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Lambda calculus cannot express this: all functions are anonymous in lambda calculus, so we can't refer by name to a value which is yet to be defined, inside the lambda term defining that same value. However, a lambda expression can receive itself as its own argument, for example in (λ x . x x ) E .
Anonymous functions originate in the work of Alonzo Church in his invention of the lambda calculus, in which all functions are anonymous, in 1936, before electronic computers. [2] In several programming languages, anonymous functions are introduced using the keyword lambda , and anonymous functions are often referred to as lambdas or lambda ...
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).
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:
Hendrik Pieter (Henk) Barendregt (born 18 December 1947, Amsterdam) [1] is a Dutch logician, known for his work in lambda calculus and type theory. Life and work
Alonzo Church (June 14, 1903 – August 11, 1995) was an American computer scientist, mathematician, logician, and philosopher who made major contributions to mathematical logic and the foundations of theoretical computer science. [2]
This is particularly important for the lambda calculus, which has anonymous unary functions, but is able to compute any recursive function. This anonymous recursion can be produced generically via fixed-point combinators .
Curry's paradox may be expressed in untyped lambda calculus, enriched by implicational propositional calculus. To cope with the lambda calculus's syntactic restrictions, m {\displaystyle m} shall denote the implication function taking two parameters, that is, the lambda term ( ( m A ) B ) {\displaystyle ((mA)B)} shall be equivalent to the usual ...
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