Advertisement

Abstract

This paper tackles the problem of deciding whether a given clause belongs to some minimally unsatisfiable subset (MUS) of a formula, where the formula is in conjunctive normal form (CNF) and unsatisfiable. Deciding MUS-membership helps the understanding of why a formula is unsatisfiable. If a clause does not belong to any MUS, then removing it will certainly not contribute to restoring the formula’s consistency. Unsatisfiable formulas and consistency restoration in particular have a number of practical applications in areas such as software verification or product configuration. The MUS-membership problem is known to be in the second level of polynomial hierarchy, more precisely it is \(\Sigma{^p_2}\) -complete. Hence, quantified Boolean formulas (QBFs) represent a possible avenue for tackling the problem. This paper develops a number of novel QBF formulations of the MUS-membership problem and evaluates their practicality using modern off-the-shelf solvers.

Keywords

Minimal Model Conjunctive Normal Form Truth Assignment Boolean Formula Membership Problem 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cadoli, M., Lenzerini, M.: The complexity of closed world reasoning and circumscription. In: AAAI Conference on Artificial Intelligence, pp. 550–555 (1990)Google Scholar
  2. 2.
    Desrosiers, C., Galinier, P., Hertz, A., Paroz, S.: Using heuristics to find minimal unsatisfiable subformulas in satisfiability problems. J. Comb. Optim. 18(2), 124–150 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Eiter, T., Gottlob, G.: Propositional circumscription and extended closed-world reasoning are \(\Pi^P_2\)-complete. Theor. Comput. Sci. 114(2), 231–245 (1993)CrossRefzbMATHGoogle Scholar
  4. 4.
    Feldmann, R., Monien, B., Schamberger, S.: A distributed algorithm to evaluate quantified Boolean formulae. In: AAAI/IAAI, pp. 285–290 (2000)Google Scholar
  5. 5.
    Giunchiglia, E., Marin, P., Narizzano, M.: An effective preprocessor for QBF pre-reasoning. In: 2nd International Workshop on Quantification in Constraint Programming, QiCP (2008)Google Scholar
  6. 6.
    Grégoire, É., Mazure, B., Piette, C.: On approaches to explaining infeasibility of sets of Boolean clauses. In: International Conference on Tools with Artificial Intelligence, pp. 74–83 (November 2008)Google Scholar
  7. 7.
    Grégoire, E., Mazure, B., Piette, C.: Does this set of clauses overlap with at least one MUS? In: Schmidt, R.A. (ed.) CADE-22. LNCS, vol. 5663, pp. 100–115. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Janota, M., Grigore, R., Marques-Silva, J.: Counterexample guided abstraction refinement algorithm for propositional circumscription. In: Proceeding of the 12th European Conference on Logics in Artificial Intelligence, JELIA (2010)Google Scholar
  9. 9.
    Janota, M., Marques-Silva, J.: Abstraction-based algorithm for 2QBF. In: Sakallah, Simon (eds.) [23]Google Scholar
  10. 10.
    Janota, M., Marques-Silva, J.: cmMUS: a circumscription-based tool for MUS membership testing. In: Delgrande, J.P., Faber, W. (eds.) LPNMR 2011. LNCS, vol. 6645, pp. 266–271. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  11. 11.
    Kullmann, O.: An application of matroid theory to the SAT problem. In: IEEE Conference on Computational Complexity, pp. 116–124 (2000)Google Scholar
  12. 12.
    Kullmann, O.: Constraint satisfaction problems in clausal form: Autarkies and minimal unsatisfiability. ECCC 14(055) (2007)Google Scholar
  13. 13.
    Liberatore, P.: Redundancy in logic I: CNF propositional formulae. Artif. Intell. 163(2), 203–232 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Liffiton, M.H., Sakallah, K.A.: Algorithms for computing minimal unsatisfiable subsets of constraints. J. Autom. Reasoning 40(1), 1–33 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lifschitz, V.: Some results on circumscription. In: NMR, pp. 151–164 (1984)Google Scholar
  16. 16.
    Marques-Silva, J., Lynce, I.: On improving MUS extraction algorithms. In: Sakallah, Simon (eds.) [23]Google Scholar
  17. 17.
    McCarthy, J.: Circumscription - a form of non-monotonic reasoning. Artif. Intell. 13(1-2), 27–39 (1980)CrossRefzbMATHGoogle Scholar
  18. 18.
    Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential space. In: Switching and Automata Theory (1972)Google Scholar
  19. 19.
    Minker, J.: On indefinite databases and the closed world assumption. In: Conference on Automated Deduction, pp. 292–308 (1982)Google Scholar
  20. 20.
    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
  21. 21.
    Papadimitriou, C.H., Wolfe, D.: The complexity of facets resolved. J. Comput. Syst. Sci. 37(1), 2–13 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Papadopoulos, A., O’Sullivan, B.: Relaxations for compiled over-constrained problems. In: Stuckey, P.J. (ed.) CP 2008. LNCS, vol. 5202, pp. 433–447. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  23. 23.
    Sakallah, K.A., Simon, L. (eds.): The 14th International Conference on Theory and Applications of Satisfiability Testing (SAT). Springer, Heidelberg (2011)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Mikoláš Janota
    • 1
  • Joao Marques-Silva
    • 1
    • 2
  1. 1.INESC-IDLisbonPortugal
  2. 2.University College DublinIreland

Personalised recommendations