DepQBF 6.0: A Search-Based QBF Solver Beyond Traditional QCDCL

We present the latest major release version 6.0 of the quantified Boolean formula (QBF) solver DepQBF, which is based on QCDCL. QCDCL is an extension of the conflict-driven clause learning (CDCL) paradigm implemented in state of the art propositional satisfiability (SAT) solvers. The Q-resolution calculus (QRES) is a QBF proof system which underlies QCDCL. QCDCL solvers can produce QRES proofs of QBFs in prenex conjunctive normal form (PCNF) as a byproduct of the solving process. In contrast to traditional QCDCL based on QRES, DepQBF 6.0 implements a variant of QCDCL which is based on a generalization of QRES. This generalization is due to a set of additional axioms and leaves the original Q-resolution rules unchanged. The generalization of QRES enables QCDCL to potentially produce exponentially shorter proofs than the traditional variant. We present an overview of the features implemented in DepQBF and report on experimental results which demonstrate the effectiveness of generalized QRES in QCDCL.Comment: 12 pages + appendix; to appear in the proceedings of CADE-26, LNCS, Springer, 201

Incremental QBF Solving

We consider the problem of incrementally solving a sequence of quantified Boolean formulae (QBF). Incremental solving aims at using information learned from one formula in the process of solving the next formulae in the sequence. Based on a general overview of the problem and related challenges, we present an approach to incremental QBF solving which is application-independent and hence applicable to QBF encodings of arbitrary problems. We implemented this approach in our incremental search-based QBF solver DepQBF and report on implementation details. Experimental results illustrate the potential benefits of incremental solving in QBF-based workflows.Comment: revision (camera-ready, to appear in the proceedings of CP 2014, LNCS, Springer

Automated Benchmarking of Incremental SAT and QBF Solvers

Incremental SAT and QBF solving potentially yields improvements when sequences of related formulas are solved. An incremental application is usually tailored towards some specific solver and decomposes a problem into incremental solver calls. This hinders the independent comparison of different solvers, particularly when the application program is not available. As a remedy, we present an approach to automated benchmarking of incremental SAT and QBF solvers. Given a collection of formulas in (Q)DIMACS format generated incrementally by an application program, our approach automatically translates the formulas into instructions to import and solve a formula by an incremental SAT/QBF solver. The result of the translation is a program which replays the incremental solver calls and thus allows to evaluate incremental solvers independently from the application program. We illustrate our approach by different hardware verification problems for SAT and QBF solvers.Comment: camera-ready version (8 pages + 2 pages appendix), to appear in the proceedings of the 20th International Conference on Logic for Programming, Artificial Intelligence and Reasoning (LPAR), LNCS, Springer, 201

QRAT+: Generalizing QRAT by a More Powerful QBF Redundancy Property

The QRAT (quantified resolution asymmetric tautology) proof system simulates virtually all inference rules applied in state of the art quantified Boolean formula (QBF) reasoning tools. It consists of rules to rewrite a QBF by adding and deleting clauses and universal literals that have a certain redundancy property. To check for this redundancy property in QRAT, propositional unit propagation (UP) is applied to the quantifier free, i.e., propositional part of the QBF. We generalize the redundancy property in the QRAT system by QBF specific UP (QUP). QUP extends UP by the universal reduction operation to eliminate universal literals from clauses. We apply QUP to an abstraction of the QBF where certain universal quantifiers are converted into existential ones. This way, we obtain a generalization of QRAT we call QRAT+. The redundancy property in QRAT+ based on QUP is more powerful than the one in QRAT based on UP. We report on proof theoretical improvements and experimental results to illustrate the benefits of QRAT+ for QBF preprocessing.Comment: preprint of a paper to be published at IJCAR 2018, LNCS, Springer, including appendi

Incrementally Computing Minimal Unsatisfiable Cores of QBFs via a Clause Group Solver API

We consider the incremental computation of minimal unsatisfiable cores (MUCs) of QBFs. To this end, we equipped our incremental QBF solver DepQBF with a novel API to allow for incremental solving based on clause groups. A clause group is a set of clauses which is incrementally added to or removed from a previously solved QBF. Our implementation of the novel API is related to incremental SAT solving based on selector variables and assumptions. However, the API entirely hides selector variables and assumptions from the user, which facilitates the integration of DepQBF in other tools. We present implementation details and, for the first time, report on experiments related to the computation of MUCs of QBFs using DepQBF's novel clause group API.Comment: (fixed typo), camera-ready version, 6-page tool paper, to appear in proceedings of SAT 2015, LNCS, Springe

