The class of partial recursive functions is the mathematical abstraction of the class of partial functions computable by an algorithm. In this chapter we present them in the form of the μ-recursive functions. We then state some basic results, the main motivation being to set the stage for the theory of effective domains. Finally we show that the partial μ-recursive functions can be obtained from some simple initial functions using substitution and the fixed point theorem for computable functional. This illuminates the central role of taking fixed points and supports the claim of Chapter 1 that the function computed by an algorithm or a program is the least fixed point of a computable functional.

Section 9.1 Partial Recursive Functions

An algorithm for a class K of problems is a method or procedure which can be described in a finite way (a finite set of instructions) and which can be followed by someone or something to yield a computation solving each problem in K. The computation should proceed in discrete steps. For a given problem in K the procedure should say exactly how to perform each step in the computation. After performing a step, the procedure should prescribe how to do the next step. This next step must only depend on the problem and on the then existing situation, that is what has been done during previous steps.