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Polynomial Calculus for Quantified Boolean Logic: Lower Bounds Through Circuits and Degree

Authors Olaf Beyersdorff , Tim Hoffmann , Kaspar Kasche , Luc Nicolas Spachmann

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Author Details

Olaf Beyersdorff
  • Friedrich Schiller University Jena, Germany
Tim Hoffmann
  • Friedrich Schiller University Jena, Germany
Kaspar Kasche
  • Friedrich Schiller University Jena, Germany
Luc Nicolas Spachmann
  • Friedrich Schiller University Jena, Germany

Funding

  • Beyersdorff, Olaf: Carl-Zeiss Foundation and DFG grant BE 4209/3-1
  • Hoffmann, Tim: Carl-Zeiss Foundation
  • Kasche, Kaspar: Carl-Zeiss Foundation

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Olaf Beyersdorff, Tim Hoffmann, Kaspar Kasche, and Luc Nicolas Spachmann. Polynomial Calculus for Quantified Boolean Logic: Lower Bounds Through Circuits and Degree. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.27

Abstract

We initiate an in-depth proof-complexity analysis of polynomial calculus (𝒬-PC) for Quantified Boolean Formulas (QBF). In the course of this we establish a tight proof-size characterisation of 𝒬-PC in terms of a suitable circuit model (polynomial decision lists). Using this correspondence we show a size-degree relation for 𝒬-PC, similar in spirit, yet different from the classic size-degree formula for propositional PC by Impagliazzo, Pudlák and Sgall (1999).
We use the circuit characterisation together with the size-degree relation to obtain various new lower bounds on proof size in 𝒬-PC. This leads to incomparability results for 𝒬-PC systems over different fields.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
Keywords
  • proof complexity
  • QBF
  • polynomial calculus
  • circuits
  • lower bounds

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