Difference between revisions of "Monad"
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Revision as of 22:09, 3 September 2012
import Control.Monad 
Monads in Haskell can be thought of as composable computation descriptions. The essence of monad is thus separation of composition timeline from the composed computation's execution timeline, as well as the ability of computation to implicitly carry extra data, as pertaining to the computation itself, in addition to its one (hence the name) output, that it will produce when run (or queried, or called upon). This lends monads to supplementing pure calculations with features like I/O, common environment or state, and to preprocessing of computations (simplification, optimization etc.).
Each monad, or computation type, provides means, subject to Monad Laws, to (a) create a description of computation action that will produce (a.k.a. "return") a given Haskell value, (b) somehow run a computation action description (possibly getting its output back into Haskell should the monad choose to allow it, if computations described by the monad are pure, or causing the prescribed side effects if it's not), and (c) combine (a.k.a. "bind") a computation action description with a reaction to it – a regular Haskell function of one argument (that will receive computationproduced value) returning another action description (using or dependent on that value, if need be) – thus creating a combined computation action description that will feed the original action's output through the reaction while automatically taking care of the particulars of the computational process itself. A monad might also define additional primitives to provide access to and/or enable manipulation of data it implicitly carries, specific to its nature.
Thus in Haskell, though it is a purelyfunctional language, side effects that will be performed by a computation can be dealt with and combined purely at the monad's composition time. Monads thus resemble programs in a particular DSL. While programs may describe impure effects and actions outside Haskell, they can still be combined and processed ("assembled") purely, inside Haskell, creating a pure Haskell value  a computation action description that describes an impure calculation. That is how Monads in Haskell separate between the pure and the impure.
The computation doesn't have to be impure and can be pure itself as well. Then monads serve to provide the benefits of separation of concerns, and automatic creation of a computational "pipeline". Because they are very useful in practice but rather mindtwisting for the beginners, numerous tutorials that deal exclusively with monads were created (see monad tutorials).
Contents
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
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
See this intuitive explanation of why they should obey the Monad laws. It basically says that monad's reactions should be associative under Kleisli composition, defined as (f >=> g) x = f x >>= g
, with return
its left and right identity element.
Any Monad can be made a Functor by defining
fmap ab ma = ma >>= (return . ab)
However, the Functor class is not a superclass of the Monad class. See Functor hierarchy proposal.
Special notation
In order to improve the look of code that uses monads Haskell provides a special syntactic sugar called do
notation. For example, 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
can be also written using the do
notation as follows:
do
x < thing1
y < func1 x
thing2
z < func2 y
return z
Code written using the do
notation is transformed by the compiler to ordinary expressions that use Monad
class functions.
When using the 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 the 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
 C++, doc
 CML.event ?
 Clean State monad
 Clojure
 JavaScript
 Java
 Joy
 LINQ, more, C#, VB (inaccessible)
 Lisp
 Miranda
 OCaml:
 Perl
 Perl6 ?
 Prolog
 Python
 Python
 here
 Twisted's Deferred monad
 Ruby:
 Scala:
 Scheme:
 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:
 Slate
 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
 Optional results
 Random values
 Read only state
 Writable state
 Unique supply
 ST  memoryonly effects
 Global state
 Undoable state effects
 Function application
 Functions which may error
 Atomic memory transactions
 Continuations
 IO  unrestricted side effects
 Nondeterministic evaluation
 List monad: computations with multiple choices
 Concurrent threads
 Backtracking computations
 Region allocation effects
 LogicT: backtracking monad transformer with fair operations and pruning
 Pi calculus as a monad
 Halfs, uses a readonly and writeonly monad for filesystem work.
 House's H monad for safe hardware access
 Commutable monads for parallel programming
 The Quantum computing monad
 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
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.