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Abstract

Minimal Unsatisfiable Subsets (MUSes) and Minimal Correction Subsets (MCSes) are essential tools for the analysis of unsatisfiable formulas. MUSes and MCSes find a growing number of applications, that include abstraction refinement in software verification, type debugging, software package management and software configuration, among many others. In some applications, there can exist preferences over which clauses to include in computed MUSes or MCSes, but also in computed Maximal Satisfiable Subsets (MSSes). Moreover, different definitions of preferred MUSes, MCSes and MSSes can be considered. This paper revisits existing definitions of preferred MUSes, MCSes and MSSes of unsatisfiable formulas, and develops a preliminary characterization of the computational complexity of computing preferred MUSes, MCSes and MSSes. Moreover, the paper investigates which of the existing algorithms and pruning techniques can be applied for computing preferred MUSes, MCSes and MSSes. Finally, the paper shows that the computation of preferred sets can have significant impact in practical performance.

Keywords

Conjunctive Normal Form Pruning Technique Polynomial Number Membership Problem Conjunctive Normal Form Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Ansótegui, C., Bonet, M.L., Levy, J.: SAT-based MaxSAT algorithms. Artif. Intell. 196, 77–105 (2013)CrossRefzbMATHGoogle Scholar
  2. 2.
    Argelich, J., Lynce, I., Marques-Silva, J.: On solving boolean multilevel optimization problems. In: Boutilier, C. (ed.) IJCAI, pp. 393–398 (2009)Google Scholar
  3. 3.
    Bailey, J., Stuckey, P.J.: Discovery of minimal unsatisfiable subsets of constraints using hitting set dualization. In: Hermenegildo, M.V., Cabeza, D. (eds.) PADL 2004. LNCS, vol. 3350, pp. 174–186. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Bakker, R.R., Dikker, F., Tempelman, F., Wognum, P.M.: Diagnosing and solving over-determined constraint satisfaction problems. In: Bajcsy, R. (ed.) IJCAI, pp. 276–281. Morgan Kaufmann (1993)Google Scholar
  5. 5.
    Belov, A., Lynce, I., Marques-Silva, J.: Towards efficient MUS extraction. AI Commun. 25(2), 97–116 (2012)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. IOS Press (2009)Google Scholar
  7. 7.
    Birnbaum, E., Lozinskii, E.L.: Consistent subsets of inconsistent systems: structure and behaviour. J. Exp. Theor. Artif. Intell. 15(1), 25–46 (2003)CrossRefzbMATHGoogle Scholar
  8. 8.
    Boutilier, C., Brafman, R.I., Domshlak, C., Hoos, H.H., Poole, D.: CP-nets: A tool for representing and reasoning with conditional ceteris paribus preference statements. J. Artif. Intell. Res. (JAIR) 21, 135–191 (2004)Google Scholar
  9. 9.
    Cayrol, C., Lagasquie-Schiex, M.-C., Schiex, T.: Nonmonotonic reasoning: From complexity to algorithms. Ann. Math. Artif. Intell. 22(3-4), 207–236 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Chen, Y., Safarpour, S., Marques-Silva, J., Veneris, A.G.: Automated design debugging with maximum satisfiability. IEEE Trans. on CAD of Integrated Circuits and Systems 29(11), 1804–1817 (2010)CrossRefGoogle Scholar
  11. 11.
    Chinneck, J.W., Dravnieks, E.W.: Locating minimal infeasible constraint sets in linear programs. INFORMS Journal on Computing 3(2), 157–168 (1991)CrossRefzbMATHGoogle Scholar
  12. 12.
    de Siqueira, J.L.,, N., Puget, J.-F.: Explanation-based generalisation of failures. In: ECAI, pp. 339–344 (1988)Google Scholar
  13. 13.
    Di Rosa, E., Giunchiglia, E.: Combining approaches for solving satisfiability problems with qualitative preferences. AI Commun. 26(4), 395–408 (2013)MathSciNetGoogle Scholar
  14. 14.
    Felfernig, A., Schubert, M., Zehentner, C.: An efficient diagnosis algorithm for inconsistent constraint sets. AI EDAM 26(1), 53–62 (2012)Google Scholar
  15. 15.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman (1979)Google Scholar
  16. 16.
    Grégoire, É., Mazure, B., Piette, C.: On approaches to explaining infeasibility of sets of Boolean clauses. In: ICTAI (1), pp. 74–83. IEEE Press (2008)Google Scholar
  17. 17.
    Hemery, F., Lecoutre, C., Sais, L., Boussemart, F.: Extracting MUCs from constraint networks. In: Brewka, G., Coradeschi, S., Perini, A., Traverso, P. (eds.) ECAI. Frontiers in Artificial Intelligence and Applications, vol. 141, pp. 113–117. IOS Press (2006)Google Scholar
  18. 18.
