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 uptodate 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 beginnerfriendly.
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.
About the Book
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 largescale 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 stronglytyped 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
License
Copyright (c) 20142017 Phil Freeman.
The text of this book is licensed under the Creative Commons AttributionNonCommercialShareAlike 3.0 Unported License: https://creativecommons.org/licenses/byncsa/3.0/deed.en_US.
Some text is derived from the PureScript Documentation Repo, which uses the same license, and is copyright various contributors.
The exercises are licensed under the MIT license.
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 triedandtrusted functions such as
map
,filter
andreduce
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 firstclass values to define callbacks.
import { readFile, writeFile } from 'fs' readFile(sourceFile, function (error, data) { if (!error) { writeFile(destFile, data, function (error) { if (!error) { console.log("File copied"); } }); } });

Libraries such as React and virtualdom 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 fastandloose programming style of JavaScript.
Types and Type Inference
The debate over statically typed languages versus dynamically typed languages is welldocumented. 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 compiletime, 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 typecorrectness 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 typedriven: 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 endtoend 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 compiletoJS 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 uptodate 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 0.15.*
of
the PureScript compiler.
About You
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.
How to Read This Book
The chapters in this book are largely self contained. A beginner with little functional programming experience would be welladvised, 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 stronglytyped 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 builtins 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 Discord server is a great place to chat about issues you may be having. The server is dedicated to chat about PureScript
 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
, purescriptnode
and purescriptcontrib
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 8bit 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.
Editor support
You can use your preferred editor to write PureScript (for example, to solve the book exercises). See Editor Support Documentation.
Note that some editors expect a
spago.dhall
file in the root of the opened project for full IDE support. For example, you should open thechapter2
directory to work on the exercises in this chapter.If you use VS Code, you can use the provided workspace to open all the chapters simultaneously.
Solving Exercises
Now that you've installed the necessary development tools, clone this book's repo.
git clone https://github.com/purescriptcontrib/purescriptbook.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 purescriptbook
./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 gettingstarted 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 testdrivendevelopment style.
Exercise
 (Medium) Write a
diagonal
function to compute the length of the diagonal (or hypotenuse) of a rightangled 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 blockcomment 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 Data.Number (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:
Uhoh, 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
 (Easy) Write a function
circleArea
which computes the area of a circle with a given radius. Use thepi
constant, which is defined in theNumbers
module. Hint: don't forget to importpi
by modifying theimport Data.Number
statement.  (Medium) Write a function
leftoverCents
which takes anInt
and returns what's leftover after dividing by100
. Use therem
function. Search Pursuit for this function to learn about usage and which module to import it from. Note: Your IDE may support autoimporting of this function if you accept the autocompletion 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 nopeeking/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 frontend of our application will be the interactive mode PSCi, but it would be possible to build on this code to write a frontend 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.List (List(..), filter, head)
import Data.Maybe (Maybe)
Here, we import several modules:
 The
Control.Plus
module, which defines theempty
value.  The
Data.List
module, which is provided by thelists
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 builtin 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 builtin 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 String
s.
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 toplevel 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 spacedelimited list of argument names. To enter a multiline declaration in PSCi, we can enter "paste mode" by using the :paste
command. In this mode, declarations are terminated using the ControlD 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 indentationsensitive, 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 Clike 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 furtherindented:
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 typeannotated 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 illtyped values result in type errors, illkinded 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 toplevel 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 sideeffect 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 an 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
AddressBook > AddressBook
As expected, the return type was a function. We can apply the resulting function to a second argument:
> :type (insertEntry entry) emptyBook
AddressBook
Note though that the parentheses here are unnecessary  the following is equivalent:
> :type insertEntry entry emptyBook
AddressBook
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.
The >
operator in function types is a type constructor for functions. It takes two type arguments, the function's argument type and the return type. The left and right operands respectively.
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 a function that takes the second.
Now consider the definition of insertEntry
:
insertEntry :: Entry > AddressBook > AddressBook
insertEntry entry book = Cons entry book
If we explicitly parenthesize the righthand 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 pointfree 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 pointfree 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" }
> entry = { firstName: "John", lastName: "Smith", address: address }
> _.lastName entry
"Smith"
> _.address.city entry
"Faketown"
Querying the Address Book
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 highlevel 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 typesafe 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 toplevel 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
The above usage works fine, but is awkward to read. A more familiar phrasing is "eight mod three", which you can achieve by wrapping a prefix function in backticks (`):
> 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 crossedout, 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 Data.Function
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 rightassociative (infixr
), low precedence (0
) operator, which lets us remove sets of parentheses for deeplynested applications.
Another parensbusting 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 + _)
> add3 2
5
To summarize, the following are equivalent definitions of a function that adds 5
to its argument:
add5 x = 5 + x
add5 x = add 5 x
add5 x = (+) 5 x
add5 x = 5 `add` x
add5 = add 5
add5 = \x > 5 + x
add5 = (5 + _)
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 righthand side of findEntry
using either operator. Using backwardscomposition, the righthand 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 righthand 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
 (Easy) Test your understanding of the
findEntry
function by writing down the types of each of its major subexpressions. For example, the type of thehead
function as used is specialized toAddressBook > Maybe Entry
. Note: There is no test for this exercise.  (Medium) Write a function
findEntryByStreet :: String > AddressBook > Maybe Entry
which looks up anEntry
given a street address. Hint reusing the existing code infindEntry
. Test your function in PSCi and by runningspago test
.  (Medium) Rewrite
findEntryByStreet
to replacefilterEntry
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.  (Medium) Write a function
isInBook
which tests whether a name appears in aAddressBook
, returning a Boolean value. Hint: Use PSCi to find the type of theData.List.null
function, which tests whether a list is empty or not.  (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 ignoringaddress
fields. Hint: Use PSCi to find the type of theData.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 theMaybe
type constructorarrays
, which defines functions for working with arraysstrings
, which defines functions for working with JavaScript stringsfoldabletraversable
, which defines functions for folding arrays and other data structuresconsole
, 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:
factorial :: Int > Int
factorial n =
if n == 0 then
1
else
n * factorial (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 then
0
else if 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 factorial
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 nonzero, in the array case, we will branch based on whether the input is nonempty. 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 nonempty 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
 (Easy) Write a recursive function
isEven
which returnstrue
if and only if its input is an even integer.  (Medium) Write a recursive function
countEven
which counts the number of even integers in an array. Hint: the functionhead
(also available inData.Array
) can be used to find the first element in a nonempty 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 shapepreserving 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 domainspecific 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 <$> 1 .. 5
["1","2","3","4","5"]
If we wanted to assign an associativity (left or right) to an infix operator, we could do so with the infixl
and infixr
keywords instead. Using infix
assigns no associativity, meaning that you must parenthesize any expression using the same operator multiple times or using multiple operators of the same precedence.
Filtering Arrays
The Data.Array
module provides another function filter
, which is commonly used together with map
. It provides the ability to create a new array from an existing array, keeping only those elements which match a predicate function.
For example, suppose we wanted to compute an array of all numbers between 1 and 10 which were even. We could do so as follows:
> import Data.Array
> filter (\n > n `mod` 2 == 0) (1 .. 10)
[2,4,6,8,10]
Exercises
 (Easy) Write a function
squared
which calculates the squares of an array of numbers. Hint: Use themap
or<$>
function.  (Easy) Write a function
keepNonNegative
which removes the negative numbers from an array of numbers. Hint: Use thefilter
function.  (Medium)
 Define an infix synonym
<$?>
forfilter
. Note: Infix synonyms may not be defined in the REPL, but you can define it in a file.  Write a
keepNonNegativeRewrite
function, which is the same askeepNonNegative
, but replacesfilter
with your new infix operator<$?>
.  Experiment with the precedence level and associativity of your operator in PSCi. Note: There are no unit tests for this step.
 Define an infix synonym
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 foldabletraversable
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 socalled 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 backwardsfacing 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.Alternative
module (from the control
package):
import Control.Alternative (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.Alternative
> :type guard
forall m. Alternative 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
 (Easy) Write a function
isPrime
which tests if its integer argument is prime or not. Hint: Use thefactors
function.  (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 elementsa
,b
, wherea
is an element of the first array, andb
is an element of the second.  (Medium) Write a function
triples :: Int > Array (Array Int)
which takes a numbern
and returns all Pythagorean triples whose components (thea
,b
andc
values) are each less than or equal ton
. A Pythagorean triple is an array of numbers[a, b, c]
such thata² + b² = c²
. Hint: Use theguard
function in an array comprehension.  (Difficult) Write a function
primeFactors
which produces the prime factorization ofn
, i.e. the array of prime integers whose product isn
. 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 Int
s 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:
> :paste
… f n =
… if n == 0
… then 0
… else 1 + f (n  1)
… ^D
> 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 socalled 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:
factorialTailRec :: Int > Int > Int
factorialTailRec n acc =
if n == 0
then acc
else factorialTailRec (n  1) (acc * n)
Notice that the recursive call to factorialTailRec
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 wellunderstood, 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
 (Easy) Write a function
allTrue
which usesfoldl
to test whether an array of boolean values are all true.  (Medium  No Test) Characterize those arrays
xs
for which the functionfoldl (==) false xs
returnstrue
. In other words, complete the sentence: "The function returnstrue
whenxs
contains ..."  (Medium) Write a function
fibTailRec
which is the same asfib
but in tail recursive form. Hint: Use an accumulator parameter.  (Medium) Write
reverse
in terms offoldl
.
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 aPath
.  The
size
function returns the file size for aPath
which represents a file.  The
isDirectory
function tests whether aPath
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.Examples
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.Examples
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.Examples
> 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

(Easy) Write a function
onlyFiles
which returns all files (not directories) in all subdirectories of a directory. 
(Medium) Write a function
whereIs
to search for a file by name. The function should return a value of typeMaybe 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.

