General default logic

Article

Abstract

In this paper, we propose a new nonmonotonic logic called general default logic. On the one hand, it generalizes Reiter’s default logic by adding to it rule-like operators used in logic programming. On the other hand, it extends logic programming by allowing arbitrary propositional formulas. We show that with this new logic, one can formalize naturally rule constraints, generalized closed world assumptions, and conditional defaults. We show that under a notion of strong equivalence, sentences of this new logic can be converted to a normal form. We also investigate the computational complexity of various reasoning tasks in the logic, and relate it to some other nonmonotonic formalisms such as Lin and Shoham’s logic of GK and Moore’s autoepistemic logic.

Keywords

Nonmonotonic logic Answer set program Default logic Autoepistemic logic 

Mathematics Subject Classification (2000)

68T27 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Antoniou, G., van Harmelen, F.: A Semantic Web Primer. MIT, Cambridge (2004)Google Scholar
  2. 2.
    Bidoit, N., Froidevaux, C.: Negation by default and unstratifiable logic programs. Theor. Comput. Sci. 78(1), 86–112 (1991)MathSciNetGoogle Scholar
  3. 3.
    Borgida, A., Brachman, R.J., McGuinness, D.L., Resnick, L.A.: CLASSIC: a structural data model for objects. In: Proceedings of the 1989 ACM SIGMOD International Conference on Management of Data, pp. 58–67 (1989)Google Scholar
  4. 4.
    Clark, K.L.: Negation as failure. In: Gallaire, H., Minker, J. (eds.) Logics and Databases, pp. 293–322. Plenum, New York (1978)Google Scholar
  5. 5.
    Delgrande, J., Schaub, T.: Compiling reasoning with and about preferences into default logic. In: Proceedings of IJCAI’97, pp. 168–174 (1997)Google Scholar
  6. 6.
    Donini, F.M., Lenzerini, M., Nardi, D., Schaerf, A.: AL-log: integrating datalog and description logics. J. Intell. Inf. Syst. 10(3), 227–252 (1998)CrossRefGoogle Scholar
  7. 7.
    Eiter, T., Gottlob, G.: Complexity results for disjunctive logic programming and application to nonmonotonic logics. In: International Logic Programming Symposium, pp. 266–278 (1993)Google Scholar
  8. 8.
    Eiter, T., Gottlob, G.: The complexity class \(\Theta^2_p\): recent results and applications in AI and modal logic. In: FCT ’97: Proceedings of the 11th International Symposium on Fundamentals of Computation Theory, pp. 1–18 (1997)Google Scholar
  9. 9.
    Eiter, T., Fink, M., Tompits, H., Woltran, S.: Simplifying logic programs under uniform and strong equivalence. In: Proceedings of LPNMR’04, pp. 87–99 (2004)Google Scholar
  10. 10.
    Ferraris, P.: Answer sets for propositional theories. In: Proceedings of LPNMR’05, pp. 119–131 (2005)Google Scholar
  11. 11.
    Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Proceedings of ICLP’88, pp. 1070–1080 (1988)Google Scholar
  12. 12.
    Gelfond, M., Lifschitz, V.: Classical negation in logic programs and disjunctive databases. New Gener. Comput. 9, 365–385 (1991)CrossRefGoogle Scholar
  13. 13.
    Gelfond, M., Lifschitz, V., Przymusinska, H., Truszczynski, M.: Disjunctive defaults. In: Proceedings of KR’91, pp. 230–237 (1991)Google Scholar
  14. 14.
    Gelfond, M.: Logic programming and reasoning with incomplete information. Ann. Math. Artif. Intell. 12(1–2), 89–116 (1994)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Gottlob, G.: Complexity results for nonmonotonic logics. J. Log. Comput. 2(3), 397–425 (1992)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Gottlob, G.: NP trees and Carnap’s modal logic. J. ACM 42(2), 421–457 (1995)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Grosof, B., Horrocks, I., Volz, R., Decker, S.: Description logic programs: combining logic programs with description logic (2003)Google Scholar
  18. 18.
    Halpern, J.Y., Moses, Y.: A guide to completeness and complexity for modal logics of knowledge and belief. Artif. Intell. 54(3), 319–379 (1992)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Horrocks, I., Patel-Schneider, P.F.: A proposal for an owl rules language. In: Proceedings of WWW’04, pp. 723–731 (2004)Google Scholar
