Universal computably enumerable sets and initial segment prefix-free complexity

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 $Δ^0_2$ and $Σ^0_1$ substructures.

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