [ D.text err ]
in
[ D.div
, children: [ D.ul_ (map renderError xs) ]
}
]


Note that since we are simply manipulating regular data structures here, we can use functions like map to build up more interesting elements:

children: [ D.ul_ (map renderError xs)]


We use the className property to define classes for CSS styling. We're using the Bootstrap stylesheet for this project, which is imported in index.html. For example, we want items in our form arranged as rows, and validation errors to be emphasized with alert-danger styling:

className: "alert alert-danger row"


A second helper function is formField, which creates a text input for a single form field:

formField :: String -> String -> String -> (String -> Effect Unit) -> R.JSX
formField name placeholder value setValue =
D.label
{ className: "form-group row"
, children:
[ D.div
{ className: "col-sm col-form-label"
, children: [ D.text name ]
}
, D.div
{ className: "col-sm"
, children:
[ D.input
{ className: "form-control"
, placeholder
, value
, onChange:
let
handleValue :: Maybe String -> Effect Unit
handleValue (Just v) = setValue v
handleValue Nothing  = pure unit
in
handler targetValue handleValue
}
]
}
]
}


Putting the input and display text in a label aids in accessibility for screen readers.

The onChange attribute allows us to describe how to respond to user input. We use the handler function, which has the following type:

handler :: forall a. EventFn SyntheticEvent a -> (a -> Effect Unit) -> EventHandler


For the first argument to handler we use we use targetValue, which provides the value of the text within the HTML input element. It matches the signature expected by handler where the type variable a in this case is Maybe String:

targetValue :: EventFn SyntheticEvent (Maybe String)


In JavaScript, the input element's onChange event is actually accompanied by a String value, but since strings in JavaScript can be null, Maybe is used for safety.

The second argument to handler, (a -> Effect Unit), must therefore have this signature:

Maybe String -> Effect Unit


It is a function that describes how to convert this Maybe String value into our desired effect. We define a custom handleValue function for this purpose and pass it to handler as follows:

onChange:
let
handleValue :: Maybe String -> Effect Unit
handleValue (Just v) = setValue v
handleValue Nothing  = pure unit
in
handler targetValue handleValue


setValue is the function we provided to each formField call that takes a string and makes the appropriate record-update call to the setPerson hook.

Note that handleValue can be substituted as:

onChange: handler targetValue $traverse_ setValue  Feel free to investigate the definition of traverse_ to see how both forms are indeed equivalent. That covers the basics of our component implementation. However, you should read the source accompanying this chapter in order to get a full understanding of the way the component works. Obviously, this user interface can be improved in a number of ways. The exercises will explore some ways in which we can make the application more usable. Exercises Modify src/Main.purs in the following exercises. There are no unit tests for these exercises. 1. (Easy) Modify the application to include a work phone number text box. 2. (Medium) Right now the application shows validation errors collected in a single "pink-alert" background. Modify to give each validation error its own pink-alert background by separating them with blank lines. Hint: Instead of using a ul element to show the validation errors in a list, modify the code to create one div with the alert and alert-danger styles for each error. 3. (Difficult, Extended) One problem with this user interface is that the validation errors are not displayed next to the form fields they originated from. Modify the code to fix this problem. Hint: the error type returned by the validator should be extended to indicate which field caused the error. You might want to use the following modified Errors type: data Field = FirstNameField | LastNameField | StreetField | CityField | StateField | PhoneField PhoneType data ValidationError = ValidationError String Field type Errors = Array ValidationError  You will need to write a function which extracts the validation error for a particular Field from the Errors structure. Conclusion This chapter has covered a lot of ideas about handling side-effects in PureScript: • We met the Monad type class, and its connection to do notation. • We introduced the monad laws, and saw how they allow us to transform code written using do notation. • We saw how monads can be used abstractly, to write code which works with different side-effects. • We saw how monads are examples of applicative functors, how both allow us to compute with side-effects, and the differences between the two approaches. • The concept of native effects was defined, and we met the Effect monad, which is used to handle native side-effects. • We used the Effect monad to handle a variety of effects: random number generation, exceptions, console IO, mutable state, and DOM manipulation using React. The Effect monad is a fundamental tool in real-world PureScript code. It will be used in the rest of the book to handle side-effects in a number of other use-cases. Asynchronous Effects Chapter Goals This chapter focuses on the Aff monad, which is similar to the Effect monad, but represents asynchronous side-effects. We'll demonstrate examples of asynchronously interacting with the filesystem and making HTTP requests. We'll also cover how to manage sequential and parallel execution of asynchronous effects. Project Setup New PureScript libraries introduced in this chapter are: • aff - defines the Aff monad. • node-fs-aff - asynchronous filesystem operations with Aff. • affjax - HTTP requests with AJAX and Aff. • parallel - parallel execution of Aff. When running outside of the browser (such as in our Node.js environment), the affjax library requires the xhr2 NPM module. Install that by running: $ npm install


Asynchronous JavaScript

A convenient way to work with asynchronous code in JavaScript is with async and await. See this article on asynchronous JavaScript for more background information.

Here is an example of using this technique to copy the contents of one file to another file:

var fsPromises = require('fs').promises;

async function copyFile(file1, file2) {
let data = await fsPromises.readFile(file1, { encoding: 'utf-8' });
fsPromises.writeFile(file2, data, { encoding: 'utf-8' });
}

copyFile('file1.txt', 'file2.txt')
.catch(e => {
console.log('There was a problem with copyFile: ' + e.message);
});


It is also possible to use callbacks or synchronous functions, but those are less desireable because:

• Callbacks lead to excessive nesting, known as "Callback Hell" or the "Pyramid of Doom".
• Synchronous functions block execution of the other code in your app.

Asynchronous PureScript

The Aff monad in PureScript offers similar ergonomics of JavaScript's async/await syntax. Here is the same copyFile example from before, but rewritten in PureScript using Aff:

import Prelude
import Data.Either (Either(..))
import Effect.Aff (Aff, attempt, message)
import Effect.Class.Console (log)
import Node.Encoding (Encoding(..))
import Node.Path (FilePath)

copyFile :: FilePath -> FilePath -> Aff Unit
copyFile file1 file2 = do
writeTextFile UTF8 file2 my_data

main :: Aff Unit
main = do
result <- attempt $copyFile "file1.txt" "file2.txt" case result of Left e -> log$ "There was a problem with copyFile: " <> message e
_ -> pure unit


It is also possible to re-write the above snippet using callbacks or synchronous functions (for example with Node.FS.Async and Node.FS.Sync respectively), but those share the same downsides as discussed earlier with JavaScript, and so that coding style is not recommended.

The syntax for working with Aff is very similar to working with Effect. They are both monads, and can therefore be written with do notation.

For example, if we look at the signature of readTextFile, we see that it returns the file contents as a String wrapped in Aff:

readTextFile :: Encoding -> FilePath -> Aff String


We can "unwrap" the returned string with a bind arrow (<-) in do notation:

my_data <- readTextFile UTF8 file1


Then pass it as the string argument to writeTextFile:

writeTextFile :: Encoding -> FilePath -> String -> Aff Unit


The only other notable feature unique to Aff in the above example is attempt, which captures errors or exceptions encountered while running Aff code and stores them in an Either:

attempt :: forall a. Aff a -> Aff (Either Error a)


You should hopefully be able to draw on your knowledge of concepts from previous chapters and combine this with the new Aff patterns learned in the above copyFile example to tackle the following exercises:

Exercises

1. (Easy) Write a concatenateFiles function which concatenates two text files.

2. (Medium) Write a function concatenateMany to concatenate multiple text files, given an array of input file names and an output file name. Hint: use traverse.

3. (Medium) Write a function countCharacters :: FilePath -> Aff (Either Error Int) that returns the number of characters in a file, or an error if one is encountered.

If you haven't already taken a look at the official Aff guide, skim through that now. It's not a direct prerequisite for completing the remaining exercises in this chapter, but you may find it helpful to lookup some functions on Pursuit.

You're also welcome to consult these supplemental resources too, but again, the exercises in this chapter don't depend on them:

A HTTP Client

The affjax library offers a convenient way to make asynchronous AJAX HTTP requests with Aff. Consult the Affjax docs for more usage information. Here is an example that makes HTTP GET requests at a provided URL and returns the response body or an error message:

import Prelude
import Affjax as AX
import Affjax.ResponseFormat as ResponseFormat
import Data.Either (Either(..))
import Effect.Aff (Aff)

getUrl :: String -> Aff String
getUrl url = do
result <- AX.get ResponseFormat.string url
pure $case result of Left err -> "GET /api response failed to decode: " <> AX.printError err Right response -> response.body  When calling this in the repl, launchAff_ is required to convert the Aff to a repl-compatible Effect: $ spago repl

> :pa
… import Prelude
… import Effect.Aff (launchAff_)
… import Effect.Class.Console (log)
… import Test.ExamplesHTTP (getUrl)
…
… launchAff_ do
…   str <- getUrl "https://reqres.in/api/users/1"
…   log str
…
unit
{"data":{"id":1,"email":"george.bluth@reqres.in","first_name":"George","last_name":"Bluth", ...}}


Exercises

1. (Easy) Write a function writeGet which makes an HTTP GET request to a provided url, and writes the response body to a file.

Parallel Computations

We've seen how to use the Aff monad and do notation to compose asynchronous computations in sequence. It would also be useful to be able to compose asynchronous computations in parallel. With Aff, we can compute in parallel simply by initiating our two computations one after the other.

The parallel package defines a type class Parallel for monads like Aff which support parallel execution. When we met applicative functors earlier in the book, we observed how applicative functors can be useful for combining parallel computations. In fact, an instance for Parallel defines a correspondence between a monad m (such as Aff) and an applicative functor f which can be used to combine computations in parallel:

class (Monad m, Applicative f) <= Parallel f m | m -> f, f -> m where
sequential :: forall a. f a -> m a
parallel :: forall a. m a -> f a


The class defines two functions:

• parallel, which takes computations in the monad m and turns them into computations in the applicative functor f, and
• sequential, which performs a conversion in the opposite direction.

The aff library provides a Parallel instance for the Aff monad. It uses mutable references to combine Aff actions in parallel, by keeping track of which of the two continuations has been called. When both results have been returned, we can compute the final result and pass it to the main continuation.

Because applicative functors support lifting of functions of arbitrary arity, we can perform more computations in parallel by using the applicative combinators. We can also benefit from all of the standard library functions which work with applicative functors, such as traverse and sequence!

We can also combine parallel computations with sequential portions of code, by using applicative combinators in a do notation block, or vice versa, using parallel and sequential to change type constructors where appropriate.

To demonstrate the difference between sequential and parallel execution, we'll create an array of 100 10-millisecond delays, then execute those delays with both techniques. You'll notice in the repl that seqDelay is much slower than parDelay. Note that parallel execution is enabled by simply by replacing sequence_ with parSequence_.

import Prelude

import Control.Parallel (parSequence_)
import Data.Array (replicate)
import Data.Foldable (sequence_)
import Effect (Effect)
import Effect.Aff (Aff, Milliseconds(..), delay, launchAff_)

delayArray :: Array (Aff Unit)
delayArray = replicate 100 $delay$ Milliseconds 10.0

seqDelay :: Effect Unit
seqDelay = launchAff_ $sequence_ delayArray parDelay :: Effect Unit parDelay = launchAff_$ parSequence_ delayArray

$spago repl > import Test.ParallelDelay > seqDelay -- This is slow unit > parDelay -- This is fast unit  Here's a more real-world example of making multiple HTTP requests in parallel. We're reusing our getUrl function to fetch information from two users in parallel. Note that parTraverse (the parallel version of traverse) is used in this case. This example would also work fine with traverse instead, but it will be slower. import Prelude import Control.Parallel (parTraverse) import Effect (Effect) import Effect.Aff (launchAff_) import Effect.Class.Console (logShow) import Test.HTTP (getUrl) fetchPar :: Effect Unit fetchPar = launchAff_$ do
let
urls = map (\n -> "https://reqres.in/api/users/" <> show n) [ 1, 2 ]
res <- parTraverse getUrl urls
logShow res

$spago repl > import Test.ParallelFetch > fetchPar unit ["{\"data\":{\"id\":1,\"email\":\"george.bluth@reqres.in\", ... }" ,"{\"data\":{\"id\":2,\"email\":\"janet.weaver@reqres.in\", ... }" ]  A full listing of available parallel functions can be found in the parallel docs on Pursuit. The aff docs section on parallel also contains more examples. Exercises 1. (Easy) Write a concatenateManyParallel function which has the same signature as the earlier concatenateMany function, but reads all input files in parallel. 2. (Medium) Write a getWithTimeout :: Number -> String -> Aff (Maybe String) function which makes an HTTP GET request at the provided URL and returns either: • Nothing: if the request takes longer than the provided timeout (in milliseconds). • The string response: if the request succeeds before the timeout elapses. 3. (Difficult) Write a recurseFiles function which takes a "root" file and returns an array of all paths listed in that file (and listed in the listed files too). Read listed files in parallel. Paths are relative to the directory of the file they appear in. Hint: The node-path module has some helpful functions for negotiating directories. For example, if starting from the following root.txt file: $ cat root.txt
a.txt
b/a.txt
c/a/a.txt

$cat a.txt b/b.txt$ cat b/b.txt
c/a.txt

$cat b/c/a.txt$ cat b/a.txt

$cat c/a/a.txt  The expected output is: ["root.txt","a.txt","b/a.txt","b/b.txt","b/c/a.txt","c/a/a.txt"]  Conclusion In this chapter we covered asynchronous effects and learned how to: • Run asynchronous code in the Aff monad with the aff library. • Make HTTP requests asynchronously with the affjax library. • Run asynchronous code in parallel with the parallel library. The Foreign Function Interface Chapter Goals This chapter will introduce PureScript's foreign function interface (or FFI), which enables communication from PureScript code to JavaScript code, and vice versa. We will cover how to: • Call pure, effectful, and asynchronous JavaScript functions from PureScript. • Work with untyped data. • Encode and parse JSON using the argonaut package. Towards the end of this chapter, we will revisit our recurring address book example. The goal of the chapter will be to add the following new functionality to our application using the FFI: • Alert the user with a popup notification. • Store the serialized form data in the browser's local storage, and reload it when the application restarts. There is also an addendum which covers some additional topics which are not as commonly sought-after. Feel free to read these sections, but don't let them stand in the way of progressing through the remainder of the book if they're less relevant to your learning objectives: • Understand the representation of PureScript values at runtime. • Call PureScript functions from JavaScript. Project Setup The source code for this module is a continuation of the source code from chapters 3, 7 and 8. As such, the source tree includes the appropriate source files from those chapters. This chapter introduces the argonaut library as a dependency. This library is used for encoding and decoding JSON. The exercises for this chapter should be written in test/MySolutions.purs and can be checked against the unit tests in test/Main.purs by running spago test. The Address Book app can be launched with parcel src/index.html --open. It uses the same workflow from Chapter 8, so refer to that chapter for more detailed instructions. A Disclaimer PureScript provides a straightforward foreign function interface to make working with JavaScript as simple as possible. However, it should be noted that the FFI is an advanced feature of the language. To use it safely and effectively, you should have an understanding of the runtime representation of the data you plan to work with. This chapter aims to impart such an understanding as pertains to code in PureScript's standard libraries. PureScript's FFI is designed to be very flexible. In practice, this means that developers have a choice, between giving their foreign functions very simple types, or using the type system to protect against accidental misuses of foreign code. Code in the standard libraries tends to favor the latter approach. As a simple example, a JavaScript function makes no guarantees that its return value will not be null. Indeed, idiomatic JavaScript code returns null quite frequently! However, PureScript's types are usually not inhabited by a null value. Therefore, it is the responsibility of the developer to handle these corner cases appropriately when designing their interfaces to JavaScript code using the FFI. Calling JavaScript From PureScript The simplest way to use JavaScript code from PureScript is to give a type to an existing JavaScript value using a foreign import declaration. Foreign import declarations should have a corresponding JavaScript declaration in a foreign JavaScript module. For example, consider the encodeURIComponent function, which can be used from JavaScript to encode a component of a URI by escaping special characters: $ node

node> encodeURIComponent('Hello World')
'Hello%20World'


This function has the correct runtime representation for the function type String -> String, since it takes non-null strings to non-null strings, and has no other side-effects.

We can assign this type to the function with the following foreign import declaration:

module Test.URI where

foreign import encodeURIComponent :: String -> String


We also need to write a foreign JavaScript module. If the module above is saved as test/URI.purs, then the foreign JavaScript module should be saved as test/URI.js:

"use strict";

exports.encodeURIComponent = encodeURIComponent;


Spago will find .js files in the src and test directories, and provide them to the compiler as foreign JavaScript modules.

JavaScript functions and values are exported from foreign JavaScript modules by assigning them to the exports object just like a regular CommonJS module. The purs compiler (wrapped by spago) treats this module like a regular CommonJS module, and simply adds it as a dependency to the compiled PureScript module. However, when bundling code for the browser with psc-bundle or spago bundle-app --to, it is very important to follow the pattern above, assigning exports to the exports object using a property assignment. This is because psc-bundle recognizes this format, allowing unused JavaScript exports to be removed from bundled code.

With these two pieces in place, we can now use the encodeURIComponent function from PureScript like any function written in PureScript. For example, if this declaration is saved as a module and loaded into PSCi, we can reproduce the calculation above:

$spago repl > import Test.URI > encodeURIComponent "Hello World" "Hello%20World"  We can also define our own functions in foreign modules. Here's an example of how to create and call a custom JavaScript function that squares a Number: test/Examples.js: "use strict"; exports.square = function (n) { return n * n; };  test/Examples.purs: module Test.Examples where foreign import square :: Number -> Number  $ spago repl

> import Test.Examples
> square 5.0
25.0


Functions of Multiple Arguments

Let's rewrite our diagonal function from Chapter 2 in a foreign module to demonstrate how to call functions of multiple arguments. Recall that this function calculates the diagonal of a right-angled triangle:

exports.diagonal = function(w, h) {
return Math.sqrt(w * w + h * h);
};


Because PureScript uses curried functions of single arguments, we cannot directly import the diagonal function of two arguments like so:

-- This will not work with above uncurried definition of diagonal
foreign import diagonal :: Number -> Number -> Number


However, there are a few solutions to this dilemma:

The first option is to import and run the function with an Fn wrapper from Data.Function.Uncurried (Fn and uncurried functions are discussed in more detail later):

foreign import diagonal :: Fn2 Number Number Number

$spago repl > import Test.Examples > import Data.Function.Uncurried > runFn2 diagonal 3.0 4.0 5.0  The second option is to wrap or rewrite the function as curried JavaScript: exports.diagonalNested = function(w) { return function (h) { return Math.sqrt(w * w + h * h); }; };  or equivalently with arrow functions (see ES6 note below): exports.diagonalArrow = w => h => Math.sqrt(w * w + h * h);  foreign import diagonalNested :: Number -> Number -> Number foreign import diagonalArrow :: Number -> Number -> Number  $ spago repl

> import Test.Examples
> diagonalNested 3.0 4.0
5.0
> diagonalArrow 3.0 4.0
5.0


PureScript's Prelude contains an interesting set of examples of foreign types. As we have covered already, PureScript's function types only take a single argument, and can be used to simulate functions of multiple arguments via currying. This has certain advantages - we can partially apply functions, and give type class instances for function types - but it comes with a performance penalty. For performance critical code, it is sometimes necessary to define genuine JavaScript functions which accept multiple arguments. The Prelude defines foreign types which allow us to work safely with such functions.

