An Algorithm to Evaluate Quantified Boolean Formulae

2005

Several propositional fragments have been considered so far as target languages for knowledge compilation and used for improving computational tasks from major AI areas (like inference, diagnosis and planning); among them are the ordered binary decision diagrams, prime implicates, prime implicants, "formulae" in decomposable negation normal form. On the other hand, the validity problem val(QPROP P S ) for Quantified Boolean Formulae (QBF) has been acknowledged for the past few years as an important issue for AI, and many solvers have been designed. In this paper, the complexity of restrictions of the validity problem for QBF obtained by imposing the matrix of the input QBF to belong to propositional fragments used as target languages for compilation, is identified. It turns out that this problem remains hard (PSPACE-complete) even under severe restrictions on the matrix of the input. Nevertheless some tractable restrictions are pointed out.

2006

Several propositional fragments have been considered so far as target languages for knowledge compilation and used for improving computational tasks from major AI areas (like inference, diagnosis and planning); among them are the ordered binary decision diagrams, prime implicates, prime implicants, "formulae" in decomposable negation normal form. On the other hand, the validity problem val(QPROP P S ) for Quantified Boolean Formulae (QBF) has been acknowledged for the past few years as an important issue for AI, and many solvers have been designed. In this paper, the complexity of restrictions of the validity problem for QBF obtained by imposing the matrix of the input QBF to belong to propositional fragments used as target languages for compilation, is identified. It turns out that this problem remains hard (PSPACE-complete) even under severe restrictions on the matrix of the input. Nevertheless some tractable restrictions are pointed out.

2001

2005

Several propositional fragments have been considered so far as target languages for knowledge compilation and used for improving computational tasks from major AI areas (like inference, diagnosis and planning); among them are the (quite influential) ordered binary decision diagrams, prime implicates, prime implicants, "formulae" in decomposable negation normal form. On the other hand, the validity problem QBF for Quantified Boolean Formulae (QBF) has been acknowledged for the past few years as an important issue for AI, and many solvers have been designed for this purpose. In this paper, the complexity of restrictions of QBF obtained by imposing the matrix of the input QBF to belong to such propositional fragments is identified. Both tractability and intractability results (PSPACE-completeness) are obtained.

2002

We present algorithms for solving quantified Boolean formulas (QBF, or sometimes QSAT) with worst case runtime asymptotically less than O(2 n ) when the clause-to-variable ratio is smaller or larger than some constant. We solve QBFs in conjunctive normal form (CNF) in O(1.709 m ) time and space, where m is the number of clauses. Extending the technique to a quantified version of constraint satisfaction problems (QCSP), we solve QCSP with domain size d = 3 in O(1.953 m ) time, and QCSPs with d ≥ 4 in O(d m/2+ ) time and space for > 0, where m is the number of constraints. For 3-CNF QBF, we describe an polynomial space algorithm with time complexity O(1. when the number of 3-CNF clauses is equal to n; the bound approaches 2 n as the clause-to-variable ratio approaches 2. For 3-CNF Π 2 -SAT (3-CNF QBFs of the form ∀u 1 · · · u j ∃x j+1 · · · x n F ), an improved polyspace algorithm has runtime varying from O(1.840 m ) to O(1.415 m ), as a particular clause-to-variable ratio increases from 1.

2019 IEEE 31st International Conference on Tools with Artificial Intelligence (ICTAI), 2019

The decision problem of quantified Boolean formulas (QBFs) is the archetypical problem for the complexity class PSPACE that contains many reasoning problems of practical relevance. Because of the availability of a rich solving infrastructure aspects QBFs provide an attractive framework for encoding and solving such reasoning problems ranging from symbolic reasoning in artificial intelligence to the formal verification and synthesis of computing systems. In this paper, we survey the different application areas that exploit QBF technology for solving their specific problems.

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.

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.

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.

2002

The availability of decision procedures for combinations of boolean and linear mathematical propositions opens the ability to solve problems arising from real-world domains such as verification of timed systems and planning with resources. In this paper we present a general and efficient approach to the problem, based on two main ingredients. The first is a DPLL-based SAT procedure, for dealing efficiently with the propositional component of the problem. The second is a tight integration, within the DPLL architecture, of a set of mathematical deciders for theories of increasing expressive power. A preliminary experimental evaluation shows the potential of the approach.