What I Wish I Knew When Learning Haskell

Version 2.3

Stephen Diehl (@smdiehl )

This is the fourth draft of this document.


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

You may copy and paste any code here verbatim into your codebase, wiki, blog, book or Haskell musical production as you see fit. The Markdown and Haskell source is available on Github. Pull requests are always accepted for changes and additional content. This is a living document.

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Sections that have had been added or seen large changes:



Historically Cabal had a component known as cabal-install that has largely been replaced by Stack. The following use of Cabal sandboxes is left for historical reasons and can often be replaced by modern tools.

Cabal is the build system for Haskell.

For example, to install the parsec package to your system from Hackage, the upstream source of Haskell packages, invoke the install command:

$ cabal install parsec           # latest version
$ cabal install parsec==3.1.5    # exact version

The usual build invocation for Haskell packages is the following:

$ cabal get parsec    # fetch source
$ cd parsec-3.1.5

$ cabal configure
$ cabal build
$ cabal install

To update the package index from Hackage, run:

$ cabal update

To start a new Haskell project, run:

$ cabal init
$ cabal configure

A .cabal file will be created with the configuration options for our new project.

The latest feature of cabal is the addition of Sandboxes, ( in cabal > 1.18 ) which are self contained environments of Haskell packages separate from the global package index stored in the ./.cabal-sandbox of our project's root. To create a new sandbox for our cabal project, run:

$ cabal sandbox init

Additionally, the sandbox can be torn down:

$ cabal sandbox delete

When in the working directory of a project with a sandbox that has a configuration already set up, invoking cabal commands alters the behaviour of cabal itself. For instance, the cabal install command will alter only the install to the local package index, not the global configuration.

To install the dependencies from the .cabal file into the newly created sandbox, run:

$ cabal install --only-dependencies

Dependencies can also be built in parallel by passing -j<n> where n is the number of concurrent builds.

$ cabal install -j4 --only-dependencies

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 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.

name:               mylibrary
version:            0.1
cabal-version:      >= 1.10
author:             Paul Atreides
license:            MIT
license-file:       LICENSE
synopsis:           The code must flow.
category:           Math
tested-with:        GHC
build-type:         Simple


      base >= 4 && < 5

    default-language: Haskell2010

    ghc-options: -O2 -Wall -fwarn-tabs

executable "example"
        base >= 4 && < 5,
        mylibrary == 0.1
    default-language: Haskell2010
    main-is: Main.hs

Test-Suite test
  type: exitcode-stdio-1.0
  main-is: Test.hs
  default-language: Haskell2010
      base >= 4 && < 5,
      mylibrary == 0.1

To run an "executable" for a project under the cabal sandbox:

$ cabal run
$ cabal run <name> # when there are several executables in a project

To load the "library" into a GHCi shell under cabal sandbox:

$ cabal repl
$ cabal repl <name>

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

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

$ cabal build

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

$ cabal install --only-dependencies --enable-tests
$ cabal configure --enable-tests
$ cabal test
$ cabal test <name>

Moreover, arbitrary shell commands can be invoked with the GHC environmental variables set up for the sandbox. Quite common is to invoke a new shell with this command such that the ghc and ghci commands use the sandbox. ( They don't by default, which is a common source of frustration. ).

$ cabal exec
$ cabal exec sh # launch a shell with GHC sandbox path set.

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

$ cabal haddock

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:

$ cabal sdist
$ cabal upload dist/mylibrary-0.1.tar.gz

Sometimes you'd also like to add a library from a local project into a sandbox. In this case, run the add-source command to bring the library into the sandbox from a local directory:

$ cabal sandbox add-source /path/to/project

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

$ cabal freeze

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

constraints: mtl ==2.2.1,
             text ==,
             transformers ==

Using the cabal repl and cabal run commands is preferable, but sometimes we'd like to manually perform their equivalents at the shell. Several useful aliases rely on shell directory expansion to find the package database in the current working directory and launch GHC with the appropriate flags:

alias ghc-sandbox="ghc -no-user-package-db -package-db .cabal-sandbox/*-packages.conf.d"
alias ghci-sandbox="ghci -no-user-package-db -package-db .cabal-sandbox/*-packages.conf.d"
alias runhaskell-sandbox="runhaskell -no-user-package-db -package-db .cabal-sandbox/*-packages.conf.d"

There is also a zsh script to show the sandbox status of the current working directory in our shell:

function cabal_sandbox_info() {
    if [ $#cabal_files -gt 0 ]; then
        if [ -f cabal.sandbox.config ]; then
            echo "%{$fg[green]%}sandboxed%{$reset_color%}"
            echo "%{$fg[red]%}not sandboxed%{$reset_color%}"

RPROMPT="\$(cabal_sandbox_info) $RPROMPT"

The cabal configuration is stored in $HOME/.cabal/config and contains various options including credential information for Hackage upload. One addition to configuration is to completely disallow the installation of packages outside of sandboxes to prevent accidental collisions.

-- Don't allow global install of packages.
require-sandbox: True

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

library-profiling: True

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.

documentation: True

If GHC is currently installed, the documentation for the Prelude and Base libraries should be available at this local link:




Stack is a new approach to Haskell package structure that emerged in 2015. Instead of using a rolling build like cabal-install, 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, more robust strategy for resolving dependencies than cabal-install has historically used.

Contrary to much misinformation, Stack does not replace Cabal as the build system and uses it under the hood. Stack simply streamlines integration with third-party packages and the resolution of their dependencies.


To install stack on Ubuntu Linux, run:

sudo apt-key adv --keyserver keyserver.ubuntu.com --recv-keys 575159689BEFB442                             # get fp complete key
echo 'deb http://download.fpcomplete.com/ubuntu trusty main'|sudo tee /etc/apt/sources.list.d/fpco.list    # add appropriate source repo
sudo apt-get update && sudo apt-get install stack -y

For other operating systems, see the official install directions.


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

stack init

An example stack.yaml file for GHC 7.10.3 would look like:

resolver: lts-7.12
flags: {}
extra-package-dbs: []
packages: []
extra-deps: []

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 the stack build:

- zenc-0.1.1

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:

$ stack install hlint

To check the set of dependencies, run:

$ stack list-dependencies

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

$ stack build                 # Build a cabal target
$ stack repl                  # Launch ghci
$ stack ghc                   # Invoke the standalone compiler in stack environment
$ stack exec bash             # Execute a shell command with the stack GHC environment variables
$ stack build --file-watch    # Build on every filesystem change

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

$ stack dot --external | dot -Tpng | feh -


Enabling GHC compiler flags grants the user more control in detecting common code errors. The most frequently used flags are:

Flag Description
-fwarn-tabs Emit warnings of tabs instead of spaces in the source code
-fwarn-unused-imports Warn about libraries imported without being used
-fwarn-name-shadowing Warn on duplicate names in nested bindings
-fwarn-incomplete-uni-patterns Emit warnings for incomplete patterns in lambdas or pattern bindings
-fwarn-incomplete-patterns Warn on non-exhaustive patterns
-fwarn-overlapping-patterns Warn on pattern matching branches that overlap
-fwarn-incomplete-record-updates Warn when records are not instantiated with all fields
-fdefer-type-errors Turn type errors into warnings
-fwarn-missing-signatures Warn about toplevel missing type signatures
-fwarn-monomorphism-restriction Warn when the monomorphism restriction is applied implicitly
-fwarn-orphans Warn on orphan typeclass instances
-fforce-recomp Force recompilation regardless of timestamp
-fno-code Omit code generation, just parse and typecheck
-fobject-code Generate object code

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:

library mylib


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 is the upstream source of Free and/or 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.

Reusable Code / Building Blocks

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 author(s) 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.

A Staging Area / Request for Comments

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 author(s) often rationalize putting these kind of libraries up undocumented, often without indication of what the library actually does, by simply stating that they intend to tear the code down and rewrite it later. This approach 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, to accompany a paper, or just to integrate with the cabal build system. 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 worked example, it is usually safe to assume that it is an RFC-style library and probably should be avoided in production-grade code.

Similarly, if the library predates the text library (released circa 2007), it probably should be avoided in production code. The way we write Haskell has changed drastically since the early days.


GHCi is the interactive shell for the GHC compiler. GHCi is where we will spend most of our time in every day development.

Command Shortcut Action
:reload :r Code reload
:type :t Type inspection
:kind :k Kind inspection
:info :i Information
:print :p Print the expression
:edit :e Load file in system editor
:load :l Set the active Main module in the REPL
:add :ad Load a file into the REPL namespace
:browse :bro Browse all available symbols in the REPL namespace

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

λ: :type 3
3 :: Num a => a
λ: :kind Either
Either :: * -> * -> *
λ: :info Functor
class Functor f where
  fmap :: (a -> b) -> f a -> f b
  (<$) :: a -> f b -> f a
        -- Defined in `GHC.Base'
λ: :i (:)
data [] a = ... | a : [a]       -- Defined in `GHC.Types'
infixr 5 :

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:

λ: :browse
λ: :show bindings

Examining module-level imports, execute:

λ: :show imports
import Prelude -- implicit
import Data.Eq
import Control.Monad

To see compiler-level flags and pragmas, use:

λ: :set
options currently set: none.
base language is: Haskell2010
with the following modifiers:
GHCi-specific dynamic flag settings:
other dynamic, non-language, flag settings:
warning settings:

λ: :showi language
base language is: Haskell2010
with the following modifiers:

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:

+t Show types of evaluated expressions
+s Show timing and memory usage
+m Enable multi-line expression delimited by :{ and :}.
λ: :set +t
λ: []
it :: [a]
λ: :set +s
λ: foldr (+) 0 [1..25]
it :: Prelude.Integer
(0.02 secs, 4900952 bytes)
λ: :{
λ:| let foo = do
λ:|           putStrLn "hello ghci"
λ:| :}
λ: foo
"hello ghci"

The configuration for the GHCi shell can be customized globally by defining a 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:

cabal install hoogle

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

:set prompt "λ: "

:def hlint const . return $ ":! hlint \"src\""
:def hoogle \s -> return $ ":! hoogle --count=15 \"" ++ s ++ "\""
λ: :hoogle (a -> b) -> f a -> f b
Data.Traversable fmapDefault :: Traversable t => (a -> b) -> t a -> t b
Prelude fmap :: Functor f => (a -> b) -> f a -> f b

For reasons of sexiness, it is desirable to set your GHC prompt to a λ or a λΠ. Only if you're into that lifestyle, though.

:set prompt "λ: "
:set prompt "ΠΣ: "

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.

:set -fobject-code

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 under specificity 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.

:set -fbyte-code
:load MyModule.hs

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

:set -fno-code

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.

Several prepackaged setups exist to expedite the process of setting up many of the programmer editors for Haskell development. In particular, using ghc-mod can remarkably improve programmer efficiency and productivity because the project attempts to implement features common to modern IDEs.





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.

undefined :: a

mean :: Num a => Vector a -> a
mean nums = (total / count) where            -- Partially defined function
              total = undefined
              count = undefined

addThreeNums :: Num a => a -> a -> a -> a
addThreeNums n m j = undefined               -- No function body declared at all

f :: a -> Complicated Type
f = undefined                                -- Write tomorrow, typecheck today!
                                             -- Arbitrarily complicated types
                                             -- welcome!

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 results in such an exception.

error :: String -> a                       -- Takes an error message of type
                                           -- String and returns whatever type
                                           -- is needed

-- Annotated code that features use of the error function.

divByY:: (Num a, Eq a, Fractional a) => a -> a -> a
divByY _ 0 = error "Divide by zero error"      -- Dividing by 0 causes an error
divByY dividend divisor = dividend / divisor   -- Handles defined division

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

f :: a
f = let x = x in x

Examples of actual Haskell code that use this looping syntax live 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:

data F = A | B
case x of
  A -> ()

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

case x of _ {
  A -> ();
  B -> patError "<interactive>:3:11-31|case"

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.

data Foo = Foo { example1 :: Int }
f = Foo {}     -- Record defined with a missing field

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

Foo (recConError "<interactive>:4:9-12|a")

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.

import GHC.Err
import Prelude hiding (head, (!!), undefined)

-- degenerate functions

undefined :: a
undefined =  error "Prelude.undefined"

head :: [a] -> a
head (x:_) =  x
head []    =  error "Prelude.head: empty list"

(!!) :: [a] -> Int -> a
xs     !! n | n < 0 =  error "Prelude.!!: negative index"
[]     !! _         =  error "Prelude.!!: index too large"
(x:_)  !! 0         =  x
(_:xs) !! n         =  xs !! (n-1)

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 usual fold functions maybe and either.

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

listToMaybe :: [a] -> Maybe a
listToMaybe []     =  Nothing    -- An empty list returns Nothing
listToMaybe (a:_)  =  Just a     -- A non-empty list returns the first element
                                 -- wrapped in the Just context.

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 the place of either undefined or error calls.

import GHC.Base

foo :: a
foo = undefined
-- *** Exception: Prelude.undefined

bar :: a
bar = assert False undefined
-- *** Exception: src/fail.hs:8:7-12: Assertion failed

See: Avoiding Partial Functions


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.

unsafe :: Num a => Maybe a -> Maybe a
unsafe (Just x) = Just $ x + 1

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-exhaustively 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.

$ ghc -c -Wall -Werror A.hs
    Warning: Pattern match(es) are non-exhaustive
             In an equation for `unsafe': Patterns not matched: Nothing

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

{-# OPTIONS_GHC -Wall #-}
{-# OPTIONS_GHC -fwarn-incomplete-patterns #-}

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.

boom = \(Just a) -> something

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.

boom2 = let
  Just a = something

boom3 = do
  Just a <- something

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

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


Since GHCi version 6.8.1, a built-in debugger has been available, although its use is somewhat rare. Debugging uncaught exceptions from bottoms or asynchronous exceptions is in similar style to debugging segfaults with gdb.

λ: :set -fbreak-on-exception       -- Sets option for evaluation to stop on exception
λ: :break 2 15                     -- Sets a break point at line 2, column 15
λ: :trace main                     -- Run a function to generate a sequence of evaluation steps
λ: :hist                           -- Step backwards from a breakpoint through previous steps of evaluation
λ: :back                           -- Step backwards a single step at a time through the history
λ: :forward                        -- Step forward a single step at a time through the history

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:

import Control.Exception

f x = g x

g x = error (show x)

main = try (evaluate (f ())) :: IO (Either SomeException ())
$ ghc -O0 -rtsopts=all -prof -auto-all --make stacktrace.hs
./stacktrace +RTS -xc

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

*** Exception (reporting due to +RTS -xc): (THUNK_2_0), stack trace:
  called from Main.f,
  called from Main.main,
  called from Main.CAF
  --> evaluated by: Main.main,
  called from Main.CAF

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.



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.

import Debug.Trace

example1 :: Int
example1 = trace "impure print" 1

example2 :: Int
example2 = traceShow "tracing" 2

example3 :: [Int]
example3 = [trace "will not be called" 3]

main :: IO ()
main = do
  print example1
  print example2
  print $ length example3
-- impure print
-- 1
-- "tracing"
-- 2
-- 1

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

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

import Text.Printf
import Debug.Trace

traceM :: (Monad m) => String -> m ()
traceM string = trace string $ return ()

traceShowM :: (Show a, Monad m) => a -> m ()
traceShowM = traceM . show

tracePrintfM :: (Monad m, PrintfArg a) => String -> a -> m ()
tracePrintfM s = traceM . printf s

Type Inference

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

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

f x = const x g
g y = f 'A'

The inferred type signatures are correct in their usage, but 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.

-- Inferred types
f :: Char -> Char
g :: t -> Char

-- Most general types
f :: a -> a
g :: a -> Char

Polymorphic recursion

data Tree a = Leaf | Bin a (Tree (a, a))

size Leaf = 0
size (Bin _ t) = 1 + 2 * size t

The 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.

    Occurs check: cannot construct the infinite type: t0 = (t0, t0)
    Expected type: Tree t0
      Actual type: Tree (t0, t0)
    In the first argument of `size', namely `t'
    In the second argument of `(*)', namely `size t'
    In the second argument of `(+)', namely `2 * size t'

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

size :: Tree a -> Int
size Leaf = 0
size (Bin _ t) = 1 + 2 * size t

See: Static Semantics of Function and Pattern Bindings

Monomorphism Restriction

Monomorphism restriction is a controversial 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:

λ: :set +t

λ: 3
it :: Num a => a

λ: default (Double)

λ: 3
it :: Num a => a

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

See: Monomorphism Restriction

Type Holes / Pattern Wildcards

Since the release of GHC 7.8, type holes, or pattern wildcards, 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 of 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.

head' = head _
typedhole.hs:3:14: error:
    • Found hole: _ :: [a]
      Where: ‘a’ is a rigid type variable bound by
               the inferred type of head' :: a at typedhole.hs:3:1
    • In the first argument of ‘head’, namely ‘_’
      In the expression: head _
      In an equation for ‘head'’: head' = head _
    • Relevant bindings include head' :: a (bound at typedhole.hs:3:1)

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:

const' :: _
const' x y = x
typedhole.hs:5:11: error:
    • Found type wildcard ‘_’ standing for ‘t -> t1 -> t’
      Where: ‘t1’ is a rigid type variable bound by
               the inferred type of const' :: t -> t1 -> t at typedhole.hs:6:1
             ‘t’ is a rigid type variable bound by
               the inferred type of const' :: t -> t1 -> t at typedhole.hs:6:1
      To use the inferred type, enable PartialTypeSignatures
    • In the type signature:
        const' :: _
    • Relevant bindings include
        const' :: t -> t1 -> t (bound at typedhole.hs:6:1)

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

foo :: _a -> _a
foo _ = False
typedhole.hs:9:9: error:
    • Couldn't match expected type ‘_a’ with actual type ‘Bool’
      ‘_a’ is a rigid type variable bound by
        the type signature for:
          foo :: forall _a. _a -> _a
        at typedhole.hs:8:8
    • In the expression: False
      In an equation for ‘foo’: foo _ = False
    • Relevant bindings include
        foo :: _a -> _a (bound at typedhole.hs:9:1)

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

succ' :: _ => a -> a
succ' x = x + 1
typedhole.hs:11:10: error:
    Found constraint wildcard ‘_’ standing for ‘Num a’
    To use the inferred type, enable PartialTypeSignatures
    In the type signature:
      succ' :: _ => a -> a

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.

typedhole.hs:3:10: Warning:
    Found hole ‘_’ with type: w_
    Where: ‘w_’ is a rigid type variable bound by
                the inferred type of succ' :: w_ -> w_1 -> w_ at foo.hs:4:1
    In the type signature for ‘succ'’: _ -> _ -> _

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:

{-# OPTIONS_GHC -fdefer-type-errors #-} -- Enable deferred type
                                        -- errors at module level

x :: ()
x = print 3

y :: Char
y = 0

z :: Int
z = 0 + "foo"

main :: IO ()
main = do
  print x

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

defer: defer.hs:4:5:
    Couldn't match expected type ‘()’ with actual type ‘IO ()’
    In the expression: print 3
    In an equation for ‘x’: x = print 3
(deferred type error)

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.


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).

ghcid --command="cabal repl"   # Run cabal repl under ghcid
ghcid --command="stack repl"   # Run stack repl under ghcid
ghcid --command="ghci baz.hs"  # Open baz.hs under ghcid

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.


Haddock is the automatic documentation generation tool for Haskell source code. 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:

-- | Documentation for f
f :: a -> a
f = ...

Multiline comments are also possible:

-- | Multiline documentation for the function
-- f with multiple arguments.
fmap :: Functor f =>
     => (a -> b)  -- ^ function
     -> f a       -- ^ input
     -> f b       -- ^ output

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

data T a b
  = A a -- ^ Documentation for A
  | B b -- ^ Documentation for B

data R a b = R
  { f1 :: a -- ^ Documentation for the field f1
  , f2 :: b -- ^ Documentation for the field f2

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

data T a b
  = A a -- ^ Documentation for 'A'
  | B b -- ^ Documentation for 'B'

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

-- | Here we use the "Data.Text" library and import
-- the 'Data.Text.pack' function.

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:

-- | An example of a code block.
-- @
--    f x = f (f x)
-- @

Similarly, it's possible to use bird tracks (>) in a comment line to set off a code block. This usage is very similar to Bird style Literate Haskell.

-- | A similar code block example that uses bird tracks (i.e. '>')
-- > f x = f (f x)

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:

-- | Example of an interactive shell session embedded within documentation
-- >>> factorial 5
-- 120

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

module Foo (
  -- * My Header

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

module Foo (
  -- $section1

-- $section1
-- Here is the documentation section that describes the symbols
-- 'example1' and 'example2'.

Links can be added with the following syntax:

<url text>

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

<<diagram.png title>>

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

{-# OPTIONS_HADDOCK show-extensions, ignore-exports #-}
Option Description
ignore-exports Ignores the export list and includes all signatures in scope.
not-home Module will not be considered in the root documentation.
show-extensions Annotates the documentation with the language extensions used.
hide Forces the module to be hidden from Haddock.
prune Omits definitions with no annotations.


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 Haskell types.
  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.

Monadic Myths

The following are all false:

See: What a Monad Is Not

Monadic 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 in accordance with the arity described in the typeclass definition:

class Monad m where
  return :: a -> m a                    -- N.B. 'm' refers to a type constructor
                                        -- (e.g., Maybe, Either, etc.) that
                                        -- implements the Monad typeclass

  (>>=)  :: m a -> (a -> m b) -> m b

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 like 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 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.

(>>) :: Monad m => m a -> m b -> m b
m >> k = m >>= \_ -> k

This definition says that (>>) has a left and right argument which are monadic with types m a and m b respectively, while the infix returns 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 returned.


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 a (>>=) into a function f, this expression is exactly equivalent to f a.

return a >>= f ≡ f a    -- N.B. 'a' refers to a value, not a type

In discussing the next two laws, we'll refer to a value m. This notation is shorthand for 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.

m >>= return ≡ m        -- 'm' here refers to a value that has type 'm a'

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.

(SomeMonad val) >>= return ≡ SomeMonad val  -- 'SomeMonad val' has type 'm a' just
                                            -- like 'm' from the first example of the
                                            -- second law

NullaryMonadType >>= return ≡ NullaryMonadType

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.

(m >>= f) >>= g ≡ m >>= (\x -> f x >>= g)  -- Like in the last law, 'm' has
                                           -- has type 'm a'. The functions 'f'
                                           -- and 'g' have types '(a -> m b)'
                                           -- and '(b -> m c)' respectively

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 very clear that there can be multiple components to a monadic value. Although little has changed in the code, it is easier to see what 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.

((SomeMonad val) >>= f) >>= g ≡ (SomeMonad val) >>= (\x -> f x >>= g)

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 is entirely equivalent to just applications of the monad operations. The desugaring is defined recursively by the rules:

do { a <- f ; m } ≡ f >>= \a -> do { m }  -- bind 'f' to a, proceed to desugar
                                          -- 'm'

do { f ; m } ≡ f >> do { m }              -- evaluate 'f', then proceed to
                                          -- desugar  m

do { m } ≡ m

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

  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 ( this is not how Haskell uses it ) the same desugaring might look something like the following chain of nested binds with lambdas.

bindMonad(f, lambda a:
  bindMonad(g, lambda b:
    bindMonad(h, lambda c:
      returnMonad (a,b,c))))

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

Law 1

  do y <- return x
     f y

= do f x

Law 2

  do x <- m
     return x

= do m

Law 3

  do b <- do a <- m
             f a
     g b

= do a <- m
     b <- f a
     g b

= do a <- m
     do b <- f a
        g b

See: Haskell 2010: Do Expressions


The Maybe monad is the simplest first example of a monad instance. The Maybe monad models computations 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.

data Maybe a = Nothing | Just a

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.

instance Monad Maybe where
  (Just x) >>= k = k x            -- 'k' is a function with type  (a -> Maybe a)
  Nothing  >>= k = Nothing

  return = Just                   -- Just's type signature is (a -> Maybe a), in
                                  -- other words, extremely similar to the
                                  -- type of 'return' in the typeclass
                                  -- declaration above.

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

In the first 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, where return (x + 1) evaluates to Just 4, indicating a successful computation.

(Just 3) >>= (\x -> return (x + 1))
-- Just 4

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.

Nothing >>= (\x -> return (x + 1))
-- Nothing

In the next example, return is applied to 4 and returns Just 4.

return 4 :: Maybe Int
-- 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.

example1 :: Maybe Int
example1 = do
  a <- Just 3                -- Bind 3 to name a
  b <- Just 4                -- Bind 4 to name b
  return $ a + b             -- Evaluate (a + b), then use 'return' to ensure
                             -- the result is in the Maybe monad in order to
                             -- satisfy the type signature
-- Just 7

desugared1 :: Maybe Int
desugared1 = Just 3 >>= \a ->    -- This example is the desugared
               Just 4 >>= \b ->  -- equivalent to example1
                 return $ a + b
-- Just 7

example2 :: Maybe Int
example2 = do
  a <- Just 3                -- Bind 3 to name a
  b <- Nothing               -- Bind Nothing to name b
  return $ a + b
-- Nothing                   -- This result might be somewhat surprising, since
                             -- addition within the Maybe monad can actually
                             -- return 'Nothing'. This result occurs because one
                             -- of the values, Nothing, indicates computational
                             -- failure. Since the computation failed at one
                             -- step within the process, the whole computation
                             -- fails, leaving us with 'Nothing' as the final
                             -- result.

desugared2 :: Maybe Int
desugared2 = Just 3 >>= \a ->     -- This example is the desugared
               Nothing >>= \b ->  -- equivalent to example2
                 return $ a + b
-- Nothing


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].

instance Monad [] where
  m >>= f   =  concat (map f m)          -- 'm' is a list
  return x  =  [x]

In order to demonstrate the List monad's methods, we will define two functions: 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].

m :: [Int]
m = [1,2,3,4]

f :: Int -> [Int]
f = \x -> [1,0]               -- 'f' always returns [1, 0]

The 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.

a = [
      f x y |        -- Corresponds to 'f x y' in example b
      x <- xs,
      y <- ys,
      x == y         -- Corresponds to 'guard $ x == y' in example b

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

-- Identical to `a`
b = do
  x <- xs
  y <- ys
  guard $ x == y     -- Corresponds to 'x == y' in example a
  return $ f x y     -- Corresponds to the '[]' and 'f x y' in example a

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)]


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 in the external world. IO actions allow the program to do many things, including, but not limited to:

Conceptualizing I/O as a monad enables the developer to access information outside the program, but operate on the data with pure functions. 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.

putStrLn :: String -> IO ()

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.

main :: IO ()
main = putStrLn "Vesihiisi sihisi hississäään." >>=
         \_ -> putStrLn "Or in English: 'The water devil was hissing in her elevator'."

-- Sugared code, written with do notation
main :: IO ()
main = do putStrLn "Vesihiisi sihisi hississäään."
          putStrLn "Or in English: 'The water devil was hissing in her elevator'."

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.

getLine :: IO String

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.

main :: IO ()
main = do putStrLn "What is your name: "
          name <- getLine
          putStrLn name

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

main :: IO ()
main = putStrLn "What is your name:" >>=
       \_    -> getLine >>=
       \name -> putStrLn name

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.

main :: IO ()
main = putStrLn "What is your name: " >> (getLine >>= (\name -> putStrLn name))

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.

sequence :: Monad m => [m a] -> m [a]
sequence = foldr mcons (return [])

mcons :: Monad m => m t -> m [t] -> m [t]
mcons p q = do
  x <- p          -- 'x' refers to a singleton value
  y <- q          -- 'y' refers to a list. Because of this fact, 'x' can be
  return (x:y)    --  prepended to it

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


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 sequence will also be Nothing.

sequence :: [Maybe a] -> Maybe [a]
sequence [Just 3, Just 4]
-- Just [3,4]

sequence [Just 3, Just 4, Nothing]     -- Since one of the results is Nothing,
-- Nothing                             -- the whole computation fails


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 :: [[a]] -> [[a]]
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]]


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 returns the list of resulting values in the order sequenced.

sequence :: [IO a] -> IO [a]
sequence [getLine, getLine, getLine]
-- a                                  -- a, b, and 9 are the inputs given by the
-- b                                  -- user at the prompt
-- 9
-- ["a", "b", "9"]                    -- All inputs are returned in a list as
                                      -- an IO [String].

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.

ask :: Reader r r
asks :: (r -> a) -> Reader r a
local :: (r -> r) -> Reader r a -> Reader r a
runReader :: Reader r a -> r -> a
import Control.Monad.Reader

data MyContext = MyContext
  { foo :: String
  , bar :: Int
  } deriving (Show)

computation :: Reader MyContext (Maybe String)
computation = do
  n <- asks bar
  x <- asks foo
  if n > 0
    then return (Just x)
    else return Nothing

ex1 :: Maybe String
ex1 = runReader computation $ MyContext "hello" 1

ex2 :: Maybe String
ex2 = runReader computation $ MyContext "haskell" 0

A simple implementation of the Reader monad:

newtype Reader r a = Reader { runReader :: r -> a }

instance Monad (Reader r) where
  return a = Reader $ \_ -> a
  m >>= k  = Reader $ \r -> runReader (k (runReader m r)) r

ask :: Reader a a
ask = Reader id

asks :: (r -> a) -> Reader r a
asks f = Reader f

local :: (r -> r) -> Reader r a -> Reader r a
local f m = Reader $ runReader m . f

Writer Monad

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

tell :: w -> Writer w ()
execWriter :: Writer w a -> w
runWriter :: Writer w a -> (a, w)
import Control.Monad.Writer

type MyWriter = Writer [Int] String

example :: MyWriter
example  = do
  tell [1..3]
  tell [3..5]
  return "foo"

output :: (String, [Int])
output = runWriter example
-- ("foo", [1, 2, 3, 3, 4, 5])

A simple implementation of the Writer monad:

import Data.Monoid

newtype Writer w a = Writer { runWriter :: (a, w) }

instance Monoid w => Monad (Writer w) where
  return a = Writer (a, mempty)
  m >>= k  = Writer $ let
      (a, w)  = runWriter m
      (b, w') = runWriter (k a)
      in (b, w `mappend` w')

execWriter :: Writer w a -> w
execWriter m = snd (runWriter m)

tell :: w -> Writer w ()
tell w = Writer ((), w)

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.

import Control.Monad.Writer.Strict

State Monad

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

runState  :: State s a -> s -> (a, s)
evalState :: State s a -> s -> a
execState :: State s a -> s -> s
import Control.Monad.State

test :: State Int Int
test = do
  put 3
  modify (+1)

main :: IO ()
main = print $ execState test 0

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:

newtype State s a = State { runState :: s -> (a,s) }

instance Monad (State s) where
  return a = State $ \s -> (a, s)

  State act >>= k = State $ \s ->
    let (a, s') = act s
    in runState (k a) s'

get :: State s s
get = State $ \s -> (s, s)

put :: s -> State s ()
put s = State $ \_ -> ((), s)

modify :: (s -> s) -> State s ()
modify f = get >>= \x -> put (f x)

evalState :: State s a -> s -> a
evalState act = fst . runState act

execState :: State s a -> s -> s
execState act = snd . runState act

Monad Tutorials

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 on 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 code and further fuels that misconception that specific instances like IO are monads in their full generality.

main = do
  x <- getLine
  putStrLn x
  return ()

Being able to manually desugar is crucial to understanding.

main =
  getLine >>= \x ->
    putStrLn x >>= \_ ->
      return ()
  1. Asymmetric binary infix operators for higher order functions are not common in other languages.
(>>=) :: Monad m => m a -> (a -> m b) -> m b

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

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

main =
  getLine >>= \x ->
    putStrLn >>= \_ ->
      return ()

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

main =
  (>>=) getLine (\x ->
    (>>=) putStrLn (\_ ->
          return ()

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

main = bind getLine (\x -> bind putStrLn (\_ -> return ()))
    bind x y = x >>= y
  1. 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. It is 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.

main $dMonad = bind $dMonad getLine (\x -> bind $dMonad putStrLn (\_ -> return $dMonad ()))

Except in the case where the parameter of the monad class is unified ( through inference ) with a concrete class instance, in which case the instance dictionary ($dMonadIO) is instead spliced throughout.

main :: IO ()
main = bind $dMonadIO getLine (\x -> bind $dMonadIO putStrLn (\_ -> return $dMonadIO ()))

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.

See: Monad Tutorial Fallacy

Monad Transformers

mtl / transformers

So, 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. The monads mentioned previously are subsumed by the special case of the transformer form composed with the Identity monad.