For example, the following foreign type declaration is taken from the Data.Function.Uncurried module:

foreign import data Fn2 :: Type -> Type -> Type -> Type


This defines the type constructor Fn2 which takes three type arguments. Fn2 a b c is a type representing JavaScript functions of two arguments of types a and b, and with return type c.

The functions package defines similar type constructors for function arities from 0 to 10.

We can create a function of two arguments by using the mkFn2 function, as follows:

import Data.Function.Uncurried

uncurriedAdd :: Fn2 Int Int Int
uncurriedAdd = mkFn2 \n m -> m + n


and we can apply a function of two arguments by using the runFn2 function:

$spago repl > import Test.Examples > import Data.Function.Uncurried > runFn2 uncurriedAdd 3 10 13  The key here is that the compiler inlines the mkFn2 and runFn2 functions whenever they are fully applied. The result is that the generated code is very compact: var uncurriedAdd = function (n, m) { return m + n | 0; };  For contrast, here is a traditional curried function: curriedAdd :: Int -> Int -> Int curriedAdd n m = m + n  and the resulting generated code, which is less compact due to the nested functions: var curriedAdd = function (n) { return function (m) { return m + n | 0; }; };  A Note About Modern JavaScript Syntax The arrow function syntax we saw earlier is an ES6 feature, and so it is incompatible with some older browsers (namely IE11). As of writing, it is estimated that arrow functions are unavailable for the 6% of users who have not yet updated their web browser. In order to be compatible with the most users, the JavaScript code generated by the PureScript compiler does not use arrow functions. It is also recommended to avoid arrow functions in public libraries for the same reason. You may still use arrow functions in your own FFI code, but then should include a tool such as Babel in your deployment workflow to convert these back to ES5 compatible functions. If you find arrow functions in ES6 more readable, you may transform JavaScript code in the compiler's output directory with a tool like Lebab: npm i -g lebab lebab --replace output/ --transform arrow,arrow-return  This operation would convert the above curriedAdd function to: var curriedAdd = n => m => m + n | 0;  The remaining examples in this book will use arrow functions instead of nested functions. Exercises 1. (Medium) Write a JavaScript function volumeFn in the Test.MySolutions module that finds the volume of a box. Use an Fn wrapper from Data.Function.Uncurried. 2. (Medium) Rewrite volumeFn with arrow functions as volumeArrow. Passing Simple Types The following data types may be passed between PureScript and JavaScript as-is: PureScriptJavaScript BooleanBoolean StringString Int, NumberNumber ArrayArray RecordObject We've already seen examples with the primitive types String and Number. We'll now take a look at the structural types Array and Record (Object in JavaScript). To demonstrate passing Arrays, here's how to call a JavaScript function which takes an Array of Int and returns the cumulative sum as another array. Recall that, since JavaScript does not have a separate type for Int, both Int and Number in PureScript translate to Number in JavaScript. foreign import cumulativeSums :: Array Int -> Array Int  exports.cumulativeSums = arr => { let sum = 0 let sums = [] arr.forEach(x => { sum += x; sums.push(sum); }); return sums; };  $ spago repl

> import Test.Examples
> cumulativeSums [1, 2, 3]
[1,3,6]


To demonstrate passing Records, here's how to call a JavaScript function which takes two Complex numbers as records, and returns their sum as another record. Note that a Record in PureScript is represented as an Object in JavaScript:

type Complex = {
real :: Number,
imag :: Number
}

foreign import addComplex :: Complex -> Complex -> Complex

exports.addComplex = a => b => {
return {
real: a.real + b.real,
imag: a.imag + b.imag
}
};

$spago repl > import Test.Examples > addComplex { real: 1.0, imag: 2.0 } { real: 3.0, imag: 4.0 } { imag: 6.0, real: 4.0 }  Note that the above techniques require trusting that JavaScript will return the expected types, as PureScript is not able to apply type checking to JavaScript code. We will describe this type safety concern in more detail later on in the JSON section, as well as cover techniques to protect against type mismatches. Exercises 1. (Medium) Write a JavaScript function cumulativeSumsComplex (and corresponding PureScript foreign import) that takes an Array of Complex numbers and returns the cumulative sum as another array of complex numbers. Beyond Simple Types We have seen examples of how to send and receive types with a native JavaScript representation, such as String, Number, Array, and Record, over FFI. Now we'll cover how to use some of the other types available in PureScript, like Maybe. Suppose we wanted to recreate the head function on arrays by using a foreign declaration. In JavaScript, we might write the function as follows: exports.head = arr => arr[0];  However, there is a problem with this function. We might try to give it the type forall a. Array a -> a, but for empty arrays, this function returns undefined. Therefore, this function does not have the correct runtime representation. We can instead return a Maybe value to handle this corner case. It is tempting to write the following: // Don't do this exports.maybeHead = arr => { if (arr.length) { return Data_Maybe.Just.create(arr[0]); } else { return Data_Maybe.Nothing.value; } }  foreign import maybeHead :: forall a. Array a -> Maybe a  But calling these Maybe constructors directly in the FFI code isn't recommended as it makes the code brittle to changes in the code generator. Additionally, doing this can cause problems when using purs bundle for dead code elimination. The recommended approach is to add extra parameters to your FFI-defined function to accept the functions you need to call as arguments: exports.maybeHeadImpl = just => nothing => arr => { if (arr.length) { return just(arr[0]); } else { return nothing; } };  foreign import maybeHeadImpl :: forall a. (forall x. x -> Maybe x) -> (forall x. Maybe x) -> Array a -> Maybe a maybeHead :: forall a. Array a -> Maybe a maybeHead arr = maybeHeadImpl Just Nothing arr  Note that we wrote: forall a. (forall x. x -> Maybe x) -> (forall x. Maybe x) -> Array a -> Maybe a  and not: forall a. ( a -> Maybe a) -> Maybe a -> Array a -> Maybe a  While both forms work, the latter is more vulnerable to unwanted inputs in place of Just and Nothing. For example, in the more vulnerable case we could call it as follows: maybeHeadImpl (\_ -> Just 1000) (Just 1000) [1,2,3]  which returns Just 1000 for any array input. This vulnerability is allowed because (\_ -> Just 1000) and Just 1000 match the signatures of (a -> Maybe a) and Maybe a respectively when a is Int (based on input array). In the more secure type signature, even when a is determined to be Int based on the input array, we still need to provide valid functions matching the signatures involving forall x. The only option for (forall x. Maybe x) is Nothing, since a Just would assume a type for x and then no longer be valid for all x. The only options for (forall x. x -> Maybe x) are Just (our desired argument) and (\_ -> Nothing), which is the only remaining vulnerability. Defining Foreign Types Suppose instead of returning a Maybe a, we wanted to return a new type Undefined a whose representation at runtime was like that for the type a, but also allowing the undefined value. We can define a foreign type using the FFI using a foreign type declaration. The syntax is similar to defining a foreign function: foreign import data Undefined :: Type -> Type  Note that the data keyword here indicates that we are defining a type, not a value. Instead of a type signature, we give the kind of the new type. In this case, we declare the kind of Undefined to be Type -> Type. In other words, Undefined is a type constructor. We can now simply reuse our original definition for head: exports.undefinedHead = arr => arr[0];  And in the PureScript module: foreign import undefinedHead :: forall a. Array a -> Undefined a  The body of the undefinedHead function returns arr[0] even if that value is undefined, and the type signature reflects the fact that our function can return an undefined value. This function has the correct runtime representation for its type, but is quite useless since we have no way to use a value of type Undefined a. But we can fix that by writing some new functions using the FFI! The most basic function we need will tell us whether a value is defined or not: foreign import isUndefined :: forall a. Undefined a -> Boolean  This is easily defined in our foreign JavaScript module as follows: exports.isUndefined = value => value === undefined;  We can now use isUndefined and undefinedHead together from PureScript to define a useful function: isEmpty :: forall a. Array a -> Boolean isEmpty = isUndefined <<< undefinedHead  Here, the foreign function we defined is very simple, which means we can benefit from the use of PureScript's typechecker as much as possible. This is good practice in general: foreign functions should be kept as small as possible, and application logic moved into PureScript code wherever possible. Exceptions Another option is to simply throw an exception in the case of an empty array. Strictly speaking, pure functions should not throw exceptions, but we have the flexibility to do so. We indicate the lack of safety in the function name: foreign import unsafeHead :: forall a. Array a -> a  In our foreign JavaScript module, we can define unsafeHead as follows: exports.unsafeHead = arr => { if (arr.length) { return arr[0]; } else { throw new Error('unsafeHead: empty array'); } };  Exercises 1. (Medium) Given a record that represents a quadratic polynomial a*x^2 + b*x + c = 0: type Quadratic = { a :: Number, b :: Number, c :: Number }  Write a JavaScript function quadraticRootsImpl and a wrapper quadraticRoots :: Quadratic -> Pair Complex that uses the quadratic formula to find the roots of this polynomial. Return the two roots as a Pair of Complex numbers. Hint: Use the quadraticRoots wrapper to pass a constructor for Pair to quadraticRootsImpl. Using Type Class Member Functions Just like our earlier guide on passing the Maybe constructor over FFI, this is another case of writing PureScript that calls JavaScript, which in turn calls PureScript functions again. Here we will explore how to pass type class member functions over the FFI. We start with writing a foreign JavaScript function which expects the appropriate instance of show to match the type of x. exports.boldImpl = show => x => show(x).toUpperCase() + "!!!";  Then we write the matching signature: foreign import boldImpl :: forall a. (a -> String) -> a -> String  and a wrapper function that passes the correct instance of show: bold :: forall a. Show a => a -> String bold x = boldImpl show x  Alternatively in point-free form: bold :: forall a. Show a => a -> String bold = boldImpl show  We can then call the wrapper: $ spago repl

> import Test.Examples
> import Data.Tuple
> bold (Tuple 1 "Hat")
"(TUPLE 1 \"HAT\")!!!"


Here's another example demonstrating passing multiple functions, including a function of multiple arguments (eq):

exports.showEqualityImpl = eq => show => a => b => {
if (eq(a)(b)) {
return "Equivalent";
} else {
return show(a) + " is not equal to " + show(b);
}
}

foreign import showEqualityImpl :: forall a. (a -> a -> Boolean) -> (a -> String) -> a -> a -> String

showEquality :: forall a. Eq a => Show a => a -> a -> String
showEquality = showEqualityImpl eq show

$spago repl > import Test.Examples > import Data.Maybe > showEquality Nothing (Just 5) "Nothing is not equal to (Just 5)"  Effectful Functions Let's extend our bold function to log to the console. Logging is an Effect, and Effects are represented in JavaScript as a function of zero arguments, () with arrow notation: exports.yellImpl = show => x => () => console.log(show(x).toUpperCase() + "!!!");  The new foreign import is the same as before, except that the return type changed from String to Effect Unit. foreign import yellImpl :: forall a. (a -> String) -> a -> Effect Unit yell :: forall a. Show a => a -> Effect Unit yell = yellImpl show  When testing this in the repl, notice that the string is printed directly to the console (instead of being quoted) and a unit value is returned. $ spago repl

> import Test.Examples
> import Data.Tuple
> yell (Tuple 1 "Hat")
(TUPLE 1 "HAT")!!!
unit


There are also EffectFn wrappers from Effect.Uncurried. These are similar to the Fn wrappers from Data.Function.Uncurried that we've already seen. These wrappers let you call uncurried effectful functions in PureScript.

You'd generally only use these if you want to call existing JavaScript library APIs directly, rather than wrapping those APIs in curried functions. So it doesn't make much sense to present an example of uncurried yell, where the JavaScript relies on PureScript type class members, since you wouldn't find that in the existing JavaScript ecosystem.

Instead, we'll modify our previous diagonal example to include logging in addition to returning the result:

exports.diagonalLog = function(w, h) {
let result = Math.sqrt(w * w + h * h);
console.log("Diagonal is " + result);
return result;
};

foreign import diagonalLog :: EffectFn2 Number Number Number

$spago repl > import Test.Examples > import Effect.Uncurried > runEffectFn2 diagonalLog 3.0 4.0 Diagonal is 5 5.0  Asynchronous Functions Promises in JavaScript translate directly to asynchronous effects in PureScript with the help of the aff-promise library. See that library's documentation for more information. We'll just go through a few examples. Suppose we want to use this JavaScript wait promise (or asynchronous function) in our PureScript project. It may be used to delay execution for ms milliseconds. const wait = ms => new Promise(resolve => setTimeout(resolve, ms));  We just need to export it wrapped as an Effect (function of zero arguments): exports.sleepImpl = ms => () => wait(ms);  Then import it as follows: foreign import sleepImpl :: Int -> Effect (Promise Unit) sleep :: Int -> Aff Unit sleep = sleepImpl >>> toAffE  We can then run this Promise in an Aff block like so: $ spago repl

> import Prelude
> import Test.Examples
> import Effect.Class.Console
> import Effect.Aff
> :pa
… launchAff_ do
…   log "waiting"
…   sleep 300
…   log "done waiting"
…
waiting
unit
done waiting


Note that asynchronous logging in the repl just waits to print until the entire block has finished executing. This code behaves more predictably when run with spago test where there is a slight delay between prints.

Let's look at another example where we return a value from a promise. This function is written with async and await, which is just syntactic sugar for promises.

async function diagonalWait(delay, w, h) {
await wait(delay);
return Math.sqrt(w * w + h * h);
}

exports.diagonalAsyncImpl = delay => w => h => () =>
diagonalWait(delay, w, h);


Since we're returning a Number, we represent this type in the Promise and Aff wrappers:

foreign import diagonalAsyncImpl :: Int -> Number -> Number -> Effect (Promise Number)

diagonalAsync :: Int -> Number -> Number -> Aff Number
diagonalAsync i x y = toAffE $diagonalAsyncImpl i x y  $ spago repl

import Prelude
import Test.Examples
import Effect.Class.Console
import Effect.Aff
> :pa
… launchAff_ do
…   res <- diagonalAsync 300 3.0 4.0
…   logShow res
…
unit
5.0


Exercises

Exercises for the above sections are still on the ToDo list. If you have any ideas for good exercises, please make a suggestion.

JSON

There are many reasons to use JSON in an application, for example, it's a common means of communicating with web APIs. This section will discuss other use-cases too, beginning with a technique to improve type safety when passing structural data over the FFI.

Let's revisit our earlier FFI functions cumulativeSums and addComplex and introduce a bug to each:

exports.cumulativeSumsBroken = arr => {
let sum = 0
let sums = []
arr.forEach(x => {
sum += x;
sums.push(sum);
});
sums.push("Broken"); // Bug
return sums;
};

exports.addComplexBroken = a => b => {
return {
real: a.real + b.real,
broken: a.imag + b.imag // Bug
}
};


We can use the original type signatures, and the code will still compile, despite the fact that the return types are incorrect.

foreign import cumulativeSumsBroken :: Array Int -> Array Int

foreign import addComplexBroken :: Complex -> Complex -> Complex


We can even execute the code, which might either produce unexpected results or a runtime error:

$spago repl > import Test.Examples > import Data.Foldable (sum) > sums = cumulativeSumsBroken [1, 2, 3] > sums [1,3,6,Broken] > sum sums 0 > complex = addComplexBroken { real: 1.0, imag: 2.0 } { real: 3.0, imag: 4.0 } > complex.real 4.0 > complex.imag + 1.0 NaN > complex.imag var str = n.toString(); ^ TypeError: Cannot read property 'toString' of undefined  For example, our resulting sums is no-longer a valid Array Int, now that a String is included in the Array. And further operations produce unexpected behavior, rather than an outright error, as the sum of these sums is 0 rather than 10. This could be a difficult bug to track down! Likewise, there are no errors when calling addComplexBroken; however, accessing the imag field of our Complex result will either produce unexpected behavior (returning NaN instead of 7.0), or a non-obvious runtime error. Let's use JSON to make our PureScript code more impervious to bugs in JavaScript code. The argonaut library contains the JSON decoding and encoding capabilities we need. That library has excellent documentation, so we will only cover basic usage in this book. If we create an alternate foreign import that defines the return type as Json: foreign import cumulativeSumsJson :: Array Int -> Json foreign import addComplexJson :: Complex -> Complex -> Json  Note that we're simply pointing to our existing broken functions: exports.cumulativeSumsJson = exports.cumulativeSumsBroken exports.addComplexJson = exports.addComplexBroken  And then write a wrapper to decode the returned foreign Json value: cumulativeSumsDecoded :: Array Int -> Either JsonDecodeError (Array Int) cumulativeSumsDecoded arr = decodeJson$ cumulativeSumsJson arr

addComplexDecoded :: Complex -> Complex -> Either JsonDecodeError Complex
addComplexDecoded a b = decodeJson $addComplexJson a b  Then any values that can't be successfully decoded to our return type appear as a Left error String: $ spago repl

> import Test.Examples

> cumulativeSumsDecoded [1, 2, 3]
(Left "Couldn't decode Array (Failed at index 3): Value is not a Number")

> addComplexDecoded { real: 1.0, imag: 2.0 } { real: 3.0, imag: 4.0 }
(Left "JSON was missing expected field: imag")


If we call the working versions, a Right value is returned.

