PureScript by Example
Table of Contents
 1. Introduction

2. Getting Started
 2.1 Chapter Goals
 2.2 Introduction
 2.3 Installing PureScript
 2.4 Installing Tools
 2.5 Hello, PureScript!
 2.6 Compiling for the Browser
 2.7 Removing Unused Code
 2.8 Compiling CommonJS Modules
 2.9 Tracking Dependencies with Bower
 2.10 Computing Diagonals
 2.11 Testing Code Using the Interactive Mode
 2.12 Conclusion

3. Functions and Records
 3.1 Chapter Goals
 3.2 Project Setup
 3.3 Simple Types
 3.4 Quantified Types
 3.5 Notes On Indentation
 3.6 Defining Our Types
 3.7 Type Constructors and Kinds
 3.8 Displaying Address Book Entries
 3.9 Test Early, Test Often
 3.10 Creating Address Books
 3.11 Curried Functions
 3.12 Querying the Address Book
 3.13 Infix Function Application
 3.14 Function Composition
 3.15 Tests, Tests, Tests …
 3.16 Conclusion

4. Recursion, Maps And Folds
 4.1 Chapter Goals
 4.2 Project Setup
 4.3 Introduction
 4.4 Recursion on Arrays
 4.5 Maps
 4.6 Infix Operators
 4.7 Filtering Arrays
 4.8 Flattening Arrays
 4.9 Array Comprehensions
 4.10 Do Notation
 4.11 Guards
 4.12 Folds
 4.13 Tail Recursion
 4.14 Accumulators
 4.15 Prefer Folds to Explicit Recursion
 4.16 A Virtual Filesystem
 4.17 Listing All Files
 4.18 Conclusion

5. Pattern Matching
 5.1 Chapter Goals
 5.2 Project Setup
 5.3 Simple Pattern Matching
 5.4 Simple Patterns
 5.5 Guards
 5.6 Array Patterns
 5.7 Record Patterns and Row Polymorphism
 5.8 Nested Patterns
 5.9 Named Patterns
 5.10 Case Expressions
 5.11 Pattern Match Failures and Partial Functions
 5.12 Algebraic Data Types
 5.13 Using ADTs
 5.14 Record Puns
 5.15 Newtypes
 5.16 A Library for Vector Graphics
 5.17 Computing Bounding Rectangles
 5.18 Conclusion
 6. Type Classes

7. Applicative Validation
 7.1 Chapter Goals
 7.2 Project Setup
 7.3 Generalizing Function Application
 7.4 Lifting Arbitrary Functions
 7.5 The Applicative Type Class
 7.6 Intuition for Applicative
 7.7 More Effects
 7.8 Combining Effects
 7.9 Applicative Validation
 7.10 Regular Expression Validators
 7.11 Traversable Functors
 7.12 Applicative Functors for Parallelism
 7.13 Conclusion

8. The Eff Monad
 8.1 Chapter Goals
 8.2 Project Setup
 8.3 Monads and Do Notation
 8.4 The Monad Type Class
 8.5 Monad Laws
 8.6 Folding With Monads
 8.7 Monads and Applicatives
 8.8 Native Effects
 8.9 SideEffects and Purity
 8.10 The Eff Monad
 8.11 Extensible Effects
 8.12 Interleaving Effects
 8.13 The Kind of Eff
 8.14 Records And Rows
 8.15 FineGrained Effects
 8.16 Handlers and Actions
 8.17 Mutable State
 8.18 DOM Effects
 8.19 An Address Book User Interface
 8.20 Conclusion
 9. Canvas Graphics

10. The Foreign Function Interface
 10.1 Chapter Goals
 10.2 Project Setup
 10.3 A Disclaimer
 10.4 Calling PureScript from JavaScript
 10.5 Understanding Name Generation
 10.6 Runtime Data Representation
 10.7 Representing ADTs
 10.8 Representing Quantified Types
 10.9 Representing Constrained Types
 10.10 Using JavaScript Code From PureScript
 10.11 Wrapping JavaScript Values
 10.12 Defining Foreign Types
 10.13 Functions of Multiple Arguments
 10.14 Representing Side Effects
 10.15 Defining New Effects
 10.16 Working With Untyped Data
 10.17 Handling Null and Undefined Values
 10.18 Generic JSON Serialization
 10.19 Conclusion

11. Monadic Adventures
 11.1 Chapter Goals
 11.2 Project Setup
 11.3 How To Play The Game
 11.4 The State Monad
 11.5 The Reader Monad
 11.6 The Writer Monad
 11.7 Monad Transformers
 11.8 The ExceptT Monad Transformer
 11.9 Monad Transformer Stacks
 11.10 Type Classes to the Rescue!
 11.11 Alternatives
 11.12 Monad Comprehensions
 11.13 Backtracking
 11.14 The RWS Monad
 11.15 Implementing Game Logic
 11.16 Running the Computation
 11.17 Handling Command Line Options
 11.18 Conclusion
 12. Callback Hell
 13. Generative Testing
 14. DomainSpecific Languages
1. Introduction
1.1 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:  Asynchronous programming in NodeJS leans heavily on functions as firstclass values to define callbacks.
 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.
1.2 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:
A more involved example shows that typecorrectness can be confirmed without type annotations, even when there exist types which are unknown to the compiler:
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.
1.3 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.
1.4 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 0.11.*
of
the PureScript compiler.
1.5 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.
1.6 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:
Commands which should be typed at the command line will be preceded by a dollar symbol:
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:
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.
1.7 Getting Help
If you get stuck at any point, there are a number of resources available online for learning PureScript:
 The PureScript IRC channel is a great place to chat about issues you may be having. Point your IRC client at irc.freenode.net, and connect to the #purescript channel.
 The PureScript website contains links to several learning resources, including code samples, videos and other resources for beginners.
 The PureScript documentation repository collects articles and examples on a wide variety of topics, written by PureScript developers and users.
 Try PureScript! is a website which allows users to compile PureScript code in the web browser, and contains several simple examples of code.
 Pursuit is a searchable database of PureScript types and functions.
If you prefer to learn by reading examples, the purescript
, purescriptnode
and purescriptcontrib
GitHub organizations contain plenty of examples of PureScript code.
1.8 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.
1.9 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.
2. Getting Started
2.1 Chapter Goals
In this chapter, the goal will be to set up a working PureScript development environment, and to write our first PureScript program.
Our first project will be a very simple PureScript library, which will provide a single function which can compute the length of the diagonal in a rightangled triangle.
2.2 Introduction
Here are the tools we will be using to set up our PureScript development environment:

