Axiomatic approach to the theory of algorithms and relativized computability

TL;DR

This paper presents an axiomatic framework that characterizes relativized computability, showing that classes of functions satisfying certain axioms and closure properties precisely correspond to relativized recursive functions.

Contribution

It establishes that any class of functions closed under key operations and satisfying specific axioms is exactly the class of functions computable relative to some oracle, providing a foundational characterization.

Findings

Relativized recursion theory can be axiomatized using three core axioms.

Classes satisfying these axioms and closure properties correspond exactly to relativized recursive functions.

The main result characterizes all such classes as relativized recursive function classes.

Abstract

It is well known that many theorems in recursion theory can be "relativized". This means that they remain true if partial recursive functions are replaced by functions that are partial recursive relative to some fixed oracle set. Uspensky formulates three "axioms" called "axiom of computation records", "axiom of programs'" and "arithmeticity axiom". Then, using these axioms (more precisely, two first ones) he proves basic results of the recursion theory. These two axioms are true also for the class of functions that are partial recursive relative to some fixed oracle set. Also this class is closed under substitution, primitive recursion and minimization ($\mu$-operator); these (intuitively obvious) closure properties are also used in the proofs. This observation made by Uspensky explains why many theorems of recursion theory can be relativized. It turns out that the reverse statement is also true: all relativizable results follow from the first two axioms and closure properties. Indeed, \emph{every class of partial functions that is closed under substitution, primitive recursion and minimization that satisfies the first two axioms is the class of functions that are partial recursive relative to some oracle set $A$}. This is the main result of the present article.