Universal computably enumerable sets and initial segment prefix-free complexity

TL;DR

This paper demonstrates the existence of Turing complete computably enumerable sets with arbitrarily low initial segment prefix-free complexity, challenging previous assumptions and revealing distinctions among various degrees of randomness.

Contribution

It proves that Turing complete c.e. sets can have low complexity and generalizes this to finite collections, providing negative answers to open questions about minimal pairs in complexity degrees.

Findings

Existence of Turing complete c.e. sets with low initial segment complexity

Negative answer to minimal pairs question in c.e. reals complexity structure

Differences among degrees of randomness and complexity notions

Abstract

We show that there are Turing complete computably enumerable sets of arbitrarily low non-trivial initial segment prefix-free complexity. In particular, given any computably enumerable set $A$ with non-trivial prefix-free initial segment complexity, there exists a Turing complete computably enumerable set $B$ with complexity strictly less than the complexity of $A$. On the other hand it is known that sets with trivial initial segment prefix-free complexity are not Turing complete. Moreover we give a generalization of this result for any finite collection of computably enumerable sets $A_i, i<k$ with non-trivial initial segment prefix-free complexity. An application of this gives a negative answer to a question from \cite[Section 11.12]{rodenisbook} and \cite{MRmerstcdhdtd} which asked for minimal pairs in the structure of the c.e.\ reals ordered by their initial segment prefix-free complexity. Further consequences concern various notions of degrees of randomness. For example, the Solovay degrees and the $K$-degrees of computably enumerable reals and computably enumerable sets are not elementarily equivalent. Also, the degrees of randomness based on plain and prefix-free complexity are not elementarily equivalent; the same holds for their $\Delta^0_2$ and $\Sigma^0_1$ substructures.