Dependency Schemes in QBF Calculi: Semantics and Soundness

We study the parametrisation of QBF resolution calculi by dependency schemes. One of the main problems in this area is to understand for which dependency schemes the resulting calculi are sound. Towards this end we propose a semantic framework for variable independence based on ‘exhibition’ by QBF models, and use it to express a property of dependency schemes called full exhibition that is known to be sufficient for soundness in Q-resolution. Introducing a generalised form of the long-distance resolution rule, we propose a complete parametrisation of classical long-distance Q-resolution, and show that full exhibition remains sufficient for soundness. We demonstrate that our approach applies to the current research frontiers by proving that the reflexive resolution path dependency scheme is fully exhibited

Evaluating QBF Solvers: Quantifier Alternations Matter

We present an experimental study of the effects of quantifier alternations on the evaluation of quantified Boolean formula (QBF) solvers. The number of quantifier alternations in a QBF in prenex conjunctive normal form (PCNF) is directly related to the theoretical hardness of the respective QBF satisfiability problem in the polynomial hierarchy. We show empirically that the performance of solvers based on different solving paradigms substantially varies depending on the numbers of alternations in PCNFs. In related theoretical work, quantifier alternations have become the focus of understanding the strengths and weaknesses of various QBF proof systems implemented in solvers. Our results motivate the development of methods to evaluate orthogonal solving paradigms by taking quantifier alternations into account. This is necessary to showcase the broad range of existing QBF solving paradigms for practical QBF applications. Moreover, we highlight the potential of combining different approaches and QBF proof systems in solvers.Comment: preprint of a paper to be published at CP 2018, LNCS, Springer, including appendi

Shortening QBF Proofs with Dependency Schemes

We provide the first proof complexity results for QBF dependency calculi. By showing that the reflexive resolution path dependency scheme admits exponentially shorter Q-resolution proofs on a known family of instances, we answer a question first posed by Slivovsky and Szeider in 2014 [30]. Further, we conceive a method of QBF solving in which dependency recomputation is utilised as a form of inprocessing. Formalising this notion, we introduce a new calculus in which a dependency scheme is applied dynamically. We demonstrate the further potential of this approach beyond that of the existing static system with an exponential separation

On QBF Proofs and Preprocessing

QBFs (quantified boolean formulas), which are a superset of propositional formulas, provide a canonical representation for PSPACE problems. To overcome the inherent complexity of QBF, significant effort has been invested in developing QBF solvers as well as the underlying proof systems. At the same time, formula preprocessing is crucial for the application of QBF solvers. This paper focuses on a missing link in currently-available technology: How to obtain a certificate (e.g. proof) for a formula that had been preprocessed before it was given to a solver? The paper targets a suite of commonly-used preprocessing techniques and shows how to reconstruct certificates for them. On the negative side, the paper discusses certain limitations of the currently-used proof systems in the light of preprocessing. The presented techniques were implemented and evaluated in the state-of-the-art QBF preprocessor bloqqer.Comment: LPAR 201

SAT-Based Synthesis Methods for Safety Specs

Automatic synthesis of hardware components from declarative specifications is an ambitious endeavor in computer aided design. Existing synthesis algorithms are often implemented with Binary Decision Diagrams (BDDs), inheriting their scalability limitations. Instead of BDDs, we propose several new methods to synthesize finite-state systems from safety specifications using decision procedures for the satisfiability of quantified and unquantified Boolean formulas (SAT-, QBF- and EPR-solvers). The presented approaches are based on computational learning, templates, or reduction to first-order logic. We also present an efficient parallelization, and optimizations to utilize reachability information and incremental solving. Finally, we compare all methods in an extensive case study. Our new methods outperform BDDs and other existing work on some classes of benchmarks, and our parallelization achieves a super-linear speedup. This is an extended version of [5], featuring an additional appendix.Comment: Extended version of a paper at VMCAI'1