Difference between revisions of "Monad"
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Revision as of 05:25, 9 March 2021
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, updatable state, etc.
Each monad, or computation type, provides means, subject to Monad Laws, to
 (a) create a description of a computation that will produce (a.k.a. "return") a given Haskell value, and
 (b) combine (a.k.a. "bind") a computation description with a reaction to it, – a pure Haskell function that is set to receive a computationproduced value (when and if that happens) and return another computation description, using or dependent on that value if need be, – creating a description of a combined computation that will feed the original computation's output through the reaction while automatically taking care of the particulars of the computational process itself.
Reactions are thus computation description constructors. A monad might also define additional primitives to provide access to and/or enable manipulation of data it implicitly carries, specific to its nature; cause some specific sideeffects; etc..
Sometimes the specific monadic type also provides the ability to somehow (c) run a computation description, getting its result back into Haskell if computations described by the monad are pure, but this is expressly not a part of the Monad interface. Officially, you can't get the a
out of M a
directly, only arrange for it to be "fed" into the next computation's constructor, the "reaction", indirectly. In case of an IO
monad value, a computation it describes runs implicitly as a part of the chain of I/O computation descriptions composed together into the value main
(of type IO ()
) in a given Haskell program, by convention.
# Monad interactions:
(a) reaction $ value ==> computation_description
(b) reaction =<< computation_description ==> computation_description
(c) reaction $ computation_description ==> ***type_mismatch***
(d) reaction <$> computation_description ==> computation_description_description
(e) join $ computation_description_description ==> computation_description
(join
is another function expressing the essence of monad; where m >>= k = k =<< m = join (k <$> m) = join (fmap k m)
; it is prefered in mathematics, over the bind; both express the same concept).
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 EDSL (embedded domainspecific language, "embedded" because the values denoting these computations are legal Haskell values, not some extraneous annotations).
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 help keep the pure and the impure apart.
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.
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.
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.