Advertisement

A sequent calculus for reasoning in four-valued Description Logics

  • Umberto Straccia
Contributed Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1227)

Abstract

Description Logics (DLs, for short) provide a logical reconstruction of the so-called frame-based knowledge representation languages. Originally, four-valued DLs have been proposed in order to develop expressively powerful DLs with tractable subsumption algorithms. Recently, four-valued DLs have been proposed as a model for (multimedia) document retrieval. In this context, the main reasoning task is instance checking. Unfortunately, the known subsumption algorithms for four-valued DLs, based on “structural” subsumption, do not work with respect to the semantics proposed in the DL-based approach to document retrieval. Moreover, they are unsuitable for solving the instance checking problem, as this latter problem is more general than the subsumption problem. We present an alternative decision procedure for four-valued DLs with the aim to solve these problems. The decision procedure is a sequent calculus for instance checking. Since in general the four-valued subsumption problem can be reduced to the four-valued instance checking problem, we obtain a decision procedure for the subsumption problem. Some related complexity results will be presented.

Keywords

Decision Procedure Description Logic Conjunctive Normal Form Sequent Calculus Proof Tree 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alan R. Anderson and Nuel D. Belnap. Entailment — the logic of relevance and necessity. Princeton University Press, Princeton, NJ. 1975.Google Scholar
  2. 2.
    Nuel D. Belnap. A useful four-valued logic. In Gunnar Epstein and J. Michael Dunn, editors, Modern uses of multiple-valued logic, pages 5–37. Reidel, Dordrecht, NL, 1977.Google Scholar
  3. 3.
    Alexander Borgida. Structural subsumption: What is it and why is it important? In AAAI Fall Symposium:Issues in Description Logics, pages 14–18, 1992.Google Scholar
  4. 4.
    Martin Buchheit, A. Manfred Jeusfeld, Werner Nutt, and Martin Staudt. Subsumption between queries to object-oriented databases. Information Systems, 19(1):33–54, 1994. Special issue on Extending Database Technology, EDBT'94.Google Scholar
  5. 5.
    Francesco M. Donini, Bernhard Hollunder, Maurizio Lenzerini, A. Marchetti Spaccamela, Daniele Nardi, and Werner Nutt. The complexity of existential quantification in concept languages. Artificial Intelligence, 2–3:309–327, 1992.Google Scholar
  6. 6.
    Francesco M. Donini, Maurizio Lenzerini, Daniele Nardi, and Werner Nutt. Tractable concept languages. In Proceedings of IJCAI-91, 12th International Joint Conference on Artificial Intelligence, pages 458–463, Sidney, Australia, 1991.Google Scholar
  7. 7.
    Francesco M. Donini, Maurizio Lenzerini, Daniele Nardi, and Andrea Schaerf. From subsumption to instance checking. Technical Report 15.92, Università degli studi di Roma “La Sapienza”. Dipartimento di informatica e sistemistica, Rome, Italy, 1992.Google Scholar
  8. 8.
    Melvin Fitting. First-Order Logic and Automated Theorem Proving. Springer-Verlag, 1990.Google Scholar
  9. 9.
    Jean H. Gallier. Logic for Computer Science: Foundations of Automatic Theorem Proving. Harper & Row Publishers, New York, 1986.Google Scholar
  10. 10.
    Michael R. Garey and David S. Johnson. Computers and intractability. A guide to the theory of NP-completeness. Freeman and Company, New York, NY, 1979.Google Scholar
  11. 11.
    G. Gentzen. Untersuchungen über das logische Schliessen. Mathematische Zeitschrift, 39:176–210,405–431, 1935.Google Scholar
  12. 12.
    C. A. Gobel, C. Haul, and S. Bechhofer. Describing and classifying multimedia using the description logic GRAIL. In Proceedings of the SPIE Conference on Storage and Retrieval for Still Images and Video Databases IV (SPIE-95), pages 132–143, San Jose, CA, February 1995.Google Scholar
