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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:
In the lambda calculus, Y g is a fixed-point of g, as it expands to: Y g (λh.(λx.h (x x)) (λx.h (x x))) g (λx.g (x x)) (λx.g (x x)) g ((λx.g (x x)) (λx.g (x x))) g (Y g) Now, to perform our recursive call to the factorial function, we would simply call (Y G) n, where n is the number we are calculating the factorial of. Given n = 4, for ...
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 ...
In the lambda calculus, which only uses functions of a single variable, this can be done via the Y combinator. First make the higher-order function of two variables be a function of a single variable, which directly returns a function, by currying:
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.
Also of note, Y combinator has a short expression in this system, as Y = BU(CBU) = BU(BWB) = B(W(WK))(BWB), where U = WI = SII is the self-application combinator. Using just two combinators, B and W , an infinite number of fixpoint combinators can be constructed, one example being B ( WW )( BW ( BBB )), discovered by R. Statman in 1986.
[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]
Pages in category "Lambda calculus" The following 48 pages are in this category, out of 48 total. ... SKI combinator calculus; Supercombinator; System F; System U; T.