Quantified maximum satisfiability

In recent years, there have been significant improvements in algorithms both for Quantified Boolean Formulas (QBF) and for Maximum Satisfiability (MaxSAT). This paper studies an optimization extension of QBF and considers the problem in a quantified MaxSAT setting. More precisely, the general QMaxSAT problem is defined for QBFs with a set of soft clausal constraints and consists in finding the largest subset of the soft constraints such that the remaining QBF is true. Two approaches are investigated. One is based on relaxing the soft clauses and performing an iterative search on the cost function. The other approach, which is the main contribution of the paper, is inspired by recent work on MaxSAT, and exploits the iterative identification of unsatisfiable cores. The paper investigates the application of these approaches to the two concrete problems of computing smallest minimal unsatisfiable subformulas (SMUS) and smallest minimal equivalent subformulas (SMES), decision versions of which are well-known problems in the second level of the polynomial hierarchy. Experimental results, obtained on representative problem instances, indicate that the core-guided approach for the SMUS and SMES problems outperforms the use of iterative search over the values of the cost function. More significantly, the core-guided approach to SMUS also outperforms the state-of-the-art SMUS extractor Digger.

[1]  Raymond Reiter,et al.  A Theory of Diagnosis from First Principles , 1986, Artif. Intell..

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

[3]  Sharad Malik,et al.  Validating SAT solvers using an independent resolution-based checker: practical implementations and other applications , 2003, 2003 Design, Automation and Test in Europe Conference and Exhibition.

[4]  Jie-Hong Roland Jiang,et al.  QBF Resolution Systems and Their Proof Complexities , 2014, SAT.

[5]  Peter J. Stuckey,et al.  Interactive type debugging in Haskell , 2003, Haskell '03.

[6]  Daniel Kroening,et al.  Word level predicate abstraction and refinement for verifying RTL Verilog , 2005, Proceedings. 42nd Design Automation Conference, 2005..

[7]  Armin Biere,et al.  Managing SAT inconsistencies with HUMUS , 2012, VaMoS '12.

[8]  Florian Lonsing,et al.  Incremental QBF Solving , 2014, CP.

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

[10]  Paolo Liberatore,et al.  Redundancy in logic I: CNF propositional formulae , 2002, Artif. Intell..

[11]  Joao Marques-Silva,et al.  SAT-Based Preprocessing for MaxSAT , 2013, LPAR.

[12]  Oliver Kullmann,et al.  Constraint Satisfaction Problems in Clausal Form II: Minimal Unsatisfiability and Conflict Structure , 2011, Fundam. Informaticae.

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

[14]  Egon Balas,et al.  Nonlinear 0–1 programming: I. Linearization techniques , 1984, Math. Program..

[15]  Peter L. Hammer,et al.  Boolean Methods in Operations Research and Related Areas , 1968 .

[16]  Joan Feigenbaum,et al.  Probabilistically Checkable Debate Systems and Nonapproximability of PSPACE-Hard Functions , 1995, Chic. J. Theor. Comput. Sci..

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

[18]  Hans Kleine Büning,et al.  Boolean Functions as Models for Quantified Boolean Formulas , 2007, Journal of Automated Reasoning.

[19]  Karem A. Sakallah,et al.  On Finding All Minimally Unsatisfiable Subformulas , 2005, SAT.

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

[21]  Daniel Le Berre,et al.  The Sat4j library, release 2.2 , 2010, J. Satisf. Boolean Model. Comput..

[22]  Inês Lynce,et al.  A branch and bound algorithm for extracting smallest minimal unsatisfiable subformulas , 2008, Constraints.

[23]  Vasco M. Manquinho,et al.  Algorithms for Weighted Boolean Optimization , 2009, SAT.

[24]  Eliezer L. Lozinskii,et al.  Consistent subsets of inconsistent systems: structure and behaviour , 2003, J. Exp. Theor. Artif. Intell..

[25]  Karem A. Sakallah,et al.  Algorithms for Computing Minimal Unsatisfiable Subsets of Constraints , 2007, Journal of Automated Reasoning.

[26]  Karem A. Sakallah,et al.  Refinement strategies for verification methods based on datapath abstraction , 2006, Asia and South Pacific Conference on Design Automation, 2006..

[27]  Ofer Strichman,et al.  Faster Extraction of High-Level Minimal Unsatisfiable Cores , 2011, SAT.

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

[29]  Nina Narodytska,et al.  Maximum Satisfiability Using Core-Guided MaxSAT Resolution , 2014, AAAI.

[30]  Mikolás Janota,et al.  Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence On Computing Minimal Correction Subsets , 2022 .

[31]  Bart Selman,et al.  The Achilles' Heel of QBF , 2005, AAAI.

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

[33]  João P. Marques Silva.  Minimal Unsatisfiability: Models, Algorithms and Applications (Invited Paper). , 2010, ISMVL 2010.

