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Journal on Data Semantics

, Volume 5, Issue 2, pp 55–75 | Cite as

Answering Fuzzy Conjunctive Queries Over Finitely Valued Fuzzy Ontologies

  • Stefan Borgwardt
  • Theofilos Mailis
  • Rafael Peñaloza
  • Anni-Yasmin Turhan
Original Article

Abstract

Fuzzy Description Logics (DLs) provide a means for representing vague knowledge about an application domain. In this paper, we study fuzzy extensions of conjunctive queries (CQs) over the DL \({\mathcal {SROIQ}}\) based on finite chains of degrees of truth. To answer such queries, we extend a well-known technique that reduces the fuzzy ontology to a classical one, and use classical DL reasoners as a black box. We improve the complexity of previous reduction techniques for finitely valued fuzzy DLs, which allows us to prove tight complexity results for answering certain kinds of fuzzy CQs. We conclude with an experimental evaluation of a prototype implementation, showing the feasibility of our approach.

Notes

Acknowledgments

This work was partially supported by the German Research Foundation (DFG) under the research Grant BA 1122/17-1 (FuzzyDL), the Collaborative Research Center 912 “Highly Adaptive Energy-Efficient Computing,” and the Cluster of Excellence “Center for Advancing Electronics Dresden”; it was developed while R. Peñaloza was affiliated with TU Dresden and the Center for Advancing Electronics Dresden, Germany. We also want to thank Fernando Bobillo for providing us with a binary of the DeLorean system, and the anonymous reviewers for their valuable comments on earlier drafts of this paper.

