Efficient Certified Resolution Proof Checking

We present a novel propositional proof tracing format that eliminates complex processing, thus enabling efficient (formal) proof checking. The benefits of this format are demonstrated by implementing a proof checker in C, which outperforms a state-of-the-art checker by two orders of magnitude. We then formalize the theory underlying propositional proof checking in Coq, and extract a correct-by-construction proof checker for our format from the formalization. An empirical evaluation using 280 unsatisfiable instances from the 2015 and 2016 SAT competitions shows that this certified checker usually performs comparably to a state-of-the-art non-certified proof checker. Using this format, we formally verify the recent 200 TB proof of the Boolean Pythagorean Triples conjecture

Certified Knowledge Compilation with Application to Verified Model Counting

Computing many useful properties of Boolean formulas, such as their weighted or unweighted model count, is intractable on general representations. It can become tractable when formulas are expressed in a special form, such as the decision-decomposable, negation normal form (dec-DNNF) . Knowledge compilation is the process of converting a formula into such a form. Unfortunately existing knowledge compilers provide no guarantee that their output correctly represents the original formula, and therefore they cannot validate a model count, or any other computed value. We present Partitioned-Operation Graphs (POGs), a form that can encode all of the representations used by existing knowledge compilers. We have designed CPOG, a framework that can express proofs of equivalence between a POG and a Boolean formula in conjunctive normal form (CNF). We have developed a program that generates POG representations from dec-DNNF graphs produced by the state-of-the-art knowledge compiler D4, as well as checkable CPOG proofs certifying that the output POGs are equivalent to the input CNF formulas. Our toolchain for generating and verifying POGs scales to all but the largest graphs produced by D4 for formulas from a recent model counting competition. Additionally, we have developed a formally verified CPOG checker and model counter for POGs in the Lean 4 proof assistant. In doing so, we proved the soundness of our proof framework. These programs comprise the first formally verified toolchain for weighted and unweighted model counting

Even shorter proofs without new variables

Proof formats for SAT solvers have diversified over the last decade, enabling new features such as extended resolution-like capabilities, very general extension-free rules, inclusion of proof hints, and pseudo-boolean reasoning. Interference-based methods have been proven effective, and some theoretical work has been undertaken to better explain their limits and semantics. In this work, we combine the subsumption redundancy notion from (Buss, Thapen 2019) and the overwrite logic framework from (Rebola-Pardo, Suda 2018). Natural generalizations then become apparent, enabling even shorter proofs of the pigeonhole principle (compared to those from (Heule, Kiesl, Biere 2017)) and smaller unsatisfiable core generation.Comment: 21 page

Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer

We solved a long-outstanding open problem in Ramsey theory, using SAT solving

A Nelson-Oppen based Proof System using Theory Specific Proof Systems

International audienceSMT solvers are nowadays pervasive in verification tools. When the verification is about a critical system, the result of the SMT solver is also critical and cannot be trusted. The SMT-LIB 2.0 is a standard interface for SMT solvers but does not specify the output of the get-proof command. We present a proof system that is geared towards SMT solvers and follows their conceptually modular architecture. Our proof system makes a clear distinction between propositional and theory reasoning. Moreover, individual theories provide specific proof systems that are combined using the Nelson-Oppen proof scheme. We propose specific proof systems for linear real arithmetic (LRA) and uninterpreted functions (EUF) and discuss proof generation and proof checking. We have evaluated the cost of generating proofs in our proof system. Our experiments on benchmarks taken from the SMT-LIB library show that the simple mechanisms used in our approach suffice for a large majority of the selected benchmarks