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RANDOM STRINGS AND TRUTHTABLE DEGREES OF TURING COMPLETE C.E. SETS
"... Abstract. We investigate the truthtable 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 prefixfree machine is Turing complete, but that truthtable completeness depends on the choice of universal machine. W ..."
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Abstract. We investigate the truthtable 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 prefixfree machine is Turing complete, but that truthtable completeness depends on the choice of universal machine. We show that for such sets of random strings, any finite set of their truthtable degrees do not meet to the degree 0, even within the c.e. truthtable degrees, but when taking the meet over all such truthtable 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 truthtable degrees form a minimal pair. 1.
Reductions to the set of random strings: The resourcebounded case
 in Proc. 37th International Symposium on Mathematical Foundations of Computer Science (MFCS ’12), 2012, Lecture Notes in Computer Science
"... ABSTRACT. This paper is motivated by a conjecture [All12, ADF+13] that BPP can be characterized in terms of polynomialtime nonadaptive reductions to the set of Kolmogorovrandom strings. In this paper we show that an approach laid out in [ADF+13] to settle this conjecture cannot succeed without sig ..."
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ABSTRACT. This paper is motivated by a conjecture [All12, ADF+13] that BPP can be characterized in terms of polynomialtime nonadaptive reductions to the set of Kolmogorovrandom strings. In this paper we show that an approach laid out in [ADF+13] to settle this conjecture cannot succeed without significant alteration, but that it does bear fruit if we consider timebounded Kolmogorov complexity instead. We show that if a setA is reducible in polynomial time to the set of timetbounded Kolmogorovrandom strings (for all large enough time bounds t), then A is in P/poly, and that if in addition such a reduction exists for any universal Turing machine one uses in the definition of Kolmogorov complexity, then A is in PSPACE. 1.
REDUCTIONS TO THE SET OF RANDOM STRINGS:
, 2012
"... Vol. 10(3:5)2014, pp. 1–18 www.lmcsonline.org ..."