Reductions to the Set of Random Strings: The Resource-Bounded Case

This paper is motivated by a conjecture [All12, ADF13] 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 [ADF13] 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 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.