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49
Building decision procedures for modal logics from propositional decision procedures  The case study of modal K(m)
, 1996
"... The goal of this paper is to propose a new technique for developing decision procedures for propositional modal logics. The basic idea is that propositional modal decision procedures should be developed on top of propositional decision procedures. As a case study, we consider satisfiability in m ..."
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Cited by 98 (29 self)
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The goal of this paper is to propose a new technique for developing decision procedures for propositional modal logics. The basic idea is that propositional modal decision procedures should be developed on top of propositional decision procedures. As a case study, we consider satisfiability in modal K(m), that is modal K with m modalities, and develop an algorithm, called Ksat, on top of an implementation of the DavisPutnamLongemannLoveland procedure. Ksat is thoroughly tested and compared with various procedures and in particular with the stateoftheart tableaubased system Kris. The experimental results show that Ksat outperforms Kris and the other systems of orders of magnitude, highlight an intrinsic weakness of tableaubased decision procedures, and provide partial evidence of a phase transition phenomenon for K(m).
Evaluating QBFs via symbolic Skolemization
 In Int. Conf. on Logic for Programming Artificial Intelligence and Reasoning (LPAR’04
"... Abstract. We describe a novel decision procedure for Quantified Boolean Formulas (QBFs) which aims to unleash the hidden potential of quantified reasoning in applications. The Skolem theorem acts like a glue holding several ingredients together: BDDbased representations for boolean functions, sear ..."
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Cited by 41 (9 self)
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Abstract. We describe a novel decision procedure for Quantified Boolean Formulas (QBFs) which aims to unleash the hidden potential of quantified reasoning in applications. The Skolem theorem acts like a glue holding several ingredients together: BDDbased representations for boolean functions, searchbased QBF decision procedure, and compilationtoSAT techniques, among the others. To leverage all these techniques at once we show how to evaluate QBFs by symbolically reasoning on a compact representation for the propositional expansion of the skolemized problem. We also report about a first implementation of the procedure, which yields very interesting experimental results. 1
Random 3SAT: The Plot Thickens
 IN PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING
, 2000
"... This paper presents an experimental investigation of the following questions: how does the averagecase complexity of random 3SAT, understood as a function of the order (number of variables) for xed density (ratio of number of clauses to order) instances, depend on the density? Is there a phase tra ..."
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Cited by 31 (2 self)
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This paper presents an experimental investigation of the following questions: how does the averagecase complexity of random 3SAT, understood as a function of the order (number of variables) for xed density (ratio of number of clauses to order) instances, depend on the density? Is there a phase transition in which the complexity shifts from polynomial to exponential in the order? Is the transition dependent or independent of the solver? Our experiment design uses three complete SAT solvers embodying dierent algorithms: GRASP, CPLEX, and CUDD. We observe new phase transitions for all three solvers, where the median running time shifts from polynomial in the order to exponential. The location of the phase transition appears to be solverdependent. While GRASP and CUDD shift from polynomial to exponential complexity at a density of about 3.8, CUDD exhibits this transition between densities of 0.1 and 0.5. This experimental result underscores the dependence between the solver and the complexity phase transition, and challenges the widely held belief that random 3SAT exhibits a phase transition in computational complexity very close to the crossover point.
Applying the DavisPutnam procedure to nonclausal formulas
 In Proc. AI*IA'99, number 1792 in Lecture Notes in Arti Intelligence
, 1999
"... . Traditionally, the satisability problem for propositional logics deals with formulas in Conjunctive Normal Form (CNF). A typical way to deal with nonCNF formulas requires (i) converting them into CNF, and (ii) applying solvers usually based on the DavisPutnam (DP) procedure. A well known problem ..."
