Random strings and tt-degrees of Turing complete C.E. sets

TL;DR

This paper studies the truth-table degrees of c.e. sets of random strings, showing that finite meets of their degrees do not reach degree 0, but the infinite meet does, confirming a conjecture and revealing minimal pairs among Turing complete sets.

Contribution

It proves that the infinite meet of truth-table degrees of random string c.e. sets is 0, and demonstrates the existence of Turing complete c.e. sets with minimal pairs in their truth-table degrees.

Findings

Finite meets of truth-table degrees of random string c.e. sets do not reach degree 0.

The infinite meet of these degrees is 0, confirming a conjecture.

Existence of Turing complete c.e. sets with minimal pairs in truth-table degrees.

Abstract

We investigate the truth-table degrees of (co-)c.e.\ sets, in particular, sets of random strings. It is known that the set of random strings with respect to any universal prefix-free machine is Turing complete, but that truth-table completeness depends on the choice of universal machine. We show that for such sets of random strings, any finite set of their truth-table degrees do not meet to the degree~0, even within the c.e. truth-table degrees, but when taking the meet over all such truth-table degrees, the infinite meet is indeed~0. The latter result proves a conjecture of Allender, Friedman and Gasarch. We also show that there are two Turing complete c.e. sets whose truth-table degrees form a minimal pair.