Count and Forget: Uniform Interpolation of \(\mathcal{SHQ}\)-Ontologies

  • Patrick Koopmann
  • Renate A. Schmidt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8562)

Abstract

We propose a method for forgetting concept symbols and non-transitive roles symbols of \(\mathcal{SHQ}\)-ontologies, or for computing uniform interpolants in \(\mathcal{SHQ}\). Uniform interpolants restrict the symbols occuring in an ontology to a specified set, while preserving all logical entailments that can be expressed using this set in the description logic under consideration. Uniform interpolation has applications in ontology reuse, information hiding and ontology analysis, but so far no method for computing uniform interpolants for expressive description logics with number restrictions has been developed. Our results are not only interesting because they allow to compute uniform interpolants of ontologies using a more expressive language. Using number restrictions also allows to preserve more information in uniform interpolants of ontologies in less complex logics, such as \(\mathcal{ALC}\) or \(\mathcal{EL}\). The presented method computes uniform interpolants on the basis of a new resolution calculus for \(\mathcal{SHQ}\). The output of our method is expressed using \(\mathcal{SHQ}\mu\), which is \(\mathcal{SHQ}\) extended with fixpoint operators, to always enable a finite representation of the uniform interpolant. If the uniform interpolant uses fixpoint operators, it can be represented in \(\mathcal{SHQ}\) without fixpoints operators using additional concept symbols or by approximation.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Artale, A., Calvanese, D., Kontchakov, R., Zakharyaschev, M.: DL-Lite in the Light of First-Order Logic. In: AAAI 2007, pp. 361–366. AAAI Press (2007)Google Scholar
  2. 2.
    Calvanese, D., Giacomo, G.D., Lenzerini, M.: Reasoning in Expressive Description Logics with Fixpoints based on Automata on Infinite Trees. In: Proc. IJCAI 1999, pp. 84–89. Morgan Kaufmann (1999)Google Scholar
  3. 3.
    Gabbay, D.M., Schmidt, R.A., Szałas, A.: Second Order Quantifier Elimination: Foundations, Computational Aspects and Applications. College Publ. (2008)Google Scholar
  4. 4.
    Grau, B.C.: Privacy in ontology-based information systems: A pending matter. Semantic Web 1(1-2), 137–141 (2010)Google Scholar
  5. 5.
    Grau, B.C., Horrocks, I., Kazakov, Y., Sattler, U.: Modular reuse of ontologies: theory and practice. J. Artif. Intell. Res. 31(1), 273–318 (2008)MATHGoogle Scholar
  6. 6.
    Horrocks, I., Sattler, U., Tobies, S.: Practical Reasoning for Very Expressive Description Logics. Logic J. IGPL 8(3), 239–264 (2000)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Konev, B., Walther, D., Wolter, F.: Forgetting and Uniform Interpolation in Large-Scale Description Logic Terminologies. In: Proc. IJCAI 2009, pp. 830–835. AAAI Press (2009)Google Scholar
  8. 8.
    Koopmann, P., Schmidt, R.A.: Forgetting Concept and Role Symbols in \(\mathcal{ALCH}\)-Ontologies. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR-19 2013. LNCS, vol. 8312, pp. 552–567. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  9. 9.
    Koopmann, P., Schmidt, R.A.: Implementation and Evaluation of Forgetting in \(\mathcal{ALC}\)-Ontologies. In: Proc. WoMO 2013. CEUR-WS.org (2013)Google Scholar
  10. 10.
    Koopmann, P., Schmidt, R.A.: Uniform Interpolation of \(\mathcal{ALC}\)-Ontologies Using Fixpoints. In: Fontaine, P., Ringeissen, C., Schmidt, R.A. (eds.) FroCoS 2013. LNCS, vol. 8152, pp. 87–102. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  11. 11.
    Koopmann, P., Schmidt, R.A.: Count and Forget: Uniform Interpolation of \(\mathcal{SHQ}\)-Ontologies—Long Version. Tech. Rep., The University of Manchester (2014), http://www.cs.man.ac.uk/~koopmanp/IJCAR_KoopmannSchmidt2014_long.pdf
  12. 12.
    Ludwig, M., Konev, B.: Towards Practical Uniform Interpolation and Forgetting for \(\mathcal{ALC}\) TBoxes. In: Proc. DL 2013, pp. 377–389. CEUR-WS.org (2013)Google Scholar
  13. 13.
    Lutz, C., Seylan, I., Wolter, F.: An Automata-Theoretic Approach to Uniform Interpolation and Approximation in the Description Logic \(\mathcal{EL}\). In: Proc. KR 2012, pp. 286–296. AAAI Press (2012)Google Scholar
  14. 14.
    Lutz, C., Wolter, F.: Foundations for Uniform Interpolation and Forgetting in Expressive Description Logics. In: Proc. IJCAI 2011, pp. 989–995. AAAI Press (2011)Google Scholar
  15. 15.
    Nikitina, N.: Forgetting in General \(\mathcal{EL}\) Terminologies. In: Proc. DL 2011, pp. 345–355. CEUR-WS.org (2011)Google Scholar
  16. 16.
    Schmidt, R.A., Hustadt, U.: A principle for incorporating axioms into the first-order translation of modal formulae. In: Baader, F. (ed.) CADE 2003. LNCS (LNAI), vol. 2741, pp. 412–426. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  17. 17.
    Tobies, S.: Complexity results and practical algorithms for logics in knowledge representation. Ph.D. thesis, RWTH-Aachen, Germany (2001)Google Scholar
  18. 18.
    Wang, K., Wang, Z., Topor, R., Pan, J.Z., Antoniou, G.: Concept and Role Forgetting in \(\mathcal{ALC}\) Ontologies. In: Bernstein, A., Karger, D.R., Heath, T., Feigenbaum, L., Maynard, D., Motta, E., Thirunarayan, K. (eds.) ISWC 2009. LNCS, vol. 5823, pp. 666–681. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  19. 19.
    Wang, K., Wang, Z., Topor, R., Pan, J.Z., Antoniou, G.: Eliminating concepts and roles from ontologies in expressive descriptive logics. Comput. Intell. (to appear)Google Scholar
  20. 20.
    Wang, Z., Wang, K., Topor, R.W., Pan, J.Z.: Forgetting for knowledge bases in DL-Lite. Ann. Math. Artif. Intell. 58(1-2), 117–151 (2010)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Patrick Koopmann
    • 1
  • Renate A. Schmidt
    • 1
  1. 1.The University of ManchesterUK

Personalised recommendations