Left computably enumerable reals and initial segment complexity

TL;DR

This paper investigates the relationship between left computably enumerable reals and their initial segment complexity, highlighting the role of a specific complexity measure in their computability and completeness.

Contribution

It introduces a new perspective on initial segment complexity and its influence on the computability and completeness of left c.e. reals, extending Chaitin's theorem.

Findings

The complexity measure $C(C(eta_n)|eta_n)$ is crucial for understanding computability.

Relativisation of Chaitin's theorem provides new insights.

Results clarify the role of initial segment complexity in left c.e. reals.

Abstract

We are interested in the computability between left c.e. reals $\alpha$ and their initial segments. We show that the quantity $C(C(\alpha_n)|\alpha_n)$ plays a crucial role in this and in their completeness. We look in particular at Chaitin's theorem and its relativisation due to Frank Stephan.