Try this yourself by modifying test/Examples.js with the following change to point to the working versions before running the next repl block.

exports.cumulativeSumsJson = exports.cumulativeSums

$spago repl > import Test.Examples > cumulativeSumsDecoded [1, 2, 3] (Right [1,3,6]) > addComplexDecoded { real: 1.0, imag: 2.0 } { real: 3.0, imag: 4.0 } (Right { imag: 6.0, real: 4.0 })  Using JSON is also the easiest way to pass other structural types, such as Map and Set through the FFI. Note that since JSON only consists of booleans, numbers, strings, arrays, and objects of other JSON values, we can't write a Map and Set directly in JSON. But we can represent these structures as arrays (assuming the keys and values can also be represented in JSON), and then decode them back to Map or Set. Here's an example of a foreign function signature that modifies a Map of String keys and Int values, along with the wrapper function that handles JSON encoding and decoding. foreign import mapSetFooJson :: Json -> Json mapSetFoo :: Map String Int -> Either JsonDecodeError (Map String Int) mapSetFoo = encodeJson >>> mapSetFooJson >>> decodeJson  Note that this is a prime use case for function composition. Both of these alternatives are equivalent to the above: mapSetFoo :: Map String Int -> Either JsonDecodeError (Map String Int) mapSetFoo = decodeJson <<< mapSetFooJson <<< encodeJson mapSetFoo :: Map String Int -> Either JsonDecodeError (Map String Int) mapSetFoo = encodeJson >>> mapSetFooJson >>> decodeJson  Here is the JavaScript implementation. Note the Array.from step which is necessary to convert the JavaScript Map into a JSON-friendly format before decoding converts it back to a PureScript Map. exports.mapSetFooJson = j => { let m = new Map(j); m.set("Foo", 42); return Array.from(m); };  Now we can send and receive a Map over the FFI: $ spago repl

> import Test.Examples
> import Data.Map
> import Data.Tuple

> myMap = fromFoldable [ Tuple "hat" 1, Tuple "cat" 2 ]

> :type myMap
Map String Int

> myMap
(fromFoldable [(Tuple "cat" 2),(Tuple "hat" 1)])

> mapSetFoo myMap
(Right (fromFoldable [(Tuple "Foo" 42),(Tuple "cat" 2),(Tuple "hat" 1)]))


Exercises

1. (Medium) Write a JavaScript function and PureScript wrapper valuesOfMap :: Map String Int -> Either JsonDecodeError (Set Int) that returns a Set of all the values in a Map. Hint: The .values() instance method for Map may be useful in your JavaScript code.

2. (Easy) Write a new wrapper for the previous JavaScript function with the signature valuesOfMapGeneric :: forall k v. Map k v -> Either JsonDecodeError (Set v) so it works with a wider variety of maps. Note that you'll need to add some type class constraints for k and v. The compiler will guide you.

3. (Medium) Rewrite the earlier quadraticRoots function as quadraticRootsSet which returns the Complex roots as a Set via JSON (instead of as a Pair).

4. (Difficult) Rewrite the earlier quadraticRoots function as quadraticRootsSafe which uses JSON to pass the Pair of Complex roots over FFI. Don't use the Pair constructor in JavaScript, but instead, just return the pair in a decoder-compatible format. Hint: You'll need to write a DecodeJson instance for Pair. Consult the argonaut docs for instruction on writing your own decode instance. Their decodeJsonTuple instance may also be a helpful reference. Note that you'll need a newtype wrapper for Pair to avoid creating an "orphan instance".

5. (Medium) Write a parseAndDecodeArray2D :: String -> Either String (Array (Array Int)) function to parse and decode a JSON string containing a 2D array, such as "[[1, 2, 3], [4, 5], [6]]". Hint: You'll need to use jsonParser to convert the String into Json before decoding.

6. (Medium) The following data type represents a binary tree with values at the leaves:

data Tree a
= Leaf a
| Branch (Tree a) (Tree a)


Derive generic EncodeJson and DecodeJson instances for the Tree type. Consult the argonaut docs for instructions on how to do this. Note that you'll also need generic instances of Show and Eq to enable unit testing for this exercise, but those should be straightforward to implement after tackling the JSON instances.

7. (Difficult) The following data type should be represented directly in JSON as either an integer or a string:

data IntOrString
= IntOrString_Int Int
| IntOrString_String String


Write instances of EncodeJson and DecodeJson for the IntOrString data type which implement this behavior. Hint: The alt operator from Control.Alt may be helpful.

In this section we will apply our newly-acquired FFI and JSON knowledge to build on our address book example from chapter 8. We will add the following features:

• A Save button at the bottom of the form that, when clicked, serializes the state of the form to JSON and saves it in local storage.
• Automatic retrieval of the JSON document from local storage upon page reload. The form fields are populated with the contents of this document.

We'll start by creating FFI wrappers for the following Web Storage APIs in our Effect.Storage module:

• setItem takes a key and a value (both strings), and returns a computation which stores (or updates) the value in local storage at the specified key.
• getItem takes a key, and attempts to retrieve the associated value from local storage. However, since the getItem method on window.localStorage can return null, the return type is not String, but Json.
foreign import setItem :: String -> String -> Effect Unit

foreign import getItem :: String -> Effect Json


Here is the corresponding JavaScript implementation of these functions in Effect/Storage.js:

exports.setItem = key => value => () =>
window.localStorage.setItem(key, value);

exports.getItem = key => () =>
window.localStorage.getItem(key);


We'll create a save button like so:

saveButton :: R.JSX
saveButton =
D.label
{ className: "form-group row col-form-label"
, children:
[ D.button
{ className: "btn-primary btn"
, onClick: handler_ validateAndSave
, children: [ D.text "Save" ]
}
]
}


And write our validated person as a JSON string with setItem in the validateAndSave function:

validateAndSave :: Effect Unit
validateAndSave = do
log "Running validators"
case validatePerson' person of
Left errs -> log $"There are " <> show (length errs) <> " validation errors." Right validPerson -> do setItem "person"$ stringify $encodeJson validPerson log "Saved"  Note that if we attempt to compile at this stage, we'll encounter the following error:  No type class instance was found for Data.Argonaut.Encode.Class.EncodeJson PhoneType  This is because PhoneType in the Person record needs an EncodeJson instance. We'll just derive a generic encode instance, and a decode instance too while we're at it. More information how this works is available in the argonaut docs: import Data.Argonaut (class DecodeJson, class EncodeJson) import Data.Argonaut.Decode.Generic (genericDecodeJson) import Data.Argonaut.Encode.Generic (genericEncodeJson) import Data.Generic.Rep (class Generic) derive instance genericPhoneType :: Generic PhoneType _ instance encodeJsonPhoneType :: EncodeJson PhoneType where encodeJson = genericEncodeJson instance decodeJsonPhoneType :: DecodeJson PhoneType where decodeJson = genericDecodeJson  Now we can save our person to local storage, but this isn't very useful unless we can retrieve the data. We'll tackle that next. We'll start with retrieving the "person" string from local storage: item <- getItem "person"  Then we'll create a helper function to handle converting the string from local storage to our Person record. Note that this string in storage may be null, so we represent it as a foreign Json until it is successfully decoded as a String. There are a number of other conversion steps along the way - each of which return an Either value, so it makes sense to organize these together in a do block. processItem :: Json -> Either String Person processItem item = do jsonString <- decodeJson item j <- jsonParser jsonString decodeJson j  Then we inspect this result to see if it succeeded. If it failed, we'll log the errors and use our default examplePerson, otherwise we'll use the person retrieved from local storage. initialPerson <- case processItem item of Left err -> do log$ "Error: " <> err <> ". Loading examplePerson"
pure examplePerson
Right p   -> pure p


Finally, we'll pass this initialPerson to our component via the props record:

-- Create JSX node from react component.
app = element addressBookApp { initialPerson }


And pick it up on the other side to use in our state hook:

mkAddressBookApp :: Effect (ReactComponent { initialPerson :: Person })
Tuple person setPerson <- useState props.initialPerson


As a finishing touch, we'll improve the quality of our error messages by appending to the String of each Left value with lmap.

processItem :: Json -> Either String Person
processItem item = do
jsonString <- lmap ("No string in local storage: " <> _) $decodeJson item j <- lmap ("Cannot parse JSON string: " <> _)$ jsonParser jsonString
lmap               ("Cannot decode Person: "       <> _) $decodeJson j  Only the first error should ever occur during normal operation of this app. You can trigger the other errors by opening your web browser's dev tools, editing the saved "person" string in local storage, and refreshing the page. How you modify the JSON string determines which error is triggered. See if you can trigger each of them. That covers local storage. Next we'll implement the alert action, which is very similar to the log action from the Effect.Console module. The only difference is that the alert action uses the window.alert method, whereas the log action uses the console.log method. As such, alert can only be used in environments where window.alert is defined, such as a web browser. foreign import alert :: String -> Effect Unit  exports.alert = msg => () => window.alert(msg);  We want this alert to appear when either: • A user attempts to save a form with validation errors. • The state cannot be retrieved from local storage. That is accomplished by simply replacing log with alert on these lines: Left errs -> alert$ "There are " <> show (length errs) <> " validation errors."

alert $"Error: " <> err <> ". Loading examplePerson"  Exercises 1. (Easy) Write a wrapper for the removeItem method on the localStorage object, and add your foreign function to the Effect.Storage module. 2. (Medium) Add a "Reset" button that, when clicked, calls the newly-created removeItem function to delete the "person" entry from local storage. 3. (Easy) Write a wrapper for the confirm method on the JavaScript Window object, and add your foreign function to the Effect.Alert module. 4. (Medium) Call this confirm function when a users clicks the "Reset" button to ask if they're sure they want to reset their address book. Conclusion In this chapter, we've learned how to work with foreign JavaScript code from PureScript and we've seen the issues involved with writing trustworthy code using the FFI: • We've seen the importance of ensuring that foreign functions have correct representations. • We learned how to deal with corner cases like null values and other types of JavaScript data, by using foreign types, or the Json data type. • We saw how to safely serialize and deserialize JSON data. For more examples, the purescript, purescript-contrib and purescript-node GitHub organizations provide plenty of examples of libraries which use the FFI. In the remaining chapters, we will see some of these libraries put to use to solve real-world problems in a type-safe way. Addendum Calling PureScript from JavaScript Calling a PureScript function from JavaScript is very simple, at least for functions with simple types. Let's take the following simple module as an example: module Test where gcd :: Int -> Int -> Int gcd 0 m = m gcd n 0 = n gcd n m | n > m = gcd (n - m) m | otherwise = gcd (m - n) n  This function finds the greatest common divisor of two numbers by repeated subtraction. It is a nice example of a case where you might like to use PureScript to define the function, but have a requirement to call it from JavaScript: it is simple to define this function in PureScript using pattern matching and recursion, and the implementor can benefit from the use of the type checker. To understand how this function can be called from JavaScript, it is important to realize that PureScript functions always get turned into JavaScript functions of a single argument, so we need to apply its arguments one-by-one: var Test = require('Test'); Test.gcd(15)(20);  Here, I am assuming that the code was compiled with spago build, which compiles PureScript modules to CommonJS modules. For that reason, I was able to reference the gcd function on the Test object, after importing the Test module using require. You might also like to bundle JavaScript code for the browser, using spago bundle-app --to file.js. In that case, you would access the Test module from the global PureScript namespace, which defaults to PS: var Test = PS.Test; Test.gcd(15)(20);  Understanding Name Generation PureScript aims to preserve names during code generation as much as possible. In particular, most identifiers which are neither PureScript nor JavaScript keywords can be expected to be preserved, at least for names of top-level declarations. If you decide to use a JavaScript keyword as an identifier, the name will be escaped with a double dollar symbol. For example, null = []  generates the following JavaScript: var$$null = [];  In addition, if you would like to use special characters in your identifier names, they will be escaped using a single dollar symbol. For example, example' = 100  generates the following JavaScript: var example$prime = 100;


Where compiled PureScript code is intended to be called from JavaScript, it is recommended that identifiers only use alphanumeric characters, and avoid JavaScript keywords. If user-defined operators are provided for use in PureScript code, it is good practice to provide an alternative function with an alphanumeric name for use in JavaScript.

Runtime Data Representation

Types allow us to reason at compile-time that our programs are "correct" in some sense - that is, they will not break at runtime. But what does that mean? In PureScript, it means that the type of an expression should be compatible with its representation at runtime.

For that reason, it is important to understand the representation of data at runtime to be able to use PureScript and JavaScript code together effectively. This means that for any given PureScript expression, we should be able to understand the behavior of the value it will evaluate to at runtime.

The good news is that PureScript expressions have particularly simple representations at runtime. It should always be possible to understand the runtime data representation of an expression by considering its type.

For simple types, the correspondence is almost trivial. For example, if an expression has the type Boolean, then its value v at runtime should satisfy typeof v === 'boolean'. That is, expressions of type Boolean evaluate to one of the (JavaScript) values true or false. In particular, there is no PureScript expression of type Boolean which evaluates to null or undefined.

A similar law holds for expressions of type Int Number and String - expressions of type Int or Number evaluate to non-null JavaScript numbers, and expressions of type String evaluate to non-null JavaScript strings. Expressions of type Int will evaluate to integers at runtime, even though they cannot not be distinguished from values of type Number by using typeof.

What about Unit? Well, since Unit has only one inhabitant (unit) and its value is not observable, it doesn't actually matter what it's represented with at runtime. Old code tends to represent it using {}. Newer code, however, tends to use undefined. So, although it doesn't really matter what you use to represent Unit, it is recommended to use undefined (not returning anything from a function also returns undefined).

What about some more complex types?

As we have already seen, PureScript functions correspond to JavaScript functions of a single argument. More precisely, if an expression f has type a -> b for some types a and b, and an expression x evaluates to a value with the correct runtime representation for type a, then f evaluates to a JavaScript function, which when applied to the result of evaluating x, has the correct runtime representation for type b. As a simple example, an expression of type String -> String evaluates to a function which takes non-null JavaScript strings to non-null JavaScript strings.

As you might expect, PureScript's arrays correspond to JavaScript arrays. But remember - PureScript arrays are homogeneous, so every element has the same type. Concretely, if a PureScript expression e has type Array a for some type a, then e evaluates to a (non-null) JavaScript array, all of whose elements have the correct runtime representation for type a.

We've already seen that PureScript's records evaluate to JavaScript objects. Just as for functions and arrays, we can reason about the runtime representation of data in a record's fields by considering the types associated with its labels. Of course, the fields of a record are not required to be of the same type.

For every constructor of an algebraic data type, the PureScript compiler creates a new JavaScript object type by defining a function. Its constructors correspond to functions which create new JavaScript objects based on those prototypes.

For example, consider the following simple ADT:

data ZeroOrOne a = Zero | One a


The PureScript compiler generates the following code:

function One(value0) {
this.value0 = value0;
};

One.create = function (value0) {
return new One(value0);
};

function Zero() {
};

Zero.value = new Zero();


Here, we see two JavaScript object types: Zero and One. It is possible to create values of each type by using JavaScript's new keyword. For constructors with arguments, the compiler stores the associated data in fields called value0, value1, etc.

The PureScript compiler also generates helper functions. For constructors with no arguments, the compiler generates a value property, which can be reused instead of using the new operator repeatedly. For constructors with one or more arguments, the compiler generates a create function, which takes arguments with the appropriate representation and applies the appropriate constructor.

What about constructors with more than one argument? In that case, the PureScript compiler also creates a new object type, and a helper function. This time, however, the helper function is curried function of two arguments. For example, this algebraic data type:

data Two a b = Two a b


generates this JavaScript code:

function Two(value0, value1) {
this.value0 = value0;
this.value1 = value1;
};

Two.create = function (value0) {
return function (value1) {
return new Two(value0, value1);
};
};


Here, values of the object type Two can be created using the new keyword, or by using the Two.create function.

The case of newtypes is slightly different. Recall that a newtype is like an algebraic data type, restricted to having a single constructor taking a single argument. In this case, the runtime representation of the newtype is actually the same as the type of its argument.

For example, this newtype representing telephone numbers:

newtype PhoneNumber = PhoneNumber String


is actually represented as a JavaScript string at runtime. This is useful for designing libraries, since newtypes provide an additional layer of type safety, but without the runtime overhead of another function call.

Representing Quantified Types

Expressions with quantified (polymorphic) types have restrictive representations at runtime. In practice, this means that there are relatively few expressions with a given quantified type, but that we can reason about them quite effectively.

Consider this polymorphic type, for example:

forall a. a -> a


What sort of functions have this type? Well, there is certainly one function with this type - namely, the identity function, defined in the Prelude:

id :: forall a. a -> a
id a = a


In fact, the identity function is the only (total) function with this type! This certainly seems to be the case (try writing an expression with this type which is not observably equivalent to identity), but how can we be sure? We can be sure by considering the runtime representation of the type.

What is the runtime representation of a quantified type forall a. t? Well, any expression with the runtime representation for this type must have the correct runtime representation for the type t for any choice of type a. In our example above, a function of type forall a. a -> a must have the correct runtime representation for the types String -> String, Number -> Number, Array Boolean -> Array Boolean, and so on. It must take strings to strings, numbers to numbers, etc.

But that is not enough - the runtime representation of a quantified type is more strict than this. We require any expression to be parametrically polymorphic - that is, it cannot use any information about the type of its argument in its implementation. This additional condition prevents problematic implementations such as the following JavaScript function from inhabiting a polymorphic type:

function invalid(a) {
if (typeof a === 'string') {
return "Argument was a string.";
} else {
return a;
}
}


Certainly, this function takes strings to strings, numbers to numbers, etc. but it does not meet the additional condition, since it inspects the (runtime) type of its argument, so this function would not be a valid inhabitant of the type forall a. a -> a.

Without being able to inspect the runtime type of our function argument, our only option is to return the argument unchanged, and so identity is indeed the only inhabitant of the type forall a. a -> a.

A full discussion of parametric polymorphism and parametricity is beyond the scope of this book. Note however, that since PureScript's types are erased at runtime, a polymorphic function in PureScript cannot inspect the runtime representation of its arguments (without using the FFI), and so this representation of polymorphic data is appropriate.

Representing Constrained Types

Functions with a type class constraint have an interesting representation at runtime. Because the behavior of the function might depend on the type class instance chosen by the compiler, the function is given an additional argument, called a type class dictionary, which contains the implementation of the type class functions provided by the chosen instance.

For example, here is a simple PureScript function with a constrained type which uses the Show type class:

shout :: forall a. Show a => a -> String
shout a = show a <> "!!!"


The generated JavaScript looks like this:

var shout = function (dict) {
return function (a) {
return show(dict)(a) + "!!!";
};
};


Notice that shout is compiled to a (curried) function of two arguments, not one. The first argument dict is the type class dictionary for the Show constraint. dict contains the implementation of the show function for the type a.

We can call this function from JavaScript by passing an explicit type class dictionary from Data.Show as the first parameter:

shout(require('Data.Show').showNumber)(42);


Exercises

1. (Easy) What are the runtime representations of these types?

forall a. a
forall a. a -> a -> a
forall a. Ord a => Array a -> Boolean


What can you say about the expressions which have these types?

