Simple types

In mathematics the definition of a particular function usually includes a statement of the kind of inputs it will accept, and the kind of outputs it will produce. For example, the squaring function accepts integers n as inputs and produces integers n2 as outputs, and the zero-test function accepts integers and produces Boolean values (‘true’ or ‘false’ according as the input is zero or not).

Corresponding to this way of defining functions, λ and CL can be modified by attaching expressions called ‘types’ to terms, like labels to denote their intended input and output sets. In fact almost all programming languages that use λ and CL use versions with types.

This chapter and the next two will describe two different approaches to attaching types to terms: (i) called Church-style or sometimes explicit or rigid, and (ii) called Curry-style or sometimes implicit. Both are used extensively in programming.

The Church-style approach originated in [Chu40], and is described in the present chapter. In it, a term's type is a built-in part of the term itself, rather like a person's fingerprint or eye-colour is a built-in part of the person's body. (In Curry's approach a term's type will be assigned after the term has been built, like a passport or identity-card may be given to a person some time after birth.)