Primitive Recursive and μ-Recursive Functions

Abstract

In Chapter 1.2 we have defined the computable functions on ℕ using register machines. This definition enables good understanding of the concept of computability. In this chapter we define the computable functions in a quite different way. We first define the primitive recursive functions and then the μ-recursive functions and finally prove that the μ-recursive functions coincide with the (register) computable functions. The primitive recursive functions form a class of total computable functions which contains all (total) functions of practical interest. However this class is too large to be of real importance outside of mathematical logic and theoretical computer science.