2. (Medium) Try using the functions defined in the arrays package, calling them from JavaScript, by compiling the library using spago build and importing modules using the require function in NodeJS. Hint: you may need to configure the output path so that the generated CommonJS modules are available on the NodeJS module path.

Representing Side Effects

The Effect monad is also defined as a foreign type. Its runtime representation is quite simple - an expression of type Effect a should evaluate to a JavaScript function of no arguments, which performs any side-effects and returns a value with the correct runtime representation for type a.

The definition of the Effect type constructor is given in the Effect module as follows:

foreign import data Effect :: Type -> Type


As a simple example, consider the random function defined in the random package. Recall that its type was:

foreign import random :: Effect Number


The definition of the random function is given here:

exports.random = Math.random;


Notice that the random function is represented at runtime as a function of no arguments. It performs the side effect of generating a random number, and returns it, and the return value matches the runtime representation of the Number type: it is a non-null JavaScript number.

As a slightly more interesting example, consider the log function defined by the Effect.Console module in the console package. The log function has the following type:

foreign import log :: String -> Effect Unit


And here is its definition:

exports.log = function (s) {
return function () {
console.log(s);
};
};


The representation of log at runtime is a JavaScript function of a single argument, returning a function of no arguments. The inner function performs the side-effect of writing a message to the console.

Expressions of type Effect a can be invoked from JavaScript like regular JavaScript methods. For example, since the main function is required to have type Effect a for some type a, it can be invoked as follows:

require('Main').main();


When using spago bundle-app --to or spago run, this call to main is generated automatically, whenever the Main module is defined.

Chapter Goals

The goal of this chapter will be to learn about monad transformers, which provide a way to combine side-effects provided by different monads. The motivating example will be a text adventure game which can be played on the console in NodeJS. The various side-effects of the game (logging, state, and configuration) will all be provided by a monad transformer stack.

Project Setup

This module's project introduces the following new dependencies:

• ordered-collections, which provides data typs for immutable maps and sets
• transformers, which provides implementations of standard monad transformers
• node-readline, which provides FFI bindings to the readline interface provided by NodeJS
• yargs, which provides an applicative interface to the yargs command line argument processing library

It is also necessary to install the yargs module using NPM:

npm install


How To Play The Game

To run the project, use spago run

By default you will see a usage message:

node ./dist/Main.js -p <player name>

Options:
-p, --player  Player name  [required]
-d, --debug   Use debug mode

Missing required arguments: p
The player name is required


Provide the player name using the -p option:

spago run -a "-p Phil"
>


From the prompt, you can enter commands like look, inventory, take, use, north, south, east, and west. There is also a debug command, which can be used to print the game state when the --debug command line option is provided.

The game is played on a two-dimensional grid, and the player moves by issuing commands north, south, east, and west. The game contains a collection of items which can either be in the player's possession (in the user's inventory), or on the game grid at some location. Items can be picked up by the player, using the take command.

For reference, here is a complete walkthrough of the game:

$spago run -a "-p Phil" > look You are at (0, 0) You are in a dark forest. You see a path to the north. You can see the Matches. > take Matches You now have the Matches > north > look You are at (0, 1) You are in a clearing. You can see the Candle. > take Candle You now have the Candle > inventory You have the Candle. You have the Matches. > use Matches You light the candle. Congratulations, Phil! You win!  The game is very simple, but the aim of the chapter is to use the transformers package to build a library which will enable rapid development of this type of game. The State Monad We will start by looking at some of the monads provided by the transformers package. The first example is the State monad, which provides a way to model mutable state in pure code. We have already seen an approach to mutable state provided by the Effect monad. State provides an alternative. The State type constructor takes two type parameters: the type s of the state, and the return type a. Even though we speak of the "State monad", the instance of the Monad type class is actually provided for the State s type constructor, for any type s. The Control.Monad.State module provides the following API: get :: forall s. State s s gets :: forall s. (s -> a) -> State s a put :: forall s. s -> State s Unit modify :: forall s. (s -> s) -> State s s modify_ :: forall s. (s -> s) -> State s Unit  Note that these API signatures are presented in a simplified form using the State type constructor for now. The actual API involves MonadState which we'll cover in the later "Type Classes" section of this chapter, so don't worry if you see different signatures in your IDE tooltips or on Pursuit. Let's see an example. One use of the State monad might be to add the values in an array of integers to the current state. We could do that by choosing Int as the state type s, and using traverse_ to traverse the array, with a call to modify for each array element: import Data.Foldable (traverse_) import Control.Monad.State import Control.Monad.State.Class sumArray :: Array Int -> State Int Unit sumArray = traverse_ \n -> modify \sum -> sum + n  The Control.Monad.State module provides three functions for running a computation in the State monad: evalState :: forall s a. State s a -> s -> a execState :: forall s a. State s a -> s -> s runState :: forall s a. State s a -> s -> Tuple a s  Each of these functions takes an initial state of type s and a computation of type State s a. evalState only returns the return value, execState only returns the final state, and runState returns both, expressed as a value of type Tuple a s. Given the sumArray function above, we could use execState in PSCi to sum the numbers in several arrays as follows: > :paste … execState (do … sumArray [1, 2, 3] … sumArray [4, 5] … sumArray [6]) 0 … ^D 21  Exercises 1. (Easy) What is the result of replacing execState with runState or evalState in our example above? 2. (Medium) A string of parentheses is balanced if it is obtained by either concatenating zero-or-more shorter balanced strings, or by wrapping a shorter balanced string in a pair of parentheses. Use the State monad and the traverse_ function to write a function testParens :: String -> Boolean  which tests whether or not a String of parentheses is balanced, by keeping track of the number of opening parentheses which have not been closed. Your function should work as follows: > testParens "" true > testParens "(()(())())" true > testParens ")" false > testParens "(()()" false  Hint: you may like to use the toCharArray function from the Data.String.CodeUnits module to turn the input string into an array of characters. The Reader Monad Another monad provided by the transformers package is the Reader monad. This monad provides the ability to read from a global configuration. Whereas the State monad provides the ability to read and write a single piece of mutable state, the Reader monad only provides the ability to read a single piece of data. The Reader type constructor takes two type arguments: a type r which represents the configuration type, and the return type a. The Control.Monad.Reader module provides the following API: ask :: forall r. Reader r r local :: forall r a. (r -> r) -> Reader r a -> Reader r a  The ask action can be used to read the current configuration, and the local action can be used to run a computation with a modified configuration. For example, suppose we were developing an application controlled by permissions, and we wanted to use the Reader monad to hold the current user's permissions object. We might choose the type r to be some type Permissions with the following API: hasPermission :: String -> Permissions -> Boolean addPermission :: String -> Permissions -> Permissions  Whenever we wanted to check if the user had a particular permission, we could use ask to retrieve the current permissions object. For example, only administrators might be allowed to create new users: createUser :: Reader Permissions (Maybe User) createUser = do permissions <- ask if hasPermission "admin" permissions then map Just newUser else pure Nothing  To elevate the user's permissions, we might use the local action to modify the Permissions object during the execution of some computation: runAsAdmin :: forall a. Reader Permissions a -> Reader Permissions a runAsAdmin = local (addPermission "admin")  Then we could write a function to create a new user, even if the user did not have the admin permission: createUserAsAdmin :: Reader Permissions (Maybe User) createUserAsAdmin = runAsAdmin createUser  To run a computation in the Reader monad, the runReader function can be used to provide the global configuration: runReader :: forall r a. Reader r a -> r -> a  Exercises In these exercises, we will use the Reader monad to build a small library for rendering documents with indentation. The "global configuration" will be a number indicating the current indentation level: type Level = Int type Doc = Reader Level String  1. (Easy) Write a function line which renders a function at the current indentation level. Your function should have the following type: line :: String -> Doc  Hint: use the ask function to read the current indentation level. The power function from Data.Monoid may be helpful too. 2. (Easy) Use the local function to write a function indent :: Doc -> Doc  which increases the indentation level for a block of code. 3. (Medium) Use the sequence function defined in Data.Traversable to write a function cat :: Array Doc -> Doc  which concatenates a collection of documents, separating them with new lines. 4. (Medium) Use the runReader function to write a function render :: Doc -> String  which renders a document as a String. You should now be able to use your library to write simple documents, as follows: render$ cat
[ line "Here is some indented text:"
, indent $cat [ line "I am indented" , line "So am I" , indent$ line "I am even more indented"
]
]


The Writer monad provides the ability to accumulate a secondary value in addition to the return value of a computation.

A common use case is to accumulate a log of type String or Array String, but the Writer monad is more general than this. It can actually be used to accumulate a value in any monoid, so it might be used to keep track of an integer total using the Additive Int monoid, or to track whether any of several intermediate Boolean values were true, using the Disj Boolean monoid.

The Writer type constructor takes two type arguments: a type w which should be an instance of the Monoid type class, and the return type a.

The key element of the Writer API is the tell function:

tell :: forall w a. Monoid w => w -> Writer w Unit


The tell action appends the provided value to the current accumulated result.

As an example, let's add a log to an existing function by using the Array String monoid. Consider our previous implementation of the greatest common divisor function:

gcd :: Int -> Int -> Int
gcd n 0 = n
gcd 0 m = m
gcd n m = if n > m
then gcd (n - m) m
else gcd n (m - n)


We could add a log to this function by changing the return type to Writer (Array String) Int:

import Control.Monad.Writer

gcdLog :: Int -> Int -> Writer (Array String) Int


We only have to change our function slightly to log the two inputs at each step:

    gcdLog n 0 = pure n
gcdLog 0 m = pure m
gcdLog n m = do
tell ["gcdLog " <> show n <> " " <> show m]
if n > m
then gcdLog (n - m) m
else gcdLog n (m - n)


We can run a computation in the Writer monad by using either of the execWriter or runWriter functions:

execWriter :: forall w a. Writer w a -> w
runWriter  :: forall w a. Writer w a -> Tuple a w


Just like in the case of the State monad, execWriter only returns the accumulated log, whereas runWriter returns both the log and the result.

We can test our modified function in PSCi:

> import Control.Monad.Writer

> runWriter (gcdLog 21 15)
Tuple 3 ["gcdLog 21 15","gcdLog 6 15","gcdLog 6 9","gcdLog 6 3","gcdLog 3 3"]


Exercises

1. (Medium) Rewrite the sumArray function above using the Writer monad and the Additive Int monoid from the monoid package.

2. (Medium) The Collatz function is defined on natural numbers n as n / 2 when n is even, and 3 * n + 1 when n is odd. For example, the iterated Collatz sequence starting at 10 is as follows:

10, 5, 16, 8, 4, 2, 1, ...


It is conjectured that the iterated Collatz sequence always reaches 1 after some finite number of applications of the Collatz function.

Write a function which uses recursion to calculate how many iterations of the Collatz function are required before the sequence reaches 1.

Modify your function to use the Writer monad to log each application of the Collatz function.

Each of the three monads above: State, Reader and Writer, are also examples of so-called monad transformers. The equivalent monad transformers are called StateT, ReaderT, and WriterT respectively.

What is a monad transformer? Well, as we have seen, a monad augments PureScript code with some type of side effect, which can be interpreted in PureScript by using the appropriate handler (runState, runReader, runWriter, etc.) This is fine if we only need to use one side-effect. However, it is often useful to use more than one side-effect at once. For example, we might want to use Reader together with Maybe to express optional results in the context of some global configuration. Or we might want the mutable state provided by the State monad together with the pure error tracking capability of the Either monad. This is the problem solved by monad transformers.

Note that we have already seen that the Effect monad provides a partial solution to this problem. Monad transformers provide another solution, and each approach has its own benefits and limitations.

A monad transformer is a type constructor which is parameterized not only by a type, but by another type constructor. It takes one monad and turns it into another monad, adding its own variety of side-effects.

Let's see an example. The monad transformer version of the State monad is StateT, defined in the Control.Monad.State.Trans module. We can find the kind of StateT using PSCi:

> import Control.Monad.State.Trans
> :kind StateT
Type -> (Type -> Type) -> Type -> Type


This looks quite confusing, but we can apply StateT one argument at a time to understand how to use it.

The first type argument is the type of the state we wish to use, as was the case for State. Let's use a state of type String:

> :kind StateT String
(Type -> Type) -> Type -> Type


The next argument is a type constructor of kind Type -> Type. It represents the underlying monad, which we want to add the effects of StateT to. For the sake of an example, let's choose the Either String monad:

> :kind StateT String (Either String)
Type -> Type


We are left with a type constructor. The final argument represents the return type, and we might instantiate it to Number for example:

> :kind StateT String (Either String) Number
Type


Finally we are left with something of kind Type, which means we can try to find values of this type.

The monad we have constructed - StateT String (Either String) - represents computations which can fail with an error, and which can use mutable state.

We can use the actions of the outer StateT String monad (get, put, and modify) directly, but in order to use the effects of the wrapped monad (Either String), we need to "lift" them over the monad transformer. The Control.Monad.Trans module defines the MonadTrans type class, which captures those type constructors which are monad transformers, as follows:

class MonadTrans t where
lift :: forall m a. Monad m => m a -> t m a


This class contains a single member, lift, which takes computations in any underlying monad m and lifts them into the wrapped monad t m. In our case, the type constructor t is StateT String, and m is the Either String monad, so lift provides a way to lift computations of type Either String a to computations of type StateT String (Either String) a. This means that we can use the effects of StateT String and Either String side-by-side, as long as we use lift every time we use a computation of type Either String a.

For example, the following computation reads the underlying state, and then throws an error if the state is the empty string:

import Data.String (drop, take)

split :: StateT String (Either String) String
split = do
s <- get
case s of
"" -> lift $Left "Empty string" _ -> do put (drop 1 s) pure (take 1 s)  If the state is not empty, the computation uses put to update the state to drop 1 s (that is, s with the first character removed), and returns take 1 s (that is, the first character of s). Let's try this in PSCi: > runStateT split "test" Right (Tuple "t" "est") > runStateT split "" Left "Empty string"  This is not very remarkable, since we could have implemented this without StateT. However, since we are working in a monad, we can use do notation or applicative combinators to build larger computations from smaller ones. For example, we can apply split twice to read the first two characters from a string: > runStateT ((<>) <$> split <*> split) "test"
(Right (Tuple "te" "st"))


We can use the split function with a handful of other actions to build a basic parsing library. In fact, this is the approach taken by the parsing library. This is the power of monad transformers - we can create custom-built monads for a variety of problems, choosing the side-effects that we need, and keeping the expressiveness of do notation and applicative combinators.

The transformers package also defines the ExceptT e monad transformer, which is the transformer corresponding to the Either e monad. It provides the following API:

class MonadError e m where
throwError :: forall a. e -> m a
catchError :: forall a. m a -> (e -> m a) -> m a

runExceptT :: forall e m a. ExceptT e m a -> m (Either e a)


The MonadError class captures those monads which support throwing and catching of errors of some type e, and an instance is provided for the ExceptT e monad transformer. The throwError action can be used to indicate failure, just like Left in the Either e monad. The catchError action allows us to continue after an error is thrown using throwError.

The runExceptT handler is used to run a computation of type ExceptT e m a.

This API is similar to that provided by the exceptions package and the Exception effect. However, there are some important differences:

• Exception uses actual JavaScript exceptions, whereas ExceptT models errors as a pure data structure.
• The Exception effect only supports exceptions of one type, namely JavaScript's Error type, whereas ExceptT supports errors of any type. In particular, we are free to define new error types.

Let's try out ExceptT by using it to wrap the Writer monad. Again, we are free to use actions from the monad transformer ExceptT e directly, but computations in the Writer monad should be lifted using lift:

import Control.Monad.Except

writerAndExceptT :: ExceptT String (Writer (Array String)) String
writerAndExceptT = do
lift $tell ["Before the error"] _ <- throwError "Error!" lift$ tell ["After the error"]
pure "Return value"


If we test this function in PSCi, we can see how the two effects of accumulating a log and throwing an error interact. First, we can run the outer ExceptT computation of type by using runExceptT, leaving a result of type Writer (Array String) (Either String String). We can then use runWriter to run the inner Writer computation:

> runWriter $runExceptT writerAndExceptT Tuple (Left "Error!") ["Before the error"]  Note that only those log messages which were written before the error was thrown actually get appended to the log. Monad Transformer Stacks As we have seen, monad transformers can be used to build new monads on top of existing monads. For some monad transformer t1 and some monad m, the application t1 m is also a monad. That means that we can apply a second monad transformer t2 to the result t1 m to construct a third monad t2 (t1 m). In this way, we can construct a stack of monad transformers, which combine the side-effects provided by their constituent monads. In practice, the underlying monad m is either the Effect monad, if native side-effects are required, or the Identity monad, defined in the Data.Identity module. The Identity monad adds no new side-effects, so transforming the Identity monad only provides the effects of the monad transformer. In fact, the State, Reader and Writer monads are implemented by transforming the Identity monad with StateT, ReaderT and WriterT respectively. Let's see an example in which three side effects are combined. We will use the StateT, WriterT and ExceptT effects, with the Identity monad on the bottom of the stack. This monad transformer stack will provide the side effects of mutable state, accumulating a log, and pure errors. We can use this monad transformer stack to reproduce our split action with the added feature of logging. type Errors = Array String type Log = Array String type Parser = StateT String (WriterT Log (ExceptT Errors Identity)) split :: Parser String split = do s <- get lift$ tell ["The state is " <> s]
case s of
"" -> lift $lift$ throwError ["Empty string"]
_ -> do
put (drop 1 s)
pure (take 1 s)


If we test this computation in PSCi, we see that the state is appended to the log for every invocation of split.

Note that we have to remove the side-effects in the order in which they appear in the monad transformer stack: first we use runStateT to remove the StateT type constructor, then runWriterT, then runExceptT. Finally, we run the computation in the Identity monad by using unwrap.

> runParser p s = unwrap $runExceptT$ runWriterT $runStateT p s > runParser split "test" (Right (Tuple (Tuple "t" "est") ["The state is test"])) > runParser ((<>) <$> split <*> split) "test"
(Right (Tuple (Tuple "te" "st") ["The state is test", "The state is est"]))


However, if the parse is unsuccessful because the state is empty, then no log is printed at all:

> runParser split ""
(Left ["Empty string"])


This is because of the way in which the side-effects provided by the ExceptT monad transformer interact with the side-effects provided by the WriterT monad transformer. We can address this by changing the order in which the monad transformer stack is composed. If we move the ExceptT transformer to the top of the stack, then the log will contain all messages written up until the first error, as we saw earlier when we transformed Writer with ExceptT.

