License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/LIPIcs.STACS.2022.54
URN: urn:nbn:de:0030-drops-158646
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15864/
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Ren, Hanlin ; Santhanam, Rahul

A Relativization Perspective on Meta-Complexity

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LIPIcs-STACS-2022-54.pdf (0.8 MB)

Abstract

Meta-complexity studies the complexity of computational problems about complexity theory, such as the Minimum Circuit Size Problem (MCSP) and its variants. We show that a relativization barrier applies to many important open questions in meta-complexity. We give relativized worlds where:
1) MCSP can be solved in deterministic polynomial time, but the search version of MCSP cannot be solved in deterministic polynomial time, even approximately. In contrast, Carmosino, Impagliazzo, Kabanets, Kolokolova [CCC'16] gave a randomized approximate search-to-decision reduction for MCSP with a relativizing proof.
2) The complexities of MCSP[2^{n/2}] and MCSP[2^{n/4}] are different, in both worst-case and average-case settings. Thus the complexity of MCSP is not "robust" to the choice of the size function.
3) Levin’s time-bounded Kolmogorov complexity Kt(x) can be approximated to a factor (2+ε) in polynomial time, for any ε > 0.
4) Natural proofs do not exist, and neither do auxiliary-input one-way functions. In contrast, Santhanam [ITCS'20] gave a relativizing proof that the non-existence of natural proofs implies the existence of one-way functions under a conjecture about optimal hitting sets.
5) DistNP does not reduce to GapMINKT by a family of "robust" reductions. This presents a technical barrier for solving a question of Hirahara [FOCS'20].

BibTeX - Entry

@InProceedings{ren_et_al:LIPIcs.STACS.2022.54,
  author =	{Ren, Hanlin and Santhanam, Rahul},
  title =	{{A Relativization Perspective on Meta-Complexity}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{54:1--54:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/15864},
  URN =		{urn:nbn:de:0030-drops-158646},
  doi =		{10.4230/LIPIcs.STACS.2022.54},
  annote =	{Keywords: meta-complexity, relativization, minimum circuit size problem}
}

Keywords: meta-complexity, relativization, minimum circuit size problem
Collection: 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)
Issue Date: 2022
Date of publication: 09.03.2022

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