Solving QBF with Counterexample Guided Refinement

Abstract This article puts forward the application of Counterexample Guided Abstraction Refinement (CEGAR) in solving the well-known PSPACE-complete problem of quantified Boolean formulas (QBF). The article studies the application of CEGAR in two scenarios. In the first scenario, CEGAR is used to expand quantifiers of the formula and subsequently a satisfiability (SAT) solver is applied. First it is shown how to do that for two levels of quantification and then it is generalized for arbitrary number of levels by recursion. It is also shown that these ideas can be generalized to non-prenex and non-CNF QBF solvers. In the second scenario, CEGAR is employed as an additional learning technique in an existing DPLL-based QBF solver. Experimental evaluation of the implemented prototypes shows that the CEGAR-driven solver outperforms existing solvers on a number of benchmark families and that the DPLL solver benefits from the additional type of learning.

[1]  Armin Biere,et al.  Bridging the gap between dual propagation and CNF-based QBF solving , 2013, 2013 Design, Automation & Test in Europe Conference & Exhibition (DATE).

[2]  Mikolás Janota,et al.  Solving QBF with Counterexample Guided Refinement , 2012, SAT.

[3]  Konstantin Korovin.  Instantiation-Based Automated Reasoning: From Theory to Practice , 2009, CADE.

[4]  Enrico Giunchiglia,et al.  Reasoning with Quantified Boolean Formulas , 2021, Handbook of Satisfiability.

[5]  Armin Biere,et al.  DepQBF: A Dependency-Aware QBF Solver (System Description) , 2010 .

[6]  Enrico Giunchiglia,et al.  Solving satisfiability problems with preferences , 2010, Constraints.

[7]  Hans Kleine Büning,et al.  Bounded Universal Expansion for Preprocessing QBF , 2007, SAT.

[8]  Mikolás Janota,et al.  Quantified Maximum Satisfiability: - A Core-Guided Approach , 2013, SAT.

[9]  Fahiem Bacchus,et al.  Exploiting QBF Duality on a Circuit Representation , 2010, AAAI.

[10]  Hans Kleine Büning,et al.  Theory of Quantified Boolean Formulas , 2021, Handbook of Satisfiability.

[11]  G. S. Tseitin.  On the Complexity of Derivation in Propositional Calculus , 1983 .

[12]  Marco Benedetti,et al.  Evaluating QBFs via Symbolic Skolemization , 2005, LPAR.

[13]  David Monniaux,et al.  Quantifier Elimination by Lazy Model Enumeration , 2010, CAV.

[14]  Mikolás Janota,et al.  Expansion-based QBF solving versus Q-resolution , 2015, Theor. Comput. Sci..

[15]  Klaus Schneider,et al.  Solving Games Using Incremental Induction , 2013, IFM.

[16]  Lintao Zhang,et al.  Solving QBF by Combining Conjunctive and Disjunctive Normal Forms , 2006, AAAI.

[17]  Fahiem Bacchus,et al.  Exploiting Circuit Representations in QBF Solving , 2010, SAT.

[18]  Moshe Y. Vardi,et al.  Symbolic Decision Procedures for QBF , 2004, CP.

[19]  David A. Basin,et al.  QUBOS: Deciding Quantified Boolean Logic Using Propositional Satisfiability Solvers , 2002, FMCAD.

[20]  Mikolás Janota,et al.  On Propositional QBF Expansions and Q-Resolution , 2013, SAT.

[21]  Radu Grigore,et al.  Counterexample Guided Abstraction Refinement Algorithm for Propositional Circumscription , 2010, JELIA.

[22]  Armin Biere,et al.  Nenofex: Expanding NNF for QBF Solving , 2008, SAT.

[23]  Mikolás Janota,et al.  How to Complete an Interactive Configuration Process? , 2010, SOFSEM.

[24]  M. Schaefer,et al.  Completeness in the Polynomial-Time Hierarchy A Compendium ∗ , 2008 .

[25]  David L. Dill,et al.  Checking Satisfiability of First-Order Formulas by Incremental Translation to SAT , 2002, CAV.

