Monad transformer

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In functional programming, a monad transformer is a type constructor which takes a monad as an argument and returns a monad as a result.

Monad transformers can be used to compose features encapsulated by monads – such as state, exception handling, and I/O – in a modular way. Typically, a monad transformer is created by generalising an existing monad; applying the resulting monad transformer to the identity monad yields a monad which is equivalent to the original monad (ignoring any necessary boxing and unboxing).

Definition[edit]

A monad transformer consists of:

  1. A type constructor t of kind (* -> *) -> * -> *
  2. Monad operations return and bind (or an equivalent formulation) for all t m where m is a monad, satisfying the monad laws
  3. An additional operation, lift :: m a -> t m a, satisfying the following laws:[1] (the notation `bind` below indicates infix application):
    1. lift . return = return
    2. lift (m `bind` k) = (lift m) `bind` (lift . k)


Examples[edit]

The option monad transformer[edit]

Given any monad , the option monad transformer (where denotes the option type) is defined by:

The exception monad transformer[edit]

Given any monad , the exception monad transformer (where E is the type of exceptions) is defined by:

The reader monad transformer[edit]

Given any monad , the reader monad transformer (where E is the environment type) is defined by:

The state monad transformer[edit]

Given any monad , the state monad transformer (where S is the state type) is defined by:

The writer monad transformer[edit]

Given any monad , the writer monad transformer (where W is endowed with a monoid operation with identity element ) is defined by:

The continuation monad transformer[edit]

Given any monad , the continuation monad transformer maps an arbitrary type R into functions of type , where R is the result type of the continuation. It is defined by:

Note that monad transformations are usually not commutative: for instance, applying the state transformer to the option monad yields a type (a computation which may fail and yield no final state), whereas the converse transformation has type (a computation which yields a final state and an optional return value).

See also[edit]

References[edit]

  1. ^ Liang, Sheng; Hudak, Paul; Jones, Mark (1995). "Monad transformers and modular interpreters" (PDF). Proceedings of the 22nd ACM SIGPLAN-SIGACT symposium on Principles of programming languages. New York, NY: ACM. pp. 333–343. doi:10.1145/199448.199528. 

External links[edit]