(Difficult) Write a function
largestSmallest
which takes aPath
and returns an array containing the single largest and single smallest files in thePath
. Note: consider the cases where there are zero or one files in thePath
by returning an empty array or a oneelement array respectively.
Conclusion
In this chapter, we covered the basics of recursion in PureScript, as a means of expressing algorithms concisely. We also introduced userdefined 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 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 Number
module, but this time using the as
keyword:
import Data.Number as Number
This makes the types and functions in that module available for use, but only by using the qualified name, like Number.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 Data.Number as N
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
andBoolean
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 booleanvalued 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. The guard in the final line uses the expression otherwise
, which might seem like a keyword, but is in fact just a regular binding in Prelude
:
> :type otherwise
Boolean
> otherwise
true
As this example demonstrates, guards appear on the left of the equals symbol, separated from the list of patterns by a pipe character (
).
Exercises
 (Easy) Write the
factorial
function using pattern matching. Hint: Consider the two corner cases of zero and nonzero inputs. Note: This is a repeat of an example from the previous chapter, but see if you can rewrite it here on your own.  (Medium) Write a function
binomial
which finds the coefficient of the x^k
th term in the polynomial expansion of (1 + x)^n
. This is the same as the number of ways to choose a subset ofk
elements from a set ofn
elements. Use the formulan! / k! (n  k)!
, where!
is the factorial function written earlier. Hint: Use pattern matching to handle corner cases. If it takes a long time to complete or crashes with an error about the call stack, try adding more corner cases.  (Medium) Write a function
pascal
which uses Pascal`s 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 welltyped. 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 this behavior is different than that of the original showPerson
. Without the row variable r
, showPerson
only accepts records with exactly a first
and last
field and no others.
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 }
type Person = { name :: String, address :: Address }
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 twoelement 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
 (Easy) Write a function
sameCity
which uses record patterns to test whether twoPerson
records belong to the same city.  (Medium) What is the most general type of the
sameCity
function, taking into account row polymorphism? What about thelivesInLA
function defined above? Note: There is no test for this exercise.  (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 typeforall a. a > Array a > a
Case Expressions
Patterns do not only appear in toplevel 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
_ > lzs (fromMaybe [] $ tail xs)
For example:
> lzs [1, 2, 3, 4]
[]
> lzs [1, 1, 2, 3]
[1, 2, 3]
This function works by case analysis. If the array is empty, our only option is to return an empty array. If the array is nonempty, we first use a case
expression to split into two cases. If the sum of the array is zero, we return the whole array. If not, we recurse on the tail of the array.
Pattern Match Failures and Partial Functions
If patterns in a case expression are tried in order, then what happens in the case when none of the patterns in a case alternatives match their inputs? In this case, the case expression will fail at runtime with a pattern match failure.
We can see this behavior with a simple example:
import Partial.Unsafe (unsafePartial)
partialFunction :: Boolean > Boolean
partialFunction = unsafePartial \true > true
This function contains only a single case, which only matches a single input, true
. If we compile this file, and test in PSCi with any other argument, we will see an error at runtime:
> partialFunction false
Failed pattern match
Functions which return a value for any combination of inputs are called total functions, and functions which do not are called partial.
It is generally considered better to define total functions where possible. If it is known that a function does not return a result for some valid set of inputs, it is usually better to return a value capable of indicating failure, such as type Maybe a
for some a
, using Nothing
when it cannot return a valid result. This way, the presence or absence of a value can be indicated in a typesafe way.
The PureScript compiler will generate an error if it can detect that your function is not total due to an incomplete pattern match. The unsafePartial
function can be used to silence these errors (if you are sure that your partial function is safe!) If we removed the call to the unsafePartial
function above, then the compiler would generate the following error:
A case expression could not be determined to cover all inputs.
The following additional cases are required to cover all inputs:
false
This tells us that the value false
is not matched by any pattern. In general, these warnings might include multiple unmatched cases.
If we also omit the type signature above:
partialFunction true = true
then PSCi infers a curious type:
> :type partialFunction
Partial => Boolean > Boolean
We will see more types which involve the =>
symbol later on in the book (they are related to type classes), but for now, it suffices to observe that PureScript keeps track of partial functions using the type system, and that we must explicitly tell the type checker when they are safe.
The compiler will also generate a warning in certain cases when it can detect that cases are redundant (that is, a case only matches values which would have been matched by a prior case):
redundantCase :: Boolean > Boolean
redundantCase true = true
redundantCase false = false
redundantCase false = false
In this case, the last case is correctly identified as redundant:
A case expression contains unreachable cases:
false
Note: PSCi does not show warnings, so to reproduce this example, you will need to save this function as a file and compile it using spago build
.
Algebraic Data Types
This section will introduce a feature of the PureScript type system called Algebraic Data Types (or ADTs), which are fundamentally related to pattern matching.
However, we'll first consider a motivating example, which will provide the basis of a solution to this chapter's problem of implementing a simple vector graphics library.
Suppose we wanted to define a type to represent some simple shapes: lines, rectangles, circles, text, etc. In an object oriented language, we would probably define an interface or abstract class Shape
, and one concrete subclass for each type of shape that we wanted to be able to work with.
However, this approach has one major drawback: to work with Shape
s abstractly, it is necessary to identify all of the operations one might wish to perform, and to define them on the Shape
interface. It becomes difficult to add new operations without breaking modularity.
Algebraic data types provide a typesafe way to solve this sort of problem, if the set of shapes is known in advance. It is possible to define new operations on Shape
in a modular way, and still maintain typesafety.
Here is how Shape
might be represented as an algebraic data type:
data Shape
= Circle Point Number
 Rectangle Point Number Number
 Line Point Point
 Text Point String
type Point =
{ x :: Number
, y :: Number
}
This declaration defines Shape
as a sum of different constructors, and for each constructor identifies the data that is included. A Shape
is either a Circle
which contains a center Point
and a radius (a number), or a Rectangle
, or a Line
, or Text
. There are no other ways to construct a value of type Shape
.
An algebraic data type is introduced using the data
keyword, followed by the name of the new type and any type arguments. The type's constructors (i.e. its data constructors) are defined after the equals symbol, and are separated by pipe characters (
). The data carried by an ADT's constructors doesn't have to be restricted to primitive types: constructors can include records, arrays, or even other ADTs.
Let's see another example from PureScript's standard libraries. We saw the Maybe
type, which is used to define optional values, earlier in the book. Here is its definition from the maybe
package:
data Maybe a = Nothing  Just a
This example demonstrates the use of a type parameter a
. Reading the pipe character as the word "or", its definition almost reads like English: "a value of type Maybe a
is either Nothing
, or Just
a value of type a
".
Note that we don't use the syntax forall a.
anywhere in our data definition. forall
syntax is necessary for functions, but is not used when defining ADTs with data
or type aliases with type
.
Data constructors can also be used to define recursive data structures. Here is one more example, defining a data type of singlylinked lists of elements of type a
:
data List a = Nil  Cons a (List a)
This example is taken from the lists
package. Here, the Nil
constructor represents an empty list, and Cons
is used to create nonempty lists from a head element and a tail. Notice how the tail is defined using the data type List a
, making this a recursive data type.
Using ADTs
It is simple enough to use the constructors of an algebraic data type to construct a value: simply apply them like functions, providing arguments corresponding to the data included with the appropriate constructor.
For example, the Line
constructor defined above required two Point
s, so to construct a Shape
using the Line
constructor, we have to provide two arguments of type Point
:
exampleLine :: Shape
exampleLine = Line p1 p2
where
p1 :: Point
p1 = { x: 0.0, y: 0.0 }
p2 :: Point
p2 = { x: 100.0, y: 50.0 }
So, constructing values of algebraic data types is simple, but how do we use them? This is where the important connection with pattern matching appears: the only way to consume a value of an algebraic data type is to use a pattern to match its constructor.
Let's see an example. Suppose we want to convert a Shape
into a String
. We have to use pattern matching to discover which constructor was used to construct the Shape
. We can do this as follows:
showShape :: Shape > String
showShape (Circle c r) =
"Circle [center: " <> showPoint c <> ", radius: " <> show r <> "]"
showShape (Rectangle c w h) =
"Rectangle [center: " <> showPoint c <> ", width: " <> show w <> ", height: " <> show h <> "]"
showShape (Line start end) =
"Line [start: " <> showPoint start <> ", end: " <> showPoint end <> "]"
showShape (Text loc text) =
"Text [location: " <> showPoint loc <> ", text: " <> show text <> "]"
showPoint :: Point > String
showPoint { x, y } =
"(" <> show x <> ", " <> show y <> ")"
Each constructor can be used as a pattern, and the arguments to the constructor can themselves be bound using patterns of their own. Consider the first case of showShape
: if the Shape
matches the Circle
constructor, then we bring the arguments of Circle
(center and radius) into scope using two variable patterns, c
and r
. The other cases are similar.
Exercises
 (Easy) Write a function
circleAtOrigin
which constructs aCircle
(of typeShape
) centered at the origin with radius10.0
.  (Medium) Write a function
doubleScaleAndCenter
which scales the size of aShape
by a factor of2.0
and centers it at the origin.  (Medium) Write a function
shapeText
which extracts the text from aShape
. It should returnMaybe String
, and use theNothing
constructor if the input is not constructed usingText
.
Newtypes
There is a special case of algebraic data types, called newtypes. Newtypes are introduced using the newtype
keyword instead of the data
keyword.
Newtypes must define exactly one constructor, and that constructor must take exactly one argument. That is, a newtype gives a new name to an existing type. In fact, the values of a newtype have the same runtime representation as the underlying type, so there is no runtime performance overhead. They are, however, distinct from the point of view of the type system. This gives an extra layer of type safety.
As an example, we might want to define newtypes as typelevel aliases for Number
, to ascribe units like volts, amps, and ohms:
newtype Volt = Volt Number
newtype Ohm = Ohm Number
newtype Amp = Amp Number
Then we define functions and values using these types:
calculateCurrent :: Volt > Ohm > Amp
calculateCurrent (Volt v) (Ohm r) = Amp (v / r)
battery :: Volt
battery = Volt 1.5
lightbulb :: Ohm
lightbulb = Ohm 500.0
current :: Amp
current = calculateCurrent battery lightbulb
This prevents us from making silly mistakes, such as attempting to calculate the current produced by two lightbulbs without a voltage source.
current :: Amp
current = calculateCurrent lightbulb lightbulb
{
TypesDoNotUnify:
current = calculateCurrent lightbulb lightbulb
^^^^^^^^^
Could not match type
Ohm
with type
Volt
}
If we instead just used Number
without newtype
, then the compiler can't help us catch this mistake:
 This also compiles, but is not as type safe.
calculateCurrent :: Number > Number > Number
calculateCurrent v r = v / r
battery :: Number
battery = 1.5
lightbulb :: Number
lightbulb = 500.0
current :: Number
current = calculateCurrent lightbulb lightbulb  uncaught mistake
Note that while a newtype can only have a single constructor, and the constructor must be of a single value, a newtype can take any number of type variables. For example, the following newtype would be a valid definition (err
and a
are the type variables, and the CouldError
constructor expects a single value of type Either err a
):
newtype CouldError err a = CouldError (Either err a)
Also note that the constructor of a newtype often has the same name as the newtype itself, but this is not a requirement. For example, unique names are also valid:
newtype Coulomb = MakeCoulomb Number
In this case, Coulomb
is the type constructor (of zero arguments) and MakeCoulomb
is the data constructor. These constructors live in different namespaces, even when the names are identical, such as with the Volt
example. This is true for all ADTs. Note that although the type constructor and data constructor can have different names, in practice it is idiomatic for them to share the same name. This is the case with Amp
and Volt
types above.
Another application of newtypes is to attach different behavior to an existing type without changing its representation at runtime. We cover that use case in the next chapter when we discuss type classes.
Exercises
 (Easy) Define
Watt
as anewtype
ofNumber
. Then define acalculateWattage
function using this newWatt
type and the above definitionsAmp
andVolt
:
calculateWattage :: Amp > Volt > Watt
A wattage in Watt
s can be calculated as the product of a given current in Amp
s and a given voltage in Volt
s.
A Library for Vector Graphics
Let's use the data types we have defined above to create a simple library for using vector graphics.
Define a type synonym for a Picture
 just an array of Shape
s:
type Picture = Array Shape
For debugging purposes, we'll want to be able to turn a Picture
into something readable. The showPicture
function lets us do that:
showPicture :: Picture > Array String
showPicture = map showShape
Let's try it out. Compile your module with spago build
and open PSCi with spago repl
:
$ spago build
$ 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
 (Medium) Extend the vector graphics library with a new operation
area
which computes the area of aShape
. For the purpose of this exercise, the area of a line or a piece of text is assumed to be zero.  (Difficult) Extend the
Shape
type with a new data constructorClipped
, which clips anotherPicture
to a rectangle. Extend theshapeBounds
function to compute the bounds of a clipped picture. Note that this makesShape
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.
If you come from an Object Oriented background, please note that the word "class" means something very different in this context than what you're used to. A type class serves a purpose more similar to an OO interface.
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 theMaybe
data type, which represents optional values.tuples
, which defines theTuple
data type, which represents pairs of values.either
, which defines theEither
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 Show Boolean where
show true = "true"
show false = "false"
This code declares a type class instance called showBoolean
 in PureScript, type class instances can be 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 Effect
s, 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)
Type class instances can be defined in one of two places: in the same module that the type class is defined, or in the same module that the type "belonging to" the type class is defined. An instance defined in any other spot is called an "orphan instance" and is not allowed by the PureScript compiler. Some of the exercises in this chapter will require you to copy the definition of a type into your MySolutions module so that you can define type class instances for that type.
Exercises

