Skip to main content
SpringerLink
Log in

Finite Lattices Do Not Make Reasoning in \({\mathcal {ALCOI}}\) Harder

  • Conference paper
  • First Online:
Uncertainty Reasoning for the Semantic Web III (URSW 2012, URSW 2011, URSW 2013)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 8816))

Abstract

We consider the fuzzy description logic \({\mathcal {ALCOI}}\) with semantics based on a finite residuated De Morgan lattice. We show that reasoning in this logic is ExpTime-complete w.r.t. general TBoxes. In the sublogics \({\mathcal {ALCI}}\) and \({\mathcal {ALCO}}\), it is PSpace-complete w.r.t. acyclic TBoxes. This matches the known complexity bounds for reasoning in classical description logics between \({\mathcal ALC} \) and \({\mathcal {ALCOI}}\).

Partially supported by the DFG under grant BA 1122/17-1, in the research training group 1763 (QuantLA), and in the Cluster of Excellence ‘cfAED’.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The paper [18] considers the fuzzy modal logic K, which can be seen as a syntactic variant of fuzzy \({\mathcal ALC} \) with only one role.

  2. 2.

    We do not consider mixed TBoxes. We could allow axioms of the form \({\langle A\sqsubseteq C\ge \ell \rangle } \) in acyclic TBoxes, as long as they do not introduce cyclic dependencies. To avoid overloading the notation, we exclude this case.

  3. 3.

    This method, called lazy unfolding, is only correct for acyclic TBoxes.

  4. 4.

    There is at most one such predecessor, namely the parent node.

References

  1. Baader, F., Borgwardt, S., Peñaloza, R.: On the decidability status of fuzzy \({\cal {ALC}}\) with general concept inclusions. J. Philos. Logic (2014). doi:10.1007/s10992-014-9329-3

  2. Baader, F., Hladik, J., Peñaloza, R.: Automata can show PSPACE results for description logics. Inf. Comput. 206(9–10), 1045–1056 (2008)

    Article  MATH  Google Scholar 

  3. Baader, F., Lutz, C., Miličić, M., Sattler, U., Wolter, F.: Integrating description logics and action formalisms for reasoning about Web services. LTCS-Report 05–02, Chair for Automata Theory, TU Dresden, Germany (2005)

    Google Scholar 

  4. Baader, F., Peñaloza, R.: Are fuzzy description logics with general concept inclusion axioms decidable? In: Proceedings of the 2011 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE’11), pp. 1735–1742. IEEE Computer Society Press (2011)

    Google Scholar 

  5. 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, vol. 6989, pp. 55–70. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  6. Baader, F., Sattler, U.: An overview of tableau algorithms for description logics. Stud. Logica 69(1), 5–40 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bobillo, F., Bou, F., Straccia, U.: On the failure of the finite model property in some fuzzy description logics. Fuzzy Sets Syst. 172(1), 1–12 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bobillo, F., Straccia, U.: A fuzzy description logic with product t-norm. In: Proceedings of the 2007 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE’07), pp. 1–6. IEEE Computer Society Press (2007)

    Google Scholar 

  9. Bobillo, F., Straccia, U.: Fuzzy description logics with general t-norms and datatypes. Fuzzy Sets Syst. 160(23), 3382–3402 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bobillo, F., Straccia, U.: Reasoning with the finitely many-valued Łukasiewicz fuzzy description logic \({\cal {SROIQ}}\). Inf. Sci. 181(4), 758–778 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Borgwardt, S., Peñaloza, R.: Finite lattices do not make reasoning in \({\cal {ALCI}}\) harder. In: Bobillo, F., et al. (eds.) Proceedings of the 7th International Workshop on Uncertainty Reasoning for the Semantic Web (URSW’11). CEUR Workshop Proceedings, vol. 778, pp. 51–62 (2011)

    Google Scholar 

  12. Borgwardt, S., Peñaloza, R.: Consistency reasoning in lattice-based fuzzy description logics. Int. J. Approximate Reasoning 55(9), 1917–1938 (2014)

    Article  MATH  Google Scholar 

  13. Borgwardt, S., Peñaloza, R.: Description logics over lattices with multi-valued ontologies. In: Walsh, T. (ed.) Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI’11), pp. 768–773. AAAI Press (2011)

    Google Scholar 

  14. Borgwardt, S., Peñaloza, R.: Fuzzy ontologies over lattices with t-norms. In: Rosati, R., Rudolph, S., Zakharyaschev, M. (eds.) Proceedings of the 2011 International Workshop on Description Logics (DL’11), pp. 70–80. CEUR Workshop Proceedings (2011)

    Google Scholar 

  15. Borgwardt, S., Peñaloza, R.: A tableau algorithm for fuzzy description logics over residuated De Morgan lattices. In: Krötzsch, M., Straccia, U. (eds.) RR 2012. LNCS, vol. 7497, pp. 9–24. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  16. Borgwardt, S., Peñaloza, R.: Undecidability of fuzzy description logics. In: Brewka, G., Eiter, T., McIlraith, S.A. (eds.) Proceedings of the 13th International Conference on Principles of Knowledge Representation and Reasoning (KR’12), pp. 232–242. AAAI Press (2012)

    Google Scholar 

  17. Borgwardt, S., Peñaloza, R.: The complexity of lattice-based fuzzy description logics. J. Data Seman. 2(1), 1–19 (2013)

