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Unexpected Power of Random Strings

Author Shuichi Hirahara

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

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Abstract

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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.ITCS.2020.41

Author Details

Shuichi Hirahara
  • National Institute of Informatics, Tokyo, Japan

Funding

  • Hirahara, Shuichi: ACT-I, JST and JSPS KAKENHI Grant Number 18H04090.

Acknowledgements

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.

References

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