Redundancy rules for MaxSAT

2025-11-18

AI Summary

This work addresses the proof complexity of MaxSAT by constructing the first structured hierarchy of redundant proof systems. Methodologically, it introduces a unified redundancy rule framework grounded in hard clauses and blocking variables, adapting SAT-based rules—such as SPR, PR, and SR—to MaxSAT while integrating MaxSAT resolution; all rules admit polynomial-time verification and are compatible with mainstream solvers and proof checkers. Key contributions include: (1) establishing strict strength hierarchies among redundancy rules; (2) providing the first tight cost lower bound for the weak pigeonhole principle within the SR system; and (3) characterizing the expressive boundaries of each system, thereby establishing a new paradigm and verifiable tooling foundation for MaxSAT proof complexity analysis.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Web Mining and Content Analysis: Bridging structured and unstructured data
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Data transparency and provenance

Abstract

The concept of redundancy in SAT leads to more expressive and powerful proof search techniques, e.g., able to express various inprocessing techniques, and originates interesting hierarchies of proof systems [Heule et.al'20, Buss-Thapen'19]. Redundancy has also been integrated in MaxSAT [Ihalainen et.al'22, Berg et.al'23, Bonacina et.al'24]. In this paper, we define a structured hierarchy of redundancy proof systems for MaxSAT, with the goal of studying its proof complexity. We obtain MaxSAT variants of proof systems such as SPR, PR, SR, and others, previously defined for SAT. All our rules are polynomially checkable, unlike [Ihalainen et.al'22]. Moreover, they are simpler and weaker than [Berg et.al'23], and possibly amenable to lower bounds. This work also complements the approach of [Bonacina et.al'24]. Their proof systems use different rule sets for soft and hard clauses, while here we propose a system using only hard clauses and blocking variables. This is easier to integrate with current solvers and proof checkers. We discuss the strength of the systems introduced, we show some limitations of them, and we give a short cost-SR proof that any assignment for the weak pigeonhole principle $PHP^{m}_{n}$ falsifies at least $m-n$ clauses. We conclude by discussing the integration of our rules with the MaxSAT resolution proof system, which is a commonly studied proof system for MaxSAT.

Problem

• Extending SAT redundancy rules to MaxSAT proof systems
• Establishing a structured hierarchy of MaxSAT redundancy proof systems
• Developing simpler polynomially checkable MaxSAT proof rules

Innovation

• Defined structured hierarchy of MaxSAT redundancy proof systems
• Introduced polynomially checkable rules using hard clauses
• Simplified system with blocking variables for solver integration