purs
 The PureScript compiler itself. 
npm
 The Node Package Manager, which will allow us to install the rest of our development tools.  Pulp  A commandline tool which automates many of the tasks associated with managing PureScript projects.
The rest of the chapter will guide you through installing and configuring these tools.
2.3 Installing PureScript
The recommended approach to installing the PureScript compiler is to download a binary release for your platform from the PureScript website.
You should verify that the PureScript compiler executables are available on your path. Try running the PureScript compiler on the command line to verify this:
Other options for installing the PureScript compiler include:
 Via NPM:
npm install g purescript
.  Building the compiler from source. Instructions can be found on the PureScript website.
2.4 Installing Tools
If you do not have a working installation of NodeJS, you should install it. This should also install the npm
package manager on your system. Make sure you have npm
installed and available on your path.
You will also need to install the Pulp command line tool, and the Bower package manager using npm
, as follows:
This will place the pulp
and bower
command line tools on your path. At this point, you will have all the tools needed to create your first PureScript project.
2.5 Hello, PureScript!
Let’s start out simple. We’ll use Pulp to compile and run a simple Hello World! program.
Begin by creating a project in an empty directory, using the pulp init
command:
Pulp has created two directories, src
and test
, and a bower.json
configuration file for us. The src
directory will contain our source files, and the test
directory will contain our tests. We will use the test
directory later in the book.
Modify the src/Main.purs
file to contain the following content:
This small sample illustrates a few key ideas:
 Every file begins with a module header. A module name consists of one or more capitalized words separated by dots. In this case, only a single word is used, but
My.First.Module
would be an equally valid module name.  Modules are imported using their full names, including dots to separate the parts of the module name. Here, we import the
Control.Monad.Eff.Console
module, which provides thelog
function.  The
main
program is defined as a function application. In PureScript, function application is indicated with whitespace separating the function name from its arguments.
Let’s build and run this code using the following command:
Congratulations! You just compiled and executed your first PureScript program.
2.6 Compiling for the Browser
Pulp can be used to turn our PureScript code into Javascript suitable for use in the web browser, by using the pulp browserify
command:
Following this, you should see a large amount of Javascript code printed to the console. This is the output of the Browserify tool, applied to a standard PureScript library called the Prelude, as well as the code in the src
directory. This Javascript code can be saved to a file, and included in a HTML document. If you try this, you should see the words “Hello, World!” printed to your browser’s console.
2.7 Removing Unused Code
Pulp provides an alternative command, pulp build
, which can be used with the O
option to apply dead code elimination, which removes unnecessary Javascript from the output. The result is much smaller:
Again, the generated code can be used in a HTML document. If you open output.js
, you should see a few compiled modules which look like this:
This illustrates a few points about the way the PureScript compiler generates Javascript code:
 Every module gets turned into an object, created by a wrapper function, which contains the module’s exported members.
 PureScript tries to preserve the names of variables wherever possible
 Function applications in PureScript get turned into function applications in JavaScript.
 The main method is run after all modules have been defined, and is generated as a simple method call with no arguments.
 PureScript code does not rely on any runtime libraries. All of the code that is generated by the compiler originated in a PureScript module somewhere which your code depended on.
These points are important, since they mean that PureScript generates simple, understandable code. In fact, the code generation process in general is quite a shallow transformation. It takes relatively little understanding of the language to predict what JavaScript code will be generated for a particular input.
2.8 Compiling CommonJS Modules
Pulp can also be used to generate CommonJS modules from PureScript code. This can be useful when using NodeJS, or just when developing a larger project which uses CommonJS modules to break code into smaller components.
To build CommonJS modules, use the pulp build
command (without the O
option):
The generated modules will be placed in the output
directory by default. Each PureScript module will be compiled to its own CommonJS module, in its own subdirectory.
2.9 Tracking Dependencies with Bower
To write the diagonal
function (the goal of this chapter), we will need to be able to compute square roots. The purescriptmath
package contains type definitions for functions defined on the JavaScript Math
object, so let’s install it:
The save
option causes the dependency to be added to the bower.json
configuration file.
The purescriptmath
library sources should now be available in the bower_components
subdirectory, and will be included when you compile your project.
2.10 Computing Diagonals
Let’s write the diagonal
function, which will be an example of using a function from an external library.
First, import the Math
module by adding the following line at the top of the src/Main.purs
file:
It’s also necessary to import the Prelude
module, which defines very basic operations such as numeric addition and multiplication:
Now define the diagonal
function as follows:
Note that there is no need to define a type for our function. The compiler is able to infer that diagonal
is a function which takes two numbers and returns a number. In general, however, it is a good practice to provide type annotations as a form of documentation.
Let’s also modify the main
function to use the new diagonal
function:
Now compile and run the project again, using pulp run
:
2.11 Testing Code Using the Interactive Mode
The PureScript compiler also ships with an interactive REPL called PSCi. This can be very useful for testing your code, and experimenting with new ideas. Let’s use PSCi to test the diagonal
function.
Pulp can load source modules into PSCi automatically, via the pulp repl
command:
You can type :?
to see a list of commands:
By pressing the Tab key, you should be able to see a list of all functions available in your own code, as well as any Bower dependencies and the Prelude modules.
Start by importing the Prelude
module:
Try evaluating a few expressions now:
Let’s try out our new diagonal
function in PSCi:
You can also use PSCi to define functions:
Don’t worry if the syntax of these examples is unclear right now  it will make more sense as you read through the book.
Finally, you can check the type of an expression by using the :type
command:
Try out the interactive mode now. If you get stuck at any point, simply use the Reset command :reset
to unload any modules which may be compiled in memory.
2.12 Conclusion
In this chapter, we set up a simple PureScript project using the Pulp tool.
We also wrote our first PureScript function, and a JavaScript program which could be compiled and executed either in the browser or in NodeJS.
We will use this development setup in the following chapters to compile, debug and test our code, so you should make sure that you are comfortable with the tools and techniques involved.
3. Functions and Records
3.1 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.
3.2 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:
Here, we import several modules:
 The
Control.Plus
module, which defines theempty
value.  The
Data.List
module, which is provided by thepurescriptlists
package which can be installed using Bower. 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 Pulp, with the following commands:
3.3 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:
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:
Character literals are wrapped in single quotes, unlike string literals which use double quotes:
Arrays correspond to JavaScript arrays, but unlike in JavaScript, all elements of a PureScript array must have the same type:
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:
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:
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:
Functions can be defined at the toplevel of a file by specifying arguments before the equals sign:
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:
Having defined this function in PSCi, we can apply it to its arguments by separating the two arguments from the function name by whitespace:
3.4 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:
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
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:
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:
3.5 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:
But this is not valid code:
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:
but this is not:
Certain PureScript keywords (such as where
, of
and let
) introduce a new block of code, in which declarations must be furtherindented:
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.
3.6 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:
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:
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:
Note that List Entry
is not the same as Array Entry
, which represents an array of entries.
3.7 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:
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:
PureScript’s kind system supports other interesting kinds, which we will see later in the book.
3.8 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:
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
:
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:
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.
3.9 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:
Next, load PSCi, and use the import
command to import your new module:
We can create an entry by using a record literal, which looks just like an anonymous object in JavaScript. Bind it to a name with a let
expression:
Now, try applying our function to the example:
Let’s also test showEntry
by creating an address book entry record containing our example address:
3.10 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.
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:
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:
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:
But List Entry
is the same as AddressBook
, so this is equivalent to
In our case, we already have the appropriate inputs: an Entry
, and a AddressBook
, so can apply Cons
and get a new AddressBook
, which is exactly what we wanted!
Here is our implementation of insertEntry
:
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.
3.11 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
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:
As expected, the return type was a function. We can apply the resulting function to a second argument:
Note though that the parentheses here are unnecessary  the following is equivalent:
This is because function application associates to the left, and this explains why we can just specify function arguments one after the other, separated by whitespace.
Note that in the rest of the book, I will talk about things like “functions of two arguments”. However, it is to be understood that this means a curried function, taking a first argument and returning another function.
Now consider the definition of insertEntry
:
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:
But now, by the same argument, we can remove entry
from both sides:
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.
3.12 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:
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 a list element 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:
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:
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
:
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.
3.13 Infix Function Application
In the code for findEntry
above, we used a different form of function application: the head
function was applied to the expression filter filterEntry book
by using the infix $
symbol.
This is equivalent to the usual application head (filter filterEntry book)
($)
is just an alias for a regular function called apply
, which is defined in the Prelude. It is defined as follows:
So apply
takes a function and a value and applies the function to the value. The infixr
keyword is used to define ($)
as an alias for apply
.
But why would we want to use $
instead of regular function application? The reason is that $
is a rightassociative, low precedence operator. This means that $
allows us to remove sets of parentheses for deeplynested applications.
For example, the following nested function application, which finds the street in the address of an employee’s boss:
becomes (arguably) easier to read when expressed using $
:
3.14 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
In this form, we can apply the eta conversion trick from earlier, to arrive at the final form of findEntry
:
An equally valid righthand side would be:
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.
3.15 Tests, Tests, Tests …
Now that we have the core of a working application, let’s try it out using PSCi.
Let’s first try looking up an entry in the empty address book (we obviously expect this to return an empty result):
An error! Not to worry, this error simply means that PSCi doesn’t know how to print a value of type Entry
as a String.
The return type of findEntry
is Maybe Entry
, which we can convert to a String
by hand.
Our showEntry
function expects an argument of type Entry
, but we have a value of type Maybe Entry
. Remember that this means that the function returns an optional value of type Entry
. What we need to do is apply the showEntry
function if the optional value is present, and propagate the missing value if not.
Fortunately, the Prelude module provides a way to do this. The map
operator can be used to lift a function over an appropriate type constructor like Maybe
(we’ll see more on this function, and others like it, later in the book, when we talk about functors):
That’s better  the return value Nothing
indicates that the optional return value does not contain a value  just as we expected.
For ease of use, we can create a function which prints an Entry
as a String, so that we don’t have to use showEntry
every time:
Now let’s create a nonempty address book, and try again. We’ll reuse our example entry from earlier:
This time, the result contained the correct value. Try defining an address book book2
with two names by inserting another name into book1
, and look up each entry by name.
3.16 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.
4. Recursion, Maps And Folds
4.1 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.
4.2 Project Setup
The source code for this chapter is contained in the two files src/Data/Path.purs
and src/FileOperations.purs
.
The Data.Path
module contains a model of a virtual filesystem. You do not need to modify the contents of this module.
The FileOperations
module contains functions which use the Data.Path
API. Solutions to the exercises can be completed in this file.
The project has the following Bower dependencies:

purescriptmaybe
, which defines theMaybe
type constructor 
purescriptarrays
, which defines functions for working with arrays 
purescriptstrings
, which defines functions for working with Javascript strings 
purescriptfoldabletraversable
, which defines functions for folding arrays and other data structures 
purescriptconsole
, which defines functions for printing to the console
4.3 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:
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 Fibonnacci function:
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.
4.4 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:
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.
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:
4.5 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:
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.
4.6 Infix Operators
The map
function can also be written between the mapping function and the array, by wrapping the function name in backticks:
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 <$>
. This operator can be used infix like any other binary operator:
Let’s look at the type of map
:
The type of map
is actually more general than we need in this chapter. For our purposes, we can treat map
as if it had the following less general type:
This type says that we can choose any two types, a
and b
, with which to apply the map
function. a
is the type of elements in the source array, and b
is the type of elements in the target array. In particular, there is no reason why map
has to preserve the type of the array elements. We can use map
or <$>
to transform integers to strings, for example:
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:
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:
We can use this operator as follows:
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:
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.
4.7 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:
4.8 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:
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:
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”.
4.9 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
:
We can test our function
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:
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
:
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
:
This code uses the product
function from the Data.Foldable
module in the purescriptfoldabletraversable
library.
Excellent! We’ve managed to find the correct set of factor pairs without duplicates.
4.10 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:
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 last line
pure [i, j]
is an example of this kind of expression.  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:
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
:
and the result would be the same.
4.11 Guards
One further change we can make to the factors
function is to move the filter inside the array comprehension. This is possible using the guard
function from the Control.MonadZero
module (from the purescriptcontrol
package):
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:
In our case, we can assume that PSCi reported the following type:
For our purposes, the following calculations tell us everything we need to know about the guard
function on arrays:
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.
4.12 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:
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:
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:
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:
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:
This illustrates the difference between the two functions. The left fold expression is equivalent to the following application:
whereas the right fold is equivalent to this:
4.13 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:
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 purescriptfree
and purescripttailrec
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:
Notice that the recursive call to fact
is the last thing that happens in this function  it is in tail position.
4.14 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 this array recursion which reverses the input array by appending elements at the head of the input array to the end of the result.
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:
In this case, we delegate to the helper function reverse'
, which performs the heavy lifting of reversing the array. Notice though that the function reverse'
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 second implementation of reverse
, note that the helper function reverse'
essentially uses the accumulator parameter to maintain an additional piece of state  the partially constructed result. The result starts out empty, and grows by one element for every element in the input array. However, because later elements are added at the front of the array, the result is the original array in reverse!
Note also that while we might think of the accumulator as “state”, there is no direct mutation going on. The accumulator is an immutable array, and we simply use function arguments to thread the state through the computation.
4.15 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, the reverse
example can be written as a fold in at least two ways. Here is a version which uses foldr
:
Writing reverse
in terms of foldl
will be left as an exercise for the reader.
4.16 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 a function is a file or a directory.
In terms of types, we have the following type definitions:
We can try out the API in PSCi:
The FileOperations
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 FileOperations
module.
4.17 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:
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:
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:
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:
Try out the new version in PSCi  you should get the same result. I’ll let you decide which version you find clearer.
4.18 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.
5. Pattern Matching
5.1 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.
5.2 Project Setup
The source code for this chapter is defined in the file src/Data/Picture.purs
.
The project uses some Bower packages which we have already seen, and adds the following new dependencies:

