Fixpoints and relative precompleteness

TL;DR

This paper explores the relationship between relative precompleteness, lowness, and divisibility in numberings, revealing that for divisible numberings, these properties align with computability, and analyzes the complexity of Skolem functions in this context.

Contribution

It introduces a divisibility notion for numberings and demonstrates its implications for lowness, precompleteness, and the complexity of Skolem functions in computability theory.

Findings

Lowness and relative precompleteness coincide with computability for divisible numberings.

Skolem functions from Arslanov's criterion can have maximal Turing degree in divisible numberings.

Standard numberings of partial computable functions and c.e. sets exhibit maximal Skolem function complexity.

Abstract

We study relative precompleteness in the context of the theory of numberings, and relate this to a notion of lowness. We introduce a notion of divisibility for numberings, and use it to show that for the class of divisible numberings, lowness and relative precompleteness coincide with being computable. We also study the complexity of Skolem functions arising from Arslanov's completeness criterion with parameters. We show that for suitably divisible numberings, these Skolem functions have the maximal possible Turing degree. In particular this holds for the standard numberings of the partial computable functions and the c.e. sets.