Finite Fuzzy Description Logics and Crisp Representations

  • Fernando Bobillo
  • Umberto Straccia
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7123)


Fuzzy Description Logics (DLs) are a formalism for the representation of structured knowledge that is imprecise or vague by nature. In fuzzy DLs, restricting to a finite set of degrees of truth has proved to be useful, both for theoretical and practical reasons. In this paper, we propose finite fuzzy DLs as a generalization of existing approaches. We assume a finite totally ordered set of linguistic terms or labels, which is very useful in practice since expert knowledge is usually expressed using linguistic terms. Then, we consider fuzzy DLs based on any smooth t-norm defined over this set. Initially we focus on the finite fuzzy DL \(\mathcal{ALCH}\), studying some logical properties, and showing the decidability of the logic by presenting a reasoning preserving reduction to the classical case. Finally, we extend our logic in two directions: by considering non-smooth t-norms and by considering additional DL constructors.


Fuzzy Logic Description Logic Linguistic Term Fuzzy Concept Optimal Pair 
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.


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  1. 1.
    Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P.F.: The Description Logic Handbook: Theory, Implementation, and Applications. Cambridge University Press (2003)Google Scholar
  2. 2.
    Cuenca-Grau, B., Horrocks, I., Motik, B., Parsia, B., Patel-Schneider, P.F., Sattler, U.: OWL 2: The next step for OWL. Journal Web Semantics 6(4), 309–322 (2008)CrossRefGoogle Scholar
  3. 3.
    Lukasiewicz, T., Straccia, U.: Managing uncertainty and vagueness in Description Logics for the Semantic Web. Journal of Web Semantics 6(4), 291–308 (2008)CrossRefGoogle Scholar
  4. 4.
    Liau, C.J.: On rough terminological logics. In: Proceedings of the 4th International Workshop on Rough Sets, Fuzzy Sets and Machine Discovery, RSFD 1996, pp. 47–54 (1996)Google Scholar
  5. 5.
    Doherty, P., Grabowski, M., Łukaszewicz, W., Szałas, A.: Towards a framework for approximate ontologies. Fundamenta Informaticae 57(2-4), 147–165 (2003)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Schlobach, S., Klein, M.C.A., Peelen, L.: Description logics with approximate definitions: Precise modeling of vague concepts. In: Proceedings of the 20th International Joint Conference on Artificial Intelligence, IJCAI 2007, pp. 557–562 (2007)Google Scholar
  7. 7.
    Jiang, Y., Wang, J., Tang, S., Xiao, B.: Reasoning with rough description logics: An approximate concepts approach. Information Sciences 179(5), 600–612 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cerami, M., Esteva, F., Bou, F.: Decidability of a Description Logic over infinite-valued product logic. In: Proceedings of the 12th International Conference on Principles of Knowledge Representation and Reasoning, KR 2010, pp. 203–213. AAAI Press (2010)Google Scholar
  9. 9.
    Baader, F., Peñaloza, R.: Are fuzzy Description Logics with General Concept Inclusion axioms decidable? In: Proceedings of the 20th IEEE International Conference on Fuzzy Systems, FUZZ-IEEE 2011, pp. 1735–1742. IEEE Press (2011)Google Scholar
  10. 10.
    Baader, F., Peñaloza, R.: GCIs make reasoning in fuzzy DL with the product t-norm undecidable. In: Proceedings of the 24th International Workshop on Description Logics, DL 2011. CEUR Workshop Proceedings, vol. 745 (2011)Google Scholar
  11. 11.
    Baader, F., Peñaloza, R.: On the Undecidability of Fuzzy Description Logics with GCIs and Product T-norm. In: Tinelli, C., Sofronie-Stokkermans, V. (eds.) FroCoS 2011. LNCS, vol. 6989, pp. 55–70. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. 12.
    Hájek, P.: Making fuzzy Description Logics more general. Fuzzy Sets and Systems 154(1), 1–15 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Bobillo, F., Bou, F., Straccia, U.: On the failure of the finite model property in some fuzzy Description Logics. Fuzzy Sets and Systems 172(1), 1–12 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Bobillo, F., Delgado, M., Gómez-Romero, J.: Crisp representations and reasoning for fuzzy ontologies. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 17(4), 501–530 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Bobillo, F., Delgado, M., Gómez-Romero, J., Straccia, U.: Fuzzy Description Logics under Gödel semantics. International Journal of Approximate Reasoning 50(3), 494–514 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Bobillo, F., Straccia, U.: Reasoning with the finitely many-valued Łukasiewicz fuzzy Description Logic \(\mathcal{SROIQ}\). Information Sciences 181(4), 758–778 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Hájek, P.: What does mathematical fuzzy logic offer to Description Logic? In: Capturing Intelligence: Fuzzy Logic and the Semantic Web, ch. 5, pp. 91–100. Elsevier (2006)Google Scholar
  18. 18.
    García-Cerdaña, A., Armengol, E., Esteva, F.: Fuzzy Description Logics and t-norm based fuzzy logics. International Journal of Approximate Reasoning 51(6), 632–655 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Cerami, M., García-Cerdaña, A., Esteva, F.: From classical Description Logic to n–graded fuzzy Description Logic. In: Proceedings of the 19th IEEE International Conference on Fuzzy Systems, FUZZ-IEEE 2010, pp. 1506–1513. IEEE Press (2010)Google Scholar
  20. 20.
    Mayor, G., Torrens, J.: On a class of operators for expert systems. International Journal of Intelligent Systems 8(7), 771–778 (1993)zbMATHCrossRefGoogle Scholar
  21. 21.
    Mayor, G., Torrens, J.: Triangular norms in discrete settings. In: Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, ch. 7, pp. 189–230. Elsevier (2005)Google Scholar
  22. 22.
    Mas, M., Monserrat, M., Torrens, J.: S-implications and R-implications on a finite chain. Kybernetika 40(1), 3–20 (2004)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Mas, M., Monserrat, M., Torrens, J.: On two types of discrete implications. International Journal of Approximate Reasoning 40(3), 262–279 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Straccia, U.: Description Logics over lattices. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 14(1), 1–16 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Jiang, Y., Tang, Y., Wang, J., Deng, P., Tang, S.: Expressive fuzzy Description Logics over lattices. Knowledge-Based Systems 23(2), 150–161 (2010)CrossRefGoogle Scholar
  26. 26.
    Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Horrocks, I., Kutz, O., Sattler, U.: The even more irresistible \(\mathcal{SROIQ}\). In: Proceedings of the 10th International Conference of Knowledge Representation and Reasoning, KR 2006, pp. 452–457. IEEE Press (2006)Google Scholar
  28. 28.
    Stoilos, G., Stamou, G., Pan, J.Z.: Handling imprecise knowledge with fuzzy Description Logic. In: Proceedings of the 2006 International Workshop on Description Logics, DL 2006, pp. 119–126 (2006)Google Scholar
  29. 29.
    Straccia, U.: Reasoning within Fuzzy Description Logics. Journal of Artificial Intelligence Research 14, 137–166 (2001)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Stoilos, G., Stamou, G., Pan, J.Z.: Fuzzy extensions of OWL: Logical properties and reduction to fuzzy Description Logics. International Journal of Approximate Reasoning 51(6), 656–679 (2010)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Fernando Bobillo
    • 1
  • Umberto Straccia
    • 2
  1. 1.Dpt. of Computer Science and Systems EngineeringUniversity of ZaragozaSpain
  2. 2.Istituto di Scienza e Tecnologie dell’Informazione (ISTI - CNR)PisaItaly

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