purescriptglobals
, which provides access to some common JavaScript values and functions. 
purescriptmath
, which provides access to the JavaScriptMath
module.
The Data.Picture
module defines a data type Shape
for simple shapes, and a type Picture
for collections of shapes, along with functions for working with those types.
The module imports the Data.Foldable
module, which provides functions for folding data structures:
The Data.Picture
module also imports the Global
and Math
modules, but this time using the as
keyword:
This makes the types and functions in those modules available for use, but only by using qualified names, like Global.infinity
and Math.max
. This can be useful to avoid overlapping imports, or just to make it clearer which modules certain things are imported from.
Note: it is not necessary to use the same module name as the original module for a qualified import. Shorter qualified names like import Math as M
are possible, and quite common.
5.3 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:
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.
5.4 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:
Try these functions in PSCi.
5.5 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:
In this case, the third line uses a guard to impose the extra condition that the first argument is strictly larger than the second.
As this example demonstrates, guards appear on the left of the equals symbol, separated from the list of patterns by a pipe character (
).
5.6 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:
Here is another function which matches arrays of length five, binding each of its five elements in a different way:
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:
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.
5.7 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:
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:
What is the type variable r
here? Well, if we try showPerson
in PSCi, we see something interesting:
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:
We can read the new type signature of showPerson
as “takes any record with first
and last
fields which are Strings
and any other fields, and returns a String
”.
This function is polymorphic in the row r
of record fields, hence the name row polymorphism.
Note that we could have also written
and PSCi would have inferred the same type.
We will see row polymorphism again later, when we discuss extensible effects.
5.8 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:
5.9 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:
This way, we save ourselves from allocating a new array if the pair is already sorted.
5.10 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):
For example:
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.
5.11 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:
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:
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 with type Maybe a
for some a
, using Nothing
to indicate failure. 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:
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:
then PSCi infers a curious type:
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):
In this case, the last case is correctly identified as redundant:
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 pulp build
.
5.12 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 shape types: 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:
The Point
type might also be defined as an algebraic data type, as follows:
The Point
data type illustrates some interesting points:
 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.
 Even though ADTs are useful for describing data with multiple constructors, they can also be useful when there is only a single constructor.
 The constructors of an algebraic data type might have the same name as the ADT itself. This is quite common, and it is important not to confuse the
Point
type constructor with thePoint
data constructor  they live in different namespaces.
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 are defined after the equals symbol, and are separated by pipe characters (
).
Let’s see another example from PureScript’s standard libraries. We saw the Maybe
type, which is used to to define optional values, earlier in the book. Here is it’s definition from the purescriptmaybe
package:
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
”.
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
:
This example is taken from the purescriptlists
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.
5.13 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
:
To construct the points p1
and p2
, we apply the Point
constructor to its single argument, which is a record.
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:
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.
showPoint
is another example of pattern matching. In this case, there is only a single case, but we use a nested pattern to match the fields of the record contained inside the Point
constructor.
5.14 Record Puns
The showPoint
function matches a record inside its argument, binding the x
and y
properties to values with the same names. In PureScript, we can simplify this sort of pattern match as follows:
Here, we only specify the names of the properties, 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 x
and y
in scope, we can construct a Point
using Point { x, y }
:
This can be useful for improving readability of code in some circumstances.
5.15 Newtypes
There is an important 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. 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 pixels and inches:
This way, it is impossible to pass a value of type Pixels
to a function which expects Inches
, but there is no runtime performance overhead.
Newtypes will become important when we cover type classes in the next chapter, since they allow us to attach different behavior to a type without changing its representation at runtime.
5.16 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:
For debugging purposes, we’ll want to be able to turn a Picture
into something readable. The showPicture
function lets us do that:
Let’s try it out. Compile your module with pulp build
and open PSCi with pulp repl
:
5.17 Computing Bounding Rectangles
The example code for this module contains a function bounds
which computes the smallest bounding rectangle for a Picture
.
The Bounds
data type defines a bounding rectangle. It is also defined as an algebraic data type with a single constructor:
bounds
uses the foldl
function from Data.Foldable
to traverse the array of Shapes
in a Picture
, and accumulate the smallest bounding rectangle:
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.
5.18 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. We will see this idea again later in the book.
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.
6. Type Classes
6.1 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.
6.2 Project Setup
The source code for this chapter is defined in the file src/Data/Hashable.purs
.Â
The project has the following Bower dependencies:

purescriptmaybe
, which defines theMaybe
data type, which represents optional values. 
purescripttuples
, which defines theTuple
data type, which represents pairs of values. 
purescripteither
, which defines theEither
data type, which represents disjoint unions. 
purescriptstrings
, which defines functions which operate on strings. 
purescriptfunctions
, which defines some helper functions for defining PureScript functions.
The module Data.Hashable
imports several modules provided by these Bower packages.
6.3 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:
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:
This code declares a type class instance called showBoolean
 in PureScript, type class instances are named to aid the readability of the generated JavaScript. We say that the Boolean
type belongs to the Show
type class.
We can try out the Show
type class in PSCi, by showing a few values with different types:
These examples demonstrate how to show
values of various primitive types, but we can also show
values with more complicated types:
If we try to show a value of type Data.Either
, we get an interesting error message:
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:
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:
6.4 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
.
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:
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
.
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:
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.
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 is beyond the scope of this chapter. The interested reader is encouraged to read the documentation for the superclasses of Field
in purescriptprelude
.
Semigroups and Monoids
The Semigroup
type class identifies those types which support an append
operation to combine two values:
Strings form a semigroup under regular string concatenation, and so do arrays. Several other standard instances are provided by the purescriptmonoid
package.
The <>
concatenation operator, which we have already seen, is provided as an alias for append
.
The Monoid
type class (provided by the purescriptmonoid
package) extends the Semigroup
type class with the concept of an empty value, called mempty
:
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:
The purescriptmonoid
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 purescriptfoldabletraversable
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:
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:
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. purescriptfoldabletraversable
also defines Foldable
instances for types like Maybe
and Tuple
, and other libraries like purescriptlists
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:
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:
How can we understand the meaning of the map
function, when it acts on many different structures, each in a different way?
Well, we can build an intuition that the map
function applies the function it is given to each element of a container, and builds a new container from the results, with the same shape as the original. But how do we make this concept precise?
Type class instances for Functor
are expected to adhere to a set of laws, called the functor laws:
map id xs = xs
map g (map f xs) = map (g <<< f) xs
The first law is the identity law. It states that lifting the identity function (the function which returns its argument unchanged) over a structure just returns the original structure. This makes sense since the identity function does not modify its input.
The second law is the composition law. It states that mapping one function over a structure, and then mapping a second, is the same thing as mapping the composition of the two functions over the structure.
Whatever “lifting” means in the general sense, it should be true that any reasonable definition of lifting a function over a data structure should obey these rules.
Many standard type classes come with their own set of similar laws. The laws given to a type class give structure to the functions of that type class and allow us to study its instances in generality. The interested reader can research the laws ascribed to the standard type classes that we have seen already.
6.5 Type Annotations
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.
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:
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
, whereas 
a => 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:
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.
6.6 Overlapping Instances
PureScript has another rule regarding type class instances, called the overlapping instances rule. Whenever a type class instance is required at a function call site, PureScript will use the information inferred by the type checker to choose the correct instance. At that time, there should be exactly one appropriate instance for that type. If there are multiple valid instances, the compiler will issue a warning.
To demonstrate this, we can try creating two conflicting type class instances for an example type. In the following code, we create two overlapping Show
instances for the type T
:
This module will compile with no warnings. However, if we use show
at type T
(requiring the compiler to to find a Show
instance), the overlapping instances rule will be enforced, resulting in a warning:
The overlapping instances rule is enforced so that automatic selection of type class instances is a predictable process. If we allowed two type class instances for a type to exist, then either could be chosen depending on the order of module imports, and that could lead to unpredictable behavior of the program at runtime, which is undesirable.
If it is truly the case that there are two valid type class instances for a type, satisfying the appropriate laws, then a common approach is to define newtypes which wrap the existing type. Since different newtypes are allowed to have different type class instances under the overlapping instances rule, there is no longer an issue. This approach is taken in PureScript’s standard libraries, for example in purescriptmaybe
, where the Maybe a
type has multiple valid instances for the Monoid
type class.
6.7 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:
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:
These two type class instances are provided in the purescriptprelude
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.
6.8 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.
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:
Try using foldStream
in PSCi for different types of Stream
and different types of Monoid
.
6.9 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:
This gives a somewhat confusing error message:
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:
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:
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:
Functional dependencies can be quite useful when using multiparameter type classes to design certain APIs.
6.10 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. We’ve seen the partial functions head
and tail
, defined in Data.Array.Partial
already:
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:
Instead, we can republish the Partial
constraint for any functions making use of partial functions:
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:
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.
6.11 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 order 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.
6.12 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:
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:
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:
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:
We can also define a simple instance for Boolean
values using pattern matching:
With an instance for hashing integers, we can create an instance for hashing Char
s by using the toCharCode
function from Data.Char
:
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:
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:
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 equals 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.
6.13 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.
7. Applicative Validation
7.1 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.
7.2 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 Bower dependencies, many of which we have seen before. There are two new dependencies:

purescriptcontrol
, which defines functions for abstracting control flow using type classes likeApplicative
. 
purescriptvalidation
, 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.
7.3 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:
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:
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:
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:
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:
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.
7.4 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
:
In the Maybe
example above, the type constructor f
is Maybe
, so that lift3
is specialized to the following type:
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 any 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:
The Apply
type class is a subclass of Functor
, and defines an additional function apply
. As <$>
was defined as an alias for map
, the Prelude
module defines <*>
as an alias for apply
. As we’ll see, these two operators are often used together.
The type of apply
looks a lot like the type of map
. The difference between map
and apply
is that map
takes a function as an argument, whereas the first argument to apply
is wrapped in the type constructor f
. We’ll see how this is used soon, but first, let’s see how to implement the Apply
type class for the Maybe
type:
This type class instance says that we can apply an optional function to an optional value, and the result is defined only if both are defined.
Now we’ll see how map
and apply
can be used together to lift functions of arbitrary number of arguments.
For functions of one argument, we can just use map
directly.
For functions of two arguments, we have a curried function g
with type a > b > c
, say. This is equivalent to the type a > (b > c)
, so we can apply map
to g
to get a new function of type f a > f (b > c)
for any type constructor f
with a Functor
instance. Partially applying this function to the first lifted argument (of type f a
), we get a new wrapped function of type f (b > c)
. If we also have an Apply
instance for f
, then we can then use apply
to apply the second lifted argument (of type f b
) to get our final value of type f c
.
Putting this all together, we see that if we have values x :: f a
and y :: f b
, then the expression (g <$> x) <*> y
has type f c
(remember, this expression is equivalent to apply (map g x) y
). The precedence rules defined in the Prelude allow us to remove the parentheses: g <$> x <*> y
.
In general, we can use <$>
on the first argument, and <*>
for the remaining arguments, as illustrated here for lift3
:
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:
Try lifting some other functions of various numbers of arguments over Maybe
in this way.
7.5 The Applicative Type Class
There is a related type class called Applicative
, defined as follows:
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
:
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.
7.6 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.
7.7 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:
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 of a parameter. We can lift fullName
over Maybe
to create an implementation of the web service which checks for missing parameters:
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
:
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:
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:
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:
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.
7.8 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 an 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
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:
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:
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:
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.
7.9 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:
where PhoneType
is defined as an algebraic data type:
These functions can be used to construct a Person
representing an address book entry. For example, the following value is defined in Data.AddressBook
:
Test this value in PSCi (this result has been formatted):
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:
In the first two lines, we use the nonEmpty
function to validate a nonempty string. nonEmpty
returns an error (indicated with the Left
constructor) if its input is empty, or a successful empty value (unit
) using the Right
constructor otherwise. We use the sequencing operator *>
to indicate that we want to perform two validations, returning the result from the validator on the right. In this case, the validator on the right simply uses pure
to return the input unchanged.
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:
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 purescriptvalidation
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:
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:
This time, we receive an array of all validation errors.
7.10 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:
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:
Again, try running this validator against some valid and invalid inputs in PSCi:
7.11 Traversable Functors
The remaining validator is validatePerson
, which combines the validators we have seen so far to validate an entire Person
structure:
There is one more interesting function here, which we haven’t seen yet  traverse
, which appears in the final line.
traverse
is defined in the Data.Traversable
module, in the Traversable
type class:
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:
Intuitively, given any applicative functor f
, and a function which takes a value of type a
and returns a value of type b
(with sideeffects tracked by f
), 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 f
).
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
This type signature says that if we have a validation function f
for a type a
, then traverse f
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:
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:
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 is given in the Data.List
module. The definition of traverse
is given 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 f
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:
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.
7.12 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 purescriptparallel
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:
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.
7.13 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!
8. The Eff Monad
8.1 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. We will build upon the address book example of the previous chapters, by using a particular monad to handle the sideeffects of building a user interface in the browser. The monad we will use is an important monad in PureScript  the Eff
monad  used to encapsulate socalled native effects.
8.2 Project Setup
The source code for this project builds on the source for the previous chapter. The modules from the previous project are included in the src
directory for this project.
The project adds the following Bower dependencies:

