Certifying Parity Reasoning Efficiently Using Pseudo-Boolean Proofs

Abstract

The dramatic improvements in combinatorial optimization algorithms over the last decades have had a major impact in artificial intelligence, operations research, and beyond, but the output of current state-of-the-art solvers is often hard to verify and is sometimes wrong. For Boolean satisfiability (SAT) solvers proof logging has been introduced as a way to certify correctness, but the methods used seem hard to generalize to stronger paradigms. What is more, even for enhanced SAT techniques such as parity (XOR) reasoning, cardinality detection, and symmetry handling, it has remained beyond reach to design practically efficient proofs in the standard DRAT format. In this work, we show how to instead use pseudo-Boolean inequalities with extension variables to concisely justify XOR reasoning. Our experimental evaluation of a SAT solver integration shows a dramatic decrease in proof logging and verification time compared to existing DRAT methods. Since our method is a strict generalization of DRAT, and readily lends itself to expressing also 0-1 programming and even constraint programming problems, we hope this work points the way towards a unified approach for efficient machine-verifiable proofs for a rich class of combinatorial optimization paradigms.

QMaxSATpb: A Certified MaxSAT Solver

“…If the solver is replaced by any other modern SAT solver with proof logging capabilities, only minor syntactic modifications are needed to make it VeriPB-compatible. Indeed, as mentioned before, redundance-based strengthening generalizes the well-known RAT rule, and moreover, VeriPB can additionally handle symmetry breaking, cardinality reasoning and XOR reasoning [25,7].…”

“…Proof. The variable v X j is a fresh variable; the constraints enforce it to be true if and only if x∈X x ≥ j. Gocht and Nordström [25] have shown how these defining constraints can be derived using redundance-based strengthening. 3 We include a full proof here to make the paper self-contained.…”

“…One promising proof format to fill this gap is VeriPB [17,25,24,22], a proof format for pseudo-Boolean satisfiability, that was recently extended to pseudo-Boolean optimization [7], which generalizes MaxSAT. VeriPB builds on top of the cutting planes proof system [12], and extends it with a generalization of the Resolution Asymmetric Tautology (RAT) rule that lies at the basis of the most successful proof formats for SAT.…”

“…VeriPB builds on top of the cutting planes proof system [12], and extends it with a generalization of the Resolution Asymmetric Tautology (RAT) rule that lies at the basis of the most successful proof formats for SAT. VeriPB naturally facilitates proof logging for advanced techniques such as XOR and cardinality reasoning [25] and symmetry breaking [7] that have been largely out of reach of other proof formats for SAT.…”

Logic-Based Explainability in Machine Learning

“…It is well-known that algorithms proved correct can be implemented incorrectly. In areas where the rigor of results is paramount, there have been efforts to devise mechanisms for ascertaining the correctness of either implemented algorithms or their computed results [6,7,75,76,79,98,[124][125][126]138,139,223,317]. A natural topic of research is to apply similar solutions in the case of the computation of explanations, but also in the case of explainability queries.…”