Difference between revisions of "Monad"
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* [[What a Monad is not]]
* [[What a Monad is not]]
* [[Monads as containers]]
* [[Monads as computation]]
Revision as of 15:44, 19 November 2011
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. 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, of (a) creating a description of computation to produce a given value (or such that will fail to produce anything at all), (b) running a computation description (CD) and returning its output to Haskell, and (c) combining a CD with a "reaction" to it, i.e. a Haskell function consuming of its output and returning another CD (using or dependent on that output, if need be), to create a combined CD. It might also define additional primitives to provide access and/or enable manipulation of data it implicitly carries, specific to its nature.
Thus in Haskell, though it is a purely-functional 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 CD 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 separate the pure from the pure in one big holiday celebration after the other. We still get the benefits of separation of concerns, and automatic creation of a computational "pipeline" carrying out our chained Haskell calculations one after another with computation's state threaded through behind the scenes.
Because they are very useful in practice but rather mind-twisting for the beginners, numerous tutorials that deal exclusively with monads were created (see monad tutorials).
Most common applications of monads include:
- Representing failure using
- Nondeterminism through backtracking using
- State using
- Read-only environment using
- I/O using
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.
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
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.
Commutative monads are monads for which the order of actions makes no difference (they commute), that is when following code:
do a <- f x b <- g y m a b
is the same as:
do b <- g y a <- f x m a b
Examples of commutative include:
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 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.
A collection of research papers about monads.
Monads in other languages
Implementations of monads in other languages.
- C++, doc
- CML.event ?
- Clean State monad
- LINQ, more, C#, VB (inaccessible)
- Perl6 ?
- 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.
And possibly there exist:
- Standard ML (via modules?)
Please add them if you know of other implementations.
A list of monads for various evaluation strategies and games:
- Identity monad
- Optional results
- Random values
- Read only state
- Writable state
- Unique supply
- ST - memory-only effects
- Global state
- Undoable state effects
- Function application
- Functions which may error
- Atomic memory transactions
- IO - unrestricted side effects
- Non-deterministic 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 read-only and write-only 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
- 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
There are many more interesting instance of the monad abstraction out there. Please add them as you come across each species.
- If you are tired of monads, you can easily get rid of them.