How powerful is the set of random strings? What can one say about a set A that is efficiently reducible to R, the set of Kolmogorov-random strings? We present the first upper bound on the class of computable sets in P R and NP R . The two most widely-studied notions of Kolmogorov complexity are the "plain" complexity C (x) and "prefix" complexity K (x); this gives rise to two common ways to define the set of random strings "R": R C and R K . (Of course, each different choice of universal Turing machine U in the definition of C and K yields another variant R CU or R KU .) Previous work on the power of "R" (for any of these variants) has shown:Since these inclusions hold irrespective of low-level details of how "R" is defined, and since BPP, PSPACE and NEXP are all in 0 1 (the class of decidable languages), we have, e.g.:Our main contribution is to present the first upper bounds on the complexity of sets that are efficiently reducible to R KU . We show:Hence, in particular, PSPACE is sandwiched between the class of sets polynomial-time Turing-and truth-table-reducible to R. As a side-product, we obtain new insight into the limits of techniques for derandomization from uniform hardness assumptions.

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Abstract. This paper is motivated by a conjecture [All12, ADF + 13] that BPP can be characterized in terms of polynomial-time 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 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.

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Abstract. This paper is motivated by a conjecture [All12, ADF+ 13] that BPP can be characterized in terms of polynomial-time 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 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.