(Easy) Define a
Show
instance forPoint
. Match the same output as theshowPoint
function from the previous chapter. Note: Point is now anewtype
(instead of atype
synonym), which allows us to customize how toshow
it. Otherwise, we'd be stuck with the defaultShow
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 nonstrict 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 foldabletraversable
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 show
ing each integer have been concatenated into a single String
.
But arrays are not the only types which are foldable. foldabletraversable
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 sideeffects 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 identity 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
}

(Easy) Define a
Show
instance forComplex
. Match the output format expected by the tests (e.g.1.2+3.4i
,5.67.8i
, etc.). 
(Easy) Derive an
Eq
instance forComplex
. Note: You may instead write this instance manually, but why do more work if you don't have to? 
(Medium) Define a
Semiring
instance forComplex
. Note: You can usewrap
andover2
fromData.Newtype
to create a more concise solution. If you do so, you will also need to importclass Newtype
fromData.Newtype
and derive aNewtype
instance forComplex
. 
(Easy) Derive (via
newtype
) aRing
instance forComplex
. 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

(Medium) Derive (via
Generic
) aShow
instance forShape
. How does the amount of code written andString
output compare toshowShape
from the previous chapter? Hint: See the Deriving fromGeneric
section of the Type Class Deriving guide.
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 typea
to typeb
, whereasa => b
applies the constrainta
to the typeb
.
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 Int
s and Number
s.
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 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 (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

(Easy) The following declaration defines a type of nonempty arrays of elements of type
a
:data NonEmpty a = NonEmpty a (Array a)
Write an
Eq
instance for the typeNonEmpty a
which reuses the instances forEq a
andEq (Array a)
. Note: you may instead derive theEq
instance. 
(Medium) Write a
Semigroup
instance forNonEmpty a
by reusing theSemigroup
instance forArray
. 
(Medium) Write a
Functor
instance forNonEmpty
. 
(Medium) Given any type
a
with an instance ofOrd
, we can add a new "infinite" value which is greater than any other value:data Extended a = Infinite  Finite a
Write an
Ord
instance forExtended a
which reuses theOrd
instance fora
. 
(Difficult) Write a
Foldable
instance forNonEmpty
. Hint: reuse theFoldable
instance for arrays. 
(Difficult) Given a type constructor
f
which defines an ordered container (and so has aFoldable
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 off
. Write aFoldable
instance forOneMore f
:instance Foldable f => Foldable (OneMore f) where ...

(Medium) Write a
dedupShapes :: Array Shape > Array Shape
function which removes duplicateShape
s from an array using thenubEq
function. 
(Medium) Write a
dedupShapesFast
function which is the same asdedupShapes
, but uses the more efficientnub
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 Stream (Array a) a where
uncons = Array.uncons
instance 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
Multiparameter 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 Char
s, 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 multiparameter 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 multiparameter type classes to design certain APIs.
Nullary Type Classes
We can even define type classes with zero type arguments! These correspond to compiletime 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 socalled 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 "isa" relationship between the two classes. That is, every member of the subclass is a member of the superclass as well.
Exercises

(Medium) Define a partial function
unsafeMaximum :: Partial => Array Int > Int
which finds the maximum of a nonempty array of integers. Test out your function in PSCi usingunsafePartial
. Hint: Use themaximum
function fromData.Foldable
. 
(Medium) The
Action
class is a multiparameter 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 are used to determine how 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)
Applying an empty action is a noop. And applying two actions in sequence is the same as applying the actions combined. That is, actions respect the operations defined by the
Monoid
class.For example, the natural numbers form a monoid under multiplication:
newtype Multiply = Multiply Int instance Semigroup Multiply where append (Multiply n) (Multiply m) = Multiply (n * m) instance Monoid Multiply where mempty = Multiply 1
Write an instance which implements this action:
instance Action Multiply Int where ...
Remember, your instance must satisfy the laws listed above.

