Derandomizing from Random Strings

2009/12/16 by Buhrman, Harry, Fortnow, Lance, Koucký, Michal +1
#Computational Complexity (cs.CC) #F.1.2 #FOS: Computer and information sciences

paper · doi:10.48550/arxiv.0912.3162

Abstract

In this paper we show that BPP is truth-table reducible to the set of Kolmogorov random strings R_K. It was previously known that PSPACE, and hence BPP is Turing-reducible to R_K. The earlier proof relied on the adaptivity of the Turing-reduction to find a Kolmogorov-random string of polynomial length using the set R_K as oracle. Our new non-adaptive result relies on a new fundamental fact about the set R_K, namely each initial segment of the characteristic sequence of R_K is not compressible by recursive means. As a partial converse to our claim we show that strings of high Kolmogorov-complexity when used as advice are not much more useful than randomly chosen strings.

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