By Lance A Leventhal

Booklet by means of Leventhal, Lance A

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All their functions have single arity. Combinators are λ calculus expressions with no free variables, which makes their meaning context-independent. The recursive combinator, Y, discovered by Haskell Curry, allows to deﬁne recursive functions in the λ calculus, proving it Turing-complete (Curry, 1941). Higher-order programming, which is natural in the λ calculus, enables us to view functions as data. Lisp is a functional programming language that uses the same syntax for programs and for data (lists).

The solution to the equation (f X) = X. , f ). (f This does not work, because f expects a function of type Z → Z, but it is taking another f , which has the more complex type (Z → Z) → (Z → Z). x. (if (= n 0) 1 (∗ n (− n 1))), which is equivalent to g(n) = 1 n ∗ (n − 1) if n = 0 if n > 0. We need to ﬁnd the correct expression X such that when f is applied to X, we get X, the recursive factorial function. (if (= n 0) 1 (∗ (n (f (− n 1))))) (x x))). (x x)), and explains why the recursive function can keep going.

The composition operation ◦ can itself be considered a function, also called a higher-order function, that takes two other functions as its input and returns a function as its output; that is, if the ﬁrst function is of type Z → Z and the second function is also of type Z → Z, then ◦ : (Z → Z) × (Z → Z) → (Z → Z). We can also deﬁne function composition in the λ calculus. x + 1. ( f (g x)). x2 x + 1). The resulting function gives the same results as f (g(x)) = (x + 1)2 . In the Scheme programming language (Sussman and Steele, 1998), as well as in most functional programming languages, we can directly use λ calculus expressions.