Concept Forgetting in \(\mathcal {ALCOI}\)-Ontologies Using an Ackermann Approach

  • Yizheng ZhaoEmail author
  • Renate A. Schmidt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9366)


We present a method for forgetting concept symbols in ontologies specified in the description logic \(\mathcal {ALCOI}\). The method is an adaptation and improvement of a second-order quantifier elimination method developed for modal logics and used for computing correspondence properties for modal axioms. It follows an approach exploiting a result of Ackermann adapted to description logics. An important feature inherited from the modal approach is that the inference rules are guided by an ordering compatible with the elimination order of the concept symbols. This provides more control over the inference process and reduces non-determinism, resulting in a smaller search space. The method is extended with a new case splitting inference rule, and several simplification rules. Compared to related forgetting and uniform interpolation methods for description logics, the method can handle inverse roles, nominals and ABoxes. Compared to the modal approach on which it is based, it is more efficient in time and improves the success rates. The method has been implemented in Java using the OWL API. Experimental results show that the order in which the concept symbols are eliminated significantly affects the success rate and efficiency.


Modal Logic Inference Rule Description Logic Correspondence Theory Elimination Order 
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 \(\ddot{u}\)ber das Eliminationsproblem der mathematischen Logik. Mathematische Annalen 110(1), 390–413 (1935)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bachmair, L., Ganzinger, H., Waldmann, U.: Refutational theorem proving for hierarchic first-order theories. Applicable Algebra in Engineering, Communication and Computing 5(3–4), 193–212 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Conradie, W., Goranko, V., Vakarelov, D.: Algorithmic correspondence and completeness in modal logic. I. The core algorithm SQEMA. Logical Methods in Computer Science 2(1) (2006)Google Scholar
  4. 4.
    Grau, B.C., Motik, B.: Reasoning over ontologies with hidden content: The importby-query approach. Journal of Artificial Intelligence Research 45, 197–255 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Doherty, P., Łukaszewicz, W., Szałas, A.: Computing circumscription revisited: A reduction algorithm. Journal of Automated Reasoning 18(3), 297–336 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Doherty, P., Łukaszewicz, W., Szałas, A.: General domain circumscription and its effective reductions. Fundamenta Informaticae 36(1), 23–55 (1998)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Gabbay, D.M., Ohlbach, H.J.: Quantifier elimination in second-order predicate logic. In: Principles of Knowledge Representation and Reasoning (KR92), pp. 425–435. Morgan Kaufmann (1992)Google Scholar
  8. 8.
    Gabbay, D.M., Schmidt, R.A., Szałas, A.: Second Order Quantifier Elimination: Foundations, Computational Aspects and Applications. College Publications (2008)Google Scholar
  9. 9.
    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
  10. 10.
    Konev, B., Lutz, C., Walther, D., Wolter, F.: Model-theoretic inseparability and modularity of description logic ontologies. Artificial Intelligence 203, 66–103 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Konev, B., Walther, D., Wolter, F.: Forgetting and uniform interpolation in extensions of the description logic \(\cal EL\). In: Proceedings of the 22nd International Workshop on Description Logics (DL 2009). CEUR Workshop Proceedings, vol. 477. (2009)Google Scholar
  12. 12.
    Koopmann, P., Schmidt, R.A.: Forgetting concept and role symbols in \(\cal ALCH\)-ontologies. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR 2013. LNCS, vol. 8312, pp. 552–567. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  13. 13.