purescripteff
, which defines theEff
monad, the subject of the second half of the chapter. 
purescriptreact
, a set of bindings to the React user interface library, which we will use to build a user interface for our address book application.
In addition to the modules from the previous chapter, this chapter’s project adds a Main
module, which provides the entry point to the application, and functions to render the user interface.
To compile this project, first install React using npm install
, and then build and bundle the JavaScript source with pulp browserify to dist/Main.js
. To run the project, open the html/index.html
file in your web browser.
8.3 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:
We can see that this function works in PSCi:
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 type applied to the type constructor Maybe
. Suppose we have some type XML
representing XML nodes, and a function
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:
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
.
8.4 The Monad Type Class
The Monad
type class is defined as follows:
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:
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:
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:
Every time the PureScript compiler sees this pattern, it replaces the code with this:
or, written infix:
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:
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.
8.5 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:
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:
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:
is equivalent to this code:
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:
8.6 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:
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:
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:
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:
Then we can use foldM
to express iterated safe division:
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.
8.7 Monads and Applicatives
Every instance of the Monad
type class is also an instance of the Applicative
type class, by virtue of the superclass relationship between the two classes.
However, there is also an implementation of the Applicative
type class which comes “for free” for any instance of Monad
, given by the ap
function:
If m
is a lawabiding member of the Monad
type class, then there is a valid Applicative
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:
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.
8.8 Native Effects
We will now look at one particular monad which is of central importance in PureScript  the Eff
monad.
The Eff
monad is defined in the Prelude, in the Control.Monad.Eff
module. It is used to manage socalled native sideeffects.
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 Eff
. 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 Eff
. So it is not the effect itself which is native, but rather how it is implemented at runtime.
8.9 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 Eff
monad will be presented is to run a computation of type Eff eff a
from JavaScript.
The Pulp 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 Eff
monad.
In this way, we know exactly what sideeffects to expect: exactly those used by main
. In addition, we can use the Eff
monad to restrict what types of sideeffects main
is allowed to have, so that we can say with certainty for example, that our application will interact with the console, but nothing else.
8.10 The Eff Monad
The goal of the Eff
monad is to provide a welltyped API for computations with sideeffects, while at the same time generating efficient Javascript. It is also called the monad of extensible effects, which will be explained shortly.
Here is an example. It uses the purescriptrandom
package, which defines functions for generating random numbers:
If this file is saved as src/Main.purs
, then it can be compiled and run using Pulp:
Running this command, you will see a randomly chosen number between 0
and 1
printed to the console.
This program uses do notation to combine two types of native effects provided by the Javascript runtime: random number generation and console IO.
8.11 Extensible Effects
We can inspect the type of main by opening the module in PSCi:
This type looks quite complicated, but is easily explained by analogy with PureScript’s records.
Consider a simple function which uses a record type:
This function creates a full name string from a record containing firstName
and lastName
properties. If you find the type of this function in PSCi as before, you will see this:
This type reads as follows: “fullName
takes a record with firstName
and lastName
fields and any other properties and returns a String
”.
That is, fullName
does not care if you pass a record with more fields, as long as the firstName
and lastName
properties are present:
Similarly, the type of main
above can be interpreted as follows: “main
is a computation with sideeffects, which can be run in any environment which supports random number generation and console IO, and any other types of side effect, and which returns a value of type Unit
”.
This is the origin of the name “extensible effects”: we can always extend the set of sideeffects, as long as we can support the set of effects that we need.
8.12 Interleaving Effects
This extensibility allows code in the Eff
monad to interleave different types of sideeffect.
The random
function which we used has the following type:
The set of effects (random :: RANDOM  eff1)
here is not the same as those appearing in main
.
However, we can instantiate the type of random
in such a way that the effects do match. If we choose eff1
to be (console :: CONSOLE  eff)
, then the two sets of effects become equal, up to reordering.
Similarly, logShow
has a type which can be specialized to match the effects of main
:
This time we have to choose eff2
to be (random :: RANDOM  eff)
.
The point is that the types of random
and logShow
indicate the sideeffects which they contain, but in such a way that other sideeffects can be mixedin, to build larger computations with larger sets of sideeffects.
Note that we don’t have to give a type for main
. The compiler will find a most general type for main
given the polymorphic types of random
and logShow
.
8.13 The Kind of Eff
The type of main
is unlike other types we’ve seen before. To explain it, we need to consider the kind of Eff
. Recall that types are classified by their kinds just like values are classified by their types. So far, we’ve only seen kinds built from Type
(the kind of types) and >
(which builds kinds for type constructors).
To find the kind of Eff
, use the :kind
command in PSCi:
There are two kinds here that we have not seen before.
Control.Monad.Eff.Effect
is the kind of effects, which represents typelevel labels for different types of sideeffects. To understand this, note that the two labels we saw in main
above both have kind Control.Monad.Eff.Effect
:
The #
kind constructor is used to construct kinds for rows, i.e. unordered, labelled sets.
So Eff
is parameterized by a row of effects, and its return type. That is, the first argument to Eff
is an unordered, labelled set of effect types, and the second argument is the return type.
We can now read the type of main
above:
The first argument to Eff
is (console :: CONSOLE, random :: RANDOM  eff)
. This is a row which contains the CONSOLE
effect and the RANDOM
effect. The pipe symbol 
separates the labelled effects from the row variable eff
which represents any other sideeffects we might want to mix in.
The second argument to Eff
is Unit
, which is the return type of the computation.
8.14 Records And Rows
Considering the kind of Eff
allows us to make a deeper connection between extensible effects and records.
Take the function we defined above:
The kind of the type on the left of the function arrow must be Type
, because only types of kind Type
have values.
The curly braces are actually syntactic sugar, and the full type as understood by the PureScript compiler is as follows:
Note that the curly braces have been removed, and there is an extra Record
constructor. Record
is a builtin type constructor defined in the Prim
module. If we find its kind, we see the following:
That is, Record
is a type constructor which takes a row of types and constructs a type. This is what allows us to write rowpolymorphic functions on records.
The type system uses the same machinery to handle extensible effects as is used for rowpolymorphic records (or extensible records). The only difference is the kind of the types appearing in the labels. Records are parameterized by a row of types, and Eff
is parameterized by a row of effects.
The same type system feature could even be used to build other types which were parameterized on rows of type constructors, or even rows of other rows!
8.15 FineGrained Effects
Type annotations are usually not required when using Eff
, since rows of effects can be inferred, but they can be used to indicate to the compiler which effects are expected in a computation.
If we annotate the previous example with a closed row of effects:
(note the lack of the row variable eff
here), then we cannot accidentally include a subcomputation which makes use of a different type of effect. In this way, we can control the sideeffects that our code is allowed to have.
8.16 Handlers and Actions
Functions such as print
and random
are called actions. Actions have the Eff
type on the right hand side of their functions, and their purpose is to introduce new effects.
This is in contrast to handlers, in which the Eff
type appears as the type of a function argument. While actions add to the set of required effects, a handler usually subtracts effects from the set.
As an example, consider the purescriptexceptions
package. It defines two functions, throwException
and catchException
:
throwException
is an action. Eff
appears on the right hand side, and introduces the new EXCEPTION
effect.
catchException
is a handler. Eff
appears as the type of the second function argument, and the overall effect is to remove the EXCEPTION
effect.
This is useful, because the type system can be used to delimit portions of code which require a particular effect. That code can then be wrapped in a handler, allowing it to be embedded inside a block of code which does not allow that effect.
For example, we can write a piece of code which throws exceptions using the Exception
effect, then wrap that code using catchException
to embed the computation in a piece of code which does not allow exceptions.
Suppose we wanted to read our application’s configuration from a JSON document. The process of parsing the document might result in an exception. The process of reading and parsing the configuration could be written as a function with this type signature:
Then, in the main
function, we could use catchException
to handle the EXCEPTION
effect, logging the error and returning a default configuration:
The purescripteff
package also defines the runPure
handler, which takes a computation with no sideeffects, and safely evaluates it as a pure value:
8.17 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:
newSTRef
is used to create a new mutable reference cell of type STRef h a
, which can be read using the readSTRef
action, and modified using the writeSTRef
and modifySTRef
actions. The type a
is the type of the value stored in the cell, and the type h
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 (using the forE
action in Control.Monad.Eff
) to update the value stored in that cell:
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 runST
function:
The thing to notice here is that the region type h
is quantified inside the parentheses on the left of the function arrow. That means that whatever action we pass to runST
has to work with any region h
whatsoever.
However, once a reference cell has been created by newSTRef
, 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 runST
. This is what allows runST
to safely remove the ST
effect!
In fact, since ST
is the only effect in our example, we can use runST
in conjunction with runPure
to turn simulate
into a pure function:
You can even try running this function in PSCi:
In fact, if we inline the definition of simulate
at the call to runST
, as follows:
then the compiler will notice that the reference cell is not allowed to escape its scope, and can safely turn it into a var
. Here is the generated JavaScript for the body of the call to runST
:
The ST
effect is a good way to generate short JavaScript when working with locallyscoped mutable state, especially when used together with actions like forE
, foreachE
, whileE
and untilE
which generate efficient loops in the Eff
monad.
8.18 DOM Effects
In the final sections of this chapter, we will apply what we have learned about effects in the Eff
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:

purescriptdom
is an extensive set of lowlevel bindings to the browser’s DOM APIs. 
purescriptjquery
is a set of bindings to the jQuery library.
There are also PureScript libraries which build abstractions on top of these libraries, such as

