Cai, Mingzhong and Downey, Rodney G and Epstein, Rachel and Lempp, Steffen and Miller, Joseph - Random strings and tt-degrees of Turing complete C.E. sets

lmcs:1126 - Logical Methods in Computer Science, September 10, 2014, Volume 10, Issue 3
Random strings and tt-degrees of Turing complete C.E. sets

Authors: Cai, Mingzhong and Downey, Rodney G and Epstein, Rachel and Lempp, Steffen and Miller, Joseph

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.


Source : oai:arXiv.org:1408.5636
DOI : 10.2168/LMCS-10(3:15)2014
Volume: Volume 10, Issue 3
Published on: September 10, 2014
Submitted on: June 25, 2015
Keywords: Computer Science - Logic in Computer Science


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