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What is Y Combinator?
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The Y combinator is an implementation of a fixed-point combinator in lambda calculus. Fixed-point combinators may also be easily defined in other functional and imperative languages. The implementation in lambda calculus is more difficult due to limitations in lambda calculus.
Aug 13, 2018 · But we call rec ‘Y’ in lambda calculus this time, because this encoding is the famous Y-combinator that lets you have recursion in any languages: Y = λf.(λx.f (x x))(λx.f (x x)) Let’s verify that it behaves like rec by giving it an input g: Y g = λf.(λx.f (x x)) (λx.f (x x)) g (replace Y with its definition)
The Y combinator is defined as Y ≜ f:( x:f (x x)) ( x:f (x x)): It was discovered by Haskell Curry, and is one of the simplest fixed-point combinators. The fixed point of the higher order function G is equal to G (G (G (G (G :::)))). Intuitively, the Y combinator unrolls this equality, as needed. Let’s see it in action, on our function G ...
The Lambda calculus is an abstract mathematical theory of computation, involving \lambda λ functions. The lambda calculus can be thought of as the theoretical foundation of functional programming.
\[KIab = (\lambda \lambda x y.x) (\lambda x.x) a b = (\lambda x.x) b = b\] Thus \(KI = \lambda a b . b = \lambda \lambda 0\). In other words, we have combined \(K\) and \(I\) to produce a closed lambda term distinct to either one.
Here is a slightly more complicated approach: (define M1 (let ([g (lambda (f) (lambda (lat) (cond [(null? lat) 0] [else (+ 1 ((f eternity) (cdr lat)))])))]) ( g g))) Note that (g eternity) is. (lambda (lat) (cond [(null? lat) 0] [else (+ 1 ((eternity eternity) (cdr lat)))]))