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Cited by 29 (6 self)
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. Traditionally, the satisability problem for propositional logics deals with formulas in Conjunctive Normal Form (CNF). A typical way to deal with nonCNF formulas requires (i) converting them into CNF, and (ii) applying solvers usually based on the DavisPutnam (DP) procedure. A well known problem of this solution is that the CNF conversion may introduce many new variables, thus greatly widening the space of assignments in which the DP procedure has to search in order to nd solutions. In this paper we present two variants of the DP procedure which overcome the problem outlined above. The idea underlying these variants is that splitting should occur only for the variables in the original formula. The CNF conversion methods employed ensure their correctness and completeness. As a consequence, we get two decision procedures for nonCNF formulas (i) which can exploit all the present and future sophisticated technology of current DP implementations, and (ii) whose space of assignments t...
Symbolic Decision Procedures for QBF
 Proceedings of 10th Int. Conf. on Principles and Practice of Constraint Programming (CP 2004
, 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 searchbased. In this work we explore an alternative approach to QBF solving, based on symb ..."
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Cited by 28 (1 self)
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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 searchbased. 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 decisiondiagram 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 multiresolution. 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 searchbased solvers. We show that QBDD is not competitive, but QMRES compares favorably with searchbased solvers on various benchmarks consisting of nonrandom formulas.
Evaluating Search Heuristics and Optimization Techniques in Propositional Satisfiability
 In Proc. of IJCAR 2001, volume 2083 of LNCS
, 2001
"... This paper is devoted to the experimental evaluation of several stateofthe art search heuristics and optimization techniques in propositional satisfiability (SAT). The test set consists of random 3CNF formulas as well as real world instances from planning, scheduling, circuit analysis, bounded ..."
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Cited by 26 (12 self)
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This paper is devoted to the experimental evaluation of several stateofthe art search heuristics and optimization techniques in propositional satisfiability (SAT). The test set consists of random 3CNF formulas as well as real world instances from planning, scheduling, circuit analysis, bounded model checking, and security protocols. All the heuristics and techniques have been implemented in a new library for SAT, called SIM. The comparison is fair because in SIM the selected heuristics and techniques are realized on a common platform. The comparison is significative because SIM as a solver performs very well when compared to other stateoftheart solvers. 1
Simplification  A general constraint propagation technique for propositional and modal tableaux
, 1998
"... . Tableau and sequent calculi are the basis for most popular interactive theorem provers for formal verification. Yet, when it comes to automatic proof search, tableaux are often slower than DavisPutnam, SAT procedures or other techniques. This is partly due to the absence of a bivalence principle ..."
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Cited by 25 (2 self)
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. Tableau and sequent calculi are the basis for most popular interactive theorem provers for formal verification. Yet, when it comes to automatic proof search, tableaux are often slower than DavisPutnam, SAT procedures or other techniques. This is partly due to the absence of a bivalence principle (viz. the cutrule) but there is another source of inefficiency: the lack of constraint propagation mechanisms. This paper proposes an innovation in this direction: the rule of simplification, which plays for tableaux the role of subsumption for resolution and of unit for the DavisPutnam procedure. The simplicity and generality of simplification make possible its extension in a uniform way from propositional logic to a wide range of modal logics. This technique gives an unifying view of a number of tableauxlike calculi such as DPLL, KE, HARP, hypertableaux, BCP, KSAT. We show its practical impact with experimental results for random 3SAT and the industrial IFIP benchmarks for hardware ve...
Extended resolution proofs for conjoining BDDs
 IN: PROC. OF THE 1ST INTL. COMPUTER SCIENCE SYMP. IN RUSSIA (CSR 2006). LNCS 3967
, 2006
"... We present a method to convert the construction of binary decision diagrams (BDDs) into extended resolution proofs. Besides in proof checking, proofs are fundamental to many applications and our results allow the use of BDDs instead—or in combination with—established proof generation techniques, ba ..."
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Cited by 25 (5 self)
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We present a method to convert the construction of binary decision diagrams (BDDs) into extended resolution proofs. Besides in proof checking, proofs are fundamental to many applications and our results allow the use of BDDs instead—or in combination with—established proof generation techniques, based for instance on clause learning. We have implemented a proof generator for propositional logic formulae in conjunctive normal form, called EBDDRES. We present details of our implementation and also report on experimental results. To our knowledge this is the first step towards a practical application of extended resolution.