What I Wish I Knew When Learning Haskell

Version
2.5

Version

This is the fifth major draft of this document since 2009.

• HTML Version
• Screen PDF
• Printable PDF
• EPUB Version
• Kindle Version

Pull requests are always accepted for changes and additional content. This is a living document. The only way this document will stay up to date is through the kindness of readers like you and community patches and pull requests on Github.

If you’d like a physical copy of the text you can either print it for yourself (see Printable PDF) or purchase one online:

• Blurb Publisher

Author

This text is authored by Stephen Diehl.

• Web: www.stephendiehl.com
• Twitter: https://twitter.com/smdiehl
• Github: https://github.com/sdiehl

Special thanks for Erik Aker for copyediting assitance.

License

Copyright © 2009-2020 Stephen Diehl

This code included in the text is dedicated to the public domain. You can copy, modify, distribute and perform the code, even for commercial purposes, all without asking permission.

You may distribute this text in its full form freely, but may not reauthor or sublicense this work. Any reproductions of major portions of the text must include attribution.

The software is provided “as is”, without warranty of any kind, express or implied, including But not limited to the warranties of merchantability, fitness for a particular purpose and noninfringement. In no event shall the authors or copyright holders be liable for any claim, damages or other liability, whether in an action of contract, tort or otherwise, Arising from, out of or in connection with the software or the use or other dealings in the software.

What is Haskell?

At its heart Haskell is a lazy, functional, statically-typed programming language with advanced type system features such as higher-rank, higher-kinded parametric polymorphism, monadic effects, generalized algebraic data types, ad-hoc polymorphism through type classes, associated type families, and more. As a programming language, Haskell pushes the frontiers of programming language design more so than any other general purpose language while still remaining practical for everyday use.

Beyond language features, Haskell remains an organic, community-driven effort, run by its userbase instead of by corporate influences. While there are some Haskell companies and consultancies, most are fairly small and none have an outsized influence on the development of the language. This is in stark contrast to ecosystems like Java and Go where Oracle and Google dominate all development. In fact, the Haskell community is a synthesis between multiple disciplines of academic computer science and industrial users from large and small firms, all of whom contribute back to the language ecosystem.

Originally, Haskell was borne out of academic research. Designed as an ML dialect, it was initially inspired by an older language called Miranda. In the early 90s, a group of academics formed the GHC committee to pursue building a research vehicle for lazy programming languages as a replacement for Miranda. This was a particularly in-vogue research topic at the time and as a result the committee attracted various talented individuals who initiated the language and ultimately laid the foundation for modern Haskell.

Over the last 30 years Haskell has evolved into a mature ecosystem, with an equally mature compiler. Even so, the language is frequently reimagined by passionate contributors who may be furthering academic research goals or merely contributing out of personal interest. Although laziness was originally the major research goal, this has largely become a quirky artifact that most users of the language are generally uninterested in. In modern times the major themes of Haskell community are:

• A vehicle for type system research
• Experimentation in the design space of typed effect systems
• Algebraic structures as a method of program synthesis
• Referential transparency as a core language feature
• Embedded domain specific languages
• Experimentation toward practical dependent types
• Stronger encoding of invariants through type-level programming
• Efficient functional compiler design
• Alternative models of parallel and concurrent programming

Although these are the major research goals, Haskell is still a fully general purpose language, and it has been applied in wildly diverse settings from garbage trucks to cryptanalysis for the defense sector and everything in-between. With a thriving ecosystem of industrial applications in web development, compiler design, machine learning, financial services, FPGA development, algorithmic trading, numerical computing, cryptography research, and cybersecurity, the language has a lot to offer to any field or software practitioner.

Haskell as an ecosystem is one that is purely organic, it takes decades to evolve, makes mistakes and is not driven by any one ideology or belief about the purpose of functional programming. This makes Haskell programming simultaneously frustrating and exciting; and therein lies the fun that has been the intellectual siren song that has drawn many talented programmers to dabble in this beautiful language at some point in their lives.

See:

• A History of Haskell
• Wearing the Hair Shirt: A Retrospective on Haskell

How to Read

This is a guide for working software engineers who have an interest in Haskell but don’t know Haskell yet. I presume you know some basics about how your operating system works, the shell, and some fundamentals of other imperative programming languages. If you are a Python or Java software engineer with no Haskell experience, this is the executive summary of Haskell theory and practice for you. We’ll delve into a little theory as needed to explain concepts but no more than necessary. If you’re looking for a purely introductory tutorial, this probably isn’t the right start for you, however this can be read as a companion to other introductory texts.

There is no particular order to this guide, other than the first chapter which describes how to get set up with Haskell and use the foundational compiler and editor tooling. After that you are free to browse the chapters in any order. Most are divided into several sections which outline different concepts, language features or libraries. However, the general arc of this guide bends toward more complex topics in later chapters.

As there is no ordering after the first chapter I will refer to concepts globally without introducing them first. If something doesn’t make sense just skip it and move on. I strongly encourage you to play around with the source code modules for yourself. Haskell cannot be learned from an armchair; instead it can only be mastered by writing a ton of code for yourself. GHC may initially seem like a cruel instructor, but in time most people grow to see it as their friend.

GHC

GHC is the Glorious Glasgow Haskell Compiler. Originally written in 1989, GHC is now the de facto standard for Haskell compilers. A few other compilers have existed along the way, but they either are quite limited or have bit rotted over the years. At this point, GHC is a massive compiler and it supports a wide variety of extensions. It’s also the only reference implementation for the Haskell language and as such, it defines what Haskell the language is by its implementation.

GHC is run at the command line with the command ghc.

GHC’s runtime is written in C and uses machinery from GCC infrastructure for its native code generator and can also use LLVM for its native code generation. GHC is supported on the following architectures:

• Linux x86
• Linux x86_64
• Linux PowerPC
• NetBSD x86
• OpenBSD x86
• FreeBSD x86
• MacOS X Intel
• MacOS X PowerPC
• Windows x86_64

GHC itself depends on the following Linux packages.

• build-essential
• libgmp-dev
• libffi-dev
• libncurses-dev
• libtinfo5

ghcup

There are two major packages that need to be installed to use Haskell:

• ghc
• cabal-install

GHC can be installed on Linux and Mac with ghcup by running the following command:

This can be used to manage multiple versions of GHC installed locally.

To select which version of GHC is available on the PATH use the set command.

This can also be used to install cabal.

To modify your shell to include ghc and cabal.

Or you can permanently add the following to your .bashrc or .zshrc file:

Package Managers

There are two major Haskell packaging tools: Cabal and Stack. Both take differing views on versioning schemes but can more or less interoperate at the package level. So, why are there two different package managers?

The simplest explanation is that Haskell is an organic ecosystem with no central authority, and as such different groups of people with different ideas and different economic interests about optimal packaging built their own solutions around two different models. The interests of an organic community don’t always result in open source convergence; however, the ecosystem has seen both package managers reach much greater levels of stability as a result of collaboration. In this article, I won’t offer a preference for which system to use: it is left up to the reader to experiment and use the system which best suits your or your company’s needs.

Project Structure

A typical Haskell project hosted on Github or Gitlab will have several executable, test and library components across several subdirectories. Each of these files will correspond to an entry in the Cabal file.

More complex projects consisting of multiple modules will include multiple project directories like those above, but these will be nested in subfolders with a cabal.project or stack.yaml in the root of the repository.

An example Cabal project file, named cabal.project above, this multi-component library repository would include these lines.

By contrast, an example Stack project stack.yaml for the above multi-component library repository would be:

Cabal

Cabal is the build system for Haskell. Cabal is also the standard build tool for Haskell source supported by GHC. Cabal can be used simultaneously with Stack or standalone with cabal new-build.

To update the package index from Hackage, run:

To start a new Haskell project, run:

This will result in a .cabal file being created with the configuration options for our new project.

Cabal can also build dependencies in parallel by passing -j where n is the number of concurrent builds.

Let’s look at an example .cabal file. There are two main entry points that any package may provide: a library and an executable. Multiple executables can be defined, but only one library. In addition, there is a special form of executable entry point Test-Suite, which defines an interface for invoking unit tests from cabal.

For a library, the exposed-modules field in the .cabal file indicates which modules within the package structure will be publicly visible when the package is installed. These modules are the user-facing APIs that we wish to expose to downstream consumers.

For an executable, the main-is field indicates the module that exports the main function responsible for running the executable logic of the application. Every module in the package must be listed in one of other-modules, exposed-modules or main-is fields.

To run an “executable” under cabal execute the command:

To load the “library” into a GHCi shell under cabal execute the command:

The metavariable is either one of the executable or library declarations in the .cabal file and can optionally be disambiguated by the prefix exe: or lib: respectively.

To build the package locally into the ./dist/build folder, execute the build command:

To run the tests, our package must itself be reconfigured with the --enable-tests flag and the build-depends options. The Test-Suite must be installed manually, if not already present.

Moreover, arbitrary shell commands can be invoked with the GHC environmental variables. It is quite common is to run a new bash shell with this command such that the ghc and ghci commands use the package environment. This can also run any system executable with the GHC_PACKAGE_PATH variable set to the libraries package database.

The haddock documentation can be generated for the local project by executing the haddock command. The documentation will be built to the ./dist folder.

When we’re finally ready to upload to Hackage ( presuming we have a Hackage account set up ), then we can build the tarball and upload with the following commands:

The current state of a local build can be frozen with all current package constraints enumerated:

This will create a file cabal.config with the constraint set.

The cabal configuration is stored in $HOME/.cabal/config and contains various options including credential information for Hackage upload.

A library can also be compiled with runtime profiling information enabled. More on this is discussed in the section on Concurrency and Profiling.

Another common flag to enable is documentation which forces the local build of Haddock documentation, which can be useful for offline reference. On a Linux filesystem these are built to the /usr/share/doc/ghc-doc/html/libraries/ directory.

Cabal can also be used to install packages globally to the system PATH. For example the parsec package to your system from Hackage, the upstream source of Haskell packages, invoke the install command:

To download the source for a package, we can use the get command to retrieve the source from Hackage.

Cabal New-Build

The interface for Cabal has seen an overhaul in the last few years and has moved more closely towards the Nix-style of local builds. Under the new system packages are separated into categories:

• Local Packages – Packages are built from a configuration file which specifies a path to a directory with a cabal file. These can be working projects as well as all of its local transitive dependencies.
• External Packages – External packages are packages retrieved from either the public Hackage or a private Hackage repository. These packages are hashed and stored locally in ~/.cabal/store to be reused across builds.

As of Cabal 3.0 the new-build commands are the default operations for build operations. So if you type cabal build using Cabal 3.0 you are already using the new-build system.

Historically these commands were separated into two different command namespaces with prefixes v1- and v2-, with v1 indicating the old sandbox build system and the v2 indicating the new-build system.

The new build commands are listed below:

Cabal also stores all of its build artifacts inside of a dist-newstyle folder stored in the project working directory. The compilation artifacts are of several categories.

• .hi – Haskell interface modules which describe the type information, public exports, symbol table, and other module guts of compiled Haskell modules.
• .hie – An extended interface file containing module symbol data.
• .hspp – A Haskell preprocessor file.
• .o – Compiled object files for each module. These are emitted by the native code generator assembler.
• .s – Assembly language source file.
• .bc – Compiled LLVM bytecode file.
• .ll – An LLVM source file.
• cabal_macros.h – Contains all of the preprocessor definitions that are accessible when using the CPP extension.
• cache – Contains all artifacts generated by solving the constraints of packages to set up a build plan. This also contains the hash repository of all external packages.
• packagedb – Database of all of the cabal metadata of all external and local packages needed to build the project. See Package Databases.

These all get stored under the dist-newstyle folder structure which is set up hierarchically under the specific CPU architecture, GHC compiler version and finally the package version.

Local Packages

Both Stack and Cabal can handle local packages built from the local filesystem, from remote tarballs, or from remote Git repositories.

Inside of the stack.yaml simply specify the git repository remote and the hash to pull.

In Cabal to add a remote create a cabal.project file and add your remote in the source-repository-package section.

Version Bounds

All Haskell packages are versioned and the numerical quantities in the version are supposed to follow the Package Versioning Policy.

As packages evolve over time there are three numbers which monotonically increase depending on what has changed in the package.

• Major version number
• Minor version number
• Patch version number

Every library’s cabal file will have a packages dependencies section which will specify the external packages which the library depends on. It will also contain the allowed versions that it is known to build against in the build-depends section. The convention is to put the upper bound to the next major unreleased version and the lower bound at the currently used version.

Individual lines in the version specification can be dependent on other variables in the cabal file.

Version bounds in cabal files can be managed automatically with a tool cabal-bounds which can automatically generate, update and format cabal files.

See:

• Package Versioning Policy

Stack

Stack is an alternative approach to Haskell’s package structure that emerged in 2015. Instead of using a rolling build like Cabal, Stack breaks up sets of packages into release blocks that guarantee internal compatibility between sets of packages. The package solver for Stack uses a different strategy for resolving dependencies than cabal-install has historically used and Stack combines this with a centralised build server called Stackage which continuously tests the set of packages in a resolver to ensure they build against each other.

Install

To install stack on Linux or Mac, run:

For other operating systems, see the official install directions.

Usage

Once stack is installed, it is possible to setup a build environment on top of your existing project’s cabal file by running:

An example stack.yaml file for GHC 8.8.1 would look like this:

Most of the common libraries used in everyday development are already in the Stackage repository. The extra-deps field can be used to add Hackage dependencies that are not in the Stackage repository. They are specified by the package and the version key. For instance, the zenc package could be added to stack build in the following way:

The stack command can be used to install packages and executables into either the current build environment or the global environment. For example, the following command installs the executable for hlint, a popular linting tool for Haskell, and places it in the PATH:

To check the set of dependencies, run:

Just as with cabal, the build and debug process can be orchestrated using stack commands:

To visualize the dependency graph, use the dot command piped first into graphviz, then piped again into your favorite image viewer:

Hpack

Hpack is an alternative package description language that uses a structured YAML format to generate Cabal files. Hpack succeeds in DRYing (Don’t Repeat Yourself) several sections of cabal files that are often quite repetitive across large projects. Hpack uses a package.yaml file which is consumed by the command line tool hpack. Hpack can be integrated into Stack and will generate resulting cabal files whenever stack build is invoked on a project using a package.yaml file. The output cabal file contains a hash of the input yaml file for consistency checking.

A small package.yaml file might look something like the following:

Base

GHC itself ships with a variety of core libraries that are loaded into all Haskell projects. The most foundational of these is base which forms the foundation for all Haskell code. The base library is split across several modules.

• Prelude – The default namespace imported in every module.
• Data – The simple data structures wired into the language
• Control – Control flow functions
• Foreign – Foreign function interface
• Numeric – Numerical tower and arithmetic operations
• System – System operations for Linux/Mac/Windows
• Text – Basic String types.
• Type – Typelevel operations
• GHC – GHC Internals
• Debug – Debug functions
• Unsafe – Unsafe “backdoor” operations

There have been several large changes to Base over the years which have resulted in breaking changes that means older versions of base are not compatible with newer versions.

• Monad Applicative Proposal (AMP)
• MonadFail Proposal (MFP)
• Semigroup Monoid Proposal (SMP)

Prelude

The Prelude is the default standard module. The Prelude is imported by default into all Haskell modules unless either there is an explicit import statement for it, or the NoImplicitPrelude extension is enabled.

The Prelude exports several hundred symbols that are the default datatypes and functions for libraries that use the GHC-issued prelude. Although the Prelude is the default import, many libraries these days do not use the standard prelude instead choosing to roll a custom one on a per-project basis or to use an off-the shelf prelude from Hackage.

The Prelude contains common datatype and classes such as List, Monad, Maybe and most associated functions for manipulating these structures. These are the most foundational programming constructs in Haskell.

Modern Haskell

There are two official language standards:

• Haskell98
• Haskell2010

And then there is what is colloquially referred to as Modern Haskell which is not an official language standard, but an ambiguous term to denote the emerging way most Haskellers program with new versions of GHC. The language features typically included in modern Haskell are not well-defined and will vary between programmers. For instance, some programmers prefer to stay quite close to the Haskell2010 standard and only include a few extensions while some go all out and attempt to do full dependent types in Haskell.

By contrast, the type of programming described by the phrase Modern Haskell typically utilizes some type-level programming, as well as flexible typeclasses, and a handful of Language Extensions.

Flags

GHC has a wide variety of flags that can be passed to configure different behavior in the compiler. Enabling GHC compiler flags grants the user more control in detecting common code errors. The most frequently used flags are:

Like most compilers, GHC takes the -Wall flag to enable all warnings. However, a few of the enabled warnings are highly verbose. For example, -fwarn-unused-do-bind and -fwarn-unused-matches typically would not correspond to errors or failures.

Any of these flags can be added to the ghc-options section of a project’s .cabal file. For example:

The flags described above are simply the most useful. See the official reference for the complete set of GHC’s supported flags.

For information on debugging GHC internals, see the commentary on GHC internals.

Hackage

Hackage is the upstream source of open source Haskell packages. With Haskell’s continuing evolution, Hackage has become many things to developers, but there seem to be two dominant philosophies of uploaded libraries.

A Repository for Production Libraries

In the first philosophy, libraries exist as reliable, community-supported building blocks for constructing higher level functionality on top of a common, stable edifice. In development communities where this method is the dominant philosophy, the authors of libraries have written them as a means of packaging up their understanding of a problem domain so that others can build on their understanding and expertise.

An Experimental Playground

In contrast to the previous method of packaging, a common philosophy in the Haskell community is that Hackage is a place to upload experimental libraries as a means of getting community feedback and making the code publicly available. Library authors often rationalize putting these kinds of libraries up without documentation, often without indication of what the library actually does or how it works. This unfortunately means a lot of Hackage namespace has become polluted with dead-end, bit-rotting code. Sometimes packages are also uploaded purely for internal use within an organisation, or to accompany an academic paper. These packages are often left undocumented as well.

For developers coming to Haskell from other language ecosystems that favor the former philosophy (e.g., Python, JavaScript, Ruby), seeing thousands of libraries without the slightest hint of documentation or description of purpose can be unnerving. It is an open question whether the current cultural state of Hackage is sustainable in light of these philosophical differences.

Needless to say, there is a lot of very low-quality Haskell code and documentation out there today, so being conservative in library assessment is a necessary skill. That said, there are also quite a few phenomenal libraries on Hackage that are highly curated by many people.

As a general rule, if the Haddock documentation for the library does not have a minimal working example, it is usually safe to assume that it is an RFC-style library and probably should be avoided for production code.

There are several heuristics you can use to answer the question Should I Use this Hackage Library:

• Check the Uploaded to see if the author has updated it in the last five years.
• Check the Maintainer email address, if the author has an academic email address and has not uploaded a package in two or more years, it is safe to assume that this is a thesis project and probably should not be used industrially.
• Check the Modules to see if the author has included toplevel Haddock docstrings. If the author has not included any documentation then the library is likely of low-quality and should not be used industrially.
• Check the Dependencies for the bound on base package. If it doesn’t include the latest base included with the latest version of GHC then the code is likely not actively maintained.
• Check the reverse Hackage search to see if the package is used by other libraries in the ecosystem. For example: https://packdeps.haskellers.com/reverse/QuickCheck

An example of a bitrotted package:

https://hackage.haskell.org/package/numeric-quest

An example of a well maintained package:

https://hackage.haskell.org/package/QuickCheck

Stackage

Stackage is an alternative opt-in packaging repository which mirrors a subset of Hackage. Packages that are included in Stackage are built in a massive continuous integration process that checks to see that given versions link successfully against each other. This can give a higher degree of assurance that the bounds of a given resolver ensure compatibility.

Stackage releases are built nightly and there are also long-term stable (LTS) releases. Nightly resolvers have a date convention while LTS releases have a major and minor version. For example:

• lts-14.22
• nightly-2020-01-30

See:

• Stackage
• Stackage FAQ

GHCi

GHCi is the interactive shell for the GHC compiler. GHCi is where we will spend most of our time in everyday development. Following is a table of useful GHCi commands.

The introspection commands are an essential part of debugging and interacting with Haskell code:

Querying the current state of the global environment in the shell is also possible. For example, to view module-level bindings and types in GHCi, run:

To examine module-level imports, execute:

Language extensions can be set at the repl.

To see compiler-level flags and pragmas, use:

Language extensions and compiler pragmas can be set at the prompt. See the Flag Reference for the vast collection of compiler flag options.

Several commands for the interactive shell have shortcuts:

.ghci.conf

The GHCi shell can be customized globally by defining a configure file ghci.conf in $HOME/.ghc/ or in the current working directory as ./.ghci.conf.

For example, we can add a command to use the Hoogle type search from within GHCi. First, install hoogle:

Then, we can enable the search functionality by adding a command to our ghci.conf:

It is common community tradition to set the prompt to a colored λ:

GHC can also be coerced into giving slightly better error messages:

GHCi can also use a pretty printing library to format all output, which is often much easier to read. For example if your project is already using the amazing pretty-simple library simply include the following line in your ghci configuration.

And the default prelude can also be disabled and swapped for something more sensible:

GHCi Performance

For large projects, GHCi with the default flags can use quite a bit of memory and take a long time to compile. To speed compilation by keeping artifacts for compiled modules around, we can enable object code compilation instead of bytecode.

Enabling object code compilation may complicate type inference, since type information provided to the shell can sometimes be less informative than source-loaded code. This underspecificity can result in breakage with some language extensions. In that case, you can temporarily reenable bytecode compilation on a per module basis with the -fbyte-code flag.

If you all you need is to typecheck your code in the interactive shell, then disabling code generation entirely makes reloading code almost instantaneous:

Editor Integration

Haskell has a variety of editor tools that can be used to provide interactive development feedback and functionality such as querying types of subexpressions, linting, type checking, and code completion. These are largely provided by the haskell-ide-engine which serves as an editor agnostic backend that interfaces with GHC and Cabal to query code.

Vim

• haskell-ide-engine
• haskell-vim
• vim-ormolu

Emacs

• haskell-mode
• haskell-ide-engine
• ormolu.el

VSCode

• haskell-ide-engine
• language-haskell
• ghcid
• hie-server
• hlint
• ghcide
• ormolu-vscode

Linux Packages

There are several upstream packages for Linux packages which are released by GHC development. The key ones of note for Linux are:

• Debian Packages
• Debian PPA

For scripts and operations tools, it is common to include commands to add the following apt repositories, and then use these to install the signed GHC and cabal-install binaries (if using Cabal as the primary build system).

It is not advisable to use a Linux system package manager to manage Haskell dependencies. Although this can be done, in practice it is better to use Cabal or Stack to create locally isolated builds to avoid incompatibilities.

Names

Names in Haskell exist within a specific namespace. Names are either unqualified of the form:

Or qualified by the module where they come from, such as:

The major namespaces are described below with their naming conventions

Modules

A module consists of a set of imports and exports and when compiled generates an interface which is linked against other Haskell modules. A module may reexport symbols from other modules.

Modules’ dependency graphs optionally may be cyclic (i.e. they import symbols from each other) through the use of a boot file, but this is often best avoided if at all possible.

Various module import strategies exist. For instance, we may:

Import all symbols into the local namespace.

Import select symbols into the local namespace:

Import into the global namespace masking a symbol:

Import symbols qualified under Data.Map namespace into the local namespace.

Import symbols qualified and reassigned to a custom namespace (M, in the example below):

You may also dump multiple modules into the same namespace so long as the symbols do not clash:

A main module is a special module which reserves the name Main and has a mandatory export of type IO () which is invoked when the executable is run.. This is the entry point from the system into a Haskell program.

Functions

Functions are the central construction in Haskell. A function f of two arguments x and y can be defined in a single line as the left-hand and right-hand side of an equation:

This line defines a function named f of two arguments, which on the right-hand side adds and yields the result. Central to the idea of functional programming is that computational functions should behave like mathematical functions. For instance, consider this mathematical definition of the above Haskell function, which, aside from the parentheses, looks the same:

f(x, y) = x + y

In Haskell, a function of two arguments need not necessarily be applied to two arguments. The result of applying only the first argument is to yield another function to which later the second argument can be applied. For example, we can define an add function and subsequently a single-argument inc function, by merely pre-applying 1 to add:

In addition to named functions Haskell also has anonymous lambda functions denoted with a backslash. For example the identity function:

Is identical to:

Functions may call themselves or other functions as arguments; a feature known as higher-order functions. For example the following function applies a given argument f, which is itself a function, to a value x twice.

Types

Typed functional programming is essential to the modern Haskell paradigm. But what are types precisely?

The syntax of a programming language is described by the constructs that define its types, and its semantics are described by the interactions among those constructs. A type system overlays additional structure on top of the syntax that imposes constraints on the formation of expressions based on the context in which they occur.

Dynamic programming languages associate types with values at evaluation, whereas statically typed languages associate types to expressions before evaluation. Dynamic languages are in a sense as statically typed as static languages, however they have a degenerate type system with only one type.

The dominant philosophy in functional programming is to “make invalid states unrepresentable” at compile-time, rather than performing massive amounts of runtime checks. To this end Haskell has developed a rich type system that is based on typed lambda calculus known as Girard’s System-F (See Rank-N Types) and has incrementally added extensions to support more type-level programming over the years.

The following ground types are quite common:

• () – The unit type
• Char – A single unicode character (“code point”)
• Text – Unicode strings
• Bool – Boolean values
• Int – Machine integers
• Integer – GMP arbitrary precision integers
• Float – Machine floating point values
• Double – Machine double floating point values

Parameterised types consist of a type and several type parameters indicated as lower case type variables. These are associated with common data structures such as lists and tuples.

• [a] – Homogeneous lists with elements of type a
• (a,b) – Tuple with two elements of types a and b
• (a,b,c) – Tuple with three elements of types a, b, and c

The type system grows quite a bit from here, but these are the foundational types you’ll first encounter. See the later chapters for all types off advanced features that can be optionally turned on.

This tutorial will only cover a small amount of the theory of type systems. For a more thorough treatment of the subject there are two canonical texts:

• Pierce, B. C., & Benjamin, C. (2002). Types and Programming Languages. MIT Press.
• Harper, R. (2016). Practical Foundations for Programming Languages. Cambridge University Press.

Type Signatures

A toplevel Haskell function consists of two lines. The value-level definition which is a function name, followed by its arguments on the left-hand side of the equals sign, and then the function body which computes the value it yields on the right-hand side:

The type-level definition is the function name followed by the type of its arguments separated by arrows, and the final term is the type of the entire function body, meaning the type of value yielded by the function itself.

Here is a simple example of a function which adds two integers.

Functions are also capable of invoking other functions inside of their function bodies:

The simplest function, called the identity function, is a function which takes a single value and simply returns it back. This is an example of a polymorphic function since it can handle values of any type. The identity function works just as well over strings as over integers.

This can alternatively be written in terms of an anonymous lambda function which is a backslash followed by a space-separated list of arguments, followed by a function body.

One of the big ideas in functional programming is that functions are themselves first class values which can be passed to other functions as arguments themselves. For example the applyTwice function takes an argument f which is of type (a -> a) and it applies that function over a given value x twice and yields the result. applyTwice is a higher-order function which will transform one function into another function.

Often to the left of a type signature you will see a big arrow => which denotes a set of constraints over the type signature. Each of these constraints will be in uppercase and will normally mention at least one of the type variables on the right hand side of the arrow. These constraints can mean many things but in the simplest form they denote that a type variable must have an implementation of a type class. The add function below operates over any two similar values x and y, but these values must have a numerical interface for adding them together.

