The power of negative reasoning
(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.
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- author
- de Rezende, Susanna F. LU ; Lauria, Massimo ; Nordström, Jakob LU and Sokolov, Dmitry LU
- organization
- publishing date
- 2021-07-01
- 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}}, }