License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2020.41
URN: urn:nbn:de:0030-drops-117262
URL: https://drops.dagstuhl.de/opus/volltexte/2020/11726/
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Hirahara, Shuichi

Unexpected Power of Random Strings

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LIPIcs-ITCS-2020-41.pdf (0.5 MB)

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.

BibTeX - Entry

@InProceedings{hirahara:LIPIcs:2020:11726,
  author =	{Shuichi Hirahara},
  title =	{{Unexpected Power of Random Strings}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{41:1--41:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Thomas Vidick},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/11726},
  URN =		{urn:nbn:de:0030-drops-117262},
  doi =		{10.4230/LIPIcs.ITCS.2020.41},
  annote =	{Keywords: Kolmogorov-Randomness, Nonadaptive Reduction, BPP, Symmetric Alternation}
}

Keywords: Kolmogorov-Randomness, Nonadaptive Reduction, BPP, Symmetric Alternation
Collection: 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)
Issue Date: 2020
Date of publication: 06.01.2020

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