purescriptthermite
, which builds onpurescriptreact
, and 
purescripthalogen
which provides a typesafe set of abstractions on top of a custom virtual DOM library.
In this chapter, we will use the purescriptreact
library to add a user interface to our address book application, but the interested reader is encouraged to explore alternative approaches.
8.19 An Address Book User Interface
Using the purescriptreact
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 purescriptreact
library uses the Eff
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 Eff
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.
The HTML file is essentially empty, except for the following line:
This line includes the JavaScript code which is generated by Pulp. We place it at the end of the file to ensure that the relevant elements are on the page before we try to access them. To rebuild the Main.js
file, Pulp can be used with the browserify
command. Make sure the dist
directory exists first, and that you have installed React as an NPM dependency:
The Main
defines the main
function, which creates the address book component, and renders it to the screen. The main
function uses the CONSOLE
and DOM
effects only, as its type signature indicates:
First, main
logs a status message to the console:
Later, main
uses the DOM API to obtain a reference (doc
) to the document body:
Note that this provides an example of interleaving effects: the log
function uses the CONSOLE
effect, and the window
and document
functions both use the DOM
effect. The type of main
indicates that it uses both effects.
main
uses the window
action to get a reference to the window object, and passes the result to the document
function using >>=
. document
takes a window object and returns a reference to its document.
Note that, by the definition of do notation, we could have instead written this as follows:
It is a matter of personal preference whether this is more or less readable. The first version is an example of pointfree form, since there are no function arguments named, unlike the second version which uses the name w
for the window object.
The Main
module defines an address book component, called addressBook
. To understand its definition, we will need to first need to understand some concepts.
In order to create a React component, we must first create a React class, which acts like a template for a component. In purescriptreact
, we can create classes using the createClass
function. createClass
requires a specification of our class, which is essentially a collection of Eff
actions which are used to handle various parts of the component’s lifecycle. The action we will be interested in is the Render
action.
Here are the types of some relevant functions provided by the React library:
There are a few interesting things to note here:
 The
Render
type synonym is provided in order to simplify some type signatures, and it represents the rendering function for a component.  A
Render
action takes a reference to the component (of typeReactThis
), and returns aReactElement
in theEff
monad. AReactElement
is a data structure describing our intended state of the DOM after rendering.  Every React component defines some type of state. The state can be changed in response to events like button clicks. In
purescriptreact
, the initial state value is provided in thespec
function.  The effect row in the
Render
type uses some interesting effects to restrict access to the React component’s state in certain functions. For example, during rendering, access to the “refs” object isDisallowed
, and access to the component state isReadOnly
.
The Main
module defines a type of states for the address book component, and an initial state:
The state contains a Person
record (which we will make editable using form components), and a collection of errors (which will be populated using our existing validation code).
Now let’s see the definition of our component:
As already indicated, addressBook
will use createClass
and spec
to create a React class. To do so, it will provide our initial state value, and a Render
action. However, what can we do in the Render
action? To answer that, purescriptreact
provides some simple actions which can be used:
The readState
and writeState
functions use extensible effects to ensure that we have access to the React state (via the ReactState
effect), but note that read and write permissions are separated further, by parameterizing the ReactState
effect on another row!
This illustrates an interesting point about PureScript’s rowbased effects: effects appearing inside rows need not be simple singletons, but can have interesting structure, and this flexibility enables some useful restrictions at compile time. If the purescriptreact
library did not make this restriction then it would be possible to get exceptions at runtime if we tried to write the state in the Render
action, for example. Instead, such mistakes are now caught at compile time.
Now we can read the definition of our addressBook
component. It starts by reading the current component state:
Note the following:
 The name
ctx
refers to theReactThis
reference, and can be used to read and write the state where appropriate.  The record inside
AppState
is matched using a record binder, including a record pun for the errors field. We explicitly name various parts of the state structure for convenience.
Recall that Render
must return a ReactElement
structure, representing the intended state of the DOM. The Render
action is defined in terms of some helper functions. One such helper function is renderValidationErrors
, which turns the Errors
structure into an array of ReactElement
s.
In purescriptreact
, ReactElement
s are typically created by applying functions like div
, which create single HTML elements. These functions usually take an array of attributes, and an array of child elements as arguments. However, names ending with a prime character (like ul'
here) omit the attribute array, and use the default attributes instead.
Note that since we are simply manipulating regular data structures here, we can use functions like map
to build up more interesting elements.
A second helper function is formField
, which creates a ReactElement
containing a text input for a single form field:
Again, note that we are composing more interesting elements from simpler elements, applying attributes to each element as we go. One attribute of note here is the onChange
attribute applied to the input
element. This is an event handler, and is used to update the component state when the user edits text in our text box. Our event handler is defined using a third helper function, updateAppState
:
updateAppState
takes a reference to the component in the form of our ReactThis
value, a function to update the Person
record, and the Event
record we are responding to. First, it extracts the new value of the text box from the change
event (using the valueOf
helper function), and uses it to create a new Person
state:
Then, it runs the validation function, and updates the component state (using writeState
) accordingly:
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.
Also try the user interface out by running pulp browserify to dist/Main.js
and then opening the html/index.html
file in your web browser. You should be able to enter some values into the form fields and see the validation errors printed onto the page.
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.
8.20 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
Eff
monad, which is used to handle native sideeffects.  We saw how the
Eff
monad supports extensible effects, and how multiple types of native effect can be interleaved into the same computation.  We saw how effects and records are handled in the kind system, and the connection between extensible records and extensible effects.
 We used the
Eff
monad to handle a variety of effects: random number generation, exceptions, console IO, mutable state, and DOM manipulation using React.
The Eff
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.
9. Canvas Graphics
9.1 Chapter Goals
This chapter will be an extended example focussing on the purescriptcanvas
package, which provides a way to generate 2D graphics from PureScript using the HTML5 Canvas API.
9.2 Project Setup
This module’s project introduces the following new Bower dependencies:

