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. Recently, it has been shown that, in the presence of general concept inclusion axioms (GCIs), some of these fuzzy DLs actually do not have the finite model property, thus throwing doubt on the correctness of tableau algorithm for which it was claimed that they can handle fuzzy DLs with GCIs.
In a previous paper, we have shown that these doubts are indeed justified, by proving that a certain fuzzy DL with product t-norm and involutive negation is undecidable. In the present paper, we show that undecidability also holds if we consider a t-norm-based fuzzy DL where disjunction and involutive negation are replaced by the constructor implication, which is interpreted as the residuum. The only condition on the t-norm is that it is a continuous t-norm “starting” with the product t-norm, which covers an uncountable family of t-norms.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P.F. (eds.): The Description Logic Handbook: Theory, Implementation, and Applications. Cambridge University Press, Cambridge (2003)
Baader, F., Peñaloza, R.: Are fuzzy description logics with general concept inclusion axioms decidable? In: Proc. of Fuzz-IEEE 2011, pp. 1735–1742. IEEE Press, Los Alamitos (2011)
Baader, F., Sattler, U.: An overview of tableau algorithms for description logics. Studia Logica 69, 5–40 (2001)
Bobillo, F., Bou, F., Straccia, U.: On the failure of the finite model property in some fuzzy description logics. CoRR, abs/1003.1588 (2010)
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)
Bobillo, F., Delgado, M., Gómez-Romero, J., Straccia, U.: Fuzzy description logics under Gödel semantics. Int. J. of Approx. Reas. 50(3), 494–514 (2009)
Bobillo, F., Straccia, U.: A fuzzy description logic with product t-norm. In: Proc. of Fuzz-IEEE 2007, pp. 1–6. IEEE, Los Alamitos (2007)
Bobillo, F., Straccia, U.: On qualified cardinality restrictions in fuzzy description logics under Łukasiewicz semantics. In: Proc. of IPMU 2008, pp. 1008–1015 (2008)
Bobillo, F., Straccia, U.: Fuzzy description logics with general t-norms and datatypes. Fuzzy Sets and Systems 160(23), 3382–3402 (2009)
Bobillo, F., Straccia, U.: Reasoning with the finitely many-valued Łukasiewicz fuzzy description logic \(\mathcal{SROIQ}\). Information Sciences 181, 758–778 (2011)
Borgwardt, S., Peñaloza, R.: Description logics over lattices with multi-valued ontologies. In: Proc. of IJCAI 2011 (to appear, 2011)
Borgwardt, S., Peñaloza, R.: Fuzzy ontologies over lattices with t-norms. In: Proc. of DL 2011, Barcelona, Spain. CEUR-WS, vol. 745 (2011)
Cerami, M., Esteva, F., Bou, F.: Decidability of a description logic over infinite-valued product logic. In: Proc. of KR 2010. AAAI Press, Menlo Park (2010)
García-Cerdaña, A., Armengol, E., Esteva, F.: Fuzzy description logics and t-norm based fuzzy logics. Int. J. of Approx. Reas. 51, 632–655 (2010)
Hájek, P.: Metamathematics of Fuzzy Logic Trends in Logic. Springer, Heidelberg (2001)
Hájek, P.: Making fuzzy description logic more general. Fuzzy Sets and Systems 154(1), 1–15 (2005)
Horrocks, I., Patel-Schneider, P.F., van Harmelen, F.: From SHIQ and RDF to OWL: The making of a web ontology language. Journal of Web Semantics 1(1), 7–26 (2003)
Lukasiewicz, T., Straccia, U.: Managing uncertainty and vagueness in description logics for the semantic web. Journal of Web Semantics 6(4), 291–308 (2008)
Molitor, R., Tresp, C.B.: Extending description logics to vague knowledge in medicine. In: Szczepaniak, P., Lisboa, P., Tsumoto, S. (eds.) Fuzzy Systems in Medicine. Studies in Fuzziness and Soft Computing, vol. 41, pp. 617–635. Springer, Heidelberg (2000)
Mostert, P.S., Shields, A.L.: On the structure of semigroups on a compact manifold with boundary. Annals of Mathematics 65, 117–143 (1957)
Post, E.: A variant of a recursively unsolvable problem. Bulletin of the American Mathematical Society 52, 264–268 (1946)
Stoilos, G., Stamou, G.B., Tzouvaras, V., Pan, J.Z., Horrocks, I.: The fuzzy description logic f-\(\mathcal{SHIN}\). In: Proc. of URSW 2005, pp. 67–76 (2005)
Straccia, U.: Reasoning within fuzzy description logics. JAIR 14, 137–166 (2001)
Straccia, U.: Description logics with fuzzy concrete domains. In: Proc. of UAI 2005, pp. 559–567. AUAI Press (2005)
Straccia, U., Bobillo, F.: Mixed integer programming, general concept inclusions and fuzzy description logics. In: Proc. of EUSFLAT 2007, pp. 213–220. Universitas Ostraviensis (2007)
Tresp, C., Molitor, R.: A description logic for vague knowledge. In: Proc. of ECAI 1998, pp. 361–365. J. Wiley and Sons, Chichester (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Baader, F., Peñaloza, R. (2011). On the Undecidability of Fuzzy Description Logics with GCIs and Product T-norm. In: Tinelli, C., Sofronie-Stokkermans, V. (eds) Frontiers of Combining Systems. FroCoS 2011. Lecture Notes in Computer Science(), vol 6989. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24364-6_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-24364-6_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-24363-9
Online ISBN: 978-3-642-24364-6
eBook Packages: Computer ScienceComputer Science (R0)