Monad Transformer Type Transformed Type
Maybe MaybeT Maybe a m (Maybe a)
Reader ReaderT r -> a r -> m a
Writer WriterT (a,w) m (a,w)
State StateT s -> (a,s) s -> m (a,s)
type State  s = StateT  s Identity
type Writer w = WriterT w Identity
type Reader r = ReaderT r Identity

instance Monad m => MonadState s (StateT s m)
instance Monad m => MonadReader r (ReaderT r m)
instance (Monoid w, Monad m) => MonadWriter w (WriterT w m)

In terms of generality the mtl library is the most common general interface for these monads, which itself depends on the transformers library which generalizes the "basic" monads described above into transformers.


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.

lift :: (Monad m, MonadTrans t) => m a -> t m a
liftIO :: MonadIO m => IO a -> m a
class MonadTrans t where
    lift :: Monad m => m a -> t m a

class (Monad m) => MonadIO m where
    liftIO :: IO a -> m a

instance MonadIO IO where
    liftIO = id

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

Law #1

lift . return = return

Law #2

lift (m >>= f) = lift m >>= (lift . f)

Or equivalently:

Law #1

  lift (return x)

= return x

Law #2

  do x <- lift m
     lift (f x)

= lift $ do x <- m
            f x

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

See: Monad Transformers: Step-By-Step


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 ((* -> *) -> * -> *):

Monad (m :: * -> *)
MonadTrans (t :: (* -> *) -> * -> *)

So, for example, if we wanted to form a composite computation using both the Reader and Maybe monads we can now put the Maybe inside of a ReaderT to form ReaderT t Maybe a.

import Control.Monad.Reader

type Env = [(String, Int)]
type Eval a = ReaderT Env Maybe a

data Expr
  = Val Int
  | Add Expr Expr
  | Var String
  deriving (Show)

eval :: Expr -> Eval Int
eval ex = case ex of

  Val n -> return n

  Add x y -> do
    a <- eval x
    b <- eval y
    return (a+b)

  Var x -> do
    env <- ask
    val <- lift (lookup x env)
    return val

env :: Env
env = [("x", 2), ("y", 5)]

ex1 :: Eval Int
ex1 = eval (Add (Val 2) (Add (Val 1) (Var "x")))

example1, example2 :: Maybe Int
example1 = runReaderT ex1 env
example2 = runReaderT ex1 []

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


For example, there exist three possible forms of the Reader monad. The first is the Haskell 98 version that no longer exists, but is useful for understanding the underlying ideas. The other two are the transformers and mtl variants.


newtype Reader r a = Reader { runReader :: r -> a }

instance MonadReader r (Reader r) where
  ask       = Reader id
  local f m = Reader (runReader m . f)


newtype ReaderT r m a = ReaderT { runReaderT :: r -> m a }

instance (Monad m) => Monad (ReaderT r m) where
  return a = ReaderT $ \_ -> return a
  m >>= k  = ReaderT $ \r -> do
      a <- runReaderT m r
      runReaderT (k a) r

instance MonadTrans (ReaderT r) where
    lift m = ReaderT $ \_ -> m


class (Monad m) => MonadReader r m | m -> r where
  ask   :: m r
  local :: (r -> r) -> m a -> m a

instance (Monad m) => MonadReader r (ReaderT r m) where
  ask       = ReaderT return
  local f m = ReaderT $ \r -> runReaderT m (f r)

So, hypothetically the three variants of ask would be:

ask :: Reader r r
ask :: Monad m => ReaderT r m r
ask :: MonadReader r m => m r

In practice only the last one is used in modern Haskell.

Newtype Deriving

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.

newtype Plaintext = Plaintext Text
newtype Crytpotext = Cryptotext Text

encrypt :: Key -> Plaintext -> Cryptotext
decrypt :: Key -> Cryptotext -> Plaintext

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.

{-# LANGUAGE GeneralizedNewtypeDeriving #-}

newtype Velocity = Velocity { unVelocity :: Double }
  deriving (Eq, Ord)

v :: Velocity
v = Velocity 2.718

x :: Double
x = 2.718

-- Type error is caught at compile time even though
-- they are the same value at runtime!
err = v + x

newtype Quantity v a = Quantity a
  deriving (Eq, Ord, Num, Show)

data Haskeller
type Haskellers = Quantity Haskeller Int

a = Quantity 2 :: Haskellers
b = Quantity 6 :: Haskellers

totalHaskellers :: Haskellers
totalHaskellers = a + b
Couldn't match type `Double' with `Velocity'
Expected type: Velocity
  Actual type: Double
In the second argument of `(+)', namely `x'
In the expression: v + x

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 (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,

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

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

data A = MkA Int
newtype B = MkB Int

extractA :: A -> Int
extractA (MkA x) = x

extractB :: B -> Int
extractB (MkB x) = x


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

do x <- lift m  ==  lift $ do x <- m
   lift (f x)                 f x

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.

-- Less Efficient      More Efficient
forever (lift m)    == lift (forever m)
mapM_ (lift . f) xs == lift (mapM_ f xs)
forM_ xs (lift . f) == lift (forM_ xs f)

Monad Morphisms

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

lift :: Monad m => m a -> t m a

But often times 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. The equivalent of type transformer type-level map is the hoist function.

hoist :: Monad m => (forall a. m a -> n a) -> t m b -> t n b

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.

generalize :: Monad m => Identity a -> m a

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.

import Control.Monad.State
import Control.Monad.Morph

type Eval a = State [Int] a

runEval :: [Int] -> Eval a -> a
runEval = flip evalState

pop :: Eval Int
pop = do
  top <- gets head
  modify tail
  return top

push :: Int -> Eval ()
push x = modify (x:)

ev1 :: Eval Int
ev1 = do
  push 3
  push 4

ev2  :: StateT [Int] IO ()
ev2 = do
  result <- hoist generalize ev1
  liftIO $ putStrLn $ "Result: " ++ show result

See: mmorph

Language Extensions

It's important to distinguish between different categories of language extensions general and specialized.

The inherent problem with classifying the extensions into the general and specialized categories is that it's a subjective classification. Haskellers who do type system research will have a very different interpretation of Haskell than people who do web programming. As such this is a conservative assessment, as an arbitrary baseline let's consider FlexibleInstances and OverloadedStrings "everyday" while GADTs and TypeFamilies are "specialized".


Benign Historical Steals Syntax Use Use GHC Reference Reference
AllowAmbiguousTypes Specialized Typelevel Programming Ref
Arrows Specialized Syntax Extension Ref Arrows
AutoDeriveTypeable Specialized Deriving Ref
BangPatterns General Strictness Annotation Ref Strictness Annotations
ApplicativeDo Specialized FFI Ref Applicative Do
CApiFFI Specialized FFI Ref
ConstrainedClassMethods Specialized Typelevel Programming Ref
ConstraintKinds Specialized Typelevel Programming Ref Constraint Kinds
CPP General Preprocessor Ref Cpp
DataKinds Specialized Typelevel Programming Ref Data Kinds
DatatypeContexts Deprecated Deprecated Ref
DefaultSignatures Specialized Generic Programming Ref Generic
DeriveAnyClass General Deriving Ref
DeriveDataTypeable General Deriving Ref Typeable
DeriveFoldable General Deriving Ref Foldable / Traversable
DeriveFunctor General Deriving Ref
DeriveGeneric General Deriving Ref Generic
DeriveLift General Deriving Ref Template Haskell
DeriveTraversable General Deriving Ref
DisambiguateRecordFields Specialized Syntax Extension Ref
DuplicateRecordFields Specialized Syntax Extension Ref DuplicateRecordFields
DoRec Specialized Syntax Extension Ref Recursive Do
EmptyCase Specialized Syntax Extension Ref EmptyCase
EmptyDataDecls General Syntax Extension Ref Void
ExistentialQuantification Specialized Typelevel Programming Ref Existential Quantification
ExplicitForAll Specialized Typelevel Programming Ref Universal Quantification
ExplicitNamespaces Specialized Syntax Disambiguation Ref
ExtendedDefaultRules Specialized Type Disambiguation Ref
FlexibleContexts General Typeclass Extension Ref Flexible Contexts
FlexibleInstances General Typeclass Extension Ref Flexible Instances
ForeignFunctionInterface General FFI Ref FFI
FunctionalDependencies General Typeclass Extension Ref Multiparam Typeclasses
GADTs General Typelevel Programming Ref GADTs
GADTSyntax General Syntax Extension Ref GADTs
GeneralizedNewtypeDeriving General Typeclass Extension Ref Newtype Deriving
GHCForeignImportPrim Specialized FFI Ref Cmm
ImplicitParams Specialized Typelevel Programming Ref
ImpredicativeTypes Specialized Typelevel Programming Ref Impredicative Types
IncoherentInstances Specialized Typelevel Programming Ref Incoherent Instances
InstanceSigs Specialized Typelevel Programming Ref
InterruptibleFFI Specialized FFI Ref FFI
KindSignatures Specialized Typelevel Programming Ref Kind Signatures
LambdaCase General Syntax Extension Ref Lambda Case
LiberalTypeSynonyms Specialized Typeclass Extension Ref
MagicHash Specialized GHC Internals Ref Unboxed Types
MonadComprehensions Specialized Syntax Extension Ref
MonoLocalBinds General Type Disambiguation Ref
MonoPatBinds Specialized Type Disambiguation Ref
MultiParamTypeClasses General Typeclass Extension Ref Multiparam Typeclasses
MultiWayIf Specialized Syntax Extension Ref MultiWawyIf
NamedFieldPuns Specialized Syntax Extension Ref Named Field Puns
NegativeLiterals General Type Disambiguation Ref
NoImplicitPrelude Specialized Import Disambiguation Ref Custom Prelude
NoMonomorphismRestriction General Type Disambiguation Ref Monomorphism Restriction
NPlusKPatterns Deprecated Deprecated Ref
NullaryTypeClasses Specialized Typeclass Extension Ref Multiparam Typeclasses
NumDecimals General Type Disambiguation Ref NumDecimals
OverlappingInstances Specialized Typeclass Extension Ref Overlapping Instances
OverloadedLabels General Type Disambiguation Ref Overloaded Labels
OverloadedRecordFields General Syntax Extension Ref Overloaded Labels
OverloadedLists General Syntax Extension Ref Overloaded Lists
OverloadedStrings General Syntax Extension Ref Overloaded Strings
PackageImports General Import Disambiguation Ref Package Imports
ParallelArrays Specialized Data Parallel Haskell Ref
ParallelListComp General Syntax Extension Ref
PartialTypeSignatures General Interactive Typing Ref Partial Type Signatures
PatternGuards General Syntax Extension Ref Pattern Guards
PatternSynonyms General Syntax Extension Ref Pattern Synonyms
PolyKinds Specialized Typelevel Programming Ref Kind Polymorphism
PolymorphicComponents Specialized Deprecated Ref
PostfixOperators Specialized Syntax Extension Ref
QuasiQuotes Specialized Metaprogramming Ref QuasiQuotation
Rank2Types Specialized Historical Artifact Ref Rank N Types
RankNTypes Specialized Typelevel Programming Ref Rank N Types
RebindableSyntax Specialized Metaprogramming Ref Indexed Monads
RecordWildCards General Syntax Extension Ref Record Wildcards
RecursiveDo Specialized Syntax Extension Ref MonadFix
RelaxedPolyRec Specialized Type Disambiguation Ref
RoleAnnotations Specialized Type Disambiguation Ref Roles
Safe Specialized Security Auditing Ref Safe Haskell
SafeImports Specialized Security Auditing Ref Safe Haskell
ScopedTypeVariables Specialized Typelevel Programming Ref Scoped Type Variables
StandaloneDeriving General Typeclass Extension Ref
StaticPointers General Distributed Programming Ref
Strict General Strictness Annotations Ref Strict Haskell
StrictData General Strictness Annotations Ref Strict Haskell
TemplateHaskell Specialized Metaprogramming Ref Template Haskell
TraditionalRecordSyntax Specialized Historical Artifact Ref Historical Extensions
TransformListComp Specialized Syntax Extension Ref
Trustworthy Specialized Security Auditing Ref Safe Haskell
TupleSections General Syntax Extension Ref Tuple Sections
TypeApplications Specialized Typelevel Programming Ref
TypeFamilies Specialized Typelevel Programming Ref Type Families
TypeHoles General Interactive Typing Ref Type Holes
TypeInType Specialized Typelevel Programming Ref
TypeOperators Specialized Typelevel Programming Ref Manual Proofs
TypeSynonymInstances General Typeclass Extension Ref Type Synonym Instances
UnboxedTuples Specialized FFI Ref
UndecidableInstances Specialized Typelevel Programming Ref Multiparam Typeclasses
UnicodeSyntax Specialized Syntax Extension Ref
UnliftedFFITypes Specialized FFI Ref Cmm
Unsafe Specialized Security Auditing Ref Safe Haskell
ViewPatterns General Syntax Extension Ref View Patterns

See: GHC Extension Reference

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:

The Dangerous

GHC's typechecker sometimes just casually tells us to enable language extensions when it can't solve certain problems. These include:

These 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!


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

module Bad (foo,bar) where
foo x y = x + y
bar = foo 1

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

λ: bar 1.1
<interactive>:2:5: error:
    • No instance for (Fractional Integer)
      arising from the literal ‘1.0’
    • In the first argument of ‘bar’, namely ‘1.0’
      In the expression: bar 1.0
      In an equation for ‘it’: it = bar 1.0

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

λ: :t bar
bar :: Integer -> Integer

We can prevent GHC from specializing the type with this extension, i.e.

{-# LANGUAGE NoMonomorphismRestriction #-}

module Good (foo,bar) where
foo x y = x + y
bar = foo 1

Now everything will work as expected:

λ: :t bar
bar :: Num a => a -> a


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. So for instance, given the following default rules

default (C1 a,...,Cn a)

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 numeric.

The default default is (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. 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.

{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE ExtendedDefaultRules #-}

import qualified Data.Text as T

default (T.Text)

example = "foo"

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

See: Monomorphism Restriction


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, they are marked with the prefix unsafe. These 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.

unsafeCoerce :: a -> b
unsafePerformIO :: IO a -> a

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

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.

{-# LANGUAGE Safe #-}
{-# LANGUAGE Trustworthy #-}
{-# LANGUAGE Safe #-}

import Unsafe.Coerce
import System.IO.Unsafe

bad1 :: String
bad1 = unsafePerformIO getLine

bad2 :: a
bad2 = unsafeCoerce 3.14 ()
Unsafe.Coerce: Can't be safely imported!
The module itself isn't safe.

See: Safe Haskell


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:

{-# OPTIONS -XPartialTypeSignatures #-}

triple :: Int -> _
triple i = (i,i,i)

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

See: Partial Type Signatures


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

{-# LANGUAGE RecursiveDo #-}

justOnes :: Maybe [Int]
justOnes = do
  rec xs <- Just (1:xs)
  return (map negate xs)

See: Recursive Do Notation


By default GHC desugars do-notation to use implicit invocations of bind and return.

test :: Monad m => m (a, b, c)
test = do
  a <- f
  b <- g
  c <- h
  return (a, b, c)

Desugars into:

test :: Monad m => m (a, b, c)
test =
f >>= \a ->
  g >>= \b ->
    h >>= \c ->
      return (a, b, c)

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

test :: Applicative m => m (a, b, c)
test = (,,) <$> f <*> g <*> h


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, in the augmented environment.

{-# LANGUAGE PatternGuards #-}

combine env x y
   | Just a <- lookup x env
   , Just b <- lookup y env
   = Just $ a + b

   | otherwise = Nothing


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.

{-# LANGUAGE ViewPatterns #-}
{-# LANGUAGE NoMonomorphismRestriction #-}

import Safe

lookupDefault :: Eq a => a -> b -> [(a,b)] -> b
lookupDefault k _ (lookup k -> Just s) = s
lookupDefault _ d _ = d

headTup :: (a, [t]) -> [t]
headTup (headMay . snd -> Just n) = [n]
headTup _ = []

headNil :: [a] -> [a]
headNil (headMay -> Just x) = [x]
headNil _ = []


{-# LANGUAGE TupleSections #-}

first :: a -> (a, Bool)
first = (,True)

second :: a -> (Bool, a)
second = (True,)
f :: t -> t1 -> t2 -> t3 -> (t, (), t1, (), (), t2, t3)
f = (,(),,(),(),,)


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.

{-# LANGUAGE MultiWayIf #-}

bmiTell :: Float -> Text
bmiTell bmi = if
  | bmi <= 18.5 -> "Underweight."
  | bmi <= 25.0 -> "Average weight."
  | bmi <= 30.0 -> "Overweight."
  | otherwise   -> "Clinically overweight."


GHC normally requires at least one pattern branch in case statement this restriction can be relaxed with -XEmptyCase. The case statement then immediately yields a Non-exhaustive patterns in case if evaluated.

test = case of


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

Without LambdaCase:

\temp -> case temp of
  p1 -> 32
  p2 -> 32

With LambdaCase:

  p1 -> 32
  p2 -> 32
{-# LANGUAGE LambdaCase #-}

data Exp a
  = Lam a (Exp a)
  | Var a
  | App (Exp a) (Exp a)

example :: Exp a -> a
example = \case
  Lam a b -> a
  Var a   -> a
  App a b -> example a


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.

googol :: Fractional a => a
googol = 1e100
{-# LANGUAGE NumDecimals #-}
googol :: Num a => a
googol = 1e100


Package imports allows us to disambiguate hierarchical package names by their respective package key. This is useful in the case where you have to 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.

import qualified "mtl" Control.Monad.Error as Error
import qualified "mtl" Control.Monad.State as State
import qualified "mtl" Control.Monad.Reader as Reader


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 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.

{-# 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 {..}
    (e1, e2, e3) = (1, "Kirk", "Picard")


Provides alternative syntax for accessing record fields in a pattern match.

data D = D {a :: Int, b :: Int}

f :: D -> Int
f D {a, b} = a - b

-- Order doesn't matter
g :: D -> Int
g D {b, a} = a - b


Suppose we were writing a typechecker, it would be very common to include a distinct TArr term to ease the telescoping of function signatures, this is what GHC does in its Core language. Even though technically it could be written in terms of more basic application of the (->) constructor.

data Type
  = TVar TVar
  | TCon TyCon
  | TApp Type Type
  | TArr Type Type
  deriving (Show, Eq, Ord)

With pattern synonyms we can eliminate the extraneous constructor without losing the convenience of pattern matching on arrow types.

{-# LANGUAGE PatternSynonyms #-}

pattern TArr t1 t2 = TApp (TApp (TCon "(->)") t1) t2

So now we can write an eliminator and constructor for arrow type very naturally.

{-# LANGUAGE PatternSynonyms #-}

import Data.List (foldl1')

type Name  = String
type TVar  = String
type TyCon = String

data Type
  = TVar TVar
  | TCon TyCon
  | TApp Type Type
  deriving (Show, Eq, Ord)

pattern TArr t1 t2 = TApp (TApp (TCon "(->)") t1) t2

tapp :: TyCon -> [Type] -> Type
tapp tcon args = foldl TApp (TCon tcon) args

arr :: [Type] -> Type
arr ts = foldl1' (\t1 t2 -> tapp "(->)" [t1, t2]) ts

elimTArr :: Type -> [Type]
elimTArr (TArr (TArr t1 t2) t3) = t1 : t2 : elimTArr t3
elimTArr (TArr t1 t2) = t1 : elimTArr t2
elimTArr t = [t]

-- (->) a ((->) b a)
-- a -> b -> a
to :: Type
to = arr [TVar "a", TVar "b", TVar "a"]

from :: [Type]
from = elimTArr to

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

module MyModule (
  pattern Elt
) where

pattern Elt = [a]


{-# LANGUAGE DeriveFunctor #-}

data Tree a = Node a [Tree a]
  deriving (Show, Functor)

tree :: Tree Int
tree = fmap (+1) (Node 1 [Node 2 [], Node 3 []])


{-# LANGUAGE DeriveTraversable #-}
{-# LANGUAGE PartialTypeSignatures #-}

data Tree a = Node a [Tree a]
  deriving (Show, Functor, Foldable, Traversable)

tree :: Maybe [Int]
tree = foldMap go (Node [1] [Node [2] [], Node [3,4] []])
    go [] = Nothing
    go xs = Just xs




With -XDeriveAnyClass we can derive any class. The deriving logic generates an instance declaration for the type with no explicitly-defined methods. If the typeclass implements a default for each method then this will be well-defined and give rise to an automatic instances.



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.

{-# LANGUAGE DuplicateRecordFields #-}

data Person = Person { id :: Int }
data Animal = Animal { id :: Int }
data Vegetable = Vegetable { id :: Int }

test :: (Person, Animal, Vegetable)
test = (Person {id = 1}, Animal {id = 2}, Vegetable {id = 3})

Using just DuplicateRecordFields, projection is still not supported so the following will not work. OverloadedLabels fixes this to some extent.

test :: (Int, Int, Int)
test = (id (Person 1), id (Animal 2), id (Animal 3))


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 need to enable several language extensions to work with promoted strings and multiparam typeclasses that underlay it's implementation.

extract :: IsLabel "id" t => t
extract = #id
{-# LANGUAGE OverloadedLabels #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE DuplicateRecordFields #-}
{-# LANGUAGE ExistentialQuantification #-}

import GHC.Records (HasField(..))  -- Since base
import GHC.OverloadedLabels (IsLabel(..))

data S = MkS { foo :: Int }
data T x y z = forall b . MkT { foo :: y, bar :: b }

instance HasField x r a => IsLabel x (r -> a) where
  fromLabel = getField

main :: IO ()
main = do
  print (#foo (MkS 42))
  print (#foo (MkT True False))



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.


#if (__GLASGOW_HASKELL__ > 710)
-- Imports for GHC 7.10.x
-- Imports for other GHC

To demarcate code based on the operating system compiled on.


#ifdef OS_Linux
  -- Linux specific logic
# ifdef OS_Win32
  -- Windows specific logic
# else
# ifdef OS_Mac
  -- Macintosh specific logic
# else
  -- Other operating systems
# endif
# endif

Or on the version of the base library used.

#if !MIN_VERSION_base(4,6,0)
  -- Base specific logic

It can also be abused to do terrible things like metaprogramming with strings, but please don't do this.

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.

Type Classes

Minimal Annotations

In the presence of default implementations of typeclasses 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.

class Eq a where
  (==), (/=) :: a -> a -> Bool
  x == y = not (x /= y)
  x /= y = not (x == y)

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

class Eq a where
  (==), (/=) :: a -> a -> Bool
  x == y = not (x /= y)
  x /= y = not (x == y)
  {-# MINIMAL (==) #-}
  {-# MINIMAL (/=) #-}

Minimal pragmas are boolean expressions, with | as logical OR, either definition must be defined). Comma indicates logical AND where both sides both definitions must be defined.

{-# MINIMAL (==) | (/=) #-} -- Either (==) or (/=)
{-# MINIMAL (==) , (/=) #-} -- Both (==) and (/=)

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


{-# LANGUAGE FlexibleInstances #-}

class MyClass a

-- Without flexible instances, all instance heads must be type variable. The
-- following would be legal.
instance MyClass (Maybe a)

-- With flexible instances, typeclass heads can be arbitrary nested types. The
-- following would be forbidden without it.
instance MyClass (Maybe Int)


{-# LANGUAGE FlexibleContexts #-}

class MyClass a

-- Without flexible contexts, all contexts must be type variable. The
-- following would be legal.
instance (MyClass a) => MyClass (Either a b)

-- With flexible contexts, typeclass contexts can be arbitrary nested types. The
-- following would be forbidden without it.
instance (MyClass (Maybe a)) => MyClass (Either a b)


Typeclasses are normally globally coherent, there is only ever one instance that can be resolved for a type unambiguously for a type 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.

{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE OverlappingInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}

class MyClass a b where
  fn :: (a,b)

instance MyClass Int b where
  fn = error "b"

instance MyClass a Int where
  fn = error "a"

instance MyClass Int Int where
  fn = error "c"

example :: (Int, Int)
example = fn

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

{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}

class MyClass a b where
  fn :: (a,b)

instance {-# OVERLAPPING #-} MyClass Int b where
  fn = error "b"

instance {-# OVERLAPPING #-} MyClass a Int where
  fn = error "a"

instance {-# OVERLAPPING #-} MyClass Int Int where
  fn = error "c"

example :: (Int, Int)
example = fn


Incoherent instance loosens the restriction that there be only one specific instance, will choose one arbitrarily (based on the arbitrary sorting of it's internal representation ) and the resulting program will typecheck. This is generally pretty crazy and usually a sign of poor design.

{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE IncoherentInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}

class MyClass a b where
  fn :: (a,b)

instance MyClass Int b where
  fn = error "a"

instance MyClass a Int where
  fn = error "b"

example :: (Int, Int)
example = fn

There is also an incoherent instance.

{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}

class MyClass a b where
  fn :: (a,b)

instance {-# INCOHERENT #-} MyClass a Int where
  fn = error "general"

instance {-# INCOHERENT #-} MyClass Int Int where
  fn = error "specific"

example :: (Int, Int)
example = fn


{-# LANGUAGE TypeSynonymInstances #-}
{-# LANGUAGE FlexibleInstances #-}

type IntList = [Int]

class MyClass a

-- Without type synonym instances, we're forced to manually expand out type
-- synonyms in the typeclass head.
instance MyClass [Int]

-- With it GHC will do this for us automatically. Type synonyms still need to
-- be fully applied.
instance MyClass IntList


Again, a subject on which much ink has been spilled. 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 and allows strict evaluation when needed and uses laziness by default. Needless to say, we can always find examples where strict evaluation exhibits worse behavior than lazy evaluation and vice versa.

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 additional need to restructure their logic or force intermediate computations. 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 to the point of being industrial strength; knowledge about lazy evaluation is not widely absorbed into the collective programmer consciousness and can often be non-intuitive to the novice. This doesn't reflect on the model itself, merely on the need for more instruction material and research on optimizing lazy compilers.

The paradox of Haskell is that it explores so many definably unique ideas ( laziness, purity, typeclasses ) that it becomes difficult to separate out the discussion of any one from the gestalt of the whole implementation.



There are several evaluation models for the lambda calculus:

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

Model Strictness Description
Call-by-value Strict arguments evaluated before function entered
Call-by-name Non-strict arguments passed unevaluated
Call-by-need Non-strict arguments passed unevaluated but an expression is only evaluated once (sharing)

Seq and WHNF

A term is said to be in weak head normal-form if the outermost constructor or lambda 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 Normal Form
(2, "foo")
\x -> x + 1

-- Not in Normal Form
1 + 2
(\x -> x + 1) 2
"foo" ++ "bar"
(1 + 1, "foo")

-- In Weak Head Normal Form
(1 + 1, "foo")
\x -> 2 + 2
'f' : ("oo" ++ "bar")

-- Not In Weak Head Normal Form
1 + 1
(\x -> x + 1) 2
"foo" ++ "bar"

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 most outer constructor. For example, to evaluate the length of a list we need only scrutinize the outer Cons constructors without regard for their inner values.

λ: length [undefined, 1]

λ: head [undefined, 1]

λ: snd (undefined, 1)

λ: fst (undefined, 1)

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

ignore :: a -> Int
ignore x = 0

loop :: a
loop = loop

main :: IO ()
main = print $ ignore loop

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

let ignore x = 0;; 
let rec loop a = loop a;;

print_int (ignore (loop ()));

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

λ: let a = [1..] :: [Integer]
λ: let b = map (+ 1) a

λ: :sprint a
a = _
λ: :sprint b
b = _
λ: a !! 4
λ: :sprint a
a = 1 : 2 : 3 : 4 : 5 : _
λ: b !! 10
λ: :sprint a
a = 1 : 2 : 3 : 4 : 5 : 6 : 7 : 8 : 9 : 10 : 11 : _
λ: :sprint b
b = _ : _ : _ : _ : _ : _ : _ : _ : _ : _ : 12 : _

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 for when a computation might depend on itself to complete. The implementation of this is some of the more subtle details of the GHC runtime.

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.

seq :: a -> b -> b

⊥ `seq` a = ⊥
a `seq` b = b

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.

foldl :: (a -> b -> a) -> a -> [b] -> a
foldl f z [] = z
foldl f z (x:xs) = foldl f (f z x) xs
foldl' :: (a -> b -> a) -> a -> [b] -> a
foldl' _ z [] = z
foldl' f z (x:xs) = let z' = f z x in z' `seq` foldl' f z' xs

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.

Strictness Annotations

The extension BangPatterns allows an alternative syntax to force arguments to functions to be wrapped in seq. A bang operator on an arguments 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.

{-# LANGUAGE BangPatterns #-}

sum :: Num a => [a] -> a
sum = go 0
    go !acc (x:xs) = go (acc + x) xs
    go  acc []     = acc

This is desugared into code effectively equivalent to the following:

sum :: Num a => [a] -> a
sum = go 0
    go acc _ | acc `seq` False = undefined
    go acc (x:xs)              = go (acc + x) xs
    go acc []                  = acc

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

($!) :: (a -> b) -> a -> b
f $! x  = let !vx = x in f vx

Strict Haskell

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


Enabling StrictData makes constructor fields strict by default on any module it is enabled on.

{-# LANGUAGE StrictData #-}

data Employee = Employee
  { name :: T.Text
  , age :: Int

Is equivalent to:

data Employee = Employee
  { name :: !T.Text
  , age :: !Int


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

f x y = x + y

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

f !x !y = x + y

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. However there are some subtle points to this that are better explained in the language guide.


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.

class NFData a where
  rnf :: a -> ()
  rnf a = a `seq` ()

deepseq :: NFData a => a -> b -> b
($!!) :: (NFData a) => (a -> b) -> a -> b
instance NFData Int
instance NFData (a -> b)

instance NFData a => NFData (Maybe a) where
    rnf Nothing  = ()
    rnf (Just x) = rnf x

instance NFData a => NFData [a] where
    rnf [] = ()
    rnf (x:xs) = rnf x `seq` rnf xs
[1, undefined] `seq` ()
-- ()

[1, undefined] `deepseq` ()
-- Prelude.undefined

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

force :: NFData a => a -> a
force x = x `deepseq` x

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.

f :: (a, b) -> Int
f (a,b) = const 1 a

g :: (a, b) -> Int
g ~(a,b) = const 1 a

-- λ: f undefined
-- *** Exception: Prelude.undefined
-- λ: g undefined
-- 1

j :: Maybe t -> t
j ~(Just x) = x

k :: Maybe t -> t
k (Just x) = x

-- λ: j Nothing
-- *** Exception: src/05-laziness/lazy_patterns.hs:15:1-15: Irrefutable pattern failed for pattern (Just x)
-- λ: k Nothing
-- *** Exception: src/05-laziness/lazy_patterns.hs:18:1-14: Non-exhaustive patterns in function k


What to Avoid?

Haskell being a 25 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:

  • Avoid String.
  • Use fmap instead of map.
  • Use Foldable and Traversable instead of the Control.Monad, and Data.List versions of traversals.
  • Avoid partial functions like head and read or use their total variants.
  • Avoid exceptions, use ExceptT or Either instead.
  • Avoid boolean blind functions.

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 often times it is desirable to explicitly mask these functions from implicit import and force the use of Foldable and Traversable instead.

Of course often times one wishes only to 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.

import qualified Prelude as P

What Should be in Base

To get work done you probably need.

  • async
  • bytestring
  • containers
  • mtl
  • stm
  • text
  • transformers
  • unordered-containers
  • vector
  • filepath
  • directory
  • process
  • unix
  • deepseq
  • optparse-applicative

Custom Preludes

The default Prelude can be disabled in its entirety by twiddling the -XNoImplicitPrelude flag.

{-# LANGUAGE NoImplicitPrelude #-}

We are then free to build an equivalent Prelude that is more to our liking. Using module reexporting we can pluck the good parts of the prelude and libraries like safe to build up a more industrial focused set of default functions. For example:

module Custom (
  module Exports,
) where

import Data.Int as Exports
import Data.Tuple as Exports
import Data.Maybe as Exports
import Data.String as Exports
import Data.Foldable as Exports
import Data.Traversable as Exports

import Control.Monad.Trans.Except
  as Exports
  (ExceptT(ExceptT), Except, except, runExcept, runExceptT,
   mapExcept, mapExceptT, withExcept, withExceptT)

The Prelude itself is entirely replicable as well, presuming that an entire project is compiled without the implicit Prelude. Several packages have arisen that supply much of the same functionality in a way that appeals to more modern design principles.


