Kolmogorov complexity and computably enumerable sets

TL;DR

This paper explores the relationship between Kolmogorov complexity and computably enumerable sets, providing a survey of existing research and a new characterization of K-trivial sets in terms of their complexity and computability.

Contribution

It offers a new characterization of K-trivial c.e. sets via their complexity bounds and computability from the halting problem, extending understanding of their structure.

Findings

K-trivial sets are characterized by their initial segment complexity.

The family of sets with complexity bounded by a K-trivial set is uniformly computable from the halting problem.

Provides an extended discussion and open problems in the field.

Abstract

We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent developments and open problems. Besides this survey, our main original result is the following characterization of the computably enumerable sets with trivial initial segment prefix-free complexity. A computably enumerable set $A$ is $K$-trivial if and only if the family of sets with complexity bounded by the complexity of $A$ is uniformly computable from the halting problem.