- Monad transformer
-
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).
Contents
Definition
A monad transformer consists of:
- A type constructor
t
of kind(* -> *) -> * -> *
- Monad operations
return
andbind
(or an equivalent formulation) for allt m
wherem
is a monad, satisfying the monad laws - An additional operation,
lift :: m a -> t m a
, satisfying the following laws:[1] (the notation`bind`
below indicates infix application):lift . return = return
lift (m `bind` k) = (lift m) `bind` (lift . k)
Examples
The option monad transformer
Given any monad
, the option monad transformer
(where A? denotes the option type) is defined by:
The exception monad transformer
Given any monad
, the exception monad transformer M(A + E) (where E is the type of exceptions) is defined by:
The reader monad transformer
Given any monad
, the reader monad transformer
(where E is the environment type) is defined by:
The state monad transformer
Given any monad
, the state monad transformer
(where S is the state type) is defined by:
The writer monad transformer
Given any monad
, the writer monad transformer
(where W is endowed with a monoid operation * with identity element ε) is defined by:
The continuation monad transformer
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 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
- Monads in functional programming
- Natural transformation - a related concept in category theory
References
- ^ 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. http://portal.acm.org/citation.cfm?id=199528.
External links
- [1] - a highly technical blog post briefly reviewing some of the literature on monad transformers and related concepts, with a focus on categorical-theoretic treatment
Categories: - A type constructor
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