License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2021.40
URN: urn:nbn:de:0030-drops-143140
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14314/
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de Rezende, Susanna F. ; Lauria, Massimo ; Nordström, Jakob ; Sokolov, Dmitry

The Power of Negative Reasoning

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LIPIcs-CCC-2021-40.pdf (0.7 MB)

Abstract

Semialgebraic proof systems have been studied extensively in proof complexity since the late 1990s to understand the power of Gröbner basis computations, linear and semidefinite programming hierarchies, and other methods. Such proof systems are defined alternately with only the original variables of the problem and with special formal variables for positive and negative literals, but there seems to have been no study how these different definitions affect the power of the proof systems. We show for Nullstellensatz, polynomial calculus, Sherali-Adams, and sums-of-squares that adding formal variables for negative literals makes the proof systems exponentially stronger, with respect to the number of terms in the proofs. These separations are witnessed by CNF formulas that are easy for resolution, which establishes that polynomial calculus, Sherali-Adams, and sums-of-squares cannot efficiently simulate resolution without having access to variables for negative literals.

BibTeX - Entry

@InProceedings{derezende_et_al:LIPIcs.CCC.2021.40,
  author =	{de Rezende, Susanna F. and Lauria, Massimo and Nordstr\"{o}m, Jakob and Sokolov, Dmitry},
  title =	{{The Power of Negative Reasoning}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{40:1--40:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14314},
  URN =		{urn:nbn:de:0030-drops-143140},
  doi =		{10.4230/LIPIcs.CCC.2021.40},
  annote =	{Keywords: Proof complexity, Polynomial calculus, Nullstellensatz, Sums-of-squares, Sherali-Adams}
}

Keywords: Proof complexity, Polynomial calculus, Nullstellensatz, Sums-of-squares, Sherali-Adams
Collection: 36th Computational Complexity Conference (CCC 2021)
Issue Date: 2021
Date of publication: 08.07.2021

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