Limits on the Computational Power of Random Strings Eric Allender ∗ Dept. of Computer Science Rutgers University New Brunswick, NJ 08855, USA allender@cs.rutgers.edu Luke Friedman ∗ Dept. of Computer Science Rutgers University New Brunswick, NJ 08855, USA lbfried@cs.rutgers.edu William Gasarch Dept. of Computer Science University of Maryland College Park, MD, 20742 gasarch@cs.umd.edu September 2, 2010 Abstract Let C(x) and K(x) denote plain and prefix Kolmogorov complexity, respectively, and let R C and R K denote the sets of strings that are “random” according to these measures; both R K and R C are undecidable. Earlier work has shown that every set in NEXP is in NP relative to both R K and R C , and that every set in BPP is polynomial-time truth-table reducible to both R K and R C [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 C U or K U , these inclusions can be stated as: • BPP ⊆ DEC ∩  U {A : A≤ p tt R CU }. • NEXP ⊆ DEC ∩  U NP RC U . • BPP ⊆ DEC ∩  U {A : A≤ p tt R KU }. • NEXP ⊆ DEC ∩  U NP RK U . (Here, “DEC” denotes the class of decidable sets.) It remains unknown whether DEC is equal to  U {A : A≤ p tt R CU }. In this paper, we present the first upper bounds on the complexity of sets that are efficiently reducible to R KU . We show: • BPP ⊆ DEC ∩  U {A : A≤ p tt R KU }⊆ PSPACE . • NEXP ⊆ DEC ∩  U NP RK U ⊆ EXPSPACE. This also provides the first quantitative limits on the applicability of uniform derandomiza- tion techniques. ∗ Supported in part by NSF Grants CCF-0830133 and CCF-0832787. 1