Living with Inconsistency and Taming Nonmonotonicity

  • Jan Małuszyński
  • Andrzej Szałas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6702)

Abstract

In this paper we consider rule-based query languages with negation in bodies and heads of rules, traditionally denoted by Datalog¬¬. Tractable and at the same time intuitive semantics for Datalog¬¬ has not been provided even though the area of deductive databases is over 30 years old. In this paper we identify sources of the problem and propose a query language, which we call 4QL.

The 4QL language supports a modular and layered architecture and provides a tractable framework for many forms of rule-based reasoning both monotonic and nonmonotonic. As the underpinning principle we assume openness of the world, which may lead to the lack of knowledge. Negation in rule heads may lead to inconsistencies. To reduce the unknown/inconsistent zones we introduce simple constructs which provide means for application-specific disambiguation of inconsistent information, the use of Local Closed World Assumption (thus also Closed World Assumption, if needed), as well as various forms of default and defeasible reasoning.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison-Wesley Pub. Co., Reading (1996)Google Scholar
  2. 2.
    Alcântara, J., Damásio, C.V., Pereira, L.M.: An encompassing framework for paraconsistent logic programs. J. Applied Logic 3(1), 67–95 (2005)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Antoniou, G., van Harmelen, F.: A Semantic Web Primer. The MIT Press, Cambridge (2004)Google Scholar
  4. 4.
    Arieli, O.: Paraconsistent declarative semantics for extended logic programs. Ann. Math. Artif. Intell. 36(4), 381–417 (2002)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Baumgartner, R., Gottlob, G.: On the complexity of model checking for propositional default logics: New results and tractable cases. In: IJCAI, pp. 64–69 (1999)Google Scholar
  6. 6.
    Belnap, N.D.: A useful four-valued logic. In: Eptein, G., Dunn, J.M. (eds.) Modern Uses of Many Valued Logic, pp. 8–37. Reidel, Dordrechtz (1977)Google Scholar
  7. 7.
    Besnard, P.: An Introduction to Default Logic. Springer, Heidelberg (1989)CrossRefMATHGoogle Scholar
  8. 8.
    Blair, H.A., Subrahmanian, V.S.: Paraconsistent logic programming. Theor. Comput. Sci. 68(2), 135–154 (1989)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bolc, L., Borowik, P.: Many-Valued Logics, 1. Theoretical Foundations. Springer, Berlin (1992)CrossRefMATHGoogle Scholar
  10. 10.
    Brewka, G.: Non-Monotonic Reasoning: Logical Foundations of Commonsense. Cambridge University Press, Cambridge (1991)MATHGoogle Scholar
  11. 11.
    Cadoli, M., Eiter, T., Gottlob, G.: Complexity of propositional nested circumscription and nested abnormality theories. ACM Trans. Comput. Log. 6(2), 232–272 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cadoli, M., Schaerf, M.: A survey on complexity results for non-monotonic logics. Journal Logic Programming 17, 127–160 (1993)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Damásio, C.V., Pereira, L.M.: A survey of paraconsistent semantics for logic programs. In: Handbook of Defeasible Reasoning and Uncertainty Management Systems, pp. 241–320 (1998)Google Scholar
  14. 14.
    de Amo, S., Pais, M.S.: A paraconsistent logic approach for querying inconsistent databases. International Journal of Approximate Reasoning 46, 366–386 (2007)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Doherty, P., Kachniarz, J., Szałas, A.: Using contextually closed queries for local closed-world reasoning in rough knowledge databases. In: Pal, S.K., Polkowski, L., Skowron, A. (eds.) Rough-Neural Computing: Techniques for Computing with Words, Cognitive Technologies, pp. 219–250. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  16. 16.
    Doherty, P., Łukaszewicz, W., Skowron, A., Szałas, A.: Knowledge representation techniques. A rough set approach. Studies in Fuziness and Soft Computing, vol. 202. Springer, Heidelberg (2006)MATHGoogle Scholar
  17. 17.
    Doherty, P., Łukaszewicz, W., Szałas, A.: Computing circumscription revisited. Journal of Automated Reasoning 18(3), 297–336 (1997); See also 14th International Joint Conference on AI (IJCAI 1995). Morgan Kaufmann Pub. Inc., San Francisco (1995)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Doherty, P., Łukaszewicz, W., Szałas, A.: Efficient reasoning using the local closed-world assumption. In: Cerri, S.A., Dochev, D. (eds.) AIMSA 2000. LNCS (LNAI), vol. 1904, pp. 49–58. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  19. 19.
