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.[1]
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 categories. For example, the simply typed lambda calculus is the language of Cartesian closed categories (CCCs)[2]
^Brandl, Helmut (27 April 2024). "Typed Lambda Calculus / Calculus of Constructions" (PDF). Calculus of Constructions. Retrieved 27 April 2024.
^Lambek, J.; Scott, P. J. (1986), Introduction to Higher Order Categorical Logic, Cambridge University Press, ISBN 978-0-521-35653-4, MR 0856915
and 22 Related for: Typed lambda calculus information
A typedlambdacalculus is a typed formalism that uses the lambda-symbol ( λ {\displaystyle \lambda } ) to denote anonymous function abstraction. In this...
simply typedlambdacalculus ( λ → {\displaystyle \lambda ^{\to }} ), a form of type theory, is a typed interpretation of the lambdacalculus with only...
cube: Typedlambdacalculus – Lambdacalculus with typed variables (and functions) System F – A typedlambdacalculus with type-variables Calculus of constructions...
{\displaystyle \mathbb {N} \to \mathbb {R} } in typedlambdacalculus. For a more concrete example, taking A to be the type of unsigned integers from 0 to 255 (the...
the predicative calculus of inductive constructions (which removes some impredicativity). The CoC is a higher-order typedlambdacalculus, initially developed...
\;\vdash \;\lambda x.t:\sigma \to \tau }}} In System F (also named λ2 for the "second-order typedlambdacalculus") there is another type of abstraction...
typedlambdacalculus. Church's theory of types helped the formal system avoid the Kleene–Rosser paradox that afflicted the original untyped lambda calculus...
applications of unary type operators. Therefore, we can view the type operators as a simply typedlambdacalculus, which has only one basic type, usually denoted...
write complex functions is the lambdacalculus. For example, one can write the meaning of sleeps as the following lambda expression, which is a function...
uninhabited types. For most typed calculi, the type inhabitation problem is very hard. Richard Statman proved that for simply typedlambdacalculus the type inhabitation...
semantics TypedlambdacalculusTyped and untyped languages Type signature Type inference Datatype Algebraic data type (generalized) Type variable First-class...
is in defining type inference in the simply typedlambdacalculus, which is the internal language of Cartesian closed categories. Typing rules specify...
first class objects. Kappa-calculus can be regarded as "a reformulation of the first-order fragment of typedlambdacalculus". Because its functions are...
under the slogan: "Abstract [data] types have existential type". The theory is a second-order typedlambdacalculus similar to System F, but with existential...
mathematical logic, System U and System U− are pure type systems, i.e. special forms of a typedlambdacalculus with an arbitrary number of sorts, axioms and...
language is the simply typedlambdacalculus. They are generalized by closed monoidal categories, whose internal language, linear type systems, are suitable...
theories with simply typedlambdacalculus at the lowest corner and the calculus of constructions at the highest. Prior to 1994, many type theorists thought...
same type system. A logical framework is based on a general treatment of syntax, rules and proofs by means of a dependently typedlambdacalculus. Syntax...