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Mingzhong Cai ; Rodney G Downey ; Rachel Epstein ; Steffen Lempp ; Joseph Miller - 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 - https://doi.org/10.2168/LMCS-10(3:15)2014
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.
Comment: 25 pages
- Source : OpenAIRE Graph
- Randomness, Dimension and Computability; Funder: National Science Foundation; Code: 1001847
- Recursion Theory and Its Applications; Funder: National Science Foundation; Code: 1266214
Classifications
- 03D25 - Recursively (computably) enumerable sets and degrees
- 03D30 - Other degrees and reducibilities in computability and recursion theory
- 03D32 - Algorithmic randomness and dimension
- 68Q15 - Complexity classes (hierarchies, relations among complexity classes, etc.)
- 68Q30 - Algorithmic information theory (Kolmogorov complexity, etc.)