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Proving Unsatisfiability with Hitting Formulas

Authors Yuval Filmus , Edward A. Hirsch , Artur Riazanov , Alexander Smal , Marc Vinyals

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ACM Subject Classification
  • Theory of computation → Proof complexity
Keywords
  • hitting formulas
  • polynomial identity testing
  • query complexity

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Abstract

A hitting formula is a set of Boolean clauses such that any two of the clauses cannot be simultaneously falsified. Hitting formulas have been studied in many different contexts at least since [Iwama, 1989] and, based on experimental evidence, Peitl and Szeider [Tomás Peitl and Stefan Szeider, 2022] conjectured that unsatisfiable hitting formulas are among the hardest for resolution. Using the fact that hitting formulas are easy to check for satisfiability we make them the foundation of a new static proof system {{rmHitting}}: a refutation of a CNF in {{rmHitting}} is an unsatisfiable hitting formula such that each of its clauses is a weakening of a clause of the refuted CNF. Comparing this system to resolution and other proof systems is equivalent to studying the hardness of hitting formulas.
Our first result is that {{rmHitting}} is quasi-polynomially simulated by tree-like resolution, which means that hitting formulas cannot be exponentially hard for resolution and partially refutes the conjecture of Peitl and Szeider. We show that tree-like resolution and {{rmHitting}} are quasi-polynomially separated, while for resolution, this question remains open. For a system that is only quasi-polynomially stronger than tree-like resolution, {{rmHitting}} is surprisingly difficult to polynomially simulate in another proof system. Using the ideas of Raz-Shpilka’s polynomial identity testing for noncommutative circuits [Raz and Shpilka, 2005] we show that {{rmHitting}} is p-simulated by {{rmExtended {{rmFrege}}}}, but we conjecture that much more efficient simulations exist. As a byproduct, we show that a number of static (semi)algebraic systems are verifiable in deterministic polynomial time.
We consider multiple extensions of {{rmHitting}}, and in particular a proof system {{{rmHitting}}(⊕)} related to the {{{rmRes}}(⊕)} proof system for which no superpolynomial-size lower bounds are known. {{{rmHitting}}(⊕)} p-simulates the tree-like version of {{{rmRes}}(⊕)} and is at least quasi-polynomially stronger. We show that formulas expressing the non-existence of perfect matchings in the graphs K_{n,n+2} are exponentially hard for {{{rmHitting}}(⊕)} via a reduction to the partition bound for communication complexity.
See the full version of the paper for the proofs. They are omitted in this Extended Abstract.

Cite As Get BibTex

Yuval Filmus, Edward A. Hirsch, Artur Riazanov, Alexander Smal, and Marc Vinyals. Proving Unsatisfiability with Hitting Formulas. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 48:1-48:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ITCS.2024.48

Author Details

Yuval Filmus
  • Technion - Israel Institute of Technology, Haifa, Israel
Edward A. Hirsch
  • Department of Computer Science, Ariel University, Israel
Artur Riazanov
  • EPFL, Lausanne, Switzerland
Alexander Smal
  • Technion - Israel Institute of Technology, Haifa, Israel
Marc Vinyals
  • University of Auckland, New Zealand

Funding

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 802020-ERC-HARMONIC.
  • Riazanov, Artur: This work was supported by the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract number MB22.00026.

Acknowledgements

We thank Jan Johannsen, Ilario Bonacina, Oliver Kullmann, and Stefan Szeider for introducing us to the topic; Zachary Chase, Susanna de Rezende, Mika Göös, Amir Shpilka, and Dmitry Sokolov for helpful discussions.

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