purescriptcanvas
, which gives types to methods from the HTML5 Canvas API 
purescriptrefs
, 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 Pulp 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.
9.3 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 Control.Monad.Eff
module, and also the Graphics.Canvas
module, which contains actions in the Eff
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:
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:
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, by using the setFillStyle
action:
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
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:
Build the rectangle example, providing Example.Rectangle
as the name of the main module:
Now, open the html/index.html
file and verify that this code renders a blue rectangle in the center of the canvas.
9.4 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:
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:
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
:
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:
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:
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:
and open html/index.html
again to see the result. You should see the three different types of shapes rendered to the canvas.
9.5 Drawing Random Circles
The Example/Random.purs
file contains an example which uses the Eff
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:
Next, the code uses the for_
function to loop over the integers between 0
and 100
:
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:
Next, for each circle, the code creates an Arc
based on these parameters and finally fills and strokes the arc with the current styles:
Build this example by specifying the Example.Random
module as the main module:
and view the result by opening html/index.html
.
9.6 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 purescriptcanvas
library supports these transformations using the following functions:
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:
The effect of this sequence of actions is that the scene is rotated, then scaled, and finally translated.
9.7 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. purescriptcanvas
wraps this functionality into the following functions:
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:
In the interest of abstracting over common use cases using higherorder functions, the purescriptcanvas
library provides the withContext
function, which performs some canvas action while preserving the original context state:
We could rewrite the rotated
function above using withContext
as follows:
9.8 Global Mutable State
In this section, we’ll use the purescriptrefs
package to demonstrate another effect in the Eff
monad.
The Control.Monad.Eff.Ref
module provides a type constructor for global mutable references, and an associated effect:
A value of type Ref a
is a mutable reference cell containing a value of type a
, much like an STRef h a
, which we saw in the previous chapter. The difference is that, while the ST
effect can be removed by using runST
, the Ref
effect does not provide a handler. Where ST
is used to track safe, local mutation, Ref
is used to track global mutation. As such, it should be used sparingly.
The Example/Refs.purs
file contains an example which uses the REF
effect to track mouse clicks on the canvas
element.
The code starts by creating a new reference containing the value 0
, by using the newRef
action:
Inside the click event handler, the modifyRef
action is used to update the click count:
The readRef
action is used to read the new click count:
In the render
function, the click count is used to determine the transformation applied to a rectangle:
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:
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.
9.9 LSystems
In this final example, we will use the purescriptcanvas
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:
If we start with the initial sequence “FRRFRRFRR” and iterate, we obtain the following sequence:
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 choice of alphabet by a choice of type. For our example, we can choose the following type:
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
:
Our production rules can be represented as a function from Alphabet
to Sentence
as follows:
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
:
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 Alphabet
, but for any type, so we should generalize our type accordingly. Let’s replace Alphabet
and Sentence
with a
and Array a
for some quantified type variable a
:
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:
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:
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:
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:
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 > Eff (canvas :: CANVAS  eff) s
. This sounds like a job for the foldM
function which we defined earlier, and which is available from the purescriptcontrol
package:
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:
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 Eff
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:
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 Alphabet
and s
to be State
, and for the monad m
we can choose Eff (canvas :: CANVAS)
. This gives us the following type:
To implement this function, we need to handle the three data constructors of the Alphabet
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:
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:
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:
Compile the Lsystem example using
and open html/index.html
. You should see the Koch curve rendered to the canvas.
9.10 Conclusion
In this chapter, we learned how to use the HTML5 Canvas API from PureScript by using the purescriptcanvas
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 Eff
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:
This approach is taken in the purescriptdrawing
package, and it brings the flexibility of being able to manipulate the scene as data in various ways before rendering.
In the next chapter, we will see how to implement libraries like purescriptcanvas
which wrap existing JavaScript functionality, by using PureScript’s foreign function interface.
10. The Foreign Function Interface
10.1 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 the following:
 How to call pure JavaScript functions from PureScript,
 How to create new effect types and actions for use with the
Eff
monad, based on existing JavaScript code,  How to call PureScript code from JavaScript,
 How to understand the representation of PureScript values at runtime,
 How to work with untyped data using the
purescriptforeign
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:
 Alerting the user with a popup notification,
 Storing the serialized form data in the browser’s local storage, and reloading it when the application restarts.
10.2 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 adds two new Bower dependencies:
 The
purescriptforeign
library, which provides a data type and functions for working with untyped data.  The
purescriptforeigngeneric
library, which adds support for datatype generic programming to thepurescriptforeign
library.
Note: to avoid browserspecific issues with local storage when the webpage is served from a local file, it might be necessary to run this chapter’s project over HTTP.
10.3 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.
10.4 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:
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:
Here, I am assuming that the code was compiled with pulp build
, which compiles PureScript modules to CommonJS modules. For that reason, I was able to reference the gcd
function on the Test
object, after importing the Test
module using require
.
You might also like to bundle JavaScript code for the browser, using pulp build O to file.js
. In that case, you would access the Test
module from the global PureScript namespace, which defaults to PS
:
10.5 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,
generates the following Javascript:
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,
generates the following Javascript:
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.
10.6 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 not be distinguished from values of type Number
by using typeof
.
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.
10.7 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:
The PureScript compiler generates the following code:
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:
generates this JavaScript code:
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:
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.
10.8 Representing Quantified Types
Expressions with quantified (polymorphic) types have restrictive representations at runtime. In practice, this means that there are relatively few expressions with a given quantified type, but that we can reason about them quite effectively.
Consider this polymorphic type, for example:
What sort of functions have this type? Well, there is certainly one function with this type  namely, the identity function id
, defined in the Prelude
:
In fact, the id
function is the only (total) function with this type! This certainly seems to be the case (try writing an expression with this type which is not observably equivalent to id
), 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:
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 id
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.
10.9 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:
The generated JavaScript looks like this:
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 the Prelude as the first parameter:
10.10 Using JavaScript Code From PureScript
The simplest way to use JavaScript code from PureScript is to give a type to an existing JavaScript value using a foreign import declaration. Foreign import declarations should have a corresponding Javascript declaration in a foreign Javascript module.
For example, consider the encodeURIComponent
function, which can be used from JavaScript to encode a component of a URI by escaping special characters:
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:
We also need to write a foreign Javascript module. If the module above is saved as src/Data/URI.purs
, then the foreign Javascript module should be saved as
src/Data/URI.js
:
Pulp will find .js
files in the src
directory, and provide them to the compiler as foreign Javascript modules.
Javascript functions and values are exported from foreign Javascript modules by assigning them to the exports
object just like a regular CommonJS module. The purs
compiler treats this module like a regular CommonJS module, and simply adds it as a dependency to the compiled
PureScript module. However, when bundling code for the browser with pscbundle
or pulp build O to
, it is very important to follow the
pattern above, assigning exports to the exports
object using a property assignment. This is because pscbundle
recognizes this format,
allowing unused Javascript exports to be removed from bundled code.
With these two pieces in place, we can now use the encodeURIComponent
function from PureScript like any function written in PureScript.
For example, if this declaration is saved as a module and loaded into PSCi, we can reproduce the calculation above:
This approach works well for simple JavaScript values, but is of limited use for more complicated examples. The reason is that most idiomatic JavaScript code does not meet the strict criteria imposed by the runtime representations of the basic PureScript types. In those cases, we have another option  we can wrap the JavaScript code in such a way that we can force it to adhere to the correct runtime representation.
10.11 Wrapping JavaScript Values
We might want to wrap Javascript values and functions for a number of reasons:
 A function takes multiple arguments, but we want to call it like a curried function.
 We might want to use the
Eff
monad to keep track of any JavaScript sideeffects.  It might be necessary to handle corner cases like
null
orundefined
, to give a function the correct runtime representation.
For example, suppose we wanted to recreate the head
function on arrays by using a foreign declaration. In JavaScript, we might write the function as follows:
However, there is a problem with this function. We might try to give it the type forall a. Array a > a
, but for empty arrays, this function returns undefined
. Therefore, this function does not have the correct runtime representation, and we should use a wrapper function to handle this corner case.
To keep things simple, we can throw an exception in the case of an empty array. Strictly speaking, pure functions should not throw exceptions, but it will suffice for demonstration purposes, and we can indicate the lack of safety in the function name:
In our foreign Javascript module, we can define unsafeHead
as follows:
10.12 Defining Foreign Types
Throwing an exception in the case of failure is less than ideal  idiomatic PureScript code uses the type system to represent sideeffects such as missing values. One example of this approach is the Maybe
type constructor. In this section, we will build another solution using the FFI.
Suppose we wanted to define a new type Undefined a
whose representation at runtime was like that for the type a
, but also allowing the undefined
value.
We can define a foreign type using the FFI using a foreign type declaration. The syntax is similar to defining a foreign function:
Note that the data
keyword here indicates that we are defining a type, not a value. Instead of a type signature, we give the kind of the new type. In this case, we declare the kind of Undefined
to be Type > Type
. In other words, Undefined
is a type constructor.
We can now simplify our original definition for head
:
And in the PureScript module:
Note the two changes: the body of the head
function is now much simpler, and returns arr[0]
even if that value is undefined, and the type signature has been changed to reflect the fact that our function can return an undefined value.
This function has the correct runtime representation for its type, but is quite useless since we have no way to use a value of type Undefined a
. But we can fix that by writing some new functions using the FFI!
The most basic function we need will tell us whether a value is defined or not:
This is easily defined in our foreign Javascript module as follows:
We can now use isUndefined
and head
together from PureScript to define a useful function:
Here, the foreign functions we defined were very simple, which meant that we were able to 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.
10.13 Functions of Multiple Arguments
PureScript’s Prelude contains an interesting set of examples of foreign types. As we have covered already, PureScript’s function types only take a single argument, and can be used to simulate functions of multiple arguments via currying. This has certain advantages  we can partially apply functions, and give type class instances for function types  but it comes with a performance penalty. For performance critical code, it is sometimes necessary to define genuine JavaScript functions which accept multiple arguments. The Prelude defines foreign types which allow us to work safely with such functions.
For example, the following foreign type declaration is taken from the Prelude in the Data.Function.Uncurried
module:
This defines the type constructor Fn2
which takes three type arguments. Fn2 a b c
is a type representing JavaScript functions of two arguments of types a
and b
, and with return type c
.
The purescriptfunctions
package defines similar type constructors for function arities from 0 to 10.
We can create a function of two arguments by using the mkFn2
function, as follows:
and we can apply a function of two arguments by using the runFn2
function:
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:
10.14 Representing Side Effects
The Eff
monad is also defined as a foreign type in the Prelude. Its runtime representation is quite simple  an expression of type Eff eff 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 Eff
type constructor is given in the Control.Monad.Eff
module as follows:
Recall that the Eff
type constructor is parameterized by a row of effects and a return type, which is reflected in its kind.
As a simple example, consider the random
function defined in the purescriptrandom
package. Recall that its type was:
The definition of the random
function is given here:
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 Control.Monad.Eff.Console
module in the purescriptconsole
package. The log
function has the following type:
And here is its definition:
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.
The effects RANDOM
and CONSOLE
are also defined as foreign types. Their kinds are defined to be Effect
, the kind of effects. For example:
In fact, it is possible to define new effects in this way, as we will soon see.
Expressions of type Eff eff a
can be invoked from JavaScript like regular JavaScript methods. For example, since the main
function is required to have type Eff eff a
for some set of effects eff
and some type a
, it can be invoked as follows:
When using pulp build O to
or pulp run
, this call to main
is generated automatically, whenever the Main
module is defined.
10.15 Defining New Effects
The source code for this chapter defines two new effects. The simplest is the ALERT
effect, defined in the Control.Monad.Eff.Alert
module. It is used to indicate that a computation might alert the user using a popup window.
The effect is defined first, using a foreign type declaration:
ALERT
is given the kind Effect
, indicating that it represents an effect, as opposed to a type.
Next, the alert
action is defined. The alert
action displays a popup, and adds the ALERT
effect to the row of effects:
The foreign Javascript module is straightforward, defining the alert
function by assigning it to the exports
variable:
The alert
action is very similar to the log
action from the Control.Monad.Eff.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.
Note that, as in the case of log
, the alert
function uses a function of no arguments to represent the computation of type Eff (alert :: ALERT  eff) Unit
.
The second effect defined in this chapter is the STORAGE
effect, which is defined in the Control.Monad.Eff.Storage
module. It is used to indicate that a computation might read or write values using the Web Storage API.
The effect is defined in the same way:
The Control.Monad.Eff.Storage
module defines two actions: getItem
, which retrieves a value from local storage, and setItem
which inserts or updates a value in local storage. The two functions have the following types:
The interested reader can inspect the source code for this module to see the definitions of these actions.
setItem
takes a key and a value (both strings), and returns a computation which stores the value in local storage at the specified key.
The type of getItem
is more interesting. It takes a key, and attempts to retrieve the associated value from local storage. However, since the getItem
method on window.localStorage
can return null
, the return type is not String
, but Foreign
which is defined by the purescriptforeign
package in the Data.Foreign
module.
Data.Foreign
provides a way to work with untyped data, or more generally, data whose runtime representation is uncertain.
10.16 Working With Untyped Data
In this section, we will see how we can use the Data.Foreign
library to turn untyped data into typed data, with the correct runtime representation for its type.
The code for this chapter builds on the address book example from chapter 8, by adding a Save button at the bottom of the form. When the Save button is clicked, the state of the form is serialized to JSON and stored in local storage. When the page is reloaded, the JSON document is retrieved from local storage and parsed.
The Main
module defines a type for the saved form data:
The problem is that we have no guarantee that the JSON will have the correct form. Put another way, we don’t know that the JSON represents the correct type of data at runtime. This is the sort of problem that is solved by the purescriptforeign
library. Here are some other examples:
 A JSON response from a web service
 A value passed to a function from JavaScript code
Let’s try the purescriptforeign
and purescriptforeigngeneric
libraries in PSCi.
Start by importing some modules:
A good way to obtain a Foreign
value is to parse a JSON document. purescriptforeigngeneric
defines the following two functions:
The type constructor F
is actually just a type synonym, defined in Data.Foreign
:
Here, Except
is an monad for handling exceptions in pure code, much like Either
. We can convert a value in the F
monad into a value in the Either
monad by using the runExcept
function.
Most of the functions in the purescriptforeign
and purescriptforeigngeneric
libraries return a value in the F
monad, which means that we can use do notation and the applicative functor combinators to build typed values.
The Decode
type class represents those types which can be obtained from untyped data. There are type class instances defined for the primitive types and arrays, and we can define our own instances as well.
Let’s try parsing some simple JSON documents using decodeJSON
in PSCi (remembering to use runExcept
to unwrap the results):
Recall that in the Either
monad, the Right
data constructor indicates success. Note however, that invalid JSON, or an incorrect type leads to an error:
The purescriptforeigngeneric
library tells us where in the JSON document the type error occurred.
10.17 Handling Null and Undefined Values
Realworld JSON documents contain null and undefined values, so we need to be able to handle those too.
purescriptforeigngeneric
defines a type constructors which solves this problem: NullOrUndefined
. It serves a similar purpose to the Undefined
type constructor that we defined earlier, but uses the Maybe
type constructor internally to represent missing values.
The module also provides a function unNullOrUndefined
to unwrap the inner value. We can lift the appropriate function over the decodeJSON
action to parse JSON documents which permit null values:
In each case, the type annotation applies to the term to the right of the <$>
operator. For example, decodeJSON "42"
has the type F (NullOrUndefined Int)
. The unNullOrUndefined
function is then lifted over F
to give the final type F (Maybe Int)
.
The type NullOrUndefined Int
represents values which are either integers, or null. What if we wanted to parse more interesting values, like arrays of integers, where each element might be null
? In that case, we could lift the function map unNullOrUndefined
over the decodeJSON
action, as follows:
In general, using newtypes to wrap an existing type is a good way to provide different serialization strategies for the same type. The NullOrUndefined
type is defined as a newtype around the Maybe
type constructor.
10.18 Generic JSON Serialization
In fact, we rarely need to write instances for the Decode
class, since the purescriptforeigngeneric
class allows us to derive instances using a technique called datatypegeneric programming. A full explanation of this technique is beyond the scope of this book, but it allows us to write functions once, and reuse them over many different data types, based on the structure of a the types themselves.
To derive a Decode
instance for our FormData
type (so that we may deserialize it from its JSON representation), we first use the derive
keyword to derive an instance of the Generic
type class, which looks like this:
Next, we simply define the decode
function using the genericDecode
function, as follows:
In fact, we can also derive an encoder in the same way:
It is important that we use the same options in the decoder and encoder, otherwise our encoded JSON documents might not get decoded correctly.
Now, when the Save button is clicked, a value of type FormData
is passed to the encode
function, serializing it as a JSON document. The FormData
type is a newtype for a record, so a value of type FormData
passed to encode
will be serialized as a JSON object. This is because we used the unwrapSingleConstructors
option when defining our JSON encoder.
Our Decode
type class instance is used with decodeJSON
to parse the JSON document when it is retrieved from local storage, as follows:
The savedData
action reads the FormData
structure in two steps: first, it parses the Foreign
value obtained from getItem
. The type of jsonOrNull
is inferred by the compiler to be Maybe String
(exercise for the reader  how is this type inferred?). The traverse
function is then used to apply decodeJSON
to the (possibly missing) element of the result of type Maybe String
. The type class instance inferred for decodeJSON
is the one we just wrote, resulting in a value of type F (Maybe FormData)
.
We need to use the monadic structure of F
, since the argument to traverse
uses the result jsonOrNull
obtained in the first line.
There are three possibilities for the result of FormData
:
 If the outer constructor is
Left
, then there was an error parsing the JSON string, or it represented a value of the wrong type. In this case, the application displays an error using thealert
action we wrote earlier.  If the outer constructor is
Right
, but the inner constructor isNothing
, thengetItem
also returnedNothing
which means that the key did not exist in local storage. In this case, the application continues quietly.  Finally, a value matching the pattern
Right (Just _)
indicates a successfully parsed JSON document. In this case, the application updates the form fields with the appropriate values.
Try out the code, by running pulp build O to dist/Main.js
, and then opening the browser to html/index.html
. You should be able to save the form fields’ contents to local storage by clicking the Save button, and then see the fields repopulated when the page is refreshed.
Note: You may need to serve the HTML and Javascript files from a HTTP server locally in order to avoid certain browserspecific issues.
10.19 Conclusion
In this chapter, we’ve learned how to work with foreign JavaScript code from PureScript, and vice versa, and we’ve seen the issues involved with writing trustworthy code using the FFI:
 We’ve seen the importance of the runtime representation of data, and ensuring that foreign functions have the correct representation.
 We learned how to deal with corner cases like null values and other types of JavaScript data, by using foreign types, or the
Foreign
data type.  We looked at some common foreign types defined in the Prelude, and how they can be used to interoperate with idiomatic JavaScript code. In particular, the representation of sideeffects in the
Eff
monad was introduced, and we saw how to use theEff
monad to capture new side effects.  We saw how to safely deserialize JSON data using the
Decode
type class.
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.
11. Monadic Adventures
11.1 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.
11.2 Project Setup
This module’s project introduces the following new Bower dependencies:

