An Algorithm to Evaluate Quantified Boolean Formulae and Its Experimental Evaluation

The high computational complexity of advanced reasoning tasks such as reasoning about knowledge and planning calls for efficient and reliable algorithms for reasoning problems harder than NP. In this paper we propose Evaluate, an algorithm for evaluating quantified Boolean formulae (QBFs). Algorithms for evaluation of QBFs are suitable for experimental analysis of problems that belong to a wide range of complexity classes, a property not easily found in other formalisms. Evaluate is a generalization of the Davis–Putnam procedure for SAT and is guaranteed to work in polynomial space. Before presenting the algorithm, we discuss several abstract properties of QBFs that we singled out to make it more efficient. We also discuss various options that were investigated about heuristics and data structures and report the main results of the experimental analysis. In particular, Evaluate is orders of magnitude more efficient than a nested backtracking procedure that resorts to a Davis–Putnam algorithm for handling the innermost set of quantifiers. Moreover, experiments show that randomly generated QBFs exhibit regular patterns such as phase transition and easy-hard-easy distribution.

[1]  Larry J. Stockmeyer,et al.  The Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

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

[3]  James M. Crawford,et al.  Experimental Results on the Crossover Point in Random 3-SAT , 1996, Artif. Intell..

[4]  Bart Selman,et al.  Pushing the Envelope: Planning, Propositional Logic and Stochastic Search , 1996, AAAI/IAAI, Vol. 2.

[5]  Klaus Schild,et al.  A Correspondence Theory for Terminological Logics: Preliminary Report , 1991, IJCAI.

[6]  John McCarthy,et al.  Circumscription - A Form of Non-Monotonic Reasoning , 1980, Artif. Intell..

[7]  Ullrich Hustadt,et al.  On Evaluating Decision Procedures for Modal Logics , 1997 .

[8]  Fausto Giunchiglia,et al.  More Evaluation of Decision Procedures for Modal Logics , 1998, KR.

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

[10]  Marco Cadoli,et al.  An Algorithm to Evaluate Quanti ed Boolean Formulae , 1998 .

[11]  Jussi Rintanen,et al.  Improvements to the Evaluation of Quantified Boolean Formulae , 1999, IJCAI.

[12]  Hector J. Levesque,et al.  A New Method for Solving Hard Satisfiability Problems , 1992, AAAI.

[13]  Marco Schaerf,et al.  A Survey of Complexity Results for Planning , 1993 .

[14]  Albert R. Meyer,et al.  Word problems requiring exponential time(Preliminary Report) , 1973, STOC.

[15]  Joseph Y. Halpern.  The Effect of Bounding the Number of Primitive Propositions and the Depth of Nesting on the Complexity of Modal Logic , 1995, Artif. Intell..

[16]  Ullrich Hustadt,et al.  On Evaluating Decision Procedures for Modal Logic , 1997, IJCAI.

[17]  Vladimir Lifschitz,et al.  Computing Circumscription , 1985, IJCAI.

[18]  Raymond Reiter,et al.  A Logic for Default Reasoning , 1987, Artif. Intell..

[19]  Peter F. Patel-Schneider,et al.  Optimising Propositional Modal Satisfiability for Description Logic Subsumption , 1998, AISC.

[20]  James M. Crawford,et al.  Experimental Results on the Crossover Point inSatis ability , 1993 .

[21]  Georg Gottlob,et al.  The Complexity of Logic-Based Abduction , 1993, STACS.

[22]  Marco Schaerf,et al.  Experimental Analysis of the Computational Cost of Evaluating Quantified Boolean Formulae , 1997, AI*IA.

[23]  H. Levesque,et al.  Generating Hard Satis ability Problems ? , 1995 .

[24]  Hector J. Levesque,et al.  Generating Hard Satisfiability Problems , 1996, Artif. Intell..

[25]  Fausto Giunchiglia,et al.  A SAT-based Decision Procedure for ALC , 1996, KR.

[26]  Robert E. Tarjan,et al.  A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean Formulas , 1979, Inf. Process. Lett..

[27]  Georg Gottlob,et al.  An Efficient Method for Eliminating Varying Predicates from a Circumscription , 1992, Artif. Intell..

[28]  Georg Gottlob,et al.  On the complexity of propositional knowledge base revision, updates, and counterfactuals , 1992, Artif. Intell..

[29]  Richard E. Ladner,et al.  The Computational Complexity of Provability in Systems of Modal Propositional Logic , 1977, SIAM J. Comput..

[30]  Ian Horrocks,et al.  Using an Expressive Description Logic: FaCT or Fiction? , 1998, KR.

[31]  Toby Walsh,et al.  Beyond NP: the QSAT phase transition , 1999, AAAI/IAAI.

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

[33]  Fausto Giunchiglia,et al.  Building Decision Procedures for Modal Logics from Propositional Decision Procedures: The Case Study of Modal K(m) , 2000, Inf. Comput..