Type signatures can also appear at the value level in the form of explicit type signatures which are denoted in parentheses.

These are sometimes needed to provide additional hints to the typechecker when specific terms are ambiguous to the typechecker, or when additional language extensions have been enabled which don’t have precise inference methods for deducing all type variables.

Currying

In other languages functions normally have an arity which prescribes the number of arguments a function can take. Some languages have fixed arity (like Fortran) others have flexible arity (like Python) where a variable of number of arguments can be passed. Haskell follows a very simple rule: all functions in Haskell take a single argument. For multi-argument functions (some of which we’ve already seen), arguments will be individually applied until the function is saturated and the function body is evaluated.

For example, the add function from above can be partially applied to produce an add1 function:

Uncurrying is the process of taking a function which takes two arguments and transforming it into a function which takes a tuple of arguments. The Haskell prelude includes both a curry and an uncurry function for transforming functions into those that take multiple arguments from those that take a tuple of arguments and vice versa:

For example, uncurry applied to the add function creates a function that takes a tuple of integers:

Algebraic Datatypes

Custom datatypes in Haskell are defined with the data keyword followed by the the type name, its parameters, and then a set of constructors. The possible constructors are either sum types or of product types. All datatypes in Haskell can expressed as sums of products. A sum type is a set of options that is delimited by a pipe.

A datatype can only ever be inhabited by only single value from a sum type and intuitively models a set of “options” a value may take. While a product type is a combination of a set of typed values, potentially named by record fields. For example the following are two definitions of a Point product type, the latter with two fields x and y.

As another example: A deck of common playing cards could be modeled by the following set of product and sum types:

A record type can use these custom datatypes to define all the parameters that define an individual playing card.

Some example values:

The problem with the definition of this datatype is that it admits several values which are malformed. For instance it is possible to instantiate a Card with a suit Hearts but with color Black which is an invalid value. The convention for preventing these kind of values in Haskell is to limit the export of constructors in a module and only provide a limited set of functions which the module exports, which can enforce these constraints. These are smart constructors and an extremely common pattern in Haskell library design. For example we can export functions for building up specific suit cards that enforce the color invariant.

Datatypes may also be recursive, in the sense that they can contain themselves as fields. The most common example is a linked list which can be defined recursively as either an empty list or a value linked to a potentially nested version of itself.

An example value would look like:

Constructors for datatypes can also be defined as infix symbols. This is somewhat rare, but is sometimes used in more math heavy libraries. For example the constructor for our list type could be defined as the infix operator :+:. When the value is printed using a Show instance, the operator will be printed in infix form.

Lists

Linked lists or cons lists are a fundamental data structure in functional programming. GHC has builtin syntactic sugar in the form of list syntax which allows us to write lists that expand into explicit invocations of the cons operator (:). The operator is right associative and an example is shown below:

This syntax also extends to the typelevel where lists are represented as brackets around the type of values they contain.

The cons operator itself has the type signature which takes a head element as its first argument and a tail argument as its second.

The Data.Listmodule from the standard Prelude defines a variety of utility functions for operations over linked lists. For example the length function returns the integral length of the number of elements in the linked list.

While the take function extracts a fixed number of elements from the list.

Both of these functions are pure and return a new list without modifying the underlying list passed as an argument.

Another function iterate is an example of a function which returns an infinite list. It takes as its first argument a function and then repeatedly applies that function to produce a new element of the linked list.

Consuming these infinite lists can be used as a control flow construct to construct loops. For example instead of writing an explicit loop, as we would in other programming languages, we instead construct a function which generates list elements. For example producing a list which produces subsequent powers of two:

We can then use the take function to evaluate this lazy stream to a desired depth.

λ: take 15 powersOfTwo
[1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384]

An equivalent loop in an imperative language would look like the following.

Pattern Matching

To unpack an algebraic datatype and extract its fields we’ll use a built in language construction known as pattern matching. This is denoted by the case syntax and scrutinizes a specific value. A case expression will then be followed by a sequence of matches which consist of a pattern on the left and an arbitrary expression on the right. The left patterns will all consist of constructors for the type of the scrutinized value and should enumerate all possible constructors. For product type patterns that are scrutinized a sequence of variables will bind the fields associated with its positional location in the constructor. The types of all expressions on the right hand side of the matches must be identical.

Pattern matches can be written in explicit case statements or in toplevel functional declarations. The latter simply expands the former in the desugaring phase of the compiler.

Following on the playing card example in the previous section, we could use a pattern to produce a function which scores the face value of a playing card.

And we can use a double pattern match to produce a function which determines which suit trumps another suit. For example we can introduce an order of suits of cards where the ranking of cards proceeds (Clubs, Diamonds, Hearts, Spaces). A _ underscore used inside a pattern indicates a wildcard pattern and matches against any constructor given. This should be the last pattern used a in match list.

And finally we can write a function which determines if another card beats another card in terms of the two pattern matching functions above. The following pattern match brings the values of the record into the scope of the function body assigning to names specified in the pattern syntax.

Functions may also invoke themselves. This is known as recursion. This is quite common in pattern matching definitions which recursively tear down or build up data structures. This kind of pattern is one of the defining modes of programming functionally.

The following two recursive pattern matches are desugared forms of each other:

Pattern matching on lists is also an extremely common pattern. It has special pattern syntax and the tail variable is typically pluralized. In the following x denotes the head variable and xs denotes the tail. For example the following function traverses a list of integers and adds (+1) to each value.

Guards

Guard statements are expressions that evaluate to boolean values that can be used to restrict pattern matches. These occur in a pattern match statements at the toplevel with the pipe syntax on the left indicating the guard condition. The special otherwise condition is just a renaming of the boolean value True exported from Prelude.

Guards can also occur in pattern case expressions.

Operators and Sections

An operator is a function that can be applied using infix syntax or partially applied using a section. Operators can be defined to use any combination of the special ASCII symbols or any unicode symbol.

! # % & * + . / < = > ? @ ^ | - ~ :

The following are reserved syntax and cannot be overloaded:

.. : :: = | <- -> @ ~ =>

Operators are of one of three fixity classes.

• Infix - Place between expressions
• Prefix - Placed before expressions
• Postfix - Placed after expressions. See Postfix Operators.

Expressions involving infix operators are disambiguated by the operator’s fixity and precedence. Infix operators are either left or right associative. Associativity determines how operators of the same precedence are grouped in the absence of parentheses.

Precedence and associativity are denoted by fixity declarations for the operator using either infixr infixl and infix. The standard operators defined in the Prelude have the following precedence table.

infixr 9  .
infixr 8  ^, ^^, **
infixl 7  *, /, `quot`, `rem`, `div`, `mod`
infixl 6  +, -
infixr 5  ++
infix  4  ==, /=, <, <=, >=, >
infixr 3  &&
infixr 2  ||
infixr 1  >>, >>=
infixr 0  $, `seq`

Sections are written as ( op e ) or ( e op ). For example:

Operators written within enclosed parens are applied like traditional functions. For example the following are equivalent:

Tuples

Tuples are heterogeneous structures which contain a fixed number of values. Some simple examples are shown below:

For two-tuples there are two functions fst and snd which extract the left and right values respectively.

GHC supports tuples to size 62.

Where & Let Clauses

Haskell syntax contains two different types of declaration syntax: let and where. A let binding is an expression and binds anywhere in its body. For example the following let binding declares x and y in the expression x+y.

A where binding is a toplevel syntax construct (i.e. not an expression) that binds variables at the end of a function. For example the following binds x and y in the function body of f which is x+y.

Where clauses following the Haskell layout rule where definitions can be listed on newlines so long as the definitions have greater indentation than the first toplevel definition they are bound to.

Conditionals

Haskell has builtin syntax for scrutinizing boolean expressions. These are first class expressions known as if statements. An if statement is of the form if cond then trueCond else falseCond. Both the True and False statements must be present.

If statements are just syntactic sugar for case expressions over boolean values. The following example is equivalent to the above example.

Function Composition

Functions are obviously at the heart of functional programming. In mathematics function composition is an operation which takes two functions and produces another function with the result of the first argument function applied to the result of the second function. This is written in mathematical notation as:

g ∘ f

The two functions operate over a domain. For example X, Y and Z.

f : X → Y  g : Y → Z

Or in Haskell notation:

Composition operation results in a new function:

g ∘ f : X → Z

In Haskell this operator is given special infix operator to appear similar to the mathematical notation. Intuitively it takes two functions of types b -> c and a -> b and composes them together to produce a new function. This is the canonical example of a higher-order function.

Haskell code will liberally use this operator to compose chains of functions. For example the following composes a chain of list processing functions sort, filter and map:

Another common higher-order function is the flip function which takes as its first argument a function of two arguments, and reverses the order of these two arguments returning a new function.

The most common operator in all of Haskell is function application operator $. This function is right associative and takes the entire expression on the right hand side of the operator and applies it to function on the left.

This is quite often used in the pattern where the left hand side is a composition of other functions applied to a single argument. This is common in point-free style of programming which attempts to minimize the number of input arguments in favour of pure higher order function composition. The flipped form of this function does the opposite and is left associative, and applies the entire left hand side expression to a function given in the second argument to the function.

For comparison consider the use of $, & and explicit parentheses.

The on function takes a function b and yields the result of applying unary function u to two arguments x and y. This is a higher order function that transforms two inputs and combines the outputs.

This is used quite often in sort functions. For example we can write a custom sort function which sorts a list of lists based on length.

λ: import Data.List
λ: sortSize = sortBy (compare `on` length)
λ: sortSize [[1,2], [1,2,3], [1]]
[[1],[1,2],[1,2,3]]

List Comprehensions

List comprehensions are a syntactic construct that first originated in the Haskell language and has now spread to other programming languages. List comprehensions provide a simple way of working with lists and sequences of values that follow patterns. List comprehension syntax consists of three components:

• Generators - Expressions which evaluate a list of values which are iteratively added to the result.
• Let bindings - Expressions which generate a constant value which is scoped on each iteration.
• Guards - Expressions which generate a boolean expression which determine whether an iteration is added to the result.

The simplest generator is simply a list itself. The following example produces a list of integral values, each element multiplied by two.

We can extend this by adding a let statement which generalizes the multiplier on each step and binds it to a variable n.

And we can also restrict the set of resulting values to only the subset of values of x that meet a condition. In this case we restrict to only values of x which are odd.

Comprehensions with multiple generators will combine each generator pairwise to produce the cartesian product of all results.

λ: [(x,y) | x <- [1,2,3], y <- [10,20,30]]
[(1,10),(1,20),(1,30),(2,10),(2,20),(2,30),(3,10),(3,20),(3,30)]

λ: [(x,y,z) | x <- [1,2], y <- [10,20], z <- [100,200]]
[(1,10,100),(1,10,200),(1,20,100),(1,20,200),(2,10,100),(2,10,200),(2,20,100),(2,20,200)]

Haskell has builtin comprehension syntax which is syntactic sugar for specific methods of the Enum typeclass.

There is an Enum instance for Integer and Char types and so we can write list comprehensions for both, which generate ranges of values.

λ: [1 .. 15]
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]

λ: ['a' .. 'z']
"abcdefghijklmnopqrstuvwxyz"

λ: [1,3 .. 15]
[1,3,5,7,9,11,13,15]

λ: [0,50..500]
[0,50,100,150,200,250,300,350,400,450,500]

These comprehensions can be used inside of function definitions and reference locally bound variables. For example the factorial function (written as n!) is defined as the product of all positive integers up to a given value.

As a more complex example consider a naive prime number sieve:

And a more complex example, consider the classic FizzBuzz interview question. This makes use of iteration and guard statements.

Single line comments begin with double dashes --:

Multiline comments begin with {- and end with -}.

Comments may also add additional structure in the form of Haddock docstrings. These comments will begin with a pipe.

Modules may also have a comment convention which describes the individual authors, copyright and stability information in the following form:

Typeclasses

Typeclasses are one of the core abstractions in Haskell. Just as we wrote polymorphic functions above which operate over all given types (the id function is one example), we can use typeclasses to provide a form of bounded polymorphism which constrains type variables to a subset of those types that implement a given class.

For example we can define an equality class which allows us to define an overloaded notion of equality depending on the data structure provided.

Then we can define this typeclass over several different types. These definitions are called typeclass instances. For example for the Bool type the equality typeclass would be defined as:

Over the unit type, where only a single value exists, the instance is trivial:

For the Ordering type, defined as:

We would have the following Equal instance:

An Equal instance for a more complex data structure like the list type relies upon the fact that the type of the elements in the list must also have a notion of equality, so we include this as a constraint in the typeclass context, which is written to the left of the fat arrow =>. With this constraint in place, we can write this instance recursively by pattern matching on the list elements and checking for equality all the way down the spine of the list:

In the above definition, we know that we can check for equality between individual list elements if those list elements satisfy the Equal constraint. Knowing that they do, we can then check for equality between two complete lists.

For tuples, we will also include the Equal constraint for their elements, and we can then check each element for equality respectively. Note that this instance includes two constraints in the context of the typeclass, requiring that both type variables a and b must also have an Equal instance.

The default prelude comes with a variety of typeclasses that are used frequently and defined over many prelude types:

• Num - Provides a basic numerical interface for values with addition, multiplication, subtraction, and negation.
• Eq - Provides an interface for values that can be tested for equality.
• Ord - Provides an interface for values that have a total ordering.
• Read - Provides an interface for values that can be read from a string.
• Show - Provides an interface for values that can be printed to a string.
• Enum - Provides an interface for values that are enumerable to integers.
• Semigroup - Provides an algebraic semigroup interface.
• Functor - Provides an algebraic functor interface. See Functors.
• Monad - Provides an algebraic monad interface. See Monads.
• Category - Provides an algebraic category interface. See Categories.
• Bounded - Provides an interface for enumerable values with bounds.
• Integral - Provides an interface for integral-like quantities.
• Real - Provides an interface for real-like quantities.
• Fractional - Provides an interface for rational-like quantities.
• Floating - Provides an interface for defining transcendental functions over real values.
• RealFrac - Provides an interface for rounding real values.
• RealFloat - Provides an interface for working with IEE754 operations.

To see the implementation for any of these typeclasses you can run the GHCi info command to see the methods and all instances in scope. For example:

Many of the default classes have instances that can be derived automatically. After the definition of a datatype you can add a deriving clause which will generate the instances for this datatype automatically. This does not work universally but for many instances which have boilerplate definitions, GHC is quite clever and can save you from writing quite a bit of code by hand.

For example for a custom list type.

Side Effects

Contrary to a common misconception, side effects are an integral part of Haskell programming. Probably the most interesting thing about Haskell’s approach to side effects is that they are encoded in the type system. This is certainly a different approach to effectful programming, and the language has various models for modeling these effects within the type system. These models range from using Monads to building algebraic models of effects that draw clear lines between effectful code and pure code. The idea of reasoning about where effects can and cannot exist is one of the key ideas of Haskell, but this certainly does not mean trying to avoid side effects altogether!

Indeed, a Hello World program in Haskell is quite simple:

Other side effects can include reading from the terminal and prompting the user for input, such as in the complete program below:

Records

Records in Haskell are fundamentally broken for several reasons:

1. The syntax is unconventional.

Most programming languages use dot or arrow syntax for field accessors like the following:

Haskell however uses function application syntax since record accessors are simply just functions. Instead or creating a privileged class of names and syntax for field accessors, Haskell instead choose to implement the simplest model and expands accessors to function during compilation.

2. Incomplete pattern matches are implicitly generated for sums of products.

The functions generated for a or b in both of these cases are partial. See Exhaustiveness checking.

3. Lack of Namespacing

Given two records defined in the same module (or imported) GHC is unable to (by default) disambiguate which field accessor to assign at a callsite that uses a.

This can be routed around with the language extension DisambiguateRecordFields but only to a certain extent. If we want to write maximally polymorphic functions which operate over arbitrary records which have a field a, then the GHC typesystem is not able to express this without some much higher-level magic.

Pragmas

At the beginning of a module there is special syntax for pragmas which direct the compiler to compile the current module in a specific way. The most common is a language extension pragma denoted like the following:

These flags alter the semantics and syntax of the module in a variety of ways. See Language Extensions for more details on all of these options.

Additionally we can pass specific GHC flags which alter the compilation behavior, enabling or disabling specific bespoke features based on our needs. These include compiler warnings, optimisation flags and extension flags.

Warning flags allow you to inform users at compile-time with a custom error message. Additionally you can mark a module as deprecated with a specific replacement message.

Newtypes

Newtypes are a form of zero-cost abstraction that allows developers to specify compile-time names for types for which the developer wishes to expose a more restrictive interface. They’re zero-cost because these newtypes end up with the same underlying representation as the things they differentiate. This allows the compiler to distinguish between different types which are representationally identical but semantically different.

For instance velocity can be represented as a scalar quantity represented as a double but the user may not want to mix doubles with other vector quantities. Newtypes allow us to distinguish between scalars and vectors at compile time so that no accidental calculations can occur.

Most importantly these newtypes disappear during compilation and the velocity type will be represented as simply just a machine double with no overhead.

See also the section on Newtype Deriving for a further discussion of tricks involved with handling newtypes.

Bottoms

The bottom is a singular value that inhabits every type. When this value is evaluated, the semantics of Haskell no longer yield a meaningful value. In other words, further operations on the value cannot be defined in Haskell. A bottom value is usually written as the symbol ⊥, ( i.e. the compiler flipping you off ). Several ways exist to express bottoms in Haskell code.

For instance, undefined is an easily called example of a bottom value. This function has type a but lacks any type constraints in its type signature. Thus, undefined is able to stand in for any type in a function body, allowing type checking to succeed, even if the function is incomplete or lacking a definition entirely. The undefined function is extremely practical for debugging or to accommodate writing incomplete programs.

Another example of a bottom value comes from the evaluation of the error function, which takes a String and returns something that can be of any type. This property is quite similar to undefined, which also can also stand in for any type.

Calling error in a function causes the compiler to throw an exception, halt the program, and print the specified error message.

In the divByY function below, passing the function 0 as the divisor results in this function returning such an exception.

A third type way to express a bottom is with an infinitely looping term:

Examples of actual Haskell code that use this looping syntax lives in the source code of the GHC.Prim module. These bottoms exist because the operations cannot be defined in native Haskell. Such operations are baked into the compiler at a very low level. However, this module exists so that Haddock can generate documentation for these primitive operations, while the looping syntax serves as a placeholder for the actual implementation of the primops.

Perhaps the most common introduction to bottoms is writing a partial function that does not have exhaustive pattern matching defined. For example, the following code has non-exhaustive pattern matching because the case expression, lacks a definition of what to do with a B:

The code snippet above is translated into the following GHC Core output where the compiler will insert an exception to account for the non-exhaustive patterns:

GHC can be made more vocal about incomplete patterns using the -fwarn-incomplete-patterns and -fwarn-incomplete-uni-patterns flags.

A similar situation can arise with records. Although constructing a record with missing fields is rarely useful, it is still possible.

When the developer omits a field’s definition, the compiler inserts an exception in the GHC Core representation:

Fortunately, GHC will warn us by default about missing record fields.

Bottoms are used extensively throughout the Prelude, although this fact may not be immediately apparent. The reasons for including bottoms are either practical or historical.

The canonical example is the head function which has type [a] -> a. This function could not be well-typed without the bottom.

Some further examples of bottoms:

It is rare to see these partial functions thrown around carelessly in production code because they cause the program to halt. The preferred method for handling exceptions is to combine the use of safe variants provided in Data.Maybe with the functions maybe and either.

Another method is to use pattern matching, as shown in listToMaybe, a safer version of head described below:

Invoking a bottom defined in terms of error typically will not generate any position information. However, assert, which is used to provide assertions, can be short-circuited to generate position information in place of either undefined or error calls.

See: Avoiding Partial Functions

Exhaustiveness

Pattern matching in Haskell allows for the possibility of non-exhaustive patterns. For example, passing Nothing to unsafe will cause the program to crash at runtime. However, this function is an otherwise valid, type-checked program.

Since unsafe takes a Maybe a value as its argument, two possible values are valid input: Nothing and Just a. Since the case of a Nothing was not defined in unsafe, we say that the pattern matching within that function is non-exhaustive. In other words, the function does not implement appropriate handling of all valid inputs. Instead of yielding a value, such a function will halt from an incomplete match.

Partial functions from non-exhaustivity are a controversial subject, and frequent use of non-exhaustive patterns is considered a dangerous code smell. However, the complete removal of non-exhaustive patterns from the language would itself be too restrictive and forbid too many valid programs.

Several flags exist that we can pass to the compiler to warn us about such patterns or forbid them entirely, either locally or globally.

The -Wall or -fwarn-incomplete-patterns flag can also be added on a per-module basis by using the OPTIONS_GHC pragma.

A more subtle case of non-exhaustivity is the use of implicit pattern matching with a single uni-pattern in a lambda expression. In a manner similar to the unsafe function above, a uni-pattern cannot handle all types of valid input. For instance, the function boom will fail when given a Nothing, even though the type of the lambda expression’s argument is a Maybe a.

Non-exhaustivity arising from uni-patterns in lambda expressions occurs frequently in let or do-blocks after desugaring, because such code is translated into lambda expressions similar to boom.

GHC can warn about these cases of non-exhaustivity with the -fwarn-incomplete-uni-patterns flag.

Generally speaking, any non-trivial program will use some measure of partial functions. It is simply a fact. Thus, there exist obligations for the programmer that cannot be manifested in the Haskell type system.

Debugger

Since GHC version 6.8.1, a built-in debugger has been available, although its use is somewhat rare. Debugging uncaught exceptions is in a similar style to debugging segfaults with gdb. Breakpoints can be set :break and the call stack stepped through with :forward and :back.

Stack Traces

With runtime profiling enabled, GHC can also print a stack trace when a diverging bottom term (error, undefined) is hit. This action, though, requires a special flag and profiling to be enabled, both of which are disabled by default. So, for example:

And indeed, the runtime tells us that the exception occurred in the function g and enumerates the call stack.

It is best to run this code without optimizations applied -O0 so as to preserve the original call stack as represented in the source. With optimizations applied, GHC will rearrange the program in rather drastic ways, resulting in what may be an entirely different call stack.

Printf Tracing

Since Haskell is a pure language it has the unique property that most code is introspectable on its own. As such, using printf to display the state of the program at critical times throughout execution is often unnecessary because we can simply open GHCi and test the function. Nevertheless, Haskell does come with an unsafe trace function which can be used to perform arbitrary print statements outside of the IO monad. You can place these statements wherever you like in your code without without IO restrictions.

Trace uses unsafePerformIO under the hood and should not be used in production code.

In addition to the trace function, several monadic trace variants are quite common.

Type Inference

While inference in Haskell is usually complete, there are cases where the principal type cannot be inferred. Three common cases are:

• Reduced polymorphism due to mutually recursive binding groups
• Undecidability due to polymorphic recursion
• Reduced polymorphism due to the monomorphism restriction

In each of these cases, Haskell needs a hint from the programmer, which may be provided by adding explicit type signatures.

Mutually Recursive Binding Groups

In this case, the inferred type signatures are correct in their usage, but they don’t represent the most general signatures. When GHC analyzes the module it analyzes the dependencies of expressions on each other, groups them together, and applies substitutions from unification across mutually defined groups. As such the inferred types may not be the most general types possible, and an explicit signature may be desired.

Polymorphic recursion

In the second case, recursion is polymorphic because the inferred type variable a in size spans two possible types (a and (a,a)). These two types won’t pass the occurs-check of the typechecker and it yields an incorrect inferred type:

Simply adding an explicit type signature corrects this. Type inference using polymorphic recursion is undecidable in the general case.

See: Static Semantics of Function and Pattern Bindings

Monomorphism Restriction

Finally Monomorphism restriction is a builtin typing rule. By default, it is turned on when compiling and off in GHCi. The practical effect of this rule is that types inferred for functions without explicit type signatures may be more specific than expected. This is because GHC will sometimes reduce a general type, such as Num to a default type, such as Double. This can be seen in the following example in GHCi:

This rule may be deactivated with the NoMonomorphicRestriction extension, see below.

See:

• Monomorphism Restriction

Type Holes

Since the release of GHC 7.8, type holes allow underscores as stand-ins for actual values. They may be used either in declarations or in type signatures.

Type holes are useful in debugging incomplete programs. By placing an underscore on any value on the right hand-side of a declaration, GHC will throw an error during type-checking. The error message describes which values may legally fill the type hole.

GHC has rightly suggested that the expression needed to finish the program is xs :: [a].

The same hole technique can be applied at the toplevel for signatures:

Pattern wildcards can also be given explicit names so that GHC will use the names when reporting the inferred type in the resulting message.

The same wildcards can be used in type contexts to dump out inferred type class constraints:

When the flag -XPartialTypeSignatures is passed to GHC and the inferred type is unambiguous, GHC will let us leave the holes in place and the compilation will proceed with a warning instead of an error.

Deferred Type Errors

Since the release of version 7.8, GHC supports the option of treating type errors as runtime errors. With this option enabled, programs will run, but they will fail when a mistyped expression is evaluated. This feature is enabled with the -fdefer-type-errors flag in three ways: at the module level, when compiled from the command line, or inside of a GHCi interactive session.

For instance, the program below will compile:

However, when a pathological term is evaluated at runtime, we’ll see a message like this:

This error tells us that while x has a declared type of (), the body of the function print 3 has a type of IO (). However, if the term is never evaluated, GHC will not throw an exception.

Name Conventions

Haskell uses short variable names as a convention. This is offputting at first but after you read enough Haskell, it ceases to be a problem. In addition there are several ad-hoc conventions that are typically adopted

Functions that end with a tick (like fold') are typically strict variants of a default lazy function.

Functions that end with a _ (like map_) are typically variants of a function which discards the output and returns void.

Variables that are pluralized xs, ys typically refer to list tails.

Records that do not export their accessors will sometimes prefix them with underscores. These are sometimes interpreted by Template Haskell logic to produce derived field accessors.

Predicates will often prefix their function names with is, as in isPositive.

Functions which result in an Applicative or Monad type will often suffix their name with a A for Applicative or M for Monad. For example:

Functions which have chirality in which they traverse a data structure (i.e. left-to-right or right-to-left) will often suffix the name with L or R for their iteration pattern. This is useful because often times these type signatures identical.

Functions working with mutable structures or monadic state will often adopt the following naming conventions:

Functions that are prefixed with with typically take a value as their first argument and a function as their second argument returning the value with the function applied over some substructure as the result.

ghcid

ghcid is a lightweight IDE hook that allows continuous feedback whenever code is updated. It can be run from the command line in the root of the cabal project directory by specifying a command to run (e.g. ghci, cabal repl, or stack repl).

When a Haskell module is loaded into ghcid, the code is evaluated in order to provide the user with any errors or warnings that would happen at compile time. When the developer edits and saves code loaded into ghcid, the program automatically reloads and evaluates the code for errors and warnings.

HLint

HLint is a source linter for Haskell that provides a variety of hints on code improvements. It can be customised and configured with custom rules, on a per-project basis. HLint is configured through a hlint.yaml file placed in the root of a project. To generate the default configuration run:

Custom errors can be added to this file in order to match and suggest custom changes of code from the left hand side match to the right hand side replacement:

HLint’s default is to warn on all possible failures. These can be disabled globally by adding ignore pragmas.