[26]  Lane A. Hemaspaandra,et al.  SIGACT news complexity theory comun 37 , 2002, SIGA.

[27]  Enrico Giunchiglia,et al.  sQueezeBF: An Effective Preprocessor for QBFs Based on Equivalence Reasoning , 2010, SAT.

[28]  Mikolás Janota,et al.  Abstraction-Based Algorithm for 2QBF , 2011, SAT.

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

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

[31]  Mikolás Janota,et al.  On Unification of QBF Resolution-Based Calculi , 2014, MFCS.

[32]  Fahiem Bacchus,et al.  Binary Clause Reasoning in QBF , 2006, SAT.

[33]  Marco Benedetti,et al.  QBF-Based Formal Verification: Experience and Perspectives , 2008, J. Satisf. Boolean Model. Comput..

[34]  Harald Ruess,et al.  Lazy Theorem Proving for Bounded Model Checking over Infinite Domains , 2002, CADE.

[35]  Youssef Hamadi,et al.  Efficiently solving quantified bit-vector formulas , 2010, Formal Methods in Computer Aided Design.

[36]  Felip Manyà,et al.  MaxSAT, Hard and Soft Constraints , 2021, Handbook of Satisfiability.

[37]  Edmund M. Clarke,et al.  A Non-prenex, Non-clausal QBF Solver with Game-State Learning , 2010, SAT.

[38]  Jie-Hong Roland Jiang,et al.  Unified QBF certification and its applications , 2012, Formal Methods Syst. Des..

[39]  Stefan Woltran,et al.  Complexity-sensitive decision procedures for abstract argumentation , 2012, Artif. Intell..

[40]  Niklas Sörensson,et al.  An Extensible SAT-solver , 2003, SAT.

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

[42]  Anja Remshagen,et al.  A SAT-based solver for Q-ALL SAT , 2006, ACM-SE 44.

[43]  Florian Lonsing,et al.  The QBF Gallery: Behind the scenes , 2016, Artif. Intell..

[44]  Fahiem Bacchus,et al.  Using SAT in QBF , 2005, CP.

[45]  Masahiro Fujita,et al.  SAT-Based Automatic Rectification and Debugging of Combinational Circuits with LUT Insertions , 2012, Asian Test Symposium.

[46]  Allen Van Gelder.  Contributions to the Theory of Practical Quantified Boolean Formula Solving , 2012, CP.

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

[48]  Helmut Veith,et al.  Counterexample-guided abstraction refinement for symbolic model checking , 2003, JACM.

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

[50]  Armin Biere,et al.  Failed Literal Detection for QBF , 2011, SAT.

[51]  Mikolás Janota,et al.  QBf-based boolean function bi-decomposition , 2012, 2012 Design, Automation & Test in Europe Conference & Exhibition (DATE).

[52]  Armin Biere,et al.  Resolve and Expand , 2004, SAT.

[53]  Lukasz Kaiser,et al.  Experiments with Reduction Finding , 2013, SAT.

[54]  Fahiem Bacchus,et al.  Preprocessing QBF , 2006, CP.

[55]  Armin Biere,et al.  Blocked Clause Elimination for QBF , 2011, CADE.

[56]  Christopher Umans.  The Minimum Equivalent DNF Problem and Shortest Implicants , 2001, J. Comput. Syst. Sci..

[57]  Leonid Ryzhyk,et al.  Solving Games without Controllable Predecessor , 2014, CAV.

[58]  Uwe Bubeck,et al.  Model-based transformations for quantified boolean formulas , 2010, DISKI.

[59]  Karem A. Sakallah,et al.  Computing Vertex Eccentricity in Exponentially Large Graphs: QBF Formulation and Solution , 2003, SAT.

[60]  Marco Benedetti,et al.  sKizzo: A Suite to Evaluate and Certify QBFs , 2005, CADE.

[61]  Mikolás Janota,et al.  Proof Complexity of Resolution-based QBF Calculi , 2015, STACS.

[62]  Jussi Rintanen,et al.  Asymptotically Optimal Encodings of Conformant Planning in QBF , 2007, AAAI.