14.1 The formal system

The beauty of the λ-calculus is that it achieves Turing completeness - all computable functions are definable in it - with a very simple syntax. The basic operators of the λ-calculus, in its pure form, are λ-abstraction, for forming functions, and application, for applying a function to an argument. A λ-abstraction has the form λx. M: in the body M of the function, the variable x is a place-holder for the argument. (We stick to the standard λ-calculus terminology, and call x a variable, rather than a name.) Letting x and y range over the set of λ-calculus variables, the set Λ of λ-terms is defined by the grammar

In λx. M, the initial x is a static binder, binding all free occurrences of x in M. As we have already met static binders in previous chapters, we omit the definitions of α-conversion, free variable, substitution, etc. We identify α-convertible terms, and therefore write M = N if M and N are α-convertible. A λ-term is closed if it contains no free variables. The set of free variables of a term M is written fv(M). The subset of A containing only the closed terms is Λ0.

To avoid too many brackets, we assume that application associates to the left, so that MNL should be read (MN)L, and that the scope of a A extends as far as possible to the right, so that λx.MN should be read λx. (MN). We also abbreviate λx1 …. λxn.M to λx1 … xn.M, or λx. M if the length of x is not important. We follow Barendregt [Bar84] and Hindlev and Seldin [HS86] in notations and terminology for the λ-calculus.

The basic computational step of the λ-calculus is β-reduction:

in which the placeholder x is replaced by the argument N in the body M of the function. An expression of the form (λx.M)N is called a β-redex, and the derivative is its contractum.