Unexpected Power of Random Strings

Author Shuichi Hirahara

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Shuichi Hirahara
  • National Institute of Informatics, Tokyo, Japan


I thank Eric Allender, Michal Koucký, and Osamu Watanabe for helpful discussions, and thank anonymous reviewers for helpful comments. I got the ideas of this work during the joint work with Osamu Watanabe.

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Shuichi Hirahara. Unexpected Power of Random Strings. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 41:1-41:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


There has been a line of work trying to characterize BPP (the class of languages that are solvable by efficient randomized algorithms) by efficient nonadaptive reductions to the set of Kolmogorov-random strings: Buhrman, Fortnow, Koucký, and Loff (CCC 2010 [Buhrman et al., 2010]) showed that every language in BPP is reducible to the set of random strings via a polynomial-time nonadaptive reduction (irrespective of the choice of a universal Turing machine used to define Kolmogorov-random strings). It was conjectured by Allender (CiE 2012 [Allender, 2012]) and others that their lower bound is tight when a reduction works for every universal Turing machine; i.e., "the only way to make use of random strings by a nonadaptive polynomial-time algorithm is to derandomize BPP." In this paper, we refute this conjecture under the plausible assumption that the exponential-time hierarchy does not collapse, by showing that the exponential-time hierarchy EXPH can be solved in exponential time by nonadaptively asking the oracle whether a string is Kolmogorov-random or not. In addition, we provide an exact characterization of S_2^{exp} in terms of exponential-time-computable nonadaptive reductions to arbitrary dense subsets of random strings.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Problems, reductions and completeness
  • Kolmogorov-Randomness
  • Nonadaptive Reduction
  • BPP
  • Symmetric Alternation


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