purescriptmaps
, which provides a data type for immutable maps 
purescriptsets
, which provides a data type for immutable sets 
purescripttransformers
, which provides implementations of standard monad transformers 
purescriptnodereadline
, which provides FFI bindings to thereadline
interface provided by NodeJS 
purescriptyargs
, which provides an applicative interface to theyargs
command line argument processing library
It is also necessary to install the yargs
module using NPM:
11.3 How To Play The Game
To run the project, use pulp run
By default you will see a usage message:
Provide the player name using the p
option:
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:
The game is very simple, but the aim of the chapter is to use the purescripttransformers
package to build a library which will enable rapid development of this type of game.
11.4 The State Monad
We will start by looking at some of the monads provided by the purescripttransformers
package.
The first example is the State
monad, which provides a way to model mutable state in pure code. We have already seen two approaches to mutable state provided by the Eff
monad, namely the REF
and ST
effects. State
provides a third alternative, but it is not implemented using the Eff
monad.
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:
This looks very similar to the API provided by the REF
and ST
effects. However, notice that we do not pass a mutable reference cell such as a Ref
or STRef
to the actions. The difference between State
and the solutions provided by the Eff
monad is that the State
monad only supports a single piece of state which is implicit  the state is implemented as a function argument hidden by the State
monad’s data constructor, so there is no explicit reference to pass around.
Let’s see an example. One use of the State
monad might be to add the values in an array of numbers to the current state. We could do that by choosing Number
as the state type s
, and using traverse_
to traverse the array, with a call to modify
for each array element:
The Control.Monad.State
module provides three functions for running a computation in the State
monad:
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:
11.5 The Reader Monad
Another monad provided by the purescripttransformers
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:
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:
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:
To elevate the user’s permissions, we might use the local
action to modify the Permissions
object during the execution of some computation:
Then we could write a function to create a new user, even if the user did not have the admin
permission:
To run a computation in the Reader
monad, the runReader
function can be used to provide the global configuration:
11.6 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:
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:
We could add a log to this function by changing the return type to Writer (Array String) Int
:
We only have to change our function slightly to log the two inputs at each step:
We can run a computation in the Writer
monad by using either of the execWriter
or runWriter
functions:
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:
11.7 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 Eff
monad provides a partial solution to this problem, since native effects can be interleaved using the approach of extensible effects. 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:
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
:
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:
We are left with a type constructor. The final argument represents the return type, and we might instantiate it to Number
for example:
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:
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:
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:
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:
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 purescriptparsing
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.
11.8 The ExceptT Monad Transformer
The purescripttransformers
package also defines the ExceptT e
monad transformer, which is the transformer corresponding to the Either e
monad. It provides the following API:
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 purescriptexceptions
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
:
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 String (Either String String)
. We can then use runWriter
to run the inner Writer
computation:
Note that only those log messages which were written before the error was thrown actually get appended to the log.
11.9 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 Eff
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.
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 runIdentity
.
However, if the parse is unsuccessful because the state is empty, then no log is printed at all:
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.
11.10 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:
In reality, the types given in the Control.Monad.State.Class
module are more general than this:
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 purescripttransformers
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. purescripttransformers
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:
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.
11.11 Alternatives
The purescriptcontrol
package defines a number of abstractions for working with computations which can fail. One of these is the Alternative
type class:
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.List
module provides two useful functions for working with type constructors in the Alternative
type class:
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:
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.
Other examples of Alternative
type constructors are Maybe
and Array
.
11.12 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
:
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
:
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:
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.
11.13 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:
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:
This parser will match characters until the case changes:
We can even use many
to fully split a string into its lower and upper case components:
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!
11.14 The RWS Monad
One particular combination of monad transformers is so common that it is provided as a single monad transformer in the purescripttransformers
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):
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:
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:
The Coords
data type represents points on a twodimensional grid, and the GameItem
data type is an enumeration of the items in the game:
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
:
11.15 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 Eff
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:
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:
Next, pickUp
looks up the set of items in the current room. It does this by using the lookup
function defined in Data.Map
:
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:
In this case, we can use put
to update the game state, and tell
to add a message to the log:
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
:
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:
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.
11.16 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: purescriptyargs
, which provides an applicative interface to the yargs
command line parsing library, and purescriptnodereadline
, 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:
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
:
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:
The CONSOLE
effect indicates that this function interacts with the user via the console (using the purescriptnodereadline
and purescriptconsole
packages). runGame
takes the game configuration as a function argument.
The purescriptnodereadline
package provides the LineHandler
type, which represents actions in the Eff
monad which handle user input from the terminal. Here is the corresponding API:
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:
The first step is to set the prompt at the console. We pass the interface
handle, and provide the prompt string and indentation level:
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:
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:
11.17 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 purescriptyargs
package.
purescriptyargs
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 purescriptyargs
package, the functor we are interested in is the Y
functor, which adds the sideeffect of reading from command line options. It provides the following handler:
This is best illustrated by example. The application’s main
function is defined using runY
as follows:
The first argument is used to configure the yargs
library. In our case, we simply provide a usage message, but the Node.Yargs.Setup
module provides several other options.
The second argument uses the map
function to lift the runGame
function over the Y
type constructor. The argument env
is constructed in a where
declaration using the applicative operators <$>
and <*>
:
Here, the gameEnvironment
function, which has the type PlayerName > Boolean > GameEnvironment
, is lifted over Y
. The two arguments specify how to read the player name and debug flag from the command line options. The first argument describes the player name option, which is specified by the p
or player
options, and the second describes the debug mode flag, which is turned on using the d
or debug
options.
This demonstrates two basic functions defined in the Node.Yargs.Applicative
module: yarg
, which defines a command line option which takes an optional argument (of type String
, Number
or Boolean
), and flag
which defines a command line flag of type Boolean
.
Notice how we were able to use the notation afforded by the applicative operators to give a compact, declarative specification of our command line interface. In addition, it is simple to add new command line arguments, simply by adding a new function argument to runGame
, and then using <*>
to lift runGame
over an additional argument in the definition of env
.
11.18 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 Eff
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 Eff
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.
In the next chapter, we will see how monad transformers can be used to give an elegant solution to a common complaint when working with asynchronous JavaScript code  the problem of callback hell.
12. Callback Hell
12.1 Chapter Goals
In this chapter, we will see how the tools we have seen so far  namely monad transformers and applicative functors  can be put to use to solve realworld problems. In particular, we will see how we can solve the problem of callback hell.
12.2 Project Setup
The source code for this chapter can be compiled and run using pulp run
. It is also necessary to install the request
module using NPM:
12.3 The Problem
Asynchronous code in JavaScript typically uses callbacks to structure program flow. For example, to read text from a file, the preferred approach is to use the readFile
function and to pass a callback  a function that will be called when the text is available:
However, if multiple asynchronous operations are involved, this can quickly lead to nested callbacks, which can result in code which is difficult to read:
One solution to this problem is to break out individual asynchronous calls into their own functions:
This solution works but has some issues:
 It is necessary to pass intermediate results to asynchronous functions as function arguments, in the same way that we passed
data
towriteCopy
above. This is fine for small functions, but if there are many callbacks involved, the data dependencies can become complex, resulting in many additional function arguments.  There is a common pattern  the callbacks
onSuccess
andonFailure
are usually specified as arguments to every asynchronous function  but this pattern has to be documented in module documentation which accompanies the source code. It is better to capture this pattern in the type system, and to use the type system to enforce its use.
Next, we will see how to use the techniques we have learned so far to solve these issues.
12.4 The Continuation Monad
Let’s translate the copyFile
example above into PureScript by using the FFI. In doing so, the structure of the computation will become apparent, and we will be led naturally to a monad transformer which is defined in the purescripttransformers
package  the continuation monad transformer ContT
.
Note: in practice, it is not necessary to write these functions by hand every time. Asynchronous file IO functions can be found in the purescriptnodefs
and purescriptnodefsaff
libraries.
First, we need to gives types to readFile
and writeFile
using the FFI. Let’s start by defining some type synonyms, and a new effect for file IO:
readFile
takes a filename and a callback which takes two arguments. If the file was read successfully, the second argument will contain the file contents, and if not, the first argument will be used to indicate the error.
In our case, we will wrap readFile
with a function which takes two callbacks: an error callback (onFailure
) and a result callback (onSuccess
), much like we did with copyFile
and writeCopy
above. Using the multipleargument function support from Data.Function
for simplicity, our wrapped function readFileImpl
might look like this:
In the foreign Javascript module, readFileImpl
would be defined as:
This type signature indicates that readFileImpl
takes three arguments: a file path, a success callback and an error callback, and returns an effectful computation which returns an empty (Unit
) result. Notice that the callbacks themselves are given types which use the Eff
monad to track their effects.
You should try to understand why this implementation has the correct runtime representation for its type.
writeFileImpl
is very similar  it is different only in that the file content is passed to the function itself, not to the callback. Its implementation looks like this:
Given these FFI declarations, we can write the implementations of readFile
and writeFile
. These will use the Data.Function.Uncurried
module to turn the multipleargument FFI bindings into regular (curried) PureScript functions, and therefore have slightly more readable types.
In addition, instead of requiring two callbacks, one for successes and one for failures, we can require only a single callback which responds to either successes or failures. That is, the new callback takes a value in the Either ErrorCode
monad as its argument:
Now we can spot an important pattern. Each of these functions takes a callback which returns a value in some monad (in this case Eff (fs :: FS  eff)
) and returns a value in the same monad. This means that when the first callback returns a result, that monad can be used to bind the result to the input of the next asynchronous function. In fact, that’s exactly what we did by hand in the copyFile
example.
This is the basis of the continuation monad transformer, which is defined in the Control.Monad.Cont.Trans
module in purescripttransformers
.
ContT
is defined as a newtype as follows:
A continuation is just another name for a callback. A continuation captures the remainder of a computation  in our case, what happens after a result has been provided after an asynchronous call.
The argument to the ContT
data constructor looks remarkably similar to the types of readFile
and writeFile
. In fact, if we take the type a
to be the type Either ErrorCode String
, r
to be Unit
and m
to be the monad Eff (fs :: FS  eff)
, we recover the righthand side of the type of readFile
.
This motivates the following type synonym, defining an Async
monad, which we will use to compose asynchronous actions like readFile
and writeFile
:
For our purposes, we will always use ContT
to transform the Eff
monad, and the type r
will always be Unit
, but this is not required.
We can treat readFile
and writeFile
as computations in the Async
monad, by simply applying the ContT
data constructor:
With that, we can write our copyfile routine by simply using do notation for the ContT
monad transformer:
Note how the asynchronous nature of readFileCont
is hidden by the monadic bind expressed using do notation  this looks just like synchronous code, but the ContT
monad is taking care of wiring our asynchronous functions together.
We can run this computation using the runContT
handler by providing a continuation. The continuation represents what to do next, i.e. what to do when the asynchronous copyfile routine completes. For our simple example, we can just choose the logShow
function as the continuation, which will print the result of type Either ErrorCode Unit
to the console:
12.5 Putting ExceptT To Work
This solution works, but it can be improved.
In the implementation of copyFileCont
, we had to use pattern matching to analyze the result of the readFileCont
computation (of type Either ErrorCode String
) to determine what to do next. However, we know that the Either
monad has a corresponding monad transformer, ExceptT
, so it is reasonable to expect that we should be able to use ExceptT
with ContT
to combine the two effects of asynchronous computation and error handling.
In fact, it is possible, and we can see why if we look at the definition of ExceptT
:
ExceptT
simply changes the result of the underlying monad from a
to Either e a
. This means that we can rewrite copyFileCont
by transforming our current monad stack with the ExceptT ErrorCode
transformer. It is as simple as applying the ExceptT
data constructor to our existing solution:
Now, our copyfile routine is much simpler, since the asynchronous error handling is hidden inside the ExceptT
monad transformer:
12.6 A HTTP Client
As another example of using ContT
to handle asynchronous functions, we’ll now look at the Network.HTTP.Client
module from this chapter’s source code. This module uses the Async
monad to support asynchronous HTTP requests using the request
module, which is available via NPM.
The request
module provides a function which takes a URL and a callback, makes a HTTP(S) request and invokes the callback when the response is available, or in the event of an error. Here is an example request:
We will recreate this simple example in PureScript using the Async
monad.
In the Network.HTTP.Client
module, the request
method is wrapped with a function getImpl
:
Again, we can use the Data.Function.Uncurried
module to turn this into a regular, curried PureScript function. As before, we turn the two callbacks into a single callback, this time accepting a value of type Either String String
, and apply the ContT
constructor to construct an action in our Async
monad:
12.7 Parallel Computations
We’ve seen how to use the ContT
monad and do notation to compose asynchronous computations in sequence. It would also be useful to be able to compose asynchronous computations in parallel.
If we are using ContT
to transform the Eff
monad, then we can compute in parallel simply by initiating our two computations one after the other.
The purescriptparallel
package defines a type class Parallel
for monads like Async
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 Async
) and an applicative functor f
which can be used to combine computations in parallel:
The class defines two functions:

