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

Property specification patterns at work: verification and inconsistency explanation

Property specification patterns (PSPs) have been proposed to ease the formalization of requirements, yet enable automated verification thereof. In particular, the internal consistency of specifications written with PSPs can be checked automatically with the use of, for example, linear temporal logic (LTL) satisfiability solvers. However, for most practical applications, the expressiveness of PSPs is too restricted to enable writing useful requirement specifications, and proving that a set of requirements is inconsistent can be worthless unless a minimal set of conflicting requirements is extracted to help designers to correct a wrong specification. In this paper, we extend PSPs by considering Boolean as well as atomic numerical assertions, we contribute an encoding from extended PSPs to LTL formulas, and we present an algorithm computing inconsistency explanations, i.e., irreducible inconsistent subsets of the original set of requirements. Our extension enables us to reason about the internal consistency of functional requirements which would not be captured by basic PSPs. Experimental results demonstrate that our approach can check and explain (in)consistencies in specifications with nearly two thousand requirements generated using a probabilistic model, and that it enables effective handling of real-world case studies

Quantifier Structure in search based procedures for QBFs

none3The best currently available solvers for quantified Boolean formulas (QBFs) process their input in prenex form, i.e., all the quan- tifiers have to appear in the prefix of the formula separated from the purely propositional part representing the matrix. However, in many QBFs derived from applications, the propositional part is intertwined with the quantifier structure. To tackle this problem, the standard approach is to convert such QBFs in prenex form, thereby losing structural information about the prefix. In the case of search-based solvers, the prenex-form conversion introduces additional constraints on the branching heuristic and reduces the benefits of the learning mechanisms. In this paper, we show that conversion to prenex form is not necessary: current search-based solvers can be naturally extended in order to handle nonprenex QBFs and to exploit the original quantifier structure. We highlight the two mentioned drawbacks of the conversion in prenex form with a simple example, and we show that our ideas can also be useful for solving QBFs in prenex form. To validate our claims, we implemented our ideas in the state-of-the-art search-based solver QUBE and conducted an extensive experimental analysis. The results show that very substantial speedups can be obtained.The IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems (TCAD) is a monthly publication. Manuscripts considered for publication should focus on algorithms, methods, techniques, and tools for the automated design of integrated circuits and systems, and on related areas. Submitted papers may be of a tutorial or research nature. Research papers must present original contributions and must show significant new material over descriptions or derivations available elsewhere. Tutorial papers should review the state-of-the-art in specific areas of CAD and, at the same time, present readers with key research perspectives and future challenges.GIUNCHIGLIA E.; NARIZZANO M.; TACCHELLA A.Giunchiglia, Enrico; Narizzano, Massimo; Tacchella, Armand

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

Quantifier Structure in search based procedures for QBFs

The best currently available solvers for quantified Boolean formulas (QBFs) process their input in prenex form, i.e., all the quan- tifiers have to appear in the prefix of the formula separated from the purely propositional part representing the matrix. However, in many QBFs derived from applications, the propositional part is intertwined with the quantifier structure. To tackle this problem, the standard approach is to convert such QBFs in prenex form, thereby losing structural information about the prefix. In the case of search-based solvers, the prenex-form conversion introduces additional constraints on the branching heuristic and reduces the benefits of the learning mechanisms. In this paper, we show that conversion to prenex form is not necessary: current search-based solvers can be naturally extended in order to handle nonprenex QBFs and to exploit the original quantifier structure. We highlight the two mentioned drawbacks of the conversion in prenex form with a simple example, and we show that our ideas can also be useful for solving QBFs in prenex form. To validate our claims, we implemented our ideas in the state-of-the-art search-based solver QUBE and conducted an extensive experimental analysis. The results show that very substantial speedups can be obtained

Quantifier Structure in search based procedures for QBFs

The best currently available solvers for quantified Boolean formulas (QBFs) process their input in prenex form, i.e., all the quan- tifiers have to appear in the prefix of the formula separated from the purely propositional part representing the matrix. However, in many QBFs derived from applications, the propositional part is intertwined with the quantifier structure. To tackle this problem, the standard approach is to convert such QBFs in prenex form, thereby losing structural information about the prefix. In the case of search-based solvers, the prenex-form conversion introduces additional constraints on the branching heuristic and reduces the benefits of the learning mechanisms. In this paper, we show that conversion to prenex form is not necessary: current search-based solvers can be naturally extended in order to handle nonprenex QBFs and to exploit the original quantifier structure. We highlight the two mentioned drawbacks of the conversion in prenex form with a simple example, and we show that our ideas can also be useful for solving QBFs in prenex form. To validate our claims, we implemented our ideas in the state-of-the-art search-based solver QUBE and conducted an extensive experimental analysis. The results show that very substantial speedups can be obtained