Uniform Interpolation of \(\mathcal{ALC}\)-Ontologies Using Fixpoints

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8152)


We present a method to compute uniform interpolants with fixpoints for ontologies specified in the description logic \(\mathcal{ALC}\). The aim of uniform interpolation is to reformulate an ontology such that it only uses a specified set of symbols, while preserving consequences that involve these symbols. It is known that in \(\mathcal{ALC}\) uniform interpolants cannot always be finitely represented. Our method computes uniform interpolants for the target language \(\mathcal{ALC}\mu\), which is \(\mathcal{ALC}\) enriched with fixpoint operators, and always computes a finite representation. If the result does not involve fixpoint operators, it is the uniform interpolant in \(\mathcal{ALC}\). The method focuses on eliminating concept symbols and combines resolution-based reasoning with an approach known from the area of second-order quantifier elimination to introduce fixpoint operators when needed. If fixpoint operators are not desired, it is possible to approximate the interpolant.


Modal Logic Description Logic Conjunctive Normal Form Domain Element Small Clause 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ackermann, W.: Untersuchungen über das Eliminationsproblem der mathematischen Logik. Mathematische Annalen 110(1), 390–413 (1935)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bachmair, L., Ganzinger, H.: Resolution theorem proving. In: Handbook of Automated Reasoning, pp. 19–99. Elsevier, MIT Press (2001)Google Scholar
  3. 3.
    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
  4. 4.
    D’Agonstino, G., Hollenberg, M.: Uniform interpolation, automata and the modal μ-calculus. In: AiML, vol. 1, pp. 73–84. CSLI Pub. (1998)Google Scholar
  5. 5.
    D’Agostino, G., Lenzi, G.: On modal μ-calculus with explicit interpolants. J. Applied Logic 4(3), 256–278 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gabbay, D., Ohlbach, H.J.: Quantifier elimination in second-order predicate logic. In: Proc. KR 1992, pp. 425–435. Morgan Kaufmann (1992)Google Scholar
  7. 7.
    Gabbay, D.M., Schmidt, R.A., Szałas, A.: Second Order Quantifier Elimination: Foundations, Computational Aspects and Applications. College Publ. (2008)Google Scholar
  8. 8.
    Goranko, V., Hustadt, U., Schmidt, R.A., Vakarelov, D.: SCAN is complete for all Sahlqvist formulae. In: Berghammer, R., Möller, B., Struth, G. (eds.) RelMiCS 2003. LNCS, vol. 3051, pp. 149–162. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  9. 9.
    Grau, B.C., Motik, B.: Reasoning over ontologies with hidden content: The import-by-query approach. J. Artificial Intelligence Research 45, 197–255 (2012)zbMATHGoogle Scholar
  10. 10.
    Herzig, A., Mengin, J.: Uniform interpolation by resolution in modal logic. In: Hölldobler, S., Lutz, C., Wansing, H. (eds.) JELIA 2008. LNCS (LNAI), vol. 5293, pp. 219–231. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  11. 11.
    Lutz, C., Piro, R., Wolter, F.: \(\mathcal{EL}\)-concepts go second-order: Greatest fixpoints and simulation quantifiers. In: Proc. DL 2010, pp. 43–54. (2010)Google Scholar
  12. 12.
    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
  13. 13.
    Nikitina, N.: Forgetting in General EL Terminologies. In: Description Logics. Proc. DL 2011. (2011)Google Scholar
  14. 14.
    Nonnengart, A., Szałas, A.: A fixpoint approach to second-order quantifier elimination with applications to correspondence theory. In: Logic at Work, pp. 307–328. Springer (1999)Google Scholar
  15. 15.
    Schmidt, R.A.: The Ackermann approach for modal logic, correspondence theory and second-order reduction. J. Appl. Logic 10(1), 52–74 (2012)CrossRefzbMATHGoogle Scholar
  16. 16.
    Simancik, F., Kazakov, Y., Horrocks, I.: Consequence-based reasoning beyond Horn ontologies. In: Proc. IJCAI 2011, pp. 1093–1098. AAAI Press (2011)Google Scholar
  17. 17.
    Szałas, A.: Second-order reasoning in description logics. J. Appl. Non-Classical Logics 16(3-4), 517–530 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Wang, Z., Wang, K., Topor, R., Zhang, X.: Tableau-based forgetting in \(\mathcal{ALC}\) ontologies. In: Proc. ECAI 2010, pp. 47–52. IOS Press (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.The University of ManchesterUK

Personalised recommendations