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 certain classes of . For example, the simply typed lambda calculus is the language of (CCCs) Various typed lambda calculi have been studied. The simply typed lambda calculus has only one type constructor, the arrow , and its only types are basic types and function types . System T extends the simply typed lambda calculus with a type of natural numbers and higher order primitive recursion; in this system all functions provably recursive in Peano arithmetic are definable. System F allows polymorphism by using universal quantification over all types; from a logical perspective it can describe all functions that are provably total in second-order logic. Lambda calculi with dependent types are the base of intuitionistic type theory, the calculus of constructions and the logical framework (LF), a pure lambda calculus with dependent types.