  20. 20.
    Janhunen, T.: On the intertranslatability of autoepistemic, default and priority logics, and parallel circumscription. In: Proceedings of JELIA’98. LNCS, vol. 1489, pp. 216–232 (1998)Google Scholar
  21. 21.
    Konolige, K.: On the relation between default and autoepistemic logic. Artif. Intell. 35(3), 343–382 (1988)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Ladner, R.E.: The computational complexity of provability in systems of modal propositional logic. SIAM J. Comput. 6(3), 467–480 (1977)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Liberatore, P., Schaerf, M.: The complexity of model checking for propositional default logics. Data Knowl. Eng. 55(2), 189–202 (2005)CrossRefGoogle Scholar
  24. 24.
    Lifschitz, V.: Nonmonotonic databases and epistemic queries. In: IJCAI’91, pp. 381–386 (1991)Google Scholar
  25. 25.
    Lifschitz, V., Tang, L.R., Turner, H.: Nested expressions in logic programs. Ann. Math. Artif. Intell. 25(3–4), 369–389 (1999)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Lifschitz, V., Pearce, D., Valverde, A.: Strongly equivalent logic programs. ACM Trans. Comput. Log. 2(4), 526–541 (2001)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Lin, F., Chen, Y.: Discovering classes of strongly equivalent logic programs. In: Proceedings of IJCAI’05, pp. 516–521 (2005)Google Scholar
  28. 28.
    Lin, F., Shoham, Y.: A logic of knowledge and justified assumptions. Artif. Intell. 57, 271–289 (1992)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Lin, F., Zhou, Y.: From answer set logic programming to circumscription via logic of GK. In: Proceedings of the IJCAI’07, 441–446 (2007)Google Scholar
  30. 30.
    Marek, V., Truszczynski, M.: Autoepistemic logic. J. ACM 38(3), 588–619 (1991)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Moore, R.: Possible-world semantics for autoepistemic logic. In: Readings in Nonmonotonic Reasoning, pp. 137–142 (1987)Google Scholar
  32. 32.
    Motik, B., Rosati, R.: A faithful integration of description logics with logic programming. In: Proceedings of IJCAI’07, pp. 477–482 (2007)Google Scholar
  33. 33.
    Niemelä, I.: Towards automatic autoepistemic reasoning. In: Proceedings of JELIA ’90, pp. 428–443 (1991)Google Scholar
  34. 34.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  35. 35.
    Pearce, D., Tompits, H., Woltran, S.: Encodings for equilibrium logic and logic programs with nested expressions. In: Proceedings of EPIA ’01, pp. 306–320 (2001)Google Scholar
  36. 36.
    Pearce, D.: A new logical characterisation of stable models and answer sets. In: Proccedings of NMELP’96. LNCS, vol. 1216, pp. 57–70 (1997)Google Scholar
  37. 37.
    Reiter, R.: A logic for default reasoning. Artif. Intell. 13, 81–132 (1980)MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Rosati, R.: Model checking for nonmonotonic logics: algorithms and complexity. In: Proceedings of IJCAI’99, pp. 76–83 (1999)Google Scholar
  39. 39.
    Stillman, J.: The complexity of propositional default logics. In: Proceedings of AAAI’92, pp. 794–799 (1992)Google Scholar
  40. 40.
    Truszczynski, M.: Strong and uniform equivalence of nonmonotonic theories—an algebraic approach. In: Proceedings of KR’06, pp. 389–399 (2006)Google Scholar
  41. 41.
    Truszczynski, M.: The modal logic S4F, the default logic, and the logic here-and-there. In: Proceedings of AAAI’07, pp. 508–513 (2007)Google Scholar
  42. 42.
    Turner, H.: Strong equivalence for logic programs and default theories (made easy). In: Proceedings of LPNMR’01, pp. 81–92 (2001)Google Scholar
  43. 43.
    Zhou, Y., Lin, F., Zhang, Y.: General default logic. In: Proceedings of LPNMR’07, pp. 241–253 (2007)Google Scholar
  44. 44.
    Zhou, Y., Lin, F., Zhang, Y.: Embedding general default logic into the logic of GK. In: Proceedings of NMR’08, pp. 76–83 (2008)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Intelligent Systems Lab, School of Computing and MathematicsUniversity of Western SydneySydneyAustralia
  2. 2.Department of Computer ScienceHong Kong University of Science and TechnologyHong KongPeople’s Republic of China

Personalised recommendations