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).

Definition[edit]

A monad transformer consists of:

A type constructor t of kind (* -> *) -> * -> *
Monad operations return and bind (or an equivalent formulation) for all t m where m 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):
1. lift . return = return
2. lift (m `bind` k) = (lift m) `bind` (lift . k)

Examples[edit]

The option monad transformer[edit]

Given any monad $\mathrm {M} \,A$ , the option monad transformer $\mathrm {M} \left(A^{?}\right)$ (where $A^{?}$ denotes the option type) is defined by:

{\begin{array}{ll}\mathrm {return} :&A\rightarrow \mathrm {M} \left(A^{?}\right)=a\mapsto \mathrm {return} (\mathrm {Just} \,a)\\\mathrm {bind} :&\mathrm {M} \left(A^{?}\right)\rightarrow \left(A\rightarrow \mathrm {M} \left(B^{?}\right)\right)\rightarrow \mathrm {M} \left(B^{?}\right)=m\mapsto f\mapsto \mathrm {bind} \,m\,\left(a\mapsto {\begin{cases}{\mbox{return Nothing}}&{\mbox{if }}a=\mathrm {Nothing} \\f\,a'&{\mbox{if }}a=\mathrm {Just} \,a'\end{cases}}\right)\\\mathrm {lift} :&\mathrm {M} (A)\rightarrow \mathrm {M} \left(A^{?}\right)=m\mapsto \mathrm {bind} \,m\,(a\mapsto \mathrm {return} (\mathrm {Just} \,a))\end{array}}

The exception monad transformer[edit]

Given any monad $\mathrm {M} \,A$ , the exception monad transformer $\mathrm {M} (A+E)$ (where $E$ is the type of exceptions) is defined by:

{\begin{array}{ll}\mathrm {return} :&A\rightarrow \mathrm {M} (A+E)=a\mapsto \mathrm {return} (\mathrm {value} \,a)\\\mathrm {bind} :&\mathrm {M} (A+E)\rightarrow (A\rightarrow \mathrm {M} (B+E))\rightarrow \mathrm {M} (B+E)=m\mapsto f\mapsto \mathrm {bind} \,m\,\left(a\mapsto {\begin{cases}{\mbox{return err }}e&{\mbox{if }}a=\mathrm {err} \,e\\f\,a'&{\mbox{if }}a=\mathrm {value} \,a'\end{cases}}\right)\\\mathrm {lift} :&\mathrm {M} \,A\rightarrow \mathrm {M} (A+E)=m\mapsto \mathrm {bind} \,m\,(a\mapsto \mathrm {return} (\mathrm {value} \,a))\\\end{array}}

The reader monad transformer[edit]

Given any monad $\mathrm {M} \,A$ , the reader monad transformer $E\rightarrow \mathrm {M} \,A$ (where $E$ is the environment type) is defined by:

{\begin{array}{ll}\mathrm {return} :&A\rightarrow E\rightarrow \mathrm {M} \,A=a\mapsto e\mapsto \mathrm {return} \,a\\\mathrm {bind} :&(E\rightarrow \mathrm {M} \,A)\rightarrow (A\rightarrow E\rightarrow \mathrm {M} \,B)\rightarrow E\rightarrow \mathrm {M} \,B=m\mapsto k\mapsto e\mapsto \mathrm {bind} \,(m\,e)\,(a\mapsto k\,a\,e)\\\mathrm {lift} :&\mathrm {M} \,A\rightarrow E\rightarrow \mathrm {M} \,A=a\mapsto e\mapsto a\\\end{array}}

The state monad transformer[edit]

Given any monad $\mathrm {M} \,A$ , the state monad transformer $S\rightarrow \mathrm {M} (A\times S)$ (where $S$ is the state type) is defined by:

{\begin{array}{ll}\mathrm {return} :&A\rightarrow S\rightarrow \mathrm {M} (A\times S)=a\mapsto s\mapsto \mathrm {return} \,(a,s)\\\mathrm {bind} :&(S\rightarrow \mathrm {M} (A\times S))\rightarrow (A\rightarrow S\rightarrow \mathrm {M} (B\times S))\rightarrow S\rightarrow \mathrm {M} (B\times S)=m\mapsto k\mapsto s\mapsto \mathrm {bind} \,(m\,s)\,((a,s')\mapsto k\,a\,s')\\\mathrm {lift} :&\mathrm {M} \,A\rightarrow S\rightarrow \mathrm {M} (A\times S)=m\mapsto s\mapsto \mathrm {bind} \,m\,(a\mapsto \mathrm {return} \,(a,s))\end{array}}

The writer monad transformer[edit]

Given any monad $\mathrm {M} \,A$ , the writer monad transformer $\mathrm {M} (W\times A)$ (where $W$ is endowed with a monoid operation $*$ with identity element $\varepsilon$ ) is defined by:

{\begin{array}{ll}\mathrm {return} :&A\rightarrow \mathrm {M} (W\times A)=a\mapsto \mathrm {return} \,(\varepsilon ,a)\\\mathrm {bind} :&\mathrm {M} (W\times A)\rightarrow (A\rightarrow \mathrm {M} (W\times B))\rightarrow \mathrm {M} (W\times B)=m\mapsto f\mapsto \mathrm {bind} \,m\,((w,a)\mapsto \mathrm {bind} \,(f\,a)\,((w',b)\mapsto \mathrm {return} \,(w*w',b)))\\\mathrm {lift} :&\mathrm {M} \,A\rightarrow \mathrm {M} (W\times A)=m\mapsto \mathrm {bind} \,m\,(a\mapsto \mathrm {return} \,(\varepsilon ,a))\\\end{array}}

The continuation monad transformer[edit]

Given any monad $\mathrm {M} \,A$ , the continuation monad transformer maps an arbitrary type $R$ into functions of type $(A\rightarrow \mathrm {M} \,R)\rightarrow \mathrm {M} \,R$ , where $R$ is the result type of the continuation. It is defined by:

{\begin{array}{ll}\mathrm {return} \colon &A\rightarrow \left(A\rightarrow \mathrm {M} \,R\right)\rightarrow \mathrm {M} \,R=a\mapsto k\mapsto k\,a\\\mathrm {bind} \colon &\left(\left(A\rightarrow \mathrm {M} \,R\right)\rightarrow \mathrm {M} \,R\right)\rightarrow \left(A\rightarrow \left(B\rightarrow \mathrm {M} \,R\right)\rightarrow \mathrm {M} \,R\right)\rightarrow \left(B\rightarrow \mathrm {M} \,R\right)\rightarrow \mathrm {M} \,R=c\mapsto f\mapsto k\mapsto c\,\left(a\mapsto f\,a\,k\right)\\\mathrm {lift} \colon &\mathrm {M} \,A\rightarrow (A\rightarrow \mathrm {M} \,R)\rightarrow \mathrm {M} \,R=\mathrm {bind} \end{array}}

Note that monad transformations are usually not commutative: for instance, applying the state transformer to the option monad yields a type $S\rightarrow \left(A\times S\right)^{?}$ (a computation which may fail and yield no final state), whereas the converse transformation has type $S\rightarrow \left(A^{?}\times S\right)$ (a computation which yields a final state and an optional return value).

References[edit]

^ 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]

A highly technical blog post briefly reviewing some of the literature on monad transformers and related concepts, with a focus on categorical-theoretic treatment

[modular-interpreters-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.

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