|
Typed versions of the lambda calculus extend the standard lambda calculus with types. By assigning to each term a type, one can insist on type compatibility during substitutions. This is a clean theoretical model of the way types operate in conventional programming to a normal The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ...
On computer science, a datatype (often simply type) is a name or label for a set of values and some operations which can be performed on that set of values. ...
form. This is because all forms of the untyped lambda calculus that do not have normal forms cannot receive valid types. For example, the famous Ω combinator This article is about a topic in theoretical computer science, and is not to be confused with combinatorial logic, a topic in electronics. ...
- Ω ≡ (λ x . x x) (λ x . x x)
cannot be simply typed.
Indeed, the normal form property holds for still more powerful typed lambda calculi, such as System F. System F is a typed lambda calculus. ...
Because of this property, a decidable equational system can be developed that considers lambda terms equal if and only if they have the same normal form (up to α-conversion). Typed lambda calculi with this property are often used to form types for typed functional programming languages, as part of the type checking process requires checking two types for equality. Notable examples in this family are Haskell, Standard ML and Caml. Functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions. ...
Haskell is a standardized functional programming language with non-strict semantics, named after the logician Haskell Curry. ...
The SML programming language is a modern descendant of the ML programming language used in the LCF theorem-proving project. ...
CAML, Categorical Abstract Machine Language, is a version of ML developed by G. Huet, G. Cousineau, Asc nder Su rez, Pierre Weis, Michel Mauny and others from both INRIA and ENS. Implemented in Lisp, it was nicknamed Heavy CAML because of its memory and CPU requirements relative to its successor...
Typed lambda calculi are typically associated to various intuitionistic logics through Curry-Howard isomorphisms. Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ...
The Curry-Howard correspondence is the close relationship between computer programs and mathematical proofs; the correspondence is also known as the Curry-Howard isomorphism, or the formulae-as-types correspondence. ...
See also
|
Feeds |
Contact