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.CCC.2016.18
URN: urn:nbn:de:0030-drops-58426
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5842/
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Hirahara, Shuichi ; Watanabe, Osamu

Limits of Minimum Circuit Size Problem as Oracle

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Abstract

The Minimum Circuit Size Problem (MCSP) is known to be hard for statistical zero knowledge via a BPP-Turing reduction (Allender and Das, 2014), whereas establishing NP-hardness of MCSP via a polynomial-time many-one reduction is difficult (Murray and Williams, 2015) in the sense that it implies ZPP != EXP, which is a major open problem in computational complexity.

In this paper, we provide strong evidence that current techniques cannot establish NP-hardness of MCSP, even under polynomial-time Turing reductions or randomized reductions: Specifically, we introduce the notion of oracle-independent reduction to MCSP, which captures all the currently known reductions. We say that a reduction to MCSP is oracle-independent if the reduction can be generalized to a reduction to MCSP^A for any oracle A, where MCSP^A denotes an oracle version of MCSP. We prove that no language outside P is reducible to MCSP via an oracle-independent polynomial-time Turing reduction. We also show that the class of languages reducible to MCSP via an oracle-independent randomized reduction that makes at most one query is contained in AM intersect coAM. Thus, NP-hardness of MCSP cannot be established via such oracle-independent reductions unless the polynomial hierarchy collapses.

We also extend the previous results to the case of more general reductions: We prove that establishing NP-hardness of MCSP via a polynomial-time nonadaptive reduction implies ZPP != EXP, and that establishing NP-hardness of approximating circuit complexity via a polynomial-time Turing reduction also implies ZPP != EXP. Along the way, we prove that approximating Levin's Kolmogorov complexity is provably not EXP-hard under polynomial-time Turing reductions, which is of independent interest.

BibTeX - Entry

@InProceedings{hirahara_et_al:LIPIcs:2016:5842,
  author =	{Shuichi Hirahara and Osamu Watanabe},
  title =	{{Limits of Minimum Circuit Size Problem as Oracle}},
  booktitle =	{31st Conference on Computational Complexity (CCC 2016)},
  pages =	{18:1--18:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-008-8},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{50},
  editor =	{Ran Raz},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5842},
  URN =		{urn:nbn:de:0030-drops-58426},
  doi =		{10.4230/LIPIcs.CCC.2016.18},
  annote =	{Keywords: minimum circuit size problem, NP-completeness, randomized reductions, resource-bounded Kolmogorov complexity, Turing reductions}
}

Keywords: minimum circuit size problem, NP-completeness, randomized reductions, resource-bounded Kolmogorov complexity, Turing reductions
Collection: 31st Conference on Computational Complexity (CCC 2016)
Issue Date: 2016
Date of publication: 19.05.2016

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