Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

The power of negative reasoning

de Rezende, Susanna F. LU orcid ; Lauria, Massimo ; Nordström, Jakob LU and Sokolov, Dmitry LU (2021) 36th Computational Complexity Conference, CCC 2021 In Leibniz International Proceedings in Informatics, LIPIcs 200.
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... (More)

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.

(Less)
Please use this url to cite or link to this publication:
author
; ; and
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
keywords
Nullstellensatz, Polynomial calculus, Proof complexity, Sherali-Adams, Sums-of-squares
host publication
36th Computational Complexity Conference, CCC 2021
series title
Leibniz International Proceedings in Informatics, LIPIcs
editor
Kabanets, Valentine
volume
200
article number
40
publisher
Schloss Dagstuhl - Leibniz-Zentrum für Informatik
conference name
36th Computational Complexity Conference, CCC 2021
conference location
Virtual, Toronto, Canada
conference dates
2021-07-20 - 2021-07-23
external identifiers
  • scopus:85115341992
ISSN
1868-8969
ISBN
9783959771931
DOI
10.4230/LIPIcs.CCC.2021.40
language
English
LU publication?
yes
id
ef942d8f-e46c-454d-a9d2-f957e12d252d
date added to LUP
2021-10-04 11:45:14
date last changed
2022-04-27 04:26:42
@inproceedings{ef942d8f-e46c-454d-a9d2-f957e12d252d,
  abstract     = {{<p>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.</p>}},
  author       = {{de Rezende, Susanna F. and Lauria, Massimo and Nordström, Jakob and Sokolov, Dmitry}},
  booktitle    = {{36th Computational Complexity Conference, CCC 2021}},
  editor       = {{Kabanets, Valentine}},
  isbn         = {{9783959771931}},
  issn         = {{1868-8969}},
  keywords     = {{Nullstellensatz; Polynomial calculus; Proof complexity; Sherali-Adams; Sums-of-squares}},
  language     = {{eng}},
  month        = {{07}},
  publisher    = {{Schloss Dagstuhl - Leibniz-Zentrum für Informatik}},
  series       = {{Leibniz International Proceedings in Informatics, LIPIcs}},
  title        = {{The power of negative reasoning}},
  url          = {{http://dx.doi.org/10.4230/LIPIcs.CCC.2021.40}},
  doi          = {{10.4230/LIPIcs.CCC.2021.40}},
  volume       = {{200}},
  year         = {{2021}},
}