Nullstellensatz size-degree trade-offs from reversible pebbling

Susanna F. De Rezende; Jakob Nordström; Or Meir; Robert Robere

doi:10.4230/LIPIcs.CCC.2019.18

Nullstellensatz size-degree trade-offs from reversible pebbling

Susanna F. De Rezende, Jakob Nordström, Or Meir, Robert Robere

Department of Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

We establish an exactly tight relation between reversible pebblings of graphs and Nullstellensatz refutations of pebbling formulas, showing that a graph G can be reversibly pebbled in time t and space s if and only if there is a Nullstellensatz refutation of the pebbling formula over G in size t + 1 and degree s (independently of the field in which the Nullstellensatz refutation is made). We use this correspondence to prove a number of strong size-degree trade-offs for Nullstellensatz, which to the best of our knowledge are the first such results for this proof system.

Original language	English
Title of host publication	34th Computational Complexity Conference, CCC 2019
Editors	Amir Shpilka
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages	18:1–18:16
ISBN (Electronic)	9783959771160
DOIs	https://doi.org/10.4230/LIPIcs.CCC.2019.18
State	Published - 1 Jul 2019
Event	34th Computational Complexity Conference, CCC 2019 - New Brunswick, United States Duration: 18 Jul 2019 → 20 Jul 2019

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	137
ISSN (Print)	1868-8969

Conference

Conference	34th Computational Complexity Conference, CCC 2019
Country/Territory	United States
City	New Brunswick
Period	18/07/19 → 20/07/19

Bibliographical note

Funding Information:
Jakob Nordstr?m: was supported by the Knut and Alice Wallenberg grant KAW 2016.0066 Approximation and Proof Complexity and by the Swedish Research Council grants 621-2012-5645 and 2016-00782. Or Meir: was supported by the Israel Science Foundation (grant No. 1445/16). Robert Robere: was supported by NSERC, and also conducted part of this work at DIMACS with support from the National Science Foundation under grant number CCF-1445755.

Funding Information:
Funding This work was mostly carried out while the authors were visiting the Simons Institute for the Theory of Computing in association with the DIMACS/Simons Collaboration on Lower Bounds in Computational Complexity, which is conducted with support from the National Science Foundation. Susanna F. de Rezende: was supported by the Knut and Alice Wallenberg grant KAW 2016.0066 Approximation and Proof Complexity. Jakob Nordström: was supported by the Knut and Alice Wallenberg grant KAW 2016.0066 Approximation and Proof Complexity and by the Swedish Research Council grants 621-2012-5645 and 2016-00782. Or Meir: was supported by the Israel Science Foundation (grant No. 1445/16). Robert Robere: was supported by NSERC, and also conducted part of this work at DIMACS with support from the National Science Foundation under grant number CCF-1445755.

Publisher Copyright:
© Susanna F. de Rezende, Jakob Nordström, Or Meir, and Robert Robere; licensed under Creative Commons License CC-BY 34th Computational Complexity Conference (CCC 2019).

Keywords

Degree
Nullstellensatz
Pebble games
Proof complexity
Size
Trade-offs

ASJC Scopus subject areas

Software

Access to Document

10.4230/LIPIcs.CCC.2019.18

Cite this

De Rezende, S. F., Nordström, J., Meir, O., & Robere, R. (2019). Nullstellensatz size-degree trade-offs from reversible pebbling. In A. Shpilka (Ed.), 34th Computational Complexity Conference, CCC 2019 (pp. 18:1–18:16). (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 137). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.CCC.2019.18

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note = "Funding Information: Jakob Nordstr?m: was supported by the Knut and Alice Wallenberg grant KAW 2016.0066 Approximation and Proof Complexity and by the Swedish Research Council grants 621-2012-5645 and 2016-00782. Or Meir: was supported by the Israel Science Foundation (grant No. 1445/16). Robert Robere: was supported by NSERC, and also conducted part of this work at DIMACS with support from the National Science Foundation under grant number CCF-1445755. Funding Information: Funding This work was mostly carried out while the authors were visiting the Simons Institute for the Theory of Computing in association with the DIMACS/Simons Collaboration on Lower Bounds in Computational Complexity, which is conducted with support from the National Science Foundation. Susanna F. de Rezende: was supported by the Knut and Alice Wallenberg grant KAW 2016.0066 Approximation and Proof Complexity. Jakob Nordstr{\"o}m: was supported by the Knut and Alice Wallenberg grant KAW 2016.0066 Approximation and Proof Complexity and by the Swedish Research Council grants 621-2012-5645 and 2016-00782. Or Meir: was supported by the Israel Science Foundation (grant No. 1445/16). Robert Robere: was supported by NSERC, and also conducted part of this work at DIMACS with support from the National Science Foundation under grant number CCF-1445755. Publisher Copyright: {\textcopyright} Susanna F. de Rezende, Jakob Nordstr{\"o}m, Or Meir, and Robert Robere; licensed under Creative Commons License CC-BY 34th Computational Complexity Conference (CCC 2019).; 34th Computational Complexity Conference, CCC 2019 ; Conference date: 18-07-2019 Through 20-07-2019",

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De Rezende, SF, Nordström, J, Meir, O & Robere, R 2019, Nullstellensatz size-degree trade-offs from reversible pebbling. in A Shpilka (ed.), 34th Computational Complexity Conference, CCC 2019. Leibniz International Proceedings in Informatics, LIPIcs, vol. 137, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, pp. 18:1–18:16, 34th Computational Complexity Conference, CCC 2019, New Brunswick, United States, 18/07/19. https://doi.org/10.4230/LIPIcs.CCC.2019.18

Nullstellensatz size-degree trade-offs from reversible pebbling. / De Rezende, Susanna F.; Nordström, Jakob; Meir, Or et al.
34th Computational Complexity Conference, CCC 2019. ed. / Amir Shpilka. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2019. p. 18:1–18:16 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 137).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Meir, Or

AU - Robere, Robert

N1 - Funding Information: Jakob Nordstr?m: was supported by the Knut and Alice Wallenberg grant KAW 2016.0066 Approximation and Proof Complexity and by the Swedish Research Council grants 621-2012-5645 and 2016-00782. Or Meir: was supported by the Israel Science Foundation (grant No. 1445/16). Robert Robere: was supported by NSERC, and also conducted part of this work at DIMACS with support from the National Science Foundation under grant number CCF-1445755. Funding Information: Funding This work was mostly carried out while the authors were visiting the Simons Institute for the Theory of Computing in association with the DIMACS/Simons Collaboration on Lower Bounds in Computational Complexity, which is conducted with support from the National Science Foundation. Susanna F. de Rezende: was supported by the Knut and Alice Wallenberg grant KAW 2016.0066 Approximation and Proof Complexity. Jakob Nordström: was supported by the Knut and Alice Wallenberg grant KAW 2016.0066 Approximation and Proof Complexity and by the Swedish Research Council grants 621-2012-5645 and 2016-00782. Or Meir: was supported by the Israel Science Foundation (grant No. 1445/16). Robert Robere: was supported by NSERC, and also conducted part of this work at DIMACS with support from the National Science Foundation under grant number CCF-1445755. Publisher Copyright: © Susanna F. de Rezende, Jakob Nordström, Or Meir, and Robert Robere; licensed under Creative Commons License CC-BY 34th Computational Complexity Conference (CCC 2019).

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KW - Degree

KW - Nullstellensatz

KW - Pebble games

KW - Proof complexity

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KW - Trade-offs

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T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 18:1–18:16

BT - 34th Computational Complexity Conference, CCC 2019

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De Rezende SF, Nordström J, Meir O, Robere R. Nullstellensatz size-degree trade-offs from reversible pebbling. In Shpilka A, editor, 34th Computational Complexity Conference, CCC 2019. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2019. p. 18:1–18:16. (Leibniz International Proceedings in Informatics, LIPIcs). doi: 10.4230/LIPIcs.CCC.2019.18

Nullstellensatz size-degree trade-offs from reversible pebbling

Abstract

Publication series

Conference

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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