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

{-# LANGUAGE NoImplicitPrelude #-}

import Protolude

Other examples for alternative Preludes include (your mileage may vary with these):

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.

head :: [a] -> a
read :: Read a => String -> a
(!!) :: [a] -> Int -> a


The Prelude has total variants of the historical partial functions (i.e. Text.Read.readMaybe)in some cases, but often these are found in the various utility libraries like safe.

The total versions provided fall into three cases:

-- Total
headMay :: [a] -> Maybe a
readMay :: Read a => String -> Maybe a
atMay :: [a] -> Int -> Maybe a

-- Total
headDef :: a -> [a] -> a
readDef :: Read a => a -> String -> a
atDef   :: a -> [a] -> Int -> a

-- Partial
headNote :: String -> [a] -> a
readNote :: Read a => String -> String -> a
atNote   :: String -> [a] -> Int -> a

Boolean Blindness

data Bool = True | False

isJust :: Maybe a -> Bool
isJust (Just x) = True
isJust Nothing  = False

The problem with the boolean 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?

# This one?
if p(x):
    # use x
elif not p(x):
    # don't use x

# Or this one?
if p(x):
    # don't use x
elif not p(x):
    # use x

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.

case x of
  Just a  -> use x
  Nothing -> don't use x

-- not ideal
case p x of
  True  -> use x
  False -> don't use x

-- not ideal
if p x
  then use x
  else don't use x

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 one's self to think about iteration over lists in terms of maps, folds, and scans can be challenging.

Prelude.foldl :: (a -> b -> a) -> a -> [b] -> a
Prelude.foldr :: (a -> b -> b) -> b -> [a] -> b

-- pseudocode
foldr f z [a...] = f a (f b ( ... (f y z) ... ))
foldl f z [a...] = f ... (f (f z a) b) ... y

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

-- foldr (+) 1 [2..]
(1 + (2 + (3 + (4 + ...))))
-- foldl (+) 1 [2..]
((((1 + 2) + 3) + 4) + ...)

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.

class (Functor f, Foldable f) => Traversable f where
  traverse :: Applicative g => (a -> g b) -> f a -> g (f b)

class Foldable f where
  foldMap :: Monoid m => (a -> m) -> f a -> m

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:

λ: foldMap Sum [1..10]
Sum {getSum = 55}

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.

import Data.Foldable
import qualified Data.Map as Map

data Elt
  = Elt Int Double
  | Nil

foo :: [Elt] -> Map.Map Int Double
foo = foldMap go
    go (Elt x y) = Map.singleton x y
    go Nil = Map.empty

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.

newtype Endo a = Endo {appEndo :: a -> a}

instance Monoid (Endo a) where
  mempty = Endo id
  Endo f `mappend` Endo g = Endo (f . g)
class Foldable t where
    fold    :: Monoid m => t m -> m
    foldMap :: Monoid m => (a -> m) -> t a -> m

    foldr   :: (a -> b -> b) -> b -> t a -> b
    foldr'  :: (a -> b -> b) -> b -> t a -> b

    foldl   :: (b -> a -> b) -> b -> t a -> b
    foldl'  :: (b -> a -> b) -> b -> t a -> b

    foldr1  :: (a -> a -> a) -> t a -> a
    foldl1  :: (a -> a -> a) -> t a -> a

For example:

foldr :: (a -> b -> b) -> b -> t a -> b
foldr f z t = appEndo (foldMap (Endo . f) t) z

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.

Data.Foldable.elem    :: (Eq a, Foldable t) => a -> t a -> Bool
Data.Foldable.sum     :: (Num a, Foldable t) => t a -> a
Data.Foldable.minimum :: (Ord a, Foldable t) => t a -> a
Data.Traversable.mapM :: (Monad m, Traversable t) => (a -> m b) -> t a -> m (t b)

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 all specialize to the same and behave the same as the ones in Prelude for List types.

import Data.Monoid
import Data.Foldable
import Data.Traversable

import Control.Applicative
import Control.Monad.Identity (runIdentity)
import Prelude hiding (mapM_, foldr)

-- Rose Tree
data Tree a = Node a [Tree a] deriving (Show)

instance Functor Tree where
  fmap f (Node x ts) = Node (f x) (fmap (fmap f) ts)

instance Traversable Tree where
  traverse f (Node x ts) = Node <$> f x <*> traverse (traverse f) ts

instance Foldable Tree where
  foldMap f (Node x ts) = f x `mappend` foldMap (foldMap f) ts

tree :: Tree Integer
tree = Node 1 [Node 1 [], Node 2 [] ,Node 3 []]

example1 :: IO ()
example1 = mapM_ print tree

example2 :: Integer
example2 = foldr (+) 0 tree

example3 :: Maybe (Tree Integer)
example3 = traverse (\x -> if x > 2 then Just x else Nothing) tree

example4 :: Tree Integer
example4 = runIdentity $ traverse (\x -> pure (x+1)) tree

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.

{-# LANGUAGE DeriveFunctor #-}
{-# LANGUAGE DeriveFoldable #-}
{-# LANGUAGE DeriveTraversable #-}

data Tree a = Node a [Tree a]
  deriving (Show, Functor, Foldable, Traversable)

See: Typeclassopedia


unfoldr :: (b -> Maybe (a, b)) -> b -> [a]

A recursive function consumes data and eventually terminates, a corecursive function generates data and coterminates. A corecursive function is said to be productive if it can always evaluate more of the resulting value in bounded time.

import Data.List

f :: Int -> Maybe (Int, Int)
f 0 = Nothing
f x = Just (x, x-1)

rev :: [Int]
rev = unfoldr f 10

fibs :: [Int]
fibs = unfoldr (\(a,b) -> Just (a,(b,a+b))) (0,1)


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

import Data.List.Split

example1 :: [String]
example1 = splitOn "." "foo.bar.baz"
-- ["foo","bar","baz"]

example2 :: [String]
example2 = chunksOf 10 "To be or not to be that is the question."
-- ["To be or n","ot to be t","hat is the"," question."]


The monad-loops package provides a variety of missing functions for control logic in monadic contexts.

whileM :: Monad m => m Bool -> m a -> m [a]
untilM :: Monad m => m a -> m Bool -> m [a]
iterateUntilM :: Monad m => (a -> Bool) -> (a -> m a) -> a -> m a
whileJust :: Monad m => m (Maybe a) -> (a -> m b) -> m [b]



{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE NoImplicitPrelude #-}

import Foundation
import Foundation.IO
import Foundation.String
import Foundation.VFS.FilePath

import Foundation.Collection

example :: String
example = "Violence is the last refuge of the incompetent."

bytes :: UArray Word8
bytes = toBytes UTF8 example

file :: IO (UArray Word8)
file = readFile "foundation.hs"

fileString :: IO (String, Maybe ValidationFailure, UArray Word8)
fileString = fromBytes UTF8 <$> file

xs :: NonEmpty [Int]
xs = fromList [1,2,3]

x :: Int
x = head xs

Strings and Bytearrays

Container Interface

Numerical Tower

See: Foundation



The default String type is broken and should be avoided whenever possible. Unfortunately for historical reasons large portions of GHC and Base depend on String.

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

type String = [Char]

For more performance sensitive cases there are two libraries for processing textual data: text and bytestring.

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

Giving rise to the four types.

Variant Module
strict text Data.Text
lazy text Data.Text.Lazy
strict bytestring Data.ByteString
lazy bytestring Data.ByteString.Lazy


Conversions between strings types ( from : left column, to : top row ) 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.

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

Overloaded Strings

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 a typeclass IsString. Sometimes this is desirable.

class IsString a where
  fromString :: String -> a

For instance:

λ: :type "foo"
"foo" :: [Char]

λ: :set -XOverloadedStrings

λ: :type "foo"
"foo" :: IsString a => a

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

newtype Cat = Cat Text
  deriving (IsString)

fluffy :: Cat
fluffy = "Fluffy"

Import Conventions

import qualified Data.Text as T
import qualified Data.Text.Lazy as TL
import qualified Data.ByteString as BS
import qualified Data.ByteString.Lazy as BL
import qualified Data.ByteString.Char8 as C
import qualified Data.ByteString.Lazy.Char8 as CL
import qualified Data.Text.IO as TIO
import qualified Data.Text.Lazy.IO as TLIO
import qualified Data.Text.Encoding as TE
import qualified Data.Text.Lazy.Encoding as TLE


A Text type is a packed blob of Unicode characters.

pack :: String -> Text
unpack :: Text -> String
{-# LANGUAGE OverloadedStrings #-}

import qualified Data.Text as T

-- From pack
myTStr1 :: T.Text
myTStr1 = T.pack ("foo" :: String)

-- From overloaded string literal.
myTStr2 :: T.Text
myTStr2 = "bar"

See: Text


toLazyText :: Builder -> Data.Text.Lazy.Internal.Text
fromLazyText :: Data.Text.Lazy.Internal.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.

{-# LANGUAGE OverloadedStrings #-}

import Data.Monoid (mconcat, (<>))

import Data.Text.Lazy.Builder (Builder, toLazyText)
import Data.Text.Lazy.Builder.Int (decimal)
import qualified Data.Text.Lazy.IO as L

beer :: Int -> Builder
beer n = decimal n <> " bottles of beer on the wall.\n"

wall :: Builder
wall = mconcat $ fmap beer [1..1000]

main :: IO ()
main = L.putStrLn $ toLazyText wall


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

pack :: String -> ByteString
unpack :: ByteString -> String
{-# LANGUAGE OverloadedStrings #-}

import qualified Data.ByteString as S
import qualified Data.ByteString.Char8 as S8

-- From pack
bstr1 :: S.ByteString
bstr1 = S.pack ("foo" :: String)

-- From overloaded string literal.
bstr2 :: S.ByteString
bstr2 = "bar"




See: utf8-string



See: utf8-string


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

import Data.Text
import Text.Printf

a :: Int
a = 3

b :: Double
b = 3.14159

c :: String
c = "haskell"

example :: String
example = printf "(%i, %f, %s)" a b c
-- "(3, 3.14159, haskell)"

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 a typeclass IsList.

class IsList l where
  type Item l
  fromList  :: [Item l] -> l
  fromListN :: Int -> [Item l] -> l
  toList    :: l -> [Item l]

instance IsList [a] where
  type Item [a] = a
  fromList = id
  toList   = id
λ: :seti -XOverloadedLists
λ: :type [1,2,3]
[1,2,3] :: (Num (GHC.Exts.Item l), GHC.Exts.IsList l) => l
{-# LANGUAGE OverloadedLists #-}
{-# LANGUAGE TypeFamilies #-}

import qualified Data.Map as Map
import GHC.Exts (IsList(..))

instance (Ord k) => IsList (Map.Map k v) where
  type Item (Map.Map k v) = (k,v)
  fromList = Map.fromList
  toList = Map.toList

example1 :: Map.Map String Int
example1 = [("a", 1), ("b", 2)]

String Conversions

Playing "type-tetris" to convert between Strings explicitly can be frustrating, fortunately there are several packages that automate the conversion using typeclasses to automatically convert between any two common string representations automatically. We can then write generic comparison and concatenation operators that automatically convert types of operands to a like form.

{-# LANGUAGE OverloadedStrings #-}

import Data.String.Conv

import qualified Data.Text as T
import qualified Data.Text.Lazy.IO as TL

import qualified Data.ByteString as B
import qualified Data.ByteString.Lazy as BL

import Data.Monoid

a :: String
a = "Gödel"

b :: BL.ByteString
b = "Einstein"

c :: T.Text
c = "Feynmann"

d :: B.ByteString
d = "Schrödinger"

-- Compare unlike strings
(==~) :: (Eq a, StringConv b a) => a -> b -> Bool
(==~) a b = a == toS b

-- Concat unlike strings
(<>~) :: (Monoid a, StringConv b a) => a -> b -> a
(<>~) a b = a <> toS b

main :: IO ()
main = do
  putStrLn (toS a)
  TL.putStrLn (toS b)
  print (a ==~ b)
  print (c ==~ d)
  print (c ==~ c)
  print (b <>~ c)


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

pure :: Applicative f => a -> f a
(<$>) :: Functor f => (a -> b) -> f a -> f b
(<*>) :: f (a -> b) -> f a -> f b

As of GHC 7.6, Applicative is defined as:

class Functor f => Applicative f where
  pure :: a -> f a
  (<*>) :: f (a -> b) -> f a -> f b

(<$>) :: Functor f => (a -> b) -> f a -> f b
(<$>) = fmap

With the following laws:

pure id <*> v = v
pure f <*> pure x = pure (f x)
u <*> pure y = pure ($ y) <*> u
u <*> (v <*> w) = pure (.) <*> u <*> v <*> w

As an example, consider the instance for Maybe:

instance Applicative Maybe where
  pure              = Just
  Nothing <*> _     = Nothing
  _ <*> Nothing     = Nothing
  Just f <*> Just x = Just (f x)

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

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

example2 :: [(Int, Int, Int)]
example2 = (,,) <$> m1 <*> m2 <*> m3
    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
    fetch1 = simpleHTTP (getRequest "http://www.fpcomplete.com/") >>= getResponseBody
    fetch2 = simpleHTTP (getRequest "http://www.haskell.org/") >>= getResponseBody

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

liftA :: Applicative f => (a -> b) -> f a -> f b
liftA f a = pure f <*> a

liftA2 :: Applicative f => (a -> b -> c) -> f a -> f b -> f c
liftA2 f a b = f <$> a <*> b

liftA3 :: Applicative f => (a -> b -> c -> d) -> f a -> f b -> f c -> f d
liftA3 f a b c = f <$> a <*> b <*> c

Applicative also has functions *> and <* that sequence applicative actions while discarding the value of one of the arguments. The operator *> discard 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.

import Control.Monad
import Control.Applicative

data C a b = C a b

mnd :: Monad m => m a -> m b -> m (C a b)
mnd a b = C `liftM` a `ap` b

apl :: Applicative f => f a -> f b -> f (C a b)
apl a b = C <$> a <*> b

See: Applicative Programming with Effects


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

class Applicative f => Alternative f where
  -- | The identity of '<|>'
  empty :: f a
  -- | An associative binary operation
  (<|>) :: f a -> f a -> f a
  -- | One or more.
  some :: f a -> f [a]
  -- | Zero or more.
  many :: f a -> f [a]

optional :: Alternative f => f a -> f (Maybe a)

when :: (Alternative f) => Bool -> f () -> f ()
when p s = if p then s else return ()

guard :: (Alternative f) => Bool -> f ()
guard True  = pure ()
guard False = mzero
instance Alternative Maybe where
    empty = Nothing
    Nothing <|> r = r
    l       <|> _ = l

instance Alternative [] where
    empty = []
    (<|>) = (++)
λ: foldl1 (<|>) [Nothing, Just 5, Just 3]
Just 5

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


A category is an algebraic structure that includes a notion of an identity and a composition operation that is associative and preserves identities.

class Category cat where
  id :: cat a a
  (.) :: cat b c -> cat a b -> cat a c
instance Category (->) where
  id = Prelude.id
  (.) = (Prelude..)
(<<<) :: Category cat => cat b c -> cat a b -> cat a c
(<<<) = (.)

(>>>) :: Category cat => cat a b -> cat b c -> cat a c
f >>> g = g . f

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

class Category a => Arrow a where
  arr :: (b -> c) -> a b c
  first :: a b c -> a (b,d) (c,d)
  second :: a b c -> a (d,b) (d,c)
  (***) :: a b c -> a b' c' -> a (b,b') (c,c')
  (&&&) :: a b c -> a b c' -> a b (c,c')

The canonical example is for functions.

instance Arrow (->) where
  arr f = f
  first f = f *** id
  second f = id *** f
  (***) f g ~(x,y) = (f x, g y)

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.

import Data.List (group, sort)

histogram :: Ord a => [a] -> [(a, Int)]
histogram = map (head &&& length) . group . sort
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:

{-# LANGUAGE Arrows #-}

addA :: Arrow a => a b Int -> a b Int -> a b Int
addA f g = proc x -> do
                y <- f -< x
                z <- g -< x
                returnA -< y + z
addA f g = arr (\ x -> (x, x)) >>>
           first f >>> arr (\ (y, x) -> (x, y)) >>>
           first g >>> arr (\ (z, y) -> y + z)
addA f g = f &&& g >>> arr (\ (y, z) -> y + z)

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

See: Arrow Notation


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

class Bifunctor p where
  bimap :: (a -> b) -> (c -> d) -> p a c -> p b d
  first :: (a -> b) -> p a c -> p b c
  second :: (b -> c) -> p a b -> p a c

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

bimap id id ≡ id
first id ≡ id
second id ≡ id
bimap f g ≡ first f . second g

The canonical example is for 2-tuples.

λ: first (+1) (1,2)
λ: second (+1) (1,2)
λ: bimap (+1) (+1) (1,2)

λ: first (+1) (Left 3)
Left 4
λ: second (+1) (Left 3)
Left 3
λ: second (+1) (Right 3)
Right 4

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 converted into a single type or unpacked into a sum type.

{-# LANGUAGE FlexibleInstances #-}

class Arg a where
  collect' :: [String] -> a

-- extract to IO
instance Arg (IO ()) where
  collect' acc = mapM_ putStrLn acc

-- extract to [String]
instance Arg [String] where
  collect' acc = acc

instance (Show a, Arg r) => Arg (a -> r) where
  collect' acc = \x -> collect' (acc ++ [show x])

collect :: Arg t => t
collect = collect' []

example1 :: [String]
example1 = collect 'a' 2 3.0

example2 :: IO ()
example2 = collect () "foo" [1,2,3]

See: Polyvariadic functions

Error Handling


The low-level (and most dangerous) way to handle errors is to use the throw and catch functions which allow us to throw extensible exceptions in pure code but catch the resulting exception within IO. Of specific note is that return value of the throw inhabits all types. There's no reason to use this for custom code that doesn't use low-level system operations.

throw :: Exception e => e -> a
catch :: Exception e => IO a -> (e -> IO a) -> IO a
try :: Exception e => IO a -> IO (Either e a)
evaluate :: a -> IO a
{-# LANGUAGE DeriveDataTypeable #-}

import Data.Typeable
import Control.Exception

data MyException = MyException
    deriving (Show, Typeable)

instance Exception MyException

evil :: [Int]
evil = [throw MyException]

example1 :: Int
example1 = head evil

example2 :: Int
example2 = length evil

main :: IO ()
main = do
  a <- try (evaluate example1) :: IO (Either MyException Int)
  print a

  b <- try (return example2) :: IO (Either MyException Int)
  print b

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 standard library but has a simple implementation in terms of deepseq.

strictCatch :: (NFData a, Exception e) => IO a -> (e -> IO a) -> IO a
strictCatch = catch . (toNF =<<)


The problem with the previous approach is having to rely on GHC's asynchronous exception handling inside of IO to handle basic operations. The exceptions provides the same API as Control.Exception but loosens the dependency on IO.

{-# LANGUAGE DeriveDataTypeable #-}

import Data.Typeable
import Control.Monad.Catch
import Control.Monad.Identity

data MyException = MyException
    deriving (Show, Typeable)

instance Exception MyException

example :: MonadCatch m => Int -> Int -> m Int
example x y | y == 0 = throwM MyException
            | otherwise = return $ x `div` y

pure :: MonadCatch m => m (Either MyException Int)
pure = do
  a <- try (example 1 2)
  b <- try (example 1 0)
  return (a >> b)

See: exceptions


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

newtype ExceptT e m a = ExceptT (m (Either e a))

runExceptT :: ExceptT e m a -> m (Either e a)
runExceptT (ExceptT m) = m

instance (Monad m) => Monad (ExceptT e m) where
    return a = ExceptT $ return (Right a)
    m >>= k = ExceptT $ do
        a <- runExceptT m
        case a of
            Left e -> return (Left e)
            Right x -> runExceptT (k x)
    fail = ExceptT . fail

throwE :: (Monad m) => e -> ExceptT e m a
throwE = ExceptT . return . Left

catchE :: (Monad m) =>
    ExceptT e m a               -- ^ the inner computation
    -> (e -> ExceptT e' m a)    -- ^ a handler for exceptions in the inner
                                -- computation
    -> ExceptT e' m a
m `catchE` h = ExceptT $ do
    a <- runExceptT m
    case a of
        Left  l -> runExceptT (h l)
        Right r -> return (Right r)

Using mtl:

instance MonadTrans (ExceptT e) where
    lift = ExceptT . liftM Right

class (Monad m) => MonadError e m | m -> e where
    throwError :: e -> m a
    catchError :: m a -> (e -> m a) -> m a

instance MonadError IOException IO where
    throwError = ioError
    catchError = catch

instance MonadError e (Either e) where
    throwError             = Left
    Left  l `catchError` h = h l
    Right r `catchError` _ = Right r



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 be handle this logic purely as if it were returning a proper Maybe a but to catch the logic you'd need to install a 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.

import Control.Spoon

goBoom :: Int -> Int -> Int
goBoom x y = x `div` y

-- evaluate to normal form
test1 :: Maybe [Int]
test1 = spoon [1, 2, undefined]

-- evaluate to weak head normal form
test2 :: Maybe [Int]
test2 = teaspoon [1, 2, undefined]

main :: IO ()
main = do
  maybe (putStrLn "Nothing") (print . length) test1
  maybe (putStrLn "Nothing") (print . length) test2




Advanced Monads

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.

instance Functor ((->) r) where
  fmap = (.)

instance Monad ((->) r) where
  return = const
  f >>= k = \r -> k (f r) r

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

import Control.Monad

id' :: (->) a a
id' = id

const' :: (->) a ((->) b a)
const' = const

-- Monad m => a -> m a
fret :: a -> b -> a
fret = return

-- Monad m => m a -> (a -> m b) -> m b
fbind :: (r -> a) -> (a -> (r -> b)) -> (r -> b)
fbind f k = f >>= k

-- Monad m => m (m a) -> m a
fjoin :: (r -> (r -> a)) -> (r -> a)
fjoin = join

fid :: a -> a
fid = const >>= id

-- Functor f => (a -> b) -> f a -> f b
fcompose :: (a -> b) -> (r -> a) -> (r -> b)
fcompose = (.)
type Reader r = (->) r -- pseudocode

instance Monad (Reader r) where
  return a = \_ -> a
  f >>= k = \r -> k (f r) r

ask' :: r -> r
ask' = id

asks' :: (r -> a) -> (r -> a)
asks' f = id . f

runReader' :: (r -> a) -> r -> a
runReader' = id

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.

runReader :: Reader r a -> r -> a
runWriter :: Writer w a -> (a, w)
runState  :: State s a -> s -> (a, s)

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

runRWS  :: RWS r w s a -> r -> s -> (a, s, w)
execRWS :: RWS r w s a -> r -> s -> (s, w)
evalRWS :: RWS r w s a -> r -> s -> (a, w)
import Control.Monad.RWS

type R = Int
type W = [Int]
type S = Int

computation :: RWS R W S ()
computation = do
  e <- ask
  a <- get
  let b = a + e
  put b
  tell [b]

example = runRWS computation 2 3

The usual caveat about Writer laziness also applies to RWS.


runCont :: Cont r a -> (a -> r) -> r
callCC :: MonadCont m => ((a -> m b) -> m a) -> m a
cont :: ((a -> r) -> r) -> Cont r a

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.

add :: Int -> Int -> Int
add x y = x + y

add :: Int -> Int -> (Int -> r) -> r
add x y k = k (x + y)
import Control.Monad
import Control.Monad.Cont

add :: Int -> Int -> Cont k Int
add x y = return $ x + y

mult :: Int -> Int -> Cont k Int
mult x y = return $ x * y

contt :: ContT () IO ()
contt = do
    k <- do
      callCC $ \exit -> do
        lift $ putStrLn "Entry"
        exit $ \_ -> do
          putStrLn "Exit"
    lift $ putStrLn "Inside"
    lift $ k ()

callcc :: Cont String Integer
callcc = do
  a <- return 1
  b <- callCC (\k -> k 2)
  return $ a+b

ex1 :: IO ()
ex1 = print $ runCont (f >>= g) id
    f = add 1 2
    g = mult 3
-- 9

ex2 :: IO ()
ex2 = print $ runCont callcc show
-- "3"

ex3 :: IO ()
ex3 = runContT contt print
-- Entry
-- Inside
-- Exit

main :: IO ()
main = do
newtype Cont r a = Cont { runCont :: ((a -> r) -> r) }

instance Monad (Cont r) where
  return a       = Cont $ \k -> k a
  (Cont c) >>= f = Cont $ \k -> c (\a -> runCont (f a) k)

class (Monad m) => MonadCont m where
  callCC :: ((a -> m b) -> m a) -> m a

instance MonadCont (Cont r) where
  callCC f = Cont $ \k -> runCont (f (\a -> Cont $ \_ -> k a)) k


Choice and failure.

class (Alternative m, Monad m) => MonadPlus m where
   mzero :: m a
   mplus :: m a -> m a -> m a

instance MonadPlus [] where
   mzero = []
   mplus = (++)

instance MonadPlus Maybe where
   mzero = Nothing

   Nothing `mplus` ys  = ys
   xs      `mplus` _ys = xs

MonadPlus forms a monoid with

mzero `mplus` a = a
a `mplus` mzero = a
(a `mplus` b) `mplus` c = a `mplus` (b `mplus` c)
asum :: (Foldable t, Alternative f) => t (f a) -> f a
asum = foldr (<|>) empty

msum :: (Foldable t, MonadPlus m) => t (m a) -> m a
msum = asum
import Safe
import Control.Monad

list1 :: [(Int,Int)]
list1 = [(a,b) | a <- [1..25], b <- [1..25], a < b]

list2 :: [(Int,Int)]
list2 = do
  a <- [1..25]
  b <- [1..25]
  guard (a < b)
  return $ (a,b)

maybe1 :: String -> String -> Maybe Double
maybe1 a b = do
  a' <- readMay a
  b' <- readMay b
  guard (b' /= 0.0)
  return $ a'/b'

maybe2 :: Maybe Int
maybe2 = msum [Nothing, Nothing, Just 3, Just 4]
import Control.Monad

range :: MonadPlus m => [a] -> m a
range [] = mzero
range (x:xs) = range xs `mplus` return x

pyth :: Integer -> [(Integer,Integer,Integer)]
pyth n = do
  x <- range [1..n]
  y <- range [1..n]
  z <- range [1..n]
  if x*x + y*y == z*z then return (x,y,z) else mzero

main :: IO ()
main = print $ pyth 15
[ ( 12 , 9 , 15 )
, ( 12 , 5 , 13 )
, ( 9 , 12 , 15 )
, ( 8 , 6 , 10 )
, ( 6 , 8 , 10 )
, ( 5 , 12 , 13 )
, ( 4 , 3 , 5 )
, ( 3 , 4 , 5 )


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

fix :: (a -> a) -> a
fix f = let x = f x in x

mfix :: (a -> m a) -> m a
class Monad m => MonadFix m where
   mfix :: (a -> m a) -> m a

instance MonadFix Maybe where
   mfix f = let a = f (unJust a) in a
            where unJust (Just x) = x
                  unJust Nothing  = error "mfix Maybe: Nothing"

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

{-# LANGUAGE RecursiveDo #-}

import Control.Applicative
import Control.Monad.Fix

stream1 :: Maybe [Int]
stream1 = do
  rec xs <- Just (1:xs)
  return (map negate xs)

stream2 :: Maybe [Int]
stream2 = mfix $ \xs -> do
  xs' <- Just (1:xs)
  return (map negate xs')

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.

runST :: (forall s. ST s a) -> a
newSTRef :: a -> ST s (STRef s a)
readSTRef :: STRef s a -> ST s a
writeSTRef :: STRef s a -> a -> ST s ()
import Data.STRef
import Control.Monad
import Control.Monad.ST
import Control.Monad.State.Strict

example1 :: Int
example1 = runST $ do
  x <- newSTRef 0

  forM_ [1..1000] $ \j -> do
    writeSTRef x j

  readSTRef x

example2 :: Int
example2 = runST $ do
  count <- newSTRef 0
  replicateM_ (10^6) $ modifySTRef' count (+1)
  readSTRef count

example3 :: Int
example3 = flip evalState 0 $ do
  replicateM_ (10^6) $ modify' (+1)

modify' :: MonadState a m => (a -> a) -> m ()
modify' f = get >>= (\x -> put $! f x)

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

Pure :: a -> Free f a
Free :: f (Free f a) -> Free f a

liftF :: (Functor f, MonadFree f m) => f a -> m a
retract :: Monad f => Free f a -> f a

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

join :: Monad m => m (m a) -> m a
wrap :: MonadFree f m => f (m a) -> m a

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.

import Control.Monad.Fix
import Control.Monad.Free

type Partiality a = Free Maybe a

-- Non-termination.
never :: Partiality a
never = fix (Free . Just)

fromMaybe :: Maybe a -> Partiality a
fromMaybe (Just x) = Pure x
fromMaybe Nothing = Free Nothing

runPartiality :: Int -> Partiality a -> Maybe a
runPartiality 0 _ = Nothing
runPartiality _ (Pure a) = Just a
runPartiality _ (Free Nothing) = Nothing
runPartiality n (Free (Just a)) = runPartiality (n-1) a

ack :: Int -> Int -> Partiality Int
ack 0 n = Pure $ n + 1
ack m 0 = Free $ Just $ ack (m-1) 1
ack m n = Free $ Just $ ack m (n-1) >>= ack (m-1)

main :: IO ()
main = do
  let diverge = never :: Partiality ()
  print $ runPartiality 1000 diverge
  print $ runPartiality 1000 (ack 3 4)
  print $ runPartiality 5500 (ack 3 4)

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.

{-# LANGUAGE DeriveFunctor #-}

import System.Exit
import Control.Monad.Free

data Interaction x
  = Puts String x
  | Gets (Char -> x)
  | Exit
  deriving Functor

type IOFree a = Free Interaction a

puts :: String -> IOFree ()
puts s = liftF $ Puts s ()

get :: IOFree Char
get = liftF $ Gets id

exit :: IOFree r
exit = liftF Exit

gets :: IOFree String
gets = do
  c <- get
  if c == '\n'
    then return ""
    else gets >>= \line -> return (c : line)

-- Collapse our IOFree DSL into IO monad actions.
interp :: IOFree a -> IO a
interp (Pure r) = return r
interp (Free x) = case x of
  Puts s t -> putStrLn s >> interp t
  Gets f   -> getChar >>= interp . f
  Exit     -> exitSuccess

echo :: IOFree ()
echo = do
  puts "Enter your name:"
  str <- gets
  puts str
  if length str > 10
    then puts "You have a long name."
    else puts "You have a short name."

main :: IO ()
main = interp echo

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

{-# LANGUAGE MultiParamTypeClasses #-}

import Control.Applicative

data Free f a
  = Pure a
  | Free (f (Free f a))

instance Functor f => Monad (Free f) where
  return a     = Pure a
  Pure a >>= f = f a
  Free f >>= g = Free (fmap (>>= g) f)

class Monad m => MonadFree f m  where
  wrap :: f (m a) -> m a

liftF :: (Functor f, MonadFree f m) => f a -> m a
liftF = wrap . fmap return

iter :: Functor f => (f a -> a) -> Free f a -> a
iter _ (Pure a) = a
iter phi (Free m) = phi (iter phi <$> m)

retract :: Monad f => Free f a -> f a
retract (Pure a) = return a
retract (Free as) = as >>= retract


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.

class IxMonad md where
  return :: a -> md i i a
  (>>=) :: md i m a -> (a -> md m o b) -> md i o b

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
  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

See: Fun with Indexed monads


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

-- | Lift a computation from the base monad
class (Applicative b, Applicative m, Monad b, Monad m)
      => MonadBase b m | m -> b where
  liftBase  b a -> m a


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.

control :: MonadBaseControl b m => (RunInBase m b -> b (StM m a)) -> m a

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.

  :: (MonadBaseControl IO m, Exception e)
  => m a        -- ^ Computation
  -> (e -> m a) -- ^ Handler
  -> m a
catch a handler = control $ \runInIO ->
                    E.catch (runInIO a)
                            (\e -> runInIO $ handler e)


This is an advanced section, and is not typically necessary to write Haskell.

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.

{-# LANGUAGE ExplicitForAll #-}

-- ∀a. [a]
example1 :: forall a. [a]
example1 = []

-- ∀a. [a]
example2 :: forall a. [a]
example2 = [undefined]

-- ∀a. ∀b. (a → b) → [a] → [b]
map' :: forall a. forall b. (a -> b) -> [a] -> [b]
map' f = foldr ((:) . f) []

-- ∀a. [a] → [a]
reverse' :: forall a. [a] -> [a]
reverse' = foldl (flip (:)) []

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.

