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The Y combinator may also be used in implementing Curry's paradox. The heart of Curry's paradox is that untyped lambda calculus is unsound as a deductive system, and the Y combinator demonstrates this by allowing an anonymous expression to represent zero, or even many values. This is inconsistent in mathematical logic.
Conversely, the use of fixed-pointed combinators may be generically referred to as "anonymous recursion", as this is a notable use of them, though they have other applications. [3] [4] This is illustrated below using Python. First, a standard named recursion:
The simplest example of a combinator is I, the identity combinator, defined by (I x) = x. for all terms x. Another simple combinator is K, which manufactures constant functions: (K x) is the function which, for any argument, returns x, so we say ((K x) y) = x. for all terms x and y. Or, following the convention for multiple application, (K x y) = x
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, [ 2 ] one example being B ( WW )( BW ( BBB )), discovered by R ...
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.
A combinator library is a software library which implements combinatory logic as combinators, for a functional programming language: "the key idea is this: a combinator library offers functions (the combinators) that combine functions together to make bigger functions". [1]
Startup accelerator Y Combinator is backing its first weapons startup — a firm that says it can make missiles smaller and cheaper than its competitors. "Ares is building a new class of anti-ship ...
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)).