(Difficult) There are actually multiple ways to implement an instance of
Action Multiply Int
. How many can you think of? Purescript does not allow multiple implementations of a same instance, so you will have to replace your original implementation. Note: the tests cover 4 implementations. 
(Medium) Write an
Action
instance which repeats an input string some number of times:instance Action Multiply String where ...
Hint: Search Pursuit for a helperfunction with the signature
String > Int > String
. Note thatString
might appear as a more generic type (such asMonoid
).Does this instance satisfy the laws listed above?

(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. 
(Difficult) Given the following newtype, write an instance for
Action m (Self m)
, where the monoidm
acts on itself usingappend
:newtype Self m = Self m
Note: The testing framework requires
Show
andEq
instances for theSelf
andMultiply
types. You may either write these instances manually, or let the compiler handle this for you withderive newtype instance
shorthand. 
(Difficult) Should the arguments of the multiparameter 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 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 065535.
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 hashequality in terms of equality of hash codes, and should read like a declarative definition of hashequality: two values are "hashequal" 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 Hashable Int where
hash = hashCode
We can also define a simple instance for Boolean
values using pattern matching:
instance Hashable Boolean where
hash false = hashCode 0
hash true = hashCode 1
With an instance for hashing integers, we can create an instance for hashing Char
s by using the toCharCode
function from Data.Char
:
instance 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 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 String
s, by turning a String
into an array of Char
s:
instance 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 nonempty 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

(Easy) Use PSCi to test the hash functions for each of the defined instances. Note: There is no provided unit test for this exercise.

(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 thehashEqual
function, then check for value equality with==
if a duplicate pair of hashes is found. Hint: thenubByEq
function inData.Array
should make this task much simpler. 
(Medium) Write a
Hashable
instance for the following newtype which satisfies the type class law:newtype Hour = Hour Int instance Eq Hour where eq (Hour n) (Hour m) = mod n 12 == mod m 12
The newtype
Hour
and itsEq
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. 
(Difficult) Prove the type class laws for the
Hashable
instances forMaybe
,Either
andTuple
. Note: There is no test for this exercise.
Conclusion
In this chapter, we've been introduced to type classes, a typeoriented 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 realworld 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 likeApplicative
.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.
Note that this apply
is different than the apply
from Data.Function
(infixed as $
). Luckily, infix notation is almost always used for the latter, so you don't need to worry about name collisions.
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 Functor Maybe where
map f (Just a) = Just (f a)
map f Nothing = Nothing
instance 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
, 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 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 sideeffects. Applicative functors allow us to work in larger "programming languages" which support some sort of sideeffect encoded by the functor f
.
As an example, the functor Maybe
represents the side effect of possiblymissing 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 sideeffect 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 (sideeffect 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 sideeffects 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 sideeffects 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 sideeffects 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 sideeffects, and return a single wrapped list, applying the sideeffects 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 nonempty, 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
 (Medium) Write versions of the numeric operators
+
,
,*
and/
which work with optional arguments (i.e. arguments wrapped inMaybe
) and return a value wrapped inMaybe
. Name these functionsaddMaybe
,subMaybe
,mulMaybe
, anddivMaybe
. Hint: Uselift2
.  (Medium) Extend the above exercise to work with all
Apply
types (not justMaybe
). Name these new functionsaddApply
,subApply
,mulApply
, anddivApply
.  (Difficult) Write a function
combineMaybe
which has typeforall a f. Applicative f => Maybe (f a) > f (Maybe a)
. This function takes an optional computation with sideeffects, and returns a sideeffecting 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 "5555555555"
, phoneNumber CellPhone "5555550000"
]
Test this value in PSCi (this result has been formatted):
> import Data.AddressBook
> examplePerson
{ firstName: "John"
, lastName: "Smith"
, homeAddress:
{ street: "123 Fake St."
, city: "FakeTown"
, state: "CA"
}
, phones:
[ { type: HomePhone
, number: "5555555555"
}
, { type: CellPhone
, number: "5555550000"
}
]
}
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 nonempty 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 String
s 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
validateAddress :: Address > V Errors Address
validateAddress a =
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 nonempty, 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
validatePhoneNumberAdo pn = ado
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 "5555555555"
pure ({ type: HomePhone, number: "5555555555" })
> validatePhoneNumber $ phoneNumber HomePhone "555.555.5555"
invalid (["Field 'Number' did not match the required format"])
Exercises
 (Easy) Write a regular expression
stateRegex :: Regex
to check that a string only contains two alphabetic characters. Hint: see the source code forphoneNumberRegex
.  (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.  (Medium) Write a function
validateAddressImproved
that is similar tovalidateAddress
, but uses the abovestateRegex
to validate thestate
field andnonEmptyRegex
to validate thestreet
andcity
fields. Hint: see the source forvalidatePhoneNumber
for an example of how to usematches
.
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 sideeffects 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 sideeffects 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 sideeffects 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 sideeffects 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 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 nonempty, 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
> import Data.AddressBook.Validation
> traverse (nonEmpty "Example") Nothing
pure (Nothing)
> traverse (nonEmpty "Example") (Just "")
invalid (["Field 'Example' cannot be empty"])
> traverse (nonEmpty "Example") (Just "Testing")
pure ((Just "Testing"))
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

(Easy) Write
Eq
andShow
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)

(Medium) Write a
Traversable
instance forTree a
, which combines sideeffects from lefttoright. Hint: There are some additional instance dependencies that need to be defined forTraversable
. 
(Medium) Write a function
traversePreOrder :: forall a m b. Applicative m => (a > m b) > Tree a > m (Tree b)
that performs a preorder traversal of the tree. This means the order of effect execution is rootleftright, instead of leftrootright as was done for the previous inorder 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. 
(Medium) Write a function
traversePostOrder
that performs a postorder traversal of the tree where effects are executed leftrightroot. 
(Medium) Create a new version of the
Person
type where thehomeAddress
field is optional (usingMaybe
). Then write a new version ofvalidatePerson
(renamed asvalidatePersonOptionalAddress
) to validate this newPerson
. Hint: Usetraverse
to validate a field of typeMaybe a
. 
(Difficult) Write a function
sequenceUsingTraverse
which behaves likesequence
, but is written in terms oftraverse
. 
(Difficult) Write a function
traverseUsingSequence
which behaves liketraverse
, but is written in terms ofsequence
.
Applicative Functors for Parallelism
In the discussion above, I chose the word "combine" to describe how applicative functors "combine sideeffects". 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 sideeffects that their computations perform. In fact, it would be valid for an applicative functor to perform its sideeffects 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 sideeffects 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 sideeffect.
 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 sideeffects.
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 sideeffects: optional values, error messages and validation. This chapter will introduce another abstraction for dealing with sideeffects 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 theEffect
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.reactbasichooks
 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 nondeterministic algorithm:
 Choose the value
x
of the first throw.  Choose the value
y
of the second throw.  If the sum of
x
andy
isn
then return the pair[x, y]
, else fail.
Array comprehensions allow us to write this nondeterministic 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 nondeterministic 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 sideeffect), but the types a
can differ (i.e. individual computations can have different result types).
Here is another example of do notation, this time 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 deeplynested 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 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 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 prof "address" >>= \addr >
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 pointfree form  but the usual warnings about readability apply.
Monad Laws
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 rightidentity 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 rightidentity law says that this is equivalent to just expr
.
The leftidentity 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
Folding With Monads
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 sideeffects, 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 sideeffects.
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 nonempty? 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.
Monads and Applicatives
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 lawabiding 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 sideeffects. 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"
addr < child prof "address"
city < child addr "city"
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 sideeffects in sequence.
Exercises

(Easy) Write a function
third
which returns the third element of an array with three or more elements. Your function should return an appropriateMaybe
type. Hint: Look up the types of thehead
andtail
functions from theData.Array
module in thearrays
package. Use do notation with theMaybe
monad to combine these functions. 
(Medium) Write a function
possibleSums
which usesfoldM
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 oneliner using
foldM
. You might want to use thenub
andsort
functions to remove duplicates and sort the result respectively. 
(Medium) Confirm that the
ap
function and theapply
operator agree for theMaybe
monad. Note: There are no tests for this exercise. 
(Medium) Verify that the monad laws hold for the
Monad
instance for theMaybe
type, as defined in themaybe
package. Note: There are no tests for this exercise. 
(Medium) Write a function
filterM
which generalizes thefilter
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)

