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Journal of Philosophical Logic

, Volume 44, Issue 2, pp 117–146 | Cite as

On the Decidability Status of Fuzzy \(\mathcal {A}\mathcal {L}\mathcal {C}\) with General Concept Inclusions

  • Franz Baader
  • Stefan Borgwardt
  • Rafael Peñaloza
Article

Abstract

The combination of Fuzzy Logics and Description Logics (DLs) has been investigated for at least two decades because such fuzzy DLs can be used to formalize imprecise concepts. In particular, tableau algorithms for crisp Description Logics have been extended to reason also with their fuzzy counterparts. It has turned out, however, that in the presence of general concept inclusion axioms (GCIs) this extension is less straightforward than thought. In fact, a number of tableau algorithms claimed to deal correctly with fuzzy DLs with GCIs have recently been shown to be incorrect. In this paper, we concentrate on fuzzy \(\mathcal {A}\mathcal {L}\mathcal {C}\), the fuzzy extension of the well-known DL \(\mathcal {A}\mathcal {L}\mathcal {C}\). We present a terminating, sound, and complete tableau algorithm for fuzzy \(\mathcal {A}\mathcal {L}\mathcal {C}\) with arbitrary continuous t-norms. Unfortunately, in the presence of GCIs, this algorithm does not yield a decision procedure for consistency of fuzzy \(\mathcal {A}\mathcal {L}\mathcal {C}\) ontologies since it uses as a sub-procedure a solvability test for a finitely represented, but possibly infinite, system of inequations over the real interval [0,1], which are built using the t-norm. In general, it is not clear whether this solvability problem is decidable for such infinite systems of inequations. This may depend on the specific t-norm used. In fact, we also show in this paper that consistency of fuzzy \(\mathcal {A}\mathcal {L}\mathcal {C}\) ontologies with GCIs is undecidable for the product t-norm. This implies, of course, that for the infinite systems of inequations produced by the tableau algorithm for fuzzy \(\mathcal {A}\mathcal {L}\mathcal {C}\) with product t-norm, solvability is in general undecidable. We also give a brief overview of recently obtained (un)decidability results for fuzzy \(\mathcal {A}\mathcal {L}\mathcal {C}\) w.r.t. other t-norms.

Keywords

Fuzzy description logics Decidability 

Notes

Acknowledgments

Partially supported by the Deutsche Forschungsgemeinschaft (DFG) under grant BA 1122/17-1 and within the Cluster of Excellence ‘Center for Advancing Electronics Dresden’.

