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Polynomial Calculus for MaxSAT

Authors Ilario Bonacina , Maria Luisa Bonet , Jordi Levy

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Ilario Bonacina
  • Polytechnic University of Catalonia, Barcelona, Spain
Maria Luisa Bonet
  • Polytechnic University of Catalonia, Barcelona, Spain
Jordi Levy
  • Artificial Intelligence Research Institute, Spanish Research Council (IIIA-CSIC), Barcelona, Spain

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Ilario Bonacina, Maria Luisa Bonet, and Jordi Levy. Polynomial Calculus for MaxSAT. In 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 271, pp. 5:1-5:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


MaxSAT is the problem of finding an assignment satisfying the maximum number of clauses in a CNF formula. We consider a natural generalization of this problem to generic sets of polynomials and propose a weighted version of Polynomial Calculus to address this problem. Weighted Polynomial Calculus is a natural generalization of MaxSAT-Resolution and weighted Resolution that manipulates polynomials with coefficients in a finite field and either weights in ℕ or ℤ. We show the soundness and completeness of these systems via an algorithmic procedure. Weighted Polynomial Calculus, with weights in ℕ and coefficients in 𝔽₂, is able to prove efficiently that Tseitin formulas on a connected graph are minimally unsatisfiable. Using weights in ℤ, it also proves efficiently that the Pigeonhole Principle is minimally unsatisfiable.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
  • Theory of computation → Automated reasoning
  • Polynomial Calculus
  • MaxSAT
  • Proof systems
  • Algebraic reasoning


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