Limits on the computational power of random strings

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 or R_KU.) Previous work on the power of "R" (for any of these variants) has shown:BPP⊆{A:A≤_tt^pR}. PSPACE⊆P^R.NEXP⊆NP^R. Since these inclusions hold irrespective of low-level details of how "R" is defined, and since BPP,PSPACE and NEXP are all in Δ₁⁰ (the class of decidable languages), we have, e.g.: NEXP⊆Δ₁ ⁰∩∩_UNP^RKU. Our main contribution is to present the first upper bounds on the complexity of sets that are efficiently reducible to R_KU. We show:BPP⊆Δ₁ ⁰∩∩_U{A:A≤_ttpR_KU} ⊆PSPACE.NEXP⊆Δ₁⁰∩∩ _UNP^RKU⊆EXPSPACE. 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.