One problem with this code is that we have to use the lift function multiple times to lift computations over multiple monad transformers: for example, the call to throwError has to be lifted twice, once over WriterT and a second time over StateT. This is fine for small monad transformer stacks, but quickly becomes inconvenient.

Fortunately, as we will see, we can use the automatic code generation provided by type class inference to do most of this "heavy lifting" for us.

Exercises

1. (Easy) Use the ExceptT monad transformer over the Identity functor to write a function safeDivide which divides two numbers, throwing an error if the denominator is zero.

2. (Medium) Write a parser

string :: String -> Parser String


which matches a string as a prefix of the current state, or fails with an error message.

Your parser should work as follows:

> runParser (string "abc") "abcdef"
(Right (Tuple (Tuple "abc" "def") ["The state is abcdef"]))


Hint: you can use the implementation of split as a starting point. You might find the stripPrefix function useful.

3. (Difficult) Use the ReaderT and WriterT monad transformers to reimplement the document printing library which we wrote earlier using the Reader monad.

Instead of using line to emit strings and cat to concatenate strings, use the Array String monoid with the WriterT monad transformer, and tell to append a line to the result.

Type Classes to the Rescue!

When we looked at the State monad at the start of this chapter, I gave the following types for the actions of the State monad:

get    :: forall s.             State s s
put    :: forall s. s        -> State s Unit
modify :: forall s. (s -> s) -> State s Unit


In reality, the types given in the Control.Monad.State.Class module are more general than this:

get    :: forall m s. MonadState s m =>             m s
put    :: forall m s. MonadState s m => s        -> m Unit
modify :: forall m s. MonadState s m => (s -> s) -> m Unit


The Control.Monad.State.Class module defines the MonadState (multi-parameter) type class, which allows us to abstract over "monads which support pure mutable state". As one would expect, the State s type constructor is an instance of the MonadState s type class, but there are many more interesting instances of this class.

In particular, there are instances of MonadState for the WriterT, ReaderT and ExceptT monad transformers, provided in the transformers package. Each of these monad transformers has an instance for MonadState whenever the underlying Monad does. In practice, this means that as long as StateT appears somewhere in the monad transformer stack, and everything above StateT is an instance of MonadState, then we are free to use get, put and modify directly, without the need to use lift.

Indeed, the same is true of the actions we covered for the ReaderT, WriterT, and ExceptT transformers. transformers defines a type class for each of the major transformers, allowing us to abstract over monads which support their operations.

In the case of the split function above, the monad stack we constructed is an instance of each of the MonadState, MonadWriter and MonadError type classes. This means that we don't need to call lift at all! We can just use the actions get, put, tell and throwError as if they were defined on the monad stack itself:

split :: Parser String
split = do
s <- get
tell ["The state is " <> show s]
case s of
"" -> throwError "Empty string"
_ -> do
put (drop 1 s)
pure (take 1 s)


This computation really looks like we have extended our programming language to support the three new side-effects of mutable state, logging and error handling. However, everything is still implemented using pure functions and immutable data under the hood.

Alternatives

The control package defines a number of abstractions for working with computations which can fail. One of these is the Alternative type class:

class Functor f <= Alt f where
alt :: forall a. f a -> f a -> f a

class Alt f <= Plus f where
empty :: forall a. f a

class (Applicative f, Plus f) <= Alternative f


Alternative provides two new combinators: the empty value, which provides a prototype for a failing computation, and the alt function (and its alias, <|>) which provides the ability to fall back to an alternative computation in the case of an error.

The Data.Array module provides two useful functions for working with type constructors in the Alternative type class:

many :: forall f a. Alternative f => Lazy (f (Array a)) => f a -> f (Array a)
some :: forall f a. Alternative f => Lazy (f (Array a)) => f a -> f (Array a)


There is also an equivalent many and some for Data.List

The many combinator uses the Alternative type class to repeatedly run a computation zero-or-more times. The some combinator is similar, but requires at least the first computation to succeed.

In the case of our Parser monad transformer stack, there is an instance of Alternative induced by the ExceptT component, which supports failure by composing errors in different branches using a Monoid instance (this is why we chose Array String for our Errors type). This means that we can use the many and some functions to run a parser multiple times:

> import Data.Array (many)

> runParser (many split) "test"
(Right (Tuple (Tuple ["t", "e", "s", "t"] "")
[ "The state is \"test\""
, "The state is \"est\""
, "The state is \"st\""
, "The state is \"t\""
]))


Here, the input string "test" has been repeatedly split to return an array of four single-character strings, the leftover state is empty, and the log shows that we applied the split combinator four times.

The Control.MonadPlus module defines a subclass of the Alternative type class, called MonadPlus. MonadPlus captures those type constructors which are both monads and instances of Alternative:

class (Monad m, Alternative m) <= MonadZero m



In particular, our Parser monad is an instance of MonadPlus.

When we covered array comprehensions earlier in the book, we introduced the guard function, which could be used to filter out unwanted results. In fact, the guard function is more general, and can be used for any monad which is an instance of MonadPlus:

guard :: forall m. MonadZero m => Boolean -> m Unit


The <|> operator allows us to backtrack in case of failure. To see how this is useful, let's define a variant of the split combinator which only matches upper case characters:

upper :: Parser String
upper = do
s <- split
guard $toUpper s == s pure s  Here, we use a guard to fail if the string is not upper case. Note that this code looks very similar to the array comprehensions we saw earlier - using MonadPlus in this way, we sometimes refer to constructing monad comprehensions. Backtracking We can use the <|> operator to backtrack to another alternative in case of failure. To demonstrate this, let's define one more parser, which matches lower case characters: lower :: Parser String lower = do s <- split guard$ toLower s == s
pure s


With this, we can define a parser which eagerly matches many upper case characters if the first character is upper case, or many lower case character if the first character is lower case:

> upperOrLower = some upper <|> some lower


This parser will match characters until the case changes:

> runParser upperOrLower "abcDEF"
(Right (Tuple (Tuple ["a","b","c"] ("DEF"))
[ "The state is \"abcDEF\""
, "The state is \"bcDEF\""
, "The state is \"cDEF\""
]))


We can even use many to fully split a string into its lower and upper case components:

> components = many upperOrLower

> runParser components "abCDeFgh"
(Right (Tuple (Tuple [["a","b"],["C","D"],["e"],["F"],["g","h"]] "")
[ "The state is \"abCDeFgh\""
, "The state is \"bCDeFgh\""
, "The state is \"CDeFgh\""
, "The state is \"DeFgh\""
, "The state is \"eFgh\""
, "The state is \"Fgh\""
, "The state is \"gh\""
, "The state is \"h\""
]))


Again, this illustrates the power of reusability that monad transformers bring - we were able to write a backtracking parser in a declarative style with only a few lines of code, by reusing standard abstractions!

Exercises

1. (Easy) Remove the calls to the lift function from your implementation of the string parser. Verify that the new implementation type checks, and convince yourself that it should.

2. (Medium) Use your string parser with the many combinator to write a parser which recognizes strings consisting of several copies of the string "a" followed by several copies of the string "b".

3. (Medium) Use the <|> operator to write a parser which recognizes strings of the letters a or b in any order.

4. (Difficult) The Parser monad might also be defined as follows:

type Parser = ExceptT Errors (StateT String (WriterT Log Identity))


What effect does this change have on our parsing functions?

One particular combination of monad transformers is so common that it is provided as a single monad transformer in the transformers package. The Reader, Writer and State monads are combined into the reader-writer-state monad, or more simply the RWS monad. This monad has a corresponding monad transformer called the RWST monad transformer.

We will use the RWS monad to model the game logic for our text adventure game.

The RWS monad is defined in terms of three type parameters (in addition to its return type):

type RWS r w s = RWST r w s Identity


Notice that the RWS monad is defined in terms of its own monad transformer, by setting the base monad to Identity which provides no side-effects.

The first type parameter, r, represents the global configuration type. The second, w, represents the monoid which we will use to accumulate a log, and the third, s is the type of our mutable state.

In the case of our game, our global configuration is defined in a type called GameEnvironment in the Data.GameEnvironment module:

type PlayerName = String

newtype GameEnvironment = GameEnvironment
{ playerName    :: PlayerName
, debugMode     :: Boolean
}


It defines the player name, and a flag which indicates whether or not the game is running in debug mode. These options will be set from the command line when we come to run our monad transformer.

The mutable state is defined in a type called GameState in the Data.GameState module:

import qualified Data.Map as M
import qualified Data.Set as S

newtype GameState = GameState
{ items       :: M.Map Coords (S.Set GameItem)
, player      :: Coords
, inventory   :: S.Set GameItem
}


The Coords data type represents points on a two-dimensional grid, and the GameItem data type is an enumeration of the items in the game:

data GameItem = Candle | Matches


The GameState type uses two new data structures: Map and Set, which represent sorted maps and sorted sets respectively. The items property is a mapping from coordinates of the game grid to sets of game items at that location. The player property stores the current coordinates of the player, and the inventory property stores a set of game items currently held by the player.

The Map and Set data structures are sorted by their keys, can be used with any key type in the Ord type class. This means that the keys in our data structures should be totally ordered.

We will see how the Map and Set structures are used as we write the actions for our game.

For our log, we will use the List String monoid. We can define a type synonym for our Game monad, implemented using RWS:

type Log = L.List String

type Game = RWS GameEnvironment Log GameState


Implementing Game Logic

Our game is going to be built from simple actions defined in the Game monad, by reusing the actions from the Reader, Writer and State monads. At the top level of our application, we will run the pure computations in the Game monad, and use the Effect monad to turn the results into observable side-effects, such as printing text to the console.

One of the simplest actions in our game is the has action. This action tests whether the player's inventory contains a particular game item. It is defined as follows:

has :: GameItem -> Game Boolean
has item = do
GameState state <- get
pure $item S.member state.inventory  This function uses the get action defined in the MonadState type class to read the current game state, and then uses the member function defined in Data.Set to test whether the specified GameItem appears in the Set of inventory items. Another action is the pickUp action. It adds a game item to the player's inventory if it appears in the current room. It uses actions from the MonadWriter and MonadState type classes. First of all, it reads the current game state: pickUp :: GameItem -> Game Unit pickUp item = do GameState state <- get  Next, pickUp looks up the set of items in the current room. It does this by using the lookup function defined in Data.Map:  case state.player M.lookup state.items of  The lookup function returns an optional result indicated by the Maybe type constructor. If the key does not appear in the map, the lookup function returns Nothing, otherwise it returns the corresponding value in the Just constructor. We are interested in the case where the corresponding item set contains the specified game item. Again we can test this using the member function:  Just items | item S.member items -> do  In this case, we can use put to update the game state, and tell to add a message to the log:  let newItems = M.update (Just <<< S.delete item) state.player state.items newInventory = S.insert item state.inventory put$ GameState state { items     = newItems
, inventory = newInventory
}
tell (L.singleton ("You now have the " <> show item))


Note that there is no need to lift either of the two computations here, because there are appropriate instances for both MonadState and MonadWriter for our Game monad transformer stack.

The argument to put uses a record update to modify the game state's items and inventory fields. We use the update function from Data.Map which modifies a value at a particular key. In this case, we modify the set of items at the player's current location, using the delete function to remove the specified item from the set. inventory is also updated, using insert to add the new item to the player's inventory set.

Finally, the pickUp function handles the remaining cases, by notifying the user using tell:

    _ -> tell (L.singleton "I don't see that item here.")


As an example of using the Reader monad, we can look at the code for the debug command. This command allows the user to inspect the game state at runtime if the game is running in debug mode:

  GameEnvironment env <- ask
if env.debugMode
then do
state <- get
tell (L.singleton (show state))
else tell (L.singleton "Not running in debug mode.")


Here, we use the ask action to read the game configuration. Again, note that we don't need to lift any computation, and we can use actions defined in the MonadState, MonadReader and MonadWriter type classes in the same do notation block.

If the debugMode flag is set, then the tell action is used to write the state to the log. Otherwise, an error message is added.

The remainder of the Game module defines a set of similar actions, each using only the actions defined by the MonadState, MonadReader and MonadWriter type classes.

Running the Computation

Since our game logic runs in the RWS monad, it is necessary to run the computation in order to respond to the user's commands.

The front-end of our game is built using two packages: yargs, which provides an applicative interface to the yargs command line parsing library, and node-readline, which wraps NodeJS' readline module, allowing us to write interactive console-based applications.

The interface to our game logic is provided by the function game in the Game module:

game :: Array String -> Game Unit


To run this computation, we pass a list of words entered by the user as an array of strings, and run the resulting RWS computation using runRWS:

data RWSResult state result writer = RWSResult state result writer

runRWS :: forall r w s a. RWS r w s a -> r -> s -> RWSResult s a w


runRWS looks like a combination of runReader, runWriter and runState. It takes a global configuration and an initial state as an argument, and returns a data structure containing the log, the result and the final state.

The front-end of our application is defined by a function runGame, with the following type signature:

runGame :: GameEnvironment -> Effect Unit


This function interacts with the user via the console (using the node-readline and console packages). runGame takes the game configuration as a function argument.

The node-readline package provides the LineHandler type, which represents actions in the Effect monad which handle user input from the terminal. Here is the corresponding API:

type LineHandler a = String -> Effect a

foreign import setLineHandler
:: forall a
. Interface
-> LineHandler a
-> Effect Unit


The Interface type represents a handle for the console, and is passed as an argument to the functions which interact with it. An Interface can be created using the createConsoleInterface function:

import Node.ReadLine as RL

runGame env = do
interface <- RL.createConsoleInterface RL.noCompletion


The first step is to set the prompt at the console. We pass the interface handle, and provide the prompt string and indentation level:

  RL.setPrompt "> " 2 interface


In our case, we are interested in implementing the line handler function. Our line handler is defined using a helper function in a let declaration, as follows:

lineHandler :: GameState -> String -> Effect Unit
lineHandler currentState input = do
case runRWS (game (split (wrap " ") input)) env currentState of
RWSResult state _ written -> do
for_ written log
RL.setLineHandler interface $lineHandler state RL.prompt interface pure unit  The let binding is closed over both the game configuration, named env, and the console handle, named interface. Our handler takes an additional first argument, the game state. This is required since we need to pass the game state to runRWS to run the game's logic. The first thing this action does is to break the user input into words using the split function from the Data.String module. It then uses runRWS to run the game action (in the RWS monad), passing the game environment and current game state. Having run the game logic, which is a pure computation, we need to print any log messages to the screen and show the user a prompt for the next command. The for_ action is used to traverse the log (of type List String) and print its entries to the console. Finally, setLineHandler is used to update the line handler function to use the updated game state, and the prompt is displayed again using the prompt action. The runGame function finally attaches the initial line handler to the console interface, and displays the initial prompt:  RL.setLineHandler interface$ lineHandler initialGameState
RL.prompt interface


Exercises

1. (Medium) Implement a new command cheat, which moves all game items from the game grid into the user's inventory.

2. (Difficult) The Writer component of the RWS monad is currently used for two types of messages: error messages and informational messages. Because of this, several parts of the code use case statements to handle error cases.

Refactor the code to use the ExceptT monad transformer to handle the error messages, and RWS to handle informational messages.

Handling Command Line Options

The final piece of the application is responsible for parsing command line options and creating the GameEnvironment configuration record. For this, we use the yargs package.

yargs is an example of applicative command line option parsing. Recall that an applicative functor allows us to lift functions of arbitrary arity over a type constructor representing some type of side-effect. In the case of the yargs package, the functor we are interested in is the Y functor, which adds the side-effect of reading from command line options. It provides the following handler:

runY :: forall a. YargsSetup -> Y (Effect a) -> Effect a


This is best illustrated by example. The application's main function is defined using runY as follows:

main = runY (usage "$0 -p <player name>")$ map runGame env


The first argument is used to configure the yargs library. In our case, we simply provide a usage message, but the Node.Yargs.Setup module provides several other options.

The second argument uses the map function to lift the runGame function over the Y type constructor. The argument env is constructed in a where declaration using the applicative operators <$> and <*>:  where env :: Y GameEnvironment env = gameEnvironment <$> yarg "p" ["player"]
(Just "Player name")
(Right "The player name is required")
false
<*> flag "d" ["debug"]
(Just "Use debug mode")


Here, the gameEnvironment function, which has the type PlayerName -> Boolean -> GameEnvironment, is lifted over Y. The two arguments specify how to read the player name and debug flag from the command line options. The first argument describes the player name option, which is specified by the -p or --player options, and the second describes the debug mode flag, which is turned on using the -d or --debug options.

This demonstrates two basic functions defined in the Node.Yargs.Applicative module: yarg, which defines a command line option which takes an optional argument (of type String, Number or Boolean), and flag which defines a command line flag of type Boolean.

Notice how we were able to use the notation afforded by the applicative operators to give a compact, declarative specification of our command line interface. In addition, it is simple to add new command line arguments, simply by adding a new function argument to runGame, and then using <*> to lift runGame over an additional argument in the definition of env.

Exercises

1. (Medium) Add a new Boolean-valued property cheatMode to the GameEnvironment record. Add a new command line flag -c to the yargs configuration which enables cheat mode. The cheat command from the previous exercise should be disallowed if cheat mode is not enabled.

Conclusion

This chapter was a practical demonstration of the techniques we've learned so far, using monad transformers to build a pure specification of our game, and the Effect monad to build a front-end using the console.

Because we separated our implementation from the user interface, it would be possible to create other front-ends for our game. For example, we could use the Effect monad to render the game in the browser using the Canvas API or the DOM.

We have seen how monad transformers allow us to write safe code in an imperative style, where effects are tracked by the type system. In addition, type classes provide a powerful way to abstract over the actions provided by a monad, enabling code reuse. We were able to use standard abstractions like Alternative and MonadPlus to build useful monads by combining standard monad transformers.

Monad transformers are an excellent demonstration of the sort of expressive code that can be written by relying on advanced type system features such as higher-kinded polymorphism and multi-parameter type classes.

Canvas Graphics

Chapter Goals

This chapter will be an extended example focussing on the canvas package, which provides a way to generate 2D graphics from PureScript using the HTML5 Canvas API.

Project Setup

This module's project introduces the following new dependencies:

• canvas, which gives types to methods from the HTML5 Canvas API
• refs, which provides a side-effect for using global mutable references

The source code for the chapter is broken up into a set of modules, each of which defines a main method. Different sections of this chapter are implemented in different files, and the Main module can be changed by modifying the Spago build command to run the appropriate file's main method at each point.

