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Local-search Extraction of MUSes

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Abstract

SAT is probably one of the most-studied constraint satisfaction problems. In this paper, a new hybrid technique based on local search is introduced in order to approximate and extract minimally unsatisfiable subformulas (in short, MUSes) of unsatisfiable SAT instances. It is based on an original counting heuristic grafted to a local search algorithm, which explores the neighborhood of the current interpretation in an original manner, making use of a critical clause concept. Intuitively, a critical clause is a falsified clause that becomes true thanks to a local search flip only when some other clauses become false at the same time. In the paper, the critical clause concept is investigated. It is shown to be the cornerstone of the efficiency of our approach, which outperforms competing ones to compute MUSes, inconsistent covers and sets of MUSes, most of the time.

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References

  1. Bailey, J., & Stuckey, P. J. (2005). Discovery of minimal unsatisfiable subsets of constraints using hitting set dualization. In International Symposium on Practical Aspects of Declarative Languages (PADL’05), pages 174–186.

  2. Boussemart, F., Hémery, F., Lecoutre, C., & Saïs, L. (2004). Boosting systematic search by weighting constraints. In European Conference on Artificial Intelligence (ECAI’04), pages 146–150.

  3. Brisoux, L., Grégoire, É., & Saïs, L. (2001). Checking depth-limited consistency and inconsistency in knowledge-based systems. Int. J. Intell. Syst. 16(3): 333–360.

    Article  Google Scholar 

  4. Bruni, R. (2003). Approximating minimal unsatisfiable subformulae by means of adaptive core search. Discrete Appl. Math. 130(2): 85–100.

    Article  MATH  Google Scholar 

  5. Bruni, R. (2005). On exact selection of minimally unsatisfiable subformulae. Ann. Math. Artif. Intell. 43(1): 35–50.

    Article  MATH  Google Scholar 

  6. Büning, H. K. (2000). On subclasses of minimal unsatisfiable formulas. Discrete Appl. Math. 107(1–3): 83–98.

    Article  MATH  Google Scholar 

  7. Davis, M., Logemann, G., & Loveland, D. (1962). A machine program for theorem proving. J. Assoc. Comput. Mach. 5(7): 394–397.

    MATH  Google Scholar 

  8. Davydov, G., Davydova, I., & Büning, H. K. (1998). An efficient algorithm for the minimal unsatisfiability problem for a subclass of CNF. Ann. Math. Artif. Intell. 23(3,4): 229–245.

    Article  MATH  Google Scholar 

  9. DIMACS (1993). Benchmarks on SAT. ftp://dimacs.rutgers.edu/pub/challenge/-satisfiability/.

  10. Eiter, T., & Gottlob, G. (1992). On the complexity of propositional knowledge base revision, updates and counterfactual. Artif. Intell. 57: 227–270.

    Article  Google Scholar 

  11. Fleischner, H., Kullman, O., & Szeider, S. (2002). Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference. Theor. Comput. Sci. 289(1): 503–516.

    Article  MATH  Google Scholar 

  12. Grégoire, É., Mazure, B., & Piette, C. (2007). Boosting a complete technique to find MSSes and MUSes thanks to a local search oracle. In International Joint Conference on Artificial Intelligence (IJCAI’07), vol. 2, pages 2300–2305.

  13. Grégoire, É., & Ansart, D. (2000). Overcoming the Christmas tree syndrome. Int. J. Artif. Intell. Tools (IJAIT) 9(2): 97–111.

    Article  Google Scholar 

  14. Grégoire, É., Mazure, B., & Saïs, L. (2002). Using failed local search for SAT as an oracle for tackling harder A.I. problems more efficiently. In International Conference on Artificial Intelligence: Methodology, Systems, Applications (AIMSA’02). LNCS, number 2443, pages 51–60. Springer.

  15. Hamscher, W., Console, L., & de Kleer, J., eds. (1992) Readings in Model-based Diagnosis. San Mateo, CA: Morgan Kaufmann.

    Google Scholar 

  16. Huang, J. (2005). MUP: A minimal unsatisfiability prover. In Asia and South Pacific Design Automation Conference (ASP-DAC’05), pages 432–437.

  17. Junker, U. (2001). QuickXPlain: Conflict detection for arbitrary constraint propagation algorithms. In IJCAI’01 Workshop on Modelling and Solving problems with constraints (CONS-1), pages 75–82.

  18. Kautz, H., Selman, B., & McAllester, D. (2004). Walksat in the SAT 2004 competition. In International Conference on Theory and Applications of Satisfiability Testing (SAT’04).

  19. Kullmann, O., Lynce, I., & Marques Silva, J. P. (2006). Categorisation of clauses in conjunctive normal forms: Minimally unsatisfiable sub-clause-sets and the lean kernel. In International Conference on Theory and Applications of Satisfiability Testing (SAT’06), pages 22–35.

  20. Liffiton, M. H., & Sakallah, K. A. (2005). On finding all minimally unsatisfiable subformulas. In International Conference on Theory and Applications of Satisfiability Testing (SAT’05), pages 173–186.

  21. Lynce, I., & Marques-Silva, J. (2004). On computing minimum unsatisfiable cores. In International Conference on Theory and Applications of Satisfiability Testing (SAT’04).

  22. Mazure, B., Saïs, L., & Grégoire, É. (1996). A powerful heuristic to locate inconsistent kernels in knowledge-based systems. In International Conference on Information Processing and Management of Uncertainty in Knowledge-based Systems (IPMU’96), pages 1265–1269.

  23. Mazure, B., Saïs, L., & Grégoire, É. (1998). Boosting complete techniques thanks to local search. Ann. Math. Artif. Intell. 22: 319–322.

    Article  MATH  Google Scholar 

  24. Mneimneh, M. N., Lynce, I., Andraus, Z. S., Marques Silva, J. P., & Sakallah, K. A. (2005). A branch-and-bound algorithm for extracting smallest minimal unsatisfiable formulas. In International Conference on Theory and Applications of Satisfiability Testing (SAT’05), pages 467–474.

  25. Oh, Y., Mneimneh, M. N., Andraus, Z. S., Sakallah, K. A., & Markov, I. L. (2004). AMUSE: a minimally-unsatisfiable subformula extractor. In Design Automation Conference (DAC’04), pages 518–523.

  26. Papadimitriou, C. H., & Wolfe, D. (1988). The complexity of facets resolved. J. Comput. Syst. Sci. 37(1): 2–13.

    Article  MATH  Google Scholar 

  27. SATLIB (2004). Benchmarks on SAT. http://www.intellektik.informatik.tu-darmstadt.de/SATLIB/benchm.html.

  28. Zhang, L., & Malik, S. (2003). Extracting small unsatisfiable cores from unsatisfiable boolean formula. In International Conference on Theory and Applications of Satisfiability Testing (SAT’03).

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Authors and Affiliations

  1. CRIL-CNRS & IRCICA, Université d’Artois, rue Jean Souvraz SP18, F-62307, Lens Cedex, France

    Éric Grégoire, Bertrand Mazure & Cédric Piette

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  1. Éric Grégoire
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  2. Bertrand Mazure
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  3. Cédric Piette
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Correspondence to Éric Grégoire.

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Grégoire, É., Mazure, B. & Piette, C. Local-search Extraction of MUSes. Constraints 12, 325–344 (2007). https://doi.org/10.1007/s10601-007-9019-7

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