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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, one example being B ( WW )( BW ( BBB )), discovered by R. Statman in 1986.
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.
For example, in the expression λy.x x y, y is a bound variable and x is a free variable. Also a variable is bound by its nearest abstraction. In the following example the single occurrence of x in the expression is bound by the second lambda: λx.y (λx.z 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]
Y Combinator released the Simple Agreement for Future Equity ("SAFE") investment instrument as an alternative to convertible debt in late 2013. [4] It was written by Carolynn Levy. This investment vehicle has since become popular in the U.S., Canada, [5] and Israel, due to its simplicity and low transaction costs.
Y Combinator, LLC (YC) is an American technology startup accelerator and venture capital firm launched in March 2005 [1] which has been used to launch more than 4,000 companies. [2] The accelerator program started in Boston and Mountain View , expanded to San Francisco in 2019, and was entirely online during the COVID-19 pandemic. [ 3 ]
A classic example using the second recursion theorem is the function (,) =. The corresponding index in this case yields a computable function that outputs its own index when applied to any value. [8] When expressed as computer programs, such indices are known as quines.