The HTML file html/index.html contains a single canvas element which will be used in each example, and a script element to load the compiled PureScript code. To test the code for each section, open the HTML file in your browser. Because most exercises target the browser, there are no unit tests for this chapter.

Simple Shapes

The Example/Rectangle.purs file contains a simple introductory example, which draws a single blue rectangle at the center of the canvas. The module imports the Effect type from the Effect module, and also the Graphics.Canvas module, which contains actions in the Effect monad for working with the Canvas API.

The main action starts, like in the other modules, by using the getCanvasElementById action to get a reference to the canvas object, and the getContext2D action to access the 2D rendering context for the canvas:

The void function takes a functor and replaces its value with Unit. In the example it is used to make main conform with its signature.

main :: Effect Unit
main = void $unsafePartial do Just canvas <- getCanvasElementById "canvas" ctx <- getContext2D canvas  Note: the call to unsafePartial here is necessary since the pattern match on the result of getCanvasElementById is partial, matching only the Just constructor. For our purposes, this is fine, but in production code, we would probably want to match the Nothing constructor and provide an appropriate error message. The types of these actions can be found using PSCi or by looking at the documentation: getCanvasElementById :: String -> Effect (Maybe CanvasElement) getContext2D :: CanvasElement -> Effect Context2D  CanvasElement and Context2D are types defined in the Graphics.Canvas module. The same module also defines the Canvas effect, which is used by all of the actions in the module. The graphics context ctx manages the state of the canvas, and provides methods to render primitive shapes, set styles and colors, and apply transformations. We continue by setting the fill style to solid blue using the setFillStyle action. The longer hex notation of #0000FF may also be used for blue, but shorthand notation is easier for simple colors:  setFillStyle ctx "#00F"  Note that the setFillStyle action takes the graphics context as an argument. This is a common pattern in the Graphics.Canvas module. Finally, we use the fillPath action to fill the rectangle. fillPath has the following type: fillPath :: forall a. Context2D -> Effect a -> Effect a  fillPath takes a graphics context and another action which builds the path to render. To build a path, we can use the rect action. rect takes a graphics context, and a record which provides the position and size of the rectangle:  fillPath ctx$ rect ctx
{ x: 250.0
, y: 250.0
, width: 100.0
, height: 100.0
}


Build the rectangle example, providing Example.Rectangle as the name of the main module:

$spago bundle-app --main Example.Rectangle --to dist/Main.js  Now, open the html/index.html file and verify that this code renders a blue rectangle in the center of the canvas. Putting Row Polymorphism to Work There are other ways to render paths. The arc function renders an arc segment, and the moveTo, lineTo and closePath functions can be used to render piecewise-linear paths. The Shapes.purs file renders three shapes: a rectangle, an arc segment and a triangle. We have seen that the rect function takes a record as its argument. In fact, the properties of the rectangle are defined in a type synonym: type Rectangle = { x :: Number , y :: Number , width :: Number , height :: Number }  The x and y properties represent the location of the top-left corner, while the w and h properties represent the width and height respectively. To render an arc segment, we can use the arc function, passing a record with the following type: type Arc = { x :: Number , y :: Number , radius :: Number , start :: Number , end :: Number }  Here, the x and y properties represent the center point, r is the radius, and start and end represent the endpoints of the arc in radians. For example, this code fills an arc segment centered at (300, 300) with radius 50. The arc completes 2/3rds of a rotation. Note that the unit circle is flipped vertically, since the y-axis increases towards the bottom of the canvas:  fillPath ctx$ arc ctx
{ x      : 300.0
, y      : 300.0
, start  : 0.0
, end    : Math.tau * 2.0 / 3.0
}


Notice that both the Rectangle and Arc record types contain x and y properties of type Number. In both cases, this pair represents a point. This means that we can write row-polymorphic functions which can act on either type of record.

For example, the Shapes module defines a translate function which translates a shape by modifying its x and y properties:

translate
:: forall r
. Number
-> Number
-> { x :: Number, y :: Number | r }
-> { x :: Number, y :: Number | r }
translate dx dy shape = shape
{ x = shape.x + dx
, y = shape.y + dy
}


Notice the row-polymorphic type. It says that translate accepts any record with x and y properties and any other properties, and returns the same type of record. The x and y fields are updated, but the rest of the fields remain unchanged.

This is an example of record update syntax. The expression shape { ... } creates a new record based on the shape record, with the fields inside the braces updated to the specified values. Note that the expressions inside the braces are separated from their labels by equals symbols, not colons like in record literals.

The translate function can be used with both the Rectangle and Arc records, as can be seen in the Shapes example.

The third type of path rendered in the Shapes example is a piecewise-linear path. Here is the corresponding code:

  setFillStyle ctx "#F00"

fillPath ctx $do moveTo ctx 300.0 260.0 lineTo ctx 260.0 340.0 lineTo ctx 340.0 340.0 closePath ctx  There are three functions in use here: • moveTo moves the current location of the path to the specified coordinates, • lineTo renders a line segment between the current location and the specified coordinates, and updates the current location, • closePath completes the path by rendering a line segment joining the current location to the start position. The result of this code snippet is to fill an isosceles triangle. Build the example by specifying Example.Shapes as the main module: $ spago bundle-app --main Example.Shapes --to dist/Main.js


and open html/index.html again to see the result. You should see the three different types of shapes rendered to the canvas.

Exercises

1. (Easy) Experiment with the strokePath and setStrokeStyle functions in each of the examples so far.

2. (Easy) The fillPath and strokePath functions can be used to render complex paths with a common style by using a do notation block inside the function argument. Try changing the Rectangle example to render two rectangles side-by-side using the same call to fillPath. Try rendering a sector of a circle by using a combination of a piecewise-linear path and an arc segment.

3. (Medium) Given the following record type:

type Point = { x :: Number, y :: Number }


which represents a 2D point, write a function renderPath which strokes a closed path constructed from a number of points:

renderPath
:: Context2D
-> Array Point
-> Effect Unit


Given a function

f :: Number -> Point


which takes a Number between 0 and 1 as its argument and returns a Point, write an action which plots f by using your renderPath function. Your action should approximate the path by sampling f at a finite set of points.

Experiment by rendering different paths by varying the function f.

Drawing Random Circles

The Example/Random.purs file contains an example which uses the Effect monad to interleave two different types of side-effect: random number generation, and canvas manipulation. The example renders one hundred randomly generated circles onto the canvas.

The main action obtains a reference to the graphics context as before, and then sets the stroke and fill styles:

  setFillStyle ctx "#F00"
setStrokeStyle ctx "#000"


Next, the code uses the for_ function to loop over the integers between 0 and 100:

  for_ (1 .. 100) \_ -> do


On each iteration, the do notation block starts by generating three random numbers distributed between 0 and 1. These numbers represent the x and y coordinates, and the radius of a circle:

    x <- random
y <- random
r <- random


Next, for each circle, the code creates an Arc based on these parameters and finally fills and strokes the arc with the current styles:

    let path = arc ctx
{ x     : x * 600.0
, y     : y * 600.0
, start : 0.0
, end   : Math.tau
}

fillPath ctx path
strokePath ctx path


Build this example by specifying the Example.Random module as the main module:

$spago bundle-app --main Example.Random --to dist/Main.js  and view the result by opening html/index.html. Transformations There is more to the canvas than just rendering simple shapes. Every canvas maintains a transformation which is used to transform shapes before rendering. Shapes can be translated, rotated, scaled, and skewed. The canvas library supports these transformations using the following functions: translate :: Context2D -> TranslateTransform -> Effect Context2D rotate :: Context2D -> Number -> Effect Context2D scale :: Context2D -> ScaleTransform -> Effect Context2D transform :: Context2D -> Transform -> Effect Context2D  The translate action performs a translation whose components are specified by the properties of the TranslateTransform record. The rotate action performs a rotation around the origin, through some number of radians specified by the first argument. The scale action performs a scaling, with the origin as the center. The ScaleTransform record specifies the scale factors along the x and y axes. Finally, transform is the most general action of the four here. It performs an affine transformation specified by a matrix. Any shapes rendered after these actions have been invoked will automatically have the appropriate transformation applied. In fact, the effect of each of these functions is to post-multiply the transformation with the context's current transformation. The result is that if multiple transformations applied after one another, then their effects are actually applied in reverse: transformations ctx = do translate ctx { translateX: 10.0, translateY: 10.0 } scale ctx { scaleX: 2.0, scaleY: 2.0 } rotate ctx (Math.tau / 4.0) renderScene  The effect of this sequence of actions is that the scene is rotated, then scaled, and finally translated. Preserving the Context A common use case is to render some subset of the scene using a transformation, and then to reset the transformation afterwards. The Canvas API provides the save and restore methods, which manipulate a stack of states associated with the canvas. canvas wraps this functionality into the following functions: save :: Context2D -> Effect Context2D restore :: Context2D -> Effect Context2D  The save action pushes the current state of the context (including the current transformation and any styles) onto the stack, and the restore action pops the top state from the stack and restores it. This allows us to save the current state, apply some styles and transformations, render some primitives, and finally restore the original transformation and state. For example, the following function performs some canvas action, but applies a rotation before doing so, and restores the transformation afterwards: rotated ctx render = do save ctx rotate (Math.tau / 3.0) ctx render restore ctx  In the interest of abstracting over common use cases using higher-order functions, the canvas library provides the withContext function, which performs some canvas action while preserving the original context state: withContext :: Context2D -> Effect a -> Effect a  We could rewrite the rotated function above using withContext as follows: rotated ctx render = withContext ctx do rotate (Math.tau / 3.0) ctx render  Global Mutable State In this section, we'll use the refs package to demonstrate another effect in the Effect monad. The Effect.Ref module provides a type constructor for global mutable references, and an associated effect: > import Effect.Ref > :kind Ref Type -> Type  A value of type Ref a is a mutable reference cell containing a value of type a, used to track global mutation. As such, it should be used sparingly. The Example/Refs.purs file contains an example which uses a Ref to track mouse clicks on the canvas element. The code starts by creating a new reference containing the value 0, by using the new action:  clickCount <- Ref.new 0  Inside the click event handler, the modify action is used to update the click count, and the updated value is returned.  count <- Ref.modify (\count -> count + 1) clickCount  In the render function, the click count is used to determine the transformation applied to a rectangle:  withContext ctx do let scaleX = Math.sin (toNumber count * Math.tau / 8.0) + 1.5 let scaleY = Math.sin (toNumber count * Math.tau / 12.0) + 1.5 translate ctx { translateX: 300.0, translateY: 300.0 } rotate ctx (toNumber count * Math.tau / 36.0) scale ctx { scaleX: scaleX, scaleY: scaleY } translate ctx { translateX: -100.0, translateY: -100.0 } fillPath ctx$ rect ctx
{ x: 0.0
, y: 0.0
, width: 200.0
, height: 200.0
}


This action uses withContext to preserve the original transformation, and then applies the following sequence of transformations (remember that transformations are applied bottom-to-top):

• The rectangle is translated through (-100, -100) so that its center lies at the origin.
• The rectangle is scaled around the origin.
• The rectangle is rotated through some multiple of 10 degrees around the origin.
• The rectangle is translated through (300, 300) so that it center lies at the center of the canvas.

Build the example:

$spago bundle-app --main Example.Refs --to dist/Main.js  and open the html/index.html file. If you click the canvas repeatedly, you should see a green rectangle rotating around the center of the canvas. Exercises 1. (Easy) Write a higher-order function which strokes and fills a path simultaneously. Rewrite the Random.purs example using your function. 2. (Medium) Use Random and Dom to create an application which renders a circle with random position, color and radius to the canvas when the mouse is clicked. 3. (Medium) Write a function which transforms the scene by rotating it around a point with specified coordinates. Hint: use a translation to first translate the scene to the origin. L-Systems In this final example, we will use the canvas package to write a function for rendering L-systems (or Lindenmayer systems). An L-system is defined by an alphabet, an initial sequence of letters from the alphabet, and a set of production rules. Each production rule takes a letter of the alphabet and returns a sequence of replacement letters. This process is iterated some number of times starting with the initial sequence of letters. If each letter of the alphabet is associated with some instruction to perform on the canvas, the L-system can be rendered by following the instructions in order. For example, suppose the alphabet consists of the letters L (turn left), R (turn right) and F (move forward). We might define the following production rules: L -> L R -> R F -> FLFRRFLF  If we start with the initial sequence "FRRFRRFRR" and iterate, we obtain the following sequence: FRRFRRFRR FLFRRFLFRRFLFRRFLFRRFLFRRFLFRR FLFRRFLFLFLFRRFLFRRFLFRRFLFLFLFRRFLFRRFLFRRFLF...  and so on. Plotting a piecewise-linear path corresponding to this set of instruction approximates a curve called the Koch curve. Increasing the number of iterations increases the resolution of the curve. Let's translate this into the language of types and functions. We can represent our alphabet of letters with the following ADT: data Letter = L | R | F  This data type defines one data constructor for each letter in our alphabet. How can we represent the initial sequence of letters? Well, that's just an array of letters from our alphabet, which we will call a Sentence: type Sentence = Array Letter initial :: Sentence initial = [F, R, R, F, R, R, F, R, R]  Our production rules can be represented as a function from Letter to Sentence as follows: productions :: Letter -> Sentence productions L = [L] productions R = [R] productions F = [F, L, F, R, R, F, L, F]  This is just copied straight from the specification above. Now we can implement a function lsystem which will take a specification in this form, and render it to the canvas. What type should lsystem have? Well, it needs to take values like initial and productions as arguments, as well as a function which can render a letter of the alphabet to the canvas. Here is a first approximation to the type of lsystem: Sentence -> (Letter -> Sentence) -> (Letter -> Effect Unit) -> Int -> Effect Unit  The first two argument types correspond to the values initial and productions. The third argument represents a function which takes a letter of the alphabet and interprets it by performing some actions on the canvas. In our example, this would mean turning left in the case of the letter L, turning right in the case of the letter R, and moving forward in the case of a letter F. The final argument is a number representing the number of iterations of the production rules we would like to perform. The first observation is that the lsystem function should work for only one type of Letter, but for any type, so we should generalize our type accordingly. Let's replace Letter and Sentence with a and Array a for some quantified type variable a: forall a. Array a -> (a -> Array a) -> (a -> Effect Unit) -> Int -> Effect Unit  The second observation is that, in order to implement instructions like "turn left" and "turn right", we will need to maintain some state, namely the direction in which the path is moving at any time. We need to modify our function to pass the state through the computation. Again, the lsystem function should work for any type of state, so we will represent it using the type variable s. We need to add the type s in three places: forall a s. Array a -> (a -> Array a) -> (s -> a -> Effect s) -> Int -> s -> Effect s  Firstly, the type s was added as the type of an additional argument to lsystem. This argument will represent the initial state of the L-system. The type s also appears as an argument to, and as the return type of the interpretation function (the third argument to lsystem). The interpretation function will now receive the current state of the L-system as an argument, and will return a new, updated state as its return value. In the case of our example, we can define use following type to represent the state: type State = { x :: Number , y :: Number , theta :: Number }  The properties x and y represent the current position of the path, and the theta property represents the current direction of the path, specified as the angle between the path direction and the horizontal axis, in radians. The initial state of the system might be specified as follows: initialState :: State initialState = { x: 120.0, y: 200.0, theta: 0.0 }  Now let's try to implement the lsystem function. We will find that its definition is remarkably simple. It seems reasonable that lsystem should recurse on its fourth argument (of type Int). On each step of the recursion, the current sentence will change, having been updated by using the production rules. With that in mind, let's begin by introducing names for the function arguments, and delegating to a helper function: lsystem :: forall a s . Array a -> (a -> Array a) -> (s -> a -> Effect s) -> Int -> s -> Effect s lsystem init prod interpret n state = go init n where  The go function works by recursion on its second argument. There are two cases: when n is zero, and when n is non-zero. In the first case, the recursion is complete, and we simply need to interpret the current sentence according to the interpretation function. We have a sentence of type Array a, a state of type s, and a function of type s -> a -> Effect s. This sounds like a job for the foldM function which we defined earlier, and which is available from the control package:  go s 0 = foldM interpret state s  What about in the non-zero case? In that case, we can simply apply the production rules to each letter of the current sentence, concatenate the results, and repeat by calling go recursively:  go s i = go (concatMap prod s) (i - 1)  That's it! Note how the use of higher order functions like foldM and concatMap allowed us to communicate our ideas concisely. However, we're not quite done. The type we have given is actually still too specific. Note that we don't use any canvas operations anywhere in our implementation. Nor do we make use of the structure of the Effect monad at all. In fact, our function works for any monad m! Here is the more general type of lsystem, as specified in the accompanying source code for this chapter: lsystem :: forall a m s . Monad m => Array a -> (a -> Array a) -> (s -> a -> m s) -> Int -> s -> m s  We can understand this type as saying that our interpretation function is free to have any side-effects at all, captured by the monad m. It might render to the canvas, or print information to the console, or support failure or multiple return values. The reader is encouraged to try writing L-systems which use these various types of side-effect. This function is a good example of the power of separating data from implementation. The advantage of this approach is that we gain the freedom to interpret our data in multiple different ways. We might even factor lsystem into two smaller functions: the first would build the sentence using repeated application of concatMap, and the second would interpret the sentence using foldM. This is also left as an exercise for the reader. Let's complete our example by implementing its interpretation function. The type of lsystem tells us that its type signature must be s -> a -> m s for some types a and s and a type constructor m. We know that we want a to be Letter and s to be State, and for the monad m we can choose Effect. This gives us the following type:  interpret :: State -> Letter -> Effect State  To implement this function, we need to handle the three data constructors of the Letter type. To interpret the letters L (move left) and R (move right), we simply have to update the state to change the angle theta appropriately:  interpret state L = pure$ state { theta = state.theta - Math.tau / 6.0 }
interpret state R = pure $state { theta = state.theta + Math.tau / 6.0 }  To interpret the letter F (move forward), we can calculate the new position of the path, render a line segment, and update the state, as follows:  interpret state F = do let x = state.x + Math.cos state.theta * 1.5 y = state.y + Math.sin state.theta * 1.5 moveTo ctx state.x state.y lineTo ctx x y pure { x, y, theta: state.theta }  Note that in the source code for this chapter, the interpret function is defined using a let binding inside the main function, so that the name ctx is in scope. It would also be possible to move the context into the State type, but this would be inappropriate because it is not a changing part of the state of the system. To render this L-system, we can simply use the strokePath action:  strokePath ctx$ lsystem initial productions interpret 5 initialState


