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arXiv · 1408.5636

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

Mingzhong Cai·Rodney G Downey·Rachel Epstein·Steffen Lempp·Joseph Miller
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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.

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Mingzhong Cai, Rodney G Downey, Rachel Epstein, Steffen Lempp, Joseph Miller. 2014-09-08. Random strings and tt-degrees of Turing complete C.E. sets. https://doi.org/10.2168/lmcs-10(3%3A15)2014

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