Or within specific modules by specifying the within option.

See:

• HLint Github

Docker Images

Haskell has stable Docker images that are widely used for deployments across Kubernetes and Docker environments. The two Dockerhub repositories of note are:

• Official Haskell Images
• Stack LTS Images

To import the official Haskell images with ghc and cabal-install include the following preamble in your Dockerfile with your desired GHC version.

FROM haskell:8.8.1

To import the stack images include the following preamble in your Dockerfile with your desired Stack resolver replaced.

FROM fpco/stack-build:lts-14.0

Continuous Integration

These days it is quite common to use cloud hosted continuous integration systems to test code from version control systems. There are many community contributed build scripts for different service providers, including the following:

• Travis CI for Cabal
• Travis CI for Stack
• Circle CI for Cabal & Stack
• Github Actions for Cabal & Stack

See also the official CI repository:

• haskell-ci

Ormolu

Ormolu is an opinionated Haskell source formatter that produces a canonical way of rendering the Haskell abstract syntax tree to text. This ensures that code shared amongst teams and checked into version control conforms to a single universal standard for whitespace and lexeme placing. This is similar to tools in other languages such as go fmt.

For example running ormolu example.hs --inplace on the following module:

Will rerender the file as:

Ormolu can be installed via a variety of mechanisms.

See:

• ormolu

Haddock

Haddock is the automatic documentation generation tool for Haskell source code, and it integrates with the usual cabal toolchain. In this section, we will explore how to document code so that Haddock can generate documentation successfully.

Several frequent comment patterns are used to document code for Haddock. The first of these methods uses -- | to delineate the beginning of a comment:

Multiline comments are also possible:

-- ^ is used to comment Constructors or Record fields:

Elements within a module (i.e. values, types, classes) can be hyperlinked by enclosing the identifier in single quotes:

Modules themselves can be referenced by enclosing them in double quotes:

haddock also allows the user to include blocks of code within the generated documentation. Two methods of demarcating the code blocks exist in haddock. For example, enclosing a code snippet in @ symbols marks it as a code block:

Similarly, it is possible to use bird tracks (>) in a comment line to set off a code block.

Snippets of interactive shell sessions can also be included in haddock documentation. In order to denote the beginning of code intended to be run in a REPL, the >>> symbol is used:

Headers for specific blocks can be added by prefacing the comment in the module block with a *:

Sections can also be delineated by $ blocks that pertain to references in the body of the module:

Links can be added with the following syntax:

Images can also be included, so long as the path is either absolute or relative to the directory in which haddock is run.

haddock options can also be specified with pragmas in the source, either at the module or project level.

Unsafe Functions

As everyone eventually finds out there are several functions within the implementation of GHC (not the Haskell language) that can be used to subvert the type-system; these functions are marked with the prefix unsafe. Unsafe functions exist only for when one can manually prove the soundness of an expression but can’t express this property in the type-system, or externalities to Haskell.

Using these functions to subvert the Haskell typesystem will cause all measure of undefined behavior with unimaginable pain and suffering, and so they are strongly discouraged. When initially starting out with Haskell there are no legitimate reasons to use these functions at all.

Monads form one of the core components for constructing Haskell programs. In their most general form monads are an algebraic building block that can give rise to ways of structuring control flow, handling data structures and orchestrating logic. Monads are a very general algebraic way of structuring code and have a certain reputation for being confusing. However their power and flexibility have become foundational to the way modern Haskell programs are structured.

There is a singular truth to keep in mind when learning monads.

A monad is just its algebraic laws. Nothing more, nothing less.

Eightfold Path to Monad Satori

Much ink has been spilled waxing lyrical about the supposed mystique of monads. Instead, I suggest a path to enlightenment:

1. Don’t read the monad tutorials.
2. No really, don’t read the monad tutorials.
3. Learn about the Haskell typesystem.
4. Learn what a typeclass is.
5. Read the Typeclassopedia.
6. Read the monad definitions.
7. Use monads in real code.
8. Don’t write monad-analogy tutorials.

In other words, the only path to understanding monads is to read the fine source, fire up GHC, and write some code. Analogies and metaphors will not lead to understanding.

Monad Myths

The following are all false:

• Monads are impure.
• Monads are about effects.
• Monads are about state.
• Monads are about imperative sequencing.
• Monads are about IO.
• Monads are dependent on laziness.
• Monads are a “back-door” in the language to perform side-effects.
• Monads are an embedded imperative language inside Haskell.
• Monads require knowing abstract mathematics.
• Monads are unique to Haskell.

Monad Methods

Monads are not complicated. They are implemented as a typeclass with two methods, return and (>>=) (pronounced “bind”). In order to implement a Monad instance, these two functions must be defined:

The first type signature in the Monad class definition is for return. Any preconceptions one might have for the word “return” should be discarded. It has an entirely different meaning in the context of Haskell and acts very differently than in languages such as C, Python, or Java. Instead of being the final arbiter of what value a function produces, return in Haskell injects a value of type a into a monadic context (e.g., Maybe, Either, etc.), which is denoted as m a.

The other function essential to implementing a Monad instance is (>>=). This infix function takes two arguments. On its left side is a value with type m a, while on the right side is a function with type (a -> m b). The bind operation results in a final value of type m b.

A third, auxiliary function ((>>)) is defined in terms of the bind operation that discards its argument.

This definition says that (>>) has a left and right argument which are monadic with types m a and m b respectively, while the infix function yields a value of type m b. The actual implementation of (>>) says that when m is passed to (>>) with k on the right, the value k will always be yielded.

Monad Laws

In addition to specific implementations of (>>=) and return, all monad instances must satisfy three laws.

Law 1

The first law says that when return a is passed through (>>=) into a function f, this expression is exactly equivalent to f a.

In discussing the next two laws, we’ll refer to a value m. This notation is shorthand for a value wrapped in a monadic context. Such a value has type m a, and could be represented more concretely by values like Nothing, Just x, or Right x. It is important to note that some of these concrete instantiations of the value m have multiple components. In discussing the second and third monad laws, we’ll see some examples of how this plays out.

Law 2

The second law states that a monadic value m passed through (>>=) into return is exactly equivalent to itself. In other words, using bind to pass a monadic value to return does not change the initial value.

A more explicit way to write the second Monad law exists. In this following example code, the first expression shows how the second law applies to values represented by non-nullary type constructors. The second snippet shows how a value represented by a nullary type constructor works within the context of the second law.

Law 3

While the first two laws are relatively clear, the third law may be more difficult to understand. This law states that when a monadic value m is passed through (>>=) to the function f and then the result of that expression is passed to >>= g, the entire expression is exactly equivalent to passing m to a lambda expression that takes one parameter x and outputs the function f applied to x. By the definition of bind, f x must return a value wrapped in the same monad. Because of this property, the resultant value of that expression can be passed through (>>=) to the function g, which also returns a monadic value.

Again, it is possible to write this law with more explicit code. Like in the explicit examples for law 2, m has been replaced by SomeMonad val in order to be make it clear that there can be multiple components to a monadic value. Although little has changed in the code, it is easier to see that value –namely, val– corresponds to the x in the lambda expression. After SomeMonad val is passed through (>>=) to f, the function f operates on val and returns a result still wrapped in the SomeMonad type constructor. We can call this new value SomeMonad newVal. Since it is still wrapped in the monadic context, SomeMonad newVal can thus be passed through the bind operation into the function g.

Monad law summary: Law 1 and 2 are identity laws (left and right identity respectively) and law 3 is the associativity law. Together they ensure that Monads can be composed and ‘do the right thing’.

See:

• Monad Laws

Do Notation

Monadic syntax in Haskell is written in a sugared form, known as do notation. The advantages of this special syntax are that it is easier to write and often easier to read, and it is entirely equivalent to simply applying the monad operations. The desugaring is defined recursively by the rules:

Thus, through the application of the desugaring rules, the following expressions are equivalent:

do
  a <- f                               -- f, g, and h are bound to the names a,
  b <- g                               -- b, and c. These names are then passed
  c <- h                               -- to 'return' to ensure that all values
  return (a, b, c)                     -- are wrapped in the appropriate monadic
                                       -- context

do {                                   -- N.B. '{}'  and ';' characters are
  a <- f;                              --  rarely used in do-notation
  b <- g;
  c <- h;
  return (a, b, c)
  }

f >>= a ->
  g >>= b ->
    h >>= c ->
      return (a, b, c)

If one were to write the bind operator as an uncurried function (which is not how Haskell uses it) the same desugaring might look something like the following chain of nested binds with lambdas.

In the do-notation, the monad laws from above are equivalently written:

Law 1

Law 2

Law 3

See:

• Haskell 2010: Do Expressions

Maybe Monad

The Maybe monad is the simplest first example of a monad instance. The Maybe monad models a computation which may fail to yield a value at any point during computation.

The Maybe type has two value constructors. The first, Just, is a unary constructor representing a successful computation, while the second, Nothing, is a nullary constructor that represents failure.

The monad instance describes the implementation of (>>=) for Maybe by pattern matching on the possible inputs that could be passed to the bind operation (i.e., Nothing or Just x). The instance declaration also provides an implementation of return, which in this case is simply Just.

The following code shows some simple operations to do within the Maybe monad.

In the above example, the value Just 3 is passed via (>>=) to the lambda function x -> return (x + 1). x refers to the Int portion of Just 3, and we can use x in the second half of the lambda expression, return (x + 1) which evaluates to Just 4, indicating a successful computation.

In the second example, the value Nothing is passed via (>>=) to the same lambda function as in the previous example. However, according to the Maybe Monad instance, whenever Nothing is bound to a function, the expression’s result will be Nothing.

Here, return is applied to 4 and results in Just 4.

The next code examples show the use of do notation within the Maybe monad to do addition that might fail. Desugared examples are provided as well.

List Monad

The List monad is the second simplest example of a monad instance. As always, this monad implements both (>>=) and return.

The definition of bind says that when the list m is bound to a function f, the result is a concatenation of map f over the list m. The return method simply takes a single value x and injects into a singleton list [x].

In order to demonstrate the List monad’s methods, we will define two values: m and f. m is a simple list, while f is a function that takes a single Int and returns a two element list [1, 0].

When applied to bind, evaluation proceeds as follows:

m >>= f
==> [1,2,3,4] >>= x -> [1,0]
==> concat (map (x -> [1,0]) [1,2,3,4])
==> concat ([[1,0],[1,0],[1,0],[1,0]])
==> [1,0,1,0,1,0,1,0]

The list comprehension syntax in Haskell can be implemented in terms of the list monad. List comprehensions can be considered syntactic sugar for more obviously monadic implementations. Examples a and b illustrate these use cases.

The first example (a) illustrates how to write a list comprehension. Although the syntax looks strange at first, there are elements of it that may look familiar. For instance, the use of <- is just like bind in a do notation: It binds an element of a list to a name. However, one major difference is apparent: a seems to lack a call to return. Not to worry, though, the [] fills this role. This syntax can be easily desugared by the compiler to an explicit invocation of return. Furthermore, it serves to remind the user that the computation takes place in the List monad.

The second example (b) shows the list comprehension above rewritten with do notation:

The final examples are further illustrations of the List monad. The functions below each return a list of 3-tuples which contain the possible combinations of the three lists that get bound the names a, b, and c. N.B.: Only values in the list bound to a can be used in a position of the tuple; the same fact holds true for the lists bound to b and c.

example :: [(Int, Int, Int)]
example = do
  a <- [1,2]
  b <- [10,20]
  c <- [100,200]
  return (a,b,c)
-- [(1,10,100),(1,10,200),(1,20,100),(1,20,200),(2,10,100),(2,10,200),(2,20,100),(2,20,200)]

desugared :: [(Int, Int, Int)]
desugared = [1, 2] >>= a ->
              [10, 20] >>= b ->
                [100, 200] >>= c ->
                  return (a, b, c)
-- [(1,10,100),(1,10,200),(1,20,100),(1,20,200),(2,10,100),(2,10,200),(2,20,100),(2,20,200)]

IO Monad

Perhaps the most (in)famous example in Haskell of a type that forms a monad is IO. A value of type IO a is a computation which, when performed, does some I/O before returning a value of type a. These computations are called actions. IO actions executed in main are the means by which a program can operate on or access information from the external world. IO actions allow the program to do many things, including, but not limited to:

• Print a String to the terminal
• Read and parse input from the terminal
• Read from or write to a file on the system
• Establish an ssh connection to a remote computer
• Take input from a radio antenna for signal processing
• Launch the missiles.

Conceptualizing I/O as a monad enables the developer to access information from outside the program, but also to use pure functions to operate on that information as data. The following examples will show how we can use IO actions and IO values to receive input from stdin and print to stdout.

Perhaps the most immediately useful function for doing I/O in Haskell is putStrLn. This function takes a String and returns an IO (). Calling it from main will result in the String being printed to stdout followed by a newline character.

Here is some code that prints a couple of lines to the terminal. The first invocation of putStrLn is executed, causing the String to be printed to stdout. The result is bound to a lambda expression that discards its argument, and then the next putStrLn is executed.

Another useful function is getLine which has type IO String. This function gets a line of input from stdin. The developer can then bind this line to a name in order to operate on the value within the program.

The code below demonstrates a simple combination of these two functions as well as desugaring IO code. First, putStrLn prints a String to stdout to ask the user to supply their name, with the result being bound to a lambda that discards it argument. Then, getLine is executed, supplying a prompt to the user for entering their name. Next, the resultant IO String is bound to name and passed to putStrLn. Finally, the program prints the name to the terminal.

The next code block is the desugared equivalent of the previous example where the uses of (>>=) are made explicit.

Our final example executes in the same way as the previous two examples. This example, though, uses the special (>>) operator to take the place of binding a result to the lambda that discards its argument.

See:

• Haskell 2010: Basic/Input Output

What’s the point?

Although it is difficult, if not impossible, to touch, see, or otherwise physically interact with a monad, this construct has some very interesting implications for programmers. For instance, consider the non-intuitive fact that we now have a uniform interface for talking about three very different, but foundational ideas for programming: Failure, Collections and Effects.

Let’s write down a new function called sequence which folds a function mcons over a list of monadic computations. We can think of mcons as analogous to the list constructor (i.e. (a : b : [])) except it pulls the two list elements out of two monadic values (p,q) by means of bind. The bound values are then joined with the list constructor :, before finally being rewrapped in the appropriate monadic context with return.

What does this function mean in terms of each of the monads discussed above?

Maybe

For the Maybe monad, sequencing a list of values within the Maybe context allows us to collect the results of a series of computations which can possibly fail. However, sequence yields the aggregated values only if each computation succeeds. In other words, if even one of the Maybe values in the initial list passed to sequenceis a Nothing, the result of evaluating sequence for the whole list will also be Nothing.

List

The bind operation for the list monad forms the pairwise list of elements from the two operands. Thus, folding the binds contained in mcons over a list of lists with sequence implements the general Cartesian product for an arbitrary number of lists.

sequence [[1,2,3],[10,20,30]]
-- [[1,10],[1,20],[1,30],[2,10],[2,20],[2,30],[3,10],[3,20],[3,30]]

IO

Applying sequence within the IO context results in still a different result. The function takes a list of IO actions, performs them sequentially, and then gives back the list of resulting values in the order sequenced.

So there we have it, three fundamental concepts of computation that are normally defined independently of each other actually all share this similar structure. This unifying pattern can be abstracted out and reused to build higher abstractions that work for all current and future implementations. If you want a motivating reason for understanding monads, this is it! These insights are the essence of what I wish I knew about monads looking back.

See:

• Control.Monad

Reader Monad

The reader monad lets us access shared immutable state within a monadic context.

A simple implementation of the Reader monad:

Writer Monad

The writer monad lets us emit a lazy stream of values from within a monadic context.

A simple implementation of the Writer monad:

This implementation is lazy, so some care must be taken that one actually wants to only generate a stream of thunks. Most often the lazy writer is not suitable for use, instead implement the equivalent structure by embedding some monomial object inside a StateT monad, or using the strict version.

State Monad

The state monad allows functions within a stateful monadic context to access and modify shared state.

The state monad is often mistakenly described as being impure, but it is in fact entirely pure and the same effect could be achieved by explicitly passing state. A simple implementation of the State monad takes only a few lines:

Why are monads confusing?

So many monad tutorials have been written that it begs the question: what makes monads so difficult when first learning Haskell? I hypothesize there are three aspects to why this is so:

1. There are several levels of indirection with desugaring.

A lot of the Haskell we write is radically rearranged and transformed into an entirely new form under the hood.

Most monad tutorials will not manually expand out the do-sugar. This leaves the beginner thinking that monads are a way of dropping into a pseudo-imperative language inside of pure code and further fuels the misconception that specific instances like IO describe monads in their full generality. When in fact the IO monad is only one among many instances.

Being able to manually desugar is crucial to understanding.

2. Infix operators for higher order functions are not common in other languages.

On the left hand side of the operator we have an m a and on the right we have a -> m b. Thus, this operator is asymmetric, utilizing a monadic value on the left and a higher order function on the right. Although some languages do have infix operators that are themselves higher order functions, it is still a rather rare occurrence.

Thus, with a function desugared, it can be confusing that (>>=) operator is in fact building up a much larger function by composing functions together.

Written in prefix form, it becomes a little bit more digestible.

Perhaps even removing the operator entirely might be more intuitive coming from other languages.

3. Ad-hoc polymorphism is not commonplace in other languages.

Haskell’s implementation of overloading can be unintuitive if one is not familiar with type inference. Indeed, newcomers to Haskell often believe they can gain an intuition for monads in a way that will unify their understanding of all monads. This is a fallacy, however, because any particular monad instance is merely an instantiation of the monad typeclass functions implemented for that particular type.

This is all abstracted away from the user, but the (>>=) or bind function is really a function of 3 arguments with the extra typeclass dictionary argument ($dMonad) implicitly threaded around.

In general, this is true for all typeclasses in Haskell and it’s true here as well, except in the case where the parameter of the monad class is unified (through inference) with a concrete class instance.

Now, all of these transformations are trivial once we understand them, they’re just typically not discussed. In my opinion the fundamental fallacy of monad tutorials is not that intuition for monads is hard to convey (nor are metaphors required!), but that novices often come to monads with an incomplete understanding of points (1), (2), and (3) and then trip on the simple fact that monads are the first example of a Haskell construct that is the confluence of all three.

Thus we make monads more difficult than they need to be. At the end of the day they are simple algebraic critters.

mtl / transformers

The descriptions of Monads in the previous chapter are a bit of a white lie. Modern Haskell monad libraries typically use a more general form of these, written in terms of monad transformers which allow us to compose monads together to form composite monads.

Imagine if you had an application that wanted to deal with a Maybe monad wrapped inside a State Monad, all wrapped inside the IO monad. This is the problem that monad transformers solve, a problem of composing different monads. At their core, monad transformers allow us to nest monadic computations in a stack with an interface to exchange values between the levels, called lift:

In production code, the monads mentioned previously may actually be their more general transformer form composed with the Identity monad.

The following table shows the relationships between these forms:

Just as the base monad class has laws, monad transformers also have several laws:

Law #1

Law #2

Or equivalently:

Law #1

Law #2

It’s useful to remember that transformers compose outside-in but are unrolled inside out.

Transformers

The lift definition provided above comes from the transformers library along with an IO-specialized form called liftIO:

These definitions rely on the following typeclass definitions, which describe composing one monad with another monad (the “t” is the transformed second monad):

Basics

The most basic use requires us to use the T-variants for each of the monad transformers in the outer layers and to explicitly lift and return values between the layers. Monads have kind (* -> *), so monad transformers which take monads to monads have ((* -> *) -> * -> *):

For example, if we wanted to form a composite computation using both the Reader and Maybe monads, using MonadTrans we could use Maybe inside of a ReaderT to form ReaderT t Maybe a.

The fundamental limitation of this approach is that we find ourselves lift.lift.lifting and return.return.returning a lot.

mtl

The mtl library is the most commonly used interface for these monad tranformers, but mtl depends on the transformers library from which it generalizes the “basic” monads described above into more general transformers, such as the following:

This solves the “lift.lift.lifting” problem introduced by transformers.

ReaderT

By way of an example there exist three possible forms of the Reader monad. The first is the primitive version which no longer exists, but which is useful for understanding the underlying ideas. The other two are the transformers and mtl variants.

Reader

ReaderT

MonadReader

So, hypothetically the three variants of ask would be:

In practice the mtl variant is the one commonly used in Modern Haskell.

Newtype Deriving

Newtype deriving is a common technique used in combination with the mtl library and as such we will discuss its use for transformers in this section.

As discussed in the newtypes section, newtypes let us reference a data type with a single constructor as a new distinct type, with no runtime overhead from boxing, unlike an algebraic datatype with a single constructor. Newtype wrappers around strings and numeric types can often drastically reduce accidental errors.

Consider the case of using a newtype to distinguish between two different text blobs with different semantics. Both have the same runtime representation as a text object, but are distinguished statically, so that plaintext can not be accidentally interchanged with encrypted text.

This is a surprisingly powerful tool as the Haskell compiler will refuse to compile any function which treats Cryptotext as Plaintext or vice versa!

The other common use case is using newtypes to derive logic for deriving custom monad transformers in our business logic. Using -XGeneralizedNewtypeDeriving we can recover the functionality of instances of the underlying types composed in our transformer stack.

Using newtype deriving with the mtl library typeclasses we can produce flattened transformer types that don’t require explicit lifting in the transform stack. For example, here is a little stack machine involving the Reader, Writer and State monads.

{-# LANGUAGE GeneralizedNewtypeDeriving #-}

import Control.Monad.Reader
import Control.Monad.Writer
import Control.Monad.State

type Stack   = [Int]
type Output  = [Int]
type Program = [Instr]

type VM a = ReaderT Program (WriterT Output (State Stack)) a

newtype Comp a = Comp { unComp :: VM a }
  deriving (Functor, Applicative, Monad, MonadReader Program, MonadWriter Output, MonadState Stack)

data Instr = Push Int | Pop | Puts

evalInstr :: Instr -> Comp ()
evalInstr instr = case instr of
  Pop    -> modify tail
  Push n -> modify (n:)
  Puts   -> do
    tos <- gets head
    tell [tos]

eval :: Comp ()
eval = do
  instr <- ask
  case instr of
    []     -> return ()
    (i:is) -> evalInstr i >> local (const is) eval

execVM :: Program -> Output
execVM = flip evalState [] . execWriterT . runReaderT (unComp eval)

program :: Program
program = [
     Push 42,
     Push 27,
     Puts,
     Pop,
     Puts,
     Pop
  ]

main :: IO ()
main = mapM_ print $ execVM program

Pattern matching on a newtype constructor compiles into nothing. For example theextractB function below does not scrutinize the MkB constructor like extractA does, because MkB does not exist at runtime; it is purely a compile-time construct.

Efficiency

The second monad transformer law guarantees that sequencing consecutive lift operations is semantically equivalent to lifting the results into the outer monad.

Although they are guaranteed to yield the same result, the operation of lifting the results between the monad levels is not without cost and crops up frequently when working with the monad traversal and looping functions. For example, all three of the functions on the left below are less efficient than the right hand side which performs the bind in the base monad instead of lifting on each iteration.

Monad Morphisms

Although the base monad transformer package provides a MonadTrans class for lifting to another monad:

But oftentimes we need to work with and manipulate our monad transformer stack to either produce new transformers, modify existing ones or extend an upstream library with new layers. The mmorph library provides the capacity to compose monad morphism transformation directly on transformer stacks. This is achieved primarily by use of the hoist function which maps a function from a base monad into a function over a transformed monad.

Hoist takes a monad morphism (a mapping from a m a to a n a) and applies in on the inner value monad of a transformer stack, transforming the value under the outer layer.

The monad morphism generalize takes an Identity monad into any another monad m.

For example, it generalizes State s a (which is StateT s Identity a) to StateT s m a.

So we can generalize an existing transformer to lift an IO layer onto it.

See:

• mmorph

Effect Systems

The mtl model has several properties which make it suboptimal from a theoretical perspective. Although it is used widely in production Haskell we will discuss its shortcomings and some future models called effect systems.

Extensibility

When you add a new custom transformer inside of our business logic we’ll typically have to derive a large number of boilerplate instances to compose it inside of existing mtl transformer stack. For example adding MonadReader instance for n number of undecidable instances that do nothing but mostly lifts. You can see this massive boilerplate all over the design of the mtl library and its transitive dependencies.

This is called the n² instance problem or the instance boilerplate problem and remains an open problem of mtl.

Composing Transformers

Effects don’t generally commute from a theoretical perspective and as such monad transformer composition is not in general commutative. For example stacking State and Except is not commutative:

In addition, the standard method of deriving mtl classes for a transformer stack breaks down when using transformer stacks with the same monad at different layers of the stack. For example stacking multiple State transformers is a pattern that shows up quite frequently.

In order to get around this you would have to handwrite the instances for this transformer stack and manually lift anytime you perform a State action. This is a suboptimal design and difficult to route around without massive boilerplate.

While these problems exist, most users of mtl don’t implement new transformers at all and can get by. However in recent years there have been written many other libraries that have explored the design space of alternative effect modeling systems. These systems are still quite early compared to the mtl but some are able to avoid some of the shortcomings of mtl in favour of newer algebraic models of effects. The two most commonly used libraries are:

• polysemy
• fused-effects

Polysemy

Polysemy is a new effect system library based on the free-monad approach to modeling effects. The library uses modern type system features to model effects on top of a Sem monad. The monad will have a members constraint type which constraints a parameter r by a type-level list of effects in the given unit of computation.

For example we seamlessly mix and match error handling, tracing, and stateful updates inside of one computation without the new to create a layered monad. This would look something like the following:

These effects can then be evaluated using an interpreter function which unrolls and potentially evaluates the effects of the Sem free monad. Some of these interpreters for tracing, state and error are similar to the evaluations for monad transformers but evaluate one layer of type-level list of the effect stack.

The resulting Sem monad with a single field can then be lowered into a single resulting monad such as IO or Either.

The library provides rich set of of effects that can replace many uses of monad transformers.

• Polysemy.Async - Asynchronous computations
• Polysemy.AtomicState - Atomic operations
• Polysemy.Error - Error handling
• Polysemy.Fail - Computations that fail
• Polysemy.IO - Monadic IO
• Polysemy.Input - Input effects
• Polysemy.Output - Output effects
• Polysemy.NonDet - Non-determinism effect
• Polysemy.Reader - Contextual state a la Reader monad
• Polysemy.Resource - Resources with finalizers
• Polysemy.State - Stateful effects
• Polysemy.Trace - Tracing effect
• Polysemy.Writer - Accumulation effect a la Writer monad

For example for a simple stateful computation with only a single effect.

And a more complex example which combines multiple effects:

Polysemy will require the following language extensions to operate:

The use of free-monads is not entirely without cost, and there are experimental GHC plugins which can abstract away some of the overhead from the effect stack. Code thats makes use of polysemy should enable the following GHC flags to enable aggressive typeclass specialisation:

• -flate-specialise
• -fspecialise-aggressively

Fused Effects

Fused-effects is an alternative approach to effect systems based on an algebraic effects model. Unlike polysemy, fused-effects does not use a free monad as an intermediate form. Fused-effects has competative performance compared with mtl and doesn’t require additional GHC plugins or extension compiler fusion rules to optimise away the abstraction overhead.

