Limits on the Computational Power of Random Strings

Let C(x) and K(x) denote plain and prefix Kolmogorov complexity, respectively, and let RC and RK denote the sets of strings that are “random” according to these measures; both RK and RC are undecidable. Earlier work has shown that every set in NEXP is in NP relative to both RK and RC , and that every set in BPP is polynomial-time truth-table reducible to both RK and RC [ABK06a, BFKL10]. (All of these inclusions hold, no matter which “universal” Turing machine one uses in the definitions of C(x) and K(x).) Since each machine U gives rise to a slightly different measure CU or KU , these inclusions can be stated as: • BPP ⊆ DEC ∩U{A : A≤ttRCU }. • NEXP ⊆ DEC ∩U NPCU . • BPP ⊆ DEC ∩U{A : A≤ttRKU }. • NEXP ⊆ DEC ∩U NPKU . (Here, “DEC” denotes the class of decidable sets.) It remains unknown whether DEC is equal to ⋂ U{A : A≤ttRCU }. In this paper, we present the first upper bounds on the complexity of sets that are efficiently reducible to RKU . We show: • BPP ⊆ DEC ∩U{A : A≤ttRKU } ⊆ PSPACE . • NEXP ⊆ DEC ∩U NPKU ⊆ EXPSPACE. This also provides the first quantitative limits on the applicability of uniform derandomization techniques. ∗Supported in part by NSF Grants CCF-0830133 and CCF-0832787.