parallel
, which takes computations in the monadm
and turns them into computations in the applicative functorf
, and 
sequential
, which performs a conversion in the opposite direction.
The purescriptparallel
library provides a Parallel
instance for the Async
monad. It uses mutable references to combine Async
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.
We can use the parallel
function to create a version of our readFileCont
action which can be combined in parallel. Here is a simple example which reads two text files in parallel, and concatenates and prints their results:
Note that, since readFileCont
returns a value of type Either ErrorCode String
, we need to lift the append
function over the Either
type constructor using lift2
to form our combining function.
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.
12.8 Conclusion
In this chapter, we have seen a practical demonstration of monad transformers:
 We saw how the common JavaScript idiom of callbackpassing can be captured by the
ContT
monad transformer.  We saw how the problem of callback hell can be solved by using do notation to express sequential asynchronous computations, and applicative functors to express parallelism.
 We used
ExceptT
to express asynchronous errors.
13. Generative Testing
13.1 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 purescriptquickcheck
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 purescriptquickcheck
to test a simple library, using Pulp to integrate our test suite into our development process.
13.2 Project Setup
This chapter’s project adds purescriptquickcheck
as a Bower dependency.
In a Pulp 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 pulp test
command.
13.3 Writing Properties
The Merge
module implements a simple function merge
, which we will use to demonstrate the features of the purescriptquickcheck
library.
merge
takes two sorted arrays of integers, and merges their elements so that the result is also sorted. For example:
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 in two properties:
 (Sortedness) If
xs
andys
are sorted, thenmerge xs ys
is also sorted.  (Subarray)
xs
andys
are both subarrays ofmerge xs ys
, and their elements appear in the same order.
purescriptquickcheck
allows us to test these properties directly, by generating random test cases. We simply state the properties that we want our code to have, as functions:
Here, isSorted
and isSubarrayOf
are implemented as helper functions with the following types:
When we run this code, purescriptquickcheck
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:
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:
As we can see, this error message is not very helpful, but it can be improved with a little work.
13.4 Improving Error Messages
To provide error messages along with our failed test cases, purescriptquickcheck
provides the <?>
operator. Simply separate the property definition from the error message using <?>
, as follows:
This time, if we modify the code to introduce a bug, we see our improved error message after the first failed test case:
Notice how the input xs
and ys
were generated as a arrays of randomlyselected integers.
13.5 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:
If we modify our original tests to use mergePoly
in place of merge
, we see the following error message:
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 a helper function to force the compiler to infer a particular type. For example, if we define a function ints
as a synonym for the identity function:
then we can modify our tests so that the compiler infers the type Array Int
for our two array arguments:
Here, xs
and ys
both have type Array Int
, since the ints
function has been used to disambiguate the unknown type.
13.6 Generating Arbitrary Data
Now we will see how the purescriptquickcheck
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:
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 purescriptquickcheck
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:
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 sortedness test for merge
:
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:
With this type constructor, we can modify our test as follows:
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:
The Tree
module defines the following API:
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:
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:
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:
Here, the argument t
is a randomlygenerated tree of type Tree Int
, where the type argument disambiguated by the identity function treeOfInt
.
13.7 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:
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:
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
:
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.
In the case of the sortedness property, 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:
Here, we use a function intToBool
to disambiguate the type of the function f
:
In the case of the subarray property, we simply have to change the name of the function to mergeWith
 we still expect our input arrays to be subarrays of the result:
In addition to being Arbitrary
, functions are also Coarbitrary
:
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.
13.8 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:
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:
If the tree is a Branch
, then we will perturb the generator using the left subtree, the value and the right subtree, using function composition to create our perturbing function:
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:
Now we are able to generate the predicate function randomly. For example, we expect the anywhere
function to respect disjunction:
Here, the treeOfInt
function is used to fix the type of values contained in the tree to the type Int
:
13.9 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:
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.
13.10 Conclusion
In this chapter, we met the purescriptquickcheck
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
pulp 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.
14. DomainSpecific Languages
14.1 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 library of combinators which we wrote for the
Async
andParallel
functors in chapter 12 could be considered an example of a domainspecific language for the domain of asynchronous programming.  The
purescriptquickcheck
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.
14.2 Project Setup
The project accompanying this chapter adds one new Bower dependency  the purescriptfree
library, which defines the free monad, one of the tools which we will be using.
We will test this chapter’s project in PSCi.
14.3 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:
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
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:
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.
14.4 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:
Next, we create smart constructors for those HTML elements we want our users to be able to create, by applying the element
function:
Finally, we update the module exports list to only export those functions which are known to construct correct data structures:
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:
Let’s apply the same technique to the Attribute
type. First, we provide a generalpurpose smart constructor for attributes. Here is a first attempt:
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:
With that, we can modify our operator as follows:
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:
Here is the final exports list for our new module. Note that we no longer export any data constructors directly:
If we try this new module in PSCi, we can already see massive improvements in the conciseness of the user code:
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.
14.5 Phantom Types
To motivate the next technique, consider the following code:
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:
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:
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:
We also provide type class instances for the String
and Int
types:
We also have to update our AttributeKey
constants so that their types reflect the new type parameter:
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:
14.6 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:
we will be able to write this:
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 purescriptfree
library, in the Control.Monad.Free
module. We can find out some basic information about it using PSCi, as follows:
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:
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:
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
:
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
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:
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.
14.7 Interpreting the Monad
The Control.Monad.Free
module provides a number of functions for interpreting a computation in a free monad:
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:
renderElement
is defined in a where block:
The definition of renderElement
is straightforward, using the tell
action from the Writer
monad to accumulate several small strings:
Next, we define the renderAttribute
function, which is equally simple:
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
:
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
:
We can implement this function by simply pattern matching on the two data constructors of ContentF
:
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:
14.8 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:
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:
We also define a new data type for hyperlinks which can appear in a
elements, as follows:
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:
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:
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:
Now we can build our new action by using the liftF
function, as before:
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:
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:
We also need to add a new case to renderContentItem
, to interpret the new NewName
data constructor:
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:
You can verify that multiple calls to newName
do in fact result in unique names.
14.9 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.