Typed lambda calculus

A typed <a href="/facts/Lambda_calculus/DaD3RuQ4">lambda calculus</a> is a typed <a href="/facts/Formalism_(mathematics)/g0690AVL">formalism</a> that uses the lambda symbol (
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
) 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 <a href="/facts/Untyped_lambda_calculus/DaD3RuQ4">untyped lambda calculus</a>, 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 <a href="/facts/Programming_languages/cWTYbgWa">programming languages</a> and are the base of typed <a href="/facts/Functional_programming_languages/cdxqPEIM">functional programming languages</a> such as <a href="/facts/ML_programming_language/RfwxASRF">ML</a> and <a href="/facts/Haskell_(programming_language)/4Htv9WLu">Haskell</a> and, more indirectly, typed <a href="/facts/Imperative_programming/6b5dUnE1">imperative programming languages</a>. Typed lambda calculi play an important role in the design of <a href="/facts/Type_systems/Tz0bAMS9">type systems</a> 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 <a href="/facts/Mathematical_logic/SaI7Y3wN">mathematical logic</a> and <a href="/facts/Proof_theory/kopBdQn9">proof theory</a> via the <a href="/facts/Curry%E2%80%93Howard_isomorphism/Q3eVu60u">Curry–Howard isomorphism</a> and they can be considered as the <a href="/facts/Internal_language/u4CIDbss">internal language</a> of certain classes of <a href="/facts/Category_theory/6zS3DXPa">categories</a>. For example, the <a href="/facts/Simply_typed_lambda_calculus/5RQlXLjy">simply typed lambda calculus</a> is the language of <a href="/facts/Cartesian_closed_category/mPXAQps2">Cartesian closed categories</a> (CCCs).

Typed lambda calculus open-in-new

Typed lambda calculus