(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 theap
function defined above. Recall thatlift2
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 socalled native sideeffects. If you are familiar with Haskell, it is the equivalent of the IO
monad.
What are native sideeffects? They are the sideeffects which distinguish JavaScript expressions from idiomatic PureScript expressions, which typically are free from sideeffects. 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 "nonnative" sideeffects:
 Optional values, as represented by the
Maybe
data type  Errors, as represented by the
Either
data type  Multifunctions, as represented by arrays or lists
Note that the distinction is subtle. It is true, for example, that an error message is a possible sideeffect of a JavaScript expression, in the form of an exception. In that sense, exceptions do represent native sideeffects, and it is possible to represent them using Effect
. However, error messages implemented using Either
are not a sideeffect 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.
SideEffects and Purity
In a pure language like PureScript, one question which presents itself is: without sideeffects, how can one write useful realworld code?
The answer is that PureScript does not aim to eliminate sideeffects. It aims to represent sideeffects in such a way that pure computations can be distinguished from computations with sideeffects in the type system. In this sense, the language is still pure.
Values with sideeffects have different types from pure values. As such, it is not possible to pass a sideeffecting argument to a function, for example, and have sideeffects performed unexpectedly.
The only way in which sideeffects 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 welltyped API for computations with sideeffects, 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 fromEffect.Class.Console
. This is interchangeable with the one fromEffect.Console
when dealing with the basicEffect
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 aMonadEffect
instance forEffect
.log :: forall m. MonadEffect m => String > m Unit
Now let's 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 theMain
module for amain
function. You may also specify an alternate module as an entry point with themain
flag, as is done in the above example. Just be sure that this alternate module also contains amain
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 nodefs
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 nonnative 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 * 1.0e3,
x: o.x + o.v * 1.0e3
};
})(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 noninlined 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 * 1.0e3,
x: o.x + o.v * 1.0e3
};
})(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 locallyscoped mutable state, especially when used together with actions like for
, foreach
, and while
which generate efficient loops.
Exercises
 (Medium) Rewrite the
safeDivide
function asexceptionDivide
and throw an exception usingthrowException
with the message"div zero"
if the denominator is zero.  (Medium) Write a function
estimatePi :: Int > Number
that usesn
terms of the Gregory Series to calculate an approximation ofpi
. Hints: You can pattern your answer like the definition ofsimulate
above. You might need to convert anInt
into aNumber
usingtoNumber :: Int > Number
fromData.Int
.  (Medium) Write a function
fibonacci :: Int > Int
to compute then
th Fibonacci number, usingST
to track the values of the previous two Fibonacci numbers. Using PSCi, compare the speed of your newST
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 opensource DOM libraries. For example:
webdom
provides type definitions and low level interface implementations for the W3C DOM spec.webhtml
provides type definitions and low level interface implementations for the W3C HTML5 spec.jquery
is a set of bindings to the jQuery library.
There are also PureScript libraries which build abstractions on top of these libraries, such as
thermite
, which builds onreact
reactbasichooks
, which builds onreactbasic
halogen
which provides a typesafe set of abstractions on top of a custom virtual DOM library.
In this chapter, we will use the reactbasichooks
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 reactbasichooks
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 reactbasichooks
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 filesave 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>
<head>
<meta charset="UTF8">
<title>Address Book</title>
<link rel="stylesheet" href="https://stackpath.bootstrapcdn.com/bootstrap/4.4.1/css/bootstrap.min.css" crossorigin="anonymous">
</head>
<body>
<div id="container"></div>
<script type="module" src="./index.js"></script>
</body>
</html>
The <script
line includes the JavaScript entry point, index.js
, which contains this single line:
import { main } from "../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
log "Rendering address book component"
 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. Passin 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" <<< toNonElementParentNode =<< document =<< window
 or, equivalently:
ctr < window >>= document >>= toNonElementParentNode >>> getElementById "container"
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 {})
mkAddressBookApp =
reactComponent
"AddressBookApp"
(\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 propstoJSX 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 {})
mkAddressBookApp = do
reactComponent "AddressBookApp" \props > R.do
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 breakup 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 of the types 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"}
NonRecord
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 hookscompatible 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 aRecord
of attributes. An array of child elements may be passed to thechildren
field of this record.div'
: Same asdiv
, but returns theReactComponent
before conversion toJSX
.
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
{ className: "alert alertdanger row"
, 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 row
s, and validation errors to be emphasized with alertdanger
styling:
className: "alert alertdanger 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: "formgroup row"
, children:
[ D.div
{ className: "colsm colformlabel"
, children: [ D.text name ]
}
, D.div
{ className: "colsm"
, children:
[ D.input
{ className: "formcontrol"
, 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 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 recordupdate 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.

(Easy) Modify the application to include a work phone number text box.

(Medium) Right now the application shows validation errors collected in a single "pinkalert" background. Modify to give each validation error its own pinkalert 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 onediv
with thealert
andalertdanger
styles for each error. 
(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 theErrors
structure.
Conclusion
This chapter has covered a lot of ideas about handling sideeffects 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 sideeffects.
 We saw how monads are examples of applicative functors, how both allow us to compute with sideeffects, 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 sideeffects.  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 realworld PureScript code. It will be used in the rest of the book to handle sideeffects in a number of other usecases.
Asynchronous Effects
Chapter Goals
This chapter focuses on the Aff
monad, which is similar to the Effect
monad, but represents asynchronous sideeffects. 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 theAff
monad.nodefsaff
 asynchronous filesystem operations withAff
.affjax
 HTTP requests with AJAX andAff
.parallel
 parallel execution ofAff
.
When running outside of the browser (such as in our Node.js environment), the affjax
library requires the xhr2
NPM module, which is listed as dependency in the package.json
of this chapter. 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:
import { promises as fsPromises } from 'fs'
async function copyFile(file1, file2) {
let data = await fsPromises.readFile(file1, { encoding: 'utf8' });
fsPromises.writeFile(file2, data, { encoding: 'utf8' });
}
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, launchAff_)
import Effect (Effect)
import Effect.Class.Console (log)
import Node.Encoding (Encoding(..))
import Node.FS.Aff (readTextFile, writeTextFile)
import Node.Path (FilePath)
main :: Effect Unit
main = launchAff_ program
program :: Aff Unit
program = do
result < attempt $ copyFile "file1.txt" "file2.txt"
case result of
Left e > log $ "There was a problem with copyFile: " <> message e
_ > pure unit
copyFile :: FilePath > FilePath > Aff Unit
copyFile file1 file2 = do
my_data < readTextFile UTF8 file1
writeTextFile UTF8 file2 my_data
Note that we have to use launchAff_
to convert the Aff
to Effect
because main
must be Effect Unit
.
It is also possible to rewrite 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

(Easy) Write a
concatenateFiles
function which concatenates two text files. 
(Medium) Write a function
concatenateMany
to concatenate multiple text files, given an array of input file names and an output file name. Hint: usetraverse
. 
(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.
Additional Aff Resources
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
. Depending on what environment you are targeting you need to use either the purescriptaffjaxweb or the purescriptaffjaxnode library.
In the rest of this chapter we will be targeting node and thus using purescriptaffjaxnode
.
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.Node as AN
import Affjax.ResponseFormat as ResponseFormat
import Data.Either (Either(..))
import Effect.Aff (Aff)
getUrl :: String > Aff String
getUrl url = do
result < AN.get ResponseFormat.string url
pure case result of
Left err > "GET /api response failed to decode: " <> AN.printError err
Right response > response.body
When calling this in the repl, launchAff_
is required to convert the Aff
to a replcompatible Effect
:
$ spago repl
> :pa
… import Prelude
… import Effect.Aff (launchAff_)
… import Effect.Class.Console (log)
… import Test.HTTP (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
 (Easy) Write a function
writeGet
which makes an HTTPGET
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 monadm
and turns them into computations in the applicative functorf
, andsequential
, 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 10millisecond 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 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 realworld 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

(Easy) Write a
concatenateManyParallel
function which has the same signature as the earlierconcatenateMany
function, but reads all input files in parallel. 
(Medium) Write a
getWithTimeout :: Number > String > Aff (Maybe String)
function which makes an HTTPGET
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.

