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

Simulating Strong Practical Proof Systems with Extended Resolution

Proof systems for propositional logic provide the basis for decision procedures that determine the satisfiability status of logical formulas. While the well-known proof system of extended resolution—introduced by Tseitin in the sixties—allows for the compact representation of proofs, modern SAT solvers (i.e., tools for deciding propositional logic) are based on different proof systems that capture practical solving techniques in an elegant way. The most popular of these proof systems is likely DRAT, which is considered the de-facto standard in SAT solving. Moreover, just recently, the proof system DPR has been proposed as a generalization of DRAT that allows for short proofs without the need of new variables. Since every extended-resolution proof can be regarded as a DRAT proof and since every DRAT proof is also a DPR proof, it was clear that both DRAT and DPR generalize extended resolution. In this paper, we show that — from the viewpoint of proof complexity — these two systems are no stronger than extended resolution. We do so by showing that (1) extended resolution polynomially simulates DRAT and (2) DRAT polynomially simulates DPR. We implemented our simulations as proof-transformation tools and evaluated them to observe their behavior in practice. Finally, as a side note, we show how Kullmann’s proof system based on blocked clauses (another generalization of extended resolution) is related to the other systems

Frying the Egg, Roasting the Chicken: Unit Deletions in DRAT Proofs

The final publication is available via https://doi.org/ 10.1145/3372885.3373821.The clausal proof format DRAT is the standard de facto to certify SAT solvers' unsatisfiability results. DRAT proofs act as logs of clause inferences and clause deletions in the solver. The non-monotonic nature of the proof system makes deletions relevant. State-of-the-art proof checkers ignore deletions of unit clauses, differing from the standard in meaningful ways that require adaptions when proofs are generated or used for purposes other than checking. On the other hand, dealing with unit deletions in the proof checker breaks many of the usual invariants used for efficiency reasons. Furthermore, many SAT solvers introduce spurious unit deletions in proofs. These deletions are never intended to be applied in the checker but are nevertheless introduced, making many proofs generated by state-of-the-art solvers incorrect. We present the first competitive DRAT checker that honors unit deletions, as well as fixes for the spurious deletion issue in proof generation. Our experimental results confirm that unit deletions can be applied with similar average performance to state-of-the-art checkers. We also confirm that a large fraction of the proofs generated during the last SAT solving competition do not respect the DRAT standard. This result was confirmed with proof incorrectness certificates that were independently validated. We find that our proof incorrectness certificates can be of help when debugging SAT solvers and DRAT checkers.Fonds zur Förderung der Wissenschaftlichen ForschungWiener Wissenschafts-, Forschungs- und Technologiefonds (WWTF