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The lambda calculus may be seen as an idealized version of a functional programming language, like Haskell or Standard ML. Under this view, β-reduction corresponds to a computational step. This step can be repeated by additional β-reductions until there are no more applications left to reduce.
The Y combinator is an implementation of a fixed-point combinator in lambda calculus. Fixed-point combinators may also be easily defined in other functional and imperative languages. The implementation in lambda calculus is more difficult due to limitations in lambda calculus. The fixed-point combinator may be used in a number of different areas:
The names "lambda abstraction", "lambda function", and "lambda expression" refer to the notation of function abstraction in lambda calculus, where the usual function f (x) = M would be written (λx. M), and where M is an expression that uses x. Compare to the Python syntax of lambda x: M.
The lambda calculus, developed in the 1930s by Alonzo Church, is a formal system of computation built from function application.In 1937 Alan Turing proved that the lambda calculus and Turing machines are equivalent models of computation, [37] showing that the lambda calculus is Turing complete.
Simply typed lambda calculus is the internal language of cartesian closed categories; and it is for this reason that pairs and lists are the primary types in the type theory of LISP, Scheme and many functional programming languages.
The lambda calculus influenced the design of Lisp and functional programming languages in general.
In computer programming, apply applies a function to a list of arguments. Eval and apply are the two interdependent components of the eval-apply cycle, which is the essence of evaluating Lisp, described in SICP. [1] Function application corresponds to beta reduction in lambda calculus.
In the untyped lambda calculus, all functions are higher-order; in a typed lambda calculus, from which most functional programming languages are derived, higher-order functions that take one function as argument are values with types of the form ().