(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: Thenodepath
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 theaff
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 soughtafter. 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 must have a corresponding JavaScript declaration exported from a foreign JavaScript module.
For example, consider the encodeURIComponent
function, which can be used in 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 nonnull strings to nonnull strings, and has no other sideeffects.
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 to import it from. A corresponding foreign JavaScript module is one of the same name but extension changed from .purs
to .js
. If the Purescript module above is saved as URI.purs
, then the foreign JavaScript module is saved as URI.js
.
Since encodeURIComponent
is already defined, we have to export it as _encodeURIComponent
:
"use strict";
export const _encodeURIComponent = encodeURIComponent;
Since version 0.15, Purescript uses the ES module system when interoperating with JavaScript. In ES modules, functions and values are exported from a module by providing the export
keyword on an object.
With these two pieces in place, we can now use the _encodeURIComponent
function from PureScript like any function written in PureScript. For example, in 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";
export const 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. This function calculates the diagonal of a rightangled triangle.
foreign import diagonal :: Number > Number > Number
Recall that functions in PureScript are curried. diagonal
is a function that takes a Number
and returns a function, that takes a Number
and returns a Number
.
export const diagonal = function (w) {
return function (h) {
return Math.sqrt(w * w + h * h);
};
};
Or with ES6 arrow syntax (see ES6 note below).
export const diagonalArrow = w => h =>
Math.sqrt(w * w + h * h);
foreign import diagonalArrow :: Number > Number > Number
$ spago repl
> import Test.Examples
> diagonal 3.0 4.0
5.0
> diagonalArrow 3.0 4.0
5.0
Uncurried Functions
Writing curried functions in JavaScript isn't always feasible, despite being scarcely idiomatic. A typical multiargument JavaScript function would be of the uncurried form:
export const diagonalUncurried = function (w, h) {
return Math.sqrt(w * w + h * h);
};
The module Data.Function.Uncurried
exports wrapper types and utility functions to work with uncurried functions.
foreign import diagonalUncurried :: Fn2 Number Number Number
Inspecting the type constructor Fn2
:
$ spago repl
> import Data.Function.Uncurried
> :kind Fn2
Type > Type > Type > Type
Fn2
takes three type arguments. Fn2 a b c
is a type representing an uncurried function of two arguments of types a
and b
, that returns a value of type c
. We used it to import diagonalUncurried
from the foreign module.
We can then call it with runFn2
which takes the uncurried function then the arguments.
$ spago repl
> import Test.Examples
> import Data.Function.Uncurried
> runFn2 diagonalUncurried 3.0 4.0
5.0
The functions
package defines similar type constructors for function arities from 0 to 10.
A Note About Uncurried Functions
PureScript's curried functions has certain advantages. It allows us to partially apply functions, and to give type class instances for function types  but it comes with a performance penalty. For performance critical code, it is sometimes necessary to define uncurried JavaScript functions which accept multiple arguments.
We can also create uncurried functions from PureScript. For a function of two arguments, we can use the mkFn2
function.
uncurriedAdd :: Fn2 Int Int Int
uncurriedAdd = mkFn2 \n m > m + n
We can apply the uncurried function of two arguments by using runFn2
as before:
uncurriedSum :: Int
uncurriedSum = runFn2 uncurriedAdd 3 10
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;
};
var uncurriedSum = uncurriedAdd(3, 10);
For contrast, here is a traditional curried function:
curriedAdd :: Int > Int > Int
curriedAdd n m = m + n
curriedSum :: Int
curriedSum = curriedAdd 3 10
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;
};
};
var curriedSum = curriedAdd(3)(10);
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,arrowreturn
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
 (Medium) Write a JavaScript function
volumeFn
in theTest.MySolutions
module that finds the volume of a box. Use anFn
wrapper fromData.Function.Uncurried
.  (Medium) Rewrite
volumeFn
with arrow functions asvolumeArrow
.
Passing Simple Types
The following data types may be passed between PureScript and JavaScript asis:
PureScript  JavaScript 

Boolean  Boolean 
String  String 
Int, Number  Number 
Array  Array 
Record  Object 
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 Array
s, 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
export const 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
export const 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
 (Medium) Write a JavaScript function
cumulativeSumsComplex
(and corresponding PureScript foreign import) that takes anArray
ofComplex
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:
export const head = arr =>
arr[0];
How would we type this function? We might try to give it the type forall a. Array a > a
, but for empty arrays, this function returns undefined
. Therefore, the type forall a. Array a > a
does not correctly represent this implementation.
We instead want to return a Maybe
value to handle this corner case:
foreign import maybeHead :: forall a. Array a > Maybe a
But how do we return a Maybe
? It is tempting to write the following:
// Don't do this
import Data_Maybe from '../Data.Maybe'
export const maybeHead = arr => {
if (arr.length) {
return Data_Maybe.Just.create(arr[0]);
} else {
return Data_Maybe.Nothing.value;
}
}
Importing and using the Data.Maybe
module directly in the foreign module isn't recommended as it makes our code brittle to changes in the code generator — create
and value
are not public APIs. Additionally, doing this can cause problems when using purs bundle
for dead code elimination.
The recommended approach is to add extra parameters to our FFIdefined function to accept the functions we need.
export const 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
value would assume a type for x
and will 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 want to actually return arr[0]
. We want a type that represents a value either of type a
or the undefined
value (but not null
). We'll call this type Undefined a
.
We can define a foreign type using a foreign type declaration. The syntax is similar to defining a foreign function:
foreign import data Undefined :: Type > Type
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
:
export const 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]
which may be undefined
, and the type signature correctly reflects that fact.
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
. Well, not exactly. We can use this type in another FFI!
We can write a function that will tell us whether a value is undefined or not:
foreign import isUndefined :: forall a. Undefined a > Boolean
This is defined in our foreign JavaScript module as follows:
export const 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:
export const unsafeHead = arr => {
if (arr.length) {
return arr[0];
} else {
throw new Error('unsafeHead: empty array');
}
};
Exercises

(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 wrapperquadraticRoots :: Quadratic > Pair Complex
that uses the quadratic formula to find the roots of this polynomial. Return the two roots as aPair
ofComplex
numbers. Hint: Use thequadraticRoots
wrapper to pass a constructor forPair
toquadraticRootsImpl
. 
(Medium) Write the function
toMaybe :: forall a. Undefined a > Maybe a
. This function convertsundefined
toNothing
anda
values toJust
s. 
(Difficult) With
toMaybe
in place, we can rewritemaybeHead
asmaybeHead :: forall a. Array a > Maybe a maybeHead = toMaybe <<< undefinedHead
Is this a better approach than our previous implementation? Note: There is no unit test for this exercise.
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
.
export const 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 pointfree 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
):
export const 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 Effect
s are represented in JavaScript as a function of zero arguments, ()
with arrow notation:
export const 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:
export const 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 affpromise
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):
export const 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);
}
export const 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 usecases 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:
export const cumulativeSumsBroken = arr => {
let sum = 0
let sums = []
arr.forEach(x => {
sum += x;
sums.push(sum);
});
sums.push("Broken"); // Bug
return sums;
};
export const 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 nolonger 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 nonobvious 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:
export const cumulativeSumsJson = cumulativeSumsBroken
export const addComplexJson = 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.
export const cumulativeSumsJson = cumulativeSums
export const addComplexJson = addComplex
$ 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 JSONfriendly format before decoding converts it back to a PureScript Map
.
export const 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

(Medium) Write a JavaScript function and PureScript wrapper
valuesOfMap :: Map String Int > Either JsonDecodeError (Set Int)
that returns aSet
of all the values in aMap
. Hint: The.values()
instance method for Map may be useful in your JavaScript code. 
(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 fork
andv
. The compiler will guide you. 
(Medium) Rewrite the earlier
quadraticRoots
function asquadraticRootsSet
which returns theComplex
roots as aSet
via JSON (instead of as aPair
). 
(Difficult) Rewrite the earlier
quadraticRoots
function asquadraticRootsSafe
which uses JSON to pass thePair
ofComplex
roots over FFI. Don't use thePair
constructor in JavaScript, but instead, just return the pair in a decodercompatible format. Hint: You'll need to write aDecodeJson
instance forPair
. 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 anewtype
wrapper forPair
to avoid creating an "orphan instance". 
(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 usejsonParser
to convert theString
intoJson
before decoding. 
(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
andDecodeJson
instances for theTree
type. Consult the argonaut docs for instructions on how to do this. Note that you'll also need generic instances ofShow
andEq
to enable unit testing for this exercise, but those should be straightforward to implement after tackling the JSON instances. 
(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
andDecodeJson
for theIntOrString
data type which implement this behavior. Hint: Thealt
operator fromControl.Alt
may be helpful.
Address book
In this section we will apply our newlyacquired 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.
 A popup alert if there is an issue saving or loading the form state.
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 thegetItem
method onwindow.localStorage
can returnnull
, the return type is notString
, butJson
.
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
:
export const setItem = key => value => () =>
window.localStorage.setItem(key, value);
export const getItem = key => () =>
window.localStorage.getItem(key);
We'll create a save button like so:
saveButton :: R.JSX
saveButton =
D.label
{ className: "formgroup row colformlabel"
, children:
[ D.button
{ className: "btnprimary 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 Generic PhoneType _
instance EncodeJson PhoneType where encodeJson = genericEncodeJson
instance 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 })
mkAddressBookApp =
reactComponent "AddressBookApp" \props > R.do
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
export const 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
 (Easy) Write a wrapper for the
removeItem
method on thelocalStorage
object, and add your foreign function to theEffect.Storage
module.  (Medium) Add a "Reset" button that, when clicked, calls the newlycreated
removeItem
function to delete the "person" entry from local storage.  (Easy) Write a wrapper for the
confirm
method on the JavaScriptWindow
object, and add your foreign function to theEffect.Alert
module.  (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
, purescriptcontrib
and purescriptnode
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 realworld problems in a typesafe 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 onebyone:
import Test from 'Test.js';
Test.gcd(15)(20);
Here, I am assuming that the code was compiled with spago build
, which compiles PureScript modules to ES modules. For that reason, I was able to reference the gcd
function on the Test
object, after importing the Test
module using import
.
You can also use the spago bundleapp
and spago bundlemodule
commands to bundle your generated JavaScript into a single file. Consult the documentation for more information.
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 toplevel 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 userdefined 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 compiletime 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 nonnull JavaScript numbers, and expressions of type String
evaluate to nonnull JavaScript strings. Expressions of type Int
will evaluate to integers at runtime, even though they cannot 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 nonnull JavaScript strings to nonnull 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 (nonnull) 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.
Representing ADTs
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, there are relatively few expressions with a given quantified type, but 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:
identity :: forall a. a > a
identity a = a
Note that the actual
identity
function defined inPrelude
has a slightly different type.
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 that 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:
import { showNumber } from 'Data.Show'
shout(showNumber)(42);
Exercises

(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?

