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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 fact computability can itself be defined via the lambda calculus: a function F: N → N of natural numbers is a computable function if and only if there exists a lambda expression f such that for every pair of x, y in N, F(x)=y if and only if f x = β y, where x and y are the Church numerals corresponding to x and y, respectively and = β ...
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.
The predecessor function must use the numeral n to apply the function n-1 times. Before implementing the predecessor function, here is a scheme that wraps the value in a container function. We will define new functions to use in place of f and x, called inc and init. The container function is called value.
A function's identity is based on its implementation. A lambda calculus function (or term) is an implementation of a mathematical function. In the lambda calculus there are a number of combinators (implementations) that satisfy the mathematical definition of a fixed-point combinator.
The purpose of β-reduction is to calculate a value. A value in lambda calculus is a function. So β-reduction continues until the expression looks like a function abstraction. A lambda expression that cannot be reduced further, by either β-redex, or η-redex is in normal form. Note that alpha-conversion may convert functions.
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.
Typed lambda calculus; Typed and untyped languages; Type signature; Type inference; Datatype; Algebraic data type (generalized) Type variable; First-class value; Polymorphism; Calculus of constructions