On The Complexity of Model Checking and Inference in Minimal Models

Extended Abstract
  • Leferis M. Kirousis
  • Phokion G. Kolaitis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2173)


Every logical formalism gives rise to two fundamental algorithmic problems: model checking and inference. In propositional logic, the model checking problem is polynomial-time solvable, while the inference problem is coNP-complete. In propositional circumscription, however, these problems have higher computational complexity, namely the model checking problem is coNP-complete, while the inference problem is П P 2 -complete. In this paper, we survey recent results on the computational complexity of restricted cases of these problems in the context of Schaefer’s framework of generalized satisfiability problems. These results establish dichotomies in the complexity of the model checking problem and the inference problem for propositional circumscription. Specifically, in each restricted case the model checking problem for propositional circumscription either is coNP-complete or is polynomial-time solvable. Furthermore, in each restricted case the inference problem for propositional circumscription either is П P 2 -complete or is in coNP. These dichotomy theorems yield a complete classification of the “hard” and the “easier” cases of the model checking problem and the inference problem for propositional circumscription. Moreover, they provide efficiently checkable criteria that tell apart the “hard” cases from the “easier” ones.


Model Check Minimal Model Dichotomy Theorem Logical Relation Truth Assignment 
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  1. Cad92.
    M. Cadoli. The complexity of model checking for circumscriptive formulae. Information Processing Letters, pages 113–118, 1992. 43, 44, 48Google Scholar
  2. Cad93.
    M. Cadoli. Two Methods for Tractable Reasoning in Artificial Intelligence: Language Restriction and Theory Approximation. PhD thesis, Universitè Degli Studi Di Roma “La Sapienza“, Rome, Italy, 1993. 43, 44, 48Google Scholar
  3. CH96.
    N. Creignou and M. Hermann. Complexity of generalized satisfiability counting problems. Information and Computation, 125(1):1–12, 1996. 46zbMATHCrossRefMathSciNetGoogle Scholar
  4. CH97.
    N. Creignou and J.-J. Herbrard. On generating all solutions of generalized satisfiability problems. Theoretical Informatics and Applications, 31(6):499–511, 1997. 46zbMATHMathSciNetGoogle Scholar
  5. CL94.
    M. Cadoli and M. Lenzerini. The complexity of closed world reasoning and circumscription. Journal of Information and System Sciences, pages 255–301, 1994. Preliminary version in Proceedings of the 8th National Conference on Artificial Intelligence-AAAI’ 90. 44, 48, 52Google Scholar
  6. Coo71.
    S. A. Cook. The complexity of theorem proving procedures. In Proc. 3rd ACM Symp. on Theory of Computing, pages 151–158, 1971. 43Google Scholar
  7. Cre95.
    N. Creignou. A dichotomy theorem for maximum generalized satisfiability problems. Journal of Computer and System Sciences, 51:511–522, 1995. 46CrossRefMathSciNetGoogle Scholar
  8. DP92.
    R. Dechter and J. Pearl. Structure identification in relational data. Artificial Intelligence, 48:237–270, 1992. 46CrossRefMathSciNetGoogle Scholar
  9. EG93.
    Th. Eiter and G. Gottlob. Propositional circumscription and extended closedworld reasoning are ?P 2-complete. Theoretical Computer Science, 114:231–245, 1993. 43, 48, 50zbMATHCrossRefMathSciNetGoogle Scholar
  10. FHW80.
    S. Fortune, J. Hopcroft, and J. Wyllie. The directed homeomorphism problem. Theoretical Computer Science, 10:111–121, 1980. 46zbMATHCrossRefMathSciNetGoogle Scholar
  11. GJ79.
    M. R. Garey and D. S. Johnson. Computers and Intractability-A Guide to the Theory of NP-Completeness. W. H. Freeman and Co., 1979. 45Google Scholar
  12. HN90.
    P. Hell and J. Ne?set?ril. On the complexity of H-coloring. Journal of Combinatorial Theory, Series B, 48:92–110, 1990. 46zbMATHCrossRefMathSciNetGoogle Scholar
  13. Joh90.
    D. S. Johnson. A catalog of complexity classes. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity, chapter 2, pages 67–161. North-Holland, Amsterdam, 1990. 51Google Scholar
  14. KK01a.
    L. M. Kirousis and Ph.G. Kolaitis. The complexity of minimal satisfiability problems. In Proc. of the 18th Annual Symposium on Theoretical Aspects of Computer Science-STACS 2001, volume 2010 of Lecture Notes in Computer Science, pages 407–418. Springer, 2001. Full version at: Electronic Colloquium on Computational Complexity-(, Report No. 82, 2000. 44, 48, 49, 50, 51MathSciNetGoogle Scholar
  15. KK01b.
    L. M. Kirousis and Ph.G. Kolaitis. A dichotomy in the complexity of propositional circumscription. In Proc. of the 16th Annual IEEE Symposium on Logic in Computer Science-LICS 2001, pages 71–80, 2001. 44, 48, 49, 50, 51Google Scholar
  16. KSW97.
    S. Khanna, M. Sudan, and D. P. Williamson. A complete classification of the approximability of maximization problems derived from boolean constraint satisfaction. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pages 11–20, 1997. 46Google Scholar
  17. Lad75.
    R. Ladner. On the structure of polynomial time reducibility. Journal of the Association for Computing Machinery, 22:155–171, 1975. 45, 46zbMATHMathSciNetGoogle Scholar
  18. McC80.
    J. McCarthy. Circumscription-a form of nonmonotonic reasoning. Artificial Intelligence, 13:27–39, 1980. 42zbMATHCrossRefMathSciNetGoogle Scholar
  19. Pap94.
    C. H. Papadimitriou. Computational Complexity. Addison-Wesley Publishing Company, 1994. 43Google Scholar
  20. Sch78.
    T. J. Schaefer. The complexity of satisfiability problems. In Proc. 10th ACM Symp. on Theory of Computing, pages 216–226, 1978. 44, 45, 46, 47, 51, 52Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Leferis M. Kirousis
    • 1
  • Phokion G. Kolaitis
    • 2
  1. 1.Department of Computer Engineering and InformaticsUniversity of PatrasPatrasGreece
  2. 2.Computer Science DepartmentUniversity of CaliforniaUSA

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