(Medium) Try using the functions defined in the
arrays
package, calling them from JavaScript, by compiling the library usingspago build
and importing modules using theimport
function in NodeJS. Hint: you may need to configure the output path so that the generated ES 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 sideeffects 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:
export const 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 nonnull 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:
export const 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 sideeffect 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:
import { main } from 'Main'
main();
When using spago bundleapp to
or spago run
, this call to main
is generated automatically, whenever the Main
module is defined.
Monadic Adventures
Chapter Goals
The goal of this chapter will be to learn about monad transformers, which provide a way to combine sideeffects provided by different monads. The motivating example will be a text adventure game which can be played on the console in NodeJS. The various sideeffects 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:
orderedcollections
, which provides data types for immutable maps and setstransformers
, which provides implementations of standard monad transformersnodereadline
, which provides FFI bindings to thereadline
interface provided by NodeJSoptparse
, which provides applicative parsers for processing command line arguments
How To Play The Game
To run the project, use spago run
By default you will see a usage message:
Monadic Adventures! A game to learn monad transformers
Usage: run.js (pplayer <player name>) [ddebug]
Play the game as <player name>
Available options:
p,player <player name>
The player's name <String>
d,debug Use debug mode
h,help Show this help text
To provide command line arguments, you can either call spago run
with the a
option to pass additional arguments directly to your application, or you can call spago bundleapp
, which will create an index.js file that can be run directly with node
.
For example, to provide the player name using the p
option:
$ spago run a "p Phil"
>
$ spago bundleapp
$ node index.js 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 twodimensional 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

(Easy) What is the result of replacing
execState
withrunState
orevalState
in our example above? 
(Medium) A string of parentheses is balanced if it is obtained by either concatenating zeroormore shorter balanced strings, or by wrapping a shorter balanced string in a pair of parentheses.
Use the
State
monad and thetraverse_
function to write a functiontestParens :: 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 theData.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

(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. Thepower
function fromData.Monoid
may be helpful too. 
(Easy) Use the
local
function to write a functionindent :: Doc > Doc
which increases the indentation level for a block of code.

(Medium) Use the
sequence
function defined inData.Traversable
to write a functioncat :: Array Doc > Doc
which concatenates a collection of documents, separating them with new lines.

(Medium) Use the
runReader
function to write a functionrender :: 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
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
import Control.Monad.Writer.Class
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
> import Control.Monad.Writer.Class
> runWriter (gcdLog 21 15)
Tuple 3 ["gcdLog 21 15","gcdLog 6 15","gcdLog 6 9","gcdLog 6 3","gcdLog 3 3"]
Exercises

(Medium) Rewrite the
sumArray
function above using theWriter
monad and theAdditive Int
monoid from themonoid
package. 
(Medium) The Collatz function is defined on natural numbers
n
asn / 2
whenn
is even, and3 * n + 1
whenn
is odd. For example, the iterated Collatz sequence starting at10
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.
Monad Transformers
Each of the three monads above: State
, Reader
and Writer
, are also examples of socalled 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 sideeffect. However, it is often useful to use more than one sideeffect 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 sideeffects.
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
sidebyside, 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 custombuilt monads for a variety of problems, choosing the sideeffects that we need, and keeping the expressiveness of do notation and applicative combinators.
The ExceptT Monad Transformer
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
instance Monad m => MonadError e (ExceptT e m)
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, whereasExceptT
models errors as a pure data structure. The
Exception
effect only supports exceptions of one type, namely JavaScript'sError
type, whereasExceptT
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
import Control.Monad.Writer
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 sideeffects provided by their constituent monads.
In practice, the underlying monad m
is either the Effect
monad, if native sideeffects are required, or the Identity
monad, defined in the Data.Identity
module. The Identity
monad adds no new sideeffects, 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 sideeffects 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 sideeffects provided by the ExceptT
monad transformer interact with the sideeffects 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

(Easy) Use the
ExceptT
monad transformer over theIdentity
functor to write a functionsafeDivide
which divides two numbers, throwing an error (as the String "Divide by zero!") if the denominator is zero. 
(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 thestripPrefix
function useful. 
(Difficult) Use the
ReaderT
andWriterT
monad transformers to reimplement the document printing library which we wrote earlier using theReader
monad.Instead of using
line
to emit strings andcat
to concatenate strings, use theArray String
monoid with theWriterT
monad transformer, andtell
to append a line to the result. Use the same names as in the original implementation but ending with an apostrophe ('
).
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
(multiparameter) 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 sideeffects 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 zeroormore 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 singlecharacter strings, the leftover state is empty, and the log shows that we applied the split
combinator four times.
Monad Comprehensions
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) <= MonadPlus 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. Alternative 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

(Easy) Remove the calls to the
lift
function from your implementation of thestring
parser. Verify that the new implementation type checks, and convince yourself that it should. 
(Medium) Use your
string
parser with thesome
combinator to write a parserasFollowedByBs
which recognizes strings consisting of several copies of the string"a"
followed by several copies of the string"b"
. 
(Medium) Use the
<>
operator to write a parserasOrBs
which recognizes strings of the lettersa
orb
in any order. 
(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?
The RWS Monad
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 readerwriterstate 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 sideeffects.
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 Data.Map as M
import 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 twodimensional 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 sideeffects, 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 :: GameState < 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 frontend of our game is built using two packages: optparse
, which provides applicative command line parsing, and nodereadline
, which wraps NodeJS' readline
module, allowing us to write interactive consolebased 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 frontend 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 nodereadline
and console
packages). runGame
takes the game configuration as a function argument.
The nodereadline
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 "> " 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 (lineHandler state) $ interface
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 (lineHandler initialGameState) interface
RL.prompt interface
Exercises

(Medium) Implement a new command
cheat
, which moves all game items from the game grid into the user's inventory. Create a functioncheat :: Game Unit
in theGame
module, and use this function fromgame
. 
(Difficult) The
Writer
component of theRWS
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, andRWS
to handle informational messages. Note: There are no tests for this exercise.
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 optparse
package.
optparse
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 sideeffect. In the case of the optparse
package, the functor we are interested in is the Parser
functor (imported from the optparse module Options.Applicative
, not to be confused with our Parser
that we defined in the Split
module), which adds the sideeffect of reading from command line options. It provides the following handler:
customExecParser :: forall a. ParserPrefs → ParserInfo a → Effect a
This is best illustrated by example. The application's main
function is defined using customExecParser
as follows:
main = OP.customExecParser prefs argParser >>= runGame
The first argument is used to configure the optparse
library. In our case, we simply configure it to show the help message when the application is run without any arguments (instead of showing a "missing argument" error) by using OP.prefs OP.showHelpOnEmpty
, but the Options.Applicative.Builder
module provides several other options.
The second argument is the complete description of our parser program:
argParser :: OP.ParserInfo GameEnvironment
argParser = OP.info (env <**> OP.helper) parserOptions
parserOptions = fold
[ OP.fullDesc
, OP.progDesc "Play the game as <player name>"
, OP.header "Monadic Adventures! A game to learn monad transformers"
]
Here OP.info
combines a Parser
with a set of options for how the help message is formatted. env <**> OP.helper
takes any command line argument Parser
named env
and adds a help
option to it automatically. Options for the help message are of type InfoMod
, which is a monoid, so we can use the fold
function to add several options together.
The interesting part of our parser is constructing the GameEnvironment
:
env :: OP.Parser GameEnvironment
env = gameEnvironment <$> player <*> debug
player :: OP.Parser String
player = OP.strOption $ fold
[ OP.long "player"
, OP.short 'p'
, OP.metavar "<player name>"
, OP.help "The player's name <String>"
]
debug :: OP.Parser Boolean
debug = OP.switch $ fold
[ OP.long "debug"
, OP.short 'd'
, OP.help "Use debug mode"
]
player
and debug
are both Parser
s, so we can use our applicative operators <$>
and <*>
to lift our gameEnvironment
function, which has the type PlayerName > Boolean > GameEnvironment
over Parser
. OP.strOption
constructs a command line option that expects a string value, and is configured via a collection of Mod
s folded together. OP.flag
works similarly, but doesn't expect an associated value. optparse
offers extensive documentation on different modifiers available to build various command line parsers.
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
 (Medium) Add a new Booleanvalued property
cheatMode
to theGameEnvironment
record. Add a new command line flagc
to theoptparse
configuration which enables cheat mode. Thecheat
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 frontend using the console.
Because we separated our implementation from the user interface, it would be possible to create other frontends 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 higherkinded polymorphism and multiparameter 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 APIrefs
, which provides a sideeffect 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 bundleapp 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 piecewiselinear 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 topleft 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 yaxis increases towards the bottom of the canvas:
fillPath ctx $ arc ctx
{ x : 300.0
, y : 300.0
, radius : 50.0
, 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 rowpolymorphic 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 rowpolymorphic 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 piecewiselinear 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 bundleapp 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