Compile the L-system example using

$spago bundle-app --main Example.LSystem --to dist/Main.js  and open html/index.html. You should see the Koch curve rendered to the canvas. Exercises 1. (Easy) Modify the L-system example above to use fillPath instead of strokePath. Hint: you will need to include a call to closePath, and move the call to moveTo outside of the interpret function. 2. (Easy) Try changing the various numerical constants in the code, to understand their effect on the rendered system. 3. (Medium) Break the lsystem function into two smaller functions. The first should build the final sentence using repeated application of concatMap, and the second should use foldM to interpret the result. 4. (Medium) Add a drop shadow to the filled shape, by using the setShadowOffsetX, setShadowOffsetY, setShadowBlur and setShadowColor actions. Hint: use PSCi to find the types of these functions. 5. (Medium) The angle of the corners is currently a constant (tau/6). Instead, it can be moved into the Letter data type, which allows it to be changed by the production rules: type Angle = Number data Letter = L Angle | R Angle | F  How can this new information be used in the production rules to create interesting shapes? 6. (Difficult) An L-system is given by an alphabet with four letters: L (turn left through 60 degrees), R (turn right through 60 degrees), F (move forward) and M (also move forward). The initial sentence of the system is the single letter M. The production rules are specified as follows: L -> L R -> R F -> FLMLFRMRFRMRFLMLF M -> MRFRMLFLMLFLMRFRM  Render this L-system. Note: you will need to decrease the number of iterations of the production rules, since the size of the final sentence grows exponentially with the number of iterations. Now, notice the symmetry between L and M in the production rules. The two "move forward" instructions can be differentiated using a Boolean value using the following alphabet type: data Letter = L | R | F Boolean  Implement this L-system again using this representation of the alphabet. 7. (Difficult) Use a different monad m in the interpretation function. You might try using Effect.Console to write the L-system onto the console, or using Effect.Random to apply random "mutations" to the state type. Conclusion In this chapter, we learned how to use the HTML5 Canvas API from PureScript by using the canvas library. We also saw a practical demonstration of many of the techniques we have learned already: maps and folds, records and row polymorphism, and the Effect monad for handling side-effects. The examples also demonstrated the power of higher-order functions and separating data from implementation. It would be possible to extend these ideas to completely separate the representation of a scene from its rendering function, using an algebraic data type, for example: data Scene = Rect Rectangle | Arc Arc | PiecewiseLinear (Array Point) | Transformed Transform Scene | Clipped Rectangle Scene | ...  This approach is taken in the drawing package, and it brings the flexibility of being able to manipulate the scene as data in various ways before rendering. For examples of games rendered to the canvas, see the "Behavior" and "Signal" recipes in the cookbook. Generative Testing Chapter Goals In this chapter, we will see a particularly elegant application of type classes to the problem of testing. Instead of testing our code by telling the compiler how to test, we simply assert what properties our code should have. Test cases can be generated randomly from this specification, using type classes to hide the boilerplate code of random data generation. This is called generative testing (or property-based testing), a technique made popular by the QuickCheck library in Haskell. The quickcheck package is a port of Haskell's QuickCheck library to PureScript, and for the most part, it preserves the types and syntax of the original library. We will see how to use quickcheck to test a simple library, using Spago to integrate our test suite into our development process. Project Setup This chapter's project adds quickcheck as a dependency. In a Spago project, test sources should be placed in the test directory, and the main module for the test suite should be named Test.Main. The test suite can be run using the spago test command. Writing Properties The Merge module implements a simple function merge, which we will use to demonstrate the features of the quickcheck library. merge :: Array Int -> Array Int -> Array Int  merge takes two sorted arrays of integers, and merges their elements so that the result is also sorted. For example: > import Merge > merge [1, 3, 5] [2, 4, 5] [1, 2, 3, 4, 5, 5]  In a typical test suite, we might test merge by generating a few small test cases like this by hand, and asserting that the results were equal to the appropriate values. However, everything we need to know about the merge function can be summarized by this property: • If xs and ys are sorted, then merge xs ys is the sorted result of both arrays appended together. quickcheck allows us to test this property directly, by generating random test cases. We simply state the properties that we want our code to have, as functions. In this case, we have a single property: main = do quickCheck \xs ys -> eq (merge (sort xs) (sort ys)) (sort$ xs <> ys)


When we run this code, quickcheck will attempt to disprove the properties we claimed, by generating random inputs xs and ys, and passing them to our functions. If our function returns false for any inputs, the property will be incorrect, and the library will raise an error. Fortunately, the library is unable to disprove our properties after generating 100 random test cases:

$spago test Installation complete. Build succeeded. 100/100 test(s) passed. ... Tests succeeded.  If we deliberately introduce a bug into the merge function (for example, by changing the less-than check for a greater-than check), then an exception is thrown at runtime after the first failed test case: Error: Test 1 failed: Test returned false  As we can see, this error message is not very helpful, but it can be improved with a little work. Improving Error Messages To provide error messages along with our failed test cases, quickcheck provides the <?> operator. Simply separate the property definition from the error message using <?>, as follows: quickCheck \xs ys -> let result = merge (sort xs) (sort ys) expected = sort$ xs <> ys
in
eq result expected <?> "Result:\n" <> show result <> "\nnot equal to expected:\n" <> show expected


This time, if we modify the code to introduce a bug, we see our improved error message after the first failed test case:

Error: Test 1 (seed 534161891) failed:
Result:
[-822215,-196136,-116841,618343,887447,-888285]
not equal to expected:
[-888285,-822215,-196136,-116841,618343,887447]


Notice how the input xs and ys were generated as arrays of randomly-selected integers.

Exercises

1. (Easy) Write a property which asserts that merging an array with the empty array does not modify the original array. Note: This new property is redundant, since this situation is already covered by our existing property. We're just trying to give you readers a simple way to practice using quickCheck.
2. (Easy) Add an appropriate error message to the remaining property for merge.

Testing Polymorphic Code

The Merge module defines a generalization of the merge function, called mergePoly, which works not only with arrays of numbers, but also arrays of any type belonging to the Ord type class:

mergePoly :: forall a. Ord a => Array a -> Array a -> Array a


If we modify our original test to use mergePoly in place of merge, we see the following error message:

No type class instance was found for

Test.QuickCheck.Arbitrary.Arbitrary t0

The instance head contains unknown type variables.


This error message indicates that the compiler could not generate random test cases, because it did not know what type of elements we wanted our arrays to have. In these sorts of cases, we can use type annotations to force the compiler to infer a particular type, such as Array Int:

quickCheck \xs ys ->
eq (mergePoly (sort xs) (sort ys) :: Array Int) (sort $xs <> ys)  We can alternatively use a helper function to specify type, which may result in cleaner code. For example, if we define a function ints as a synonym for the identity function: ints :: Array Int -> Array Int ints = id  then we can modify our test so that the compiler infers the type Array Int for our two array arguments: quickCheck \xs ys -> eq (ints$ mergePoly (sort xs) (sort ys)) (sort $xs <> ys)  Here, xs and ys both have type Array Int, since the ints function has been used to disambiguate the unknown type. Exercises 1. (Easy) Write a function bools which forces the types of xs and ys to be Array Boolean, and add additional properties which test mergePoly at that type. 2. (Medium) Choose a pure function from the core libraries (for example, from the arrays package), and write a QuickCheck property for it, including an appropriate error message. Your property should use a helper function to fix any polymorphic type arguments to either Int or Boolean. Generating Arbitrary Data Now we will see how the quickcheck library is able to randomly generate test cases for our properties. Those types whose values can be randomly generated are captured by the Arbitrary type class: class Arbitrary t where arbitrary :: Gen t  The Gen type constructor represents the side-effects of deterministic random data generation. It uses a pseudo-random number generator to generate deterministic random function arguments from a seed value. The Test.QuickCheck.Gen module defines several useful combinators for building generators. Gen is also a monad and an applicative functor, so we have the usual collection of combinators at our disposal for creating new instances of the Arbitrary type class. For example, we can use the Arbitrary instance for the Int type, provided in the quickcheck library, to create a distribution on the 256 byte values, using the Functor instance for Gen to map a function from integers to bytes over arbitrary integer values: newtype Byte = Byte Int instance arbitraryByte :: Arbitrary Byte where arbitrary = map intToByte arbitrary where intToByte n | n >= 0 = Byte (n mod 256) | otherwise = intToByte (-n)  Here, we define a type Byte of integral values between 0 and 255. The Arbitrary instance uses the map function to lift the intToByte function over the arbitrary action. The type of the inner arbitrary action is inferred as Gen Int. We can also use this idea to improve our test for merge: quickCheck \xs ys -> eq (numbers$ mergePoly (sort xs) (sort ys)) (sort $xs <> ys)  In this test, we generated arbitrary arrays xs and ys, but had to sort them, since merge expects sorted input. On the other hand, we could create a newtype representing sorted arrays, and write an Arbitrary instance which generates sorted data: newtype Sorted a = Sorted (Array a) sorted :: forall a. Sorted a -> Array a sorted (Sorted xs) = xs instance arbSorted :: (Arbitrary a, Ord a) => Arbitrary (Sorted a) where arbitrary = map (Sorted <<< sort) arbitrary  With this type constructor, we can modify our test as follows: quickCheck \xs ys -> eq (ints$ mergePoly (sorted xs) (sorted ys)) (sort $sorted xs <> sorted ys)  This may look like a small change, but the types of xs and ys have changed to Sorted Int, instead of just Array Int. This communicates our intent in a clearer way - the mergePoly function takes sorted input. Ideally, the type of the mergePoly function itself would be updated to use the Sorted type constructor. As a more interesting example, the Tree module defines a type of sorted binary trees with values at the branches: data Tree a = Leaf | Branch (Tree a) a (Tree a)  The Tree module defines the following API: insert :: forall a. Ord a => a -> Tree a -> Tree a member :: forall a. Ord a => a -> Tree a -> Boolean fromArray :: forall a. Ord a => Array a -> Tree a toArray :: forall a. Tree a -> Array a  The insert function is used to insert a new element into a sorted tree, and the member function can be used to query a tree for a particular value. For example: > import Tree > member 2$ insert 1 $insert 2 Leaf true > member 1 Leaf false  The toArray and fromArray functions can be used to convert sorted trees to and from arrays. We can use fromArray to write an Arbitrary instance for trees: instance arbTree :: (Arbitrary a, Ord a) => Arbitrary (Tree a) where arbitrary = map fromArray arbitrary  We can now use Tree a as the type of an argument to our test properties, whenever there is an Arbitrary instance available for the type a. For example, we can test that the member test always returns true after inserting a value: quickCheck \t a -> member a$ insert a $treeOfInt t  Here, the argument t is a randomly-generated tree of type Tree Int, where the type argument disambiguated by the identity function treeOfInt. Exercises 1. (Medium) Create a newtype for String with an associated Arbitrary instance which generates collections of randomly-selected characters in the range a-z. Hint: use the elements and arrayOf functions from the Test.QuickCheck.Gen module. 2. (Difficult) Write a property which asserts that a value inserted into a tree is still a member of that tree after arbitrarily many more insertions. Testing Higher-Order Functions The Merge module defines another generalization of the merge function - the mergeWith function takes an additional function as an argument which is used to determine the order in which elements should be merged. That is, mergeWith is a higher-order function. For example, we can pass the length function as the first argument, to merge two arrays which are already in length-increasing order. The result should also be in length-increasing order: > import Data.String > mergeWith length ["", "ab", "abcd"] ["x", "xyz"] ["","x","ab","xyz","abcd"]  How might we test such a function? Ideally, we would like to generate values for all three arguments, including the first argument which is a function. There is a second type class which allows us to create randomly-generated functions. It is called Coarbitrary, and it is defined as follows: class Coarbitrary t where coarbitrary :: forall r. t -> Gen r -> Gen r  The coarbitrary function takes a function argument of type t, and a random generator for a function result of type r, and uses the function argument to perturb the random generator. That is, it uses the function argument to modify the random output of the random generator for the result. In addition, there is a type class instance which gives us Arbitrary functions if the function domain is Coarbitrary and the function codomain is Arbitrary: instance arbFunction :: (Coarbitrary a, Arbitrary b) => Arbitrary (a -> b)  In practice, this means that we can write properties which take functions as arguments. In the case of the mergeWith function, we can generate the first argument randomly, modifying our tests to take account of the new argument. We cannot guarantee that the result will be sorted - we do not even necessarily have an Ord instance - but we can expect that the result be sorted with respect to the function f that we pass in as an argument. In addition, we need the two input arrays to be sorted with respect to f, so we use the sortBy function to sort xs and ys based on comparison after the function f has been applied: quickCheck \xs ys f -> let result = map f$
mergeWith (intToBool f)
(sortBy (compare on f) xs)
(sortBy (compare on f) ys)
expected =
map f $sortBy (compare on f)$ xs <> ys
in
eq result expected


Here, we use a function intToBool to disambiguate the type of the function f:

intToBool :: (Int -> Boolean) -> Int -> Boolean
intToBool = id


In addition to being Arbitrary, functions are also Coarbitrary:

instance coarbFunction :: (Arbitrary a, Coarbitrary b) => Coarbitrary (a -> b)


This means that we are not limited to just values and functions - we can also randomly generate higher-order functions, or functions whose arguments are higher-order functions, and so on.

Writing Coarbitrary Instances

Just as we can write Arbitrary instances for our data types by using the Monad and Applicative instances of Gen, we can write our own Coarbitrary instances as well. This allows us to use our own data types as the domain of randomly-generated functions.

Let's write a Coarbitrary instance for our Tree type. We will need a Coarbitrary instance for the type of the elements stored in the branches:

instance coarbTree :: Coarbitrary a => Coarbitrary (Tree a) where


We have to write a function which perturbs a random generator given a value of type Tree a. If the input value is a Leaf, then we will just return the generator unchanged:

  coarbitrary Leaf = id


If the tree is a Branch, then we will perturb the generator using the left subtree, the value, and the right subtree. We use function composition to create our perturbing function:

  coarbitrary (Branch l a r) =
coarbitrary l <<<
coarbitrary a <<<
coarbitrary r


Now we are free to write properties whose arguments include functions taking trees as arguments. For example, the Tree module defines a function anywhere, which tests if a predicate holds on any subtree of its argument:

anywhere :: forall a. (Tree a -> Boolean) -> Tree a -> Boolean


Now we are able to generate the predicate function randomly. For example, we expect the anywhere function to respect disjunction:

quickCheck \f g t ->
anywhere (\s -> f s || g s) t ==
anywhere f (treeOfInt t) || anywhere g t


Here, the treeOfInt function is used to fix the type of values contained in the tree to the type Int:

treeOfInt :: Tree Int -> Tree Int
treeOfInt = id


Testing Without Side-Effects

For the purposes of testing, we usually include calls to the quickCheck function in the main action of our test suite. However, there is a variant of the quickCheck function, called quickCheckPure which does not use side-effects. Instead, it is a pure function which takes a random seed as an input, and returns an array of test results.

We can test quickCheckPure using PSCi. Here, we test that the merge operation is associative:

> import Prelude
> import Merge
> import Test.QuickCheck
> import Test.QuickCheck.LCG (mkSeed)

> :paste
… quickCheckPure (mkSeed 12345) 10 \xs ys zs ->
…   ((xs merge ys) merge zs) ==
…     (xs merge (ys merge zs))
… ^D

Success : Success : ...


quickCheckPure takes three arguments: the random seed, the number of test cases to generate, and the property to test. If all tests pass, you should see an array of Success data constructors printed to the console.

quickCheckPure might be useful in other situations, such as generating random input data for performance benchmarks, or generating sample form data for web applications.

Exercises

1. (Easy) Write Coarbitrary instances for the Byte and Sorted type constructors.

2. (Medium) Write a (higher-order) property which asserts associativity of the mergeWith f function for any function f. Test your property in PSCi using quickCheckPure.

3. (Medium) Write Arbitrary and Coarbitrary instances for the following data type:

data OneTwoThree a = One a | Two a a | Three a a a


Hint: Use the oneOf function defined in Test.QuickCheck.Gen to define your Arbitrary instance.

4. (Medium) Use all to simplify the result of the quickCheckPure function - your new function should have type List Result -> Boolean and should return true if every test passes and false otherwise.

5. (Medium) As another approach to simplifying the result of quickCheckPure, try writing a function squashResults :: List Result -> Result. Consider using the First monoid from Data.Maybe.First with the foldMap function to preserve the first error in case of failure.

Conclusion

In this chapter, we met the quickcheck package, which can be used to write tests in a declarative way using the paradigm of generative testing. In particular:

• We saw how to automate QuickCheck tests using spago test.
• We saw how to write properties as functions, and how to use the <?> operator to improve error messages.
• We saw how the Arbitrary and Coarbitrary type classes enable generation of boilerplate testing code, and how they allow us to test higher-order properties.
• We saw how to implement custom Arbitrary and Coarbitrary instances for our own data types.

Domain-Specific Languages

Chapter Goals

In this chapter, we will explore the implementation of domain-specific languages (or DSLs) in PureScript, using a number of standard techniques.

A domain-specific language is a language which is well-suited to development in a particular problem domain. Its syntax and functions are chosen to maximize readability of code used to express ideas in that domain. We have already seen a number of examples of domain-specific languages in this book:

• The Game monad and its associated actions, developed in chapter 11, constitute a domain-specific language for the domain of text adventure game development.
• The quickcheck package, covered in chapter 13, is a domain-specific language for the domain of generative testing. Its combinators enable a particularly expressive notation for test properties.

This chapter will take a more structured approach to some of standard techniques in the implementation of domain-specific languages. It is by no means a complete exposition of the subject, but should provide you with enough knowledge to build some practical DSLs for your own tasks.

Our running example will be a domain-specific language for creating HTML documents. Our aim will be to develop a type-safe language for describing correct HTML documents, and we will work by improving a naive implementation in small steps.

Project Setup

The project accompanying this chapter adds one new dependency - the free library, which defines the free monad, one of the tools which we will be using.

We will test this chapter's project in PSCi.

A HTML Data Type

The most basic version of our HTML library is defined in the Data.DOM.Simple module. The module contains the following type definitions:

newtype Element = Element
{ name         :: String
, attribs      :: Array Attribute
, content      :: Maybe (Array Content)
}

data Content
= TextContent String
| ElementContent Element

newtype Attribute = Attribute
{ key          :: String
, value        :: String
}


The Element type represents HTML elements. Each element consists of an element name, an array of attribute pairs and some content. The content property uses the Maybe type to indicate that an element might be open (containing other elements and text) or closed.

