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.

[1]  J. A. Robinson,et al.  A Machine-Oriented Logic Based on the Resolution Principle , 1965, JACM.

[2]  Moshe Y. Vardi,et al.  Optimizing a BDD-Based Modal Solver , 2003, CADE.

[3]  David A. Plaisted,et al.  A Structure-Preserving Clause Form Translation , 1986, J. Symb. Comput..

[4]  David A. Basin,et al.  Bounded Model Construction for Monadic Second-Order Logics , 2000, CAV.

[5]  Marco Schaerf,et al.  An Algorithm to Evaluate Quantified Boolean Formulae and Its Experimental Evaluation , 2002, Journal of Automated Reasoning.

[6]  Roberto J. Bayardo,et al.  Using CSP Look-Back Techniques to Solve Real-World SAT Instances , 1997, AAAI/IAAI.

[7]  Reinhold Letz,et al.  Lemma and Model Caching in Decision Procedures for Quantified Boolean Formulas , 2002, TABLEAUX.

[8]  Harald Ganzinger,et al.  Resolution Theorem Proving , 2001, Handbook of Automated Reasoning.

[9]  Daniel P. Miranker,et al.  A Complexity Analysis of Space-Bounded Learning Algorithms for the Constraint Satisfaction Problem , 1996, AAAI/IAAI, Vol. 1.

[10]  Thierry Boy de la Tour.  Minimizing the Number of Clauses by Renaming , 1990, CADE.

[11]  Sharad Malik,et al.  Towards a Symmetric Treatment of Satisfaction and Conflicts in Quantified Boolean Formula Evaluation , 2002, CP.

[12]  Bernd Becker,et al.  Checking equivalence for partial implementations , 2001, Proceedings of the 38th Design Automation Conference (IEEE Cat. No.01CH37232).

[13]  Joao Marques-Silva,et al.  GRASP-A new search algorithm for satisfiability , 1996, Proceedings of International Conference on Computer Aided Design.

[14]  Armando Tacchella,et al.  QBF Reasoning on Real-World Instances , 2004, SAT.

[15]  Ian P. Gent,et al.  Solution Learning and Solution Directed Backjumping, Revisited , 2004 .

[16]  Sharad Malik,et al.  Conflict driven learning in a quantified Boolean Satisfiability solver , 2002, ICCAD 2002.

[17]  Armando Tacchella,et al.  Challenges in the QBF Arena: the SAT'03 Evaluation of QBF Solvers , 2003, SAT.

[18]  Sharad Malik,et al.  Efficient conflict driven learning in a Boolean satisfiability solver , 2001, IEEE/ACM International Conference on Computer Aided Design. ICCAD 2001. IEEE/ACM Digest of Technical Papers (Cat. No.01CH37281).

[19]  Patrick Prosser,et al.  HYBRID ALGORITHMS FOR THE CONSTRAINT SATISFACTION PROBLEM , 1993, Comput. Intell..

[20]  J. A. Robinson,et al.  The Generalized Resolution Principle , 1983 .

[21]  Sloman,et al.  Automation of Reasoning , 1983, Symbolic Computation.

[22]  Marco Schaerf,et al.  An Algorithm to Evaluate Quantified Boolean Formulae , 1998, AAAI/IAAI.

[23]  Armando Tacchella,et al.  Watched Data Structures for QBF Solvers , 2003, SAT.

[24]  Jussi Rintanen,et al.  Constructing Conditional Plans by a Theorem-Prover , 1999, J. Artif. Intell. Res..

[25]  Armando Tacchella,et al.  SAT-based planning in complex domains: Concurrency, constraints and nondeterminism , 2003, Artif. Intell..

[26]  Donald W. Loveland,et al.  A machine program for theorem-proving , 2011, CACM.

[27]  Armando Tacchella,et al.  Learning for quantified boolean logic satisfiability , 2002, AAAI/IAAI.

[28]  Hans Kleine Büning,et al.  Resolution for Quantified Boolean Formulas , 1995, Inf. Comput..

[29]  Christian G. Fermüller,et al.  Resolution Decision Procedures , 2001, Handbook of Automated Reasoning.

[30]  Armando Tacchella,et al.  Backjumping for Quantified Boolean Logic satisfiability , 2001, Artif. Intell..

[31]  Rina Dechter,et al.  Enhancement Schemes for Constraint Processing: Backjumping, Learning, and Cutset Decomposition , 1990, Artif. Intell..

[32]  Armando Tacchella,et al.  QuBE++: An Efficient QBF Solver , 2004, FMCAD.

[33]  Graham Wrightson,et al.  Automation of reasoning--classical papers on computational logic , 2012 .

[34]  Armando Tacchella,et al.  Monotone Literals and Learning in QBF Reasoning , 2004, CP.

[35]  Sharad Malik,et al.  Chaff: engineering an efficient SAT solver , 2001, Proceedings of the 38th Design Automation Conference (IEEE Cat. No.01CH37232).

[36]  Matthew L. Ginsberg,et al.  Dynamic Backtracking , 1993, J. Artif. Intell. Res..

[37]  Karem A. Sakallah,et al.  GRASP—a new search algorithm for satisfiability , 1996, ICCAD 1996.