Monad
Hint: if you're just looking for an introduction to monads, see Merely monadic or one of the other monad tutorials.
import Control.Monad |
The Monad
class
Monads can be viewed as a standard programming interface to various data or control structures, which is captured by Haskell's Monad
class. All the 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
In addition to implementing the class functions, all instances of Monad
should satisfy 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 the monad laws should be satisfied, 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. For more information, see the 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 would otherwise end up being the same.
Common monads
These include:
- Representing failure using
Maybe
monad - Nondeterminism using
List
monad to represent carrying multiple values - State using
State
monad - Read-only 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 form of 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
can also be written using do
-notation:
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 in Haskell 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
For monads which are commutative the order of actions makes no difference (i.e. they commute), so the 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 monads include:
Reader
monadMaybe
monad
Monad tutorials
Monads are known for being quite confusing to many 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 Henk-Jan 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 first-order intuitionistic linear logic where all right synchronous connectives are restricted to a monad.
- Collection of links to monad implementations in various languages. on Lambda The Ultimate.
Unfinished:
- Parsing, Maybe and Error in Tcl
And possibly there exists:
- Standard ML (via modules?)
(If you know of other implementations, please add them here.)
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 read-only 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 - memory-only, locally-encapsulated 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.
- non-determinism - interleave computations with suspension.
- stepwise computation - encode non-deterministic 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 straight-line code via co-routines
- QIO - The Quantum computing monad
- Pi calculus - a monad for Pi-calculus 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
- Continuation-based queues as monads
- Typed network protocol monad
- Non-Determinism Monad for Level-Wise 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 instances 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.