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Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling

Authors Susanna F. de Rezende, Jakob Nordström, Or Meir, Robert Robere

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Author Details

Susanna F. de Rezende
  • KTH Royal Institute of Technology, Stockholm, Sweden
Jakob Nordström
  • University of Copenhagen, Denmark
  • KTH Royal Institute of Technology, Stockholm, Sweden
Or Meir
  • University of Haifa, Israel
Robert Robere
  • DIMACS, New Brunswick, U.S.A.

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.
  • de Rezende, Susanna F.: was supported by the Knut and Alice Wallenberg grant KAW 2016.0066 Approximation and Proof Complexity.
  • Nordström, Jakob: 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.
  • Meir, Or: was supported by the Israel Science Foundation (grant No. 1445/16).
  • Robere, Robert: 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.

Acknowledgements

We are grateful for many interesting discussions about matters pebbling-related (and not-so-pebbling-related) with Arkadev Chattopadhyay, Toniann Pitassi, and Marc Vinyals.

Cite AsGet BibTex

Susanna F. de Rezende, Jakob Nordström, Or Meir, and Robert Robere. Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.CCC.2019.18

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
Keywords
  • proof complexity
  • Nullstellensatz
  • pebble games
  • trade-offs
  • size
  • degree

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