How Fuzzy Is My Fuzzy Description Logic?

  • Stefan Borgwardt
  • Felix Distel
  • Rafael Peñaloza
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7364)


Fuzzy Description Logics (DLs) with t-norm semantics have been studied as a means for representing and reasoning with vague knowledge. Recent work has shown that even fairly inexpressive fuzzy DLs become undecidable for a wide variety of t-norms. We complement those results by providing a class of t-norms and an expressive fuzzy DL for which ontology consistency is linearly reducible to crisp reasoning, and thus has its same complexity. Surprisingly, in these same logics crisp models are insufficient for deciding fuzzy subsumption.


Description Logic Zero Divisor Residual Negation Fuzzy Ontology Existential Restriction 
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.
    Baader, F., Peñaloza, R.: Are fuzzy description logics with general concept inclusion axioms decidable? In: Proc. of the 2011 IEEE Int. Conf. on Fuzzy Systems (FUZZ-IEEE 2011), pp. 1735–1742. IEEE Press (2011)Google Scholar
  3. 3.
    Baader, F., Peñaloza, R.: GCIs make reasoning in fuzzy DLs with the product t-norm undecidable. In: Rosati, R., Rudolph, S., Zakharyaschev, M. (eds.) Proc. of the 24th Int. Workshop on Description Logics (DL 2011), Barcelona, Spain. CEUR Workshop Proceedings, vol. 745 (2011)Google Scholar
  4. 4.
    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 (LNAI), vol. 6989, pp. 55–70. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  5. 5.
    Bobillo, F., Bou, F., Straccia, U.: On the failure of the finite model property in some fuzzy description logics. Fuzzy Sets and Systems 172(23), 1–12 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    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
  7. 7.
    Bobillo, F., Straccia, U.: A fuzzy description logic with product t-norm. In: Proc. of the 2007 IEEE Int. Conf. on Fuzzy Systems FUZZ-IEEE 2007, pp. 1–6. IEEE Press (2007)Google Scholar
  8. 8.
    Bobillo, F., Straccia, U.: On qualified cardinality restrictions in fuzzy description logics under Łukasiewicz semantics. In: Proc. of the 12th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2008), pp. 1008–1015 (2008)Google Scholar
  9. 9.
    Bobillo, F., Straccia, U.: Fuzzy description logics with general t-norms and datatypes. Fuzzy Sets and Systems 160(23), 3382–3402 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Borgwardt, S., Peñaloza, R.: Undecidability of fuzzy description logics. In: Proc. of the 13th Int. Conf. on Principles of Knowledge Representation and Reasoning (KR 2012), Rome, Italy. AAAI Press (to appear, 2012)Google Scholar
  11. 11.
    Cerami, M., Straccia, U.: On the undecidability of fuzzy description logics with GCIs with Łukasiewicz t-norm. Technical report, Computing Research Repository (2011), arXiv:1107.4212v3 [cs.LO]Google Scholar
  12. 12.
    García-Cerdaña, Á., Armengol, E., Esteva, F.: Fuzzy description logics and t-norm based fuzzy logics. International Journal of Approximate Reasoning 51, 632–655 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hájek, P.: Metamathematics of Fuzzy Logic (Trends in Logic). Springer (2001)Google Scholar
  14. 14.
    Hájek, P.: Making fuzzy description logic more general. Fuzzy Sets and Systems 154(1), 1–15 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Hladik, J.: A tableau system for the description logic \(\mathcal{SHIO}\). In: Proceedings of the Doctoral Programme of IJCAR 2004. CEUR Worksop Proceedings, vol. 106, pp. 21–25 (2004)Google Scholar
  16. 16.
    Horrocks, I., Sattler, U., Tobies, S.: A PSpace-algorithm for deciding \(\mathcal{ALCNI}_{R^+}\)-satisfiability. LTCS-Report 98-08, RWTH Aachen, Germany (1998)Google Scholar
  17. 17.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Springer (2000)Google Scholar
  18. 18.
    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
  19. 19.
    Lutz, C., Areces, C., Horrocks, I., Sattler, U.: Keys, nominals, and concrete domains. Journal of Artificial Intelligence Research 23, 667–726 (2004)MathSciNetGoogle Scholar
  20. 20.
    Molitor, R., Tresp, C.B.: Extending Description Logics to Vague Knowledge in Medicine. In: Szczepaniak, P., Lisboa, P.J.G., Tsumoto, S. (eds.) Fuzzy Systems in Medicine. STUDFUZZ, vol. 41, pp. 617–635. Springer (2000)Google Scholar
  21. 21.
    Mostert, P.S., Shields, A.L.: On the structure of semigroups on a compact manifold with boundary. Annals of Mathematics 65, 117–143 (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Stoilos, G., Stamou, G.B.: A framework for reasoning with expressive continuous fuzzy description logics. In: Grau, B.C., Horrocks, I., Motik, B., Sattler, U. (eds.) Proc. of the 22nd Int. Workshop on Description Logics (DL 2009). CEUR Workshop Proceedings, vol. 477 (2009)Google Scholar
  23. 23.
    Stoilos, G., Stamou, G.B., Tzouvaras, V., Pan, J.Z., Horrocks, I.: The fuzzy description logic f-\(\mathcal{SHIN}\). In: Proc. of the 1st Int. Workshop on Uncertainty Reasoning for the Semantic Web (URSW 2005), pp. 67–76 (2005)Google Scholar
  24. 24.
    Stoilos, G., Straccia, U., Stamou, G.B., Pan, J.Z.: General concept inclusions in fuzzy description logics. In: Proc. of the 17th Eur. Conf. on Artificial Intelligence (ECAI 2006). Frontiers in Artificial Intelligence and Applications, vol. 141, pp. 457–461. IOS Press (2006)Google Scholar
  25. 25.
    Straccia, U.: Reasoning within fuzzy description logics. Journal of Artificial Intelligence Research 14, 137–166 (2001)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Straccia, U., Bobillo, F.: Mixed integer programming, general concept inclusions and fuzzy description logics. In: Proc. of the 5th EUSFLAT Conf (EUSFLAT 2007), pp. 213–220. Universitas Ostraviensis (2007)Google Scholar
  27. 27.
    Tresp, C.B., Molitor, R.: A description logic for vague knowledge. In: Proc. of the 13th Eur. Conf. on Artificial Intelligence (ECAI 1998), Brighton, UK, pp. 361–365. J. Wiley and Sons (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Stefan Borgwardt
    • 1
  • Felix Distel
    • 1
  • Rafael Peñaloza
    • 1
  1. 1.Theoretical Computer ScienceTU DresdenGermany

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