A first step towards a unified proof checker for QBF

ACM Transactions on Computation Theory, 2019

Modern QBF solvers typically use two different paradigms, conflict-driven clause learning (CDCL) solving or expansion solving. Proof systems for quantified Boolean formulas (QBFs) provide a theoretical underpinning for the performance of these solvers, with Q-Resolution and its extensions relating to CDCL solving and ∀Exp+Res relating to expansion solving. This article defines two novel calculi, which are resolution-based and enable unification of some of the principal existing resolution-based QBF calculi, namely Q-resolution, long-distance Q-resolution and the expansion-based calculus ∀Exp+Res. However, the proof complexity of the QBF resolution proof systems is currently not well understood. In this article, we completely determine the relative power of the main QBF resolution systems, settling in particular the relationship between the two different types of resolution-based QBF calculi: proof systems for CDCL-based solvers (Q-resolution, universal, and long-distance Q-resolutio...

Proceedings of the AAAI Conference on Artificial Intelligence

Many computer science problems can be naturally and compactly expressed using quantified Boolean formulas (QBFs). Evaluating thetruth or falsity of a QBF is an important task, and constructing the corresponding model or countermodel can be as important and sometimes even more useful in practice. Modern search and learning based QBF solvers rely fundamentally on resolution and can be instrumented to produce resolution proofs, from which in turn Skolem-function models and Herbrand-function countermodels can be extracted. These (counter)models are the key enabler of various applications. Not until recently the superiority of long-distanceresolution (LQ-resolution) to short-distance resolution(Q-resolution) was demonstrated. While a polynomial algorithm exists for (counter)model extraction from Q-resolution proofs, it remains open whether it exists forLQ-resolution proofs. This paper settles this open problem affirmatively by constructing a linear-time extraction procedure. Experimental...

Lecture Notes in Computer Science, 2011

A resoution proof is a certificate of the unsatisfiability of a Boolean formula. Resolution proofs, as generated by modern SAT solvers, find application in many verification techniques. For efficiency smaller proofs are preferable over larger ones. This paper presents a new approach to proof reduction, situated among the purely post-processing methods. The main idea is to reduce the proof size by eliminating redundancies of occurrences of pivots along the proof paths. This is achieved by matching and rewriting local contexts into simpler ones. In our approach, rewriting can be easily customized in the way local contexts are matched, in the amount of transformations to be performed, or in the different application of the rewriting rules. We provide an extensive experimental evaluation of our technique on a set of benchmarks, which shows considerable reduction in the proofs size.

Lecture Notes in Computer Science, 2017

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.

Artificial Intelligence, 2019

After a break of about five years, in 2016 the classical QBFEVAL has been revived. QBFEVAL is a competitive evaluation of solvers for quantified Boolean formulas (QBF), the extension of propositional formulas with existential and universal quantifiers over the propositional variables. Due to the enormous interest in QBFEVAL'16, more recently, QBFEVAL'17 was organized. Both competitions were affiliated to the respective editions of the International Conference on Theory and Applications of Satisfiability Testing (SAT'16 and SAT'17), the major conference in research on SAT and related areas. In this paper we report about the 2016 and 2017 competitive evaluations of QBF solvers (QBF-EVAL'16 and QBFEVAL'17), the two most recent events in a series of competitions established with the aim of assessing the advancements in reasoning about QBFs. This report gives an overview of the setup of these two events, on their participants and on the results of the experiments that were performed for evaluating the participating systems.

2003

The implementation of effective reasoning tools for deciding the satisfiability of Quantified Boolean Formulas (QBFs) is an important issue in several research fields such as Formal Verification, Planning, and Reasoning about Knowledge. Several QBF solvers have been implemented in the last few years, most of them extending the well-known Davis, Putnam, Logemann, Loveland procedure (DPLL) for propositional satisfiability (SAT). At the same time, a substantial breed of QBF benchmarks emerged, both in the form of statistical models for the generation of random formulas, and in the form of real-world instances. In this paper we report about the – first ever – evaluation of QBF solvers that was run as a joint event to SAT’03 Conference on Theory and Applications of Satisfiability Testing. Owing to the relative youngness of QBF tools and applications, we decided to run the comparison on a non-competitive basis, using the same technology that powered SAT’02 and SAT’03 competitions of SAT solvers. Running the evaluation enabled us to collect all sorts of data regarding the relative strength of different solvers and methods, the quality of the benchmarks, and to understand some of the current challenges for researchers involved in the QBF arena.

2003

Mathematical Foundations of Computer Science 2014, 2014

Several calculi for quantified Boolean formulas (QBFs) exist, but relations between them are not yet fully understood. This paper defines a novel calculus, which is resolution-based and enables unification of the principal existing resolution-based QBF calculi, namely Qresolution, long-distance Q-resolution and the expansion-based calculus ∀Exp+Res. All these calculi play an important role in QBF solving. This paper shows simulation results for the new calculus and some of its variants. Further, we demonstrate how to obtain winning strategies for the universal player from proofs in the calculus. We believe that this new proof system provides an underpinning necessary for formal analysis of modern QBF solvers.

Lecture Notes in Computer Science, 2006

Symbolic SAT solving is an approach where the clauses of a CNF formula are represented using BDDs. These BDDs are then conjoined, and finally checking satisfiability is reduced to the question of whether the final BDD is identical to false. We present a method combining symbolic SAT solving with BDD quantification (variable elimination) and generation of extended resolution proofs. Proofs are fundamental to many applications, and our results allow the use of BDDs instead of-or in combination with-established proof generation techniques like clause learning. We have implemented a symbolic SAT solver with variable elimination that produces extended resolution proofs. We present details of our implementation, called EBDDRES, which is an extension of the system presented in , and also report on experimental results.

2010

In this paper we report about QBFEVAL’10, the seventh in a series of events established with the aim of assessing the advancements in reasoning about quantified Boolean formulas (QBFs). The paper discusses the results obtained and the experimental setup, from the criteria used to select QBF instances to the evaluation infrastructure. We also discuss the current state-of-the-art in light of past challenges and we envision future research directions that are motivated by the results of QBFEVAL’10.