Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling

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

Research output: Contribution to journalArticlepeer-review


We establish an exactly tight relation between reversiblepebblings 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 formulaover G in size t + 1 and degree s (independently of the field in whichthe Nullstellensatz refutation is made). We use this correspondenceto prove a number of strong size-degree trade-offs for Nullstellensatz,which to the best of our knowledge are the first such results for thisproof system.

Original languageEnglish
Article number4
JournalComputational Complexity
Issue number1
StatePublished - Jun 2021

Bibliographical note

Funding Information:
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.

Publisher Copyright:
© 2021, Springer Nature Switzerland AG.


  • 68Q17
  • Nullstellensatz
  • Pebbling
  • Proof complexity
  • Trade-offs

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Mathematics (all)
  • Computational Theory and Mathematics
  • Computational Mathematics


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