    Article  Google Scholar 

  18. Bou, F., Cerami, M., Esteva, F.: Finite-valued Łukasiewicz modal logic is PSPACE-complete. In: Walsh, T. (ed.) Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI’11), pp. 774–779. AAAI Press (2011)

    Google Scholar 

  19. Cerami, M., García-Cerdaña, À., Esteva, F.: On finitely-valued fuzzy description logics. Int. J. Approximate Reasoning 55(9), 1890–1916 (2014)

    Article  MATH  Google Scholar 

  20. Cerami, M., Straccia, U.: On the (un)decidability of fuzzy description logics under Łukasiewicz t-norm. Inf. Sci. 227, 1–21 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  21. De Cooman, G., Kerre, E.E.: Order norms on bounded partially ordered sets. J. Fuzzy Math. 2, 281–310 (1993)

    Google Scholar 

  22. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Studies in Logic and the Foundations of Mathematics, vol. 151. Elsevier, Amsterdam (2007)

    Google Scholar 

  23. Hájek, P.: Metamathematics of Fuzzy Logic (Trends in Logic). Springer, London (2001)

    Google Scholar 

  24. Hájek, P.: Making fuzzy description logic more general. Fuzzy Sets Syst. 154(1), 1–15 (2005)

    Article  MATH  Google Scholar 

  25. Horrocks, I., Sattler, U.: A description logic with transitive and inverse roles and role hierarchies. J. Logic Comput. 9(3), 385–410 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  26. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Trends in Logic, Studia Logica Library. Springer, New York (2000)

    Book  MATH  Google Scholar 

  27. Lukasiewicz, T., Straccia, U.: Managing uncertainty and vagueness in description logics for the semantic web. J. Web Seman. 6(4), 291–308 (2008)

    Article  Google Scholar 

  28. Motik, B., Shearer, R., Horrocks, I.: Hypertableau reasoning for description logics. J. Artif. Intell. Res. 36, 165–228 (2009)

    MATH  MathSciNet  Google Scholar 

  29. Schaerf, A.: Reasoning with individuals in concept languages. Data Knowl. Eng. 13(2), 141–176 (1994)

    Article  Google Scholar 

  30. Schild, K.: A correspondence theory for terminological logics: preliminary report. In: Mylopoulos, J., Reiter, R. (eds.) Proceedings of the 12th International Joint Conference on Artificial Intelligence (IJCAI’91), pp. 466–471. Morgan Kaufmann (1991)

    Google Scholar 

  31. Schmidt-Schauß, M., Smolka, G.: Attributive concept descriptions with complements. Artif. Intell. 48(1), 1–26 (1991)

    Article  MATH  Google Scholar 

  32. Steigmiller, A., Liebig, T., Glimm, B.: Extended caching, backjumping and merging for expressive description logics. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS, vol. 7364, pp. 514–529. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  33. Straccia, U.: Description logics over lattices. Int. J. Uncertainty Fuzziness Knowl.-Based Syst. 14(1), 1–16 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  34. Straccia, U.: Foundations of Fuzzy Logic and Semantic Web Languages. Studies in Informatics. CRC Press, Hoboken (2013)

    Google Scholar 

  35. Tobies, S.: The complexity of reasoning with cardinality restrictions and nominals in expressive description logics. J. Artif. Intell. Res. 12, 199–217 (2000)

    MATH  MathSciNet  Google Scholar 

  36. Tobies, S.: Complexity results and practical algorithms for logics in knowledge representation. Ph.D. thesis, RWTH Aachen, Germany (2001)

    Google Scholar 

  37. Vardi, M.Y., Wolper, P.: Automata-theoretic techniques for modal logics of programs. J. Comput. Syst. Sci. 32(2), 183–221 (1986)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Theoretical Computer Science, TU Dresden, Dresden, Germany

    Stefan Borgwardt & Rafael Peñaloza

  2. Center for Advancing Electronics Dresden, Dresden, Germany

    Rafael Peñaloza

Authors
  1. Stefan Borgwardt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rafael Peñaloza
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefan Borgwardt .

Editor information

Editors and Affiliations

  1. University of Zaragoza, Zaragoza, Spain

    Fernando Bobillo

  2. Universidade de Brasília, Brasília, Brazil

    Rommel N. Carvalho

  3. George Mason University, Fairfax, Virginia, USA

    Paulo C.G. Costa

  4. Università degli Studi di Bari, Bari, Italy

    Claudia d'Amato

  5. Università degli Studi di Bari, Bari, Italy

    Nicola Fanizzi

  6. George Mason University, Fairfax, Virginia, USA

    Kathryn B. Laskey

  7. MITRE Corporation, McLean, Virginia, USA

    Kenneth J. Laskey

  8. University of Oxford, Oxford, United Kingdom

    Thomas Lukasiewicz

  9. National University of Ireland, Galway, Ireland

    Matthias Nickles

  10. Goldman Sachs, Washington, District of Columbia, USA

    Michael Pool

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Borgwardt, S., Peñaloza, R. (2014). Finite Lattices Do Not Make Reasoning in \({\mathcal {ALCOI}}\) Harder. In: Bobillo, F., et al. Uncertainty Reasoning for the Semantic Web III. URSW URSW URSW 2012 2011 2013. Lecture Notes in Computer Science(), vol 8816. Springer, Cham. https://doi.org/10.1007/978-3-319-13413-0_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-13413-0_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-13412-3

  • Online ISBN: 978-3-319-13413-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation