Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling

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.

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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. We would also like to thank the anonymous reviewers for suggestions that improved the presentation of this work, and in particular for suggesting the perspective of totally unimodular matrices.

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. 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. Susanna F. de Rezende and Jakob Nordström were supported by the Knut and Alice Wallenberg grant KAW 2016.0066 Approximation and Proof Complexity. Jakob Nordström was also supported by the Swedish Research Council grant 2016-00782 and by the Independent Research Fund Denmark grant 9040-00389B. A preliminary version (de Rezende et al. 2019) of this work appeared in CCC 2019.

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De Rezende, S.F., Meir, O., Nordström, J. et al. Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling. comput. complex. 30, 4 (2021). https://doi.org/10.1007/s00037-020-00201-y

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Keywords

  • Proof complexity
  • Nullstellensatz
  • Trade-offs
  • Pebbling

Subject classification

  • 68Q17