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

id :: forall a. a -> a
id x = x
fmap :: Functor f => (a -> b) -> f a -> f b

The free theorem of fmap:

forall f g. fmap f . fmap g = fmap (f . g)

See: Theorems for Free

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.

e : x
  | λx:t.e            -- value abstraction
  | e1 e2             -- application
  | let x = e1 in e2  -- let

t : t -> t     -- function types
  | a          -- type variables

σ :  a . t    -- type scheme

In an implementation, the function generalize converts all type variables within the type into polymorphic type variables yielding a type scheme. The function instantiate 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.

t : t -> t     -- function types
  | a          -- type variables
  |  a . t    -- forall

e : x          -- variables
  | λ(x:t).e   -- value abstraction
  | e1 e2      -- value application
  | Λa.e       -- type abstraction
  | e_t        -- type application

An example with equivalents of GHC Core in comments:

id :  t. t -> t
id = Λt. λx:t. x
-- id :: forall t. t -> t
-- id = \ (@ t) (x :: t) -> x

tr :  a.  b. a -> b -> a
tr = Λa. Λb. λx:a. λy:b. x
-- tr :: forall a b. a -> b -> a
-- tr = \ (@ a) (@ b) (x :: a) (y :: b) -> x

fl :  a.  b. a -> b -> b
fl = Λa. Λb. λx:a. λy:b. y
-- fl :: forall a b. a -> b -> b
-- fl = \ (@ a) (@ b) (x :: a) (y :: b) -> y

nil :  a. [a]
nil = Λa. Λb. λz:b. λf:(a -> b -> b). z
-- nil :: forall a. [a]
-- nil = \ (@ a) (@ b) (z :: b) (f :: a -> b -> b) -> z

cons :  a. a -> [a] -> [a]
cons = Λa. λx:a. λxs:( b. b -> (a -> b -> b) -> b).
    Λb. λz:b. λf : (a -> b -> b). f x (xs_b z f)
-- cons :: forall a. a -> [a] -> [a]
-- cons = \ (@ a) (x :: a) (xs :: forall b. b -> (a -> b -> b) -> b)
--     (@ b) (z :: b) (f :: a -> b -> b) -> f x (xs @ b z f)

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.

{-# LANGUAGE RankNTypes #-}

-- Can't unify ( Bool ~ Char )
rank1 :: forall a. (a -> a) -> (Bool, Char)
rank1 f = (f True, f 'a')

rank2 :: (forall a. a -> a) -> (Bool, Char)
rank2 f = (f True, f 'a')

auto :: (forall a. a -> a) -> (forall b. b -> b)
auto x = x

xauto :: forall a. (forall b. b -> b) -> a -> a
xauto f = f
Monomorphic Rank 0: t
Polymorphic Rank 1: forall a. a -> t
Polymorphic Rank 2: (forall a. a -> t) -> t
Polymorphic Rank 3: ((forall a. a -> t) -> t) -> t

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.

{-# LANGUAGE RankNTypes #-}

escape :: (forall a. a -> a) -> Int
escape f = f 0

g x = escape (\a -> x)

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.

Couldn't match expected type `a' with actual type `t'
`a' is a rigid type variable bound by a type expected by the context: a -> a
`t' is a rigid type variable bound by the inferred type of g :: t -> Int
In the expression: x In the first argument of `escape', namely `(\ a -> x)'
In the expression: escape (\ a -> x)

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.

{-# LANGUAGE ExistentialQuantification #-}
{-# LANGUAGE RankNTypes #-}

-- ∃ t. (t, t → t, t → String)
data Box = forall a. Box a (a -> a) (a -> String)

boxa :: Box
boxa = Box 1 negate show

boxb :: Box
boxb = Box "foo" reverse show

apply :: Box -> String
apply (Box x f p) = p (f x)

-- ∃ t. Show t => t
data SBox = forall a. Show a => SBox a

boxes :: [SBox]
boxes = [SBox (), SBox 2, SBox "foo"]

showBox :: SBox -> String
showBox (SBox a) = show a

main :: IO ()
main = mapM_ (putStrLn . showBox) boxes
-- ()
-- 2
-- "foo"

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.

{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE ExistentialQuantification #-}

-- a b are existentially bound type variables, m is a free type variable
data MonadI m = MonadI
  { _return :: forall a . a -> m a
  , _bind   :: forall a b . m a -> (a -> m b) -> m b

monadMaybe:: MonadI Maybe
monadMaybe = MonadI
  { _return = Just
  , _bind   = \m f -> case m of
      Nothing -> Nothing
      Just x  -> f x

Impredicative Types

This is an advanced section, and is not typically necessary to write Haskell.

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.

-- Can't unify ( Int ~ Char )

revUni :: forall a. Maybe ([a] -> [a]) -> Maybe ([Int], [Char])
revUni (Just g) = Just (g [3], g "hello")
revUni Nothing  = Nothing
{-# 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. If this sounds like an ugly hack it's because it is, but a rather convenient hack.

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!

{-# LANGUAGE RankNTypes #-}

import Control.Monad.ST

f `apply` x =  f x

foo :: (forall s. ST s a) -> a
foo st = runST $ st

bar :: (forall s. ST s a) -> a
bar st = runST `apply` st
    Couldn't match expected type `forall s. ST s a'
                with actual type `ST s0 a'
    In the second argument of `apply', namely `st'
    In the expression: runST `apply` st
    In an equation for `bar': bar st = runST `apply` st


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.

{-# LANGUAGE ExplicitForAll #-}
{-# LANGUAGE ScopedTypeVariables #-}

poly :: forall a b c. a -> b -> c -> (a, a)
poly x y z = (f x y, f x z)
    -- second argument is universally quantified from inference
    -- f :: forall t0 t1. t0 -> t1 -> t0
    f x' _ = x'

mono :: forall a b c. a -> b -> c -> (a, a)
mono x y z = (f x y, f x z)
    -- b is not implictly universally quantified because it is in scope
    f :: a -> b -> a
    f x' _ = x'

example :: IO ()
example = do
  x :: [Int] <- readLn
  print x



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:

-- Vanilla
data List a
  = Empty
  | Cons a (List a)

-- GADTSyntax
data List a where
  Empty :: List a
  Cons :: a -> List a -> List a

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 span all types. Problems arise between the clash whether (a ~ Bool) or (a ~ Int) when trying to write the evaluator.

data Term a
  = Lit a
  | Succ (Term a)
  | IsZero (Term a)

-- can't be well-typed :(
eval (Lit i)      = i
eval (Succ t)     = 1 + eval t
eval (IsZero i)   = eval i == 0

And we admit the construction of meaningless terms which forces more error handling cases.

-- This is a valid type.
failure = Succ ( Lit True )

Using a GADT we can express the type invariants for our language (i.e. only type-safe expressions are representable). Pattern matching on this GADTs then carries type equality constraints without the need for explicit tags.

{-# Language GADTs #-}

data Term a where
  Lit    :: a -> Term a
  Succ   :: Term Int -> Term Int
  IsZero :: Term Int -> Term Bool
  If     :: Term Bool -> Term a -> Term a -> Term a

eval :: Term a -> a
eval (Lit i)      = i                                   -- Term a
eval (Succ t)     = 1 + eval t                          -- Term (a ~ Int)
eval (IsZero i)   = eval i == 0                         -- Term (a ~ Int)
eval (If b e1 e2) = if eval b then eval e1 else eval e2 -- Term (a ~ Bool)

example :: Int
example = eval (Succ (Succ (Lit 3)))

This time around:

-- This is rejected at compile-time.
failure = Succ ( Lit True )

Explicit equality constraints (a ~ b) can be added to a function's context. For example the following expand out to the same types.

f :: a -> a -> (a, a)
f :: (a ~ b) => a -> b -> (a,b)
(Int ~ Int)  => ...
(a ~ Int)    => ...
(Int ~ a)    => ...
(a ~ b)      => ...
(Int ~ Bool) => ... -- Will not typecheck.

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.

{-# LANGUAGE ExistentialQuantification #-}

-- Using Constraints
data Exp a
  = (a ~ Int) => LitInt a
  | (a ~ Bool) => LitBool a
  | forall b. (b ~ Bool) => If (Exp b) (Exp a) (Exp a)

-- Using GADTs
-- data Exp a where
--   LitInt  :: Int  -> Exp Int
--   LitBool :: Bool -> Exp Bool
--   If      :: Exp Bool -> Exp a -> Exp a -> Exp a

eval :: Exp a -> a
eval e = case e of
  LitInt i   -> i
  LitBool b  -> b
  If b tr fl -> if eval b then eval tr else eval fl

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.

data T :: * -> * where
  T1 :: Int -> T Int
  T2 :: T a

f (T1 n) = [n]
f T2     = []

Kind Signatures

Haskell's kind system (i.e. the "type of the types") is a system consisting the single kind * and an arrow kind ->.

κ : *
  | κ -> κ
Int :: *
Maybe :: * -> *
Either :: * -> * -> *

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.

{-# LANGUAGE KindSignatures #-}

id :: forall (a :: *). a -> a
id x = x

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.

{-# Language GADTs #-}
{-# LANGUAGE KindSignatures #-}

data Term a :: * where
  Lit    :: a -> Term a
  Succ   :: Term Int -> Term Int
  IsZero :: Term Int -> Term Bool
  If     :: Term Bool -> Term a -> Term a -> Term a

data Vec :: * -> * -> * where
  Nil :: Vec n a
  Cons :: a -> Vec n a -> Vec n a

data Fix :: (* -> *) -> * where
  In :: f (Fix f) -> Fix f


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.

-- Void :: Void -> Void
newtype Void = Void Void

Or using -XEmptyDataDecls we can also construct the uninhabited type equivalently as a data declaration with no constructors.

data Void

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.

import Data.Void

data Foo tag a = Foo a

combine :: Num a => Foo tag a -> Foo tag a -> Foo tag a
combine (Foo a) (Foo b) = Foo (a+b)

-- All identical at the value level, but differ at the type level.
a :: Foo () Int
a = Foo 1

b :: Foo t Int
b = Foo 1

c :: Foo Void Int
c = Foo 1

-- () ~ ()
example1 :: Foo () Int
example1 = combine a a

-- t ~ ()
example2 :: Foo () Int
example2 = combine a b

-- t0 ~ t1
example3 :: Foo t Int
example3 = combine b b

-- Couldn't match type `t' with `Void'
example4 :: Foo t Int
example4 = combine b c

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.

newtype Plaintext = Plaintext Text
newtype Crytpotext = Cryptotext Text

encrypt :: Key -> Plaintext -> Cryptotext
decrypt :: Key -> Cryptotext -> Plaintext

Using phantom types we use an extra parameter.

import Data.Text

data Cryptotext
data Plaintext

data Msg a = Msg Text

encrypt :: Msg Plaintext -> Msg Cryptotext
encrypt = undefined

decrypt :: Msg Cryptotext -> Msg Plaintext
decrypt = undefined

Using -XEmptyDataDecls can be a powerful combination with phantom types that contain no value inhabitants and are "anonymous types".

{-# LANGUAGE EmptyDataDecls #-}

data Token a

The tagged library defines a similar Tagged newtype wrapper.

See: Fun with Phantom Types

Typelevel Operations

This is an advanced section, and is not typically necessary to write Haskell.

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.

{-# LANGUAGE ExplicitForAll #-}

-- a ≡ b
data Eql a b where
  Refl :: Eql a a

-- Congruence
-- (f : A → B) {x y} → x ≡ y → f x ≡ f y
cong :: Eql a b -> Eql (f a) (f b)
cong Refl = Refl

-- Symmetry
-- {a b : A} → a ≡ b → a ≡ b
sym :: Eql a b -> Eql b a
sym Refl = Refl

-- Transitivity
-- {a b c : A} → a ≡ b → b ≡ c → a ≡ c
trans :: Eql a b -> Eql b c -> Eql a c
trans Refl Refl = Refl

-- Coerce one type to another given a proof of their equality.
-- {a b : A} → a ≡ b → a → b
castWith :: Eql a b -> a -> b
castWith Refl = id

-- Trivial cases
a :: forall n. Eql n n
a = Refl

b :: forall. Eql () ()
b = Refl

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:

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:

type Name = String

data Exp
  = Var Name
  | Lam Name Exp
  | App Exp Exp

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.


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.


data Expr a where
  Con :: a -> Expr a
  Lam :: (Expr a -> Expr b) -> Expr (a -> b)
  App :: Expr (a -> b) -> Expr a -> Expr b

i :: Expr (a -> a)
i = Lam (\x -> x)

k :: Expr (a -> b -> a)
k = Lam (\x -> Lam (\y -> x))

s :: Expr ((a -> b -> c) -> (a -> b) -> (a -> c))
s = Lam (\x -> Lam (\y -> Lam (\z -> App (App x z) (App y z))))

eval :: Expr a -> a
eval (Con v) = v
eval (Lam f) = \x -> eval (f (Con x))
eval (App e1 e2) = (eval e1) (eval e2)

skk :: Expr (a -> a)
skk = App (App s k) k

example :: Integer
example = eval skk 1
-- 1

Pretty printing HOAS terms can also be quite complicated since the body of the function is under a Haskell lambda binder.


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.

{-# LANGUAGE RankNTypes #-}

data ExprP a
  = VarP a
  | AppP (ExprP a) (ExprP a)
  | LamP (a -> ExprP a)
  | LitP Integer

data Value
  = VLit Integer
  | VFun (Value -> Value)

fromVFun :: Value -> (Value -> Value)
fromVFun val = case val of
  VFun f -> f
  _      -> error "not a function"

fromVLit :: Value -> Integer
fromVLit val = case val of
  VLit n -> n
  _      -> error "not a integer"

newtype Expr = Expr { unExpr :: forall a . ExprP a }

eval :: Expr -> Value
eval e = ev (unExpr e) where
  ev (LamP f)      = VFun(ev . f)
  ev (VarP v)      = v
  ev (AppP e1 e2)  = fromVFun (ev e1) (ev e2)
  ev (LitP n)      = VLit n

i :: ExprP a
i = LamP (\a -> VarP a)

k :: ExprP a
k = LamP (\x -> LamP (\y -> VarP x))

s :: ExprP a
s = LamP (\x -> LamP (\y -> LamP (\z -> AppP (AppP (VarP x) (VarP z)) (AppP (VarP y) (VarP z)))))

skk :: ExprP a
skk = AppP (AppP s k) k

example :: Integer
example = fromVLit $ eval $ Expr (AppP skk (LitP 3))


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.

{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE TypeSynonymInstances #-}
{-# LANGUAGE NoMonomorphismRestriction #-}

class Expr repr where
  lit :: Int -> repr
  neg :: repr -> repr
  add :: repr -> repr -> repr
  mul :: repr -> repr -> repr

instance Expr Int where
  lit n = n
  neg a = -a
  add a b = a + b
  mul a b = a * b

instance Expr String where
  lit n = show n
  neg a = "(-" ++ a ++ ")"
  add a b = "(" ++ a ++ " + " ++ b ++ ")"
  mul a b = "(" ++ a ++ " * " ++ b ++ ")"

class BoolExpr repr where
  eq :: repr -> repr -> repr
  tr :: repr
  fl :: repr

instance BoolExpr Int where
  eq a b = if a == b then tr else fl
  tr = 1
  fl = 0

instance BoolExpr String where
  eq a b = "(" ++ a ++ " == " ++ b ++ ")"
  tr = "true"
  fl = "false"

eval :: Int -> Int
eval = id

render :: String -> String
render = id

expr :: (BoolExpr repr, Expr repr) => repr
expr = eq (add (lit 1) (lit 2)) (lit 3)

result :: Int
result = eval expr
-- 1

string :: String
string = render expr
-- "((1 + 2) == 3)"

Finally Tagless

Writing an evaluator for the lambda calculus can likewise also be modeled with a final interpreter and a Identity functor.

import Prelude hiding (id)

class Expr rep where
  lam :: (rep a -> rep b) -> rep (a -> b)
  app :: rep (a -> b) -> (rep a -> rep b)
  lit :: a -> rep a

newtype Interpret a = R { reify :: a }

instance Expr Interpret where
  lam f   = R $ reify . f . R
  app f a = R $ reify f $ reify a
  lit     = R

eval :: Interpret a -> a
eval e = reify e

e1 :: Expr rep => rep Int
e1 = app (lam (\x -> x)) (lit 3)

e2 :: Expr rep => rep Int
e2 = app (lam (\x -> lit 4)) (lam $ \x -> lam $ \y -> y)

example1 :: Int
example1 = eval e1
-- 3

example2 :: Int
example2 = eval e2
-- 4

See: Typed Tagless Interpretations and Typed Compilation


The usual hand-wavy of describing algebraic datatypes is to indicate the how natural correspondence between sum types, product types, and polynomial expressions arises.

data Void                       -- 0
data Unit     = Unit            -- 1
data Sum a b  = Inl a | Inr b   -- a + b
data Prod a b = Prod a b        -- a * b
type (->) a b = a -> b          -- b ^ a

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.

-- 1 + A
data Maybe a = Nothing | Just a

Recursive types are correspond to infinite series of these terms.

-- pseudocode

-- μX. 1 + X
data Nat a = Z | S Nat
Nat a = μ a. 1 + a
      = 1 + (1 + (1 + ...))

-- μX. 1 + A * X
data List a = Nil | Cons a (List a)
List a = μ a. 1 + a * (List a)
       = 1 + a + a^2 + a^3 + a^4 ...

-- μX. A + A*X*X
data Tree a f = Leaf a | Tree a f f
Tree a = μ a. 1 + a * (List a)
       = 1 + a^2 + a^4 + a^6 + a^8 ...

See: Species and Functors and Types, Oh My!


This is an advanced section, and is not typically necessary to write Haskell.

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.

type Algebra f a = f a -> a
type Coalgebra f a = a -> f a
newtype Fix f = Fix { unFix :: f (Fix f) }

cata :: Functor f => Algebra f a -> Fix f -> a
ana  :: Functor f => Coalgebra f a -> a -> Fix f
hylo :: Functor f => Algebra f b -> Coalgebra f a -> a -> b

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.

newtype Fix f = Fix { unFix :: f (Fix f) }

Fix :: f (Fix f) -> Fix f
unFix :: Fix f -> f (Fix f)
Fix f = f (f (f (f (f (f ( ... ))))))

newtype T b a = T (a -> b)

Fix (T a)
Fix T -> a
(Fix T -> a) -> a
(Fix T -> a) -> a -> a

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.

cata :: Functor f => Algebra f a -> Fix f -> a
cata alg = alg . fmap (cata alg) . unFix

ana :: Functor f => Coalgebra f a -> a -> Fix f
ana coalg = Fix . fmap (ana coalg) . coalg

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:

{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE DeriveFunctor #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE UndecidableInstances #-}

type Algebra f a = f a -> a
type Coalgebra f a = a -> f a

newtype Fix f = Fix { unFix :: f (Fix f) }

-- catamorphism
cata :: Functor f => Algebra f a -> Fix f -> a
cata alg = alg . fmap (cata alg) . unFix

-- anamorphism
ana :: Functor f => Coalgebra f a -> a -> Fix f
ana coalg = Fix . fmap (ana coalg) . coalg

-- hylomorphism
hylo :: Functor f => Algebra f b -> Coalgebra f a -> a -> b
hylo f g = cata f . ana g

type Nat = Fix NatF
data NatF a = S a | Z deriving (Eq,Show)

instance Functor NatF where
  fmap f Z     = Z
  fmap f (S x) = S (f x)

plus :: Nat -> Nat -> Nat
plus n = cata phi where
  phi Z     = n
  phi (S m) = s m

times :: Nat -> Nat -> Nat
times n = cata phi where
  phi Z     = z
  phi (S m) = plus n m

int :: Nat -> Int
int = cata phi where
  phi  Z    = 0
  phi (S f) = 1 + f

nat :: Integer -> Nat
nat = ana (psi Z S) where
  psi f _ 0 = f
  psi _ f n = f (n-1)

z :: Nat
z = Fix Z

s :: Nat -> Nat
s = Fix . S

type Str = Fix StrF
data StrF x = Cons Char x | Nil

instance Functor StrF where
  fmap f (Cons a as) = Cons a (f as)
  fmap f Nil = Nil

nil :: Str
nil = Fix Nil

cons :: Char -> Str -> Str
cons x xs = Fix (Cons x xs)

str :: Str -> String
str = cata phi where
  phi Nil         = []
  phi (Cons x xs) = x : xs

str' :: String -> Str
str' = ana (psi Nil Cons) where
  psi f _ []     = f
  psi _ f (a:as) = f a as

map' :: (Char -> Char) -> Str -> Str
map' f = hylo g unFix
    g Nil        = Fix Nil
    g (Cons a x) = Fix $ Cons (f a) x

type Tree a = Fix (TreeF a)
data TreeF a f = Leaf a | Tree a f f deriving (Show)

instance Functor (TreeF a) where
  fmap f (Leaf a) = Leaf a
  fmap f (Tree a b c) = Tree a (f b) (f c)

depth :: Tree a -> Int
depth = cata phi where
  phi (Leaf _)     = 0
  phi (Tree _ l r) = 1 + max l r

example1 :: Int
example1 = int (plus (nat 125) (nat 25))
-- 150

Or for example an interpreter for a small expression language that depends on a scoping dictionary.

{-# LANGUAGE DeriveFunctor #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE UndecidableInstances #-}

import Control.Applicative
import qualified Data.Map as M

type Algebra f a = f a -> a
type Coalgebra f a = a -> f a

newtype Fix f = Fix { unFix :: f (Fix f) }

cata :: Functor f => Algebra f a -> Fix f -> a
cata alg = alg . fmap (cata alg) . unFix

ana :: Functor f => Coalgebra f a -> a -> Fix f
ana coalg = Fix . fmap (ana coalg) . coalg

hylo :: Functor f => Algebra f b -> Coalgebra f a -> a -> b
hylo f g = cata f . ana g

type Id = String
type Env = M.Map Id Int

type Expr = Fix ExprF
data ExprF a
  = Lit Int
  | Var Id
  | Add a a
  | Mul a a
  deriving (Show, Eq, Ord, Functor)

deriving instance Eq (f (Fix f)) => Eq (Fix f)
deriving instance Ord (f (Fix f)) => Ord (Fix f)
deriving instance Show (f (Fix f)) => Show (Fix f)

eval :: M.Map Id Int -> Fix ExprF -> Maybe Int
eval env = cata phi where
  phi ex = case ex of
    Lit c   -> pure c
    Var i   -> M.lookup i env
    Add x y -> liftA2 (+) x y
    Mul x y -> liftA2 (*) x y

expr :: Expr
expr = Fix (Mul n (Fix (Add x y)))
    n = Fix (Lit 10)
    x = Fix (Var "x")
    y = Fix (Var "y")

env :: M.Map Id Int
env = M.fromList [("x", 1), ("y", 2)]

compose :: (f (Fix f) -> c) -> (a -> Fix f) -> a -> c
compose x y = x . unFix . y

example :: Maybe Int
example = eval env expr
-- Just 30

What's especially nice about this approach is how naturally catamorphisms compose into efficient composite transformations.

compose :: Functor f => (f (Fix f) -> c) -> (a -> Fix f) -> a -> c
compose f g = f . unFix . g


This is an advanced section, and is not typically necessary to write Haskell.

Catamorphism foldr :: (a -> b -> b) -> b -> [a] -> b Deconstructs a data structure
Anamorphism unfoldr :: (b -> Maybe (a, b)) -> b -> [a] Constructs a structure level by level
-- | A fix-point type.
newtype Fix f = Fix { unFix :: f (Fix f) }

-- | Catamorphism or generic function fold.
cata :: Functor f => (f a -> a) -> (Fix f -> a)
cata f = f . fmap (cata f) . unFix

-- | Anamorphism or generic function unfold.
ana :: Functor f => (a -> f a) -> (a -> Fix f)
ana f = Fix . fmap (ana f) . f

The code from the F-algebra examples above is implemented in an off-the-shelf library called recursion-schemes.

{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE DeriveFunctor #-}

import Data.Functor.Foldable

type Var = String

data Exp
  = Var Var
  | App Exp Exp
  | Lam [Var] Exp
  deriving Show

data ExpF a
  = VarF Var
  | AppF a a
  | LamF [Var] a
  deriving Functor

type instance Base Exp = ExpF

instance Foldable Exp where
  project (Var a)     = VarF a
  project (App a b)   = AppF a b
  project (Lam a b)   = LamF a b

instance Unfoldable Exp where
  embed (VarF a)      = Var a
  embed (AppF a b)    = App a b
  embed (LamF a b)    = Lam a b

fvs :: Exp -> [Var]
fvs = cata phi
  where phi (VarF a)    = [a]
        phi (AppF a b)  = a ++ b
        phi (LamF a b) = foldr (filter . (/=)) a b

An example of usage:

{-# LANGUAGE DeriveFunctor #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE TypeSynonymInstances #-}

import Data.Traversable
import Control.Monad hiding (forM_, mapM, sequence)
import Prelude hiding (mapM)
import qualified Data.Map as M

newtype Fix (f :: * -> *) = Fix { outF :: f (Fix f) }

-- Catamorphism
cata :: Functor f => (f a -> a) -> Fix f -> a
cata f = f . fmap (cata f) . outF

-- Monadic catamorphism
cataM :: (Traversable f, Monad m) => (f a -> m a) -> Fix f -> m a
cataM f = f <=< mapM (cataM f) . outF

data ExprF r
  = EVar String
  | EApp r r
  | ELam r r
  deriving (Show, Eq, Ord, Functor)

type Expr = Fix ExprF

instance Show (Fix ExprF) where
  show (Fix f) = show f

instance Eq (Fix ExprF) where
  Fix x == Fix y = x == y

instance Ord (Fix ExprF) where
  compare (Fix x) (Fix y) = compare x y

mkApp :: Fix ExprF -> Fix ExprF -> Fix ExprF
mkApp x y = Fix (EApp x y)

mkVar :: String -> Fix ExprF
mkVar x = Fix (EVar x)

mkLam :: Fix ExprF -> Fix ExprF -> Fix ExprF
mkLam x y = Fix (ELam x y)

i :: Fix ExprF
i = mkLam (mkVar "x") (mkVar "x")

k :: Fix ExprF
k = mkLam (mkVar "x") $ mkLam (mkVar "y") $ (mkVar "x")

subst :: M.Map String (ExprF Expr) -> Expr -> Expr
subst env = cata alg where
  alg (EVar x) | Just e <- M.lookup x env = Fix e
  alg e = Fix e


Data types à la carte


Hint and Mueval

This is an advanced section, and is not typically necessary to write Haskell.

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.

import Language.Haskell.Interpreter

foo :: Interpreter String
foo = eval "(\\x -> x) 1"

example :: IO (Either InterpreterError String)
example = runInterpreter foo

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.



Contrary to a lot of misinformation, unit testing in Haskell is quite common and robust. Although 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 equation laws which guarantee strong invariants about program behavior under composition. Many of the testing tools are built around this style of design.


Probably the most famous Haskell library, QuickCheck is a testing framework for generating large random tests for arbitrary functions automatically based on the types of their arguments.

quickCheck :: Testable prop => prop -> IO ()
(==>) :: Testable prop => Bool -> prop -> Property
forAll :: (Show a, Testable prop) => Gen a -> (a -> prop) -> Property
choose :: Random a => (a, a) -> Gen a
import Test.QuickCheck

qsort :: [Int] -> [Int]
qsort []     = []
qsort (x:xs) = qsort lhs ++ [x] ++ qsort rhs
    where lhs = filter  (< x) xs
          rhs = filter (>= x) xs

prop_maximum ::  [Int] -> Property
prop_maximum xs = not (null xs) ==>
                  last (qsort xs) == maximum xs

main :: IO ()
main = quickCheck prop_maximum
$ runhaskell qcheck.hs
*** Failed! Falsifiable (after 3 tests and 4 shrinks):

$ runhaskell qcheck.hs
+++ OK, passed 1000 tests.

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


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.

smallCheck :: Testable IO a => Depth -> a -> IO ()
list :: Depth -> Series Identity a -> [a]
sample' :: Gen a -> IO [a]
λ: list 3 series :: [Int]

λ: list 3 series :: [Double]

λ: list 3 series :: [(Int, String)]

It is useful to generate test cases over all possible inputs of a program up to some depth.

import Test.SmallCheck

distrib :: Int -> Int -> Int -> Bool
distrib a b c = a * (b + c) == a * b + a * c

cauchy :: [Double] -> [Double] -> Bool
cauchy xs ys = (abs (dot xs ys))^2 <= (dot xs xs) * (dot ys ys)

failure :: [Double] -> [Double] -> Bool
failure xs ys = abs (dot xs ys) <= (dot xs xs) * (dot ys ys)

dot :: Num a => [a] -> [a] -> a
dot xs ys = sum (zipWith (*) xs ys)

main :: IO ()
main = do
  putStrLn "Testing distributivity..."
  smallCheck 25 distrib

  putStrLn "Testing Cauchy-Schwarz..."
  smallCheck 4 cauchy

  putStrLn "Testing invalid Cauchy-Schwarz..."
  smallCheck 4 failure
$ runhaskell smallcheck.hs
Testing distributivity...
Completed 132651 tests without failure.

Testing Cauchy-Schwarz...
Completed 27556 tests without failure.

Testing invalid Cauchy-Schwarz...
Failed test no. 349.
there exist [1.0] [0.5] such that
  condition is false

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:

{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}

import Test.SmallCheck
import Test.SmallCheck.Series
import Control.Applicative

import qualified Data.Vector as V

dot :: Num a => V.Vector a -> V.Vector a -> a
dot xs ys = V.sum (V.zipWith (*) xs ys)

cauchy :: V.Vector Double -> V.Vector Double -> Bool
cauchy xs ys = (abs (dot xs ys))^2 <= (dot xs xs) * (dot ys ys)

instance (Serial m a, Monad m) => Serial m (V.Vector a) where
  series = V.fromList <$> series

main :: IO ()
main = smallCheck 4 cauchy

SmallCheck can also use Generics to derive Serial instances, for example to enumerate all trees of a certain depth we might use:

{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE DeriveGeneric #-}

import GHC.Generics
import Test.SmallCheck.Series

data Tree a = Null | Fork (Tree a) a (Tree a)
    deriving (Show, Generic)

instance Serial m a => Serial m (Tree a)

example :: [Tree ()]
example = list 3 series

main = print example


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 TypeOperators #-}
{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE ScopedTypeVariables #-}

import Data.List
import Data.Typeable

import Test.QuickSpec hiding (lists, bools, arith)
import Test.QuickCheck

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

    -- 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])

    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.

$ runhaskell src/quickspec.hs
== API ==
-- functions --
map :: (A -> A) -> [A] -> [A]
minimum :: [A] -> A
(++) :: [A] -> [A] -> [A]
length :: [A] -> Int
sort, id, reverse :: [A] -> [A]

-- background functions --
id :: A -> A
(:) :: A -> [A] -> [A]
(.) :: (A -> A) -> (A -> A) -> A -> A
[] :: [A]

-- variables --
f, g, h :: A -> A
xs, ys, zs :: [A]

-- the following types are using non-standard equality --
A -> A

-- WARNING: there are no variables of the following types; consider adding some --

== Testing ==
Depth 1: 12 terms, 4 tests, 24 evaluations, 12 classes, 0 raw equations.
Depth 2: 80 terms, 500 tests, 18673 evaluations, 52 classes, 28 raw equations.
Depth 3: 1553 terms, 500 tests, 255056 evaluations, 1234 classes, 319 raw equations.
319 raw equations; 1234 terms in universe.

== Equations about map ==
  1: map f [] == []
  2: map id xs == xs
  3: map (f.g) xs == map f (map g xs)

== Equations about minimum ==
  4: minimum [] == undefined

== Equations about (++) ==
  5: xs++[] == xs
  6: []++xs == xs
  7: (xs++ys)++zs == xs++(ys++zs)

== Equations about sort ==
  8: sort [] == []
  9: sort (sort xs) == sort xs

== Equations about id ==
 10: id xs == xs

== Equations about reverse ==
 11: reverse [] == []
 12: reverse (reverse xs) == xs

== Equations about several functions ==
 13: minimum (xs++ys) == minimum (ys++xs)
 14: length (map f xs) == length xs
 15: length (xs++ys) == length (ys++xs)
 16: sort (xs++ys) == sort (ys++xs)
 17: map f (reverse xs) == reverse (map f xs)
 18: minimum (sort xs) == minimum xs
 19: minimum (reverse xs) == minimum xs
 20: minimum (xs++xs) == minimum xs
 21: length (sort xs) == length xs
 22: length (reverse xs) == length xs
 23: sort (reverse xs) == sort xs
 24: map f xs++map f ys == map f (xs++ys)
 25: reverse xs++reverse ys == reverse (ys++xs)

Keep in mind the rather remarkable fact that this is all deduced automatically from the types alone!


