Random strings and tt-degrees of Turing complete C.E. sets
Logic in Computer Science
2015-07-01 v2
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.
Cite
@article{arxiv.1408.5636,
title = {Random strings and tt-degrees of Turing complete C.E. sets},
author = {Mingzhong Cai and Rodney G Downey and Rachel Epstein and Steffen Lempp and Joseph Miller},
journal= {arXiv preprint arXiv:1408.5636},
year = {2015}
}
Comments
25 pages