References

  1. 1.
    Baader, F., Bürckert, H.J., Nebel, B., Nutt, W., Smolka, G. (1993). On the expressivity of feature logics with negation, functional uncertainty, and sort equations. Journal of Logic Language and Information, 2, 1–18.CrossRefGoogle Scholar
  2. 2.
    Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P.F. (Eds.) (2003). The description logic handbook: theory, implementation, and applications. Cambridge University Press.Google Scholar
  3. 3.
    Baader, F., & Peñaloza, R. (2011). Are fuzzy description logics with general concept inclusion axioms decidable? In Proceedings of the 2011 IEEE international conference on fuzzy systems (FUZZ-IEEE 2011) (pp. 1735–1742). IEEE Press.Google Scholar
  4. 4.
    Baader, F., & Peñaloza, R. (2011). On the undecidability of fuzzy description logics with GCIs and product t-norm. In: C. Tinelli, & V. Sofronie-Stokkermans (Eds.) In Proceedings of 8th international symposium on frontiers of combining systems (FroCoS 2011), lecture notes in computer science (Vol. 6989, pp. 55–70). Springer-Verlag.Google Scholar
  5. 5.
    Baader, F., & Sattler, U. (2001). An overview of tableau algorithms for description logics. Studia Logica, 69, 5–40.CrossRefGoogle Scholar
  6. 6.
    Bobillo, F., Bou, F., Straccia, U. (2011). On the failure of the finite model property in some fuzzy description logics. Fuzzy Sets and Systems, 172(1), 1–12. doi: 10.1016/j.fss.2011.02.012.CrossRefGoogle Scholar
  7. 7.
    Bobillo, F., & Straccia, U. (2007). A fuzzy description logic with product t-norm. In Proceedings of the 2007 IEEE international conference on fuzzy systems (FUZZ-IEEE 2007) (pp. 1–6). IEEE Press.Google Scholar
  8. 8.
    Bobillo, F., & Straccia, U. (2008). On qualified cardinality restrictions in fuzzy description logics under Lukasiewicz semantics. In Proceedings of the 12th international conference on information processing and managment of uncertainty in knowledge-based systems (IPMU 2008) (pp. 1008–1015).Google Scholar
  9. 9.
    Bobillo, F., & Straccia, U. (2009). Fuzzy description logics with general t-norms and datatypes. Fuzzy Sets and Systems, 160(23), 3382–3402.CrossRefGoogle Scholar
  10. 10.
    Borgwardt, S., Distel, F., Peñaloza, R. (2014). Decidable Gödel description logics without the finitely-valued model property. In C. Baral, G. De Giacomo, T. Eiter (Eds.), Proceedings of the 14th international conference on principles of knowledge representation and reasoning (KR’14) (pp. 228–237). AAAI Press.Google Scholar
  11. 11.
    Borgwardt, S., Distel, F., Peñaloza, R. (2012). Gödel negation makes unwitnessed consistency crisp In Y. Kazakov, D. Lembo, F. Wolter (Eds.), Proceedings of the 2012 international workshop on description logics (DL 2012), CEUR workshop proceedings (Vol. 846, pp. 103–113).Google Scholar
  12. 12.
    Borgwardt, S., Distel, F., Peñaloza, R. (2012). How fuzzy is my fuzzy description logic? In B. Gramlich, D. Miller, U. Sattler (Eds.), Proceedings of the 6th international joint conference on automated reasoning (IJCAR 2012), lecture notes in artificial intelligence (Vol. 7364, pp. 82–96). Springer-Verlag.Google Scholar
  13. 13.
    Borgwardt, S., & Peñaloza, R. (2012). Non-Gödel negation makes unwitnessed consistency undecidable In Y. Kazakov, D. Lembo, F. Wolter (Eds.), Proceedings of the 2012 international workshop on description logics (DL 2012), CEUR workshop proceedings (Vol. 846, pp. 411–421).Google Scholar
  14. 14.
    Borgwardt, S., & Peñaloza, R. (2012). Undecidability of fuzzy description logics In G. Brewka, T. Eiter, S. McIlraith (Eds.), Proceedings of the 13th international conference on principles of knowledge representation and reasoning (KR 2012) (pp. 232–242). AAAI Press.Google Scholar
  15. 15.
    Buchheit, M., Donini, F.M., Schaerf, A. (1993). Decidable reasoning in terminological knowledge representation systems. In Proceedings of the 13th international joint conference on artificial intelligence (IJCAI 1993) (pp. 704–709). Los Altos: Morgan Kaufmann.Google Scholar
  16. 16.