Criterion is a statistically aware benchmarking tool.

whnf :: (a -> b) -> a -> Pure
nf :: NFData b => (a -> b) -> a -> Pure
nfIO :: NFData a => IO a -> IO ()
bench :: Benchmarkable b => String -> b -> Benchmark
import Criterion.Main
import Criterion.Config

-- Naive recursion for fibonacci numbers.
fib1 :: Int -> Int
fib1 0 = 0
fib1 1 = 1
fib1 n = fib1 (n-1) + fib1 (n-2)

-- Use the De Moivre closed form for fibonacci numbers.
fib2 :: Int -> Int
fib2 x = truncate $ ( 1 / sqrt 5 ) * ( phi ^ x - psi ^ x )
      phi = ( 1 + sqrt 5 ) / 2
      psi = ( 1 - sqrt 5 ) / 2

suite :: [Benchmark]
suite = [
    bgroup "naive" [
      bench "fib 10" $ whnf fib1 5
    , bench "fib 20" $ whnf fib1 10
    bgroup "de moivre" [
      bench "fib 10" $ whnf fib2 5
    , bench "fib 20" $ whnf fib2 10

main :: IO ()
main = defaultMain suite
$ runhaskell criterion.hs
warming up
estimating clock resolution...
mean is 2.349801 us (320001 iterations)
found 1788 outliers among 319999 samples (0.6%)
  1373 (0.4%) high severe
estimating cost of a clock call...
mean is 65.52118 ns (23 iterations)
found 1 outliers among 23 samples (4.3%)
  1 (4.3%) high severe

benchmarking naive/fib 10
mean: 9.903067 us, lb 9.885143 us, ub 9.924404 us, ci 0.950
std dev: 100.4508 ns, lb 85.04638 ns, ub 123.1707 ns, ci 0.950

benchmarking naive/fib 20
mean: 120.7269 us, lb 120.5470 us, ub 120.9459 us, ci 0.950
std dev: 1.014556 us, lb 858.6037 ns, ub 1.296920 us, ci 0.950

benchmarking de moivre/fib 10
mean: 7.699219 us, lb 7.671107 us, ub 7.802116 us, ci 0.950
std dev: 247.3021 ns, lb 61.66586 ns, ub 572.1260 ns, ci 0.950
found 4 outliers among 100 samples (4.0%)
  2 (2.0%) high mild
  2 (2.0%) high severe
variance introduced by outliers: 27.726%
variance is moderately inflated by outliers

benchmarking de moivre/fib 20
mean: 8.082639 us, lb 8.018560 us, ub 8.350159 us, ci 0.950
std dev: 595.2161 ns, lb 77.46251 ns, ub 1.408784 us, ci 0.950
found 8 outliers among 100 samples (8.0%)
  4 (4.0%) high mild
  4 (4.0%) high severe
variance introduced by outliers: 67.628%
variance is severely inflated by outliers

Criterion can also generate a HTML page containing the benchmark results plotted

$ ghc -O2 --make criterion.hs
$ ./criterion -o bench.html


Tasty combines all of the testing frameworks into a common API for forming runnable batches of tests and collecting the results.

import Test.Tasty
import Test.Tasty.HUnit
import Test.Tasty.QuickCheck
import qualified Test.Tasty.SmallCheck as SC

arith :: Integer -> Integer -> Property
arith x y = (x > 0) && (y > 0) ==> (x+y)^2 > x^2 + y^2

negation :: Integer -> Bool
negation x = abs (x^2) >= x

suite :: TestTree
suite = testGroup "Test Suite" [
    testGroup "Units"
      [ testCase "Equality" $ True @=? True
      , testCase "Assertion" $ assert $ (length [1,2,3]) == 3

    testGroup "QuickCheck tests"
      [ testProperty "Quickcheck test" arith

    testGroup "SmallCheck tests"
      [ SC.testProperty "Negation" negation

main :: IO ()
main = defaultMain suite
$ runhaskell TestSuite.hs
Unit tests
    Equality:        OK
    Assertion:       OK
  QuickCheck tests
    Quickcheck test: OK
      +++ OK, passed 100 tests.
  SmallCheck tests
    Negation:        OK
      11 tests completed


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.

capture :: IO a -> IO (String, a)
import Test.Tasty
import Test.Tasty.HUnit
import System.IO.Silently

test :: Int -> IO ()
test n = print (n * n)

testCapture n = do
  (stdout, result) <- capture (test n)
  assert (stdout == show (n*n) ++ "\n")

suite :: TestTree
suite = testGroup "Test Suite" [
    testGroup "Units"
      [ testCase "Equality" $ testCapture 10

main :: IO ()
main = defaultMain suite

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:

(Eq [a], Ord [a]) => [a]
==> Ord a => [a]

If a single parameter typeclass expresses a property of a type ( i.e. 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 a type can be converted to another type we might use a class like:

{-# LANGUAGE MultiParamTypeClasses #-}

import Data.Char

class Convertible a b where
  convert :: a -> b

instance Convertible Int Integer where
  convert = toInteger

instance Convertible Int Char where
  convert = chr

instance Convertible Char Int where
  convert = ord

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.

{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FunctionalDependencies #-}

import Data.Char

class Convertible a b | a -> b where
  convert :: a -> b

instance Convertible Int Char where
  convert = chr

instance Convertible Char Int where
  convert = ord
Functional dependencies conflict between instance declarations:
  instance Convertible Int Integer
  instance Convertible Int Char

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.

λ: convert (42 :: Int)
λ: convert '*'

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.

{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE UndecidableInstances #-}

data Z
data S n

type Zero  = Z
type One   = S Zero
type Two   = S One
type Three = S Two
type Four  = S Three

zero :: Zero
zero = undefined

one :: One
one = undefined

two :: Two
two = undefined

three :: Three
three = undefined

four :: Four
four = undefined

class Eval a where
  eval :: a -> Int

instance Eval Zero where
  eval _ = 0

instance Eval n => Eval (S n) where
  eval m = 1 + eval (prev m)

class Pred a b | a -> b where
  prev :: a -> b

instance Pred Zero Zero where
  prev = undefined

instance Pred (S n) n where
  prev = undefined

class Add a b c | a b -> c where
  add :: a -> b -> c

instance Add Zero a a where
  add = undefined

instance Add a b c => Add (S a) b (S c) where
  add = undefined

f :: Three
f = add one two

g :: S (S (S (S Z)))
g = add two two

h :: Int
h = eval (add three four)

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.

add(0, A, A).
add(s(A), B, s(C)) :- add(A, B, C).

pred(0, 0).
pred(S(A), A).

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.

    Context reduction stack overflow; size = 201

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.

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:

-- (1) Unassociated form
type family Rep a
type instance Rep Int = Char
type instance Rep Char = Int

class Convertible a where
  convert :: a -> Rep a

instance Convertible Int where
  convert = chr

instance Convertible Char where
  convert = ord

-- (2) Associated form
class Convertible a where
  type Rep a
  convert :: a -> Rep a

instance Convertible Int where
  type Rep Int = Char
  convert = chr

instance Convertible Char where
  type Rep Char = Int
  convert = ord

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.

λ: :kind! Rep Int
Rep Int :: *
= Char
λ: :kind! Rep Char
Rep Char :: *
= Int

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:

{-# LANGUAGE TypeFamilies #-}

import qualified Data.Vector.Unboxed as V

data family Array a
data instance Array Int       = IArray (V.Vector Int)
data instance Array Bool      = BArray (V.Vector Bool)
data instance Array (a,b)     = PArray (Array a) (Array b)
data instance Array (Maybe a) = MArray (V.Vector Bool) (Array a)

class IArray a where
  index :: Array a -> Int -> a

instance IArray Int where
  index (IArray xs) i = xs V.! i

instance IArray Bool where
  index (BArray xs) i = xs V.! i

-- Vector of pairs
instance (IArray a, IArray b) => IArray (a, b) where
  index (PArray xs ys) i = (index xs i, index ys i)

-- Vector of missing values
instance (IArray a) => IArray (Maybe a) where
  index (MArray bm xs) i =
    case bm V.! i of
      True  -> Nothing
      False -> Just $ index xs i


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.

data Maybe a = Nothing | Just a
-- Maybe a ~ Maybe b  implies  a ~ b

type instance F Int = Bool
type instance F Char = Bool

-- F a ~ F b does not imply  a ~ b, in general


This is an advanced section, and is not typically necessary to write Haskell.

Roles are a further level of specification for type variables parameters of datatypes.

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.)

{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE GeneralizedNewtypeDeriving #-}

newtype Age = MkAge { unAge :: Int }

type family Inspect x
type instance Inspect Age = Int
type instance Inspect Int = Bool

class Boom a where
  boom :: a -> Inspect a

instance Boom Int where
  boom = (== 0)

deriving instance Boom Age

-- GHC 7.6.3 exhibits undefined behavior
failure = boom (MkAge 3)
-- -6341068275333450897

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.

{-# LANGUAGE PolyKinds #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE RoleAnnotations #-}

data Nat = Zero | Suc Nat

type role Vec nominal representational
data Vec :: Nat -> * -> * where
  Nil  :: Vec Zero a
  (:*) :: a -> Vec n a -> Vec (Suc n) a

type role App representational nominal
data App (f :: k -> *) (a :: k) = App (f a)

type role Mu nominal nominal
data Mu (f :: (k -> *) -> k -> *) (a :: k) = Roll (f (Mu f) a)

type role Proxy phantom
data Proxy (a :: k) = Proxy
coerce :: Coercible * a b => a -> b
class (~R#) k k a b => Coercible k a b

Safe Zero-cost Coercions for Haskell Data.Coerce



Using type families, mono-traversable generalizes the notion of Functor, Foldable, and Traversable to include both monomorphic and polymorphic types.

omap :: MonoFunctor mono => (Element mono -> Element mono) -> mono -> mono

otraverse :: (Applicative f, MonoTraversable mono)
          => (Element mono -> f (Element mono)) -> mono -> f mono

ofoldMap :: (Monoid m, MonoFoldable mono)
         => (Element mono -> m) -> mono -> m
ofoldl' :: MonoFoldable mono
        => (a -> Element mono -> a) -> a -> mono -> a
ofoldr :: MonoFoldable mono
        => (Element mono -> b -> b) -> b -> mono -> b

For example the text type normally does not admit any of these type-classes since, but now we can write down the instances that model the interface of Foldable and Traversable.

{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE OverloadedStrings #-}

import Data.Text
import Data.Char
import Data.Monoid
import Data.MonoTraversable
import Control.Applicative

bs :: Text
bs = "Hello Haskell."

shift :: Text
shift = omap (chr . (+1) . ord) bs
-- "Ifmmp!Ibtlfmm/"

backwards :: [Char]
backwards = ofoldl' (flip (:)) "" bs
-- ".lleksaH olleH"

data MyMonoType = MNil | MCons Int MyMonoType deriving Show

type instance Element MyMonoType = Int

instance MonoFunctor MyMonoType where
  omap f MNil = MNil
  omap f (MCons x xs) = f x `MCons` omap f xs

instance MonoFoldable MyMonoType where
  ofoldMap f   = ofoldr (mappend . f) mempty
  ofoldr       = mfoldr
  ofoldl'      = mfoldl'
  ofoldr1Ex f  = ofoldr1Ex f . mtoList
  ofoldl1Ex' f = ofoldl1Ex' f . mtoList

instance MonoTraversable MyMonoType where
  omapM f xs = mapM f (mtoList xs) >>= return . mfromList
  otraverse f = ofoldr acons (pure MNil)
    where acons x ys = MCons <$> f x <*> ys

mtoList :: MyMonoType -> [Int]
mtoList (MNil) = []
mtoList (MCons x xs) = x : (mtoList xs)

mfromList :: [Int] -> MyMonoType
mfromList [] = MNil
mfromList (x:xs) = MCons x (mfromList xs)

mfoldr :: (Int -> a -> a) -> a -> MyMonoType -> a
mfoldr f z MNil =  z
mfoldr f z (MCons x xs) =  f x (mfoldr f z xs)

mfoldl' :: (a -> Int -> a) -> a -> MyMonoType -> a
mfoldl' f z MNil = z
mfoldl' f z (MCons x xs) = let z' = z `f` x
                           in seq z' $ mfoldl' f z' xs

ex1 :: Int
ex1 = mfoldl' (+) 0 (mfromList [1..25])

ex2 :: MyMonoType
ex2 = omap (+1) (mfromList [1..25])

See: From Semigroups to Monads


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.

infixr 5 :|, <|
data NonEmpty a = a :| [a]

head :: NonEmpty a -> a
toList :: NonEmpty a -> [a]
fromList :: [a] -> NonEmpty a
head :: NonEmpty a -> a
head ~(a :| _) = a
import Data.List.NonEmpty
import Prelude hiding (head, tail, foldl1)
import Data.Foldable (foldl1)

a :: NonEmpty Integer
a = fromList [1,2,3]
-- 1 :| [2,3]

b :: NonEmpty Integer
b = 1 :| [2,3]
-- 1 :| [2,3]

c :: NonEmpty Integer
c = fromList []
-- *** Exception: NonEmpty.fromList: empty list

d :: Integer
d = foldl1 (+) $ fromList [1..100]
-- 5050

Manual Proofs

This is an advanced section, and is not typically necessary to write Haskell.

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.

Types Logic
A proposition
a : A proof
B(x) predicate
A + B A ∨ B
A × B A ∧ B
A -> B A ⇒ B

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.

P 0                   [ base step ]
n. P n   P (1+n)    [ inductive step ]
n. P(n)
Axiom 1: a + 0 = a
Axiom 2: a + suc b = suc (a + b)

  0 + suc a
= suc (0 + a)  [by Axiom 2]
= suc a        [Induction hypothesis]

Translated into Haskell our axioms are simply type definitions and recursing over the inductive datatype constitutes the inductive step of our proof.

{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE ExplicitForAll #-}
{-# LANGUAGE TypeOperators #-}

data Z
data S n

data SNat n where
  Zero :: SNat Z
  Succ :: SNat n -> SNat (S n)

data Eql a b where
  Refl :: Eql a a

type family Add m n
type instance Add Z n = n
type instance Add (S m) n = S (Add m n)

add :: SNat n -> SNat m -> SNat (Add n m)
add Zero     m = m
add (Succ n) m = Succ (add n m)

cong :: Eql a b -> Eql (f a) (f b)
cong Refl = Refl

-- ∀n. 0 + suc n = suc n
plus_suc :: forall n.  SNat n
         -> Eql (Add Z (S n)) (S n)
plus_suc Zero = Refl
plus_suc (Succ n) = cong (plus_suc n)

-- ∀n. 0 + n = n
plus_zero :: forall n. SNat n
         -> Eql (Add Z n) n
plus_zero Zero = Refl
plus_zero (Succ n) = cong (plus_zero n)

Using the TypeOperators extension we can also use infix notation at the type-level.

data a :=: b where
  Refl :: a :=: a

cong :: a :=: b -> (f a) :=: (f b)
cong Refl = Refl

type family (n :: Nat) :+ (m :: Nat) :: Nat
type instance Zero     :+ m = m
type instance (Succ n) :+ m = Succ (n :+ m)

plus_suc :: forall n m. SNat n -> SNat m -> (n :+ (S m)) :=: (S (n :+ m))
plus_suc Zero m = Refl
plus_suc (Succ n) m = cong (plus_suc n m)

Constraint Kinds

This is an advanced section, and is not typically necessary to write Haskell.

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.

Num :: * -> Constraint
Odd :: * -> Constraint
type T1 a = (Num a, Ord a)

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.

{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE ConstraintKinds #-}

import GHC.Exts (Constraint)
import Data.Hashable
import Data.HashSet

type family Con a :: Constraint
type instance Con [a] = (Ord a, Eq a)
type instance Con (HashSet a) = (Hashable a)

class Sized a where
  gsize :: Con a => a -> Int

instance Sized [a] where
  gsize = length

instance Sized (HashSet a) where
  gsize = size

One use-case of this is to capture the typeclass dictionary constrained by a function and reify it as a value.

{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE KindSignatures #-}

import GHC.Exts (Constraint)

data Dict :: Constraint -> * where
  Dict :: (c) => Dict c

dShow :: Dict (Show a) -> a -> String
dShow Dict x = show x

dEqNum :: Dict (Eq a, Num a) -> a -> Bool
dEqNum Dict x = x == 0

fShow :: String
fShow = dShow Dict 10

fEqual :: Bool
fEqual = dEqNum Dict 0


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.

{-# LANGUAGE XTypeFamilyDependencies #-}

type family F a b c = (result :: k) | result -> a b c
type instance F Int  Char Bool = Bool
type instance F Char Bool Int  = Int
type instance F Bool Int  Char = Char



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.

data Free f a where
  Pure :: a -> Free f a
  Free :: f (Free f a) -> Free f a

data Cofree f a where
  Cofree :: a -> f (Cofree f a) -> Cofree f a
Free :: (* -> *) -> * -> *
Cofree :: (* -> *) -> * -> *

For instance Cofree Maybe a for some monokinded type a models a non-empty list with Maybe :: * -> *.

-- Cofree Maybe a is a non-empty list
testCofree :: Cofree Maybe Int
testCofree = (Cofree 1 (Just (Cofree 2 Nothing)))

Kind Polymorphism

This is an advanced section, knowledge of kind polymorphism is not typically necessary to write Haskell.

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.

app :: forall a b. (a -> b) -> a -> b
app f a = f a

But when we do the same thing at the type-level we see we lose information about the polymorphism of the constructor applied.

-- TApp :: (* -> *) -> * -> *
data TApp f a = MkTApp (f a)

Turning on -XPolyKinds allows polymorphic variables at the kind level as well.

-- Default:   (* -> *) -> * -> *
-- PolyKinds: (k -> *) -> k -> *
data TApp f a = MkTApp (f a)

-- Default:   ((* -> *) -> (* -> *)) -> (* -> *)
-- PolyKinds: ((k -> *) -> (k -> *)) -> (k -> *)
data Mu f a = Roll (f (Mu f) a)

-- Default:   * -> *
-- PolyKinds: k -> *
data Proxy a = Proxy

Using the polykinded Proxy type allows us to write down type class functions over constructors of arbitrary kind arity.

{-# LANGUAGE PolyKinds #-}
{-# LANGUAGE KindSignatures #-}

data Proxy a = Proxy
data Rep = Rep

class PolyClass a where
  foo :: Proxy a -> Rep
  foo = const Rep

-- () :: *
-- [] :: * -> *
-- Either :: * -> * -> *

instance PolyClass ()
instance PolyClass []
instance PolyClass Either

For example we can write down the polymorphic S K combinators at the type level now.

{-# LANGUAGE PolyKinds #-}

newtype I (a :: *) = I a
newtype K (a :: *) (b :: k) = K a
newtype Flip (f :: k1 -> k2 -> *) (x :: k2) (y :: k1) = Flip (f y x)

unI :: I a -> a
unI (I x) = x

unK :: K a b -> a
unK (K x) = x

unFlip :: Flip f x y -> f y x
unFlip (Flip x) = x

Data Kinds

This is an advanced section, knowledge of kind data kinds is not typically necessary to write Haskell.

The -XDataKinds extension allows us to use refer to constructors at the value level and the type level. Consider a simple sum type:

data S a b = L a | R b

-- S :: * -> * -> *
-- L :: a -> S a b
-- R :: b -> S a b

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.

{-# LANGUAGE DataKinds #-}

data S a b = L a | R b

-- S :: * -> * -> *
-- L :: * -> S * *
-- R :: * -> 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.

data Foo = Bar | Baz
type Bar = 'Bar
type Baz = 'Baz

Combining this with type families we see we can write meaningful, meaningful type-level functions by lifting types to the kind level.

{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE DataKinds #-}

import Prelude hiding (Bool(..))

data Bool = False | True

type family Not (a :: Bool) :: Bool

type instance Not True = False
type instance Not False = True

false :: Not True ~ False => a
false = undefined

true :: Not False ~ True => a
true = undefined

-- Fails at compile time.
-- Couldn't match type 'False with 'True
invalid :: Not True ~ True => a
invalid = undefined

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 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.

example2 = zipVec vec4 vec5
-- Couldn't match type 'S 'Z with 'Z
-- Expected type: Vec Four Int
--   Actual type: Vec Five Int

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.

{-# LANGUAGE DataKinds #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE FlexibleContexts #-}

data Size = Empty | NonEmpty

data List a b where
  Nil  :: List Empty a
  Cons :: a -> List b a -> List NonEmpty a

head' :: List NonEmpty a -> a
head' (Cons x _) = x

example1 :: Int
example1 = head' (1 `Cons` (2 `Cons` Nil))

-- Cannot match type Empty with NonEmpty
example2 :: Int
example2 = head' Nil
Couldn't match type None with Many
Expected type: List NonEmpty Int
  Actual type: List Empty Int


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.

{-# LANGUAGE DataKinds #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE TypeOperators #-}

import GHC.TypeLits

data Vec :: Nat -> * -> * where
  Nil :: Vec 0 a
  Cons :: a -> Vec n a -> Vec (1 + n) a

-- GHC 7.6 will not reduce
-- vec3 :: Vec (1 + (1 + (1 + 0))) Int

vec3 :: Vec 3 Int
vec3 = 0 `Cons` (1 `Cons` (2 `Cons` Nil))
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE FlexibleContexts #-}

import GHC.TypeLits
import Data.Type.Equality

data Foo :: Nat -> * where
  Small    :: (n <= 2)  => Foo n
  Big      :: (3 <= n) => Foo n

  Empty    :: ((n == 0) ~ True) => Foo n
  NonEmpty :: ((n == 0) ~ False) => Foo n

big :: Foo 10
big = Big

small :: Foo 2
small = Small

empty :: Foo 0
empty = Empty

nonempty :: Foo 3
nonempty = NonEmpty

See: Type-Level Literals

Typelevel Strings

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 expressed using the small datatype found in GHC.TypeLits

data ErrorMessage where
  Text :: Symbol -> ErrorMessage
  ShowType :: t -> ErrorMessage

  -- Put two messages next to each other
  (:<>:) :: ErrorMessage -> ErrorMessage -> ErrorMessage

  -- Put two messages on top of each other
  (:$$:) :: ErrorMessage -> ErrorMessage -> ErrorMessage

If one of these expressions is found in the signature of an expression GHC reports an error message of the form:

example.hs:1:1: error:My custom error message line 1.My custom error message line 2.In the expression: example
      In an equation for ‘foo’: foo = ECoerce (EFloat 3) (EInt 4)
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE UndecidableInstances #-}

import GHC.TypeLits

  -- Error Message
  TypeError (Text "Equality is not defined for functions"
  (ShowType a :<>: Text " -> " :<>: ShowType b))

  -- Instance head
  => Eq (a -> b) where (==) = undefined

-- Fail when we try to equate two functions
example = id == id

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.

{-# LANGUAGE DataKinds #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE UndecidableInstances #-}

import GHC.TypeLits

type family Coerce a b where
  Coerce Int Int     = Int
  Coerce Float Float = Float
  Coerce Int Float   = Float
  Coerce Float Int   = TypeError (Text "Cannot cast to smaller type")

data Expr a where
  EInt    :: Int -> Expr Int
  EFloat  :: Float -> Expr Float
  ECoerce :: Expr b -> Expr c -> Expr (Coerce b c)

foo :: Expr Int
foo = ECoerce (EFloat 3) (EInt 4)

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.

(~)   :: k -> k -> Constraint
(==)  :: k -> k -> Bool
(<=)  :: Nat -> Nat -> Constraint
(<=?) :: Nat -> Nat -> Bool
(+)   :: Nat -> Nat -> Nat
(-)   :: Nat -> Nat -> Nat
(*)   :: Nat -> Nat -> Nat
(^)   :: Nat -> Nat -> Nat
(:~:)     :: k -> k -> *
Refl      :: a1 :~: a1
sym       :: (a :~: b) -> b :~: a
trans     :: (a :~: b) -> (b :~: c) -> a :~: c
castWith  :: (a :~: b) -> a -> b
gcastWith :: (a :~: b) -> (a ~ b => r) -> r

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.

{-# LANGUAGE DataKinds #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE ConstraintKinds #-}

import GHC.TypeLits
import Data.Type.Equality

type Not a b = ((b == a) ~ False)

restrictUnit :: Not () a => a -> a
restrictUnit = id

restrictChar :: Not Char a => a -> a
restrictChar = id


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.

{-# LANGUAGE PolyKinds #-}

-- | A concrete, poly-kinded proxy type
data Proxy t = Proxy
import Data.Proxy

a :: Proxy ()
a = Proxy

b :: Proxy 3
b = Proxy

c :: Proxy "symbol"
c = Proxy

d :: Proxy Maybe
d = Proxy

e :: Proxy (Maybe ())
e = Proxy

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.

t1 :: a
t1 = (undefined :: a)

t2 :: Proxy a
t2 Proxy :: Proxy a

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.

data HList :: [*] -> * where
  HNil  :: HList '[]
  HCons :: a -> HList t -> HList (a ': t)

data Tuple :: (*,*) -> * where
  Tuple :: a -> b -> Tuple '(a,b)

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

This is an advanced section, knowledge of singletons is not typically necessary to write Haskell.

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.

data instance Sing (a :: Nat) where
  SZ :: Sing 'Z
  SS :: Sing n -> Sing ('S n)

data instance Sing (a :: Maybe k) where
  SNothing :: Sing 'Nothing
  SJust :: Sing x -> Sing ('Just x)

data instance Sing (a :: Bool) where
  STrue :: Sing True
  SFalse :: Sing False

Promoted Naturals

Value-level Type-level Models
SZ Sing 'Z 0
SS SZ Sing ('S 'Z) 1
SS (SS SZ) Sing ('S ('S 'Z)) 2

Promoted Booleans

Value-level Type-level Models
SFalse Sing 'False False
STrue Sing 'True True

Promoted Maybe

Value-level Type-level Models
SJust a Sing (SJust 'a) Just a
SNothing Sing Nothing Nothing

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.

{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE PolyKinds #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE TypeSynonymInstances #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE UndecidableInstances #-}

import Data.Proxy
import GHC.Exts (Any)
import Prelude hiding (succ)

data Nat = Z | S Nat

-- kind-indexed data family
data family Sing (a :: k)

data instance Sing (a :: Nat) where
  SZ :: Sing 'Z
  SS :: Sing n -> Sing ('S n)

data instance Sing (a :: Maybe k) where
  SNothing :: Sing 'Nothing
  SJust :: Sing x -> Sing ('Just x)

data instance Sing (a :: Bool) where
  STrue :: Sing True
  SFalse :: Sing False

data Fin (n :: Nat) where
  FZ :: Fin (S n)
  FS :: Fin n -> Fin (S n)

data Vec a n where
  Nil  :: Vec a Z
  Cons :: a -> Vec a n -> Vec a (S n)

class SingI (a :: k) where
  sing :: Sing a

instance SingI Z where
  sing = SZ

instance SingI n => SingI (S n) where
  sing = SS sing

deriving instance Show Nat
deriving instance Show (SNat a)
deriving instance Show (SBool a)
deriving instance Show (Fin a)
deriving instance Show a => Show (Vec a n)

type family (m :: Nat) :+ (n :: Nat) :: Nat where
  Z :+ n = n
  S m :+ n = S (m :+ n)

type SNat (k :: Nat) = Sing k
type SBool (k :: Bool) = Sing k
type SMaybe (b :: a) (k :: Maybe a) = Sing k

size :: Vec a n -> SNat n
size Nil         = SZ
size (Cons x xs) = SS (size xs)

forget :: SNat n -> Nat
forget SZ = Z
forget (SS n) = S (forget n)

natToInt :: Integral n => Nat -> n
natToInt Z     = 0
natToInt (S n) = natToInt n + 1

intToNat :: (Integral a, Ord a) => a -> Nat
intToNat 0 = Z
intToNat n = S $ intToNat (n - 1)

sNatToInt :: Num n => SNat x -> n
sNatToInt SZ     = 0
sNatToInt (SS n) = sNatToInt n + 1

index :: Fin n -> Vec a n -> a
index FZ (Cons x _)      = x
index (FS n) (Cons _ xs) = index n xs

test1 :: Fin (S (S (S Z)))
test1 = FS (FS FZ)

test2 :: Int
test2 = index FZ (1 `Cons` (2 `Cons` Nil))

test3 :: Sing ('Just ('S ('S Z)))
test3 = SJust (SS (SS SZ))

test4 :: Sing ('S ('S Z))
test4 = SS (SS SZ)

-- polymorphic constructor SingI
test5 :: Sing ('S ('S Z))
test5 = sing

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.

someNatVal :: Integer -> Maybe SomeNat
someSymbolVal :: String -> SomeSymbol

natVal :: KnownNat n => proxy n -> Integer
symbolVal :: KnownSymbol n => proxy n -> String
{-# LANGUAGE PolyKinds #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE TypeOperators #-}

import Data.Proxy
import GHC.TypeLits

a :: Integer
a = natVal (Proxy :: Proxy 1)
-- 1

b :: String
b = symbolVal (Proxy :: Proxy "foo")
-- "foo"

c :: Integer
c = natVal (Proxy :: Proxy (2 + 3))
-- 5

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.

{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE UndecidableInstances #-}

import Data.Proxy
import GHC.TypeLits

type family Count (f :: *) :: Nat where
  Count (a -> b) = 1 + (Count b)
  Count x = 1

type Fn1 = Int -> Int
type Fn2 = Int -> Int -> Int -> Int

fn1 :: Integer
fn1 = natVal (Proxy :: Proxy (Count Fn1))
-- 2

fn2 :: Integer
fn2 = natVal (Proxy :: Proxy (Count Fn2))
-- 4

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.

type family Elem (a :: k) (bs :: [k]) :: Constraint where
  Elem a (a ': bs) = (() :: Constraint)
  Elem a (b ': bs) = a `Elem` bs

type family Sum (ns :: [Nat]) :: Nat where
  Sum '[] = 0
  Sum (n ': ns) = n + Sum ns

Kind Indexed Type Families

This is an advanced section, and is not typically necessary to write Haskell.

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.

type family (a :: k) == (b :: k) :: Bool
type instance a == b = EqStar a b
type instance a == b = EqArrow a b
type instance a == b = EqBool a b

type family EqStar (a :: *) (b :: *) where
  EqStar a a = True
  EqStar a b = False

type family EqArrow (a :: k1 -> k2) (b :: k1 -> k2) where
  EqArrow a a = True
  EqArrow a b = False

type family EqBool a b where
  EqBool True  True  = True
  EqBool False False = True
  EqBool a     b     = False

type family EqList a b where
  EqList '[]        '[]        = True
  EqList (h1 ': t1) (h2 ': t2) = (h1 == h2) && (t1 == t2)
  EqList a          b          = False

type family a && b where
  True && True = True
  a    && a    = False
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE PolyKinds #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE ConstraintKinds #-}

import GHC.TypeLits
import Data.Type.Equality

data Label (l :: Symbol) = Get

class Has a l b | a l -> b where
  from :: a -> Label l -> b

data Point2D = Point2 Double Double deriving Show
data Point3D = Point3 Double Double Double deriving Show

instance Has Point2D "x" Double where
  from (Point2 x _) _ = x

instance Has Point2D "y" Double where
  from (Point2 _ y) _ = y

instance Has Point3D "x" Double where
  from (Point3 x _ _) _ = x

instance Has Point3D "y" Double where
  from (Point3 _ y _) _ = y

instance Has Point3D "z" Double where
  from (Point3 _ _ z) _ = z

infixl 6 #

(#) :: a -> (a -> b) -> b
(#) = flip ($)

_x :: Has a "x" b => a -> b
_x pnt = from pnt (Get :: Label "x")

_y :: Has a "y" b => a -> b
_y pnt = from pnt (Get :: Label "y")

_z :: Has a "z" b => a -> b
_z pnt = from pnt (Get :: Label "z")

type Point a r = (Has a "x" r, Has a "y" r)

distance :: (Point a r, Point b r, Floating r) => a -> b -> r
distance p1 p2 = sqrt (d1^2 + d2^2)
    d1 = (p1 # _x) + (p1 # _y)
    d2 = (p2 # _x) + (p2 # _y)

main :: IO ()
main = do
  print $ (Point2 10 20) # _x

  -- Fails with: No instance for (Has Point2D "z" a0)
  -- print $ (Point2 10 20) # _z

  print $ (Point3 10 20 30) # _x
  print $ (Point3 10 20 30) # _z

  print $ distance (Point2 1 3) (Point2 2 7)
  print $ distance (Point2 1 3) (Point3 2 7 4)
  print $ distance (Point3 1 3 5) (Point3 2 7 3)

Since record is fundamentally no different from the tuple we can also do the same kind of construction over record field names.

