Difference between revisions of "Monad"
m (Added suggestion for reading 'Merely monadic' and monad tutorial.) 
(Various formatting changes) 

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{{Standard classMonadmodule=Control.Monadmoduledoc=ControlMonadpackage=base}} 
{{Standard classMonadmodule=Control.Monadmoduledoc=ControlMonadpackage=base}} 

−  == Monad class == 
+  == The <code>Monad</code> class == 
−  Monads can be viewed as a standard programming interface to various data or control structures, which is captured by the < 
+  Monads can be viewed as a standard programming interface to various data or control structures, which is captured by the <code>Monad</code> class. All common monads are members of it: 
<haskell> 
<haskell> 

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</haskell> 
</haskell> 

−  In addition to implementing the class functions, all instances of Monad should obey the following equations, or '''''Monad Laws''''': 
+  In addition to implementing the class functions, all instances of <code>Monad</code> should obey the following equations, or '''''Monad Laws''''': 
<haskell> 
<haskell> 

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For more information, including an intuitive explanation of why they should be obeyed, see [[Monad laws]]. 
For more information, including an intuitive explanation of why they should be obeyed, see [[Monad laws]]. 

−  As of GHC 7.10, the Applicative typeclass is a superclass of Monad, and the Functor typeclass is a superclass of Applicative. This means that all monads are applicatives, all applicatives are functors, and, therefore, all monads are also functors. See [[Functor hierarchy proposal]]. 
+  As of GHC 7.10, the <code>Applicative</code> typeclass is a superclass of <code>Monad</code>, and the <code>Functor</code> typeclass is a superclass of <code>Applicative</code>. This means that all monads are applicatives, all applicatives are functors, and, therefore, all monads are also functors. See [[Functor hierarchy proposal]]. 
−  If the Monad definitions are preferred, Functor and Applicative instances can be defined from them with 
+  If the <code>Monad</code> definitions are preferred, <code>Functor</code> and <code>Applicative</code> instances can be defined from them with 
<haskell> 
<haskell> 

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</haskell> 
</haskell> 

−  although the recommended order is to define 
+  although the recommended order is to define <code>return</code> as <code>pure</code>, if the two are the same. 
== Common monads == 
== Common monads == 

Most common applications of monads include: 
Most common applications of monads include: 

−  * Representing failure using < 
+  * Representing failure using <code>Maybe</code> monad 
−  * Nondeterminism using < 
+  * Nondeterminism using <code>List</code> monad to represent carrying multiple values 
−  * State using < 
+  * State using <code>State</code> monad 
−  * Readonly environment using < 
+  * Readonly environment using <code>Reader</code> monad 
−  * I/O using < 
+  * I/O using <code>IO</code> monad 
−  == '''< 
+  == '''<code>do</code>'''notation == 
−  In order to improve the look of code that uses monads Haskell provides a special [[syntactic sugar]] called < 
+  In order to improve the look of code that uses monads Haskell provides a special [[syntactic sugar]] called <code>do</code>notation. For example, the following expression: 
<haskell> 
<haskell> 

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</haskell> 
</haskell> 

−  This can also be written using the < 
+  This can also be written using the <code>do</code>notation as follows: 
<haskell> 
<haskell> 

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(the curly braces and the semicolons are optional, when the indentation rules are observed). 
(the curly braces and the semicolons are optional, when the indentation rules are observed). 

−  Code written using < 
+  Code written using <code>do</code>notation is transformed by the compiler to ordinary expressions that use the functions from the <code>Monad</code> class (i.e. the two varieties of bind, <code>(>>=)</code> and <code>(>>)</code>). 
−  When using < 
+  When using <code>do</code>notation and a monad like <code>State</code> or <code>IO</code> programs look very much like programs written in an imperative language as each line contains a statement that can change the simulated global state of the program and optionally binds a (local) variable that can be used by the statements later in the code block. 
−  It is possible to intermix the < 
+  It is possible to intermix the <code>do</code>notation with regular notation. 
−  More on < 
+  More on <code>do</code>notation can be found in a section of [[Monads as computation#Do notationMonads as computation]] and in other [[Monad#Monad tutorialstutorials]]. 
== Commutative monads == 
== Commutative monads == 

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Examples of commutative include: 
Examples of commutative include: 

−  * < 
+  * <code>Reader</code> monad 
−  * < 
+  * <code>Maybe</code> monad 
== Monad tutorials == 
== Monad tutorials == 

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== Monad reference guides == 
== Monad reference guides == 

