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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.
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 :
In fact computability can itself be defined via the lambda calculus: a function F: N → N of natural numbers is a computable function if and only if there exists a lambda expression f such that for every pair of x, y in N, F(x)=y if and only if f x = β y, where x and y are the Church numerals corresponding to x and y, respectively and = β ...
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 ...
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.
Paul Graham, cofounder of Y Combinator, urged moderates to vote for Kamala Harris in an open letter on X, citing character over policy.
Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic.It was introduced by Moses Schönfinkel [1] and Haskell Curry, [2] and has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages.
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 ...