The fused-effects library exposes a constraint kind called Has which annotates a type signature that contains effectful logic. In this signature m is called the carrier for the sig effect signature containing the eff effect.

For example the traditional State effect is modeled by the following datatype with three parameters. The s parameter is the state object, the m is the effect parameter. This exposes the same interface as Control.Monad.State except for the Has constraint instead.

The Carrier for the State effect is defined as StateC and the evaluators for the state carrier are defined in the same interface as mtl except they evaluate into a result containing the effect parameter m.

The evaluators for the effect lift monadic actions from an effectful computation.

Fused-effects requires the following language extensions to operate.

Minimal Example

A minimal example using the State effect to track stateful updates to a single integral value.

The evaluation of this monadic state block results in a m Integer with the Algebra and Effect context. This can then be evaluated into either Identity or IO using run.

Composite Effects

Consider a more complex example which combines exceptions with Throw effect with State. Importantly note that functions runThrow and evalState cannot infer the state type from the signature alone and thus require additional annotations. This differs from mtl which typically has more optimal inference.

Philosophy

Haskell takes a drastically different approach to language design than most other languages as a result of being the synthesis of input from industrial and academic users. GHC allows the core language itself to be extended with a vast range of opt-in flags which change the semantics of the language on a per-module or per-project basis. While this does add a lot of complexity at first, it also adds a level of power and flexibility for the language to evolve at a pace that is unrivaled in the broader space of programming language design.

Classes

It’s important to distinguish between different classes of GHC language extensions: general and specialized.

The inherent problem with classifying extensions into general and specialized categories is that it is a subjective classification. Haskellers who do theorem proving research will have a very different interpretation of Haskell than people who do web programming. Thus, we will use the following classifications:

• Benign implies both that importing the extension won’t change the semantics of the module if not used and that enabling it makes it no easier to shoot yourself in the foot.
• Historical implies that one shouldn’t use this extension, it is in GHC purely for backwards compatibility. Sometimes these are dangerous to enable.
• Steals syntax means that enabling this extension causes certain code, that is valid in vanilla Haskell, to be no longer be accepted. For example, f $(a) is the same as f $ (a) in Haskell98, but TemplateHaskell will interpret $(a) as a splice.

The golden source of truth for language extensions is the official GHC user’s guide which contains a plethora of information on the details of these extensions.

See: GHC Extension Reference

Extension Dependencies

Some language extensions will implicitly enable other language extensions for their operation. The table below shows the dependencies between various extensions and which sets are implied.

The Benign

It’s not obvious which extensions are the most common but it’s fairly safe to say that these extensions are benign and are safely used extensively:

• NoImplicitPrelude
• OverloadedStrings
• LambdaCase
• FlexibleContexts
• FlexibleInstances
• GeneralizedNewtypeDeriving
• TypeSynonymInstances
• MultiParamTypeClasses
• FunctionalDependencies
• NoMonomorphismRestriction
• GADTs
• BangPatterns
• DeriveGeneric
• DeriveAnyClass
• DerivingStrategies
• ScopedTypeVariables

The Advanced

These extensions are typically used by advanced projects that push the limits of what is possible with Haskell to enforce complex invariants and very type-safe APIs.

• PolyKinds
• DataKinds
• DerivingVia
• GADTs
• RankNTypes
• ExistentialQuantification
• TypeFamilies
• TypeOperators
• TypeApplications
• UndecidableInstances

The Lowlevel

These extensions are typically used by low-level libraries that are striving for optimal performance or need to integrate with foreign functions and native code. Most of these are used to manipulate base machine types and interface directly with the low-level byte representations of data structures.

• CPP
• BangPatterns
• CApiFFI
• Strict
• StrictData
• RoleAnnotations
• ForeignFunctionInterface
• InterruptibleFFI
• UnliftedFFITypes
• MagicHash
• UnboxedSums
• UnboxedTuples

The Dangerous

GHC’s typechecker sometimes casually tells us to enable language extensions when it can’t solve certain problems. Unless you know what you’re doing, these extensions almost always indicate a design flaw and shouldn’t be turned on to remedy the error at hand, as much as GHC might suggest otherwise!

• AllowAmbigiousTypes
• DatatypeContexts
• OverlappingInstances
• IncoherentInstances
• ImpredicativeTypes

NoMonomorphismRestriction

The NoMonomorphismRestriction allows us to disable the monomorphism restriction typing rule GHC uses by default. See monomorphism restriction.

For example, if we load the following module into GHCi

And then we attempt to call the function bar with a Double, we get a type error:

The problem is that GHC has inferred an overly specific type:

We can prevent GHC from specializing the type with this extension:

Now everything will work as expected:

ExtendedDefaultRules

In the absence of explicit type signatures, Haskell normally resolves ambiguous literals using several defaulting rules. When an ambiguous literal is typechecked, if at least one of its typeclass constraints is numeric and all of its classes are standard library classes, the module’s default list is consulted, and the first type from the list that will satisfy the context of the type variable is instantiated. For instance, given the following default rules

The following set of heuristics is used to determine what to instantiate the ambiguous type variable to.

1. The type variable a appears in no other constraints
2. All the classes Ci are standard.
3. At least one of the classes Ci is numerical.

The standard default definition is implicitly defined as (Integer, Double)

This is normally fine, but sometimes we’d like more granular control over defaulting. The -XExtendedDefaultRules loosens the restriction that we’re constrained with working on Numerical typeclasses and the constraint that we can only work with standard library classes. For example, if we’d like to have our string literals (using -XOverloadedStrings) automatically default to the more efficient Text implementation instead of String we can twiddle the flag and GHC will perform the right substitution without the need for an explicit annotation on every string literal.

For code typed at the GHCi prompt, the -XExtendedDefaultRules flag is always on, and cannot be switched off.

See: Monomorphism Restriction

Safe Haskell

The Safe Haskell language extensions allow us to restrict the use of unsafe language features using -XSafe which restricts the import of modules which are themselves marked as Safe. It also forbids the use of certain language extensions (-XTemplateHaskell) which can be used to produce unsafe code. The primary use case of these extensions is security auditing of codebases for compliance purposes.

See: Safe Haskell

PartialTypeSignatures

Normally a function is either given a full explicit type signature or none at all. The partial type signature extension allows something in between.

Partial types may be used to avoid writing uninteresting pieces of the signature, which can be convenient in development:

If the -Wpartial-type-signatures GHC option is set, partial types will still trigger warnings.

See:

• Partial Type Signatures

RecursiveDo

Recursive do notation allows for the use of self-reference expressions on both sides of a monadic bind. For instance the following example uses lazy evaluation to generate an infinite list. This is sometimes used to instantiate a cyclic datatype inside a monadic context where the datatype needs to hold a reference to itself.

See: Recursive Do Notation

ApplicativeDo

By default GHC desugars do-notation to use implicit invocations of bind and return. With normal monad sugar the following…

… desugars into:

With ApplicativeDo this instead desugars into use of applicative combinators and a laxer Applicative constraint.

Which is equivalent to the traditional notation.

PatternGuards

Pattern guards are an extension to the pattern matching syntax. Given a <- pattern qualifier, the right hand side is evaluated and matched against the pattern on the left. If the match fails then the whole guard fails and the next equation is tried. If it succeeds, then the appropriate binding takes place, and the next qualifier is matched.

ViewPatterns

View patterns are like pattern guards that can be nested inside of other patterns. They are a convenient way of pattern-matching against values of algebraic data types.

TupleSections

The TupleSections syntax extension allows tuples to be constructed similar to how operator sections. With this extension enabled, tuples of arbitrary size can be “partially” specified with commas and values given for specific positions in the tuple. For example for a 2-tuple:

An example for a 7-tuple where three values are specified in the section.

f :: t -> t1 -> t2 -> t3 -> (t, (), t1, (), (), t2, t3)
f = (,(),,(),(),,)

Postfix Operators

The postfix operators extensions allows user-defined operators that are placed after expressions. For example, using this extension, we could define a postfix factorial function.

MultiWayIf

Multi-way if expands traditional if statements to allow pattern match conditions that are equivalent to a chain of if-then-else statements. This allows us to write “pattern matching predicates” on a value. This alters the syntax of Haskell language.

EmptyCase

GHC normally requires at least one pattern branch in a case statement; this restriction can be relaxed with the EmptyCase language extension. The case statement then immediately yields a Non-exhaustive patterns in case if evaluated. For example, the following will compile using this language pragma:

LambdaCase

For case statements, the language extension LambdaCase allows the elimination of redundant free variables introduced purely for the case of pattern matching on.

Without LambdaCase:

With LambdaCase:

NumDecimals

The extension NumDecimals allows the use of exponential notation for integral literals that are not necessarily floats. Without it, any use of exponential notation induces a Fractional class constraint.

PackageImports

The syntax language extension PackageImports allows us to disambiguate hierarchical package names by their respective package key. This is useful in the case where you have two imported packages that expose the same module. In practice most of the common libraries have taken care to avoid conflicts in the namespace and this is not usually a problem in most modern Haskell.

For example we could explicitly ask GHC to resolve that Control.Monad.Error package be drawn from the mtl library.

RecordWildCards

Record wild cards allow us to expand out the names of a record as variables scoped as the labels of the record implicitly. The extension can be used to extract variables names into a scope and/or to assign to variables in a record drawing(?), aligning the record’s labels with the variables in scope for the assignment. The syntax introduced is the {..} pattern selector as in the following example:

{-# LANGUAGE RecordWildCards #-}
{-# LANGUAGE OverloadedStrings #-}

import Data.Text

data  Example = Example
  { e1 :: Int
  , e2 :: Text
  , e3 :: Text
  } deriving (Show)

-- Extracting from a record using wildcards.
scope :: Example -> (Int, Text, Text)
scope Example {..} = (e1, e2, e3)

-- Assign to a record using wildcards.
assign :: Example
assign = Example {..}
  where
    (e1, e2, e3) = (1, "Kirk", "Picard")

NamedFieldPuns

NamedFieldPuns provides alternative syntax for accessing record fields in a pattern match.

PatternSynonyms

Suppose we were writing a typechecker, and we needed to parse type signatures. One common solution would to include a TArr to pattern match on type function signatures. Even though, technically it could be written in terms of more basic application of the (->) constructor.

With pattern synonyms we can eliminate the extraneous constructor without losing the convenience of pattern matching on arrow types. We introduce a new pattern using the pattern keyword.

So now we can write a deconstructor and constructor for the arrow type very naturally.

Pattern synonyms can be exported from a module like any other definition by prefixing them with the prefix pattern.

• Pattern Synonyms in GHC 8

DeriveFunctor

Many instances of functors over datatypes with parameters and trivial constructors are the result of trivially applying a function over the single constructor’s argument. GHC can derive this boilerplate automatically in deriving clauses if DeriveFunctor is enabled.

DeriveFoldable

Similar to how Functors can be automatically derived, many instances of Foldable for types of kind * -> * have instances that derive the functions:

• foldMap
• foldr
• null

For instance if we have a custom rose tree and binary tree implementation we can automatically derive the fold functions for these datatypes automatically for us.

These will generate the following instances:

DeriveTraversable

Just as with Functor and Foldable, many Traversable instances for single-paramater datatypes of kind * -> * have trivial implementations of the traverse function which can also be derived automatically. By enabling DeriveTraversable we can use stock deriving to derive these instances for us.

DeriveGeneric

Data types in Haskell can derived by GHC with the DeriveGenerics extension which is able to define the entire structure of the Generic instance and associated type families. See Generics for more details on what these types mean.

For example the simple custom List type deriving Generic:

Will generate the following Generic instance:

DeriveAnyClass

With -XDeriveAnyClass we can derive any class. The deriving logic generates an instance declaration for the type with no explicitly-defined methods or with all instances having a specific default implementation given. These are used extensively with Generics when instances provide empty Minimal Annotations which are all derived from generic logic.

A contrived example of a class with an empty minimal set might be the following:

DuplicateRecordFields

GHC 8.0 introduced the DuplicateRecordFields extensions which loosens GHC’s restriction on records in the same module with identical accessors. The precise type that is being projected into is now deferred to the callsite.

Using just DuplicateRecordFields, projection is still not supported so the following will not work.

OverloadedLabels

GHC 8.0 also introduced the OverloadedLabels extension which allows a limited form of polymorphism over labels that share the same name.

To work with overloaded label types we also need to enable several language extensions that allow us to use the promoted strings and multiparam typeclasses that underlay its implementation.

This is used in more advanced libraries like Selda which do object relational mapping between Haskell datatype fields and database columns.

See:

• OverloadedRecordFields revived

CPP

The C++ preprocessor is the fallback whenever we really need to separate out logic that has to span multiple versions of GHC and language changes while maintaining backwards compatibility. It can dispatch on the version of GHC being used to compile a module.

It can also demarcate code based on the operating system compiled on.

For another example, it can distinguish the version of the base library used.

One can also use the CPP extension to emit Haskell source at compile-time. This is used in some libraries which have massive boilerplate obligations. Of course, this can be abused quite easily and doing this sort of compile-time string-munging should be a last resort.

TypeApplications

The type system extension TypeApplications allows you to use explicit annotations for subexpressions. For example if you have a subexpression which has the inferred type a -> b -> a you can name the types of a and b by explicitly stating @Int @Bool to assign a to Int and b to Bool. This is particularly useful when working with typeclasses where type inference cannot deduce the types of all subexpressions from the toplevel signature and results in an overly specific default. This is quite common when working with roundtrips of read and show. For example:

DerivingVia

DerivingVia is an extension of GeneralizedNewtypeDeriving. Just as newtype deriving allows us to derive instances in terms of instances for the underlying representation of the newtype, DerivingVia allows deriving instances by specifying a custom type which has a runtime representation equal to the desired behavior we’re deriving the instance for. The derived instance can then be coerced to behave as if it were operating over the given type. This is a powerful new mechanism that allows us to derive many typeclasses in terms of other typeclasses.

DerivingStrategies

Deriving has proven a powerful mechanism to add typeclass instances and as such there have been a variety of bifurcations in its use. Since GHC 8.2 there are now four different algorithms that can be used to derive typeclass instances. These are enabled by different extensions and now have specific syntax for invoking each algorithm specifically. Turning on DerivingStrategies allows you to disambiguate which algorithm GHC should use for individual class derivations.

• stock - Standard GHC builtin deriving (i.e. Eq, Ord, Show)
• anyclass - Deriving via minimal annotations with DeriveAnyClass.
• newtype - Deriving with [GeneralizedNewtypeDeriving].
• via - Deriving with DerivingVia.

These can be stacked and combined on top of a data or newtype declaration.

Historical Extensions

Several language extensions have either been absorbed into the core language or become deprecated in favor of others. Others are just considered misfeatures.

• Rank2Types - Rank2Types has been subsumed by RankNTypes
• XPolymorphicComponents - Was an implementation detail of higher-rank polymorphism that no longer exists.
• NPlusKPatterns - These were largely considered an ugly edge-case of pattern matching language that was best removed.
• TraditionalRecordSyntax - Traditional record syntax was an extension to the Haskell 98 specification for what we now consider standard record syntax.
• OverlappingInstances - Subsumed by explicit OVERLAPPING pragmas.
• IncoherentInstances - Subsumed by explicit INCOHERENT pragmas.
• NullaryTypeClasses - Subsumed by explicit Multiparameter Typeclasses with no parameters.
• TypeInType - Is deprecated in favour of the combination of PolyKinds and DataKinds and extensions to the GHC typesystem after GHC 8.0.

Typeclasses are the bread and butter of abstractions in Haskell, and even out of the box in Haskell 98 they are quite powerful. However classes have grown quite a few extensions, additional syntax and enhancements over the years to augment their utility.

Standard Hierarchy

In the course of writing Haskell there are seven core instances you will use and derive most frequently. Each of them are lawful classes with several equations associated with their methods.

• Semigroup
• Monoid
• Functor
• Applicative
• Monad
• Foldable
• Traversable

Instance Search

Whenever a typeclass method is invoked at a callsite, GHC will perform an instance search over all available instances defined for the given typeclass associated with the method. This instance search is quite similar to backward chaining in logic programming languages. The search is performed during compilation after all types in all modules are known and is performed globally across all modules and all packages available to be linked. The instance search can either result in no instances, a single instance or multiple instances which satisfy the conditions of the call site.

Orphan Instances

Normally typeclass definitions are restricted to be defined in one of two places:

1. In the same module as the declaration of the datatype in the instance head.
2. In the same module as the class declaration.

These two restrictions restrict the instance search space to a system where a solution (if it exists) can always be found. If we allowed instances to be defined in any modules then we could potentially have multiple class instances defined in multiple modules and the search would be ambiguous.

This restriction can however be disabled with the -fno-warn-orphans flag.

This will allow you to define orphan instances in the current module. But beware this will make the instance search contingent on your import list and may result in clashes in your codebase where the linker will fail because there are multiple modules which define the same instance head.

When used appropriately this can be the way to route around the fact that upstream modules may define datatypes that you use, but they have not defined the instances for other downstream libraries that you also use. You can then write these instances for your codebase without modifying either upstream library.

Minimal Annotations

In the presence of default implementations for typeclass methods, there may be several ways to implement a typeclass. For instance Eq is entirely defined by either defining when two values are equal or not equal by implying taking the negation of the other. We can define equality in terms of non-equality and vice-versa.

Before 7.6.1 there was no way to specify what was the “minimal” definition required to implement a typeclass

Minimal pragmas are boolean expressions. For instance, with | as logical OR, either definition of the above functions must be defined. Comma indicates logical AND where both definitions must be defined.

Compiling the -Wmissing-methods will warn when an instance is defined that does not meet the minimal criterion.

TypeSynonymInstances

Normally type class definitions are restricted to being defined only over fully expanded types with all type synonym indirections removed. Type synonyms introduce a “naming indirection” that can be included in the instance search to allow you to write synonym instances for multiple synonyms which expand to concrete types.

This is used quite often in modern Haskell.

FlexibleInstances

Normally the head of a typeclass instance must contain only a type constructor applied to any number of type variables. There can be no nesting of other constructors or non-type variables in the head. The FlexibleInstances extension loosens this restriction to allow arbitrary nesting and non-type variables to be mentioned in the head definition. This extension also implicitly enables TypeSynonymInstances.

FlexibleContexts

Just as with instances, contexts normally are also constrained to consist entirely of constraints where a class is applied to just type variables. The FlexibleContexts extension lifts this restriction and allows any type of type variable and nesting to occur the class constraint head. There is however still a global restriction that all class hierarchies must not contain cycles.

OverlappingInstances

Typeclasses are normally globally coherent, there is only ever one instance that can be resolved for a type unambiguously at any call site in the program. There are however extensions to loosen this restriction and perform more manual direction of the instance search.

Overlapping instances loosens the coherent condition (there can be multiple instances) but introduces a criterion that it will resolve to the most specific one.

Historically enabling on the module-level was not the best idea, since generally we define multiple classes in a module only a subset of which may be incoherent. As of GHC 7.10 we now have the capacity to just annotate instances with the OVERLAPPING and INCOHERENT inline pragmas.

IncoherentInstances

Incoherent instances loosens the restriction that there be only one specific instance, it will be chosen based on a more complex search procedure which tries to identify a prime instance based on information incorporated form OVERLAPPING pragmas on instances in the search tree. Unless one is doing very advanced type-level programming use class constraints, this is usually a poor design decision and a sign to rethink the class hierarchy.

An example with INCOHERENT annotations:

Haskell is a unique language that explores an alternative evaluation model called lazy evaluation. Lazy evaluation implies that expressions will be evaluated only when needed. In truth, this evaluation may even be indefinitely deferred. Consider the example in Haskell of defining an infinite list:

The primary advantage of lazy evaluation in the large is that algorithms that operate over both unbounded and bounded data structures can inhabit the same type signatures and be composed without any additional need to restructure their logic or force intermediate computations.

Still, it’s important to recognize that this is another subject on which much ink has been spilled. In fact, there is an ongoing discussion in the land of Haskell about the compromises between lazy and strict evaluation, and there are nuanced arguments for having either paradigm be the default.

Haskell takes a hybrid approach where it allows strict evaluation when needed while it uses laziness by default. Needless to say, we can always find examples where strict evaluation exhibits worse behavior than lazy evaluation and vice versa. These days Haskell can be both as lazy or as strict as you like, giving you options for however you prefer to program.

Languages that attempt to bolt laziness on to a strict evaluation model often bifurcate classes of algorithms into ones that are hand-adjusted to consume unbounded structures and those which operate over bounded structures. In strict languages, mixing and matching between lazy vs. strict processing often necessitates manifesting large intermediate structures in memory when such composition would “just work” in a lazy language.

By virtue of Haskell being the only language to actually explore this point in the design space, knowledge about lazy evaluation is not widely absorbed into the collective programmer consciousness and can often be non-intuitive to the novice. Some time is often needed to fully grok how lazy evaluation works

Strictness

For a more strict definition of strictnees, consider that there are several evaluation models for the lambda calculus:

• Strict - Evaluation is said to be strict if all arguments are evaluated before the body of a function.
• Non-strict - Evaluation is non-strict if the arguments are not necessarily evaluated before entering the body of a function.

These ideas give rise to several models, Haskell itself uses the call-by-need model.

Seq and WHNF

On the subject of laziness and evaluation, we have names for how fully evaluated an expression is. A term is said to be in weak head normal-form if the outermost constructor or lambda expression cannot be reduced further. A term is said to be in normal form if it is fully evaluated and all sub-expressions and thunks contained within are evaluated.

In Haskell, normal evaluation only occurs at the outer constructor of case-statements in Core. If we pattern match on a list, we don’t implicitly force all values in the list. An element in a data structure is only evaluated up to the outermost constructor. For example, to evaluate the length of a list we need only scrutinize the outer Cons constructors without regard for their inner values:

For example, in a lazy language the following program terminates even though it contains diverging terms.

In a strict language like OCaml (ignoring its suspensions for the moment), the same program diverges.

Thunks

In Haskell a thunk is created to stand for an unevaluated computation. Evaluation of a thunk is called forcing the thunk. The result is an update, a referentially transparent effect, which replaces the memory representation of the thunk with the computed value. The fundamental idea is that a thunk is only updated once (although it may be forced simultaneously in a multi-threaded environment) and its resulting value is shared when referenced subsequently.

The GHCi command :sprint can be used to introspect the state of unevaluated thunks inside an expression without forcing evaluation. For instance:

While a thunk is being computed its memory representation is replaced with a special form known as blackhole which indicates that computation is ongoing and allows for a short circuit when a computation might depend on itself to complete.

The seq function introduces an artificial dependence on the evaluation of order of two terms by requiring that the first argument be evaluated to WHNF before the evaluation of the second. The implementation of the seq function is an implementation detail of GHC.

For one example where laziness can bite you, the infamous foldl is well-known to leak space when used carelessly and without several compiler optimizations applied. The strict foldl’ variant uses seq to overcome this.

In practice, a combination between the strictness analyzer and the inliner on -O2 will ensure that the strict variant of foldl is used whenever the function is inlinable at call site so manually using foldl' is most often not required.

Of important note is that GHCi runs without any optimizations applied so the same program that performs poorly in GHCi may not have the same performance characteristics when compiled with GHC.

BangPatterns

The extension BangPatterns allows an alternative syntax to force arguments to functions to be wrapped in seq. A bang operator on an argument forces its evaluation to weak head normal form before performing the pattern match. This can be used to keep specific arguments evaluated throughout recursion instead of creating a giant chain of thunks.

This is desugared into code effectively equivalent to the following:

Function application to seq’d arguments is common enough that it has a special operator.

StrictData

As of GHC 8.0 strictness annotations can be applied to all definitions in a module automatically. In previous versions to make definitions strict it was necessary to use explicit syntactic annotations at call sites.

Enabling StrictData makes constructor fields strict by default on any module where the pragma is enabled:

Is equivalent to:

Strict

Strict implies -XStrictData and extends strictness annotations to all arguments of functions.

Is equivalent to the following function declaration with explicit bang patterns:

On a module-level this effectively makes Haskell a call-by-value language with some caveats. All arguments to functions are now explicitly evaluated and all data in constructors within this module are in head normal form by construction.

Deepseq

There are often times when for performance reasons we need to deeply evaluate a data structure to normal form leaving no terms unevaluated. The deepseq library performs this task.

The typeclass NFData (Normal Form Data) allows us to seq all elements of a structure across any subtypes which themselves implement NFData.

To force a data structure itself to be fully evaluated we share the same argument in both positions of deepseq.

Irrefutable Patterns

A lazy pattern doesn’t require a match on the outer constructor, instead it lazily calls the accessors of the values as needed. In the presence of a bottom, we fail at the usage site instead of the outer pattern match.

The Debate

Laziness is a controversial design decision in Haskell. It is difficult to write production Haskell code that operates in constant memory without some insight into the evaluation model and the runtime. A lot of industrial codebases have a policy of marking all constructors as strict by default or enabling StrictData to prevent space leaks. If Haskell were being designed from scratch it probably would not choose laziness as the default model. Future implementations of Haskell compilers would not choose this point in the design space if given the option of breaking with the language specification.

There is a lot of fear, uncertainty and doubt spread about lazy evaluation that unfortunately loses the forest for the trees and ignores 30 years of advanced research on the type system. In industrial programming a lot of software is sold on the meme of being of fast instead of being correct, and lazy evaluation is an intellectually easy talking point about these upside-down priorities. Nevertheless the colloquial perception of laziness being “evil” is a meme that will continue to persist regardless of any underlying reality because software is intrinsically a social process.

What to Avoid?

Haskell being a 30 year old language has witnessed several revolutions in the way we structure and compose functional programs. Yet as a result several portions of the Prelude still reflect old schools of thought that simply can’t be removed without breaking significant parts of the ecosystem.

Currently it really only exists in folklore which parts to use and which not to use, although this is a topic that almost all introductory books don’t mention and instead make extensive use of the Prelude for simplicity’s sake.

The short version of the advice on the Prelude is:

The instances of Foldable for the list type often conflict with the monomorphic versions in the Prelude which are left in for historical reasons. So oftentimes it is desirable to explicitly mask these functions from implicit import and force the use of Foldable and Traversable instead.

Of course oftentimes one wishes to only use the Prelude explicitly and one can explicitly import it qualified and use the pieces as desired without the implicit import of the whole namespace.

What Should be in Prelude

To get work done on industrial projects you probably need the following libraries:

• text
• containers
• unordered-containers
• mtl
• transformers
• vector
• filepath
• directory
• process
• bytestring
• optparse-applicative
• unix
• aeson

Custom Preludes

The default Prelude can be disabled in its entirety by twiddling the -XNoImplicitPrelude flag which allows us to replace the default import entirely with a custom prelude. Many industrial projects will roll their own Prologue.hs module which replaces the legacy prelude.

For example if we wanted to build up a custom project prelude we could construct a Prologue module and dump the relevant namespaces we want from base into our custom export list. Using the module reexport feature allows us to create an Exports namespace which contains our Prelude’s symbols. Every subsequent module in our project will then have import Prologue as the first import.

Preludes

There are many approaches to custom preludes. The most widely used ones are all available on Hackage.

• base-prelude
• rio
• protolude
• relude
• foundation
• rebase
• classy-prelude
• basic-prelude

Different preludes take different approaches to defining what the Haskell standard library should be. Some are interoperable with existing code and others require an “all-in” approach that creates an ecosystem around it. Some projects are more community efforts and others are developed by consulting companies or industrial users wishing to standardise their commercial code.

