The implementation of effective reasoning tools for deciding the satisfiability of Quantified Boolean Formulas (QBFs) is an important research issue in Artificial Intelligence. Many decision procedures have been proposed in the last few years, most of them based on the Davis, Logemann, Loveland procedure (DLL) for propositional satisfiability (SAT). In this paper we show how it is possible to extend the conflict-directed backjumping schema for SAT to QBF: when applicable, it allows to jump over existentially quantified literals while backtracking. We introduce solution-directed backjumping, which allows the same for universally quantified literals. Then, we show how it is possible to incorporate both conflict-directed and solution-directed backjumping in a DLL-based decision procedure for QBF satisfiability. We also implement and test the procedure: The experimental analysis shows that, because of backjumping, significant speed-ups can be obtained. While there have been several proposals for backjumping in SAT, this is the first time --as far as we know-- this idea has been proposed, implemented and experimented for QBFs.

Within the verification community, there has been a recent increase in interest in Quantified Boolean Formula evaluation (QBF) as many interesting sequential circuit verification problems can be formulated as QBF instances. A closely related research area to QBF is Boolean Satisfiability (SAT). Recent advances in SAT research have resulted in some very efficient SAT solvers. One of the critical techniques employed in these solvers is Conflict Driven Learning. In this paper, we adapt conflict driven learning for application in a QBF setting. We show that conflict driven learning can be regarded as a resolution process on the clauses. We prove that under certain conditions, tautology clauses obtained from resolution in QBF also obey the rules for implication and conflicts of regular (nontautology) clauses; and therefore they can be treated as regular clauses and used in future search. We have implemented this idea in a new QBF solver called Quaffle and our initial experiments show that conflict driven learning can greatly speed up the solution process for most of the benchmarks we tested. 1.

Abstract. We presentsKizzo, a system designed to evaluate and certify Quantified Boolean Formulas (QBFs) by means of propositional skolemization and symbolic reasoning. 1

In this paper, we describe a new framework for evaluating Quantified Boolean Formulas (QBF). The new framework is based on the Davis-Putnam (DPLL) search algorithm. In existing DPLL based QBF algorithms, the problem database is represented in Conjunctive Normal Form (CNF) as a set of clauses, implications are generated from these clauses, and backtracking in the search tree is chronological. In this work, we augment the basic DPLL algorithm with conflict driven learning as well as satisfiability directed implication and learning. In addition to the traditional clause database, we add a cube database to the data structure. We show that cubes can be used to generate satisfiability directed implications similar to conflict directed implications generated by the clauses. We show that in a QBF setting, conflicting leaves and satisfying leaves of the search tree both provide valuable information to the solver in a symmetric way. We have implemented our algorithm in the new QBF solver Quaffle. Experimental results show that for some test cases, satisfiability directed implication and learning significantly prunes the search.

Much recent work has gone into adapting techniques that were originally developed for SAT solving to QBF solving. In particular, QBF solvers are often based on SAT solvers. Most competitive QBF solvers are search-based. In this work we explore an alternative approach to QBF solving, based on symbolic quantifier elimination. We extend some recent symbolic approaches for SAT solving to symbolic QBF solving, using various decision-diagram formalisms such as OBDDs and ZDDs. In both approaches, QBF formulas are solved by eliminating all their quantifiers. Our first solver, QMRES, maintains a set of clauses represented by a ZDD and eliminates quantifiers via multi-resolution. Our second solver, QBDD, maintains a set of OBDDs, and eliminate quantifier by applying them to the underlying OBDDs. We compare our symbolic solvers to several competitive search-based solvers. We show that QBDD is not competitive, but QMRES compares favorably with search-based solvers on various benchmarks consisting of non-random formulas.

Abstract. Binary clause reasoning has found some successful applications in SAT, and it is natural to investigate its use in various extensions of SAT. In this paper we investigate the use of binary clause reasoning in the context of solving Quantified Boolean Formulas (QBF). We develop a DPLL based QBF solver that employs extended binary clause reasoning (hyper-binary resolution) to infer new binary clauses both before and during search. These binary clauses are used to discover additional forced literals, as well as to perform equality reduction. Both of these transformations simplify the theory by removing one of its variables. When applied during DPLL search this stronger inference can offer significant decreases in the size of the search tree, but it can also be costly to apply. We are able to show empirically that despite the extra costs, binary clause reasoning can improve our ability to solve QBF. 1

We describe BDD-based decision procedures for the modal logic K. Our approach is inspired by the automata-theoretic approach, but we avoid explicit automata construction. Instead, we compute certain fixpoints of a set of types---which can be viewed as an on-the-fly emptiness of the automaton. We use BDDs to represent and manipulate such type sets, and investigate different kinds of representations as well as a "level-based" representation scheme. The latter turns out to speed up construction and reduce memory consumption considerably. We also study the effect of formula simplification on our decision procedures. To proof the viability of our approach, we compare our approach with a representative selection of other approaches, including a translation of to QBF. Our results indicate that the BDD-based approach dominates for modally heavy formulae, while search-based approaches dominate for propositionally heavy formulae.

Nested logic programs have recently been introduced in order to allow for arbitrarily nested formulas in the heads and the bodies of logic program rules under the answer sets semantics. Previous results show that nested logic programs can be transformed into standard (unnested) disjunctive logic programs in an elementary way, applying the negation-as-failure operator to body literals only. This is of great practical relevance since it allows us to evaluate nested logic programs by means of off-the-shelf disjunctive logic programming systems, like DLV. However, it turns out that this straightforward transformation results in an exponential blow-up in the worst-case, despite the fact that complexity results indicate that there is a polynomial translation among both formalisms. In this paper, we take up this challenge and provide a polynomial translation of logic programs with nested expressions into disjunctive logic programs. Moreover, we show that this translation is modular and (strongly) faithful. We have implemented both the straightforward as well as our advanced transformation; the resulting compiler serves as a front-end to DLV and is publicly available on the Web. 1

Abstract. We introduce a simple and practically efficient method for repairing inconsistent databases. The idea is to properly represent the underlying problem, and then use off-the-shelf applications for efficiently computing the corresponding solutions. Given a possibly inconsistent database, we represent the possible ways to restore its consistency in terms of signed formulae. Then we show how the ‘signed theory ’ that is obtained can be used by a variety of computational models for processing quantified Boolean formulae, or by constraint logic program solvers, in order to rapidly and efficiently compute desired solutions, i.e., consistent repairs of the database. 1

We describe Qubos (QUantified BOolean Solver), a decision procedure for quantified Boolean logic. The procedure is based on nonclausal simplification techniques that reduce formulae to a propositional clausal form after which o#-the-shelf satisfiability solvers can be employed. We show that there are domains exhibiting structure for which this procedure is very e#ective and we report on experimental results.