(Easy) Experiment with the
strokePath
andsetStrokeStyle
functions in each of the examples so far. 
(Easy) The
fillPath
andstrokePath
functions can be used to render complex paths with a common style by using a do notation block inside the function argument. Try changing theRectangle
example to render two rectangles sidebyside using the same call tofillPath
. Try rendering a sector of a circle by using a combination of a piecewiselinear path and an arc segment. 
(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
between0
and1
as its argument and returns aPoint
, write an action which plotsf
by using yourrenderPath
function. Your action should approximate the path by samplingf
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 sideeffect: 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
, radius : r * 50.0
, start : 0.0
, end : Number.tau
, useCounterClockwise: false
}
fillPath ctx path
strokePath ctx path
Build this example by specifying the Example.Random
module as the main module:
$ spago bundleapp 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 postmultiply 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 higherorder 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 = Number.sin (toNumber count * Number.tau / 8.0) + 1.5
let scaleY = Number.sin (toNumber count * Number.tau / 12.0) + 1.5
translate ctx { translateX: 300.0, translateY: 300.0 }
rotate ctx (toNumber count * Number.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 bottomtotop):
 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 bundleapp 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
 (Easy) Write a higherorder function which strokes and fills a path simultaneously. Rewrite the
Random.purs
example using your function.  (Medium) Use
Random
andDom
to create an application which renders a circle with random position, color and radius to the canvas when the mouse is clicked.  (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.
LSystems
In this final example, we will use the canvas
package to write a function for rendering Lsystems (or Lindenmayer systems).
An Lsystem 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 Lsystem 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 piecewiselinear 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 Lsystem.
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 Lsystem 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 nonzero.
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 nonzero 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 sideeffects 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 Lsystems which use these various types of sideeffect.
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  Number.tau / 6.0 }
interpret state R = pure $ state { theta = state.theta + Number.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 + Number.cos state.theta * 1.5
y = state.y + Number.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 Lsystem, we can simply use the strokePath
action:
strokePath ctx $ lsystem initial productions interpret 5 initialState
Compile the Lsystem example using
$ spago bundleapp main Example.LSystem to dist/Main.js
and open html/index.html
. You should see the Koch curve rendered to the canvas.
Exercises

(Easy) Modify the Lsystem example above to use
fillPath
instead ofstrokePath
. Hint: you will need to include a call toclosePath
, and move the call tomoveTo
outside of theinterpret
function. 
(Easy) Try changing the various numerical constants in the code, to understand their effect on the rendered system.

(Medium) Break the
lsystem
function into two smaller functions. The first should build the final sentence using repeated application ofconcatMap
, and the second should usefoldM
to interpret the result. 
(Medium) Add a drop shadow to the filled shape, by using the
setShadowOffsetX
,setShadowOffsetY
,setShadowBlur
andsetShadowColor
actions. Hint: use PSCi to find the types of these functions. 
(Medium) The angle of the corners is currently a constant (
tau/6
). Instead, it can be moved into theLetter
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?

(Difficult) An Lsystem is given by an alphabet with four letters:
L
(turn left through 60 degrees),R
(turn right through 60 degrees),F
(move forward) andM
(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 Lsystem. 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
andM
in the production rules. The two "move forward" instructions can be differentiated using aBoolean
value using the following alphabet type:data Letter = L  R  F Boolean
Implement this Lsystem again using this representation of the alphabet.

(Difficult) Use a different monad
m
in the interpretation function. You might try usingEffect.Console
to write the Lsystem onto the console, or usingEffect.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 sideeffects.
The examples also demonstrated the power of higherorder 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 propertybased 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
andys
are sorted, thenmerge 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 lessthan check for a greaterthan 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 randomlyselected integers.
Exercises
 (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.
 (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.
Consider adding a type annotation.
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
 (Easy) Write a function
bools
which forces the types ofxs
andys
to beArray Boolean
, and add additional properties which testmergePoly
at that type.  (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 eitherInt
orBoolean
.
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 sideeffects of deterministic random data generation. It uses a pseudorandom 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 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 (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 (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 randomlygenerated tree of type Tree Int
, where the type argument disambiguated by the identity function treeOfInt
.
Exercises
 (Medium) Create a newtype for
String
with an associatedArbitrary
instance which generates collections of randomlyselected characters in the rangeaz
. Hint: use theelements
andarrayOf
functions from theTest.QuickCheck.Gen
module.  (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 HigherOrder 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 higherorder function.
For example, we can pass the length
function as the first argument, to merge two arrays which are already in lengthincreasing order. The result should also be in lengthincreasing 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 randomlygenerated 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 (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 (Arbitrary a, Coarbitrary b) => Coarbitrary (a > b)
This means that we are not limited to just values and functions  we can also randomly generate higherorder functions, or functions whose arguments are higherorder 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 randomlygenerated 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 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 SideEffects
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 sideeffects. 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

(Easy) Write
Coarbitrary
instances for theByte
andSorted
type constructors. 
(Medium) Write a (higherorder) property which asserts associativity of the
mergeWith f
function for any functionf
. Test your property in PSCi usingquickCheckPure
. 
(Medium) Write
Arbitrary
andCoarbitrary
instances for the following data type:data OneTwoThree a = One a  Two a a  Three a a a
Hint: Use the
oneOf
function defined inTest.QuickCheck.Gen
to define yourArbitrary
instance. 
(Medium) Use
all
to simplify the result of thequickCheckPure
function  your new function should have typeList Result > Boolean
and should returntrue
if every test passes andfalse
otherwise. 
(Medium) As another approach to simplifying the result of
quickCheckPure
, try writing a functionsquashResults :: List Result > Result
. Consider using theFirst
monoid fromData.Maybe.First
with thefoldMap
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
andCoarbitrary
type classes enable generation of boilerplate testing code, and how they allow us to test higherorder properties.  We saw how to implement custom
Arbitrary
andCoarbitrary
instances for our own data types.
DomainSpecific Languages
Chapter Goals
In this chapter, we will explore the implementation of domainspecific languages (or DSLs) in PureScript, using a number of standard techniques.
A domainspecific language is a language which is wellsuited 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 domainspecific languages in this book:
 The
Game
monad and its associated actions, developed in chapter 11, constitute a domainspecific language for the domain of text adventure game development.  The
quickcheck
package, covered in chapter 13, is a domainspecific 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 domainspecific 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 domainspecific language for creating HTML documents. Our aim will be to develop a typesafe 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 domainspecific 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 socalled 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 generalpurpose 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

(Easy) Use the
Data.DOM.Smart
module to experiment by creating new HTML documents usingrender
. 
(Medium) Some HTML attributes such as
checked
anddisabled
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 ofattribute
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 socalled 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 righthand side of the definition. The type a
only exists to provide more information at compiletime. Any value of type AttributeKey a
is simply a string at runtime, but at compiletime, 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 IsValue String where
toValue = id
instance 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

(Easy) Create a data type which represents either pixel or percentage lengths. Write an instance of
IsValue
for your type. Modify thewidth
andheight
attributes to use your new type. 
(Difficult) By defining typelevel representatives for the Boolean values
true
andfalse
, we can use a phantom type to encode whether anAttributeKey
represents an empty attribute such asdisabled
orchecked
.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.
The Free Monad
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 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
 (Medium) Add a new data constructor to the
ContentF
type to support a new actioncomment
, which renders a comment in the generated HTML. Implement the new action usingliftF
. Update the interpretationrenderContentItem
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 nontrivial 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 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 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 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 ] $
… text "Back to top"
… ^D
<p><a name="name0">Top</a><a href="#name0">Back to top</a></p>
unit
You can verify that multiple calls to newName
do in fact result in unique names.
Exercises

(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
andimg
(with return typeElement
) with theelem
action to create new actions with return typeContent Unit
.  Change the
render
function to accept an argument of typeContent Unit
instead ofElement
.
 Combine functions like

(Medium) Hide the implementation of the
Content
monad by using anewtype
instead of a type synonym. You should not export the data constructor for yournewtype
. 
(Difficult) Modify the
ContentF
type to support a new actionisMobile :: 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 theReaderT
monad transformer to interpret this action. Alternatively, you might prefer to use theRWS
monad.
Conclusion
In this chapter, we developed a domainspecific 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 correctbyconstruction.
 We used an userdefined 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 domainspecific language.
The implementation of domainspecific 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.