Abstract
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.
Partially supported by the DFG under grant BA 1122/17-1 and in the Collaborative Research Center 912 “Highly Adaptive Energy-Efficient Computing”.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
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.
References
Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P.F.: The Description Logic Handbook: Theory, Implementation, and Applications. Cambridge University Press (2003)
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)
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)
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)
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. International Journal of Approximate Reasoning 50(3), 494–514 (2009)
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)
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)
Bobillo, F., Straccia, U.: Fuzzy description logics with general t-norms and datatypes. Fuzzy Sets and Systems 160(23), 3382–3402 (2009)
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)
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]
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)
Hájek, P.: Metamathematics of Fuzzy Logic (Trends in Logic). Springer (2001)
Hájek, P.: Making fuzzy description logic more general. Fuzzy Sets and Systems 154(1), 1–15 (2005)
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)
Horrocks, I., Sattler, U., Tobies, S.: A PSpace-algorithm for deciding \(\mathcal{ALCNI}_{R^+}\)-satisfiability. LTCS-Report 98-08, RWTH Aachen, Germany (1998)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Springer (2000)
Lukasiewicz, T., Straccia, U.: Managing uncertainty and vagueness in description logics for the semantic web. Journal of Web Semantics 6(4), 291–308 (2008)
Lutz, C., Areces, C., Horrocks, I., Sattler, U.: Keys, nominals, and concrete domains. Journal of Artificial Intelligence Research 23, 667–726 (2004)
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)
Mostert, P.S., Shields, A.L.: On the structure of semigroups on a compact manifold with boundary. Annals of Mathematics 65, 117–143 (1957)
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)
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)
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)
Straccia, U.: Reasoning within fuzzy description logics. Journal of Artificial Intelligence Research 14, 137–166 (2001)
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)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Borgwardt, S., Distel, F., Peñaloza, R. (2012). How Fuzzy Is My Fuzzy Description Logic?. In: Gramlich, B., Miller, D., Sattler, U. (eds) Automated Reasoning. IJCAR 2012. Lecture Notes in Computer Science(), vol 7364. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31365-3_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-31365-3_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31364-6
Online ISBN: 978-3-642-31365-3
eBook Packages: Computer ScienceComputer Science (R0)