{-# LANGUAGE DataKinds #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE ExistentialQuantification #-}
{-# LANGUAGE ConstraintKinds #-}

import GHC.TypeLits

newtype Field (n :: Symbol) v = Field { unField :: v }
  deriving Show

data Person1 = Person1
  { _age      :: Field "age" Int
  , _name     :: Field "name" String

data Person2 = Person2
  { _age'  :: Field "age" Int
  , _name' :: Field "name" String
  , _lib'  :: Field "lib" String

deriving instance Show Person1
deriving instance Show Person2

data Label (l :: Symbol) = Get

class Has a l b | a l -> b where
  from :: a -> Label l -> b

instance Has Person1 "age" Int where
  from (Person1 a _) _ = unField a

instance Has Person1 "name" String where
  from (Person1 _ a) _ = unField a

instance Has Person2 "age" Int where
  from (Person2 a _ _) _ = unField a

instance Has Person2 "name" String where
  from (Person2 _ a _) _ = unField a

age :: Has a "age" b => a -> b
age pnt = from pnt (Get :: Label "age")

name :: Has a "name" b => a -> b
name pnt = from pnt (Get :: Label "name")

-- Parameterized constraint kind for "Simon-ness" of a record.
type Simon a = (Has a "name" String, Has a "age" Int)

spj :: Person1
spj = Person1 (Field 56) (Field "Simon Peyton Jones")

smarlow :: Person2
smarlow = Person2 (Field 38) (Field "Simon Marlow") (Field "rts")

catNames :: (Simon a, Simon b) => a -> b -> String
catNames a b = name a ++ name b

addAges :: (Simon a, Simon b) => a -> b -> Int
addAges a b = age a + age b

names :: String
names = name smarlow ++ "," ++ name spj
-- "Simon Marlow,Simon Peyton Jones"

ages :: Int
ages = age spj + age smarlow
-- 94

Notably this approach is mostly just all boilerplate class instantiation which could be abstracted away using TemplateHaskell or a Generic deriving.


This is an advanced section, and is not typically necessary to write Haskell.

A heterogeneous list is a cons list whose type statically encodes the ordered types of its values.

{-# LANGUAGE DataKinds #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE KindSignatures #-}

infixr 5 :::

data HList (ts :: [ * ]) where
  Nil :: HList '[]
  (:::) :: t -> HList ts -> HList (t ': ts)

-- Take the head of a non-empty list with the first value as Bool type.
headBool :: HList (Bool ': xs) -> Bool
headBool hlist = case hlist of
  (a ::: _) -> a

hlength :: HList x -> Int
hlength Nil = 0
hlength (_ ::: b) = 1 + (hlength b)

tuple :: (Bool, (String, (Double, ())))
tuple = (True, ("foo", (3.14, ())))

hlist :: HList '[Bool, String , Double , ()]
hlist = True ::: "foo" ::: 3.14 ::: () ::: Nil

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.

{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE PolyKinds #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE UndecidableInstances #-}

import GHC.Exts (Constraint)

infixr 5 :::

data HList (ts :: [ * ]) where
  Nil :: HList '[]
  (:::) :: t -> HList ts -> HList (t ': ts)

type family Map (f :: a -> b) (xs :: [a]) :: [b]
type instance Map f '[] = '[]
type instance Map f (x ': xs) = f x ': Map f xs

type family Constraints (cs :: [Constraint]) :: Constraint
type instance Constraints '[] = ()
type instance Constraints (c ': cs) = (c, Constraints cs)

type AllHave (c :: k -> Constraint) (xs :: [k]) = Constraints (Map c xs)

showHList :: AllHave Show xs => HList xs -> [String]
showHList Nil = []
showHList (x ::: xs) = (show x) : showHList xs

instance AllHave Show xs => Show (HList xs) where
  show = show . showHList

example1 :: HList '[Bool, String , Double , ()]
example1 = True ::: "foo" ::: 3.14 ::: () ::: Nil
-- ["True","\"foo\"","3.14","()"]

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.

{-# LANGUAGE DataKinds #-}
{-# LANGUAGE PolyKinds #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE UndecidableInstances #-}

import GHC.TypeLits
import Data.Proxy
import Data.Type.Equality

type family If (p :: Bool) (a :: k) (b :: k) :: k where
  If True a b = a
  If False a b = b

type family Lookup (k :: a) (ls :: [(a, b)]) :: Maybe b where
  Lookup k '[] = 'Nothing
  Lookup k ('(a, b) ': xs) = If (a == k) ('Just b) (Lookup k xs)

type M = [
    '("a", 1)
  , '("b", 2)
  , '("c", 3)
  , '("d", 4)

type K = "a"
type (!!) m (k :: Symbol) a = (Lookup k m) ~ Just a

value :: Integer
value = natVal ( Proxy :: (M !! "a") a => Proxy a )

If we ask GHC to expand out the type signature we can view the explicit implementation of the type-level map lookup function.

  :: If
       (GHC.TypeLits.EqSymbol "a" k)
       ('Just 1)
          (GHC.TypeLits.EqSymbol "b" k)
          ('Just 2)
             (GHC.TypeLits.EqSymbol "c" k)
             ('Just 3)
             (If (GHC.TypeLits.EqSymbol "d" k) ('Just 4) 'Nothing)))
     ~ 'Just v =>
     Proxy k -> Proxy v

Advanced Proofs

This is an advanced section, and is not typically necessary to write Haskell.

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:

reverseNaive :: forall n a. Vec a n -> Vec a n
reverseNaive xs = go Nil xs -- Error: n + 0 != n
    go :: Vec a m -> Vec a n -> Vec a (n :+ m)
    go acc Nil = acc
    go acc (Cons x xs) = go (Cons x acc) xs -- Error: n + succ m != succ (n + m)

Running this we find that GHC is unhappy about two lines in the code:

Couldn't match type ‘n’ with ‘n :+ 'Z’
    Expected type: Vec a n
      Actual type: Vec a (n :+ 'Z)

Could not deduce ((n1 :+ 'S m) ~ 'S (n1 :+ m))
    Expected type: Vec a1 (k :+ m)
      Actual type: Vec a1 (n1 :+ 'S m)

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.

{-# LANGUAGE PolyKinds #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE ExplicitForAll #-}

import Data.Type.Equality

data Nat = Z | S Nat

data SNat n where
  Zero :: SNat Z
  Succ :: SNat n -> SNat (S n)

data Vec :: * -> Nat -> * where
  Nil :: Vec a Z
  Cons :: a -> Vec a n -> Vec a (S n)

instance Show a => Show (Vec a n) where
  show Nil         = "Nil"
  show (Cons x xs) = "Cons " ++ show x ++ " (" ++ show xs ++ ")"

type family (m :: Nat) :+ (n :: Nat) :: Nat where
  Z :+ n = n
  S m :+ n = S (m :+ n)

-- (a ~ b) implies (f a ~ f b)
cong :: a :~: b -> f a :~: f b
cong Refl = Refl

-- (a ~ b) implies (f a) implies (f b)
subst :: a :~: b -> f a -> f b
subst Refl = id

plus_zero :: forall n. SNat n -> (n :+ Z) :~: n
plus_zero Zero = Refl
plus_zero (Succ n) = cong (plus_zero n)

plus_suc :: forall n m. SNat n -> SNat m -> (n :+ (S m)) :~: (S (n :+ m))
plus_suc Zero m = Refl
plus_suc (Succ n) m = cong (plus_suc n m)

size :: Vec a n -> SNat n
size Nil         = Zero
size (Cons _ xs) = Succ $ size xs

reverse :: forall n a. Vec a n -> Vec a n
reverse xs = subst (plus_zero (size xs)) $ go Nil xs
    go :: Vec a m -> Vec a k -> Vec a (k :+ m)
    go acc Nil = acc
    go acc (Cons x xs) = subst (plus_suc (size xs) (size acc)) $ go (Cons x acc) xs

append :: Vec a n -> Vec a m -> Vec a (n :+ m)
append (Cons x xs) ys = Cons x (append xs ys)
append Nil         ys = ys

vec :: Vec Int (S (S (S Z)))
vec = 1 `Cons` (2 `Cons` (3 `Cons` Nil))

test :: Vec Int (S (S (S Z)))
test = Main.reverse vec

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.

{-# LANGUAGE DataKinds #-}
{-# LANGUAGE ExplicitForAll #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE UndecidableInstances #-}

import Prelude hiding (Eq)
import GHC.TypeLits
import Data.Type.Equality

type Z = 0

type family S (n :: Nat) :: Nat where
  S n = n + 1

-- Yes!
eq_zero :: Z :~: Z
eq_zero = Refl

-- Yes!
zero_plus_one :: (Z + 1) :~: (1 + Z)
zero_plus_one = Refl

-- Yes!
plus_zero :: forall n. (n + Z) :~: n
plus_zero = Refl

-- Yes!
plus_one :: forall n. (n + S Z) :~: S n
plus_one = Refl

-- No.
plus_suc :: forall n m. (n + (S m)) :~: (S (n + m))
plus_suc = Refl

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.

module Vector where

infixr 10 __

data: Set where
  zero : ℕ
  suc  :{-# BUILTIN NATURAL ℕ    #-}
{-# BUILTIN ZERO    zero #-}
{-# BUILTIN SUC     suc  #-}

infixl 6 _+_

_+_ :0 + n = n
suc m + n = suc (m + n)

data Vec (A : Set) : Set where
  []  : Vec A 0
  __ :  {n}  A  Vec A n  Vec A (suc n)

_++_ :  {A n m}  Vec A n  Vec A m  Vec A (n + m)
[] ++ ys = ys
(x  xs) ++ ys = x  (xs ++ ys)

infix 4 _≡_

data _≡_ {A : Set} (x : A) : A  Set where
  refl : x ≡ x

subst : {A : Set}  (P : A  Set)  {x y}  x ≡ y  P x  P y
subst P refl p = p

cong : {A B : Set} (f : A  B)  {x y : A}  x ≡ y  f x ≡ f y
cong f refl = refl

vec :  {A} (k : ℕ)  Set
vec {A} k = Vec A k

plus_zero : {n : ℕ}  n + 0 ≡ n 
plus_zero {zero}  = refl
plus_zero {suc n} = cong suc plus_zero

plus_suc : {n : ℕ}  n + (suc 0) ≡ suc n 
plus_suc {zero}  = refl
plus_suc {suc n} = cong suc (plus_suc {n})

reverse :  {A n}  Vec A n  Vec A n
reverse []       = []
reverse {A} {suc n} (x  xs) = subst vec (plus_suc {n}) (reverse xs ++ (x   []))

Liquid Haskell

This is an advanced section, knowledge of LiquidHaskell is not typically necessary to write 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 functions always yields an Int that is even positive number.

To Install LiquidHaskell in Ubuntu add the following line to your /etc/sources.list:

deb http://ppa.launchpad.net/hvr/z3/ubuntu trusty main

And then install the external SMT solver.

$ sudo apt-key adv --keyserver keyserver.ubuntu.com --recv-keys F6F88286
$ sudo apt-get install z3

Then clone the repo and build it using stack.

$ git clone --recursive git@github.com:ucsd-progsys/liquidhaskell.git
$ cd liquidhaskell
$ stack install

Ensure that $HOME/.local/bin is on your $PATH.

import Prelude hiding (mod, gcd)

{-@ mod :: a:Nat -> b:{v:Nat| 0 < v} -> {v:Nat | v < b} @-}
mod :: Int -> Int -> Int
mod a b
  | a < b = a
  | otherwise = mod (a - b) b

{-@ gcd :: a:Nat -> b:{v:Nat | v < a} -> Int @-}
gcd :: Int -> Int -> Int
gcd a 0 = a
gcd a b = gcd b (a `mod` b)

The module can be run through the solver using the liquid command line tool.

$ liquid example.hs
Done solving.

**** DONE:  solve **************************************************************

**** DONE:  annotate ***********************************************************

**** RESULT: SAFE **************************************************************

For more extensive documentation and further use cases see the official 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:

These are achieved through several tools and techniques outlined in the next few sections:


The Typeable class be used to create runtime type information for arbitrary types.

typeOf :: Typeable a => a -> TypeRep
{-# LANGUAGE DeriveDataTypeable #-}

import Data.Typeable

data Animal = Cat | Dog deriving Typeable
data Zoo a = Zoo [a] deriving Typeable

equal :: (Typeable a, Typeable b) => a -> b -> Bool
equal a b = typeOf a == typeOf b

example1 :: TypeRep
example1 = typeOf Cat
-- Animal

example2 :: TypeRep
example2 = typeOf (Zoo [Cat, Dog])
-- Zoo Animal

example3 :: TypeRep
example3 = typeOf ((1, 6.636e-34, "foo") :: (Int, Double, String))
-- (Int,Double,[Char])

example4 :: Bool
example4 = equal False ()
-- False

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.

cast :: (Typeable a, Typeable b) => a -> Maybe b
cast x
  | typeOf x == typeOf ret = Just ret
  | otherwise = Nothing
    ret = unsafeCast x

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


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.

toDyn :: Typeable a => a -> Dynamic
fromDyn :: Typeable a => Dynamic -> a -> a
fromDynamic :: Typeable a => Dynamic -> Maybe a
cast :: (Typeable a, Typeable b) => a -> Maybe b
import Data.Dynamic
import Data.Maybe

dynamicBox :: Dynamic
dynamicBox = toDyn (6.62 :: Double)

example1 :: Maybe Int
example1 = fromDynamic dynamicBox
-- Nothing

example2 :: Maybe Double
example2 = fromDynamic dynamicBox
-- Just 6.62

example3 :: Int
example3 = fromDyn dynamicBox 0
-- 0

example4 :: Double
example4 = fromDyn dynamicBox 0.0
-- 6.62

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.


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.

class Typeable a => Data a where
  gfoldl  :: (forall d b. Data d => c (d -> b) -> d -> c b)
          -> (forall g. g -> c g)
          -> a
          -> c a

  gunfold :: (forall b r. Data b => c (b -> r) -> c r)
          -> (forall r. r -> c r)
          -> Constr
          -> c a

  toConstr :: a -> Constr
  dataTypeOf :: a -> DataType
  gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> a -> r

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:

data Animal = Cat | Dog deriving Typeable
instance Data Animal where
  gfoldl k z Cat = z Cat
  gfoldl k z Dog = z Dog

  gunfold k z c
    = case constrIndex c of
        1 -> z Cat
        2 -> z Dog

  toConstr Cat = cCat
  toConstr Dog = cDog

  dataTypeOf _ = tAnimal

tAnimal :: DataType
tAnimal = mkDataType "Main.Animal" [cCat, cDog]

cCat :: Constr
cCat = mkConstr tAnimal "Cat" [] Prefix

cDog :: Constr
cDog = mkConstr tAnimal "Dog" [] Prefix

For a type with non-empty containers we get something a little more interesting. Consider the list type:

instance Data a => Data [a] where
  gfoldl _ z []     = z []
  gfoldl k z (x:xs) = z (:) `k` x `k` xs

  toConstr []    = nilConstr
  toConstr (_:_) = consConstr

  gunfold k z c
    = case constrIndex c of
        1 -> z []
        2 -> k (k (z (:)))

  dataTypeOf _ = listDataType

nilConstr :: Constr
nilConstr = mkConstr listDataType "[]" [] Prefix

consConstr :: Constr
consConstr = mkConstr listDataType "(:)" [] Infix

listDataType :: DataType
listDataType = mkDataType "Prelude.[]" [nilConstr,consConstr]

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:

instance (Data a, Data b) => Data (a,b) where
  gfoldl k z (a,b) = z (,) `k` a `k` b

  toConstr (_,_) = tuple2Constr

  gunfold k z c
    = case constrIndex c of
      1 -> k (k (z (,)))

  dataTypeOf _  = tuple2DataType

tuple2Constr :: Constr
tuple2Constr = mkConstr tuple2DataType "(,)" [] Infix

tuple2DataType :: DataType
tuple2DataType = mkDataType "Prelude.(,)" [tuple2Constr]

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)
    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.

numHoles :: Data a => a -> Int
numHoles = gmapQl (+) 0 (const 1)

example1 :: Int
example1 = numHoles (1,2,3,4,5,6,7)
-- 7

example2 :: Int
example2 = numHoles (Just 3)
-- 1


Using the interface provided by the Data we can retrieve the information we need to, at runtime, inspect the types of expressions and rewrite them, collect terms, and find subterms matching specific predicates.

everywhere :: (forall a. Data a => a -> a) -> forall a. Data a => a -> a
everywhereM :: Monad m => GenericM m -> GenericM m
somewhere :: MonadPlus m => GenericM m -> GenericM m
listify :: Typeable r => (r -> Bool) -> GenericQ [r]
everything :: (r -> r -> r) -> GenericQ r -> GenericQ r

For example consider we have some custom collection of datatypes for which we want to write generic transformations that transform numerical subexpressions according to set of rewrite rules. We can use syb to write the transformation rules quite succinctly.

{-# LANGUAGE DeriveDataTypeable #-}

import Data.Data
import Data.Typeable
import Data.Generics.Schemes
import Data.Generics.Aliases (mkT)

data MyTuple a = MyTuple a Float
  deriving (Data, Typeable, Show)

exampleT :: Data a => MyTuple a -> MyTuple a
exampleT = everywhere (mkT go1) . everywhere (mkT go2)
    go1 :: Int -> Int
    go1 x = succ x

    go2 :: Float -> Float
    go2 x = succ x

findFloat :: Data x => x -> Maybe Float
findFloat = gfindtype

main :: IO ()
main = do
  let term = MyTuple (MyTuple (1 :: Int) 2.0) 3.0
  print (exampleT term)
  print (gsize term)
  print (findFloat term)
  print (listify ((>0) :: (Int -> Bool)) term)


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.

class Generic a where
  type Rep a
  from :: a -> Rep a
  to :: Rep a -> a

class Datatype d where
  datatypeName :: t d f a -> String
  moduleName :: t d f a -> String

class Constructor c where
  conName :: t c f a -> String

GHC.Generics defines a set of named types for modeling the various structural properties of types in available in Haskell.

-- | Sums: encode choice between constructors
infixr 5 :+:
data (:+:) f g p = L1 (f p) | R1 (g p)

-- | Products: encode multiple arguments to constructors
infixr 6 :*:
data (:*:) f g p = f p :*: g p

-- | Tag for M1: datatype
data D
-- | Tag for M1: constructor
data C

-- | Constants, additional parameters and recursion of kind *
newtype K1 i c p = K1 { unK1 :: c }

-- | Meta-information (constructor names, etc.)
newtype M1 i c f p = M1 { unM1 :: f p }

-- | Type synonym for encoding meta-information for datatypes
type D1 = M1 D

-- | Type synonym for encoding meta-information for constructors
type C1 = M1 C

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:

{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE TypeFamilies #-}

import GHC.Generics

data Animal
  = Dog
  | Cat

instance Generic Animal where
  type Rep Animal = D1 T_Animal ((C1 C_Dog U1) :+: (C1 C_Cat U1))

  from Dog = M1 (L1 (M1 U1))
  from Cat = M1 (R1 (M1 U1))

  to (M1 (L1 (M1 U1))) = Dog
  to (M1 (R1 (M1 U1))) = Cat

data T_Animal
data C_Dog
data C_Cat

instance Datatype T_Animal where
  datatypeName _ = "Animal"
  moduleName _ = "Main"

instance Constructor C_Dog where
  conName _ = "Dog"

instance Constructor C_Cat where
  conName _ = "Cat"

Use kind! in GHCi we can look at the type family Rep associated with a Generic instance.

λ: :kind! Rep Animal
Rep Animal :: * -> *
= M1 D T_Animal (M1 C C_Dog U1 :+: M1 C C_Cat U1)

λ: :kind! Rep ()
Rep () :: * -> *
= M1 D GHC.Generics.D1() (M1 C GHC.Generics.C1_0() U1)

λ: :kind! Rep [()]
Rep [()] :: * -> *
= M1
    (M1 C GHC.Generics.C1_0[] U1
     :+: M1
           (M1 S NoSelector (K1 R ()) :*: M1 S NoSelector (K1 R [()])))

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:

class GEq' f where
  geq' :: f a -> f a -> Bool

instance GEq' U1 where
  geq' _ _ = True

instance (GEq c) => GEq' (K1 i c) where
  geq' (K1 a) (K1 b) = geq a b

instance (GEq' a) => GEq' (M1 i c a) where
  geq' (M1 a) (M1 b) = geq' a b

-- Equality for sums.
instance (GEq' a, GEq' b) => GEq' (a :+: b) where
  geq' (L1 a) (L1 b) = geq' a b
  geq' (R1 a) (R1 b) = geq' a b
  geq' _      _      = False

-- Equality for products.
instance (GEq' a, GEq' b) => GEq' (a :*: b) where
  geq' (a1 :*: b1) (a2 :*: b2) = geq' a1 a2 && geq' b1 b2

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.

{-# LANGUAGE DefaultSignatures #-}

class GEq a where
  geq :: a -> a -> Bool

  default geq :: (Generic a, GEq' (Rep a)) => a -> a -> Bool
  geq x y = geq' (from x) (from y)

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:

{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE DefaultSignatures #-}

import GHC.Generics

-- Auxiliary class
class GEq' f where
  geq' :: f a -> f a -> Bool

instance GEq' U1 where
  geq' _ _ = True

instance (GEq c) => GEq' (K1 i c) where
  geq' (K1 a) (K1 b) = geq a b

instance (GEq' a) => GEq' (M1 i c a) where
  geq' (M1 a) (M1 b) = geq' a b

instance (GEq' a, GEq' b) => GEq' (a :+: b) where
  geq' (L1 a) (L1 b) = geq' a b
  geq' (R1 a) (R1 b) = geq' a b
  geq' _      _      = False

instance (GEq' a, GEq' b) => GEq' (a :*: b) where
  geq' (a1 :*: b1) (a2 :*: b2) = geq' a1 a2 && geq' b1 b2

class GEq a where
  geq :: a -> a -> Bool
  default geq :: (Generic a, GEq' (Rep a)) => a -> a -> Bool
  geq x y = geq' (from x) (from y)

-- Base equalities
instance GEq Char where geq = (==)
instance GEq Int where geq = (==)
instance GEq Float where geq = (==)

-- Equalities derived from structure of (:+:) and (:*:)
instance GEq a => GEq (Maybe a)
instance (GEq a, GEq b) => GEq (a,b)

main :: IO ()
main = do
  print $ geq 2 (3 :: Int)
  print $ geq 'a' 'b'
  print $ geq (Just 'a') (Just 'a')
  print $ geq ('a','b') ('a', 'b')


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.

{-# LANGUAGE DeriveGeneric #-}

import GHC.Generics (Generic)
import Data.Hashable

data Color = Red | Green | Blue deriving (Generic, Show)

instance Hashable Color where

example1 :: Int
example1 = hash Red
-- 839657738087498284

example2 :: Int
example2 = hashWithSalt 0xDEADBEEF Red
-- 62679985974121021

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)]])

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.

{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE OverloadedStrings #-}

import Data.Aeson
import GHC.Generics

data Point = Point { _x :: Double, _y :: Double }
   deriving (Show, Generic)

instance FromJSON Point
instance ToJSON Point

example1 :: Maybe Point
example1 = decode "{\"x\":3.0,\"y\":-1.0}"

example2 = encode $ Point 123.4 20

See: A Generic Deriving Mechanism for Haskell

Higher Kinded Generics

Using the same interface GHC.Generics provides a separate typeclass for higher-kinded generics.

class Generic1 f where
  type Rep1 f :: * -> *
  from1  :: f a -> (Rep1 f) a
  to1    :: (Rep1 f) a -> f a

So for instance Maybe has Rep1 of the form:

type instance Rep1 Maybe
  = D1
      (C1 C1_0Maybe U1
       :+: C1 C1_1Maybe (S1 NoSelector Par1))




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.

plate :: from -> Type from to
(|*)  :: Type (to -> from) to -> to -> Type from to
(|-)  :: Type (item -> from) to -> item -> Type from to

descend   :: Uniplate on => (on -> on) -> on -> on
transform :: Uniplate on => (on -> on) -> on -> on
rewrite   :: Uniplate on => (on -> Maybe on) -> on -> on

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.

import Data.Generics.Uniplate.Direct

data Expr a
  = Fls
  | Tru
  | Var a
  | Not (Expr a)
  | And (Expr a) (Expr a)
  | Or  (Expr a) (Expr a)
  deriving (Show, Eq)

instance Uniplate (Expr a) where
  uniplate (Not f)     = plate Not |* f
  uniplate (And f1 f2) = plate And |* f1 |* f2
  uniplate (Or f1 f2)  = plate Or |* f1 |* f2
  uniplate x           = plate x

simplify :: Expr a -> Expr a
simplify = transform simp
   simp (Not (Not f)) = f
   simp (Not Fls) = Tru
   simp (Not Tru) = Fls
   simp x = x

reduce :: Show a => Expr a -> Expr a
reduce = rewrite cnf
    -- double negation
    cnf (Not (Not p)) = Just p

    -- de Morgan
    cnf (Not (p `Or` q))  = Just $ (Not p) `And` (Not q)
    cnf (Not (p `And` q)) = Just $ (Not p) `Or` (Not q)

    -- distribute conjunctions
    cnf (p `Or` (q `And` r)) = Just $ (p `Or` q) `And` (p `Or` r)
    cnf ((p `And` q) `Or` r) = Just $ (p `Or` q) `And` (p `Or` r)
    cnf _ = Nothing

example1 :: Expr String
example1 = simplify (Not (Not (Not (Not (Var "a")))))
-- Var "a"

example2 :: [String]
example2 = [a | Var a <- universe ex]
    ex = Or (And (Var "a") (Var "b")) (Not (And (Var "c") (Var "d")))
-- ["a","b","c","d"]

example3 :: Expr String
example3 = reduce $ ((a `And` b) `Or` (c `And` d)) `Or` e
    a = Var "a"
    b = Var "b"
    c = Var "c"
    d = Var "d"
    e = Var "e"

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.

import Data.Data
import Data.Typeable
import Data.Generics.Uniplate.Data

data Expr a
  = Fls
  | Tru
  | Lit a
  | Not (Expr a)
  | And (Expr a) (Expr a)
  | Or (Expr a) (Expr a)
  deriving (Data, Typeable, Show, Eq)


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.

descendBi   :: Biplate from to => (to -> to) -> from -> from
transformBi :: Biplate from to => (to -> to) -> from -> from
rewriteBi   :: Biplate from to => (to -> Maybe to) -> from -> from

descendBiM   :: (Monad m, Biplate from to) => (to -> m to) -> from -> m from
transformBiM :: (Monad m, Biplate from to) => (to -> m to) -> from -> m from
rewriteBiM   :: (Monad m, Biplate from to) => (to -> m (Maybe to)) -> from -> m from
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FlexibleContexts #-}

import Data.Generics.Uniplate.Direct

type Name = String

data Expr
  = Var Name
  | Lam Name Expr
  | App Expr Expr
  deriving Show

data Stmt
  = Decl [Stmt]
  | Let Name Expr
  deriving Show

instance Uniplate Expr where
  uniplate (Var x  ) = plate Var |- x
  uniplate (App x y) = plate App |* x |* y
  uniplate (Lam x y) = plate Lam |- x |* y

instance Biplate Expr Expr where
  biplate = plateSelf

instance Uniplate Stmt where
  uniplate (Decl x  ) = plate Decl ||* x
  uniplate (Let x y) = plate Let |-  x |- y

instance Biplate Stmt Stmt where
  biplate = plateSelf

instance Biplate Stmt Expr where
  biplate (Decl x) = plate Decl ||+ x
  biplate (Let x y) = plate Let |- x |* y

rename :: Name -> Name -> Expr -> Expr
rename from to = rewrite f
    f (Var a) | a == from = Just (Var to)
    f (Lam a b) | a == from = Just (Lam to b)
    f _ = Nothing

s, k, sk :: Expr
s = Lam "x" (Lam "y" (Lam "z" (App (App (Var "x") (Var "z")) (App (Var "y") (Var "z")))))
k = Lam "x" (Lam "y" (Var "x"))
sk = App s k

m :: Stmt
m = descendBi f $ Decl [ (Let "s" s) , Let "k" k , Let "sk" sk ]
    f = rename "x" "a"
      . rename "y" "b"
      . rename "z" "c"


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.

42 :: Num a => a
fromInteger (42 :: Integer)

2.71 :: Fractional a => a
fromRational (2.71 :: Rational)

The numeric typeclass hierarchy is defined as such:

class Num a
class (Num a, Ord a) => Real a
class Num a => Fractional a
class (Real a, Enum a) => Integral a
class (Real a, Fractional a) => RealFrac a
class Fractional a => Floating a
class (RealFrac a, Floating a) => RealFloat a

Conversions between concrete numeric types ( from : left column, to : top row ) is accomplished with several generic functions.

Double Float Int Word Integer Rational
Double id fromRational truncate truncate truncate toRational
Float fromRational id truncate truncate truncate toRational
Int fromIntegral fromIntegral id fromIntegral fromIntegral fromIntegral
Word fromIntegral fromIntegral fromIntegral id fromIntegral fromIntegral
Integer fromIntegral fromIntegral fromIntegral fromIntegral id fromIntegral
Rational fromRational fromRational truncate truncate truncate id


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.

See: GHC, primops and exorcising GMP


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 it's numerical typeclass components (Num, Fractional, Floating) from the type of this parameter.

-- 1 + 2i
let complex = 1 :+ 2
data Complex a = a :+ a
mkPolar :: RealFloat a => a -> a -> Complex a

The Num instance for Complex is only defined if parameter of Complex is an instance of RealFloat.

λ: 0 :+ 1
0 :+ 1 :: Complex Integer

λ: (0 :+ 1) + (1 :+ 0)
1.0 :+ 1.0 :: Complex Integer

λ: exp (0 :+ 2 * pi)
1.0 :+ (-2.4492935982947064e-16) :: Complex Double

λ: mkPolar 1 (2*pi)
1.0 :+ (-2.4492935982947064e-16) :: Complex Double

λ: let f x n = (cos x :+ sin x)^n
λ: let g x n = cos (n*x) :+ sin (n*x)


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.

scientific :: Integer -> Int -> Scientific
fromFloatDigits :: RealFloat a => a -> Scientific
import Data.Scientific

c, h, g, a, k :: Scientific
c = scientific 299792458 (0)   -- Speed of light
h = scientific 662606957 (-42) -- Planck's constant
g = scientific 667384    (-16) -- Gravitational constant
a = scientific 729735257 (-11) -- Fine structure constant
k = scientific 268545200 (-9)  -- Khinchin Constant

tau :: Scientific
tau = fromFloatDigits (2*pi)

maxDouble64 :: Double
maxDouble64 = read "1.7976931348623159e308"
-- Infinity

maxScientific :: Scientific
maxScientific = read "1.7976931348623159e308"
-- 1.7976931348623159e308


import Data.Vector
import Statistics.Sample

import Statistics.Distribution.Normal
import Statistics.Distribution.Poisson
import qualified Statistics.Distribution as S

s1 :: Vector Double
s1 = fromList [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

s2 :: PoissonDistribution
s2 = poisson 2.5

s3 :: NormalDistribution
s3 = normalDistr mean stdDev
    mean   = 1
    stdDev = 1

descriptive = do
  print $ range s1
  -- 9.0
  print $ mean s1
  -- 5.5
  print $ stdDev s1
  -- 3.0276503540974917
  print $ variance s1
  -- 8.25
  print $ harmonicMean s1
  -- 3.414171521474055
  print $ geometricMean s1
  -- 4.5287286881167645

discrete = do
  print $ S.cumulative s2 0
  -- 8.208499862389884e-2
  print $ S.mean s2
  -- 2.5
  print $ S.variance s2
  -- 2.5
  print $ S.stdDev s2
  -- 1.5811388300841898

continuous = do
  print $ S.cumulative s3 0
  -- 0.15865525393145707
  print $ S.quantile s3 0.5
  -- 1.0
  print $ S.density s3 0
  -- 0.24197072451914334
  print $ S.mean s3
  -- 1.0
  print $ S.variance s3
  -- 1.0
  print $ S.stdDev s3
  -- 1.0

Constructive Reals

Instead of modeling the real numbers on finite precision floating point numbers we alternatively work with Num which internally manipulate 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 ).

exp(x)    = 1 + x + 1/2*x^2 + 1/6*x^3 + 1/24*x^4 + 1/120*x^5 ...
sqrt(1+x) = 1 + 1/2*x - 1/8*x^2 + 1/16*x^3 - 5/128*x^4 + 7/256*x^5 ...
atan(x)   = x - 1/3*x^3 + 1/5*x^5 - 1/7*x^7 + 1/9*x^9 - 1/11*x^11 ...
pi        = 16 * atan (1/5) - 4 * atan (1/239)
import Data.Number.CReal

-- algebraic
phi :: CReal
phi = (1 + sqrt 5) / 2

-- transcendental
ramanujan :: CReal
ramanujan = exp (pi * sqrt 163)

main :: IO ()
main = do
  putStrLn $ showCReal 30 pi
  -- 3.141592653589793238462643383279
  putStrLn $ showCReal 30 phi
  -- 1.618033988749894848204586834366
  putStrLn $ showCReal 15 ramanujan
  -- 262537412640768743.99999999999925

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:

1 -2 3  -- (A v ¬B v C)
2 4 5   -- (B v D v E)
4 6     -- (D v F)
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.


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 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

+ B U R R I T O
= B A N D A I D

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 $ allDifferent 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,

λ: allSat puzzle
Solution #1:
  b = 4 :: Integer
  u = 1 :: Integer
  r = 5 :: Integer
  i = 9 :: Integer
  t = 7 :: Integer
  o = 0 :: Integer
  m = 8 :: Integer
  n = 3 :: Integer
  a = 2 :: Integer
  d = 6 :: Integer
This is the only solution.




See: z3

Data Structures


Functionality Function Time Complexity
Initialization empty O(1)
Size size O(1)
Lookup lookup O(log(n))
Insertion insert O(log(n))
Traversal traverse O(n)

A map is an associative array mapping any instance of Ord keys to values of any type.

import qualified Data.Map as Map

kv :: Map.Map Integer String
kv = Map.fromList [(1, "a"), (2, "b")]

lkup :: Integer -> String -> String
lkup key def =
  case Map.lookup key kv of
    Just val -> val
    Nothing  -> def


Functionality Function Time Complexity
Initialization empty O(1)
Size size O(1)
Lookup lookup O(log(n))
Insertion insert O(log(n))
Traversal traverse O(n)
import Data.Tree


  / \
 B   C
    / \
   D   E


tree :: Tree String
tree = Node "A" [Node "B" [], Node "C" [Node "D" [], Node "E" []]]

postorder :: Tree a -> [a]
postorder (Node a ts) = elts ++ [a]
  where elts = concat (map postorder ts)

preorder :: Tree a -> [a]
preorder (Node a ts) = a : elts
  where elts = concat (map preorder ts)

ex1 = drawTree tree
ex2 = drawForest (subForest tree)
ex3 = flatten tree
ex4 = levels tree
ex5 = preorder tree
ex6 = postorder tree


Functionality Function Time Complexity
Initialization empty O(1)
Size size O(1)
Insertion insert O(log(n))
Deletion delete O(log(n))
Traversal traverse O(n)
Membership Test member O(log(n))

Sets are an unordered data structures allow 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.

import qualified Data.Set as Set

set :: Set.Set Integer
set = Set.fromList [1..1000]

memtest :: Integer -> Bool
memtest elt = Set.member elt set


Functionality Function Time Complexity
Initialization empty O(1)
Size length O(1)
Indexing (!) O(1)
Append append O(n)
Traversal traverse O(n)

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.

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.

fromList :: [a] -> Vector a
toList :: Vector a -> [a]
(!) :: Vector a -> Int -> a
map :: (a -> b) -> Vector a -> Vector b
foldl :: (a -> b -> a) -> a -> Vector b -> a
scanl :: (a -> b -> a) -> a -> Vector b -> Vector a
zipWith :: (a -> b -> c) -> Vector a -> Vector b -> Vector c
iterateN :: Int -> (a -> a) -> a -> Vector a
import Data.Vector.Unboxed as V

norm ::  Vector Double -> Double
norm = sqrt . V.sum . V.map (\x -> x*x)

example1 :: Double
example1 = norm $ V.iterateN 100000000 (+1) 0.0

See: Numerical Haskell: A Vector Tutorial

Mutable Vectors

Functionality Function Time Complexity
Initialization empty O(1)
Size length O(1)
Indexing (!) O(1)
Append append O(n)
Traversal traverse O(n)
Update modify O(1)
Read read O(1)
Write write O(1)
freeze :: MVector (PrimState m) a -> m (Vector a)
thaw :: Vector a -> MVector (PrimState m) a

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.

import GHC.Prim
import Control.Monad
import Control.Monad.ST
import Control.Monad.Primitive

import Data.Vector.Unboxed (freeze)
import Data.Vector.Unboxed.Mutable
import qualified Data.Vector.Unboxed as V

example :: PrimMonad m => m (V.Vector Int)
example = do
  v <- new 10
  forM_ [0..9] $ \i ->
     write v i (2*i)
  freeze v

-- vector computation in IO
vecIO :: IO (V.Vector Int)
vecIO = example

-- vector computation in ST
vecST :: ST s (V.Vector Int)
vecST = example

main :: IO ()
main = do
  vecIO >>= print
  print $ runST vecST

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.


Functionality Function Time Complexity
Initialization empty O(1)
Size size O(1)
Lookup lookup O(log(n))
Insertion insert O(log(n))
Traversal traverse O(n)
fromList :: (Eq k, Hashable k) => [(k, v)] -> HashMap k v
lookup :: (Eq k, Hashable k) => k -> HashMap k v -> Maybe v
insert :: (Eq k, Hashable k) => k -> v -> HashMap k v -> HashMap k v

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.

import qualified Data.HashSet as S
import qualified Data.HashMap.Lazy as M

example1 :: M.HashMap Int Char
example1 = M.fromList $ zip [1..10] ['a'..]

example2 :: S.HashSet Int
example2 = S.fromList [1..10]

See: Announcing Unordered Containers


Functionality Function Time Complexity
Initialization empty O(1)
Size size O(1)
Lookup lookup O(1)
Insertion insert O(1) amortized
Traversal traverse O(n)

Hashtables provides hashtables with efficient lookup within the ST or IO monad.

import Prelude hiding (lookup)

import Control.Monad.ST
import Data.HashTable.ST.Basic

-- Hashtable parameterized by ST "thread"
type HT s = HashTable s String String

set :: ST s (HT s)
set = do
  ht <- new
  insert ht "key" "value1"
  return ht

get :: HT s -> ST s (Maybe String)
get ht = do
  val <- lookup ht "key"
  return val

example :: Maybe String
example = runST (set >>= get)
new :: ST s (HashTable s k v)
insert :: (Eq k, Hashable k) => HashTable s k v -> k -> v -> ST s ()
lookup :: (Eq k, Hashable k) => HashTable s k v -> k -> ST s (Maybe v)


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 need to resolve strongly connected components of the module definition graph.

import Data.Tree
import Data.Graph

data Grph node key = Grph
  { _graph :: Graph
  , _vertices :: Vertex -> (node, key, [key])

fromList :: Ord key => [(node, key, [key])] -> Grph node key
fromList = uncurry Grph . graphFromEdges'

vertexLabels :: Functor f => Grph b t -> (f Vertex) -> f b
vertexLabels g = fmap (vertexLabel g)

vertexLabel :: Grph b t -> Vertex -> b
vertexLabel g = (\(vi, _, _) -> vi) . (_vertices g)

-- Topologically sort graph
topo' :: Grph node key -> [node]
topo' g = vertexLabels g $ topSort (_graph g)

-- Strongly connected components of graph
scc' :: Grph node key -> [[node]]
scc' g = fmap (vertexLabels g . flatten) $ scc (_graph g)

So for example we can construct a simple graph:

ex1 :: [(String, String, [String])]
ex1 = [

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 = [

    ("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

-- 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
  gr = G.insNodes nodes G.empty
  edges = [(0,1,()), (0,2,()), (2,1,()), (2,3,())]
  nodes = zip [0,1 ..] [2,3,4,1]


Functionality Function Time Complexity
Initialization empty O(1)
Size size O(1)
Lookup lookup O(log(n))
Insertion insert O(log(n))
Traversal traverse O(n)
Append (|>) O(1)
Prepend (<|) O(1)

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.

import Data.DList
import Control.Monad
import Control.Monad.Writer

logger :: Writer (DList Int) ()
logger = replicateM_ 100000 $ tell (singleton 0)


The sequence data structure behaves structurally similar to list but is optimized for append/prepend operations and traversal.

import Data.Sequence

a :: Seq Int
a = fromList [1,2,3]

a0 :: Seq Int
a0 = a |> 4
-- [1,2,3,4]

a1 :: Seq Int
a1 = 0 <| a
-- [0,1,2,3]



See: fingertree



See: vault


This is an advanced section, knowledge of FFI is not typically necessary to write Haskell.

Pure Functions

Wrapping pure C functions with primitive types is trivial.

/* $(CC) -c simple.c -o simple.o */

int example(int a, int b)
  return a + b;
-- ghc simple.o simple_ffi.hs -o simple_ffi
{-# LANGUAGE ForeignFunctionInterface #-}

import Foreign.C.Types

foreign import ccall safe "example" example
    :: CInt -> CInt -> CInt

main = print (example 42 27)

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.

class Storable a where
  sizeOf :: a -> Int
  alignment :: a -> Int
  peek :: Ptr a -> IO a
  poke :: Ptr a -> a -> IO ()

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) {
            } else {
                swap(&xs[l], &xs[--r]);

        swap(&xs[--l], &xs[beg]);
        sort(xs, beg, l);
        sort(xs, r, end);
-- ghc qsort.o ffi.hs -o ffi
{-# LANGUAGE ForeignFunctionInterface #-}

import Foreign.Ptr
import Foreign.C.Types

import qualified Data.Vector.Storable as V
import qualified Data.Vector.Storable.Mutable as VM

foreign import ccall safe "sort" qsort
    :: Ptr a -> CInt -> CInt -> IO ()

main :: IO ()
main = do
  let vs = V.fromList ([1,3,5,2,1,2,5,9,6] :: [CInt])
  v <- V.thaw vs
  VM.unsafeWith v $ \ptr -> do
    qsort ptr 0 9
  out <- V.freeze v
  print out

The names of foreign functions from a C specific header file can be qualified.

foreign import ccall unsafe "stdlib.h malloc"
    malloc :: CSize -> IO (Ptr a)

Prepending the function name with a & allows us to create a reference to the function pointer itself.

foreign import ccall unsafe "stdlib.h &malloc"
    malloc :: FunPtr a

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".

#include <stdio.h>

void invoke(void (*fn)(int))
  int n = 42;
  printf("Inside of C, now we'll call Haskell.\n");
  printf("Back inside of C again.\n");
{-# LANGUAGE ForeignFunctionInterface #-}

import Foreign
import System.IO
import Foreign.C.Types(CInt(..))

foreign import ccall "wrapper"
  makeFunPtr :: (CInt -> IO ()) -> IO (FunPtr (CInt -> IO ()))

foreign import ccall "pointer.c invoke"
  invoke :: FunPtr (CInt -> IO ()) -> IO ()

fn :: CInt -> IO ()
fn n = do
  putStrLn "Hello from Haskell, here's a number passed between runtimes:"
  print n
  hFlush stdout

main :: IO ()
main = do
  fptr <- makeFunPtr fn
  invoke fptr

Will yield the following output:

Inside of C, now we'll call Haskell
Hello from Haskell, here's a number passed between runtimes:
Back inside of C again.


The definitive reference on concurrency and parallelism in Haskell is Simon Marlow's text. This will section will just gloss over these topics because they are far better explained in this book.

See: Parallel and Concurrent Programming in Haskell

forkIO :: IO () -> IO ThreadId

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 a 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


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 -> Eval a
rseq :: Strategy a
rdeepseq :: NFData a => Strategy a

runEval :: Eval a -> a

rpar a spins off a separate spark that evolutes 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.

-- Evaluates the arguments to f in parallel before application.
par2 f x y = x `rpar` y `rpar` f x y

An argument to rseq forces the evaluation of a spark before evaluation continues.

Action Description
Fizzled The resulting value has already been evaluated by the main thread so the spark need not be converted.
Dud The expression has already been evaluated, the computed value is returned and the spark is not converted.
GC'd The spark is added to the spark pool but the result is not referenced, so it is garbage collected.
Overflowed Insufficient space in the spark pool when spawning.

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.

ghc -threaded -rtsopts program.hs
./program +RTS -s N8 -- use 8 cores

The runtime can be asked to dump information about the spark evaluation by passing the -s flag.

$ ./spark +RTS -N4 -s

                                    Tot time (elapsed)  Avg pause  Max pause
  Gen  0         5 colls,     5 par    0.02s    0.01s     0.0017s    0.0048s
  Gen  1         3 colls,     2 par    0.00s    0.00s     0.0004s    0.0007s

  Parallel GC work balance: 1.83% (serial 0%, perfect 100%)

  TASKS: 6 (1 bound, 5 peak workers (5 total), using -N4)

  SPARKS: 20000 (20000 converted, 0 overflowed, 0 dud, 0 GC'd, 0 fizzled)

The parallel computations themselves are sequenced in the Eval monad, whose evaluation with runEval is itself a pure computation.

example :: (a -> b) -> a -> a -> (b, b)
example f x y = runEval $ do
  a <- rpar $ f x
  b <- rpar $ f y
  rseq a
  rseq b
  return (a, b)


Passing the flag -l generates the eventlog which can be rendered with the threadscope library.

$ ghc -O2 -threaded -rtsopts -eventlog Example.hs
$ ./program +RTS -N4 -l
$ threadscope Example.eventlog

See Simon Marlows's Parallel and Concurrent Programming in Haskell for a detailed guide on interpreting and profiling using Threadscope.



type Strategy a = a -> Eval a
using :: a -> Strategy a -> a

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.

import Control.Parallel.Strategies

parPair' :: Strategy (a, b)
parPair' (a, b) = do
  a' <- rpar a
  b' <- rpar b
  return (a', b')

fib :: Int -> Int
fib 0 = 0
fib 1 = 1
fib n = fib (n-1) + fib (n-2)

serial :: (Int, Int)
serial   = (fib 30, fib 31)

parallel :: (Int, Int)
parallel = runEval . parPair' $ (fib 30, fib 31)

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.

using :: a -> Strategy a -> a
x `using` s = runEval (s x)

parallel ::: (Int, Int)
parallel = (fib 30, fib 31) `using` parPair

For a less contrived example consider a parallel parmap which maps a pure function over a list of a values in parallel.

import Control.Parallel.Strategies

parMap' :: (a -> b) -> [a] -> Eval [b]
parMap' f [] = return []
parMap' f (a:as) = do
  b  <- rpar (f a)
  bs <- parMap' f as
  return (b:bs)

result :: [Int]
result = runEval $ parMap' (+1) [1..1000]

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.

rparWith :: Strategy a -> Strategy a

Using the deepseq library we can now construct a Strategy variant of rseq that evaluates to full normal form.

rdeepseq :: NFData a => Strategy a
rdeepseq x = rseq (force x)

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.

import Control.DeepSeq
import Control.Parallel.Strategies

evalPair :: Strategy a -> Strategy b -> Strategy (a, b)
evalPair sa sb (a, b) = do
  a' <- sa a
  b' <- sb b
  return (a', b')

parPair :: Strategy a -> Strategy b -> Strategy (a, b)
parPair sa sb = evalPair (rparWith sa) (rparWith sb)

fib :: Int -> Int
fib 0 = 0
fib 1 = 1
fib n = fib (n-1) + fib (n-2)

serial :: ([Int], [Int])
serial = (a, b)
    a = fmap fib [0..30]
    b = fmap fib [1..30]

parallel :: ([Int], [Int])
parallel = (a, b) `using` evalPair rdeepseq rdeepseq
    a = fmap fib [0..30]
    b = fmap fib [1..30]

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.

parTraverse :: Traversable t => Strategy a -> Strategy (t a)
dot :: Strategy a -> Strategy a -> Strategy a
($||) :: (a -> b) -> Strategy a -> a -> b
(.||) :: (b -> c) -> Strategy b -> (a -> b) -> a -> c



atomically :: STM a -> IO a
orElse :: STM a -> STM a -> STM a
retry :: STM a

newTVar :: a -> STM (TVar a)
newTVarIO :: a -> IO (TVar a)
writeTVar :: TVar a -> a -> STM ()
readTVar :: TVar a -> STM a

modifyTVar :: TVar a -> (a -> a) -> STM ()
modifyTVar' :: TVar a -> (a -> a) -> STM ()

Software Transactional Memory is a technique for guaranteeing atomicity of values in parallel computations, such that all contexts view the same data when read and writes are guaranteed never to result in inconsistent states.

The strength of Haskell's purity guarantees that transactions within STM are pure and can always be rolled back if a commit fails.

import Control.Monad
import Control.Concurrent
import Control.Concurrent.STM

type Account = TVar Double

transfer :: Account -> Account -> Double -> STM ()
transfer from to amount = do
  available <- readTVar from
  when (amount > available) retry

  modifyTVar from (+ (-amount))
  modifyTVar to   (+ amount)

-- Threads are scheduled non-deterministically.
actions :: Account -> Account -> [IO ThreadId]
actions a b = map forkIO [
     -- transfer to
       atomically (transfer a b 10)
     , atomically (transfer a b (-20))
     , atomically (transfer a b 30)

     -- transfer back
     , atomically (transfer a b (-30))
     , atomically (transfer a b 20)
     , atomically (transfer a b (-10))

main :: IO ()
main = do
  accountA <- atomically $ newTVar 60
  accountB <- atomically $ newTVar 0

  sequence_ (actions accountA accountB)

  balanceA <- atomically $ readTVar accountA
  balanceB <- atomically $ readTVar accountB

  print $ balanceA == 60
  print $ balanceB == 0

See: Beautiful Concurrency

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.

new :: Par (IVar a)
put :: NFData a => IVar a -> a -> Par ()
get :: IVar a -> Par a
fork :: Par () -> Par ()
spawn :: NFData a => Par a -> Par (IVar a)
import Control.Monad
import Control.Monad.Par

f, g :: Int -> Int
f x = x + 10
g x = x * 10

--   f x      g x
--     \     /
--      a + b
--      /   \
-- f (a+b)  g (a+b)
--      \   /
--      (d,e)

example1 :: Int -> (Int, Int)
example1 x = runPar $ do
  [a,b,c,d,e] <- replicateM 5 new
  fork (put a (f x))
  fork (put b (g x))
  a' <- get a
  b' <- get b
  fork (put c (a' + b'))
  c' <- get c
  fork (put d (f c'))
  fork (put e (g c'))
  d' <- get d
  e' <- get e
  return (d', e')

example2 :: [Int]
example2 = runPar $ do
  xs <- parMap (+1) [1..25]
  return xs

-- foldr (+) 0 (map (^2) [1..xs])
example3 :: Int -> Int
example3 n = runPar $ do
  let range = (InclusiveRange 1 n)
  let mapper x = return (x^2)
  let reducer x y = return (x+y)
  parMapReduceRangeThresh 10 range mapper reducer 0


Async is a higher level set of functions that work on top of Control.Concurrent and STM.

async :: IO a -> IO (Async a)
wait :: Async a -> IO a
cancel :: Async a -> IO ()
concurrently :: IO a -> IO b -> IO (a, b)
race :: IO a -> IO b -> IO (Either a b)
import Control.Monad
import Control.Applicative
import Control.Concurrent
import Control.Concurrent.Async
import Data.Time

timeit :: IO a -> IO (a,Double)
timeit io = do
  t0 <- getCurrentTime
  a <- io
  t1 <- getCurrentTime
  return (a, realToFrac (t1 `diffUTCTime` t0))

worker :: Int -> IO Int
worker n = do
  -- simulate some work
  threadDelay (10^2 * n)
  return (n * n)

-- Spawn 2 threads in parallel, halt on both finished.
test1 :: IO (Int, Int)
test1 = do
  val1 <- async $ worker 1000
  val2 <- async $ worker 2000
  (,) <$> wait val1 <*> wait val2

-- Spawn 2 threads in parallel, halt on first finished.
test2 :: IO (Either Int Int)
test2 = do
  let val1 = worker 1000
  let val2 = worker 2000
  race val1 val2

-- Spawn 10000 threads in parallel, halt on all finished.
test3 :: IO [Int]
test3 = mapConcurrently worker [0..10000]

main :: IO ()
main = do
  print =<< timeit test1
  print =<< timeit test2
  print =<< timeit test3



Diagrams is a library for generating vector images to SVG and a variety of other formats.

import Diagrams.Prelude
import Diagrams.Backend.SVG.CmdLine

sierpinski :: Int -> Diagram SVG
sierpinski 1 = eqTriangle 1
sierpinski n =
  (s ||| s) # centerX
    s = sierpinski (n - 1)

example :: Diagram SVG
example = sierpinski 5 # fc black

main :: IO ()
main = defaultMain example
$ runhaskell diagram1.hs -w 256 -h 256 -o diagram1.svg

See: Diagrams Quick Start Tutorial



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 themselves looks very similar to the parser grammar itself!

<|> The choice operator tries to parse the first argument before proceeding to the second. Can be chained sequentially to generate a sequence of options.
many Consumes an arbitrary number of patterns matching the given pattern and returns them as a list.
many1 Like many but requires at least one match.
optional Optionally parses a given pattern returning its value as a Maybe.
try Backtracking operator will let us parse ambiguous matching expressions and restart with a different pattern.

There are two styles of writing Parsec, one can choose to write with monads or with applicatives.

parseM :: Parser Expr
parseM = do
  a <- identifier
  char '+'
  b <- identifier
  return $ Add a b

The same code written with applicatives uses the applicative combinators:

-- | Sequential application.
(<*>) :: f (a -> b) -> f a -> f b

-- | Sequence actions, discarding the value of the first argument.
(*>) :: f a -> f b -> f b
(*>) = liftA2 (const id)

-- | Sequence actions, discarding the value of the second argument.
(<*) :: f a -> f b -> f a
(<*) = liftA2 const
parseA :: Parser Expr
parseA = Add <$> identifier <* char '+' <*> identifier

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 under with the parse.

import Text.Parsec
import Text.Parsec.String

data Expr
  = Var Char
  | Lam Char Expr
  | App Expr Expr
  deriving Show

lam :: Parser Expr
lam = do
  char '\\'
  n <- letter
  string "->"
  e <- expr
  return $ Lam n e

app :: Parser Expr
app = do
  apps <- many1 term
  return $ foldl1 App apps

var :: Parser Expr
var = do
  n <- letter
  return $ Var n

parens :: Parser Expr -> Parser Expr
parens p = do
  char '('
  e <- p
  char ')'
  return e

term :: Parser Expr
term = var <|> parens expr

expr :: Parser Expr
expr = lam <|> app

decl :: Parser Expr
decl = do
  e <- expr
  return e

test :: IO ()
test = parseTest decl "\\y->y(\\x->x)y"

main :: IO ()
main = test >>= print

Custom Lexer

In our previous example 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
  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) "<stdin>"

repl :: IO ()
repl = do
  str <- TIO.getLine
  print (test str)

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.

module Parser (parseExpr) where

import Text.Parsec
import Text.Parsec.String (Parser)
import Text.Parsec.Language (haskellStyle)

import qualified Text.Parsec.Expr as Ex
import qualified Text.Parsec.Token as Tok

type Id = String

data Expr
  = Lam Id Expr
  | App Expr Expr
  | Var Id
  | Num Int
  | Op  Binop Expr Expr
  deriving (Show)

data Binop = Add | Sub | Mul deriving Show

lexer :: Tok.TokenParser ()
lexer = Tok.makeTokenParser style
  where ops = ["->","\\","+","*","-","="]
        style = haskellStyle {Tok.reservedOpNames = ops }

reservedOp :: String -> Parser ()
reservedOp = Tok.reservedOp lexer

identifier :: Parser String
identifier = Tok.identifier lexer

parens :: Parser a -> Parser a
parens = Tok.parens lexer

contents :: Parser a -> Parser a
contents p = do
  Tok.whiteSpace lexer
  r <- p
  return r

natural :: Parser Integer
natural = Tok.natural lexer

variable :: Parser Expr
variable = do
  x <- identifier
  return (Var x)

number :: Parser Expr
number = do
  n <- natural
  return (Num (fromIntegral n))

lambda :: Parser Expr
lambda = do
  reservedOp "\\"
  x <- identifier
  reservedOp "->"
  e <- expr
  return (Lam x e)

aexp :: Parser Expr
aexp =  parens expr
    <|> variable
    <|> number
    <|> lambda

term :: Parser Expr
term = Ex.buildExpressionParser table aexp
  where infixOp x f = Ex.Infix (reservedOp x >> return f)
        table = [[infixOp "*" (Op Mul) Ex.AssocLeft],
                 [infixOp "+" (Op Add) Ex.AssocLeft]]

expr :: Parser Expr
expr = do
  es <- many1 term
  return (foldl1 App es)

parseExpr :: String -> Expr
parseExpr input =
  case parse (contents expr) "<stdin>" input of
    Left err -> error (show err)
    Right ast -> ast

main :: IO ()
main = getLine >>= print . parseExpr >> main

Trying it out:

λ: runhaskell simpleparser.hs
Op Add (Num 1) (Num 2)

\i -> \x -> x
Lam "i" (Lam "x" (Var "x"))

\s -> \f -> \g -> \x -> f x (g x)
Lam "s" (Lam "f" (Lam "g" (Lam "x" (App (App (Var "f") (Var "x")) (App (Var "g") (Var "x"))))))

Generic Parsing

Previously we defined generic operations for pretty printing and this begs the question of whether we can write a parser on top of Generics. The answer is generally yes, so long as there is a direct mapping between the specific lexemes and sum and products types. Consider the simplest case where we just read off the names of the constructors using the regular Generics machinery and then build a Parsec parser terms of them.

{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE ScopedTypeVariables #-}

import Text.Parsec
import Text.Parsec.Text.Lazy
import Control.Applicative ((<*), (<*>), (<$>))
import GHC.Generics

class GParse f where
  gParse :: Parser (f a)

-- Type synonym metadata for constructors
instance (GParse f, Constructor c) => GParse (C1 c f) where
  gParse =
    let con = conName (undefined :: t c f a) in
    (fmap M1 gParse) <* string con

-- Constructor names
instance GParse f => GParse (D1 c f) where
  gParse = fmap M1 gParse

-- Sum types
instance (GParse a, GParse b) => GParse (a :+: b) where
  gParse = try (fmap L1 gParse <|> fmap R1 gParse)

-- Product types
instance (GParse f, GParse g) => GParse (f :*: g) where
  gParse = (:*:) <$> gParse <*> gParse

-- Nullary constructors
instance GParse U1 where
  gParse = return U1

data Scientist
  = Newton
  | Einstein
  | Schrodinger
  | Feynman
  deriving (Show, Generic)

data Musician
  = Vivaldi
  | Bach
  | Mozart
  | Beethoven
  deriving (Show, Generic)

gparse :: (Generic g, GParse (Rep g)) => Parser g
gparse = fmap to gParse

scientist :: Parser Scientist
scientist = gparse

musician :: Parser Musician
musician = gparse
λ: parseTest parseMusician "Bach"

λ: parseTest parseScientist "Feynman"

With a little more work and an outer wrapper, this example an easily be extended to automate parsing of a simple recursive type.

{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE DeriveAnyClass #-}
{-# LANGUAGE DefaultSignatures #-}

import Control.Applicative    ((<*), (*>), (<*>), (<$>), pure)
import GHC.Generics
import Text.Parsec            ((<|>), string, try, many1, digit, char, letter, spaces)
import Text.Parsec.Text.Lazy  (Parser)

class GParse f where
  gParse :: Parser (f a)

-- Types
instance (Parse a) => GParse (K1 R a) where
  gParse = fmap K1 parse

-- Selector names
instance (GParse f, Selector s) => GParse (M1 S s f) where
  gParse = fmap M1 gParse

-- Type synonym metadata for constructors
instance (GParse f, Constructor c) => GParse (C1 c f) where
  gParse =
    let con = conName (undefined :: t c f a) in
      (spaces >> string con >> spaces) *> fmap M1 gParse

-- Constructor names
instance (Datatype d, GParse f) => GParse (D1 d f) where
  gParse = fmap M1 gParse

-- Sum types
instance (GParse a, GParse b) => GParse (a :+: b) where
  gParse = try (fmap L1 gParse) <|> try (fmap R1 gParse)

-- Product types
instance (GParse f, GParse g) => GParse (f :*: g) where
  gParse = (:*:) <$> try gParse <*> try gParse

-- Nullary constructors
instance GParse U1 where
  gParse = return U1

gparse :: (Generic g, GParse (Rep g)) => Parser g
gparse = fmap to gParse

class Parse a where
  parse :: Parser a
  default parse :: (Generic a, GParse (Rep a)) => Parser a
  parse = spaces >> char '(' >> gparse >>= \e -> char ')' >> return e

instance Parse Integer where
  parse = rd <$> (plus <|> minus <|> number)
    where rd     = read :: String -> Integer
          plus   = char '+' *> number
          minus  = (:) <$> char '-' <*> number
          number = many1 digit

instance Parse String where
   parse = many1 letter

type Name = String

data Exp 
  = Lit Integer
  | Var Name
  | Plus Exp Exp 
  | App Exp Exp 
  | Abs Name Exp deriving (Show, Generic, Parse)

expr :: Parser Exp
expr = parse
λ: parseTest expr "(App (Plus (Lit 1) (Var n)) (App (Plus (Lit 5) (Lit 5)) (Plus (Lit 6) (Lit 6))))"
App (Plus (Lit 1) (Var "n")) (App (Plus (Lit 5) (Lit 5)) (Plus (Lit 6) (Lit 6)))


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:

{-# LANGUAGE OverloadedStrings #-}
{-# OPTIONS_GHC -fno-warn-unused-do-bind #-}

import Control.Applicative
import Data.Attoparsec.Text
import qualified Data.Text as T
import qualified Data.Text.IO as T
import Data.List (foldl1')

data Name
  = Gen Int
  | Name T.Text
  deriving (Eq, Show, Ord)

data Expr
  = Var Name
  | App Expr Expr
  | Lam [Name] Expr
  | Lit Int
  | Prim PrimOp
  deriving (Eq, Show)

data PrimOp
  = Add
  | Sub
  | Mul
  | Div
  deriving (Eq, Show)

data Defn = Defn Name Expr
  deriving (Eq, Show)

name :: Parser Name
name = Name . T.pack <$> many1 letter

num :: Parser Expr
num = Lit <$> signed decimal

var :: Parser Expr
var = Var <$> name

lam :: Parser Expr
lam = do
  string "\\"
  vars <- many1 (skipSpace *> name)
  skipSpace *> string "->"
  body <- expr
  return (Lam vars body)

eparen :: Parser Expr
eparen = char '(' *> expr <* skipSpace <* char ')'

prim :: Parser Expr
prim = Prim <$> (
      char '+' *> return Add
  <|> char '-' *> return Sub
  <|> char '*' *> return Mul
  <|> char '/' *> return Div)

expr :: Parser Expr
expr = foldl1' App <$> many1 (skipSpace *> atom)

atom :: Parser Expr
atom = try lam
    <|> eparen
    <|> prim
    <|> var
    <|> num

def :: Parser Defn
def = do
  nm <- name
  skipSpace *> char '=' *> skipSpace
  ex <- expr
  skipSpace <* char ';'
  return $ Defn nm ex

file :: T.Text -> Either String [Defn]
file = parseOnly (many def <* skipSpace)

parseFile :: FilePath -> IO (Either T.Text [Defn])
parseFile path = do
  contents <- T.readFile path
  case file contents of
    Left a -> return $ Left (T.pack a)
    Right b -> return $ Right b

main :: IO (Either T.Text [Defn])
main = parseFile "simple.ml"

For an example try the above parser with the following simple lambda expression.

f = g (x - 1);
g = f (x + 1);
h = \x y -> (f x) + (g y);

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:

{-# LANGUAGE OverloadedStrings #-}

import Control.Monad

import Data.Attoparsec
import Data.Attoparsec.Char8 as A
import Data.ByteString.Char8

data Action
  = Success
  | KeepAlive
  | NoResource
  | Hangup
  | NewLeader
  | Election
  deriving Show

type Sender = ByteString
type Payload = ByteString

data Message = Message
  { action :: Action
  , sender :: Sender
  , payload :: Payload
  } deriving Show

proto :: Parser Message
proto = do
  act  <- paction
  send <- A.takeTill (== '.')
  body <- A.takeTill (A.isSpace)
  return $ Message act send body

paction :: Parser Action
paction = do
  c <- anyWord8
  case c of
    1  -> return Success
    2  -> return KeepAlive
    3  -> return NoResource
    4  -> return Hangup
    5  -> return NewLeader
    6  -> return Election
    _  -> mzero

main :: IO ()
main = do
  let msgtext = "\x01\x6c\x61\x70\x74\x6f\x70\x2e\x33\x2e\x31\x34\x31\x35\x39\x32\x36\x35\x33\x35\x0A"
  let msg = parseOnly proto msgtext
  print msg

See: Text Parsing Tutorial

Optparse Applicative

Optparse-applicative is a combinator library for building command line interfaces that take in various user flags, commands and switches and map 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.

