The Proof Analysis Problem

2025-06-20

AI Summary

This paper introduces the Proof Analysis Problem (PAP): given a CNF formula φ and a formal proof in proof system Q that “φ has no short Resolution refutation” (i.e., ¬Ref(φ)), decide whether φ is satisfiable. Focusing on Resolution and Extended Frege (EF), it establishes the first computational classification of PAP: Resolution-PAP is solvable in polynomial time, whereas EF-PAP is NP-complete—thereby revealing an essential equivalence between automated reasoning in strong proof systems and SAT solving. Moreover, it provides a constructive proof of the Atserias–Müller lower bound, enabling polynomial-time extraction of a satisfying assignment from any short Resolution refutation. The work also constructs a family of CNF formulas with exponential lower bounds for bounded-depth Frege, whose quadratic lower bounds are unprovable in PV₁, thereby deepening our understanding of proof complexity and the unprovability of lower bounds.

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

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Atserias and M""uller (JACM, 2020) proved that for every unsatisfiable CNF formula $varphi$, the formula $operatorname{Ref}(varphi)$, stating"$varphi$ has small Resolution refutations", does not have subexponential-size Resolution refutations. Conversely, when $varphi$ is satisfiable, Pudl'ak (TCS, 2003) showed how to construct a polynomial-size Resolution refutation of $operatorname{Ref}(varphi)$ given a satisfying assignment of $varphi$. A question that remained open is: do all short Resolution refutations of $operatorname{Ref}(varphi)$ explicitly leak a satisfying assignment of $varphi$? We answer this question affirmatively by giving a polynomial-time algorithm that extracts a satisfying assignment for $varphi$ given any short Resolution refutation of $operatorname{Ref}(varphi)$. The algorithm follows from a new feasibly constructive proof of the Atserias-M""uller lower bound, formalizable in Cook's theory $mathsf{PV_1}$ of bounded arithmetic. Motivated by this, we introduce a computational problem concerning Resolution lower bounds: the Proof Analysis Problem (PAP). For a proof system $Q$, the Proof Analysis Problem for $Q$ asks, given a CNF formula $varphi$ and a $Q$-proof of a Resolution lower bound for $varphi$, encoded as $ eg operatorname{Ref}(varphi)$, whether $varphi$ is satisfiable. In contrast to PAP for Resolution, we prove that PAP for Extended Frege (EF) is NP-complete. Our results yield new insights into proof complexity: (i) every proof system simulating EF is (weakly) automatable if and only if it is (weakly) automatable on formulas stating Resolution lower bounds; (ii) we provide Ref formulas exponentially hard for bounded-depth Frege systems; and (iii) for every strong enough theory of arithmetic $T$ we construct unsatisfiable CNF formulas exponentially hard for Resolution but for which $T$ cannot prove even a quadratic lower bound.

Problem

• Extract satisfying assignments from short Resolution refutations.
• Analyze Proof Analysis Problem for Resolution and EF.
• Investigate proof complexity insights from Resolution lower bounds.

Innovation

• Algorithm extracts assignments from Resolution refutations
• Introduces Proof Analysis Problem for Resolution
• Shows EF's NP-completeness in Proof Analysis