Reductions to the set of random strings: the resource-bounded case

Authors:
Eric Allender;Harry Buhrman;Luke Friedman;Bruno Loff
Affiliations:
Department of Computer Science, Rutgers University, Piscataway, NJ;CWI and University of Amsterdam, The Netherlands;Department of Computer Science, Rutgers University, Piscataway, NJ;CWI, The Netherlands
Venue:
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
Year:
2012

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Abstract

This paper is motivated by a conjecture [1,5] that BPP can be characterized in terms of polynomial-time nonadaptive reductions to the set of Kolmogorov-random strings. In this paper we show that an approach laid out in [5] to settle this conjecture cannot succeed without significant alteration, but that it does bear fruit if we consider time-bounded Kolmogorov complexity instead. We show that if a set A is reducible in polynomial time to the set of time-t-bounded Kolmogorov-random 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.