In this chapter we develop the basics of recursive function theory, or as it is more generally known, computability theory. Its history goes back to the seminal works of Turing, Kleene and others in the 1930s.

A computable function is one defined by a program whose operational semantics tell an idealized computer what to do to its storage locations as it proceeds deterministically from input to output, without any prior restrictions on storage space or computation time. We shall be concerned with various program styles and the relationships between them, but the emphasis throughout this chapter and in part 2 will be on one underlying data type, namely the natural numbers, since it is there that the most basic foundational connections between proof theory and computation are to be seen in their clearest light. This is not to say that computability over more general and abstract data types is less important. Quite the contrary. For example, from a logical point of view, Stoltenberg-Hansen and Tucker [1999], Tucker and Zucker [2000], [2006] and Moschovakis [1997] give excellent presentations of a more abstract approach, and our part 3 develops a theory in higher types from a completely general standpoint.

The two best-known models of machine computation are the Turing Machine and the (Unlimited) Register Machine of Shepherdson and Sturgis [1963]. We base our development on the latter since it affords the quickest route to the results we want to establish (see also Cutland [1980]).