A Symbolic Search Based Approach for Quantified Boolean Formulas

2004

Solving Quantified Boolean Formulas (QBF) has become an attractive research area in Artificial intelligence. Many important artificial intelligence problems (planning, non monotonic reasoning, formal verification, etc.) can be reduced to QBFs. In this paper, a new DLL-based method is proposed that integrates binary decision diagram (BDD) to set free the variable ordering heuristics that are traditionally constrained by the static order of the QBF quantifiers. BDD is used to represent in a compact form the set of models of the boolean formula. Interesting reduction operators are proposed in order to dynamically reduce the BDD size and to answer the validity of the QBF. Experimental results on instances from the QBF'03 evaluation show that our approach can efficiently solve instances that are very hard for current QBF solvers.

2005

Quantified Boolean formulas (QBFs) play an important role in theoretical computer science. QBF extends propositional logic in such a way that many advanced forms of reasoning can be easily formulated and evaluated. In this dissertation we present our ZQSAT, which is an algorithm for evaluating quantified Boolean formulas. ZQSAT is based on ZBDD: Zero-Suppressed Binary Decision Diagram , which is a variant of BDD, and an adopted version of the DPLL algorithm. It has been implemented in C using the CUDD: Colorado University Decision Diagram package. The capability of ZBDDs in storing sets of subsets efficiently enabled us to store the clauses of a QBF very compactly and let us to embed the notion of memoization to the DPLL algorithm. These points led us to implement the search algorithm in such a way that we could store and reuse the results of all previously solved subformulas with a little overheads. ZQSAT can solve some sets of standard QBF benchmark problems (known to be hard for ...

2008

In this paper we introduce QuBIS an (in)complete solver for quantified Boolean formulas (QBFs). The particularity of QuBIS is that it is not inherently incomplete, but it has the ability to surrender upon realizing that its deduction mechanism is becoming ineffective. Whenever this happens, QuBIS outputs a partial result which can be fed to a complete QBF solver for further processing. As our experiments show, not only QuBIS is competitive as an incomplete solver, but providing the output of QuBIS as an input to complete solvers can boost their performances on several instances.

2002

Learning, i.e., the ability to record and exploit some information which is unveiled during the search, proved to be a very effective AI technique for problem solving and, in particular, for constraint satisfaction. We introduce learning as a general purpose technique to improve the performances of decision procedures for Quantified Boolean Formulas (QBFs). Since many of the recently proposed decision procedures for QBFs solve the formula using search methods, the addition of learning to such procedures has the potential of reducing useless explorations of the search space. To show the applicability of learning for QBF satisfiability we have implemented it in QUBE, a state-of-the-art QBF solver. While the backjumping engine embedded in QUBE provides a good starting point for our task, the addition of learning required us to devise new data structures and led to the definition and implementation of new pruning strategies. We report some experimental results that witness the effectiveness of learning. Noticeably, QUBE augmented with learning is able to solve instances that were previously out if its reach. To the extent of our knowledge, this is the first time that learning is proposed, implemented and tested for QBFs satisfiability.

2004

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.

2002

Learning, i.e., the ability to record and exploit some information which is unveiled during the search, proved to be a very effective AI technique for problem solving and, in particular, for constraint satisfaction. In we have introduced learning as a general purpose technique to improve the performances of decision procedures for Quantified Boolean Formulas (QBFs). We have added learning techniques to QUBE, a state-of-the-art QBF solver. Embedding learning techniques in QUBE proved to be a challenging task, which also led to the definition and the implementation of new pruning strategies. In this paper, we report some experimental results that witness the effectiveness of learning and the associated pruning strategies. Noticeably, QUBE augmented with learning is able to solve instances that were previously out if its reach. On the other hand, the additional complexity of learning does impose some overhead on QUBE. We present results that highlight the presence of such overhead, we discuss its causes, and suggest possible remedies in QUBE.

2009

In this paper we approach the problem of reasoning with quantified Boolean formulas (QBFs) by combining search and resolution, and by switching between them according to structural properties of QBFs. We provide empirical evidence that QBFs which cannot be solved by search or resolution alone, can be solved by combining them, and that our approach makes a proof-of-concept implementation competitive with current QBF solvers.

Progress in Artificial Intelligence, 2019

This paper proposes to study the synthesis of unordered binary decision diagrams (BDDs) using solvers for Quantified Boolean Formulas (QBF). The synthesis of a BDD falls naturally in the realm of quantified formulas as we are typically looking for a BDD satisfying a certain specification. This means that we ask whether there exists a BDD such that for all inputs the specification is satisfied. We show that this query can be encoded naturally into QBF and experimentally evaluate these queries for the minority function. The short paper should be seen as a challenge for further research on QBF solving.

Artificial Intelligence, 2003

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.

1998

The high computational complexity of advanced reasoning tasks such as belief revision and planning calls for efficient and reliable algorithms for reasoning problems harder than NP. In this paper we propose Evaluate, an algorithm for evaluating Quantified Boolean Formulae, a language that extends propositional logic in a way such that many advanced forms of propositional reasoning, e.g., reasoning about knowledge, can be easily formulated as evaluation of a QBF. Algorithms for evaluation of QBFs are suitable for the experimental analysis on a wide range of complexity classes, a property not easily found in other formalisms. Evaluate is based on a generalization of the Davis-Putnam procedure for SAT, and is guaranteed to work in polynomial space. Before presenting Evaluate, we discuss all the abstract properties of QBFs that we singled out to make the algorithm more efficient. We also briefly mention the main results of the experimental analysis, which is reported elsewhere.