In Modern Haskell there are many different perspectives on Prelude design and the degree to which more advanced ideas should be used. Which one is right for you is a matter of personal preference and constraints in your company.

Protolude

Protolude is a minimalist Prelude which provides many sensible defaults for writing modern Haskell and is compatible with existing code.

Protolude is one of the more conservative preludes and is developed by the author of this document.

See:

• Protolude Hackage
• Protolude Github

Partial Functions

A partial function is a function which doesn’t terminate and yield a value for all given inputs. Conversely a total function terminates and is always defined for all inputs. As mentioned previously, certain historical parts of the Prelude are full of partial functions.

The difference between partial and total functions is the compiler can’t reason about the runtime safety of partial functions purely from the information specified in the language and as such the proof of safety is left to the user to guarantee. They are safe to use in the case where the user can guarantee that invalid inputs cannot occur, but like any unchecked property its safety or not-safety is going to depend on the diligence of the programmer. This very much goes against the overall philosophy of Haskell and as such they are discouraged when not necessary.

A list of partial functions in the default prelude:

Partial for all inputs

• error
• undefined
• fail – For Monad IO

Partial for empty lists

• head
• init
• tail
• last
• foldl
• foldr
• foldl'
• foldr'
• foldr1
• foldl1
• cycle
• maximum
• minimum

Partial for Nothing

• fromJust

Partial for invalid strings lists

• read

Partial for infinite lists

• sum
• product
• reverse

Partial for negative or unbounded numbers

• (!)
• (!!)
• toEnum
• genericIndex

Replacing Partiality

The Prelude has total variants of the historical partial functions (e.g. Text.Read.readMaybe) in some cases, but often these are found in the various replacement preludes

The total versions provided fall into three cases:

• May - return Nothing when the function is not defined for the inputs
• Def - provide a default value when the function is not defined for the inputs
• Note - call error with a custom error message when the function is not defined for the inputs. This is not safe, but slightly easier to debug!

Boolean Blindness

Boolean blindness is a common problem found in many programming languages. Consider the following two definitions which deconstruct a Maybe value into a boolean. Is there anything wrong with the definitions and below and why is this not caught in the type system?

The problem with the Bool type is that there is effectively no difference between True and False at the type level. A proposition taking a value to a Bool takes any information given and destroys it. To reason about the behavior we have to trace the provenance of the proposition we’re getting the boolean answer from, and this introduces a whole slew of possibilities for misinterpretation. In the worst case, the only way to reason about safe and unsafe use of a function is by trusting that a predicate’s lexical name reflects its provenance!

For instance, testing some proposition over a Bool value representing whether the branch can perform the computation safely in the presence of a null is subject to accidental interchange. Consider that in a language like C or Python testing whether a value is null is indistinguishable to the language from testing whether the value is not null. Which of these programs encodes safe usage and which segfaults?

From inspection we can’t tell without knowing how p is defined, the compiler can’t distinguish the two either and thus the language won’t save us if we happen to mix them up. Instead of making invalid states unrepresentable we’ve made the invalid state indistinguishable from the valid one!

The more desirable practice is to match on terms which explicitly witness the proposition as a type (often in a sum type) and won’t typecheck otherwise.

To be fair though, many popular languages completely lack the notion of sum types (the source of many woes in my opinion) and only have product types, so this type of reasoning sometimes has no direct equivalence for those not familiar with ML family languages.

In Haskell, the Prelude provides functions like isJust and fromJust both of which can be used to subvert this kind of reasoning and make it easy to introduce bugs and should often be avoided.

Foldable / Traversable

If coming from an imperative background retraining oneself to think about iteration over lists in terms of maps, folds, and scans can be challenging.

For a concrete example consider the simple arithmetic sequence over the binary operator (+):

Foldable and Traversable are the general interface for all traversals and folds of any data structure which is parameterized over its element type ( List, Map, Set, Maybe, …). These two classes are used everywhere in modern Haskell and are extremely important.

A foldable instance allows us to apply functions to data types of monoidal values that collapse the structure using some logic over mappend.

A traversable instance allows us to apply functions to data types that walk the structure left-to-right within an applicative context.

The foldMap function is extremely general and non-intuitively many of the monomorphic list folds can themselves be written in terms of this single polymorphic function.

foldMap takes a function of values to a monoidal quantity, a functor over the values and collapses the functor into the monoid. For instance for the trivial Sum monoid:

For instance if we wanted to map a list of some abstract element types into a hashtable of elements based on pattern matching we could use it.

The full Foldable class (with all default implementations) contains a variety of derived functions which themselves can be written in terms of foldMap and Endo.

For example:

Most of the operations over lists can be generalized in terms of combinations of Foldable and Traversable to derive more general functions that work over all data structures implementing Foldable.

Unfortunately for historical reasons the names exported by Foldable quite often conflict with ones defined in the Prelude, either import them qualified or just disable the Prelude. The operations in the Foldable class all specialize to the same and behave the same as the ones in Prelude for List types.

The instances we defined above can also be automatically derived by GHC using several language extensions. The automatic instances are identical to the hand-written versions above.

The string situation in Haskell is a sad affair. The default String type is defined as linked list of pointers to characters which is an extremely pathological and inefficient way of representing textual data. Unfortunately for historical reasons large portions of GHC and Base depend on String.

The String problem is intrinsically linked to the fact that the default GHC Prelude provides a set of broken defaults that are difficult to change because GHC and the entire ecosystem historically depend on it. There are however high performance string libraries that can swapped in for the broken String type and we will discuss some ways of working with high-performance and memory efficient replacements.

String

The default Haskell string type is implemented as a naive linked list of characters, this is hilariously terrible for most purposes but no one knows how to fix it without rewriting large portions of all code that exists, and simply nobody wants to commit the time to fix it. So it remains broken, likely forever.

However, fear not as there are are two replacement libraries for processing textual data: text and bytestring.

• text - Used for handling unicode data.
• bytestring - Used for handling ASCII data that needs to interchange with C code or network protocols.

For each of these there are two variants for both text and bytestring.

• lazy - Lazy text objects are encoded as lazy lists of strict chunks of bytes.
• strict - Byte vectors are encoded as strict Word8 arrays of bytes or code points

Giving rise to the Cartesian product of the four common string types:

String Conversions

Conversions between strings types are done with several functions across the bytestring and text libraries. The mapping between text and bytestring is inherently lossy so there is some degree of freedom in choosing the encoding. We’ll just consider utf-8 for simplicity.

(From : left column, To : top row) Data.Text Data.Text.Lazy Data.ByteString Data.ByteString.Lazy ——————— ——— ————– ————— —————— Data.Text id fromStrict encodeUtf8 encodeUtf8 Data.Text.Lazy toStrict id encodeUtf8 encodeUtf8 Data.ByteString decodeUtf8 decodeUtf8 id fromStrict Data.ByteString.Lazy decodeUtf8 decodeUtf8 toStrict id

Be careful with the functions (decodeUtf8, decodeUtf16LE, etc.) as they are partial and will throw errors if the byte array given does not contain unicode code points. Instead use one of the following functions which will allow you to explicitly handle the error case:

OverloadedStrings

With the -XOverloadedStrings extension string literals can be overloaded without the need for explicit packing and can be written as string literals in the Haskell source and overloaded via the typeclass IsString. Sometimes this is desirable.

For instance:

We can also derive IsString for newtypes using GeneralizedNewtypeDeriving, although much of the safety of the newtype is then lost if it is used interchangeable with other strings.

Import Conventions

Since there are so many modules that provide string datatypes, and these modules are used ubiquitously, some conventions are often adopted to import these modules as specific agreed-upon qualified names. In many Haskell projects you will see the following social conventions used for distinguish text types.

For datatypes:

For IO operations:

For encoding operations:

In addition many libraries and alternative preludes will define the following type synonyms:

Text

The Text type is a packed blob of Unicode characters.

See: Text

Text.Builder

The Text.Builder allows the efficient monoidal construction of lazy Text types without having to go through inefficient forms like String or List types as intermediates.

ByteString

ByteStrings are arrays of unboxed characters with either strict or lazy evaluation.

Printf

Haskell also has a variadic printf function in the style of C.

Overloaded Lists

It is ubiquitous for data structure libraries to expose toList and fromList functions to construct various structures out of lists. As of GHC 7.8 we now have the ability to overload the list syntax in the surface language with the typeclass IsList.

For example we could write an overloaded list instance for hash tables that simply converts to the hash table using fromList. Some math libraries that use vector-like structures will use overloaded lists in this fashion.

Regex

regex-tdfa implements POSIX extended regular expressions. These can operate over any of the major string types and with OverloadedStrings enabled allows you to write well-typed regex expressions as strings.

Escaping Text

Haskell uses C-style single-character escape codes

String Splitting

The split package provides a variety of missing functions for splitting list and string types.

Like monads Applicatives are an abstract structure for a wide class of computations that sit between functors and monads in terms of generality.

As of GHC 7.6, Applicative is defined as:

With the following laws:

As an example, consider the instance for Maybe:

As a rule of thumb, whenever we would use m >>= return . f what we probably want is an applicative functor, and not a monad.

import Control.Applicative ((<$>), (<*>))
import Network.HTTP

example1 :: Maybe Integer
example1 = (+) <$> m1 <*> m2
  where
    m1 = Just 3
    m2 = Nothing

-- Nothing

example2 :: [(Int, Int, Int)]
example2 = (,,) <$> m1 <*> m2 <*> m3
  where
    m1 = [1, 2]
    m2 = [10, 20]
    m3 = [100, 200]

-- [(1,10,100),(1,10,200),(1,20,100),(1,20,200),(2,10,100),(2,10,200),(2,20,100),(2,20,200)]

example3 :: IO String
example3 = (++) <$> fetch1 <*> fetch2
  where
    fetch1 = simpleHTTP (getRequest "http://www.python.org/") >>= getResponseBody
    fetch2 = simpleHTTP (getRequest "http://www.haskell.org/") >>= getResponseBody

The pattern f <$> a <*> b ... shows up so frequently that there is a family of functions to lift applicatives of a fixed number arguments. This pattern also shows up frequently with monads (liftM, liftM2, liftM3).

Applicative also has functions *> and <* that sequence applicative actions while discarding the value of one of the arguments. The operator *> discards the left while <* discards the right. For example in a monadic parser combinator library the *> would parse with first parser argument but return the second.

The Applicative functions <$> and <*> are generalized by liftM and ap for monads.

See: Applicative Programming with Effects

Alternative

Alternative is an extension of the Applicative class with a zero element and an associative binary operation respecting the zero.

These instances show up very frequently in parsers where the alternative operator can model alternative parse branches.

Arrows

A category is an algebraic structure that includes a notion of an identity and a composition operation that is associative and preserves identities. In practice arrows are not often used in modern Haskell and are often considered a code smell.

Arrows are an extension of categories with the notion of products.

The canonical example is for functions.

In this form, functions of multiple arguments can be threaded around using the arrow combinators in a much more pointfree form. For instance a histogram function has a nice one-liner.

λ: histogram "Hello world"
[(' ',1),('H',1),('d',1),('e',1),('l',3),('o',2),('r',1),('w',1)]

Arrow notation

GHC has builtin syntax for composing arrows using proc notation. The following are equivalent after desugaring:

In practice this notation is not often used and may become deprecated in the future.

See: Arrow Notation

Bifunctors

Bifunctors are a generalization of functors to include types parameterized by two parameters and include two map functions for each parameter.

The bifunctor laws are a natural generalization of the usual functor laws. Namely they respect identities and composition in the usual way:

The canonical example is for 2-tuples.

Polyvariadic Functions

One surprising application of typeclasses is the ability to construct functions which take an arbitrary number of arguments by defining instances over function types. The arguments may be of arbitrary type, but the resulting collected arguments must either be converted into a single type or unpacked into a sum type.

There are a plethora of ways of handling errors in Haskell. While Haskell’s runtime supports throwing and handling exceptions, it is important to use the right method in the right context.

Either Monad

In keeping with the Haskell tradition it is always preferable to use pure logic when possible. In many simple cases error handling can be done quite simply by using the Monad instance of Either. Monadic bind simply threads a Right value through the monad and “short-circuits” evaluation when a Left is introduced. This is simple enough error handling which privileges the Left constructor to hold the error. Many simple functions which can fail can simply use the Either Error a in the result type to encode simple error handling.

The downside to this is that it forces every consumer of the function to pattern match on the result to handle the error case. It also assumes that all Error types can be encoded inside of the sum type holding the possible failures.

ExceptT

When using the transformers style effect stacks it is quite common to need to have a layer of the stack which can fail. When using the style of composing effects a monad transformer (which is a wrapper around Either monad) can be added which lifts the error handling into an ExceptT effect layer.

As of mtl 2.2 or higher, the ErrorT class has been replaced by ExceptT at the transformers level.

And also this can be extended to the mtl MonadError instance for which we can write instances for IO and Either themselves:

See:

• Control.Monad.Except

Control.Exception

GHC has a builtin system for propagating errors up at the runtime level, below the business logic level. These are used internally for all sorts of concurrency and system interfaces. The runtime provides builtin operations throw and catch functions which allow us to throw exceptions in pure code and catch the resulting exception within IO. Note that the return value of throw inhabits all types.

Because a value will not be evaluated unless needed, if one desires to know for sure that an exception is either caught or not it can be deeply forced into head normal form before invoking catch. The strictCatch is not provided by the standard library but has a simple implementation in terms of deepseq.

Exceptions

The problem with the previous approach is having to rely on GHC’s asynchronous exception handling inside of IO to handle basic operations and the bifurcation of APIs which need to expose different APIs for any monad that has failure (IO, STM, ExceptT, etc.).

The exceptions package provides the same API as Control.Exception but loosens the dependency on IO. It instead provides a granular set of typeclasses which can operate over different monads which require a precise subset of error handling methods.

• MonadThrow - Monads which expose an interface for throwing exceptions.
• MonadCatch - Monads which expose an interface for handling exceptions.
• MonadMask - Monads which expose an interface for masking asynchronous exceptions.

There are three core primitives that are used in handling runtime exceptions:

• finally - For handling guaranteed finalisation of code in the presence of exceptions.
• onException - For handing exception case only if an exception is thrown.
• bracket - For implementing resource handling with custom acquisition and finalizer logic, in the presence of exceptions.

finally takes an IO action to run as a computation and a secondary function to run after the evaluation of the first.

onException has a similar signature but the second function is run only if an exception is raised.

The bracket function takes two functions, an acquisition function and a finalizer function which “bracket” the evaluation of the third. The finaliser will be run if the computation throwns an exception and unwinds.

A simple example of usage is bracket logic that handles file descriptors which need to be explicitly closed after evaluation is done. The initialiser in this case will return a file descriptor to the body and then run hClose on the file descriptor after the body is done with evaluation.

In addition the exceptions library exposes several functions for explicitly handling a variety of exceptions of various forms. Toplevel handlers that need to “catch em’ all” should use catchAny for wildcard error handling.

A simple example of usage:

See: exceptions

Spoon

Sometimes you’ll be forced to deal with seemingly pure functions that can throw up at any point. There are many functions in the standard library like this, and many more on Hackage. You’d like to handle this logic purely as if it were returning a proper Maybe a but to catch the logic you’d need to install an IO handler inside IO to catch it. Spoon allows us to safely (and “purely”, although it uses a referentially transparent invocation of unsafePerformIO) to catch these exceptions and put them in Maybe where they belong.

The spoon function evaluates its argument to head normal form, while teaspoon evaluates to weak head normal form.

When working with the wider library you will find there a variety of “advanced monads” which are higher-level constructions on top of of the monadic interface which enrich the structure with additional rules or build APIs for combining different types of monads. Some of the most-used cases are mentioned in this section.

Function Monad

If one writes Haskell long enough one might eventually encounter the curious beast that is the ((->) r) monad instance. It generally tends to be non-intuitive to work with, but is quite simple when one considers it as an unwrapped Reader monad.

This just uses a prefix form of the arrow type operator.

RWS Monad

The RWS monad combines the functionality of the three monads discussed above, the Reader, Writer, and State. There is also a RWST transformer.

These three eval functions are now combined into the following functions:

The usual caveat about Writer laziness also applies to RWS.

Cont

In continuation passing style, composite computations are built up from sequences of nested computations which are terminated by a final continuation which yields the result of the full computation by passing a function into the continuation chain.

• MonadCont Under the Hood

MonadPlus

Choice and failure.

MonadPlus forms a monoid with

MonadFail

Before the great awakening, Monads used to be defined as the following class.

This was eventually deemed not to be an great design and in particular the fail function was a misplaced lawless entity that would generate bottoms. It was also necessary to define fail for all monads, even those without a notion of failure. This was considered quite ugly and eventually a breaking change to base (landed in 4.9) was added which split out MonadFail into a separate class where it belonged.

Some of the common instances of MonadFail are shown below:

MonadFix

The fixed point of a monadic computation. mfix f executes the action f only once, with the eventual output fed back as the input.

The regular do-notation can also be extended with -XRecursiveDo to accommodate recursive monadic bindings.

ST Monad

The ST monad models “threads” of stateful computations which can manipulate mutable references but are restricted to only return pure values when evaluated and are statically confined to the ST monad of a s thread.

Using the ST monad we can create a class of efficient purely functional data structures that use mutable references in a referentially transparent way.

Free Monads

Free monads are monads which instead of having a join operation that combines computations, instead forms composite computations from application of a functor.

One of the best examples is the Partiality monad which models computations which can diverge. Haskell allows unbounded recursion, but for example we can create a free monad from the Maybe functor which can be used to fix the call-depth of, for example the Ackermann function.

The other common use for free monads is to build embedded domain-specific languages to describe computations. We can model a subset of the IO monad by building up a pure description of the computation inside of the IOFree monad and then using the free monad to encode the translation to an effectful IO computation.

An implementation such as the one found in free might look like the following:

Indexed Monads

Indexed monads are a generalisation of monads that adds an additional type parameter to the class that carries information about the computation or structure of the monadic implementation.

The canonical use-case is a variant of the vanilla State which allows type-changing on the state for intermediate steps inside of the monad. This indeed turns out to be very useful for handling a class of problems involving resource management since the extra index parameter gives us space to statically enforce the sequence of monadic actions by allowing and restricting certain state transitions on the index parameter at compile-time.

To make this more usable we’ll use the somewhat esoteric -XRebindableSyntax allowing us to overload the do-notation and if-then-else syntax by providing alternative definitions local to the module.

{-# LANGUAGE RebindableSyntax #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE NoMonomorphismRestriction #-}

import Data.IORef
import Data.Char
import Prelude hiding (fmap, (>>=), (>>), return)
import Control.Applicative

newtype IState i o a = IState { runIState :: i -> (a, o) }

evalIState :: IState i o a -> i -> a
evalIState st i = fst $ runIState st i

execIState :: IState i o a -> i -> o
execIState st i = snd $ runIState st i

ifThenElse :: Bool -> a -> a -> a
ifThenElse b i j = case b of
  True -> i
  False -> j

return :: a -> IState s s a
return a = IState $ s -> (a, s)

fmap :: (a -> b) -> IState i o a -> IState i o b
fmap f v = IState $ i -> let (a, o) = runIState v i
                          in (f a, o)

join :: IState i m (IState m o a) -> IState i o a
join v = IState $ i -> let (w, m) = runIState v i
                        in runIState w m

(>>=) :: IState i m a -> (a -> IState m o b) -> IState i o b
v >>= f = IState $ i -> let (a, m) = runIState v i
                         in runIState (f a) m

(>>) :: IState i m a -> IState m o b -> IState i o b
v >> w = v >>= _ -> w

get :: IState s s s
get = IState $ s -> (s, s)

gets :: (a -> o) -> IState a o a
gets f = IState $ s -> (s, f s)

put :: o -> IState i o ()
put o = IState $ _ -> ((), o)

modify :: (i -> o) -> IState i o ()
modify f = IState $ i -> ((), f i)



data Locked = Locked
data Unlocked = Unlocked

type Stateful a = IState a Unlocked a

acquire :: IState i Locked ()
acquire = put Locked

-- Can only release the lock if it's held, try release the lock
-- that's not held is a now a type error.
release :: IState Locked Unlocked ()
release = put Unlocked

-- Statically forbids improper handling of resources.
lockExample :: Stateful a
lockExample = do
  ptr <- get  :: IState a a a
  acquire     :: IState a Locked ()
  -- ...
  release     :: IState Locked Unlocked ()
  return ptr

-- Couldn't match type `Locked' with `Unlocked'
-- In a stmt of a 'do' block: return ptr
failure1 :: Stateful a
failure1 = do
  ptr <- get
  acquire
  return ptr -- didn't release

-- Couldn't match type `a' with `Locked'
-- In a stmt of a 'do' block: release
failure2 :: Stateful a
failure2 = do
  ptr <- get
  release -- didn't acquire
  return ptr

-- Evaluate the resulting state, statically ensuring that the
-- lock is released when finished.
evalReleased :: IState i Unlocked a -> i -> a
evalReleased f st = evalIState f st

example :: IO (IORef Integer)
example = evalReleased <$> pure lockExample <*> newIORef 0

Lifted Base

The default prelude predates a lot of the work on monad transformers and as such many of the common functions for handling errors and interacting with IO are bound strictly to the IO monad and not to functions implementing stacks on top of IO or ST. The lifted-base provides generic control operations such as catch can be lifted from IO or any other base monad.

monad-base

Monad base provides an abstraction over liftIO and other functions to explicitly lift into a “privileged” layer of the transformer stack. It’s implemented as a multiparameter typeclass with the “base” monad as the parameter b.

monad-control

Monad control builds on top of monad-base to extended lifting operation to control operations like catch and bracket can be written generically in terms of any transformer with a base layer supporting these operations. Generic operations can then be expressed in terms of a MonadBaseControl and written in terms of the combinator control which handles the bracket and automatic handler lifting.

For example the function catch provided by Control.Exception is normally locked into IO.

catch :: Exception e => IO a -> (e -> IO a) -> IO a

By composing it in terms of control we can construct a generic version which automatically lifts inside of any combination of the usual transformer stacks that has MonadBaseControl instance.

In logic a predicate is a statement about a subject. For instance the statement: Socrates is a man, can be written as:

Man(Socrates)

A predicate assigned to a variable Man(x) has a truth value if the predicate holds for the subject. The domain of a variable is the set of all variables that may be assigned to the variable. A quantifier turns predicates into propositions by assigning values to all variables. For example the statement: All men are mortal. This is an example of a universal quantifier which describe a predicate that holds forall inhabitants of the domain of variables.

Forall x. If Man(x) then Mortal(x)

The truth value that that Socrates is mortal can be derived from above relation. Programming with quantifiers in Haskell follows this same kind of logical convention except we will be working with types and constraints on types.

Universal Quantification

Universal quantification the primary mechanism of encoding polymorphism in Haskell. The essence of universal quantification is that we can express functions which operate the same way for a set of types and whose function behavior is entirely determined only by the behavior of all types in this span. These are represented at the type-level by in the introduction of a universal quantifier (forall or ∀) over a set of the type variables in the signature.

Normally quantifiers are omitted in type signatures since in Haskell’s vanilla surface language it is unambiguous to assume to that free type variables are universally quantified. So the following two are equivalent:

Free Theorems

A universally quantified type-variable actually implies quite a few rather deep properties about the implementation of a function that can be deduced from its type signature. For instance the identity function in Haskell is guaranteed to only have one implementation since the only information that the information that can present in the body:

These so called free theorems are properties that hold for any well-typed inhabitant of a universally quantified signature.

For example a free theorem of fmap is that every implementation of functor can only ever have the property that composition of maps of functions is the same as maps of the functions composed together.

Type Systems

Hindley-Milner type system

The Hindley-Milner type system is historically important as one of the first typed lambda calculi that admitted both polymorphism and a variety of inference techniques that could always decide principal types.

In an type checker implementation, a generalize function converts all type variables within the type into polymorphic type variables yielding a type scheme. While a instantiate function maps a scheme to a type, but with any polymorphic variables converted into unbound type variables.

Rank-N Types

System-F is the type system that underlies Haskell. System-F subsumes the HM type system in the sense that every type expressible in HM can be expressed within System-F. System-F is sometimes referred to in texts as the Girald-Reynolds polymorphic lambda calculus or second-order lambda calculus.

An example with equivalents of GHC Core in comments:

Normally when Haskell’s typechecker infers a type signature it places all quantifiers of type variables at the outermost position such that no quantifiers appear within the body of the type expression, called the prenex restriction. This restricts an entire class of type signatures that would otherwise be expressible within System-F, but has the benefit of making inference much easier.

-XRankNTypes loosens the prenex restriction such that we may explicitly place quantifiers within the body of the type. The bad news is that the general problem of inference in this relaxed system is undecidable in general, so we’re required to explicitly annotate functions which use RankNTypes or they are otherwise inferred as rank 1 and may not typecheck at all.

Of important note is that the type variables bound by an explicit quantifier in a higher ranked type may not escape their enclosing scope. The typechecker will explicitly enforce this by enforcing that variables bound inside of rank-n types (called skolem constants) will not unify with free meta type variables inferred by the inference engine.

In this example in order for the expression to be well typed, f would necessarily have (Int -> Int) which implies that a ~ Int over the whole type, but since a is bound under the quantifier it must not be unified with Int and so the typechecker must fail with a skolem capture error.

This can actually be used for our advantage to enforce several types of invariants about scope and use of specific type variables. For example the ST monad uses a second rank type to prevent the capture of references between ST monads with separate state threads where the s type variable is bound within a rank-2 type and cannot escape, statically guaranteeing that the implementation details of the ST internals can’t leak out and thus ensuring its referential transparency.

Existential Quantification

An existential type is a pair of a type and a term with a special set of packing and unpacking semantics. The type of the value encoded in the existential is known by the producer but not by the consumer of the existential value.

The existential over SBox gathers a collection of values defined purely in terms of their Show interface and an opaque pointer, no other information is available about the values and they can’t be accessed or unpacked in any other way.

Passing around existential types allows us to hide information from consumers of data types and restrict the behavior that functions can use. Passing records around with existential variables allows a type to be “bundled” with a fixed set of functions that operate over its hidden internals.

Impredicative Types

Although extremely brittle, GHC also has limited support for impredicative polymorphism which allows instantiating type variable with a polymorphic type. Implied is that this loosens the restriction that quantifiers must precede arrow types and now they may be placed inside of type-constructors.

{-# LANGUAGE ImpredicativeTypes #-}

-- Uses higher-ranked polymorphism.
f :: (forall a. [a] -> a) -> (Int, Char)
f get = (get [1,2], get ['a', 'b', 'c'])

-- Uses impredicative polymorphism.
g :: Maybe (forall a. [a] -> a) -> (Int, Char)
g Nothing = (0, '0')
g (Just get) = (get [1,2], get ['a','b','c'])

Use of this extension is very rare, and there is some consideration that -XImpredicativeTypes is fundamentally broken. Although GHC is very liberal about telling us to enable it when one accidentally makes a typo in a type signature!

Some notable trivia, the ($) operator is wired into GHC in a very special way as to allow impredicative instantiation of runST to be applied via ($) by special-casing the ($) operator only when used for the ST monad.

For example if we define a function apply which should behave identically to ($) we’ll get an error about polymorphic instantiation even though they are defined identically!