−  An explanation of the basic Monad functions, with examples, can be found in the reference guide [http://members.chello.nl/hjgtuyl/tourdemonad.html A tour of the Haskell Monad functions], by HenkJan van Tuyl. 
+  An explanation of the basic <code>Monad</code> functions, with examples, can be found in the reference guide [http://members.chello.nl/hjgtuyl/tourdemonad.html A tour of the Haskell Monad functions], by HenkJan van Tuyl. 
== Monad research == 
== Monad research == 
Revision as of 02:40, 16 March 2021
Hint: if you're just looking for an introduction to monads, see Merely monadic or one of the monad tutorials.
import Control.Monad 
Contents
The Monad
class
Monads can be viewed as a standard programming interface to various data or control structures, which is captured by the Monad
class. All common monads are members of it:
class Monad m where
(>>=) :: m a > ( a > m b) > m b
(>>) :: m a > m b > m b
return :: a > m a
fail :: String > m a
In addition to implementing the class functions, all instances of Monad
should obey the following equations, or Monad Laws:
return a >>= k = k a
m >>= return = m
m >>= (\x > k x >>= h) = (m >>= k) >>= h
For more information, including an intuitive explanation of why they should be obeyed, see Monad laws.
As of GHC 7.10, the Applicative
typeclass is a superclass of Monad
, and the Functor
typeclass is a superclass of Applicative
. This means that all monads are applicatives, all applicatives are functors, and, therefore, all monads are also functors. See Functor hierarchy proposal.
If the Monad
definitions are preferred, Functor
and Applicative
instances can be defined from them with
fmap fab ma = do { a < ma ; return (fab a) }
 ma >>= (return . fab)
pure a = do { return a }
 return a
mfab <*> ma = do { fab < mfab ; a < ma ; return (fab a) }
 mfab >>= (\ fab > ma >>= (return . fab))
 mfab `ap` ma
although the recommended order is to define return
as pure
, if the two are the same.
Common monads
Most common applications of monads include:
 Representing failure using
Maybe
monad  Nondeterminism using
List
monad to represent carrying multiple values  State using
State
monad  Readonly environment using
Reader
monad  I/O using
IO
monad
do
notation
In order to improve the look of code that uses monads Haskell provides a special syntactic sugar called do
notation. For example, the following expression:
thing1 >>= (\x > func1 x >>= (\y > thing2
>>= (\_ > func2 y >>= (\z > return z))))
which can be written more clearly by breaking it into several lines and omitting parentheses:
thing1 >>= \x >
func1 x >>= \y >
thing2 >>= \_ >
func2 y >>= \z >
return z
This can also be written using the do
notation as follows:
do {
x < thing1 ;
y < func1 x ;
thing2 ;
z < func2 y ;
return z
}
(the curly braces and the semicolons are optional, when the indentation rules are observed).
Code written using do
notation is transformed by the compiler to ordinary expressions that use the functions from the Monad
class (i.e. the two varieties of bind, (>>=)
and (>>)
).
When using do
notation and a monad like State
or IO
programs look very much like programs written in an imperative language as each line contains a statement that can change the simulated global state of the program and optionally binds a (local) variable that can be used by the statements later in the code block.
It is possible to intermix the do
notation with regular notation.
More on do
notation can be found in a section of Monads as computation and in other tutorials.
Commutative monads
Commutative monads are monads for which the order of actions makes no difference (they commute), that is when following code:
do
a < actA
b < actB
m a b
is the same as:
do
b < actB
a < actA
m a b
Examples of commutative include:

Reader
monad 
Maybe
monad
Monad tutorials
Monads are known for being deeply confusing to lots of people, so there are plenty of tutorials specifically related to monads. Each takes a different approach to Monads, and hopefully everyone will find something useful.
See the Monad tutorials timeline for a comprehensive list of monad tutorials.
Monad reference guides
An explanation of the basic Monad
functions, with examples, can be found in the reference guide A tour of the Haskell Monad functions, by HenkJan van Tuyl.
Monad research
A collection of research papers about monads.
Monads in other languages
Implementations of monads in other languages.
 C
 Clojure
 CML.event ?
 Clean State monad
 JavaScript
 Java
 Joy
 LINQ
 Lisp
 Miranda
 OCaml:
 Perl6 ?
 Prolog
 Python
 Python
 Twisted's Deferred monad
 Ruby:
 Scheme:
 Swift
 Tcl
 The Unix Shell
 More monads by Oleg
 CLL: a concurrent language based on a firstorder intuitionistic linear logic where all right synchronous connectives are restricted to a monad.
Unfinished:
 Parsing, Maybe and Error in Tcl
And possibly there exist:
 Standard ML (via modules?)
Please add them if you know of other implementations.
Collection of links to monad implementations in various languages. on Lambda The Ultimate.
Interesting monads
A list of monads for various evaluation strategies and games:
 Identity monad  the trivial monad.
 Optional results from computations  error checking without null.
 Random values  run code in an environment with access to a stream of random numbers.
 Read only variables  guarantee readonly access to values.
 Writable state  i.e. log to a state buffer
 A supply of unique values  useful for e.g. guids or unique variable names
 ST  memoryonly, locallyencapsulated mutable variables. Safely embed mutable state inside pure functions.
 Global state  a scoped, mutable state.
 Undoable state effects  roll back state changes
 Function application  chains of function application.
 Functions which may error  track location and causes of errors.
 Atomic memory transactions  software transactional memory
 Continuations  computations which can be interrupted and resumed.
 IO  unrestricted side effects on the world
 Search monad  bfs and dfs search environments.
 nondeterminism  interleave computations with suspension.
 stepwise computation  encode nondeterministic choices as stepwise deterministic ones
 Backtracking computations
 Region allocation effects
 LogicT  backtracking monad transformer with fair operations and pruning
 concurrent events and threads  refactor event and callback heavy programs into straightline code via coroutines
 QIO  The Quantum computing monad
 Pi calculus  a monad for Picalculus style concurrent programming
 Commutable monads for parallel programming
 Simple, Fair and Terminating Backtracking Monad
 Typed exceptions with call traces as a monad
 Breadth first list monad
 Continuationbased queues as monads
 Typed network protocol monad
 NonDeterminism Monad for LevelWise Search
 Transactional state monad
 A constraint programming monad
 A probability distribution monad
 Sets  Set computations
 HTTP  http connections as a monadic environment
 Memoization  add memoization to code
There are many more interesting instance of the monad abstraction out there. Please add them as you come across each species.
Fun
 If you are tired of monads, you can easily get rid of them.
Help
Because they are very useful in practice but rather mindtwisting for beginners, there are some tutorials about monads which are available for those new to Haskell: