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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. Statman in 1986.
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
The application process is open to anyone but highly competitive. For instance, Y Combinator and TechStars have application acceptance rates between 1% and 3%. Seed investment in startups may be made, in exchange for equity. Typically, the investment is between US$20,000 to US$50,000 in the US, or £10,000 to £50,000 in Europe. [3]
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:
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 ...
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.
Y Combinator—the startup incubator that helped launch DoorDash, Airbnb, Reddit, and Instacart—is backing a weapons maker for the first time, betting that it could shake up the defense industry ...
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.