Ad
related to: lambda calculus y combinator
Search results
Results From The WOW.Com Content Network
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:
Lambda calculus (also written as ... Or self-application a-la that which leads to Y combinator could be used. Recursion and fixed points Recursion is the definition ...
Combinatory logic is a model of computation equivalent to lambda calculus, but without abstraction. The advantage of this is that evaluating expressions in lambda calculus is quite complicated because the semantics of substitution must be specified with great care to avoid variable capture problems.
A Calculus of Mobile Processes, Part I (PostScript) (by Milner, Parrow, and Walker) shows a scheme for combinator graph reduction for the SKI calculus in pages 25–28. the Nock programming language may be seen as an assembly language based on SK combinator calculus in the same way that traditional assembly language is based on Turing machines.
The term λx. λy. x, sometimes called the K combinator, is written as λ λ 2 with de Bruijn indices. The binder for the occurrence x is the second λ in scope. The term λx. λy. λz. x z (y z) (the S combinator), with de Bruijn indices, is λ λ λ 3 1 (2 1). The term λz. (λy. y (λx. x)) (λx. z x) is λ (λ 1 (λ 1)) (λ 2 1). See the ...
For example, a list of three elements x, y and z can be encoded by a higher-order function that when applied to a combinator c and a value n returns c x (c y (c z n)). Equivalently, it is an application of the chain of functional compositions of partial applications, (c x ∘ c y ∘ c z) n.
A version of the Y combinator, called the Y* poly-variadic fix-point combinator [5] is then used to calculate fixed point of all the functions at the same time. The result is a mutually recursive implementation of the let expression.
[4] [5] The result of a Scott continuous λ-calculus topology is a function space built upon a programming semantic allowing fixed point combinatorics, such as the Y combinator, and data types. [6] [7] By 1971, λ-calculus was equipped to define any sequential computation and could be easily adapted to parallel computations. [8]