cmMUS: A Tool for Circumscription-Based MUS Membership Testing

  • Mikoláš Janota
  • Joao Marques-Silva
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6645)


This article presents cmMUS—a tool for deciding whether a clause belongs to some minimal unsatisfiable subset (MUS) of a given formula. While MUS-membership has a number of practical applications, related with understanding the causes of unsatisfiability, it is computationally challenging—it is \(\Sigma_2^P\)-complete. The presented tool cmMUS solves the problem by translating it to propositional circumscription, a well-known problem from the area of non-monotonic reasoning. The tool constantly outperforms other approaches to the problem, which is demonstrated on a variety of benchmarks.


Conjunctive Normal Form Boolean Formula Conjunctive Normal Form Formula Stable Model Semantic Disjunctive Logic 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Drescher, C., Gebser, M., Grote, T., Kaufmann, B., König, A., Ostrowski, M., Schaub, T.: Conflict-driven disjunctive answer set solving. In: Brewka, G., Lang, J. (eds.) KR, pp. 422–432. AAAI Press, Menlo Park (2008)Google Scholar
  2. 2.
    Eiter, T., Gottlob, G.: Propositional circumscription and extended closed-world reasoning are \({\pi}^{\rm P}_{2}\)-complete. Theor. Comput. Sci. 114(2), 231–245 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Eiter, T., Gottlob, G.: On the computational cost of disjunctive logic programming: Propositional case. Annals of Mathematics and Artificial Intelligence 15, 289–323 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gelfond, M., Lifschitz, V.: Classical negation in logic programs and disjunctive databases. New Generation Computing 9(3), 365–385 (1991)CrossRefzbMATHGoogle Scholar
  5. 5.
    Gelfond, M.: Answer Sets. In: Handbook of Knowledge Representation. Elsevier, Amsterdam (2008)Google Scholar
  6. 6.
    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
  7. 7.
    Grégoire, É., Mazure, B., Piette, C.: Does this set of clauses overlap with at least one MUS? In: Schmidt, R.A. (ed.) CADE 2009. LNCS, vol. 5663, pp. 100–115. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Janhunen, T., Oikarinen, E.: Capturing parallel circumscription with disjunctive logic programs. In: European Conf. on Logics in Artif. Intell., pp. 134–146 (2004)Google Scholar
  9. 9.
    Janota, M., Grigore, R., Marques-Silva, J.: Counterexample guided abstraction refinement algorithm for propositional circumscription. In: Janhunen, T., Niemelä, I. (eds.) JELIA 2010. LNCS, vol. 6341, pp. 195–207. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Janota, M., Marques-Silva, J.: Models and algorithms for MUS membership testing. Tech. Rep. TR-07/2011, INESC-ID (January 2011)Google 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. In: Electronic Colloquium on Computational Complexity (ECCC), vol.  14(055) (2007)Google Scholar
  13. 13.
    Liffiton, M.H., Sakallah, K.A.: Algorithms for computing minimal unsatisfiable subsets of constraints. J. Autom. Reasoning 40(1), 1–33 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    McCarthy, J.: Circumscription - a form of non-monotonic reasoning. Artif. Intell. 13(1-2), 27–39 (1980)CrossRefzbMATHGoogle Scholar
  15. 15.
    Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential space. In: IEEE Conference Record of 13th Annual Symposium on Switching and Automata Theory (October 1972)Google Scholar
  16. 16.
    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
  17. 17.
    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

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

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

Personalised recommendations