References

  1. 1.
    Baader F, Borgwardt S, Peñaloza R (2015) On the decidability status of fuzzy \({\cal ALC}\) with general concept inclusions. Journal of Philosophical Logic 44(2):117–146. doi: 10.1007/s10992-014-9329-3 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bobillo F (2015) The impact of crispness in fuzzy tractable ontologies: The case of fuzzy OWL 2 EL. IEEE Transactions on Fuzzy Systems doi: 10.1109/TFUZZ.2015.2505329. In press
  3. 3.
    Bobillo F, Delgado M, Gómez-Romero J (2008) Optimizing the crisp representation of the fuzzy Description Logic \({\cal SROIQ\it }\). In: Uncertainty Reasoning for the Semantic Web I, pp. 189–206. SpringerGoogle Scholar
  4. 4.
    Bobillo F, Delgado M, Gómez-Romero J (2009) Crisp representations and reasoning for fuzzy ontologies. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 17(4):501–530. doi: 10.1142/S0218488509006121 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bobillo F, Delgado M, Gómez-Romero J (2012) DeLorean: A reasoner for fuzzy OWL 2. Expert Systems with Applications 39(1):258–272. doi: 10.1016/j.eswa.2011.07.016 CrossRefGoogle Scholar
  6. 6.
    Bobillo F, Delgado M, Gómez-Romero J (2013) Reasoning in fuzzy OWL 2 with DeLorean. In: Bobillo F, da Costa PCG, d’Amato C, Fanizzi N, Laskey K, Laskey K, Lukasiewicz T,  Nickles M, Pool M (eds.) Uncertainty Reasoning for the Semantic Web II, Lecture Notes in Computer Science, vol. 7123, pp. 119–138. Springer doi: 10.1007/978-3-642-35975-0-7
  7. 7.
    Bobillo F, Delgado M, Gómez-Romero J, Straccia U (2009) Fuzzy Description Logics under Gödel semantics. International Journal of Approximate Reasoning 50(3):494–514MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bobillo F, Delgado M, Gómez-Romero J, Straccia U (2012) Joining Gödel and Zadeh fuzzy logics in fuzzy description logics. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 20(4):475–508. doi: 10.1142/S0218488512500249 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bobillo F, Delgado M, Sanchez-Sanchez JC (2013) Parallel algorithms for fuzzy ontology reasoning. IEEE Transactions on Fuzzy Systems 21(4):775–781. doi: 10.1109/TFUZZ.2012.2230266 CrossRefGoogle Scholar
  10. 10.
    Bobillo F, Straccia U (2011) Reasoning with the finitely many-valued Łukasiewicz fuzzy Description Logic \({\cal SROIQ}\). Information Sciences 181(4):758–778MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bobillo F, Straccia U (2013) Finite fuzzy Description Logics and crisp representations. In: Uncertainty Reasoning for the Semantic Web II, pp. 99–118. SpringerGoogle Scholar
  12. 12.
    Borgwardt S, Distel F, Peñaloza R (2015) The limits of decidability in fuzzy description logics with general concept inclusions. Artificial Intelligence 218:23–55. doi: 10.1016/j.artint.2014.09.001 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Borgwardt S, Peñaloza R (2013) The complexity of lattice-based fuzzy description logics. Journal on Data Semantics 2(1):1–19CrossRefGoogle Scholar
  14. 14.
    Borgwardt S, Peñaloza R (2014) Consistency reasoning in lattice-based fuzzy description logics. International Journal of Approximate Reasoning 55(9):1917–1938. doi: 10.1016/j.ijar.2013.07.006 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Calvanese D, De Giacomo G, Lenzerini M (1998) On the decidability of query containment under constraints. In: Proceedings of the seventeenth ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems, pp. 149–158. ACMGoogle Scholar
  16. 16.
    Calvanese D, Eiter T, Ortiz, M (2009) Regular path queries in expressive description logics with nominals. In: Boutilier C (ed.) Proc. of the 21st Int. Joint Conf. on Artificial Intelligence (IJCAI’09), pp. 714–720. AAAI Press. http://ijcai.org/papers09/Papers/IJCAI09-124.pdf
  17. 17.
    Cheng J, Ma Z, Zhang F, Wang X (2009) Deciding query entailment for fuzzy \({\cal SHIN\it }\) ontologies. In: Proceedings of the 4th Asian Conference on The Semantic Web, ASWC ’09, pp. 120–134. SpringerGoogle Scholar
  18. 18.
    Cheng J, Ma Z, Zhang F, Wang X (2009) Deciding query entailment in fuzzy Description Logic knowledge bases. In: Database and Expert Systems Applications, pp. 830–837. SpringerGoogle Scholar
  19. 19.
    Cimiano P, Haase P, Ji Q, Mailis T, Stamou G, Stoilos G, Tran DT, Tzouvaras V (2008) Reasoning with large A-Boxes in fuzzy Description Logics using DL reasoners: an experimental evaluation. In: Proceedings of the ESWC Workshop on Advancing Reasoning on the Web: Scalability and CommonsenseGoogle Scholar
  20. 20.
    Eiter T, Lutz C, Ortiz M, Šimkus M (2009) Query answering in description logics with transitive roles. In: Boutilier C (ed.) Proc. of the 21st Int. Joint Conf. on Artificial Intelligence (IJCAI’09), pp. 759–764. AAAI Press . http://ijcai.org/papers09/Papers/IJCAI09-131.pdf
  21. 21.
    Glimm B, Horrocks I, Motik B, Stoilos G, Wang Z (2014) HermiT: An OWL 2 reasoner. Journal of Automated Reasoning 53(3):245–269CrossRefzbMATHGoogle Scholar
  22. 22.
    Glimm B, Horrocks I, Sattler U (2008) Unions of conjunctive queries in \({\cal SHOQ\it }\). In: Brewka G, Lang J (eds.) Proc. of the 11th Int. Conf. on Principles of Knowledge Representation and Reasoning (KR’08), pp. 252–262. AAAI Press. http://www.aaai.org/Library/KR/2008/kr08-025.php
  23. 23.
    Guo Y, Pan Z, Heflin J (2005) LUBM: A benchmark for OWL knowledge base systems. Web Semantics: Science, Services and Agents on the World Wide Web 3(2):158–182CrossRefGoogle Scholar