See:

• SPJ Notes on $

Scoped Type Variables

Normally the type variables used within the toplevel signature for a function are only scoped to the type-signature and not the body of the function and its rigid signatures over terms and let/where clauses. Enabling -XScopedTypeVariables loosens this restriction allowing the type variables mentioned in the toplevel to be scoped within the value-level body of a function and all signatures contained therein.

Generalized Algebraic Data types (GADTs) are an extension to algebraic datatypes that allow us to qualify the constructors to datatypes with type equality constraints, allowing a class of types that are not expressible using vanilla ADTs.

-XGADTs implicitly enables an alternative syntax for datatype declarations ( -XGADTSyntax ) such that the following declarations are equivalent:

For an example use consider the data type Term, we have a term in which we Succ which takes a Term parameterized by a which spans all types. Problems arise between the clash whether (a ~ Bool) or (a ~ Int) when trying to write the evaluator.

And we admit the construction of meaningless terms which forces more error handling cases.

Using a GADT we can express the type invariants for our language (i.e. only type-safe expressions are representable). Pattern matching on this GADT then carries type equality constraints without the need for explicit tags.

This time around:

Explicit equality constraints (a ~ b) can be added to a function’s context. For example the following expand out to the same types.

This is effectively the implementation detail of what GHC is doing behind the scenes to implement GADTs ( implicitly passing and threading equality terms around ). If we wanted we could do the same setup that GHC does just using equality constraints and existential quantification. Indeed, the internal representation of GADTs is as regular algebraic datatypes that carry coercion evidence as arguments.

In the presence of GADTs inference becomes intractable in many cases, often requiring an explicit annotation. For example f can either have T a -> [a] or T a -> [Int] and neither is principal.

Kind Signatures

Haskell’s kind system (i.e. the “type of the types”) is a system consisting the single kind * and an arrow kind ->.

There are in fact some extensions to this system that will be covered later ( see: PolyKinds and Unboxed types in later sections ) but most kinds in everyday code are simply either stars or arrows.

With the KindSignatures extension enabled we can now annotate top level type signatures with their explicit kinds, bypassing the normal kind inference procedures.

On top of default GADT declaration we can also constrain the parameters of the GADT to specific kinds. For basic usage Haskell’s kind inference can deduce this reasonably well, but combined with some other type system extensions that extend the kind system this becomes essential.

Void

The Void type is the type with no inhabitants. It unifies only with itself.

Using a newtype wrapper we can create a type where recursion makes it impossible to construct an inhabitant.

Or using -XEmptyDataDecls we can also construct the uninhabited type equivalently as a data declaration with no constructors.

The only inhabitant of both of these types is a diverging term like (undefined).

Phantom Types

Phantom types are parameters that appear on the left hand side of a type declaration but which are not constrained by the values of the types inhabitants. They are effectively slots for us to encode additional information at the type-level.

Notice the type variable tag does not appear in the right hand side of the declaration. Using this allows us to express invariants at the type-level that need not manifest at the value-level. We’re effectively programming by adding extra information at the type-level.

Consider the case of using newtypes to statically distinguish between plaintext and cryptotext.

Using phantom types we use an extra parameter.

Using -XEmptyDataDecls can be a powerful combination with phantom types that contain no value inhabitants and are “anonymous types”.

The tagged library defines a similar Tagged newtype wrapper.

Typelevel Operations

With a richer language for datatypes we can express terms that witness the relationship between terms in the constructors, for example we can now express a term which expresses propositional equality between two types.

The type Eql a b is a proof that types a and b are equal, by pattern matching on the single Refl constructor we introduce the equality constraint into the body of the pattern match.

As of GHC 7.8 these constructors and functions are included in the Prelude in the Data.Type.Equality module.

The lambda calculus forms the theoretical and practical foundation for many languages. At the heart of every calculus is three components:

• Var - A variable
• Lam - A lambda abstraction
• App - An application

There are many different ways of modeling these constructions and data structure representations, but they all more or less contain these three elements. For example, a lambda calculus that uses String names on lambda binders and variables might be written like the following:

A lambda expression in which all variables that appear in the body of the expression are referenced in an outer lambda binder is said to be closed while an expression with unbound free variables is open.

HOAS

Higher Order Abstract Syntax (HOAS) is a technique for implementing the lambda calculus in a language where the binders of the lambda expression map directly onto lambda binders of the host language ( i.e. Haskell ) to give us substitution machinery in our custom language by exploiting Haskell’s implementation.

Pretty printing HOAS terms can also be quite complicated since the body of the function is under a Haskell lambda binder.

PHOAS

A slightly different form of HOAS called PHOAS uses lambda datatype parameterized over the binder type. In this form evaluation requires unpacking into a separate Value type to wrap the lambda expression.

See:

• PHOAS
• Encoding Higher-Order Abstract Syntax with Parametric Polymorphism

Final Interpreters

Using typeclasses we can implement a final interpreter which models a set of extensible terms using functions bound to typeclasses rather than data constructors. Instances of the typeclass form interpreters over these terms.

For example we can write a small language that includes basic arithmetic, and then retroactively extend our expression language with a multiplication operator without changing the base. At the same time our interpreter logic remains invariant under extension with new expressions.

Finally Tagless

Writing an evaluator for the lambda calculus can likewise also be modeled with a final interpreter and a Identity functor.

See: Typed Tagless Interpretations and Typed Compilation

Datatypes

The usual hand-wavy way of describing algebraic datatypes is to indicate the how natural correspondence between sum types, product types, and polynomial expressions arises.

Intuitively it follows the notion that the cardinality of set of inhabitants of a type can always be given as a function of the number of its holes. A product type admits a number of inhabitants as a function of the product (i.e. cardinality of the Cartesian product), a sum type as the sum of its holes and a function type as the exponential of the span of the domain and codomain.

Recursive types correspond to infinite series of these terms.

F-Algebras

The initial algebra approach differs from the final interpreter approach in that we now represent our terms as algebraic datatypes and the interpreter implements recursion and evaluation occurs through pattern matching.

In Haskell a F-algebra is a functor f a together with a function f a -> a. A coalgebra reverses the function. For a functor f we can form its recursive unrolling using the recursive Fix newtype wrapper.

In this form we can write down a generalized fold/unfold function that are datatype generic and written purely in terms of the recursing under the functor.

We call these functions catamorphisms and anamorphisms. Notice especially that the types of these two functions simply reverse the direction of arrows. Interpreted in another way they transform an algebra/coalgebra which defines a flat structure-preserving mapping between Fix f f into a function which either rolls or unrolls the fixpoint. What is particularly nice about this approach is that the recursion is abstracted away inside the functor definition and we are free to just implement the flat transformation logic!

For example a construction of the natural numbers in this form:

Or for example an interpreter for a small expression language that depends on a scoping dictionary.

What is especially elegant about this approach is how naturally catamorphisms compose into efficient composite transformations.

Recursion Schemes & The Morphism Zoo

Recursion schemes are a generally way of classifying a families of traversal algorithms that modify data structures recursively. Recursion schemes give rise to a rich set of algebraic structures which can be composed to devise all sorts of elaborate term rewrite systems. Most applications of recursion schemes occur in the context of graph rewriting or abstract syntax tree manipulation.

Several basic recursion schemes form the foundation of these rules. Grossly, a anamorphism is an unfolding of a data structure into a list of terms, while a catamorphism is a is the refolding of a data structure from a list of terms.

For a Fix point type over a type with a Functor instance for the parameter f we can write down the recursion schemes as the following definitions:

One can also construct monadic versions of these functions which have a result type inside of a monad. Instead of using function composition we use Kleisi composition.

The library recursion-schemes implements these basic recursion schemes as well as whole family of higher-order combinators off the shelf. These are implemented in terms of two typeclases Recursive and Corecursive which extend an instance of Functor with default methods for catamorphisms and anamorphisms. For the Fix type above these functions expand into the following definitions:

The canonical example of a catamorphism is the factorial function which is a composition of a coalgebra which creates a list from n to 1 and an algebra which multiplies the resulting list to a single result:

Another example is unfolding of lambda calculus to perform a substitution over a variable. We can define a catamoprhism for traversing over the AST.

Another use case would be to collect the free variables inside of the AST. This example use the recursion-schemes library.

See:

• recursion-schemes

Hint and Mueval

GHC itself can actually interpret arbitrary Haskell source on the fly by hooking into the GHC’s bytecode interpreter ( the same used for GHCi ). The hint package allows us to parse, typecheck, and evaluate arbitrary strings into arbitrary Haskell programs and evaluate them.

This is generally not a wise thing to build a library around, unless of course the purpose of the program is itself to evaluate arbitrary Haskell code ( something like an online Haskell shell or the likes ).

Both hint and mueval do effectively the same task, designed around slightly different internals of the GHC Api.

See:

• hint
• mueval

Unit testing frameworks are an important component in the Haskell ecosystem. Program correctness is a central philosophical concept and unit testing forms the third part of the ecosystem that includes strong type system and property testing. Generally speaking unit tests tend to be of less importance in Haskell since the type system makes an enormous amount of invalid programs completely inexpressible by construction. Unit tests tend to be written later in the development lifecycle and generally tend to be about the core logic of the program and not the intermediate plumbing.

A prominent school of thought on Haskell library design tends to favor constructing programs built around strong equational laws which guarantee strong invariants about program behavior under composition. Many of the testing tools are built around this style of design.

QuickCheck

Probably the most famous Haskell library, QuickCheck is a testing framework. This is a framework for generating large random tests for arbitrary functions automatically based on the types of their arguments.

The test data generator can be extended with custom types and refined with predicates that restrict the domain of cases to test.

import Test.QuickCheck

data Color = Red | Green | Blue deriving Show

instance Arbitrary Color where
  arbitrary = do
    n <- choose (0,2) :: Gen Int
    return $ case n of
      0 -> Red
      1 -> Green
      2 -> Blue

example1 :: IO [Color]
example1 = sample' arbitrary
-- [Red,Green,Red,Blue,Red,Red,Red,Blue,Green,Red,Red]

See: QuickCheck: An Automatic Testing Tool for Haskell

SmallCheck

Like QuickCheck, SmallCheck is a property testing system but instead of producing random arbitrary test data it instead enumerates a deterministic series of test data to a fixed depth.

λ: list 3 series :: [Int]
[0,1,-1,2,-2,3,-3]

λ: list 3 series :: [Double]
[0.0,1.0,-1.0,2.0,0.5,-2.0,4.0,0.25,-0.5,-4.0,-0.25]

λ: list 3 series :: [(Int, String)]
[(0,""),(1,""),(0,"a"),(-1,""),(0,"b"),(1,"a"),(2,""),(1,"b"),(-1,"a"),(-2,""),(-1,"b"),(2,"a"),(-2,"a"),(2,"b"),(-2,"b")]

It is useful to generate test cases over all possible inputs of a program up to some depth.

Just like for QuickCheck we can implement series instances for our custom datatypes. For example there is no default instance for Vector, so let’s implement one:

SmallCheck can also use Generics to derive Serial instances, for example to enumerate all trees of a certain depth we might use:

QuickSpec

Using the QuickCheck arbitrary machinery we can also rather remarkably enumerate a large number of combinations of functions to try and deduce algebraic laws from trying out inputs for small cases. Of course the fundamental limitation of this approach is that a function may not exhibit any interesting properties for small cases or for simple function compositions. So in general case this approach won’t work, but practically it still quite useful.

{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeOperators #-}

import Data.List
import Data.Typeable
import QuickSpec hiding (arith, bools, lists)
import Test.QuickCheck.Arbitrary

type Var k a = (Typeable a, Arbitrary a, CoArbitrary a, k a)

listCons :: forall a. Var Ord a => a -> Sig
listCons a =
  background
    [ "[]" `fun0` ([] :: [a]),
      ":" `fun2` ((:) :: a -> [a] -> [a])
    ]

lists :: forall a. Var Ord a => a -> [Sig]
lists a =
  [ -- Names to print arbitrary variables
    funs',
    funvars',
    vars',
    -- Ambient definitions
    listCons a,
    -- Expressions to deduce properties of
    "sort" `fun1` (sort :: [a] -> [a]),
    "map" `fun2` (map :: (a -> a) -> [a] -> [a]),
    "id" `fun1` (id :: [a] -> [a]),
    "reverse" `fun1` (reverse :: [a] -> [a]),
    "minimum" `fun1` (minimum :: [a] -> a),
    "length" `fun1` (length :: [a] -> Int),
    "++" `fun2` ((++) :: [a] -> [a] -> [a])
  ]
  where
    funs' = funs (undefined :: a)
    funvars' = vars ["f", "g", "h"] (undefined :: a -> a)
    vars' = ["xs", "ys", "zs"] `vars` (undefined :: [a])

tvar :: A
tvar = undefined

main :: IO ()
main = quickSpec (lists tvar)

Running this we rather see it is able to deduce most of the laws for list functions.

Keep in mind the rather remarkable fact that this is all deduced automatically from the types alone!

Tasty

Tasty is the commonly used unit testing framework. It combines all of the testing frameworks (Quickcheck, SmallCheck, HUnit) into a common API for forming runnable batches of tests and collecting the results.

Silently

Often in the process of testing IO heavy code we’ll need to redirect stdout to compare it some known quantity. The silently package allows us to capture anything done to stdout across any library inside of IO block and return the result to the test runner.

Type families are a powerful extension the Haskell type system, developed in 2005, that provide type-indexed data types and named functions on types. This allows a whole new level of computation to occur at compile-time and opens an entire arena of type-level abstractions that were previously impossible to express. Type families proved to be nearly as fruitful as typeclasses and indeed, many previous approaches to type-level programming using classes are achieved much more simply with type families.

MultiParam Typeclasses

Resolution of vanilla Haskell 98 typeclasses proceeds via very simple context reduction that minimizes interdependency between predicates, resolves superclasses, and reduces the types to head normal form. For example:

If a single parameter typeclass expresses a property of a type ( i.e. whether it’s in a class or not in class ) then a multiparameter typeclass expresses relationships between types. For example if we wanted to express the relation that a type can be converted to another type we might use a class like:

Of course now our instances for Convertible Int are not unique anymore, so there no longer exists a nice procedure for determining the inferred type of b from just a. To remedy this let’s add a functional dependency a -> b, which tells GHC that an instance a uniquely determines the instance that b can be. So we’ll see that our two instances relating Int to both Integer and Char conflict.

Now there’s a simpler procedure for determining instances uniquely and multiparameter typeclasses become more usable and inferable again. Effectively a functional dependency | a -> b says that we can’t define multiple multiparamater typeclass instances with the same a but different b.

Now let’s make things not so simple. Turning on UndecidableInstances loosens the constraint on context reduction that can only allow constraints of the class to become structural smaller than its head. As a result implicit computation can now occur within in the type class instance search. Combined with a type-level representation of Peano numbers we find that we can encode basic arithmetic at the type-level.

If the typeclass contexts look similar to Prolog you’re not wrong, if one reads the contexts qualifier (=>) backwards as turnstiles :- then it’s precisely the same equations.

This is kind of abusing typeclasses and if used carelessly it can fail to terminate or overflow at compile-time. UndecidableInstances shouldn’t be turned on without careful forethought about what it implies.

Type Families

Type families allows us to write functions in the type domain which take types as arguments which can yield either types or values indexed on their arguments which are evaluated at compile-time in during typechecking. Type families come in two varieties: data families and type synonym families.

• type families are named function on types
• data families are type-indexed data types

First let’s look at type synonym families, there are two equivalent syntactic ways of constructing them. Either as associated type families declared within a typeclass or as standalone declarations at the toplevel. The following forms are semantically equivalent, although the unassociated form is strictly more general:

Using the same example we used for multiparameter + functional dependencies illustration we see that there is a direct translation between the type family approach and functional dependencies. These two approaches have the same expressive power.

An associated type family can be queried using the :kind! command in GHCi.

Data families on the other hand allow us to create new type parameterized data constructors. Normally we can only define typeclasses functions whose behavior results in a uniform result which is purely a result of the typeclasses arguments. With data families we can allow specialized behavior indexed on the type.

For example if we wanted to create more complicated vector structures ( bit-masked vectors, vectors of tuples, … ) that exposed a uniform API but internally handled the differences in their data layout we can use data families to accomplish this:

Injectivity

The type level functions defined by type-families are not necessarily injective, the function may map two distinct input types to the same output type. This differs from the behavior of type constructors ( which are also type-level functions ) which are injective.

For example for the constructor Maybe, Maybe t1 = Maybe t2 implies that t1 = t2.

Roles

Roles are a further level of specification for type variables parameters of datatypes.

• nominal
• representational
• phantom

They were added to the language to address a rather nasty and long-standing bug around the correspondence between a newtype and its runtime representation. The fundamental distinction that roles introduce is there are two notions of type equality. Two types are nominally equal when they have the same name. This is the usual equality in Haskell or Core. Two types are representationally equal when they have the same representation. (If a type is higher-kinded, all nominally equal instantiations lead to representationally equal types.)

• nominal - Two types are the same.
• representational - Two types have the same runtime representation.

Roles are normally inferred automatically, but with the RoleAnnotations extension they can be manually annotated. Except in rare cases this should not be necessary although it is helpful to know what is going on under the hood.

With:

See:

• Data.Coerce
• Roles
• Roles: A New Feature of GHC

NonEmpty

Rather than having degenerate (and often partial) cases of many of the Prelude functions to accommodate the null case of lists, it is sometimes preferable to statically enforce empty lists from even being constructed as an inhabitant of a type.

Manual Proofs

One of most deep results in computer science, the Curry–Howard correspondence, is the relation that logical propositions can be modeled by types and instantiating those types constitute proofs of these propositions. Programs are proofs and proofs are programs.

In dependently typed languages we can exploit this result to its full extent, in Haskell we don’t have the strength that dependent types provide but can still prove trivial results. For example, now we can model a type level function for addition and provide a small proof that zero is an additive identity.

Translated into Haskell our axioms are simply type definitions and recursing over the inductive datatype constitutes the inductive step of our proof.

Using the TypeOperators extension we can also use infix notation at the type-level.

Constraint Kinds

GHC’s implementation also exposes the predicates that bound quantifiers in Haskell as types themselves, with the -XConstraintKinds extension enabled. Using this extension we work with constraints as first class types.

The empty constraint set is indicated by () :: Constraint.

For a contrived example if we wanted to create a generic Sized class that carried with it constraints on the elements of the container in question we could achieve this quite simply using type families.

One use-case of this is to capture the typeclass dictionary constrained by a function and reify it as a value.

TypeFamilyDependencies

Type families historically have not been injective, i.e. they are not guaranteed to maps distinct elements of its arguments to the same element of its result. The syntax is similar to the multiparmater typeclass functional dependencies in that the resulting type is uniquely determined by a set of the type families parameters.

See:

• Injective type families for Haskell

Higher Kinded Types

What are higher kinded types?

The kind system in Haskell is unique by contrast with most other languages in that it allows datatypes to be constructed which take types and type constructor to other types. Such a system is said to support higher kinded types.

All kind annotations in Haskell necessarily result in a kind * although any terms to the left may be higher-kinded (* -> *).

The common example is the Monad which has kind * -> *. But we have also seen this higher-kindedness in free monads.

For instance Cofree Maybe a for some monokinded type a models a non-empty list with Maybe :: * -> *.

Kind Polymorphism

The regular value level function which takes a function and applies it to an argument is universally generalized over in the usual Hindley-Milner way.

But when we do the same thing at the type-level we see we lose information about the polymorphism of the constructor applied.

Turning on -XPolyKinds allows polymorphic variables at the kind level as well.

Using the polykinded Proxy type allows us to write down type class functions over constructors of arbitrary kind arity.

For example we can write down the polymorphic S K combinators at the type level now.

Data Kinds

The -XDataKinds extension allows us to refer to constructors at the value level and the type level. Consider a simple sum type:

With the extension enabled we see that our type constructors are now automatically promoted so that L or R can be viewed as both a data constructor of the type S or as the type L with kind S.

Promoted data constructors can referred to in type signatures by prefixing them with a single quote. Also of importance is that these promoted constructors are not exported with a module by default, but type synonym instances can be created for the ticked promoted types and exported directly.

Combining this with type families we see we can write meaningful, type-level functions by lifting types to the kind level.

Size-Indexed Vectors

Using this new structure we can create a Vec type which is parameterized by its length as well as its element type now that we have a kind language rich enough to encode the successor type in the kind signature of the generalized algebraic datatype.

{-# LANGUAGE GADTs #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE FlexibleContexts #-}

data Nat = Z | S Nat deriving (Eq, Show)

type Zero  = Z
type One   = S Zero
type Two   = S One
type Three = S Two
type Four  = S Three
type Five  = S Four

data Vec :: Nat -> * -> * where
  Nil :: Vec Z a
  Cons :: a -> Vec n a -> Vec (S n) a

instance Show a => Show (Vec n a) where
  show Nil         = "Nil"
  show (Cons x xs) = "Cons " ++ show x ++ " (" ++ show xs ++ ")"

class FromList n where
  fromList :: [a] -> Vec n a

instance FromList Z where
  fromList [] = Nil

instance FromList n => FromList (S n) where
  fromList (x:xs) = Cons x $ fromList xs


lengthVec :: Vec n a -> Nat
lengthVec Nil = Z
lengthVec (Cons x xs) = S (lengthVec xs)

zipVec :: Vec n a -> Vec n b -> Vec n (a,b)
zipVec Nil Nil = Nil
zipVec (Cons x xs) (Cons y ys) = Cons (x,y) (zipVec xs ys)

vec4 :: Vec Four Int
vec4 = fromList [0, 1, 2, 3]

vec5 :: Vec Five Int
vec5 = fromList [0, 1, 2, 3, 4]


example1 :: Nat
example1 = lengthVec vec4
-- S (S (S (S Z)))

example2 :: Vec Four (Int, Int)
example2 = zipVec vec4 vec4
-- Cons (0,0) (Cons (1,1) (Cons (2,2) (Cons (3,3) (Nil))))

So now if we try to zip two Vec types with the wrong shape then we get an error at compile-time about the off-by-one error.

The same technique we can use to create a container which is statically indexed by an empty or non-empty flag, such that if we try to take the head of an empty list we’ll get a compile-time error, or stated equivalently we have an obligation to prove to the compiler that the argument we hand to the head function is non-empty.

See:

• Giving Haskell a Promotion

Typelevel Numbers

GHC’s type literals can also be used in place of explicit Peano arithmetic.

GHC 7.6 is very conservative about performing reduction, GHC 7.8 is much less so and will can solve many typelevel constraints involving natural numbers but sometimes still needs a little coaxing.

See: Type-Level Literals

Typelevel Strings

Since GHC 8.0 we have been able to work with typelevel strings values represented at the typelevel as Symbol with kind Symbol. The GHC.TypeLits module defines a set of a typeclases for lifting these values to and from the value level and comparing and computing over the values at typelevel.

These can be used to tag specific data at the typelevel with compile-time information encoded in the strings. For example we can construct a simple unit system which allows us to attach units to numerical quantities and perform basic dimensional analysis.

Custom Errors

As of GHC 8.0 we have the capacity to provide custom type error using type families. The messages themselves hook into GHC and are expressed using the small datatype found in GHC.TypeLits

If one of these expressions is found in the signature of an expression GHC reports an error message of the form:

A less contrived example would be creating a type-safe embedded DSL that enforces invariants about the semantics at the type-level. We’ve been able to do this sort of thing using GADTs and type-families for a while but the error reporting has been horrible. With 8.0 we can have type-families that emit useful type errors that reflect what actually goes wrong and integrate this inside of GHC.

Type Equality

Continuing with the theme of building more elaborate proofs in Haskell, GHC 7.8 recently shipped with the Data.Type.Equality module which provides us with an extended set of type-level operations for expressing the equality of types as values, constraints, and promoted booleans.

With this we have a much stronger language for writing restrictions that can be checked at a compile-time, and a mechanism that will later allow us to write more advanced proofs.

Proxies

Using kind polymorphism with phantom types allows us to express the Proxy type which is inhabited by a single constructor with no arguments but with a polykinded phantom type variable which carries an arbitrary type.

In cases where we’d normally pass around a undefined as a witness of a typeclass dictionary, we can instead pass a Proxy object which carries the phantom type without the need for the bottom. Using scoped type variables we can then operate with the phantom parameter and manipulate wherever is needed.

We’ve seen constructors promoted using DataKinds, but just like at the value-level GHC also allows us some syntactic sugar for list and tuples instead of explicit cons’ing and pair’ing. This is enabled with the -XTypeOperators extension, which introduces list syntax and tuples of arbitrary arity at the type-level.

Using this we can construct all variety of composite type-level objects.

λ: :kind 1
1 :: Nat

λ: :kind "foo"
"foo" :: Symbol

λ: :kind [1,2,3]
[1,2,3] :: [Nat]

λ: :kind [Int, Bool, Char]
[Int, Bool, Char] :: [*]

λ: :kind Just [Int, Bool, Char]
Just [Int, Bool, Char] :: Maybe [*]

λ: :kind '("a", Int)
(,) Symbol *

λ: :kind [ '("a", Int), '("b", Bool) ]
[ '("a", Int), '("b", Bool) ] :: [(,) Symbol *]

Singleton Types

A singleton type is a type with a single value inhabitant. Singleton types can be constructed in a variety of ways using GADTs or with data families.

Promoted Naturals

Promoted Booleans

Promoted Maybe

Singleton types are an integral part of the small cottage industry of faking dependent types in Haskell, i.e. constructing types with terms predicated upon values. Singleton types are a way of “cheating” by modeling the map between types and values as a structural property of the type.

The builtin singleton types provided in GHC.TypeLits have the useful implementation that type-level values can be reflected to the value-level and back up to the type-level, albeit under an existential.

Closed Type Families

In the type families we’ve used so far (called open type families) there is no notion of ordering of the equations involved in the type-level function. The type family can be extended at any point in the code resolution simply proceeds sequentially through the available definitions. Closed type-families allow an alternative declaration that allows for a base case for the resolution allowing us to actually write recursive functions over types.

For example consider if we wanted to write a function which counts the arguments in the type of a function and reifies at the value-level.

The variety of functions we can now write down are rather remarkable, allowing us to write meaningful logic at the type level.

{-# LANGUAGE DataKinds #-}
{-# LANGUAGE PolyKinds #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE UndecidableInstances #-}

import GHC.TypeLits
import Data.Proxy
import Data.Type.Equality

-- Type-level functions over type-level lists.

type family Reverse (xs :: [k]) :: [k] where
  Reverse '[] = '[]
  Reverse xs = Rev xs '[]

type family Rev (xs :: [k]) (ys :: [k]) :: [k] where
  Rev '[] i = i
  Rev (x ': xs) i = Rev xs (x ': i)

type family Length (as :: [k]) :: Nat where
  Length '[] = 0
  Length (x ': xs) = 1 + Length xs

type family If (p :: Bool) (a :: k) (b :: k) :: k where
  If True a b = a
  If False a b = b

type family Concat (as :: [k]) (bs :: [k]) :: [k] where
  Concat a '[] = a
  Concat '[] b = b
  Concat (a ': as) bs = a ': Concat as bs

type family Map (f :: a -> b) (as :: [a]) :: [b] where
  Map f '[] = '[]
  Map f (x ': xs) = f x ': Map f xs

type family Sum (xs :: [Nat]) :: Nat where
  Sum '[] = 0
  Sum (x ': xs) = x + Sum xs

ex1 :: Reverse [1,2,3] ~ [3,2,1] => Proxy a
ex1 = Proxy

ex2 :: Length [1,2,3] ~ 3 => Proxy a
ex2 = Proxy

ex3 :: (Length [1,2,3]) ~ (Length (Reverse [1,2,3])) => Proxy a
ex3 = Proxy

-- Reflecting type level computations back to the value level.
ex4 :: Integer
ex4 = natVal (Proxy :: Proxy (Length (Concat [1,2,3] [4,5,6])))
-- 6

ex5 :: Integer
ex5 = natVal (Proxy :: Proxy (Sum [1,2,3]))
-- 6

-- Couldn't match type ‘2’ with ‘1’
ex6 :: Reverse [1,2,3] ~ [3,1,2] => Proxy a
ex6 = Proxy

The results of type family functions need not necessarily be kinded as (*) either. For example using Nat or Constraint is permitted.