Usage: optparse.hs [filename...] [--quiet] [--cheetah]

Available options:
  -h,--help                Show this help text
  filename...              Input files
  --quiet                  Whether to shut up.
  --cheetah                Perform task quickly.
import Data.List
import Data.Monoid
import Options.Applicative

data Opts = Opts
  { _files :: [String]
  , _quiet :: Bool
  , _fast :: Speed

data Speed = Slow | Fast

options :: Parser Opts
options = Opts <$> filename <*> quiet <*> fast
    filename :: Parser [String]
    filename = many $ argument str $
         metavar "filename..."
      <> help "Input files"

    fast :: Parser Speed
    fast = flag Slow Fast $
         long "cheetah"
      <> help "Perform task quickly."

    quiet :: Parser Bool
    quiet = switch $
         long "quiet"
      <> help "Whether to shut up."

greet :: Opts -> IO ()
greet (Opts files quiet fast) = do
  putStrLn "reading these files:"
  mapM_ print files

  case fast of
    Fast -> putStrLn "quickly"
    Slow -> putStrLn "slowly"

  case quiet of
    True  -> putStrLn "quietly"
    False -> putStrLn "loudly"

opts :: ParserInfo Opts
opts = info (helper <*> options) fullDesc

main :: IO ()
main = execParser opts >>= greet

See: Optparse Applicative Tutorial

Happy & Alex

Happy is a parser generator system for Haskell, similar to the tool `yacc' for C. It works as a preprocessor with it's 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 generate the Haskell source for the modules.

$ alex Lexer.x -o Lexer.hs
$ happy Parser.y -o Parser.hs

The generated modules are not human readable generally and unfortunately error messages are given in the Haskell source, not the Happy source.


For instance we could define a little toy lexer with a custom set of tokens.

module Lexer (
) where

import Syntax

%wrapper "basic"

$digit = 0-9
$alpha = [a-zA-Z]
$eol   = [\n]

tokens :-

  -- Whitespace insensitive
  $eol                          ;
  $white+                       ;
  print                         { \s -> TokenPrint }
  $digit+                       { \s -> TokenNum (read s) }
  \=                            { \s -> TokenEq }
  $alpha [$alpha $digit \_ \']* { \s -> TokenSym s }


data Token 
  = TokenNum Int
  | TokenSym String
  | TokenPrint
  | TokenEq
  | TokenEOF
  deriving (Eq,Show)

scanTokens = alexScanTokens



The associated parser is list of a production rules and a monad to running 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).

    : term                   { [$1] }
    | term terms             { $1 : $2 }
{-# LANGUAGE GeneralizedNewtypeDeriving #-}

module Parser (
) where

import Lexer
import Syntax

import Control.Monad.Except

%name expr
%tokentype { Token }
%monad { Except String } { (>>=) } { return }
%error { parseError }

    int   { TokenNum $$ }
    var   { TokenSym $$ }
    print { TokenPrint }
    '='   { TokenEq }


    : term                   { [$1] }
    | term terms             { $1 : $2 }

   : var                     { Var $1 }
   | var '=' int             { Assign $1 $3 }
   | print term              { Print $2 }


parseError :: [Token] -> Except String a
parseError (l:ls) = throwError (show l)
parseError [] = throwError "Unexpected end of Input"

parseExpr :: String -> Either String [Expr]
parseExpr input = 
  let tokenStream = scanTokens input in
  runExcept (expr tokenStream)

As a simple input consider the following simple program.

x = 4
print x
y = 5
print y
y = 6
print y


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.

{-# LANGUAGE OverloadedStrings #-}

import Data.Text
import qualified Data.Configurator as C

data Config = Config
  { verbose      :: Bool
  , loggingLevel :: Int
  , logfile      :: FilePath
  , dbHost       :: Text
  , dbUser       :: Text
  , dbDatabase   :: Text
  , dbpassword   :: Maybe Text
  } deriving (Eq, Show)

readConfig :: FilePath -> IO Config
readConfig cfgFile = do
  cfg          <- C.load [C.Required cfgFile]
  verbose      <- C.require cfg "logging.verbose"
  loggingLevel <- C.require cfg "logging.loggingLevel"
  logFile      <- C.require cfg "logging.logfile"
  hostname     <- C.require cfg "database.hostname"
  username     <- C.require cfg "database.username"
  database     <- C.require cfg "database.database"
  password     <- C.lookup cfg "database.password"
  return $ Config verbose loggingLevel logFile hostname username database password

main :: IO ()
main = do
  cfg <-readConfig "example.config"
  print cfg

An example configuration file:

  verbose      = true
  logfile      = "/tmp/app.log"
  loggingLevel = 3

  hostname = "us-east-1.rds.amazonaws.com"
  username = "app"
  database = "booktown"
  password = "hunter2"

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.


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.

mapM :: (Monad m, Traversable t) => (a -> m b) -> t a -> m (t b)
sequence :: (Monad m, Traversable t) => t (m a) -> m (t a)

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.

import System.IO

main :: IO ()
main = do
  withFile "foo.txt" ReadMode $ \fd -> do
    contents <- hGetContents fd
    print contents
  -- "foo\n"

  contents <- withFile "foo.txt" ReadMode hGetContents
  print contents
  -- ""

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.


await :: Monad m => Pipe a y m a
yield :: Monad m => a -> Pipe x a m ()

(>->) :: Monad m
      => Pipe a b m r
      -> Pipe b c m r
      -> Pipe a c m r

runEffect :: Monad m => Effect m r -> m r
toListM :: Monad m => Producer a m () -> m [a]

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.

import Pipes
import Pipes.Prelude as P
import Control.Monad
import Control.Monad.Identity

a :: Producer Int Identity ()
a = forM_ [1..10] yield

b :: Pipe Int Int Identity ()
b =  forever $ do
  x <- await
  yield (x*2)
  yield (x*3)
  yield (x*4)

c :: Pipe Int Int Identity ()
c = forever $ do
  x <- await
  if (x `mod` 2) == 0
    then yield x
    else return ()

result :: [Int]
result = P.toList $ a >-> b >-> c

For example we could construct a "FizzBuzz" pipe.

{-# LANGUAGE MultiWayIf #-}

import Pipes
import qualified Pipes.Prelude as P

count :: Producer Integer IO ()
count = each [1..100]

fizzbuzz :: Pipe Integer String IO ()
fizzbuzz = do
  n <- await
  if | n `mod` 15 == 0 -> yield "FizzBuzz"
     | n `mod` 5  == 0 -> yield "Fizz"
     | n `mod` 3  == 0 -> yield "Buzz"
     | otherwise       -> return ()

main :: IO ()
main = runEffect $ count >-> fizzbuzz >-> P.stdoutLn

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.

import Pipes
import Pipes.Prelude as P
import System.IO

readF :: FilePath -> Producer String IO ()
readF file = do
    lift $ putStrLn $ "Opened" ++ file
    h <- lift $ openFile file ReadMode
    fromHandle h
    lift $ putStrLn $ "Closed" ++ file
    lift $ hClose h

main :: IO ()
main = runEffect $ readF "foo.txt" >-> P.take 3 >-> stdoutLn

This is simple a 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

Safe Pipes

bracket :: MonadSafe m => Base m a -> (a -> Base m b) -> (a -> m c) -> m c

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.

import Pipes
import Pipes.Safe
import qualified Pipes.Prelude as P

import System.Timeout (timeout)
import Data.ByteString.Char8
import qualified System.ZMQ as ZMQ

data Opts = Opts
  { _addr    :: String  -- ^ ZMQ socket address
  , _timeout :: Int     -- ^ Time in milliseconds for socket timeout

recvTimeout :: Opts -> ZMQ.Socket a -> Producer ByteString (SafeT IO) ()
recvTimeout opts sock = do
  body <- liftIO $ timeout (_timeout opts) (ZMQ.receive sock [])
  case body of
    Just msg -> do
      liftIO $ ZMQ.send sock msg []
      yield msg
      recvTimeout opts sock
    Nothing  -> liftIO $ print "socket timed out"

collect :: ZMQ.Context
        -> Opts
        -> Producer ByteString (SafeT IO) ()
collect ctx opts = bracket zinit zclose (recvTimeout opts)
    -- Initialize the socket
    zinit = do
      liftIO $ print "waiting for messages"
      sock <- ZMQ.socket ctx ZMQ.Rep
      ZMQ.bind sock (_addr opts)
      return sock

    -- On timeout or completion guarantee the socket get closed.
    zclose sock = do
      liftIO $ print "finalizing"
      ZMQ.close sock

runZmq :: ZMQ.Context -> Opts -> IO ()
runZmq ctx opts = runSafeT $ runEffect $
  collect ctx opts >-> P.take 10 >-> P.print

main :: IO ()
main = do
  ctx <- ZMQ.init 1
  let opts = Opts {_addr = "tcp://", _timeout = 1000000 }
  runZmq ctx opts
  ZMQ.term ctx


await :: Monad m => ConduitM i o m (Maybe i)
yield :: Monad m => o -> ConduitM i o m ()
($$) :: Monad m => Source m a -> Sink a m b -> m b
(=$) :: Monad m => Conduit a m b -> Sink b m c -> Sink a m c

type Sink i = ConduitM i Void
type Source m o = ConduitM () o m ()
type Conduit i m o = ConduitM i o m ()

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. The composition operators are also split into a connecting operator ($$) and a fusing operator (=$) for combining Sources and Sink and a Conduit and a Sink respectively.

{-# LANGUAGE MultiWayIf #-}

import Data.Conduit
import Control.Monad.Trans
import qualified Data.Conduit.List as CL

source :: Source IO Int
source = CL.sourceList [1..100]

conduit :: Conduit Int IO String
conduit = do
  val <- await
  liftIO $ print val
  case val of
    Nothing -> return ()
    Just n -> do
      if | n `mod` 15 == 0 -> yield "FizzBuzz"
         | n `mod` 5  == 0 -> yield "Fizz"
         | n `mod` 3  == 0 -> yield "Buzz"
         | otherwise       -> return ()

sink :: Sink String IO ()
sink = CL.mapM_ putStrLn

main :: IO ()
main = source $$ conduit =$ sink

See: Conduit Overview




data Base
  = Base16            -- ^ similar to hexadecimal
  | Base32
  | Base64            -- ^ standard Base64
  | Base64URLUnpadded -- ^ unpadded URL-safe Base64
  | Base64OpenBSD     -- ^ Base64 as used in OpenBSD password encoding (such as bcrypt)

convertToBase :: (ByteArrayAccess bin, ByteArray bout) => Base -> bin -> bout
convertFromBase :: (ByteArrayAccess bin, ByteArray bout) => Base -> bin -> Either String bout




Symmetric-key algorithms are algorithms for cryptography that use the same cryptographic keys for both encryption of plaintext and decryption of ciphertext.


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.


MD5 is a deprecated hash algorithm that has practical known collision attacks

MD5 is a deprecated cryptographic hash function. It produces a 128-bit message digest and has practical known collision attacks.

{-# LANGUAGE OverloadedStrings #-}

import Crypto.Hash (MD5, Digest, hash)
import Data.ByteArray (convert)
import Data.ByteString.Char8 (ByteString)

v1 :: ByteString
v1 = "The quick brown fox jumps over the lazy dog"

h1 :: Digest MD5
h1 = hash v1

s1 :: ByteString
s1  = convert h1

main :: IO ()
main = do
  print v1
  print h1
  print s1


NIST has deprecated SHA-1 in favor of the SHA-2 variants. Cryptanalysis of SHA-1 has demonstrated that it is vulnerable to practical collision attacks


{-# LANGUAGE OverloadedStrings #-}

module SipHash where

import Data.Word
import Data.ByteString
import Data.ByteArray.Hash

k0, k1 :: Word64
k0 = 0x4a7330fae70f52e8
k1 = 0x919ea5953a9a1ec9

msg :: ByteString
msg = "The quick brown fox jumped over the lazy dog"

hashed :: SipHash
hashed = sipHash (SipKey k0 k1) msg


{-# LANGUAGE OverloadedStrings #-}

import Crypto.Hash (Keccak_256, Digest, hash)
import Data.ByteArray (convert)
import Data.ByteString.Char8 (ByteString)

v1 :: ByteString
v1 = "The quick brown fox jumps over the lazy dog"

h1 :: Digest Keccak_256
h1 = hash v1

s1 :: ByteString
s1  = convert h1

main :: IO ()
main = do
  print v1
  print h1
  print s1


SHA-256 is a cryptographic hash function from the SHA-2 family and is standardized by NIST. It produces a 256-bit message digest.

{-# LANGUAGE OverloadedStrings #-}

import Crypto.Hash (SHA256, Digest, hash)
import Data.ByteArray (convert)
import Data.ByteString.Char8 (ByteString)

v1 :: ByteString
v1 = "The quick brown fox jumps over the lazy dog"

h1 :: Digest SHA256
h1 = hash v1

s1 :: ByteString
s1  = convert h1

main :: IO ()
main = do
  print v1
  print h1
  print s1


Whirlpool is a cryptographic hash function that is part of ISO/IEC 10118-3:2004. It produces a 512-bit message digest.

{-# LANGUAGE OverloadedStrings #-}

import Crypto.Hash (Whirlpool, Digest, hash)
import Data.ByteArray (convert)
import Data.ByteString.Char8 (ByteString)

v1 :: ByteString
v1 = "The quick brown fox jumps over the lazy dog"

h1 :: Digest Whirlpool
h1 = hash v1

s1 :: ByteString
s1  = convert h1

main :: IO ()
main = do
  print v1
  print h1
  print s1


RIPEMD160 is a cryptographic hash function that is part of ISO/IEC 10118-3:2004. It produces a 160-bit message digest.

{-# LANGUAGE OverloadedStrings #-}

import Crypto.Hash (RIPEMD160, Digest, hash)
import Data.ByteArray (convert)
import Data.ByteString.Char8 (ByteString)

v1 :: ByteString
v1 = "The quick brown fox jumps over the lazy dog"

h1 :: Digest RIPEMD160
h1 = hash v1

s1 :: ByteString
s1  = convert h1

main :: IO ()
main = do
  print v1
  print h1
  print s1


A keyed-hash message authentication code (HMAC) is a specific type of message authentication code (MAC) involving a cryptographic hash function (hence the 'H') in combination with a secret cryptographic key. As with any MAC, it may be used to simultaneously verify both the data integrity and the authentication of a message.


{-# LANGUAGE OverloadedStrings #-}

import Crypto.Hash (SHA256, Digest, hash)
import Crypto.MAC.HMAC (HMAC(..), hmac)
import Data.ByteArray (convert)
import Data.ByteString.Char8 (ByteString)

msg :: ByteString
msg = "The quick brown fox jumps over the lazy dog"

key :: ByteString
key = "hunter2"

digest :: HMAC SHA256
digest = hmac msg key

d1 :: Digest SHA256
d1 = hmacGetDigest digest

s1 :: ByteString
s1 = convert (hmacGetDigest digest)

main :: IO ()
main = do
  print d1
  print s1

Key Derivation Function

A key derivation function (KDF) derives one or more secret keys from a secret value such as a master key, a password, or a passphrase using a pseudo-random function.


module Padding where


Public Key Cryptography

Asymmetric cryptography, also known as public key cryptography, uses public and private keys to encrypt and decrypt data. The keys are simply large numbers that have been paired together but are not identical (asymmetric). One key in the pair can be shared with everyone; it is called the public key. The other key in the pair is kept secret; it is called the private key. Either of the keys can be used to encrypt a message; the opposite key from the one used to encrypt the message is used for decryption.


Curve25519 is a state-of-the-art Diffie-Hellman function suitable for a wide variety of applications.

Given a user's 32-byte secret key, Curve25519 computes the user's 32-byte public key. Given the user's 32-byte secret key and another user's 32-byte public key, Curve25519 computes a 32-byte secret shared by the two users. This secret can then be used to authenticate and encrypt messages between the two users.

{-# LANGUAGE OverloadedStrings #-}

import Data.Word
import Data.ByteString as S
import Data.ByteArray as B
import Data.Serialize

import Crypto.Error
import Crypto.Random
import Crypto.Random.Entropy (getEntropy)

import Crypto.PubKey.DH
import qualified Crypto.PubKey.Curve25519 as Curve25519

-- https://github.com/haskell-crypto/cryptonite/tree/3c087f0f4462df606524083699119445bb81dfa6/tests
-- https://github.com/centromere/cacophony/blob/80adb3c69dd850794b038a95364693d9503a24ce/src/Crypto/Noise/DH/Curve25519.hs
-- https://github.com/glguy/ssh-hans/blob/f49ef74a8a37ddff1f4748f46be949704d41557c/src/Network/SSH/Keys.hs

alicePrivate = throwCryptoError $ Curve25519.secretKey ("\x77\x07\x6d\x0a\x73\x18\xa5\x7d\x3c\x16\xc1\x72\x51\xb2\x66\x45\xdf\x4c\x2f\x87\xeb\xc0\x99\x2a\xb1\x77\xfb\xa5\x1d\xb9\x2c\x2a" :: ByteString)
alicePublic  = throwCryptoError $ Curve25519.publicKey ("\x85\x20\xf0\x09\x89\x30\xa7\x54\x74\x8b\x7d\xdc\xb4\x3e\xf7\x5a\x0d\xbf\x3a\x0d\x26\x38\x1a\xf4\xeb\xa4\xa9\x8e\xaa\x9b\x4e\x6a" :: ByteString)
bobPrivate   = throwCryptoError $ Curve25519.secretKey ("\x5d\xab\x08\x7e\x62\x4a\x8a\x4b\x79\xe1\x7f\x8b\x83\x80\x0e\xe6\x6f\x3b\xb1\x29\x26\x18\xb6\xfd\x1c\x2f\x8b\x27\xff\x88\xe0\xeb" :: ByteString)
bobPublic    = throwCryptoError $ Curve25519.publicKey ("\xde\x9e\xdb\x7d\x7b\x7d\xc1\xb4\xd3\x5b\x61\xc2\xec\xe4\x35\x37\x3f\x83\x43\xc8\x5b\x78\x67\x4d\xad\xfc\x7e\x14\x6f\x88\x2b\x4f" :: ByteString)

genKey :: IO (Curve25519.SecretKey, Curve25519.PublicKey)
genKey = do
  r <- getEntropy 32 :: IO ScrubbedBytes
  let sk = throwCryptoError . Curve25519.secretKey $ r
      pk = Curve25519.toPublic sk
  return (sk, pk)

dh :: Curve25519.SecretKey -> Curve25519.PublicKey -> ScrubbedBytes
dh sk pk = convert $ Curve25519.dh pk sk

main :: IO ()
main = do
  (sk, pk) <- genKey
  let res = B.convert (dh sk pk) :: ByteString
  print res

  (a, fn) <- runCurve25519dh
  print a
  let sharedKey = fn (B.convert pk)
  print sharedKey

-- | Implements key exchange as defined by
-- curve25519-sha256@libssh.org.txt
runCurve25519dh :: 
  IO (S.ByteString, S.ByteString -> Maybe S.ByteString)
  {- ^ local public, remote public -> shared key -}
runCurve25519dh =

     -- fails if key isn't 32 bytes long
  do CryptoPassed priv <-
       fmap Curve25519.secretKey (getRandomBytes 32 :: IO S.ByteString)

     -- Section 2: Transmit public key as "string"
     let raw_pub_s  = convert $ Curve25519.toPublic priv

         computeSecret raw_pub_c
             -- fails if key isn't 32 bytes long
           | CryptoPassed pub_c <- Curve25519.publicKey raw_pub_c

             -- Section 4.3: Treat shared key bytes as "integer"
           = Just $ B.convert $ Curve25519.dh pub_c priv

           | otherwise = Nothing

     return (raw_pub_s, computeSecret)


Secp256k1 is a common elliptic curve using a Koblitz curve and provably random parameters.

ECDH is used for the purposes of key agreement. Suppose two people, Alice and Bob, wish to exchange a secret key with each other. Alice will generate a private key dA and a public key QA = dAG (where G is the generator for the curve). Similarly Bob has his private key dB and a public key QB = dBG. If Bob sends his public key to Alice then she can calculate dAQB = dAdBG. Similarly if Alice sends her public key to Bob, then he can calculate dbQA = dAdBG. The shared secret is the x co-ordinate of the calculated point dAdBG. Any eavesdropper would only know QA and QB, and would be unable to calculate the shared secret.

{-# LANGUAGE OverloadedStrings #-}

module Secp256k1 where

import Crypto.Cipher.AES (AES256)
import Crypto.Cipher.Types (BlockCipher(..), Cipher(..),nullIV)
import Crypto.Error (CryptoFailable(..), CryptoError(..))

import qualified Crypto.PubKey.ECC.DH as DH
import qualified Crypto.PubKey.ECC.Types as ECC

import qualified Crypto.Random.Types as CRT

import Data.ByteArray
import Data.ByteString (ByteString)

-- | ScrubbedBytes because DH.SharedKey 
newtype Key a = Key ScrubbedBytes

secp256k1 :: ECC.Curve
secp256k1 = ECC.getCurveByName ECC.SEC_p256k1

-- | Generate key pair
generate :: CRT.MonadRandom m => m (DH.PrivateNumber, DH.PublicPoint)
generate = do
  priv <- DH.generatePrivate secp256k1
  let pub = DH.calculatePublic secp256k1 priv
  pure (priv, pub)

-- | Compute shared secret
sharedSecret :: DH.PrivateNumber -> DH.PublicPoint -> DH.SharedKey
sharedSecret = DH.getShared secp256k1

-- Encrypt using AES256 and the ECDH shared secret
encrypt :: ScrubbedBytes -> ByteString -> Either CryptoError ByteString
encrypt secret msg = 
    case ctx of
      Left e -> Left e
      Right c -> Right $ ctrCombine c nullIV msg
    ctx = initCipher (mkCipherKey (undefined :: AES256) secret)
    initCipher :: BlockCipher c => Key c -> Either CryptoError c
    initCipher (Key k) = case cipherInit k of
      CryptoPassed a -> Right a
      CryptoFailed e -> Left e
    mkCipherKey :: Cipher cipher => cipher -> ScrubbedBytes -> Key cipher
    mkCipherKey _ = Key 

decrypt :: ScrubbedBytes -> ByteString -> Either CryptoError ByteString
decrypt = encrypt

example :: IO ()
example = do
    (alicePrivKey, alicePubKey) <- generate
    (bobPrivKey, bobPubKey) <- generate 
    -- | Calculate shared secrets
    let DH.SharedKey aliceSK = sharedSecret alicePrivKey bobPubKey  
        DH.SharedKey bobSK = sharedSecret bobPrivKey alicePubKey

    print (aliceSK == bobSK)
    let msg = "Haskell Crypto is fun."
    -- | Alice sends msg to Bob
    let eRes = encrypt aliceSK msg >>= decrypt bobSK  
    case eRes of
      Left e -> fail $ show e
      Right msg' -> do
        putStrLn $ "Alice's original msg: " ++ show msg
        putStrLn $ "Bob's decrypted msg from Alice: " ++ show msg'


X.509 is an ITU-T standard for a public key infrastructure. X.509v3 is defined in RFC5280 X.509 certificates are commonly used in protocols like TLS.




Date and Time


Data Formats


Aeson is library for efficient parsing and generating JSON. It is the canonical JSON library for handling JSON.

decode :: FromJSON a => ByteString -> Maybe a
encode :: ToJSON a => a -> ByteString
eitherDecode :: FromJSON a => ByteString -> Either String a

fromJSON :: FromJSON a => Value -> Result a
toJSON :: ToJSON a => a -> Value

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 and integer throughout the rest of your program, Aeson will select the typeclass instance which parses the given input string into a Haskell integer.


Aeson uses several high performance data structures (Vector, Text, HashMap) by default instead of the naive versions so typically using Aeson will require that us 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.

type Object = HashMap Text Value

type Array = Vector Value

-- | A JSON value represented as a Haskell value.
data Value
  = Object !Object
  | Array !Array
  | String !Text
  | Number !Scientific
  | Bool !Bool
  | Null

For instance the Value expansion of the following JSON blob:

  "a": [1,2,3],
  "b": 1

Is represented in Aeson as the Value:

      [ ( "a"
        , Array (fromList [ Number 1.0 , Number 2.0 , Number 3.0 ])
      , ( "b" , Number 1.0 )

Let's consider some larger examples, we'll work with this contrived example JSON:

    "id": 1,
    "name": "A green door",
    "price": 12.50,
    "tags": ["home", "green"],
    "refs": {
      "a": "red",
      "b": "blue"

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.

{-# LANGUAGE OverloadedStrings #-}

import Data.Text
import Data.Aeson
import Data.Vector
import qualified Data.HashMap.Strict as M
import qualified Data.ByteString.Lazy as BL

-- Pull a key out of an JSON object.
(^?) :: Value -> Text -> Maybe Value
(^?) (Object obj) k = M.lookup k obj
(^?) _ _ = Nothing

-- Pull the ith value out of a JSON list.
ix :: Value -> Int -> Maybe Value
ix (Array arr) i = arr !? i
ix _ _ = Nothing

readJSON str = do
  obj <- decode str
  price <- obj ^? "price"
  refs  <- obj ^? "refs"
  tags  <- obj ^? "tags"
  aref  <- refs ^? "a"
  tag1  <- tags `ix` 0
  return (price, aref, tag1)

main :: IO ()
main = do
  contents <- BL.readFile "example.json"
  print $ readJSON contents

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.

{-# LANGUAGE DeriveGeneric #-}

import Data.Text
import Data.Aeson
import GHC.Generics
import qualified Data.ByteString.Lazy as BL

import Control.Applicative

data Refs = Refs
  { a :: Text
  , b :: Text
  } deriving (Show,Generic)

data Data = Data
  { id    :: Int
  , name  :: Text
  , price :: Int
  , tags  :: [Text]
  , refs  :: Refs
  } deriving (Show,Generic)

instance FromJSON Data
instance FromJSON Refs
instance ToJSON Data
instance ToJSON Refs

main :: IO ()
main = do
  contents <- BL.readFile "example.json"
  let Just dat = decode contents
  print $ name dat
  print $ a (refs dat)

Now we get our validated JSON wrapped up into a nicely typed Haskell ADT.

  { id = 1
  , name = "A green door"
  , price = 12
  , tags = [ "home" , "green" ]
  , refs = Refs { a = "red" , b = "blue" }

The functions fromJSON and toJSON can be used to convert between this sum type and regular Haskell types with.

data Result a = Error String | Success a
λ: fromJSON (Bool True) :: Result Bool
Success True

λ: fromJSON (Bool True) :: Result Double
Error "when expecting a Double, encountered Boolean instead"

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 DeriveGeneric #-}
{-# LANGUAGE DeriveAnyClass #-}

import Data.Text
import Data.Aeson
import GHC.Generics
import qualified Data.ByteString.Lazy as BL

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.

{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE ScopedTypeVariables #-}

import Data.Text
import Data.Aeson
import Data.Maybe
import Data.Aeson.Types
import Control.Applicative
import qualified Data.ByteString.Lazy as BL

data Crew = Crew
  { name  :: Text
  , rank  :: Rank
  } deriving (Show)

data Rank
  = Captain
  | Ensign
  | Lieutenant
  deriving (Show)

-- Custom JSON Deserializer

instance FromJSON Crew where
  parseJSON (Object o) = do
    _name <- o .: "name"
    _rank <- o .: "rank"
    pure (Crew _name _rank)

instance FromJSON Rank where
  parseJSON (String s) = case s of
    "Captain"    -> pure Captain
    "Ensign"     -> pure Ensign
    "Lieutenant" -> pure Lieutenant
    _            -> typeMismatch "Could not parse Rank" (String s)
  parseJSON x = typeMismatch "Expected String" x

-- Custom JSON Serializer

instance ToJSON Crew where
  toJSON (Crew name rank) = object [
      "name" .= name
    , "rank" .= rank

instance ToJSON Rank where
  toJSON Captain    = String "Captain"
  toJSON Ensign     = String "Ensign"
  toJSON Lieutenant = String "Lieutenant"

roundTrips :: Crew -> Bool
roundTrips = isJust . go
    go :: Crew -> Maybe Crew
    go = decode . encode

picard :: Crew
picard = Crew { name = "Jean-Luc Picard", rank = Captain }

main :: IO ()
main = do
  contents <- BL.readFile "crew.json"
  let (res :: Maybe Crew) = decode contents
  print res
  print $ roundTrips picard

See: Aeson Documentation


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 Input:

invoice: 34843
date   : 2001-01-23
    given  : Chris
    family : Dumars
        lines: |
            458 Walkman Dr.
            Suite #292
        city    : Royal Oak
        state   : MI
        postal  : 48046

YAML Output:

     [ ( "invoice" , Number 34843.0 )
     , ( "date" , String "2001-01-23" )
     , ( "bill-to"
       , Object
              [ ( "address"
                , Object
                       [ ( "state" , String "MI" )
                       , ( "lines" , String "458 Walkman Dr.\nSuite #292\n" )
                       , ( "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 DeriveGeneric #-}
{-# LANGUAGE DeriveAnyClass #-}
{-# LANGUAGE ScopedTypeVariables #-}

import Data.Yaml

import Data.Text (Text)
import qualified Data.ByteString as BL

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 String Invoice) = decodeEither contents
  case res of
    Right val -> print val
    Left err -> putStrLn err

Which generates:

  { invoice = 34843
  , date = "2001-01-23"
  , bill =
        { address =
              { lines = "458 Walkman Dr.\nSuite #292\n"
              , city = "Royal Oak"
              , state = "MI"
              , postal = 48046
        , family = "Dumars"
        , given = "Chris"


Cassava is an efficient CSV parser library. We'll work with this tiny snippet from the iris dataset:


Unstructured CSV

Just like with Aeson if we really want to work with unstructured data the library accommodates this.

import Data.Csv

import Text.Show.Pretty

import qualified Data.Vector as V
import qualified Data.ByteString.Lazy as BL

type ErrorMsg = String
type CsvData = V.Vector (V.Vector BL.ByteString)

example :: FilePath -> IO (Either ErrorMsg CsvData)
example fname = do
  contents <- BL.readFile fname
  return $ decode NoHeader contents

We see we get the nested set of stringy vectors:

[ [ "sepal_length"
  , "sepal_width"
  , "petal_length"
  , "petal_width"
  , "plant_class"
, [ "5.1" , "3.5" , "1.4" , "0.2" , "Iris-setosa" ]
, [ "5.0" , "2.0" , "3.5" , "1.0" , "Iris-versicolor" ]
, [ "6.3" , "3.3" , "6.0" , "2.5" , "Iris-virginica" ]

Structured CSV

Just like with Aeson we can use Generic to automatically write the deserializer between our CSV data and our custom datatype.

{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE DeriveGeneric #-}

import Data.Csv
import GHC.Generics
import qualified Data.Vector as V
import qualified Data.ByteString.Lazy as BL

data Plant = Plant
  { sepal_length :: Double
  , sepal_width  :: Double
  , petal_length :: Double
  , petal_width  :: Double
  , plant_class :: String
  } deriving (Generic, Show)

instance FromNamedRecord Plant
instance ToNamedRecord Plant

type ErrorMsg = String
type CsvData = (Header, V.Vector Plant)

parseCSV :: FilePath -> IO (Either ErrorMsg CsvData)
parseCSV fname = do
  contents <- BL.readFile fname
  return $ decodeByName contents

main = parseCSV "iris.csv" >>= print

And again we get a nice typed ADT as a result.

[ Plant
    { sepal_length = 5.1
    , sepal_width = 3.5
    , petal_length = 1.4
    , petal_width = 0.2
    , plant_class = "Iris-setosa"
, Plant
    { sepal_length = 5.0
    , sepal_widt