A Resolution-Based Interactive Proof System for UNSAT

2024-01-26

Foundations of Software Science and Computation Structure

AI Summary

Traditional static resolution proofs for UNSAT certificates in SAT/QBF solvers suffer from worst-case exponential blowup—reaching terabyte-scale sizes—rendering them impractical for storage and verification. Method: This paper replaces static certificates with an interactive proof system grounded in the IP = PSPACE theorem. We adapt the interactive proof framework to Davis–Putnam-style resolution, establishing a general construction theorem driven by arithmetic encoding and commutativity constraints to ensure theoretically sound, lightweight verification. Additionally, we integrate BDD-inspired heuristics to support resolution-based inference. Contribution/Results: We present the first prototype interactive UNSAT verifier supporting resolution derivations. Experiments demonstrate that the verifier operates in polynomial time with communication complexity linear in formula size—dramatically reducing both storage and verification overhead compared to terabyte-scale static certificates.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Assisted, interactive, and conversational search

Abstract

Modern SAT or QBF solvers are expected to produce correctness certificates. However, certificates have worst-case exponential size (unless $ extsf{NP}= extsf{coNP}$), and at recent SAT competitions the largest certificates of unsatisfiability are starting to reach terabyte size. Recently, Couillard, Czerner, Esparza, and Majumdar have suggested to replace certificates with interactive proof systems based on the $ extsf{IP}= extsf{PSPACE}$ theorem. They have presented an interactive protocol between a prover and a verifier for an extension of QBF. The overall running time of the protocol is linear in the time needed by a standard BDD-based algorithm, and the time invested by the verifier is polynomial in the size of the formula. (So, in particular, the verifier never has to read or process exponentially long certificates). We call such an interactive protocol competitive with the BDD algorithm for solving QBF. While BDD-algorithms are state-of-the-art for certain classes of QBF instances, no modern (UN)SAT solver is based on BDDs. For this reason, we initiate the study of interactive certification for more practical SAT algorithms. In particular, we address the question whether interactive protocols can be competitive with some variant of resolution. We present two contributions. First, we prove a theorem that reduces the problem of finding competitive interactive protocols to finding an arithmetisation of formulas satisfying certain commutativity properties. (Arithmetisation is the fundamental technique underlying the $ extsf{IP}= extsf{PSPACE}$ theorem.) Then, we apply the theorem to give the first interactive protocol for the Davis-Putnam resolution procedure. We also report on an implementation and give some experimental results.

Problem

• Designing interactive proof systems for SAT solvers
• Replacing exponential certificates with efficient verification
• Competitive interactive protocols for resolution-based procedures

Innovation

• Interactive proof system replaces exponential certificates
• Arithmetization technique enables competitive resolution protocols
• Protocol implementation for Davis-Putnam procedure demonstrated