Conditional computability of real functions with respect to a class of operators

TL;DR

This paper introduces generalized notions of uniform and conditional computability for real functions based on classes of operators, exploring their properties and relationships with existing computability concepts.

Contribution

It defines new computability notions for real functions relative to operator classes and analyzes their preservation and local/global computability properties.

Findings

Conditional computability is preserved under substitution.

All conditionally computable functions with compact domains are uniformly computable.

The notions differ from existing non-uniform computability by Katrin Tent and Martin Ziegler.

Abstract

For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect to this class. These two computability notions are natural generalizations of certain notions introduced in a previous paper co-authored by Andreas Weiermann and in another previous paper by the same authors, respectively. Under certain weak assumptions about the class in question, we show that conditional computability is preserved by substitution, that all conditionally computable real functions are locally uniformly computable, and that the ones with compact domains are uniformly computable. The introduced notions have some similarity with the uniform computability and its non-uniform extension considered by Katrin Tent and Martin Ziegler, however, there are also essential differences between the conditional computability and the non-uniform computability in question.