Theory of computation | Logic in computer science | Lambda calculus | Type theory

Typed lambda calculus

A typed lambda calculus is a typed formalism that uses the lambda-symbol to denote anonymous function abstraction. In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a type depends on the calculus considered (see kinds below). From a certain point of view, typed lambda calculi can be seen as refinements of the untyped lambda calculus, but from another point of view, they can also be considered the more fundamental theory and untyped lambda calculus a special case with only one type. Typed lambda calculi are foundational programming languages and are the base of typed functional programming languages such as ML and Haskell and, more indirectly, typed imperative programming languages. Typed lambda calculi play an important role in the design of type systems for programming languages; here, typability usually captures desirable properties of the program (e.g., the program will not cause a memory access violation). Typed lambda calculi are closely related to mathematical logic and proof theory via the Curry–Howard isomorphism and they can be considered as the internal language of classes of categories; e.g., the simply typed lambda calculus is the language of Cartesian closed categories (CCCs). (Wikipedia).

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Related pages

Pure type system | Type constructor | Intuitionistic type theory | System F | Calculus of constructions | Dialectica interpretation | Lambda calculus | Function type | Untyped lambda calculus | Lambda cube | Subtyping | Proof theory | Girard's paradox | Kappa calculus | Second-order logic | Category theory | Mathematical logic | Cartesian closed category | Simply typed lambda calculus