Limits on the Computational Power of Random Strings

Abstract

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 PR and NPR. 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”: RC and RK. (Of course, each different choice of universal Turing machine U in the definition of C andK yields another variantRCU or RKU.) Previous work on the power of “R” (for any of these variants) has shown • BPP ⊆ {A: A≤pttR}. • PSPACE ⊆ PR. • NEXP ⊆ NPR. Since these inclusions hold irrespective of low-level details of how “R ” is defined, and since BPP,PSPACE and NEXP are all in ∆01 (the class of decidable languages), we have, e.g.: NEXP ⊆ ∆01 ∩ U NP RKU. Our main contribution is to present the first upper bounds on the complexity of sets that are efficiently reducible to RKU. We show: • BPP ⊆ ∆01