The key function of our library is a function

render :: Element -> String


which renders HTML elements as HTML strings. We can try out this version of the library by constructing values of the appropriate types explicitly in PSCi:

$spago repl > import Prelude > import Data.DOM.Simple > import Data.Maybe > import Effect.Console > :paste … log$ render $Element … { name: "p" … , attribs: [ … Attribute … { key: "class" … , value: "main" … } … ] … , content: Just [ … TextContent "Hello World!" … ] … } … ^D <p class="main">Hello World!</p> unit  As it stands, there are several problems with this library: • Creating HTML documents is difficult - every new element requires at least one record and one data constructor. • It is possible to represent invalid documents: • The developer might mistype the element name • The developer can associate an attribute with the wrong type of element • The developer can use a closed element when an open element is correct In the remainder of the chapter, we will apply certain techniques to solve these problems and turn our library into a usable domain-specific language for creating HTML documents. Smart Constructors The first technique we will apply is simple but can be very effective. Instead of exposing the representation of the data to the module's users, we can use the module exports list to hide the Element, Content and Attribute data constructors, and only export so-called smart constructors, which construct data which is known to be correct. Here is an example. First, we provide a convenience function for creating HTML elements: element :: String -> Array Attribute -> Maybe (Array Content) -> Element element name attribs content = Element { name: name , attribs: attribs , content: content }  Next, we create smart constructors for those HTML elements we want our users to be able to create, by applying the element function: a :: Array Attribute -> Array Content -> Element a attribs content = element "a" attribs (Just content) p :: Array Attribute -> Array Content -> Element p attribs content = element "p" attribs (Just content) img :: Array Attribute -> Element img attribs = element "img" attribs Nothing  Finally, we update the module exports list to only export those functions which are known to construct correct data structures: module Data.DOM.Smart ( Element , Attribute(..) , Content(..) , a , p , img , render ) where  The module exports list is provided immediately after the module name inside parentheses. Each module export can be one of three types: • A value (or function), indicated by the name of the value, • A type class, indicated by the name of the class, • A type constructor and any associated data constructors, indicated by the name of the type followed by a parenthesized list of exported data constructors. Here, we export the Element type, but we do not export its data constructors. If we did, the user would be able to construct invalid HTML elements. In the case of the Attribute and Content types, we still export all of the data constructors (indicated by the symbol .. in the exports list). We will apply the technique of smart constructors to these types shortly. Notice that we have already made some big improvements to our library: • It is impossible to represent HTML elements with invalid names (of course, we are restricted to the set of element names provided by the library). • Closed elements cannot contain content by construction. We can apply this technique to the Content type very easily. We simply remove the data constructors for the Content type from the exports list, and provide the following smart constructors: text :: String -> Content text = TextContent elem :: Element -> Content elem = ElementContent  Let's apply the same technique to the Attribute type. First, we provide a general-purpose smart constructor for attributes. Here is a first attempt: attribute :: String -> String -> Attribute attribute key value = Attribute { key: key , value: value } infix 4 attribute as :=  This representation suffers from the same problem as the original Element type - it is possible to represent attributes which do not exist or whose names were entered incorrectly. To solve this problem, we can create a newtype which represents attribute names: newtype AttributeKey = AttributeKey String  With that, we can modify our operator as follows: attribute :: AttributeKey -> String -> Attribute attribute (AttributeKey key) value = Attribute { key: key , value: value }  If we do not export the AttributeKey data constructor, then the user has no way to construct values of type AttributeKey other than by using functions we explicitly export. Here are some examples: href :: AttributeKey href = AttributeKey "href" _class :: AttributeKey _class = AttributeKey "class" src :: AttributeKey src = AttributeKey "src" width :: AttributeKey width = AttributeKey "width" height :: AttributeKey height = AttributeKey "height"  Here is the final exports list for our new module. Note that we no longer export any data constructors directly: module Data.DOM.Smart ( Element , Attribute , Content , AttributeKey , a , p , img , href , _class , src , width , height , attribute, (:=) , text , elem , render ) where  If we try this new module in PSCi, we can already see massive improvements in the conciseness of the user code: $ spago repl

> import Prelude
> import Data.DOM.Smart
> import Effect.Console
> log $render$ p [ _class := "main" ] [ text "Hello World!" ]

<p class="main">Hello World!</p>
unit


Note, however, that no changes had to be made to the render function, because the underlying data representation never changed. This is one of the benefits of the smart constructors approach - it allows us to separate the internal data representation for a module from the representation which is perceived by users of its external API.

Exercises

1. (Easy) Use the Data.DOM.Smart module to experiment by creating new HTML documents using render.

2. (Medium) Some HTML attributes such as checked and disabled do not require values, and may be rendered as empty attributes:

<input disabled>


Modify the representation of an Attribute to take empty attributes into account. Write a function which can be used in place of attribute or := to add an empty attribute to an element.

Phantom Types

To motivate the next technique, consider the following code:

> log $render$ img
[ src    := "cat.jpg"
, width  := "foo"
, height := "bar"
]

<img src="cat.jpg" width="foo" height="bar" />
unit


The problem here is that we have provided string values for the width and height attributes, where we should only be allowed to provide numeric values in units of pixels or percentage points.

To solve this problem, we can introduce a so-called phantom type argument to our AttributeKey type:

newtype AttributeKey a = AttributeKey String


The type variable a is called a phantom type because there are no values of type a involved in the right-hand side of the definition. The type a only exists to provide more information at compile-time. Any value of type AttributeKey a is simply a string at runtime, but at compile-time, the type of the value tells us the desired type of the values associated with this key.

We can modify the type of our attribute function to take the new form of AttributeKey into account:

attribute :: forall a. IsValue a => AttributeKey a -> a -> Attribute
attribute (AttributeKey key) value = Attribute
{ key: key
, value: toValue value
}


Here, the phantom type argument a is used to ensure that the attribute key and attribute value have compatible types. Since the user cannot create values of type AttributeKey a directly (only via the constants we provide in the library), every attribute will be correct by construction.

Note that the IsValue constraint ensures that whatever value type we associate to a key, its values can be converted to strings and displayed in the generated HTML. The IsValue type class is defined as follows:

class IsValue a where
toValue :: a -> String


We also provide type class instances for the String and Int types:

instance stringIsValue :: IsValue String where
toValue = id

instance intIsValue :: IsValue Int where
toValue = show


We also have to update our AttributeKey constants so that their types reflect the new type parameter:

href :: AttributeKey String
href = AttributeKey "href"

_class :: AttributeKey String
_class = AttributeKey "class"

src :: AttributeKey String
src = AttributeKey "src"

width :: AttributeKey Int
width = AttributeKey "width"

height :: AttributeKey Int
height = AttributeKey "height"


Now we find it is impossible to represent these invalid HTML documents, and we are forced to use numbers to represent the width and height attributes instead:

> import Prelude
> import Data.DOM.Phantom
> import Effect.Console

> :paste
… log $render$ img
…   [ src    := "cat.jpg"
…   , width  := 100
…   , height := 200
…   ]
… ^D

<img src="cat.jpg" width="100" height="200" />
unit


Exercises

1. (Easy) Create a data type which represents either pixel or percentage lengths. Write an instance of IsValue for your type. Modify the width and height attributes to use your new type.

2. (Difficult) By defining type-level representatives for the Boolean values true and false, we can use a phantom type to encode whether an AttributeKey represents an empty attribute such as disabled or checked.

data True
data False


Modify your solution to the previous exercise to use a phantom type to prevent the user from using the attribute operator with an empty attribute.

In our final set of modifications to our API, we will use a construction called the free monad to turn our Content type into a monad, enabling do notation. This will allow us to structure our HTML documents in a form in which the nesting of elements becomes clearer - instead of this:

p [ _class := "main" ]
[ elem $img [ src := "cat.jpg" , width := 100 , height := 200 ] , text "A cat" ]  we will be able to write this: p [ _class := "main" ]$ do
elem $img [ src := "cat.jpg" , width := 100 , height := 200 ] text "A cat"  However, do notation is not the only benefit of a free monad. The free monad allows us to separate the representation of our monadic actions from their interpretation, and even support multiple interpretations of the same actions. The Free monad is defined in the free library, in the Control.Monad.Free module. We can find out some basic information about it using PSCi, as follows: > import Control.Monad.Free > :kind Free (Type -> Type) -> Type -> Type  The kind of Free indicates that it takes a type constructor as an argument, and returns another type constructor. In fact, the Free monad can be used to turn any Functor into a Monad! We begin by defining the representation of our monadic actions. To do this, we need to create a Functor with one data constructor for each monadic action we wish to support. In our case, our two monadic actions will be elem and text. In fact, we can simply modify our Content type as follows: data ContentF a = TextContent String a | ElementContent Element a instance functorContentF :: Functor ContentF where map f (TextContent s x) = TextContent s (f x) map f (ElementContent e x) = ElementContent e (f x)  Here, the ContentF type constructor looks just like our old Content data type - however, it now takes a type argument a, and each data constructor has been modified to take a value of type a as an additional argument. The Functor instance simply applies the function f to the value of type a in each data constructor. With that, we can define our new Content monad as a type synonym for the Free monad, which we construct by using our ContentF type constructor as the first type argument: type Content = Free ContentF  Instead of a type synonym, we might use a newtype to avoid exposing the internal representation of our library to our users - by hiding the Content data constructor, we restrict our users to only using the monadic actions we provide. Because ContentF is a Functor, we automatically get a Monad instance for Free ContentF. We have to modify our Element data type slightly to take account of the new type argument on Content. We will simply require that the return type of our monadic computations be Unit: newtype Element = Element { name :: String , attribs :: Array Attribute , content :: Maybe (Content Unit) }  In addition, we have to modify our elem and text functions, which become our new monadic actions for the Content monad. To do this, we can use the liftF function, provided by the Control.Monad.Free module. Here is its type: liftF :: forall f a. f a -> Free f a  liftF allows us to construct an action in our free monad from a value of type f a for some type a. In our case, we can simply use the data constructors of our ContentF type constructor directly: text :: String -> Content Unit text s = liftF$ TextContent s unit

elem :: Element -> Content Unit
elem e = liftF $ElementContent e unit  Some other routine modifications have to be made, but the interesting changes are in the render function, where we have to interpret our free monad. Interpreting the Monad The Control.Monad.Free module provides a number of functions for interpreting a computation in a free monad: runFree :: forall f a . Functor f => (f (Free f a) -> Free f a) -> Free f a -> a runFreeM :: forall f m a . (Functor f, MonadRec m) => (f (Free f a) -> m (Free f a)) -> Free f a -> m a  The runFree function is used to compute a pure result. The runFreeM function allows us to use a monad to interpret the actions of our free monad. Note: Technically, we are restricted to using monads m which satisfy the stronger MonadRec constraint. In practice, this means that we don't need to worry about stack overflow, since m supports safe monadic tail recursion. First, we have to choose a monad in which we can interpret our actions. We will use the Writer String monad to accumulate a HTML string as our result. Our new render method starts by delegating to a helper function, renderElement, and using execWriter to run our computation in the Writer monad: render :: Element -> String render = execWriter <<< renderElement  renderElement is defined in a where block:  where renderElement :: Element -> Writer String Unit renderElement (Element e) = do  The definition of renderElement is straightforward, using the tell action from the Writer monad to accumulate several small strings:  tell "<" tell e.name for_ e.attribs$ \x -> do
tell " "
renderAttribute x
renderContent e.content


Next, we define the renderAttribute function, which is equally simple:

    where
renderAttribute :: Attribute -> Writer String Unit
renderAttribute (Attribute x) = do
tell x.key
tell "=\""
tell x.value
tell "\""


The renderContent function is more interesting. Here, we use the runFreeM function to interpret the computation inside the free monad, delegating to a helper function, renderContentItem:

      renderContent :: Maybe (Content Unit) -> Writer String Unit
renderContent Nothing = tell " />"
renderContent (Just content) = do
tell ">"
runFreeM renderContentItem content
tell "</"
tell e.name
tell ">"


The type of renderContentItem can be deduced from the type signature of runFreeM. The functor f is our type constructor ContentF, and the monad m is the monad in which we are interpreting the computation, namely Writer String. This gives the following type signature for renderContentItem:

      renderContentItem :: ContentF (Content Unit) -> Writer String (Content Unit)


We can implement this function by simply pattern matching on the two data constructors of ContentF:

      renderContentItem (TextContent s rest) = do
tell s
pure rest
renderContentItem (ElementContent e rest) = do
renderElement e
pure rest


In each case, the expression rest has the type Content Unit, and represents the remainder of the interpreted computation. We can complete each case by returning the rest action.

That's it! We can test our new monadic API in PSCi, as follows:

> import Prelude
> import Data.DOM.Free
> import Effect.Console

> :paste
… log $render$ p [] $do … elem$ img [ src := "cat.jpg" ]
…   text "A cat"
… ^D

<p><img src="cat.jpg" />A cat</p>
unit


Exercises

1. (Medium) Add a new data constructor to the ContentF type to support a new action comment, which renders a comment in the generated HTML. Implement the new action using liftF. Update the interpretation renderContentItem to interpret your new constructor appropriately.

Extending the Language

A monad in which every action returns something of type Unit is not particularly interesting. In fact, aside from an arguably nicer syntax, our monad adds no extra functionality over a Monoid.

Let's illustrate the power of the free monad construction by extending our language with a new monadic action which returns a non-trivial result.

Suppose we want to generate HTML documents which contain hyperlinks to different sections of the document using anchors. We can accomplish this already, by generating anchor names by hand and including them at least twice in the document: once at the definition of the anchor itself, and once in each hyperlink. However, this approach has some basic issues:

• The developer might fail to generate unique anchor names.
• The developer might mistype one or more instances of the anchor name.

In the interest of protecting the developer from their own mistakes, we can introduce a new type which represents anchor names, and provide a monadic action for generating new unique names.

The first step is to add a new type for names:

newtype Name = Name String

runName :: Name -> String
runName (Name n) = n


Again, we define this as a newtype around String, but we must be careful not to export the data constructor in the module's export lists.

Next, we define an instance for the IsValue type class for our new type, so that we are able to use names in attribute values:

instance nameIsValue :: IsValue Name where
toValue (Name n) = n


We also define a new data type for hyperlinks which can appear in a elements, as follows:

data Href
= URLHref String
| AnchorHref Name

instance hrefIsValue :: IsValue Href where
toValue (URLHref url) = url
toValue (AnchorHref (Name nm)) = "#" <> nm


With this new type, we can modify the value type of the href attribute, forcing our users to use our new Href type. We can also create a new name attribute, which can be used to turn an element into an anchor:

href :: AttributeKey Href
href = AttributeKey "href"

name :: AttributeKey Name
name = AttributeKey "name"


The remaining problem is that our users currently have no way to generate new names. We can provide this functionality in our Content monad. First, we need to add a new data constructor to our ContentF type constructor:

data ContentF a
= TextContent String a
| ElementContent Element a
| NewName (Name -> a)


The NewName data constructor corresponds to an action which returns a value of type Name. Notice that instead of requiring a Name as a data constructor argument, we require the user to provide a function of type Name -> a. Remembering that the type a represents the rest of the computation, we can see that this function provides a way to continue computation after a value of type Name has been returned.

We also need to update the Functor instance for ContentF, taking into account the new data constructor, as follows:

instance functorContentF :: Functor ContentF where
map f (TextContent s x) = TextContent s (f x)
map f (ElementContent e x) = ElementContent e (f x)
map f (NewName k) = NewName (f <<< k)


Now we can build our new action by using the liftF function, as before:

newName :: Content Name
newName = liftF $NewName id  Notice that we provide the id function as our continuation, meaning that we return the result of type Name unchanged. Finally, we need to update our interpretation function, to interpret the new action. We previously used the Writer String monad to interpret our computations, but that monad does not have the ability to generate new names, so we must switch to something else. The WriterT monad transformer can be used with the State monad to combine the effects we need. We can define our interpretation monad as a type synonym to keep our type signatures short: type Interp = WriterT String (State Int)  Here, the state of type Int will act as an incrementing counter, used to generate unique names. Because the Writer and WriterT monads use the same type class members to abstract their actions, we do not need to change any actions - we only need to replace every reference to Writer String with Interp. We do, however, need to modify the handler used to run our computation. Instead of just execWriter, we now need to use evalState as well: render :: Element -> String render e = evalState (execWriterT (renderElement e)) 0  We also need to add a new case to renderContentItem, to interpret the new NewName data constructor: renderContentItem (NewName k) = do n <- get let fresh = Name$ "name" <> show n
put $n + 1 pure (k fresh)  Here, we are given a continuation k of type Name -> Content a, and we need to construct an interpretation of type Content a. Our interpretation is simple: we use get to read the state, use that state to generate a unique name, then use put to increment the state. Finally, we pass our new name to the continuation to complete the computation. With that, we can try out our new functionality in PSCi, by generating a unique name inside the Content monad, and using it as both the name of an element and the target of a hyperlink: > import Prelude > import Data.DOM.Name > import Effect.Console > :paste … render$ p [ ] $do … top <- newName … elem$ a [ name := top ] $… text "Top" … elem$ a [ href := AnchorHref top ] \$
… ^D

unit


You can verify that multiple calls to newName do in fact result in unique names.

Exercises

1. (Medium) We can simplify the API further by hiding the Element type from its users. Make these changes in the following steps:

• Combine functions like p and img (with return type Element) with the elem action to create new actions with return type Content Unit.
• Change the render function to accept an argument of type Content Unit instead of Element.
2. (Medium) Hide the implementation of the Content monad by using a newtype instead of a type synonym. You should not export the data constructor for your newtype.

3. (Difficult) Modify the ContentF type to support a new action

isMobile :: Content Boolean


which returns a boolean value indicating whether or not the document is being rendered for display on a mobile device.

Hint: use the ask action and the ReaderT monad transformer to interpret this action. Alternatively, you might prefer to use the RWS monad.

Conclusion

In this chapter, we developed a domain-specific language for creating HTML documents, by incrementally improving a naive implementation using some standard techniques:

• We used smart constructors to hide the details of our data representation, only permitting the user to create documents which were correct-by-construction.
• We used an user-defined infix binary operator to improve the syntax of the language.
• We used phantom types to encode additional information in the types of our data, preventing the user from providing attribute values of the wrong type.
• We used the free monad to turn our array representation of a collection of content into a monadic representation supporting do notation. We then extended this representation to support a new monadic action, and interpreted the monadic computations using standard monad transformers.

These techniques all leverage PureScript's module and type systems, either to prevent the user from making mistakes or to improve the syntax of the domain-specific language.

The implementation of domain-specific languages in functional programming languages is an area of active research, but hopefully this provides a useful introduction some simple techniques, and illustrates the power of working in a language with expressive types.