Intelligent search strategies based on adaptive Constraint Handling Rules

The most advanced implementation of adaptive constraint processing with Constraint Handling Rules (CHR) allows the application of intelligent search strategies to solve Constraint Satisfaction Problems (CSP). This presentation compares an improved version of conflict-directed backjumping and two variants of dynamic backtracking with respect to chronological backtracking on some of the AIM instances which are a benchmark set of random 3-SAT problems. A CHR implementation of a Boolean constraint solver combined with these different search strategies in Java is thus being compared with a CHR implementation of the same Boolean constraint solver combined with chronological backtracking in SICStus Prolog. This comparison shows that the addition of ``intelligence'' to the search process may reduce the number of search steps dramatically. Furthermore, the runtime of their Java implementations is in most cases faster than the implementations of chronological backtracking. More specifically, conflict-directed backjumping is even faster than the SICStus Prolog implementation of chronological backtracking, although our Java implementation of CHR lacks the optimisations made in the SICStus Prolog system. To appear in Theory and Practice of Logic Programming (TPLP).Comment: Number of pages: 27 Number of figures: 14 Number of Tables:

Clause/Term Resolution and Learning in the Evaluation of Quantified Boolean Formulas

Resolution is the rule of inference at the basis of most procedures for automated reasoning. In these procedures, the input formula is first translated into an equisatisfiable formula in conjunctive normal form (CNF) and then represented as a set of clauses. Deduction starts by inferring new clauses by resolution, and goes on until the empty clause is generated or satisfiability of the set of clauses is proven, e.g., because no new clauses can be generated. In this paper, we restrict our attention to the problem of evaluating Quantified Boolean Formulas (QBFs). In this setting, the above outlined deduction process is known to be sound and complete if given a formula in CNF and if a form of resolution, called Q-resolution, is used. We introduce Q-resolution on terms, to be used for formulas in disjunctive normal form. We show that the computation performed by most of the available procedures for QBFs --based on the Davis-Logemann-Loveland procedure (DLL) for propositional satisfiability-- corresponds to a tree in which Q-resolution on terms and clauses alternate. This poses the theoretical bases for the introduction of learning, corresponding to recording Q-resolution formulas associated with the nodes of the tree. We discuss the problems related to the introduction of learning in DLL based procedures, and present solutions extending state-of-the-art proposals coming from the literature on propositional satisfiability. Finally, we show that our DLL based solver extended with learning, performs significantly better on benchmarks used in the 2003 QBF solvers comparative evaluation

Backjumping for Quantified Boolean Logic Satisfiability

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

Backjumping for Quantified Boolean Logic satisfiability

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 the satisfiability of QBFs: When applicable, conflict-directed backjumping allows search to skip over existentially quantified literals while backtracking. We introduce solution-directed backjumping, which allows the same behavior for universally quantified literals. We show how it is possible to incorporate both conflict- directed and solution-directed backjumping in a DLL-based decision procedure for satisfiability of QBFs. We also implement and test the procedure: The experimental analysis shows that, because of backjumping, significant speed-ups can be obtained. Summing up: We present the first algorithm that applies conflict and solution directed backjumping to QBF, and demonstrate the performance of this algorithm via an empirical study

Backjumping for quantified Boolean logic satisfiability

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.