    Koopmann, P., Schmidt, R.A.: Implementation and evaluation of forgetting in ALC-ontologies. In: Proceedings of the 7th International Workshop on Modular Ontologies (WoMo 2013). CEUR Workshop Proceedings, vol. 1081, pp. 1–12. (2013)Google Scholar
  14. 14.
    Koopmann, P., Schmidt, R.A.: Uniform interpolation of \(\cal ALC\)-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
  15. 15.
    Koopmann, P., Schmidt, R.A.: Count and forget: uniform interpolation of \(\cal SHQ\)-ontologies. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS, vol. 8562, pp. 434–448. Springer, Heidelberg (2014) Google Scholar
  16. 16.
    Koopmann, P., Schmidt, R.A.: LETHE: A saturation-based tool for non-classical reasoning (2015). Manuscript, submittedGoogle Scholar
  17. 17.
    Koopmann, P., Schmidt, R.A.: Saturated-based forgetting in the description logic SIF. In: Proceedings of the 28th International Workshop on Description Logics (DL 2015). CEUR Workshop Proceedings, vol. 1350. (2015)Google Scholar
  18. 18.
    Koopmann, P., Schmidt, R.A.: Uniform interpolation and forgetting for \(\cal ALC\)-ontologies with ABoxes. In: Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, pp. 175–181. AAAI Press (2015)Google Scholar
  19. 19.
    Ludwig, M., Konev, B.: Towards practical uniform interpolation and forgetting for \(\cal ALC\) TBoxes. In: Proceedings of the 26th International Workshop on Description Logics (DL 2013). CEUR Workshop Proceedings, vol. 1014, pp. 377–389. (2013)Google Scholar
  20. 20.
    Lutz, C., Seylan, I., Wolter, F.: An automata-theoretic approach to uniform interpolation and approximation in the description logic \(\cal EL\). In: Principles of Knowledge Representation and Reasoning: KR 2012, pp. 286–297. AAAI Press (2012)Google Scholar
  21. 21.
    Lutz, C., Wolter, F.: Foundations for uniform interpolation and forgetting in expressive description logics. In: Proceedings of IJCAI 2011, pp. 989–995. IJCAI/AAAI (2011)Google Scholar
  22. 22.
    Nikitina, N.: Forgetting in general \(\cal EL\) terminologies. In: Proceedings of the 24th International Workshop on Description Logics (DL 2011). CEUR Workshop Proceedings, vol. 745. (2011)Google Scholar
  23. 23.
    Nonnengart, A., Szałas, A.: A fixpoint approach to second-order quantifier elimination with applications to correspondence theory. In: Orlowska, E. (ed.) Logic at Work: Essays Dedicated to the Memory of Helena Rasiowa, pp. 307–328. Springer (1999)Google Scholar
  24. 24.
    Ohlbach, H.J.: SCAN–elimination of predicate quantifiers. In: McRobbie, M.A., Slaney, J.K. (eds.) CADE 1996. LNCS, vol. 1104, pp. 161–165. Springer, Heidelberg (1996) CrossRefGoogle Scholar
  25. 25.
    Schild, K.: A correspondence theory for terminological logics: preliminary report. In: Proceedings of IJCAI 1991, pp. 466–471. Morgan Kaufmann (1991)Google Scholar
  26. 26.
    Schmidt, R.A.: The Ackermann approach for modal logic, correspondence theory and second-order reduction. Journal of Applied Logic 10(1), 52–74 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Szałas, A.: On the correspondence between modal and classical logic: An automated approach. Journal of Logic and Computation 3, 605–620 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Szałas, A.: Second-order reasoning in description logics. Journal of Applied Non-Classical Logics 16(3–4), 517–530 (2006)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Wang, K., Wang, Z., Topor, R., Pan, J.Z., Antoniou, G.: Concept and role forgetting in \(\cal ALC\) 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
  30. 30.
    Wang, K., Wang, Z., Topor, R., Pan, J.Z., Antoniou, G.: Eliminating concepts and roles from ontologies in expressive description logics. Computational Intelligence 30(2), 205–232 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Wang, Z., Wang, K., Topor, R., Pan, J.Z.: Forgetting concepts in DL-Lite. In: Bechhofer, S., Hauswirth, M., Hoffmann, J., Koubarakis, M. (eds.) ESWC 2008. LNCS, vol. 5021, pp. 245–257. Springer, Heidelberg (2008)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.The University of ManchesterManchesterUK

Personalised recommendations