  13. 13.
    Bernhard Hollunder, Werner Nutt, and Manfred Schmidt-Schauß. Subsumption algorithms for concept description languages. In Proc. of ECAI-90, 9th European Conference on Artificial Intelligence, pages 348–353, Stockholm, Sweden, 1990.Google Scholar
  14. 14.
    Maurizio Lenzerini and Andrea Schaerf. Concept languages as query languages. In Proc. of the 9th Nat. Conf. on Artificial Intelligence (AAAI-91), pages 471–476, 1991.Google Scholar
  15. 15.
    Hector J. Levesque. A logic of implicit and explicit belief. In Proc. of the 4th Nat. Conf. on Artificial Intelligence (AAAI-84), pages 198–202, Austin, TX, 1984.Google Scholar
  16. 16.
    Hector J. Levesque and Ronald J. Brachman. Expressiveness and tractability in knowledge representation and reasoning. Computational Intelligence, 3:78–93, 1987.Google Scholar
  17. 17.
    Carlo Meghini, Fabrizio Sebastiani, Umberto Straccia, and Costantino Thanos. A model of information retrieval based on a terminological logic. In Proceedings of SIGIR-93, 16th International Conference on Research and Development in Information Retrieval, pages 298–307, Pittsburgh, PA, 1993.Google Scholar
  18. 18.
    Carlo Meghini and Umberto Straccia. Extending a description logic to cope with the completeness of multimedia documents. In Proc. of the 12th European Conf. on Artificial Intelligence (ECAI-96): Workshop on Knowledge Representation for Interactive Multimedia Systems, pages 42–50, Budapest, Hungary, 1996.Google Scholar
  19. 19.
    Carlo Meghini and Umberto Straccia. A relevance terminological logic for information retrieval. In Proceedings of SIGIR-96, 19th International Conference on Research and Development in Information Retrieval, pages 197–205, Zurich, Switzerland, 1996.Google Scholar
  20. 20.
    Bernhard Nebel. Reasoning and revision in hybrid representation systems. Springer, Heidelberg, FRG, 1990.Google Scholar
  21. 21.
    Peter F. Patel-Schneider. A four-valued semantics for frame-based description languages. In Proceedings of AAAI-86, 5th Conference of the American Association for Artificial Intelligence, pages 344–348, Philadelphia, PA, 1986.Google Scholar
  22. 22.
    Peter F. Patel-Schneider. A hybrid, decidable, logic-based knowledge representation system. Computational Intelligence, 3:64–77, 1987.Google Scholar
  23. 23.
    Peter F. Patel-Schneider. A four-valued semantics for terminological logics. Artificial Intelligence, 38:319–351, 1989.Google Scholar
  24. 24.
    Andrea Schaerf. Reasoning with individuals in concept languages. Data and Knowledge Engineering, 13(2):141–176, 1994.Google Scholar
  25. 25.
    Manfred Schmidt-Schauß and Gert Smolka. Attributive concept descriptions with complements. Artificial Intelligence, 48:1–26, 1991.Google Scholar
  26. 26.
    Fabrizio Sebastiani. A probabilistic terminological logic for modelling information retrieval. In Proceedings of SIGIR-94, 17th ACM International Conference on Research and Development in Information Retrieval, pages 122–130, Dublin, IRL, 1994. Published by Springer Verlag, Heidelberg, FRG.Google Scholar
  27. 27.
    Fabrizio Sebastiani and Umberto Straccia. A computationally tractable terminological logic. In Proceedings of SCAI-91, 3rd Scandinavian Conference on Artificial Intelligence, pages 307–315, Roskilde, Denmark, 1991.Google Scholar
  28. 28.
    Gerd Wagner. Ex contradictione nihil sequitur. In Proc. of the 12th Int. Joint Conf. on Artificial Intelligence (IJCAI-91), pages 538–543, Sydney, Australia, 1991.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Umberto Straccia
    • 1
  1. 1.Istituto di Elaborazione della InformazionePisaItaly

Personalised recommendations