Lambda calculus | Type theory | Polymorphism (computer science)

System F

System F (also polymorphic lambda calculus or second-order lambda calculus) is a typed lambda calculus that introduces, to simply typed lambda calculus, a mechanism of universal quantification over types. System F formalizes parametric polymorphism in programming languages, thus forming a theoretical basis for languages such as Haskell and ML. It was discovered independently by logician Jean-Yves Girard (1972) and computer scientist John C. Reynolds (1974). Whereas simply typed lambda calculus has variables ranging over terms, and binders for them, System F additionally has variables ranging over types, and binders for them. As an example, the fact that the identity function can have any type of the form A → A would be formalized in System F as the judgment where is a type variable. The upper-case is traditionally used to denote type-level functions, as opposed to the lower-case which is used for value-level functions. (The superscripted means that the bound x is of type ; the expression after the colon is the type of the lambda expression preceding it.) As a term rewriting system, System F is strongly normalizing. However, type inference in System F (without explicit type annotations) is undecidable. Under the Curry–Howard isomorphism, System F corresponds to the fragment of second-order intuitionistic logic that uses only universal quantification. System F can be seen as part of the lambda cube, together with even more expressive typed lambda calculi, including those with dependent types. According to Girard, the "F" in System F was picked by chance. (Wikipedia).

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Higher-order function | Type variable | Parametric polymorphism | Record (computer science) | System U | Type inference | Fixed-point combinator | Intuitionistic logic | Universal quantification | Decision problem | Church encoding | Kind (type theory) | Simply typed lambda calculus | Typed lambda calculus | Subtyping | Lambda cube