    Cerami, M., & Straccia, U. (2013). On the (un)decidability of fuzzy description logics under Łukasiewicz t-norm. Information Sciences, 227, 1–21. doi: 10.1016/j.ins.2012.11.019.CrossRefGoogle Scholar
  17. 17.
    García-Cerdaña, A., Armengol, E., Esteva, F. (2010). Fuzzy description logics and t-norm based fuzzy logics. International Journal of Approximate Reasoning, 51, 632–655.CrossRefGoogle Scholar
  18. 18.
    Hájek, P. (2001). Metamathematics of fuzzy logic (Trends in Logic). Springer-Verlag.Google Scholar
  19. 19.
    Hájek, P. (2005). Making fuzzy description logic more general. Fuzzy Sets and Systems, 154(1), 1–15.CrossRefGoogle Scholar
  20. 20.
    Horrocks, I., Patel-Schneider, P.F., van Harmelen, F. (2003). From SHIQ and RDF to OWL: The making of a web ontology language. Journal of Web Semantics, 1(1), 7–26.CrossRefGoogle Scholar
  21. 21.
    Horrocks, I., & Sattler, U. (2005). A tableaux decision procedure for \(\mathcal {S}\mathcal {H}\mathcal {O}\mathcal {I}\mathcal {Q}\) In L.P. Kaelbling, & A. Saffiotti (Eds.), Proceedings of the 19th international joint conference on artificial intelligence (IJCAI 2005) (pp. 448–453). Professional Book Center.Google Scholar
  22. 22.
    Klement, E.P., Mesiar, R., Pap, E. (2000). Triangular norms. Springer-Verlag.Google Scholar
  23. 23.
    Lukasiewicz, T., & Straccia, U. (2008). Managing uncertainty and vagueness in description logics for the semantic web. Journal of Web Semantics, 6(4), 291–308. doi: 10.1016/j.websem.2008.04.001.CrossRefGoogle Scholar
  24. 24.
    Lutz, C. (2003). Description logics with concrete domains—a survey In P. Balbiani, N.Y. Suzuki, F. Wolter, M. Zakharyaschev (Eds.), Advances in modal logics (Vol. 4, pp. 265–296). King’s College Publications.Google Scholar
  25. 25.
    McCallum, S. (1993). Solving polynomial strict inequalities using cylindrical algebraic decomposition. The Computer Journal, 36(5), 432–438.CrossRefGoogle Scholar
  26. 26.
    Molitor, R., & Tresp, C. (2000). Extending description logics to vague knowledge in medicine. In P. Szczepaniak, P. Lisboa, S. Tsumoto (Eds.), Fuzzy systems in medicine, studies in fuzziness and soft computing (Vol. 41, pp. 617–635). Springer-Verlag.Google Scholar
  27. 27.
    Mostert, P.S., & Shields, A.L. (1957). On the structure of semigroups on a compact manifold with boundary. Annals of Mathematics, 65, 117–143.CrossRefGoogle Scholar
  28. 28.
    Post, E. (1946). A variant of a recursively unsolvable problem. Bulletin of the American Mathematical Society, 52, 264–268.CrossRefGoogle Scholar
  29. 29.
    Schmidt-Schauß, M. Smolka (1991). Attributive concept descriptions with complements. Artificial Intelligence, 48(1), 1–26.Google Scholar
  30. 30.
    Stoilos, G., Stamou, G.B., Tzouvaras, V., Pan, J.Z., Horrocks, I. (2005). The fuzzy description logic f-\(\mathcal {S}\mathcal {H}\mathcal {I}\mathcal {N}\). In Proceedings of the 1st international workshop on uncertainty reasoning for the semantic web (URSW 2005) (pp. 67–76).Google Scholar
  31. 31.
    Straccia, U. (2001). Reasoning within fuzzy description logics. Journal of Artificial Intelligence Research, 14, 137–166.Google Scholar
  32. 32.
    Straccia, U. (2005). Description logics with fuzzy concrete domains. In Proceedings of the 21st conference in uncertainty in artificial intelligence (UAI 2005) (pp. 559–567). AUAI Press.Google Scholar
  33. 33.
    Straccia, U., & Bobillo, F. (2007). Mixed integer programming, general concept inclusions and fuzzy description logics. In Proceedings of the 5th EUSFLAT conference (pp. 213–220). Universitas Ostraviensis.Google Scholar
  34. 34.
    Tresp, C.B., & Molitor, R. (1998). A description logic for vague knowledge. In Proceedings of the 13th European conference on artificial intelligence (ECAI 1998) (pp. 361–365). Wiley.Google Scholar
  35. 35.
    Zadeh, L.A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Franz Baader
    • 1
    • 2
  • Stefan Borgwardt
    • 1
  • Rafael Peñaloza
    • 1
    • 2
  1. 1.Institute of Theoretical Computer ScienceTechnische Universität DresdenDresdenGermany
  2. 2.Center for Advancing Electronics DresdenDresdenGermany

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