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.

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 statement

where ${\displaystyle \alpha }$ is a type variable. The upper-case ${\displaystyle \Lambda }$ is traditionally used to denote type-level functions, as opposed to the lower-case ${\displaystyle \lambda }$ which is used for value-level functions. (The superscripted ${\displaystyle \alpha }$ means that the bound variable x is of type ${\displaystyle \alpha }$; 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 second-order propositional intuitionistic logic. 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.

The typing rules of System F are those of simply typed lambda calculus with the addition of the following:

where ${\displaystyle \sigma ,\tau }$ are types, ${\displaystyle \alpha }$ is a type variable, and ${\displaystyle \alpha ~{\text{type}}}$ in the context indicates that ${\displaystyle \alpha }$ is bound. The first rule is that of application, and the second is that of abstraction.

The ${\displaystyle {\mathsf {Boolean}}}$ type is defined as: ${\displaystyle \forall \alpha .\alpha \to \alpha \to \alpha }$, where ${\displaystyle \alpha }$ is a type variable. This means: ${\displaystyle {\mathsf {Boolean}}}$ is the type of all functions which take as input a type α and two expressions of type α, and produce as output an expression of type α (note that we consider ${\displaystyle \to }$ to be right-associative.)