### System F ⇝

## polymorphic lambda-calculus

<*language, types*> (Or "second order typed lambda-calculus",
"System F", "Lambda-2"). An extension of typed
lambda-calculus allowing functions which take types as
parameters. E.g. the polymorphic function "twice" may be
written:

twice = /\ t . \ (f :: t -> t) . \ (x :: t) . f (f x)(where "/\" is an upper case Greek lambda and "(v :: T)" is usually written as v with subscript T). The parameter t will be bound to the type to which twice is applied, e.g.:

twice Inttakes and returns a function of type Int -> Int. (Actual type arguments are often written in square brackets [ ]). Function twice itself has a higher type:

twice :: Delta t . (t -> t) -> (t -> t)(where Delta is an upper case Greek delta). Thus /\ introduces an object which is a function of a type and Delta introduces a type which is a function of a type.

Polymorphic lambda-calculus was invented by Jean-Yves Girard in 1971 and independently by John C. Reynolds in 1974.

["Proofs and Types", J-Y. Girard, Cambridge U Press 1989].

Last updated: 2005-03-07

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**Nearby terms:**

**polymorphic lambda-calculus**» polymorphism » polynomial » polynomial-time