Kind Indexed Type Families

Just as typeclasses are normally indexed on types, type families can also be indexed on kinds with the kinds given as explicit kind signatures on type variables.

HLists

A heterogeneous list is a cons list whose type statically encodes the ordered types of its values.

Of course this immediately begs the question of how to print such a list out to a string in the presence of type-heterogeneity. In this case we can use type-families combined with constraint kinds to apply the Show over the HLists parameters to generate the aggregate constraint that all types in the HList are Showable, and then derive the Show instance.

Typelevel Dictionaries

Much of this discussion of promotion begs the question whether we can create data structures at the type-level to store information at compile-time. For example a type-level association list can be used to model a map between type-level symbols and any other promotable types. Together with type-families we can write down type-level traversal and lookup functions.

If we ask GHC to expand out the type signature we can view the explicit implementation of the type-level map lookup function.

Advanced Proofs

Now that we have the length-indexed vector let’s go write the reverse function, how hard could it be?

So we go and write down something like this:

Running this we find that GHC is unhappy about two lines in the code:

As we unfold elements out of the vector we’ll end up doing a lot of type-level arithmetic over indices as we combine the subparts of the vector backwards, but as a consequence we find that GHC will run into some unification errors because it doesn’t know about basic arithmetic properties of the natural numbers. Namely that forall n. n + 0 = 0 and forall n m. n + (1 + m) = 1 + (n + m). Which of course it really shouldn’t be given that we’ve constructed a system at the type-level which intuitively models arithmetic but GHC is just a dumb compiler, it can’t automatically deduce the isomorphism between natural numbers and Peano numbers.

So at each of these call sites we now have a proof obligation to construct proof terms. Recall from our discussion of propositional equality from GADTs that we actually have such machinery to construct this now.

One might consider whether we could avoid using the singleton trick and just use type-level natural numbers, and technically this approach should be feasible although it seems that the natural number solver in GHC 7.8 can decide some properties but not the ones needed to complete the natural number proofs for the reverse functions.

Caveat should be that there might be a way to do this in GHC 7.6 that I’m not aware of. In GHC 7.10 there are some planned changes to solver that should be able to resolve these issues. In particular there are plans to allow pluggable type system extensions that could outsource these kind of problems to third party SMT solvers which can solve these kind of numeric relations and return this information back to GHC’s typechecker.

As an aside this is a direct transliteration of the equivalent proof in Agda, which is accomplished via the same method but without the song and dance to get around the lack of dependent types.

Liquid Haskell

LiquidHaskell is an extension to GHC’s typesystem that adds the capacity for refinement types using the annotation syntax. The type signatures of functions can be checked by the external for richer type semantics than default GHC provides, including non-exhaustive patterns and complex arithmetic properties that require external SMT solvers to verify. For instance LiquidHaskell can statically verify that a function that operates over a Maybe a is always given a Just or that an arithmetic function always yields an Int that is an even positive number.

LiquidHaskell analyses the modules and discharges proof obligations to an SMT solver to see if the conditions are satisfiable. This allows us to prove the absence of a family of errors around memory safety, arithmetic exceptions and information flow.

You will need either the Microsoft Research Z3 SMT solver or Stanford CVC4 SMT solver.

For Linux:

For Mac:

Then install LiquidHaskell either with Cabal or Stack:

Then with the LiquidHaskell framework installed you can annotate your Haskell modules with refinement types and run the liquid

The module can be run through the solver using the liquid command line tool.

To run Liquid Haskell over a Cabal project you can include the cabal directory by passing cabaldir flag and then including the source directory which contains your application code. You can specify additional specification for external modules by including a spec folder containing special LH modules with definitions.

An example specification module.

To run the checker over your project:

For more extensive documentation and further use cases see the official documentation:

• Liquid Haskell Documentation

Haskell has several techniques for automatic generation of type classes for a variety of tasks that consist largely of boilerplate code generation such as:

• Pretty Printing
• Equality
• Serialization
• Ordering
• Traversals

Generic

The most modern method of doing generic programming uses type families to achieve a better method of deriving the structural properties of arbitrary type classes. Generic implements a typeclass with an associated type Rep ( Representation ) together with a pair of functions that form a 2-sided inverse ( isomorphism ) for converting to and from the associated type and the derived type in question.

GHC.Generics defines a set of named types for modeling the various structural properties of types in available in Haskell.

Using the deriving mechanics GHC can generate this Generic instance for us mechanically, if we were to write it by hand for a simple type it might look like this:

Use kind! in GHCi we can look at the type family Rep associated with a Generic instance.

Now the clever bit, instead writing our generic function over the datatype we instead write it over the Rep and then reify the result using from. So for an equivalent version of Haskell’s default Eq that instead uses generic deriving we could write:

To accommodate the two methods of writing classes (generic-deriving or custom implementations) we can use the DefaultSignatures extension to allow the user to leave typeclass functions blank and defer to Generic or to define their own.

Now anyone using our library need only derive Generic and create an empty instance of our typeclass instance without writing any boilerplate for GEq.

Here is a complete example for deriving equality generics:

See:

• Cooking Classes with Datatype Generic Programming
• Datatype-generic Programming in Haskell
• generic-deriving

Generic Deriving

Using Generics many common libraries provide a mechanisms to derive common typeclass instances. Some real world examples:

The hashable library allows us to derive hashing functions.

The cereal library allows us to automatically derive a binary representation.

{-# LANGUAGE DeriveGeneric #-}

import Data.Word
import Data.ByteString
import Data.Serialize

import GHC.Generics

data Val = A [Val] | B [(Val, Val)] | C
  deriving (Generic, Show)

instance Serialize Val where

encoded :: ByteString
encoded = encode (A [B [(C, C)]])
-- "NULNULNULNULNULNULNULNULSOHSOHNULNULNULNULNULNULNULSOHSTXSTX"

bytes :: [Word8]
bytes = unpack encoded
-- [0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,2,2]

decoded :: Either String Val
decoded = decode encoded

The aeson library allows us to derive JSON representations for JSON instances.

See: A Generic Deriving Mechanism for Haskell

Higher Kinded Generics

Using the same interface GHC.Generics provides a separate typeclass for higher-kinded generics.

So for instance Maybe has Rep1 of the form:

Typeable

The Typeable class be used to create runtime type information for arbitrary types.

Using the Typeable instance allows us to write down a type safe cast function which can safely use unsafeCast and provide a proof that the resulting type matches the input.

Of historical note is that writing our own Typeable classes is currently possible of GHC 7.6 but allows us to introduce dangerous behavior that can cause crashes, and shouldn’t be done except by GHC itself. As of 7.8 GHC forbids hand-written Typeable instances. As of 7.10 -XAutoDeriveTypeable is enabled by default.

See: Typeable and Data in Haskell

Dynamic Types

Since we have a way of querying runtime type information we can use this machinery to implement a Dynamic type. This allows us to box up any monotype into a uniform type that can be passed to any function taking a Dynamic type which can then unpack the underlying value in a type-safe way.

In GHC 7.8 the Typeable class is poly-kinded so polymorphic functions can be applied over functions and higher kinded types.

Use of Dynamic is somewhat rare, except in odd cases that have to deal with foreign memory and FFI interfaces. Using it for business logic is considered a code smell. Consider a more idiomatic solution.

Data

Just as Typeable lets us create runtime type information, the Data class allows us to reflect information about the structure of datatypes to runtime as needed.

The types for gfoldl and gunfold are a little intimidating ( and depend on RankNTypes ), the best way to understand is to look at some examples. First the most trivial case a simple sum type Animal would produce the following code:

For a type with non-empty containers we get something a little more interesting. Consider the list type:

Looking at gfoldl we see the Data has an implementation of a function for us to walk an applicative over the elements of the constructor by applying a function k over each element and applying z at the spine. For example look at the instance for a 2-tuple as well:

This is pretty neat, now within the same typeclass we have a generic way to introspect any Data instance and write logic that depends on the structure and types of its subterms. We can now write a function which allows us to traverse an arbitrary instance of Data and twiddle values based on pattern matching on the runtime types. So let’s write down a function over which increments a Value type for both for n-tuples and lists.

{-# LANGUAGE DeriveDataTypeable #-}

import Data.Data
import Control.Monad.Identity
import Control.Applicative

data Animal = Cat | Dog deriving (Data, Typeable)

newtype Val = Val Int deriving (Show, Data, Typeable)

incr :: Typeable a => a -> a
incr = maybe id id (cast f)
  where f (Val x) = Val (x * 100)

over :: Data a => a -> a
over x = runIdentity $ gfoldl cont base (incr x)
  where
    cont k d = k <*> (pure $ over d)
    base = pure


example1 :: Constr
example1 = toConstr Dog
-- Dog

example2 :: DataType
example2 = dataTypeOf Cat
-- DataType {tycon = "Main.Animal", datarep = AlgRep [Cat,Dog]}

example3 :: [Val]
example3 = over [Val 1, Val 2, Val 3]
-- [Val 100,Val 200,Val 300]

example4 :: (Val, Val, Val)
example4 = over (Val 1, Val 2, Val 3)
-- (Val 100,Val 200,Val 300)

We can also write generic operations, for example to count the number of parameters in a data type.

Uniplate

Uniplate is a generics library for writing traversals and transformation for arbitrary data structures. It is extremely useful for writing AST transformations and rewriting systems.

The descend function will apply a function to each immediate descendant of an expression and then combines them up into the parent expression.

The transform function will perform a single pass bottom-up transformation of all terms in the expression.

The rewrite function will perform an exhaustive transformation of all terms in the expression to fixed point, using Maybe to signify termination.

Alternatively Uniplate instances can be derived automatically from instances of Data without the need to explicitly write a Uniplate instance. This approach carries a slight amount of overhead over an explicit hand-written instance.

Biplate

Biplates generalize plates where the target type isn’t necessarily the same as the source, it uses multiparameter typeclasses to indicate the type sub of the sub-target. The Uniplate functions all have an equivalent generalized biplate form.

Numeric Tower

Haskell’s numeric tower is unusual and the source of some confusion for novices. Haskell is one of the few languages to incorporate statically typed overloaded literals without a mechanism for “coercions” often found in other languages.

To add to the confusion numerical literals in Haskell are desugared into a function from a numeric typeclass which yields a polymorphic value that can be instantiated to any instance of the Num or Fractional typeclass at the call-site, depending on the inferred type.

To use a blunt metaphor, we’re effectively placing an object in a hole and the size and shape of the hole defines the object you place there. This is very different than in other languages where a numeric literal like 2.718 is hard coded in the compiler to be a specific type ( double or something ) and you cast the value at runtime to be something smaller or larger as needed.

The numeric typeclass hierarchy is defined as such:

Conversions between concrete numeric types ( from : left column, to : top row ) is accomplished with several generic functions.

GMP Integers

The Integer type in GHC is implemented by the GMP (libgmp) arbitrary precision arithmetic library. Unlike the Int type, the size of Integer values is bounded only by the available memory.

Most notably libgmp is one of the few libraries that compiled Haskell binaries are dynamically linked against. An alternative library integer-simple can be linked in place of libgmp.

Complex Numbers

Haskell supports arithmetic with complex numbers via a Complex datatype from the Data.Complex module. The first argument is the real part, while the second is the imaginary part. The type has a single parameter and inherits its numerical typeclass components (Num, Fractional, Floating) from the type of this parameter.

The Num instance for Complex is only defined if parameter of Complex is an instance of RealFloat.

Decimal & Scientific Types

Scientific provides arbitrary-precision numbers represented using scientific notation. The constructor takes an arbitrarily sized Integer argument for the digits and an Int for the exponent. Alternatively the value can be parsed from a String or coerced from either Double/Float.

Polynomial Arithmetic

The standard library for working with symbolic polynomials is the poly library. It exposes a interface for working with univariate polynomials which are backed by an efficient vector library. This allows us to efficiently manipulate and perform arithmetic operations over univariate polynomails.

For example we can instantiate symbolic polynomials, write recurrence rules and generators over them and factor them.

See: poly

Combinatorics

Combinat is the standard Haskell library for doing combinatorial calculations. It provides a variety of functions for computing:

• Permutations & Combinations
• Braid Groups
• Integer Partitions
• Young’s Tableux
• Lattice Paths

See: combinat

Number Theory

Arithmoi is the standard number theory library for Haskell. It provides functions for calculing common number theory operations used in combinators and cryptography applications in Haskell. Including:

• Modular square roots
• Möbius Inversions
• Primarily Testing
• Riemann Zeta Functions
• Pollard’s Rho Algorithm
• Jacobi symbols
• Meijer-G Functions

See: arithmoi

Stochastic Calculus

HQuantLib provides a variety of functions for working with stochastic processes. This primarily applies to stochastic calculus applied to pricing financial products such as the Black-Scholes pricing engine and routines for calculating volatility smiles of options products.

See: HQuantLib

Differential Equations

There are several Haskell libraries for finding numerical solutions to systems of differential equations. These kind of problems show up quite frequently in scientific computing problems.

For example a simple differential equation is Van der Pol oscillator which occurs frequently in physics. This is a second order differential equation which relates the position of a oscillator x in terms of time, acceleration ${d^{2}x over dt^{2}}$, and the velocity $dx over dt$ a scalar parameter μ. It is given by the equation.

$$
{displaystyle {d^{2}x over dt^{2}}-mu (1-x^{2}){dx over dt}+x=0,}
$$

For example this equation can be solved for a fixed μ and set of boundary conditions for the time parameter t. The solution is returned as an HMatrix vector.

Statistics & Probability

Haskell has a basic statistics library for calculating descriptive statistics, generating and sampling probability distributions and performing statistical tests.

Constructive Reals

Instead of modeling the real numbers on finite precision floating point numbers we alternatively work with Num which internally manipulates the power series expansions for the expressions when performing operations like arithmetic or transcendental functions without losing precision when performing intermediate computations. Then we simply slice off a fixed number of terms and approximate the resulting number to a desired precision. This approach is not without its limitations and caveats ( notably that it may diverge ).

SAT Solvers

A collection of constraint problems known as satisfiability problems show up in a number of different disciplines from type checking to package management. Simply put a satisfiability problem attempts to find solutions to a statement of conjoined conjunctions and disjunctions in terms of a series of variables. For example:

(A v ¬B v C) ∧ (B v D v E) ∧ (D v F)

To use the picosat library to solve this, it can be written as zero-terminated lists of integers and fed to the solver according to a number-to-variable relation:

import Picosat

main :: IO [Int]
main = do
  solve [[1, -2, 3], [2,4,5], [4,6]]
  -- Solution [1,-2,3,4,5,6]

The SAT solver itself can be used to solve satisfiability problems with millions of variables in this form and is finely tuned.

See:

• picosat

SMT Solvers

A generalization of the SAT problem to include predicates other theories gives rise to the very sophisticated domain of “Satisfiability Modulo Theory” problems. The existing SMT solvers are very sophisticated projects ( usually bankrolled by large institutions ) and usually have to be called out to via foreign function interface or via a common interface called SMT-lib. The two most common of use in Haskell are cvc4 from Stanford and z3 from Microsoft Research.

The SBV library can abstract over different SMT solvers to allow us to express the problem in an embedded domain language in Haskell and then offload the solving work to the third party library.

As an example, here’s how you can solve a simple cryptarithm

using SBV library:

import Data.Foldable
import Data.SBV

-- | val [4,2] == 42
val :: [SInteger] -> SInteger
val = foldr1 (d r -> d + 10*r) . reverse

puzzle :: Symbolic SBool
puzzle = do
  ds@[b,u,r,i,t,o,m,n,a,d] <- sequenceA [ sInteger [v] | v <- "buritomnad" ]
  constrain $ distinct ds
  for_ ds $ d -> constrain $ inRange d (0,9)
  pure $    val [b,u,r,r,i,t,o]
          + val     [m,o,n,a,d]
        .== val [b,a,n,d,a,i,d]

Let’s look at all possible solutions,

Map

A map is an associative array mapping any instance of Ord keys to values of any type.

Tree

A tree is directed graph with a single root.

Set

Sets are unordered data structures containing Ord values of any type and guaranteeing uniqueness with in the structure. They are not identical to the mathematical notion of a Set even though they share the same namesake.

Vector

Vectors are high performance single dimensional arrays that come come in six variants, two for each of the following types of a mutable and an immutable variant.

• Data.Vector
• Data.Vector.Storable
• Data.Vector.Unboxed

The most notable feature of vectors is constant time memory access with ((!)) as well as variety of efficient map, fold and scan operations on top of a fusion framework that generates surprisingly optimal code.

Mutable Vectors

Mutable vectors are variants of vectors which allow inplace updates.

Within the IO monad we can perform arbitrary read and writes on the mutable vector with constant time reads and writes. When needed a static Vector can be created to/from the MVector using the freeze/thaw functions.

The vector library itself normally does bounds checks on index operations to protect against memory corruption. This can be enabled or disabled on the library level by compiling with boundschecks cabal flag.

Unordered Containers

Both the HashMap and HashSet are purely functional data structures that are drop in replacements for the containers equivalents but with more efficient space and time performance. Additionally all stored elements must have a Hashable instance. These structures have different time complexities for insertions and lookups.

See: Announcing Unordered Containers

Hashtables

Hashtables provides hashtables with efficient lookup within the ST or IO monad. These have constant time lookup like most languages:

Graphs

The Graph module in the containers library is a somewhat antiquated API for working with directed graphs. A little bit of data wrapping makes it a little more straightforward to use. The library is not necessarily well-suited for large graph-theoretic operations but is perfectly fine for example, to use in a typechecker which needs to resolve strongly connected components of the module definition graph.

So for example we can construct a simple graph:

ex1 :: [(String, String, [String])]
ex1 = [
    ("a","a",["b"]),
    ("b","b",["c"]),
    ("c","c",["a"])
  ]

ts1 :: [String]
ts1 = topo' (fromList ex1)
-- ["a","b","c"]

sc1 :: [[String]]
sc1 = scc' (fromList ex1)
-- [["a","b","c"]]

Or with two strongly connected subgraphs:

ex2 :: [(String, String, [String])]
ex2 = [
    ("a","a",["b"]),
    ("b","b",["c"]),
    ("c","c",["a"]),

    ("d","d",["e"]),
    ("e","e",["f", "e"]),
    ("f","f",["d", "e"])
  ]


ts2 :: [String]
ts2 = topo' (fromList ex2)
-- ["d","e","f","a","b","c"]

sc2 :: [[String]]
sc2 = scc' (fromList ex2)
-- [["d","e","f"],["a","b","c"]]

See: GraphSCC

Graph Theory

The fgl library provides a more efficient graph structure and a wide variety of common graph-theoretic operations. For example calculating the dominance frontier of a graph shows up quite frequently in control flow analysis for compiler design.

import qualified Data.Graph.Inductive as G

cyc3 :: G.Gr Char String
cyc3 = G.buildGr
       [([("ca",3)],1,'a',[("ab",2)]),
                ([],2,'b',[("bc",3)]),
                ([],3,'c',[])]

-- Loop query
ex1 :: Bool
ex1 = G.hasLoop x

-- Dominators
ex2 :: [(G.Node, [G.Node])]
ex2 = G.dom x 0

x :: G.Gr Int ()
x = G.insEdges edges gr
  where
  gr = G.insNodes nodes G.empty
  edges = [(0,1,()), (0,2,()), (2,1,()), (2,3,())]
  nodes = zip [0,1 ..] [2,3,4,1]

DList

A dlist is a list-like structure that is optimized for O(1) append operations, internally it uses a Church encoding of the list structure. It is specifically suited for operations which are append-only and need only access it when manifesting the entire structure. It is particularly well-suited for use in the Writer monad.

Sequence

The sequence data structure behaves structurally similar to list but is optimized for append/prepend operations and traversal.

Haskell does not exist in a vacuum and will quite often need to interact with or offload computation to another programming language. Since GHC itself is built on the GCC ecosystem interfacing with libraries that can be linked via a C ABI is quite natural. Indeed many high performance libraries will call out to Fortran, C, or C++ code to perform numerical computations that can be linked seamlessly into the Haskell runtime. There are several approaches to combining Haskell with other languages in the via the Foreign Function Interface or FFI.

Pure Functions

Wrapping pure C functions with primitive types is trivial.

Storable Arrays

There exists a Storable typeclass that can be used to provide low-level access to the memory underlying Haskell values. Ptr objects in Haskell behave much like C pointers although arithmetic with them is in terms of bytes only, not the size of the type associated with the pointer ( this differs from C).

The Prelude defines Storable interfaces for most of the basic types as well as types in the Foreign.Storable module.

To pass arrays from Haskell to C we can again use Storable Vector and several unsafe operations to grab a foreign pointer to the underlying data that can be handed off to C. Once we’re in C land, nothing will protect us from doing evil things to memory!

/* $(CC) -c qsort.c -o qsort.o */
void swap(int *a, int *b)
{
    int t = *a;
    *a = *b;
    *b = t;
}

void sort(int *xs, int beg, int end)
{
    if (end > beg + 1) {
        int piv = xs[beg], l = beg + 1, r = end;

        while (l < r) {
            if (xs[l] <= piv) {
                l++;
            } else {
                swap(&xs[l], &xs[--r]);
            }
        }

        swap(&xs[--l], &xs[beg]);
        sort(xs, beg, l);
        sort(xs, r, end);
    }
}

The names of foreign functions from a C specific header file can be qualified.

Prepending the function name with a & allows us to create a reference to the function pointer itself.

Function Pointers

Using the above FFI functionality, it’s trivial to pass C function pointers into Haskell, but what about the inverse passing a function pointer to a Haskell function into C using foreign import ccall "wrapper".

Will yield the following output:

hsc2hs

When doing socket level programming, when handling UDP packets there is a packed C struct with a set of fields defined by the Linux kernel. These fields are defined in the following C pseudocode.

If we want to marshall packets to and from Haskell datatypes we need to be able to be able to take a pointer to memory holding the packet message header and scan the memory into native Haskell types. This involves knowing some information about the memory offsets for the packet structure. GHC ships with a tool known as hsc2hs which can be used to read information from C header files to automatically generate the boilerplate instances of Storable to perform this marshalling. The hsc2hs library acts a preprocessor over .hsc files and can fill in information as specific by several macros to generate Haskell source.

For example the following module from the network library must introspect the msghdr struct from .

#include 
#include 

import Network.Socket.Imports
import Network.Socket.Internal (zeroMemory)
import Network.Socket.Types (SockAddr)

import Network.Socket.ByteString.IOVec (IOVec)

data MsgHdr = MsgHdr
    { msgName    :: !(Ptr SockAddr)
    , msgNameLen :: !CUInt
    , msgIov     :: !(Ptr IOVec)
    , msgIovLen  :: !CSize
    }

instance Storable MsgHdr where
  sizeOf _    = (#const sizeof(struct msghdr))
  alignment _ = alignment (undefined :: CInt)

  peek p = do
    name       <- (#peek struct msghdr, msg_name)       p
    nameLen    <- (#peek struct msghdr, msg_namelen)    p
    iov        <- (#peek struct msghdr, msg_iov)        p
    iovLen     <- (#peek struct msghdr, msg_iovlen)     p
    return $ MsgHdr name nameLen iov iovLen

  poke p mh = do
    zeroMemory p (#const sizeof(struct msghdr))
    (#poke struct msghdr, msg_name)       p (msgName       mh)
    (#poke struct msghdr, msg_namelen)    p (msgNameLen    mh)
    (#poke struct msghdr, msg_iov)        p (msgIov        mh)
    (#poke struct msghdr, msg_iovlen)     p (msgIovLen     mh)

Running the command line tool over this module we get the following Haskell output Example.hs. This can also be run as part of a Cabal build step by including hsc2hs in your build-tools.

GHC Haskell has an extremely advanced parallel runtime that embraces several different models of concurrency to adapt to adapt to needs for different domains. Unlike other languages Haskell does not have any Global Interpreter Lock or equivalent. Haskell code can be executed in a multi-threaded context and have shared mutable state and communication channels between threads.

A thread in Haskell is created by forking off from the main process using the forkIO command. This is performed within the IO monad and yields a ThreadId which can be used to communicate with the new thread.

Haskell threads are extremely cheap to spawn, using only 1.5KB of RAM depending on the platform and are much cheaper than a pthread in C. Calling forkIO 10⁶ times completes just short of 1s. Additionally, functional purity in Haskell also guarantees that a thread can almost always be terminated even in the middle of a computation without concern.

See:

• The Scheduler
• Parallel and Concurrent Programming in Haskell

Sparks

The most basic “atom” of parallelism in Haskell is a spark. It is a hint to the GHC runtime that a computation can be evaluated to weak head normal form in parallel.

rpar a spins off a separate spark that evaluates a to weak head normal form and places the computation in the spark pool. When the runtime determines that there is an available CPU to evaluate the computation it will evaluate ( convert ) the spark. If the main thread of the program is the evaluator for the spark, the spark is said to have fizzled. Fizzling is generally bad and indicates that the logic or parallelism strategy is not well suited to the work that is being evaluated.

The spark pool is also limited ( but user-adjustable ) to a default of 8000 (as of GHC 7.8.3 ). Sparks that are created beyond that limit are said to overflow.

An argument to rseq forces the evaluation of a spark before evaluation continues.

The parallel runtime is necessary to use sparks, and the resulting program must be compiled with -threaded. Additionally the program itself can be specified to take runtime options with -rtsopts such as the number of cores to use.

The runtime can be asked to dump information about the spark evaluation by passing the -s flag.

The parallel computations themselves are sequenced in the Eval monad, whose evaluation with runEval is itself a pure computation.

Threads

For fine-grained concurrency and parallelism, Haskell has a lightweight thread system that schedules logical threads on the available operating system threads. These lightweight threads are called unbound threads, while native operating systems are called bound threads since they are bound to a single operating system thread. The functions to spawn an run tasks inside these threads all live in the IO monad. The number of possible simultaneous threads is given by the getNumCapabilities functions based on the system environment.

Managed threads work with the runtime system’s IO manager which will schedule and manage cooperative multitaksing and polling. When a individual unbound thread is blocked polling on a file description or lock it will yield to another runnable thread managed by the runtime. This yield action can also be explicitly invoked with the yield function. A thread can also schedule a wait using threadDelay to yield to the scheduler for a fixed interval given in microseconds.

Once a thread is forked the fork action will give back a ThreadId which can be used to call actions and kill the thread from another context. Inside of a running thread the current ThreadId can be queried with myThreadId.

An exception can also be raised in a given ThreadId given an instance of Exception typeclass.

When individually polling on file descriptors there are several functions that can schedule the thread to wake up again when the given file is given a wake event from the kernel. The following functions will yield the current thread waiting on either a read or write event on the given file description Fd.