[34]  James Bailey,et al.  Discovery of Minimal Unsatisfiable Subsets of Constraints Using Hitting Set Dualization , 2005, PADL.

[35]  Joao Marques-Silva,et al.  Improvements to Core-Guided Binary Search for MaxSAT , 2012, SAT.

[36]  Maria Luisa Bonet,et al.  Solving (Weighted) Partial MaxSAT through Satisfiability Testing , 2009, SAT.

[37]  Fahiem Bacchus,et al.  Recovering and Utilizing Partial Duality in QBF , 2013, SAT.

[38]  Hubie Chen,et al.  Optimization, Games, and Quantified Constraint Satisfaction , 2004, MFCS.

[39]  Sharad Malik,et al.  On Solving the Partial MAX-SAT Problem , 2006, SAT.

[40]  Inês Lynce,et al.  Towards efficient MUS extraction , 2012, AI Commun..

[41]  Wolfgang Küchlin,et al.  Formal methods for the validation of automotive product configuration data , 2003, Artificial Intelligence for Engineering Design, Analysis and Manufacturing.

[42]  Lintao Zhang,et al.  Solving QBF with combined conjunctive and disjunctive normal form , 2006, AAAI 2006.

[43]  Eugene Goldberg,et al.  Verification of proofs of unsatisfiability for CNF formulas , 2003, 2003 Design, Automation and Test in Europe Conference and Exhibition.

[44]  Maria Luisa Bonet,et al.  SAT-based MaxSAT algorithms , 2013, Artif. Intell..

[45]  Sikun Li,et al.  Extracting Minimum Unsatisfiable Cores with a Greedy Genetic Algorithm , 2006, Australian Conference on Artificial Intelligence.

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

[47]  Mikolás Janota,et al.  On QBF Proofs and Preprocessing , 2013, LPAR.

[48]  S. Malik,et al.  Validating the result of a quantified Boolean formula (QBF) solver: theory and practice , 2005, Proceedings of the ASP-DAC 2005. Asia and South Pacific Design Automation Conference, 2005..

[49]  Maria Luisa Bonet,et al.  A New Algorithm for Weighted Partial MaxSAT , 2010, AAAI.

[50]  Bernd Becker,et al.  QBF with Soft Variables , 2014, Electronic Communication of The European Association of Software Science and Technology.

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

[52]  Vasco M. Manquinho,et al.  Progression in Maximum Satisfiability , 2014, ECAI.

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

[54]  Inês Lynce,et al.  A Branch-and-Bound Algorithm for Extracting Smallest Minimal Unsatisfiable Formulas , 2005, SAT.

[55]  Joao Marques-Silva,et al.  Core-Guided Binary Search Algorithms for Maximum Satisfiability , 2011, AAAI.

[56]  Joao Marques-Silva,et al.  Core-Guided MaxSAT with Soft Cardinality Constraints , 2014, International Conference on Principles and Practice of Constraint Programming.

[57]  Joao Marques-Silva,et al.  Iterative and core-guided MaxSAT solving: A survey and assessment , 2013, Constraints.

[58]  Jie-Hong Roland Jiang,et al.  Resolution Proofs and Skolem Functions in QBF Evaluation and Applications , 2011, CAV.

[59]  Inês Lynce,et al.  On Computing Minimum Unsatisfiable Cores , 2004, SAT.

[60]  Mikolás Janota,et al.  On Computing Minimal Equivalent Subformulas , 2012, CP.

[61]  Florian Lonsing,et al.  Long-Distance Resolution: Proof Generation and Strategy Extraction in Search-Based QBF Solving , 2013, LPAR.

[62]  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).

[63]  Marco Benedetti,et al.  Quantified Constraint Optimization , 2008, CP.

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

[65]  Edmund M. Clarke,et al.  Learning abstractions for model checking , 2006 .

[66]  Armin Biere,et al.  Resolution-Based Certificate Extraction for QBF - (Tool Presentation) , 2012, SAT.

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

[68]  Armin Biere,et al.  A Unified Proof System for QBF Preprocessing , 2014, IJCAR.

[69]  Karem A. Sakallah,et al.  Reveal: A Formal Verification Tool for Verilog Designs , 2008, LPAR.

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

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

[72]  Peter J. Stuckey,et al.  There Are No CNF Problems , 2013, SAT.

[73]  Vasco M. Manquinho,et al.  Pseudo-Boolean and Cardinality Constraints , 2021, Handbook of Satisfiability.

[74]  Shie-Jue Lee,et al.  Deriving minimal conflict sets by CS-trees with mark set in diagnosis from first principles , 1999, IEEE Trans. Syst. Man Cybern. Part B.

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

[76]  Mikolás Janota,et al.  Algorithms for computing minimal equivalent subformulas , 2014, Artif. Intell..

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

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