# New PDF release: Computer Shortcuts & Special characters

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X simply returns whatever is given to it. x 5) ⇒ 5. The identity combinator in the Oz programming language2 can be written fun {I X} X end or equivalently, but closer to the λ calculus syntax:3 I = fun {\$ X} X end Contrast it to, for example, a Circumference function: Circumference = fun {\$ Radius} 2*PI*Radius end The semantics of the Circumference function depends on the deﬁnitions of PI and *. The Circumference function is, therefore, not a combinator. (f x), and allows you to evaluate a function with an argument.

9. We use this notation here for pedagogical purposes only. If you would rather follow the syntax precisely, think of x2 as (s x) and x + 1 as (i x) where s and i denote square and increment functions. 2. , Scheme, C, Oz) to illustrate that the concepts in the λ calculus are ubiquituous and apply to many different sequential programming languages. 3. The \$ notation in Oz converts a language statement into an expression (with a return value), which we use here to create an anonymous function. 4.

The problem is that this function deﬁnition uses a free variable f , which is the very factorial function that we are trying to deﬁne. (if (= n 0) 1 (∗ n (f (− n 1)))). Before we can input an integer to the function, we must input a function to satisfy f so that the returned function computes the desired factorial value. Let us call this function X. , X = f . The function f will return the proper recursive function f with the type Z → Z, but only when supplied with the correct function X. We can write the type of f as f : (Z → Z) → (Z → Z).