IORef

IORef is a mutable reference that can be read and writen to within the IO monad. It is the simplest most low-level mutable reference provided by the base library.

For example we could construct two IORefs which mutably hold the balances for two imaginary bank accounts. These references can be passed to another IO function which can update the values in place.

There are also several atomic functions to update IORef when working with the threaded runtime.

The atomic modify function atomicModifyIORef reads the value of r and applies the function f to r giving back (a',b). Then value r is updated with the new value a' and b is the return value. Both the read and the write are done atomically so it is not possible that any value will alter the underlying IORef between the read and write.

Normally IORef is garbage collected like any other value. Once it is out of scope and the runtime has no more references to it, the runtime will collect the thunk holding the IORef as well as the value the underlying pointer points at. Sometimes when working with these references will require adding additional finalisation logic.

The mkWeakIORef attaches a finalizer function in the second argument which is run when the value is garbage collected.

MVars

MVars are mutable references like IORefs that can be used to share mutable state between threads. An MVar has two states empty and full. Reading from an empty MVar will block the current thread. Writing to a full MVar will also block the current thread. Thus only one value can be held inside the MVar allowing us to synchronize the value across threads. MVars are building blocks for many higher concurrent primitives which use them under the hood.

An MVar can either be initialised in an empty state or with a supplied value.

The function takeMVar operates like a read returning the value, but once the value is read the state of the underlying MVar is left empty. This read is performed once for the first thread to wake up polling for the read.

As an example consider a multithreaded scenario where a second thread is created which polls on atomically on an MVar update.

If a thread is left sleeping waiting on an MVar and the runtime no longer has any references to code which can write to the MRef (i.e. all references to the MVar are garbage collected) the thread will be thrown the exception BlockedIndefinitelyOnMVar since no value can subsequently be written to it.

TVar

TVars are transactional mutable variables which can be read and written to within in the STM monad. The STM monad provides support for Software Transactional Memory which is a higher level abstraction for concurrent communication that doesn’t require explict thread maintenance and has lovely easy compositional nature.

The STM monad magically hooks into the runtime system and provides two key operations atomically and retry which allow monadic blocks of STM actions to be performed atomically and passed around symbolically. In the event that the runtime fails to commit a transaction, the retry function can rerun the logic contained in a STM a.

TVars can be created just like IORefs but instead of being in IO they can also be created with the STM monad.

Read, writes and updates proceed exactly like IORef updates but inside of STM.

As an example consider the IORef account transfers from above, but instead the two modifyTVar actions are performed atomically inside of the transfer function.

There is an additional TMVar which behaves precisely like the traditional MVar (i.e. it has an empty and full state) but which is embedded in IO. It is has precisely the same semantics as MVar but emits values within STM.

Chans

Channels are unbounded queues to which an unbounded number of values can be written an unbounded number of times . Channels are implemented using MVars and can be consumed by any number of other threads which read data off of the Chan. Channels are created, read from and written to using a simple new, read and write interface just as we’ve seen with other concurrency primitives.

An example in which a channel is created between a producer and consumer threads is shown below. This can be used to share data between threads and create work queue background processing systems.

There is also an STM variant of Chan called TChan.

Semaphores

Semaphores are a concurrency primitive used to control access to a common resource used by multiple threads. A semaphore is a variable containing an integral value that can be incremented or decremented by concurrent processes. A semaphore will restrict concurrency to a integral count of consumers called the limit. The QSem provides an interface for a simple lock semaphore that can be created in IO and polled on using waitQSem.

A simple example of usage:

QSem also have a variant QSemN which allows a resource to be acquired and released in a fixed quantity other than one. The waitQSemN function then takes an integral quantity to wait for.

There is also an STM variant of QSem called TSem which has the same semantics.

Threadscope

Passing the flag -l generates the eventlog which can be rendered with the threadscope library.

See:

• Performance profiling with ghc-events-analyze

Strategies

Sparks themselves form the foundation for higher level parallelism constructs known as strategies which adapt spark creation to fit the computation or data structure being evaluated. For instance if we wanted to evaluate both elements of a tuple in parallel we can create a strategy which uses sparks to evaluate both sides of the tuple.

This pattern occurs so frequently the combinator using can be used to write it equivalently in operator-like form that may be more visually appealing to some.

For a less contrived example consider a parallel parmap which maps a pure function over a list of a values in parallel.

The functions above are quite useful, but will break down if evaluation of the arguments needs to be parallelized beyond simply weak head normal form. For instance if the arguments to rpar is a nested constructor we’d like to parallelize the entire section of work in evaluated the expression to normal form instead of just the outer layer. As such we’d like to generalize our strategies so the evaluation strategy for the arguments can be passed as an argument to the strategy.

Control.Parallel.Strategies contains a generalized version of rpar which embeds additional evaluation logic inside the rpar computation in Eval monad.

Using the deepseq library we can now construct a Strategy variant of rseq that evaluates to full normal form.

We now can create a “higher order” strategy that takes two strategies and itself yields a computation which when evaluated uses the passed strategies in its scheduling.

These patterns are implemented in the Strategies library along with several other general forms and combinators for combining strategies to fit many different parallel computations.

See:

• Control.Concurent.Strategies

STM

Software transactional memory is a technique for demarcating blocks of atomic transactions that are guaranteed by the runtime to have several properties:

• No parallel processes can read from the atomic block until the transaction commits.
• The current process is isolated cannot see any changes made by other parallel processes.

This is similar to the atomicity that databases guarantee. The stm library provides a lovely compositional interface for building up higher level primitives that can be composed in atomic blocks to build safe concurrent logic without worrying about deadlocks and memory corruption from threaded and mutable reference approaches to building parallel algorithms.

The strength of Haskell’s purity guarantees that transactions within STM are pure and can always be rolled back if a commit fails. An example of usage is shown below.

Monad Par

Using the Par monad we express our computation as a data flow graph which is scheduled in order of the connections between forked computations which exchange resulting computations with IVar.

Async

Async is a higher level set of functions that work on top of Control.Concurrent and STM.

Parser combinators were originally developed in the Haskell programming language and the last 10 years have seen a massive amount of refinement and improvements on parser combinator libraries. Today Haskell has an amazing parser ecosystem.

Parsec

For parsing in Haskell it is quite common to use a family of libraries known as Parser Combinators which let us write code to generate parsers which construct themselves from an abstract description of the grammar described with combinators.

<|> can be chained sequentially to generate a sequence of options.

There are two styles of writing Parsec, one can choose to write with monads or with applicatives.

The same code written with applicatives uses the applicative combinators:

Now for instance if we want to parse simple lambda expressions we can encode the parser logic as compositions of these combinators which yield the string parser when evaluated with parse.

Custom Lexer

In our previous example a lexing pass was not necessary because each lexeme mapped to a sequential collection of characters in the stream type. If we wanted to extend this parser with a non-trivial set of tokens, then Parsec provides us with a set of functions for defining lexers and integrating these with the parser combinators. The simplest example builds on top of the builtin Parsec language definitions which define a set of most common lexical schemes.

For instance we’ll build on top of the empty language grammar on top of the haskellDef grammar that uses the Text token instead of string.

{-# LANGUAGE OverloadedStrings #-}

import Text.Parsec
import Text.Parsec.Text
import qualified Text.Parsec.Token as Tok
import qualified Text.Parsec.Language as Lang

import Data.Functor.Identity (Identity)
import qualified Data.Text as T
import qualified Data.Text.IO as TIO

data Expr
  = Var T.Text
  | App Expr Expr
  | Lam T.Text Expr
  deriving (Show)

lexer :: Tok.GenTokenParser T.Text () Identity
lexer = Tok.makeTokenParser style

style :: Tok.GenLanguageDef T.Text () Identity
style = Lang.emptyDef
  { Tok.commentStart    = "{-"
  , Tok.commentEnd      = "-}"
  , Tok.commentLine     = "--"
  , Tok.nestedComments  = True
  , Tok.identStart      = letter
  , Tok.identLetter     = alphaNum <|> oneOf "_'"
  , Tok.opStart         = Tok.opLetter style
  , Tok.opLetter        = oneOf ":!#$%&*+./<=>?@\^|-~"
  , Tok.reservedOpNames = []
  , Tok.reservedNames   = []
  , Tok.caseSensitive   = True
  }

parens :: Parser a -> Parser a
parens = Tok.parens lexer

reservedOp :: T.Text -> Parser ()
reservedOp op = Tok.reservedOp lexer (T.unpack op)

ident :: Parser T.Text
ident = T.pack <$> Tok.identifier lexer

contents :: Parser a -> Parser a
contents p = do
  Tok.whiteSpace lexer
  r <- p
  eof
  return r

var :: Parser Expr
var = do
  var <- ident
  return (Var var )

app :: Parser Expr
app = do
  e1 <- expr
  e2 <- expr
  return (App e1 e2)

fun :: Parser Expr
fun = do
  reservedOp "\"
  binder <- ident
  reservedOp "."
  rhs <- expr
  return (Lam binder rhs)

expr :: Parser Expr
expr = do
  es <- many1 aexp
  return (foldl1 App es)

aexp :: Parser Expr
aexp = fun <|> var <|> (parens expr)

test :: T.Text -> Either ParseError Expr
test = parse (contents expr) ""

repl :: IO ()
repl = do
  str <- TIO.getLine
  print (test str)
  repl

main :: IO ()
main = repl

See: Text.Parsec.Language

Simple Parsing

Putting our lexer and parser together we can write down a more robust parser for our little lambda calculus syntax.

Trying it out:

Megaparsec

Megaparsec is a generalisation of parsec which can work with the several input streams.

• Text (strict and lazy)
• ByteString (strict and lazy)
• String = [Char]

Megaparsec is an expanded and optimised form of parsec which can be used to write much larger complex parsers with custom lexers and Clang-style error message handling.

An example below for the lambda calculus is quite concise:

Attoparsec

Attoparsec is a parser combinator like Parsec but more suited for bulk parsing of large text and binary files instead of parsing language syntax to ASTs. When written properly Attoparsec parsers can be efficient.

One notable distinction between Parsec and Attoparsec is that backtracking operator (try) is not present and reflects on attoparsec’s different underlying parser model.

For a simple little lambda calculus language we can use attoparsec much in the same we used parsec:

For an example try the above parser with the following simple lambda expression.

Attoparsec adapts very well to binary and network protocol style parsing as well, this is extracted from a small implementation of a distributed consensus network protocol:

Configurator

Configurator is a library for configuring Haskell daemons and programs. It uses a simple, but flexible, configuration language, supporting several of the most commonly needed types of data, along with interpolation of strings from the configuration or the system environment.

An example configuration file:

Configurator also includes an import directive allows the configuration of a complex application to be split across several smaller files, or configuration data to be shared across several applications.

Optparse Applicative

Optparse-applicative is a combinator library for building command line interfaces that take in various user flags, commands and switches and maps them into Haskell data structures that can handle the input. The main interface is through the applicative functor Parser and various combinators such as strArgument and flag which populate the option parsing table with some monadic action which returns a Haskell value. The resulting sequence of values can be combined applicatively into a larger Config data structure that holds all the given options. The --help header is also automatically generated from the combinators.

Optparse Generic

Many optparse-applicative command line parsers can also be generated using Generics from descriptions of records. This approach is not foolproof but works well enough for simple command line applications with a few options. For more complex interfaces with subcommands and help information you’ll need to go back to the optparse-applicative level. For example:

Happy & Alex

Happy is a parser generator system for Haskell, similar to the tool `yacc’ for C. It works as a preprocessor with its own syntax that generates a parse table from two specifications, a lexer file and parser file. Happy does not have the same underlying parser implementation as parser combinators and can effectively work with left-recursive grammars without explicit factorization. It can also easily be modified to track position information for tokens and handle offside parsing rules for indentation-sensitive grammars. Happy is used in GHC itself for Haskell’s grammar.

1. Lexer.x
2. Parser.y

Running the standalone commands will take Alex/Happy source files from stdin and generate and output Haskell modules. Alex and Happy files can contain arbitrary Haskell code that can be escaped to the output.

The generated modules are not human readable generally and unfortunately error messages are given in the Haskell source, not the Happy source. Anything enclosed in braces is interpreted as literal Haskell while the code outside the braces is interpeted as parser grammar.

Happy and Alex can be integrated into a cabal file simply by including the Parser.y and Lexer.x files inside of the exposed modules and adding them to the build-tools pragma.

Lexer

For instance we could define a little toy lexer with a custom set of tokens.

Parser

The associated parser is list of a production rules and a monad to run the parser in. Production rules consist of a set of options on the left and generating Haskell expressions on the right with indexed metavariables ($1, $2, …) mapping to the ordered terms on the left (i.e. in the second term term ~ $1, term ~ $2).

An example parser module:

As a simple input consider the following simple program.

Lazy IO

The problem with using the usual monadic approach to processing data accumulated through IO is that the Prelude tools require us to manifest large amounts of data in memory all at once before we can even begin computation.

Reading from the file creates a thunk for the string that forced will then read the file. The problem is then that this method ties the ordering of IO effects to evaluation order which is difficult to reason about in the large.

Consider that normally the monad laws ( in the absence of seq ) guarantee that these computations should be identical. But using lazy IO we can construct a degenerate case.

So what we need is a system to guarantee deterministic resource handling with constant memory usage. To that end both the Conduits and Pipes libraries solved this problem using different ( though largely equivalent ) approaches.

Pipes

Pipes is a stream processing library with a strong emphasis on the static semantics of composition. The simplest usage is to connect “pipe” functions with a (>->) composition operator, where each component can await and yield to push and pull values along the stream.

For example we could construct a “FizzBuzz” pipe.

To continue with the degenerate case we constructed with Lazy IO, consider than we can now compose and sequence deterministic actions over files without having to worry about effect order.

This is a simple sampling of the functionality of pipes. The documentation for pipes is extensive and great deal of care has been taken make the library extremely thorough. pipes is a shining example of an accessible yet category theoretic driven design.

See: Pipes Tutorial

ZeroMQ

As a motivating example, ZeroMQ is a network messaging library that abstracts over traditional Unix sockets to a variety of network topologies. Most notably it isn’t designed to guarantee any sort of transactional guarantees for delivery or recovery in case of errors so it’s necessary to design a layer on top of it to provide the desired behavior at the application layer.

In Haskell we’d like to guarantee that if we’re polling on a socket we get messages delivered in a timely fashion or consider the resource in an error state and recover from it. Using pipes-safe we can manage the life cycle of lazy IO resources and can safely handle failures, resource termination and finalization gracefully. In other languages this kind of logic would be smeared across several places, or put in some global context and prone to introduce errors and subtle race conditions. Using pipes we instead get a nice tight abstraction designed exactly to fit this kind of use case.

For instance now we can bracket the ZeroMQ socket creation and finalization within the SafeT monad transformer which guarantees that after successful message delivery we execute the pipes function as expected, or on failure we halt the execution and finalize the socket.

Conduits

Conduits are conceptually similar though philosophically different approach to the same problem of constant space deterministic resource handling for IO resources.

The first initial difference is that await function now returns a Maybe which allows different handling of termination.

Since 1.2.8 the separate connecting and fusing operators are deprecated in favor of a single fusing operator (.|).

Recently Haskell has seen quite a bit of development of cryptography libraries as it serves as an excellent language for working with and manipulating algebraic structures found in cryptographic primitives. In addition to most of the basic hashing, elliptic curve and cipher suites libraries, Haskell has a excellent standard cryptography library called cryptonite which provides the standard kitchen sink of most modern primitives. These include hash functions, elliptic curve cryptography, digital signature algorithms, ciphers, one time passwords, entropy generation and safe memory handling.

SHA Hashing

A cryptographic hash function is a special class of hash function that has certain properties which make it suitable for use in cryptography. It is a mathematical algorithm that maps data of arbitrary size to a bit string of a fixed size (a hash function) which is designed to also be a one-way function, that is, a function which is infeasible to invert.

SHA-256 is a cryptographic hash function from the SHA-2 family and is standardized by NIST. It produces a 256-bit message digest.

Password Hashing

Modern applications should use one of either the Blake2 or Argon2 hashing algorithms for storing passwords in a database as part of an authentication workflow.

To use Argon2:

To use Blake2:

Curve25519 Diffie-Hellman

Curve25519 is a widely used Diffie-Hellman function suitable for a wide variety of applications. Private and public keys using Curve25519 are 32 bytes each. Elliptic curve Diffie-Hellman is a protocol in which two parties can exchange their public keys in the clear and generate a shared secret which can be used to share information across a secure channel.

A private key is a large integral value which is multiplied by the base point on the curve to generate the public key. Going to backwards from a public key requires one to solve the elliptic curve discrete logarithm which is believed to be computationally infeasible.

Diffie-Hellman key exchange be performed by executing the function dh over the private and public keys for Alice and Bob.

An example is shown below:

See:

• curve25519

Ed25519 EdDSA

EdDSA is a digital signature scheme based on Schnorr signature using the twisted Edwards curve Ed25519 and SHA-512 (SHA-2). It generates succinct (64 byte) signatures and has fast verification times.

See Also:

• ed25519

Merkle Trees

Merkle trees are a type of authenticated data structure that consits of a sequence of data that is divided into an even number of partitions which are incrementally hashed in a binary tree, with each level of the tree hashing to produce the hash of the next level until the root of the tree is reached. The root hash is called the Merkle root and uniquely identifies the data included under it. Any change to the leaves, or any reorordering of the nodes will produce a different hash.

A Merkle tree admits an efficient “proof of inclusion” where to produce evidence that a single node is included in the set can be done by simply tracing the roots of a single node up to the binary tree to the root. This is a logarithmic order set of hashes and is quite efficient.

Secure Memory Handling

When using Haskell for cryptography work and even inside web services, some care must be taken to ensure that the primitives you are using don’t accidentally expose secrets or user data accidentally. This can occur in many ways through the mishandling of keys, timing attacks against interactive protocols, and the insecure wiping of memory.

When using Haskell integers be aware that arithmetic operations are not constant time and are simply backed by GMP integers. This may or may not be appropriate for your code if you expect arithmetic operations to be branch-free or have constant time addition or multiplication. If you need constant arithmetic you will likely have to drop down to C or Assembly and link the resulting code into your Haskell logic. Many Haskell cryptography libraries do just this.

With regards to timing attacks, take note of which functions are marked as vulnerable to timing attacks as most of these are marked in public API documentation.

When comparing hashes and unencrypted data for equality also make sure to use an equality test which is constant time. The default derived instance for Eq does not have this property. The securemem library provides a SecureMem datatype which can hold an arbitrary sized ByteString and can only be compared against other SecureMem ByteStrings by a constant time algorithm.

This data structure will also automatically scrub its bytes with a runtime integrated finalizer on the pointer to the underlying memory. This ensures that as soon as the value is garbage collected, its underlying memory is wiped to zero values and does not linger on the process’s memory.

AES Encryption

AES (Advanced Encryption Standard) is a symmetric block cipher standardized by NIST. The cipher block size is fixed at 16 bytes and it is encrypted using a key of 128, 192 or 256 bits. AES is common cipher standard for symmetric encryption and used heavily in internet protocols.

An example of encrypting and decrypting data using the cryptonite library is shown below:

Galois Fields

Many modern cryptographic protocols require the use of finite field arithmetic. Finite fields are algebraic structures that have algebraic field structure (addition, multiplication, division) and closure

See:

• galois-field

Elliptic Curves

Elliptic curves are a type of algebraic structure that are used heavily in cryptography. Most generally elliptic curves are families of curves to second order plane curves in two variables defined over finite fields. These elliptic curves admit a group construction over the curve points which has multiplication and addition. For finite fields with large order computing inversions is quite computationally difficult and gives rise to a trapdoor function which is easy to compute in one direction but difficult in reverse.

There are many types of plane curves with different coefficients that can be defined. The widely studied groups are one of the four classes. These are defined in the elliptic-curve library as lifted datatypes which are used at the type-level to distinguish curve operations.

• Binary
• Edwards
• Montgomery
• Weierstrass

On top of these curves there is an additional degree of freedom in the choice of coordinate system used. There are many ways to interpret the Cartesian plane in terms of coordinates and some of these coordinate systems admit more efficient operations for multiplication and addition of points.

• Affine
• Jacobian
• Projective

For example the common Ed25519 curve can be defined as the following group structure defined as a series of type-level constructions:

Operations on this can be executed by several type classes functions.

See: elliptic-curve

Pairing Cryptography

Cryptographic pairings are a novel technique that allows us to construct bilinear mappings of the form:

e : 𝔾₁ × 𝔾₂ → 𝔾_T

These are bilinear over group addition and multiplication.

e(g₁ + g₂, h) = e(g₁, h)e(g₂, h)

e(g, h₁ + h₂) = e(g, h₁)e(g, h₂)

There are many types of pairings that can be computed. The pairing library implements the Ate pairing over several elliptic curve groups including the Barreto-Naehrig family and the BLS12-381 curve. These types of pairings are used quite frequently in modern cryptographic protocols such as the construction of zkSNARKs.

See

• Pairing
• Optimal Ate Pairing

zkSNARKs

zkSNARKS (zero knowledge succinct non-interactive arguments of knowledge) are a modern cryptographic construction that enable two parties called the Prover and Verifier to convince the verifier that a general computational statement is true without revealing anything else.

Haskell has a variety of libraries for building zkSNARK protocols including libraries to build circuit representations of embedded domain specific languages and produce succinct pairing based zero knowledge proofs.

• zkp - Implementation of the Groth16 protocol based on bilinear pairings.
• bulletproofs - Implementation of the Bulletproofs protocol.
• arithmetic-circuits Generic data structures for construction arithmetic circuits and Rank-1 constraint systems (R1CS) in Haskell.

time

Haskell’s datetime library is unambiguously called time it exposes six core data structure which hold temporal quantities of various precisions.

• Day - Datetime triple of day, month, year in the Gregorian calendar system
• TimeOfDay - A clock time measure in hours, minutes and seconds
• UTCTime - A unix time measured in seconds since the Unix epoch.
• TimeZone - A ISO8601 timezone
• LocalTime - A Day and TimeOfDay combined into a aggregate type.
• ZonedTime - A LocalTime combined with TimeZone.

There are several delta types that correspond to changes in time measured in various units of days or seconds.

• NominalDiffTime - Time delta measured in picoseconds.
• CalendarDiffDays - Calendar delta measured in months and days offset.
• CalendarDiffTime - Time difference measured in months and picoseconds.

ISO8601

The ISO standard for rendering and parsing datetimes can work with the default temporal datatypes. These work bidirectionally for both parsing and pretty printing. Simple use case is shown below:

JSON

Aeson is a library for efficient parsing and generating JSON. It is the canonical JSON library for handling JSON.

A point of some subtlety to beginners is that the return types for Aeson functions are polymorphic in their return types meaning that the resulting type of decode is specified only in the context of your programs use of the decode function. So if you use decode in a point your program and bind it to a value x and then use x as if it were an integer throughout the rest of your program, Aeson will select the typeclass instance which parses the given input string into a Haskell integer.

• Aeson Library

Value

Aeson uses several high performance data structures (Vector, Text, HashMap) by default instead of the naive versions so typically using Aeson will require that we import them and use OverloadedStrings when indexing into objects.

The underlying Aeson structure is called Value and encodes a recursive tree structure that models the semantics of untyped JSON objects by mapping them onto a large sum type which embodies all possible JSON values.

For instance the Value expansion of the following JSON blob:

Is represented in Aeson as the Value:

Let’s consider some larger examples, we’ll work with this contrived example JSON:

Unstructured or Dynamic JSON

In dynamic scripting languages it’s common to parse amorphous blobs of JSON without any a priori structure and then handle validation problems by throwing exceptions while traversing it. We can do the same using Aeson and the Maybe monad.

Structured JSON

This isn’t ideal since we’ve just smeared all the validation logic across our traversal logic instead of separating concerns and handling validation in separate logic. We’d like to describe the structure before-hand and the invalid case separately. Using Generic also allows Haskell to automatically write the serializer and deserializer between our datatype and the JSON string based on the names of record field names.

Now we get our validated JSON wrapped up into a nicely typed Haskell ADT.

The functions fromJSON and toJSON can be used to convert between this sum type and regular Haskell types with.

As of 7.10.2 we can use the new -XDeriveAnyClass to automatically derive instances of FromJSON and TOJSON without the need for standalone instance declarations. These are implemented entirely in terms of the default methods which use Generics under the hood.

{-# LANGUAGE DeriveAnyClass #-}
{-# LANGUAGE DeriveGeneric #-}

import Data.Aeson
import Data.ByteString.Lazy.Char8 as BL
import Data.Text
import GHC.Generics

data Refs
  = Refs
      { a :: Text,
        b :: Text
      }
  deriving (Show, Generic, FromJSON, ToJSON)

data Data
  = Data
      { id :: Int,
        name :: Text,
        price :: Int,
        tags :: [Text],
        refs :: Refs
      }
  deriving (Show, Generic, FromJSON, ToJSON)

main :: IO ()
main = do
  contents <- BL.readFile "example.json"
  let Just dat = decode contents
  print $ name dat
  print $ a (refs dat)
  BL.putStrLn $ encode dat

Hand Written Instances

While it’s useful to use generics to derive instances, sometimes you actually want more fine grained control over serialization and de serialization. So we fall back on writing ToJSON and FromJSON instances manually. Using FromJSON we can project into hashmap using the (.:) operator to extract keys. If the key fails to exist the parser will abort with a key failure message. The ToJSON instances can never fail and simply require us to pattern match on our custom datatype and generate an appropriate value.

The law that the FromJSON and ToJSON classes should maintain is that encode . decode and decode . encode should map to the same object. Although in practice there many times when we break this rule and especially if the serialize or de serialize is one way.

See: Aeson Documentation

Yaml

Yaml is a textual serialization format similar to JSON. It uses an indentation sensitive structure to encode nested maps of keys and values. The Yaml interface for Haskell is a precise copy of Data.Aeson

• Yaml Library

YAML Input:

YAML Output:

Object
  (fromList
     [ ( "invoice" , Number 34843.0 )
     , ( "date" , String "2001-01-23" )
     , ( "bill-to"
       , Object
           (fromList
              [ ( "address"
                , Object
                    (fromList
                       [ ( "state" , String "MI" )
                       , ( "lines" , String "458 Walkman Dr.nSuite #292n" )
                       , ( "city" , String "Royal Oak" )
                       , ( "postal" , Number 48046.0 )
                       ])
                )
              , ( "family" , String "Dumars" )
              , ( "given" , String "Chris" )
              ])
       )
     ])

To parse this file we use the following datatypes and functions:

{-# LANGUAGE DeriveAnyClass #-}
{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE ScopedTypeVariables #-}

import qualified Data.ByteString as BL
import Data.Text (Text)
import Data.Yaml
import GHC.Generics

data Invoice
  = Invoice
      { invoice :: Int,
        date :: Text,
        bill :: Billing
      }
  deriving (Show, Generic, FromJSON)

data Billing
  = Billing
      { address :: Address,
        family :: Text,
        given :: Text
      }
  deriving (Show, Generic, FromJSON)

data Address
  = Address
      { lines :: Text,
        city :: Text,
        state :: Text,
        postal :: Int
      }
  deriving (Show, Generic, FromJSON)

main :: IO ()
main = do
  contents <- BL.readFile "example.yaml"
  let (res :: Either ParseException Invoice) = decodeEither' contents
  case res of
    Left err -> print err
    Right val -> print val