  24. 24.
    Gutiérrez-Basulto V, Ibañez-García Y, Kontchakov R, Kostylev EV (2013) Conjunctive queries with negation over DL-Lite: A closer look. In: Faber W, Lembo D (eds.) Proc. of the 7th Int. Conf. on Web Reasoning and Rule Systems (RR’13), Lecture Notes in Computer Science, vol. 7994, pp. 109–122. Springer-Verlag doi: 10.1007/978-3-642-39666-3_9
  25. 25.
    Horrocks I, Kutz O, Sattler U (2006) The even more irresistible \({\cal SROIQ\it }\). In: Proc. of the 10th International Conference of Knowledge Representation and Reasoning (KR-2006), 6, pp. 57–67Google Scholar
  26. 26.
    Li Y, Xu B, Lu J, Kang D, Wang P (2005) A family of extended fuzzy Description Logics. In: Computer Software and Applications Conference, vol. 1, pp. 221–226. IEEEGoogle Scholar
  27. 27.
    Lutz C (2008) The complexity of conjunctive query answering in expressive description logics. In: Proc. of the 4th Int. Joint Conf. on Automated Reasoning (IJCAR’08), Lecture Notes in Artificial Intelligence, vol. 5195, pp. 179–193. Springer-Verlag doi: 10.1007/978-3-540-71070-7_16
  28. 28.
    Mailis T, Peñaloza R, Turhan AY (2014) Conjunctive query answering in finitely-valued fuzzy description logics. In: Kontchakov R, Mugnier ML (eds.) Proceedings of the 8th International Conference on Web Reasoning and Rule Systems (RR 2014), vol. 8741, pp. 124–139. SpringerGoogle Scholar
  29. 29.
    Mailis T, Stoilos G, Stamou G (2010) Expressive reasoning with Horn rules and fuzzy Description Logics. Knowledge and Information Systems 25(1):105–136CrossRefGoogle Scholar
  30. 30.
    Straccia, U.: Foundations of Fuzzy Logic and Semantic Web Languages. CRC Press, UK (2013)Google Scholar
  31. 31.
    Möller R, Neuenstadt C, Özçep ÖL, Wandelt S (2013) Advances in accessing big data with expressive ontologies. In: Timm IJ, Thimm M (eds.) KI 2013: Advances in Artificial Intelligence - 36th Annual German Conference on AI, Koblenz, Germany, September 16-20, 2013. Proceedings, Lecture Notes in Computer Science, vol. 8077, pp. 118–129. Springer (2013)Google Scholar
  32. 32.
    Pan, J.Z., Stamou, G.B., Stoilos, G., Thomas, E.: Expressive querying over fuzzy DL-Lite ontologies. In: 20th International Workshop on Description Logics (2007)Google Scholar
  33. 33.
    Rosati R (2007) The limits of querying ontologies. In: Schwentick T, Suciu D (eds.) Proc. of the 11th Int. Conf. on Database Theory (ICDT’07), Lecture Notes in Computer Science, vol. 4353, pp. 164–178. Springer-Verlag (2007). doi: 10.1007/11965893_12
  34. 34.
    Rudolph S, Krötzsch M, Hitzler P (2008) Cheap Boolean role constructors for description logics. In: Hölldobler S, Lutz C,  Wansing H (eds.) Proceedings of the 11th European Conference on Logics in Artificial Intelligence (JELIA’08), Lecture Notes in Computer Science, vol. 5293, pp. 362–374. Springer doi: 10.1007/978-3-540-87803-2_30
  35. 35.
    Simou N, Stoilos G, Tzouvaras V, Stamou GB, Kollias SD (2008) Storing and querying fuzzy knowledge in the Semantic Web. In: 7th International Workshop on Uncertainty Reasoning For the Semantic Web. Karlsruhe, GermanyGoogle Scholar
  36. 36.
    Stoilos G, Stamou GB (2007) Extending fuzzy description logics for the semantic web. In: Golbreich C, Kalyanpur A, Parsia B (eds.) Proc. of the 3rd Int. Workshop on OWL: Experiences and Directions (OWLED’07), CEUR Workshop Proceedings, vol. 258 http://ceur-ws.org/Vol-258/paper12.pdf
  37. 37.
    Straccia U (2004) Transforming fuzzy description logics into classical description logics. In: Proc. of the 9th Eur. Conf. on Logics in Artificial Intelligence (JELIA’04), pp. 385–399. SpringerGoogle Scholar
  38. 38.
    Straccia U (2006) Answering vague queries in fuzzy DL-Lite. In: Proceedings of the 11th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems,(IPMU-06), pp. 2238–2245Google Scholar
  39. 39.
    Straccia U (2006) Towards top-\(k\) query answering in Description Logics: the case of DL-Lite. In: Logics in Artificial Intelligence, Lecture Notes in Computer Science, pp. 439–451. SpringerGoogle Scholar
  40. 40.
    Straccia U (2013) Foundations of Fuzzy Logic and Semantic Web Languages. CRC Press, USAzbMATHGoogle Scholar
  41. 41.
    Straccia U (2014) On the top-k retrieval problem for ontology-based access to databases. In: Flexible Approaches in Data, Information and Knowledge Management, pp. 95–114. SpringerGoogle Scholar
  42. 42.
    Zhou Y, Grau BC, Nenov Y, Horrocks I (2015) PAGOdA: Pay-as-you-go ABox reasoning. In: Calvanese D, Konev B (eds.) Proceedings of the 28th International Workshop on Description Logics., CEUR Workshop Proceedings, vol. 1350. CEUR-WS.orgGoogle Scholar
  43. 43.
    Zhou Y, Nenov Y, Grau BC, Horrocks I (2014) Pay-as-you-go OWL query answering using a triple store. In: Proc. of the 28th AAAI Conf. on Artificial Intelligence (AAAI’14), pp. 1142–1148. AAAI Press (2014). http://www.aaai.org/ocs/index.php/AAAI/AAAI14/paper/view/8232

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Stefan Borgwardt
    • 1
  • Theofilos Mailis
    • 2
  • Rafael Peñaloza
    • 3
  • Anni-Yasmin Turhan
    • 1
  1. 1.Chair for Automata Theory, Institute for Theoretical Computer ScienceTechnische Universität DresdenDresdenGermany
  2. 2.Department of Informatics and TelecommunicationsNational and Kapodistrian University of AthensAthensGreece
  3. 3.KRDB Research CentreFree University of Bozen-BolzanoBolzanoItaly

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