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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 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.
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:
Function application applied to this form should give another expression in the same form. In this way any expression on functions of multiple values may be treated as if it had one value. It is not sufficient for the form to represent only the set of values. Each value must have a condition that determines when the expression takes the value.
The positive impact alternative meat products -- like plant-based meat or cultivated meat -- can have on the environment is striking. The Good Food Institute (GFI) estimates that alternative meat ...
The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. [2]By contrast, the Brouwer fixed-point theorem (1911) is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, [3] but it doesn ...
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 ...