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The Y Combinator is an elegant way to express recursive functions. To express recursion via the Y Combinator requires two steps: express a recursive function as the fixed point of a non-recursive function; and . create a function – the Y Combinator – to find the fixed point of another function, without using recursion.
Church encoding. In mathematics, Church encoding is a means of representing data and operators in the lambda calculus. The Church numerals are a representation of the natural numbers using lambda notation. The method is named for Alonzo Church, who first encoded data in the lambda calculus this way.
The Y combinator is defined as Y , f:( x:f(xx)) ( x:f(xx)): It was discovered by Haskell Curry, and is one of the simplest fixed-point combinators. The fixed point of the higher order function Gis equal to G(G(G(G(G:::)))). Intuitively, the Y com-binator unrolls this equality, as needed.
Church numerals Church numerals encode the natural number n as a function that takes f and x, and applies f to x n times. 0 , f: x:x 1 = f: x:f x 2 = f: x:f (f x) SUCC , n: f: x:f (n f x) 9/33
Notice that applying the \(Y\) combinator to \(S\) does not yield a church numeral in any finite number of \(\beta\) or \(\eta\) reductions. Instead, we get something like this: \[ S\Big(S\big(S(S\ldots)\big)\Big).
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.
Write a Racket function (nat->church n), which consumes a natural number n, and produces a quoted, church-encoded number n (nat->church 0) ;; (lambda (f) (lambda (x) x)) (nat->church 1) ;; (lambda (f) (lambda (x) (f x)));; hint (define (nat->church n) (define (gen n) (if (= n 0) 'x 'todo)) `(λ (f) (λ (x) ,(gen n))))
