Counterexample Guided Abstraction Refinement Algorithm for Propositional Circumscription

  • Mikoláš Janota
  • Radu Grigore
  • Joao Marques-Silva
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6341)

Abstract

Circumscription is a representative example of a nonmonotonic reasoning inference technique. Circumscription has often been studied for first order theories, but its propositional version has also been the subject of extensive research, having been shown equivalent to extended closed world assumption (ECWA). Moreover, entailment in propositional circumscription is a well-known example of a decision problem in the second level of the polynomial hierarchy. This paper proposes a new Boolean Satisfiability (SAT)-based algorithm for entailment in propositional circumscription that explores the relationship of propositional circumscription to minimal models. The new algorithm is inspired by ideas commonly used in SAT-based model checking, namely counterexample guided abstraction refinement. In addition, the new algorithm is refined to compute the theory closure for generalized close world assumption (GCWA). Experimental results show that the new algorithm can solve problem instances that other solutions are unable to solve.

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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.
    Castell, T., Cayrol, C., Cayrol, M., Berre, D.L.: Using the Davis and Putnam procedure for an efficient computation of preferred models. In: European Conference on Artificial Intelligence, pp. 350–354 (1996)Google Scholar
  3. 3.
    Clarke, E.M., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, ch. 1855, pp. 154–169. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  4. 4.
    Dix, J., Furbach, U., Niemelä, I.: Nonmonotonic reasoning: Towards efficient calculi and implementations. In: Voronkov, A., Robinson, A. (eds.) Handbook of Automated Reasoning, vol. 19, pp. 1241–1354. North-Holland, Amsterdam (2001)CrossRefGoogle Scholar
  5. 5.
    Egly, U., Eiter, T., Tompits, H., Woltran, S.: Solving advanced reasoning tasks using quantified boolean formulas. In: AAAI Conference on Artificial Intelligence, pp. 417–422 (2000)Google Scholar
  6. 6.
    Eiter, T., Gottlob, G.: Propositional circumscription and extended closed-world reasoning are \(\Pi^P_2\)-complete. Theor. Comput. Sci. 114(2), 231–245 (1993)CrossRefMATHGoogle Scholar
  7. 7.
    Eiter, T., Ianni, G., Lukasiewicz, T., Schindlauer, R., Tompits, H.: Combining answer set programming with description logics for the Semantic Web. Artif. Intell. 172(12-13), 1495–1539 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Flanagan, C., Qadeer, S.: Predicate abstraction for software verification. In: Principles of programming languages (POPL), pp. 191–202. ACM, New York (2002)Google Scholar
  9. 9.
    Gelfond, M., Przymusinska, H., Przymusinski, T.C.: On the relationship between circumscription and negation as failure. Artif. Intell. 38(1), 75–94 (1989)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Giunchiglia, E., Maratea, M.: Solving optimization problems with DLL. In: European Conference on Artificial Intelligence, pp. 377–381 (2006)Google Scholar
  11. 11.
    Janhunen, T., Niemelä, I., Seipel, D., Simons, P., You, J.H.: Unfolding partiality and disjunctions in stable model semantics. ACM Trans. Comput. Log. 7(1), 1–37 (2006)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    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
  13. 13.
    Janota, M., Botterweck, G., Grigore, R., Marques-Silva, J.: How to complete an interactive configuration process? In: Conference on Current Trends in Theory and Practice of Computer Science, pp. 528–539 (2010)Google Scholar
  14. 14.
    Janota, M., Grigore, R., Marques-Silva, J.: Counterexample guided abstraction refinement algorithm for propositional circumscription. Tech. Rep. TR-32-2010, INESC-ID Lisboa (2010)Google Scholar
  15. 15.
    Lifschitz, V.: Foundations of logic programming. In: Principles of Knowledge Representation, pp. 69–127 (1996)Google Scholar
  16. 16.
    McCarthy, J.: Circumscription - a form of non-monotonic reasoning. Artif. Intell. 13(1-2), 27–39 (1980)CrossRefMATHGoogle Scholar
  17. 17.
    McCarthy, J.: Applications of circumscription to formalizing common-sense knowledge. Artif. Intell. 28(1), 89–116 (1986)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Minker, J.: On indefinite databases and the closed world assumption. In: Loveland, D.W. (ed.) CADE 1982. LNCS, vol. 138, pp. 292–308. Springer, Heidelberg (1982)CrossRefGoogle Scholar
  19. 19.
    Niemelä, I.: Implementing circumscription using a tableau method. In: European Conference on Artificial Intelligence, pp. 80–84 (1996)Google Scholar
  20. 20.
    Oikarinen, E., Janhunen, T.: circ2dlp — translating circumscription into disjunctive logic programming. In: Baral, C., Greco, G., Leone, N., Terracina, G. (eds.) LPNMR 2005. LNCS (LNAI), vol. 3662, pp. 405–409. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  21. 21.
    Przymusinski, T.C.: An algorithm to compute circumscription. Artif. Intell. 38(1), 49–73 (1989)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Rosa, E.D., Giunchiglia, E., Maratea, M.: Solving satisfiability problems with preferences. Constraints. An International Journal (2010) (in press)Google Scholar
  23. 23.
  24. 24.
    Tseitin, G.S.: On the complexity of derivation in propositional calculus. Studies in constructive mathematics and mathematical logic 2(115-125), 10–13 (1968)Google Scholar
  25. 25.
    Yahya, A.H., Henschen, L.J.: Deduction in non-Horn databases. Journal of Automated Reasoning 1(2), 141–160 (1985)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Mikoláš Janota
    • 1
  • Radu Grigore
    • 2
  • Joao Marques-Silva
    • 3
  1. 1.INESC-IDLisbonPortugal
  2. 2.Queen MaryUniversity of LondonUK
  3. 3.University College DublinIreland

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