Basic Recursive Function Theory

Abstract

In the last section, we examined some reasonable models of computation. These models were found to be equivalent in the sense that they captured the same class of “computable” functions. Now we proceed to define our “generic” programming system syntactically. Prom our experience of coding each RAM program as a single natural number, we can use the natural numbers to serve as names for our programs. This will also make it easier to describe functions that take programs as arguments and produce programs as output. We will see that any way of effectively presenting the computable functions (via a list of all RAM programs, for example) will induce some structure in the way the programs appear. This structure will be evidenced by certain functional relationships among the various programs. Our generic programming system will be characterized by the specific relationships that hold for the system. The specific relationships are akin to control structures in that they specify how to modify and combine some programs to produce other programs. We will then discuss solvability (and partial solvability) of several fundamental problems. As a notational simplification, the use of pairing functions will be implicit, so we will write f(x, y) instead of the more formal f(<x,y>). Since we will have several occasions to give proofs by contradiction, we will note the arrival at a contradiction by the symbol (⇒⇒).