    Dubois, D.: On ignorance and contradiction considered as truth-values. Logic Journal of the IGPL 16(2), 195–216 (2008)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Eiter, T., Gottlob, G.: Propositional circumscription and extended closed-world reasoning are ΠP 2-complete. Theoretical Computer Science 114(2), 231–245 (1993)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Etzioni, O., Golden, K., Weld, D.S.: Sound and efficient closed-world reasoning for planning. Artificial Intelligence 89, 113–148 (1997)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Fages, F.: Consistency of Clark’s completion and existence of stable models. Methods of Logic in Computer Science 1, 51–60 (1994)Google Scholar
  23. 23.
    Fitting, M.C.: Fixpoint semantics for logic programming. A survey. Theoretical Computer Science 278(1-2), 25–51 (2002)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Gabbay, D.M., Schmidt, R., Szałas, A.: Second-Order Quantifier Elimination. Foundations, Computational Aspects and Applications. Studies in Logic, vol. 12. College Publications (2008)Google Scholar
  25. 25.
    Gelfond, M., Lifschitz, V.: Classical negation in logic programs and disjunctive databases. New Generation Comput. 9(3/4), 365–386 (1991)CrossRefMATHGoogle Scholar
  26. 26.
    Ginsberg, M.: Multi-valued logics. In: Proceedings of AAAI 1986, Fifth National Conference on Artificial Intelligence, pp. 243–247 (1986)Google Scholar
  27. 27.
    Gottlob, G.: Complexity results for nonmonotonic logics. Journal of Logic and Computation 2(3), 397–425 (1992)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Kifer, M., Lozinski, E.L.: A logic for reasoning with inconsistency. J. Autom. Reasoning 9(2), 179–215 (1992)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Lifschitz, V.: Circumscription. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A. (eds.) Handbook of Artificial Intelligence and Logic Programming, vol. 3, pp. 297–352. Oxford University Press, Oxford (1991)Google Scholar
  30. 30.
    Łukaszewicz, W.: Non-Monotonic Reasoning - Formalization of Commonsense Reasoning. Ellis Horwood Series in Artificial Intelligence. Ellis Horwood, England (1990)Google Scholar
  31. 31.
    Małuszyński, J., Szałas, A.: Logical foundations and complexity of 4QL, a query language with unrestricted negation (2010) (to appear); Journal of Applied Non-Classical Logics, http://arxiv.org/abs/1011.5105
  32. 32.
    Małuszyński, J., Szałas, A., Vitória, A.: Paraconsistent logic programs with four-valued rough sets. In: Chan, C.-C., Grzymala-Busse, J.W., Ziarko, W.P. (eds.) RSCTC 2008. LNCS (LNAI), vol. 5306, pp. 41–51. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  33. 33.
    Marek, V.W., Truszczyński, M.: Nonmonotonic Logic. Springer, Heidelberg (1993)CrossRefMATHGoogle Scholar
  34. 34.
    McCarthy, J.: Circumscription: A form of non-monotonic reasoning. Artificial Intelligence Journal 13, 27–39 (1980)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Moore, R.C.: Possible-world semantics for autoepistemic logic. In: Proc. 1st Nonmonotonic Reasoning Workshop, New Paltz, NY, pp. 344–354 (1984)Google Scholar
  36. 36.
    Nute, D.: Defeasible logic. In: Handbook of Logic in Artificial Intelligence and Logic Programming, pp. 353–395 (1994)Google Scholar
  37. 37.
    Pawlak, Z.: Rough Sets. Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht (1991)MATHGoogle Scholar
  38. 38.
    Reiter, R.: On closed world data bases. In: Gallaire, H., Minker, J. (eds.) Logic and Data Bases, pp. 55–76. Plenum Press, New York (1978)CrossRefGoogle Scholar
  39. 39.
    Reiter, R.: A logic for default reasoning. Artificial Intelligence Journal 13, 81–132 (1980)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Sakama, C., Inoue, K.: Paraconsistent stable semantics for extended disjunctive programs. J. Log. Comput. 5(3), 265–285 (1995)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Vitória, A.: Reasoning with Rough Sets and Paraconsistent Rough Sets. University of Linköping, Ph.D. Thesis (2010), http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-60794
  42. 42.
    Vitória, A., Małuszyński, J., Szałas, A.: Modeling and reasoning with paraconsistent rough sets. Fundamenta Informaticae 97(4), 405–438 (2009)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Jan Małuszyński
    • 1
  • Andrzej Szałas
    • 1
    • 2
  1. 1.Department of Computer and Information ScienceLinköping UniversityLinköpingSweden
  2. 2.Institute of InformaticsWarsaw UniversityWarsawPoland

Personalised recommendations