    Ignatiev, A., Janota, M., Marques-Silva, J.: Towards efficient optimization in package management systems. In: ICSE (May 2014)Google Scholar
  19. 19.
    Jampel, M., Freuder, E.C., Maher, M.J.: CP-WS 1995. LNCS, vol. 1106. Springer (1996)Google Scholar
  20. 20.
    Janota, M., Botterweck, G., Marques-Silva, J.: On lazy and eager interactive reconfiguration. In: Collet, P., Wasowski, A., Weyer, T. (eds.) VaMoS. ACM (2014)Google Scholar
  21. 21.
    Jose, M., Majumdar, R.: Cause clue clauses: error localization using maximum satisfiability. In: Hall, M.W., Padua, D.A. (eds.) PLDI, pp. 437–446. ACM (2011)Google Scholar
  22. 22.
    Junker, U.: QUICKXPLAIN: Preferred explanations and relaxations for over-constrained problems. In: McGuinness, D.L., Ferguson, G. (eds.) AAAI, pp. 167–172. AAAI Press / The MIT Press (2004)Google Scholar
  23. 23.
    Komuravelli, A., Gurfinkel, A., Chaki, S., Clarke, E.M.: Automatic abstraction in SMT-based unbounded software model checking. In: Sharygina and Veith [37], pp. 846–862Google Scholar
  24. 24.
    Lagniez, J.-M., Biere, A.: Factoring out assumptions to speed up MUS extraction. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 276–292. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  25. 25.
    Liberatore, P.: Redundancy in logic I: CNF propositional formulae. Artif. Intell. 163(2), 203–232 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Marques-Silva, J., Heras, F., Janota, M., Previti, A., Belov, A.: On computing minimal correction subsets. In: Rossi, F. (ed.) IJCAI. IJCAI/AAAI (2013)Google Scholar
  27. 27.
    Marques-Silva, J., Janota, M., Belov, A.: Minimal sets over monotone predicates in boolean formulae. In: Sharygina and Veith [37], pp. 592–607Google Scholar
  28. 28.
    Marques-Silva, J., Lynce, I.: On improving MUS extraction algorithms. In: Sakallah, K.A., Simon, L. (eds.) SAT 2011. LNCS, vol. 6695, pp. 159–173. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  29. 29.
    Meseguer, P., Bouhmala, N., Bouzoubaa, T., Irgens, M., Sánchez, M.: Current approaches for solving over-constrained problems. Constraints 8(1), 9–39 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Morgado, A., Heras, F., Liffiton, M.H., Planes, J., Marques-Silva, J.: Iterative and core-guided MaxSAT solving: A survey and assessment. Constraints 18(4), 478–534 (2013)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Nadel, A., Ryvchin, V., Strichman, O.: Efficient MUS extraction with resolution. In: FMCAD, pp. 197–200. IEEE (2013)Google Scholar
  32. 32.
    Nöhrer, A., Biere, A., Egyed, A.: Managing SAT inconsistencies with HUMUS. In: Eisenecker, U.W., Apel, S., Gnesi, S. (eds.) VaMoS, pp. 83–91. ACM (2012)Google Scholar
  33. 33.
    O’Callaghan, B., O’Sullivan, B., Freuder, E.C.: Generating corrective explanations for interactive constraint satisfaction. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 445–459. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  34. 34.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley (1993)Google Scholar
  35. 35.
    Reiter, R.: A theory of diagnosis from first principles. Artif. Intell. 32(1), 57–95 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Rossi, F., Venable, K.B., Walsh, T.: A Short Introduction to Preferences: Between Artificial Intelligence and Social Choice. Morgan & Claypool Publishers (2011)Google Scholar
  37. 37.
    Sharygina, N., Veith, H. (eds.): CAV 2013. LNCS, vol. 8044. Springer, Heidelberg (2013)zbMATHGoogle Scholar
  38. 38.
    van Hoeve, W.-J.: Over-Constrained Problems. In: Hybrid Optimization: The 10 Years of CPAIOR, pp. 191–225. Springer (2011)Google Scholar
  39. 39.
    Wallner, J.P., Weissenbacher, G., Woltran, S.: Advanced SAT techniques for abstract argumentation. In: Leite, J., Son, T.C., Torroni, P., van der Torre, L., Woltran, S. (eds.) CLIMA XIV 2013. LNCS (LNAI), vol. 8143, pp. 138–154. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  40. 40.
    Zhu, C.S., Weissenbacher, G., Malik, S.: Post-silicon fault localisation using maximum satisfiability and backbones. In: Bjesse, P., Slobodová, A. (eds.) FMCAD, pp. 63–66. FMCAD Inc. (2011)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Joao Marques-Silva
    • 1
    • 2
  • Alessandro Previti
    • 1
  1. 1.CASLUniversity College DublinIreland
  2. 2.IST/INESC-IDTechnical University of LisbonPortugal

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