Unification in the Description Logic \(\mathcal{EL}\)

  • Franz Baader
  • Barbara Morawska
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5595)


The Description Logic \(\mathcal{EL}\) has recently drawn considerable attention since, on the one hand, important inference problems such as the subsumption problem are polynomial. On the other hand, \(\mathcal{EL}\) is used to define large biomedical ontologies. Unification in Description Logics has been proposed as a novel inference service that can, for example, be used to detect redundancies in ontologies. The main result of this paper is that unification in \(\mathcal{EL}\) is decidable. More precisely, \(\mathcal{EL}\)-unification is NP-complete, and thus has the same complexity as \(\mathcal{EL}\)-matching. We also show that, w.r.t. the unification type, \(\mathcal{EL}\) is less well-behaved: it is of type zero, which in particular implies that there are unification problems that have no finite complete set of unifiers.


Modal Logic Description Logic Concept Variable Concept Constructor Concept Term 
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.
    Baader, F.: Unification in commutative theories. J. of Symbolic Computation 8(5) (1989)Google Scholar
  2. 2.
    Baader, F.: Terminological cycles in KL-ONE-based knowledge representation languages. In: Proc. AAAI 1990 (1990)Google Scholar
  3. 3.
    Baader, F.: Terminological cycles in a description logic with existential restrictions. In: Proc. IJCAI 2003 (2003)Google Scholar
  4. 4.
    Baader, F., Brandt, S., Lutz, C.: Pushing the \(\mathcal{EL}\) envelope. In: Proc. IJCAI 2005 (2005)Google Scholar
  5. 5.
    Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P.F. (eds.): The Description Logic Handbook: Theory, Implementation, and Applications. Cambridge University Press, Cambridge (2003)zbMATHGoogle Scholar
  6. 6.
    Baader, F., Küsters, R.: Matching in description logics with existential restrictions. In: Proc. KR 2000 (2000)Google Scholar
  7. 7.
    Baader, F., Küsters, R.: Unification in a description logic with transitive closure of roles. In: Nieuwenhuis, R., Voronkov, A. (eds.) LPAR 2001. LNCS (LNAI), vol. 2250, p. 217. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Baader, F., Küsters, R., Borgida, A., McGuinness, D.L.: Matching in description logics. J. of Logic and Computation 9(3) (1999)Google Scholar
  9. 9.
    Baader, F., Narendran, P.: Unification of concepts terms in description logics. J. of Symbolic Computation 31(3) (2001)Google Scholar
  10. 10.
    Baader, F., Nutt, W.: Basic description logics. In: [5] (2003)Google Scholar
  11. 11.
    Baader, F., Sertkaya, B., Turhan, A.-Y.: Computing the least common subsumer w.r.t. a background terminology. J. of Applied Logic 5(3) (2007)Google Scholar
  12. 12.
    Baader, F., Snyder, W.: Unification theory. In: Handbook of Automated Reasoning, vol. I. Elsevier Science Publishers, Amsterdam (2001)Google Scholar
  13. 13.
    Brandt, S.: Polynomial time reasoning in a description logic with existential restrictions, GCI axioms, and—what else. In: Proc. ECAI 2004 (2004)Google Scholar
  14. 14.
    Ghilardi, S.: Best solving modal equations. Ann. Pure Appl. Logic 102(3) (2000)Google Scholar
  15. 15.
    Horrocks, I., Patel-Schneider, P.F., van Harmelen, F.: From SHIQ and RDF to OWL: The making of a web ontology language. Journal of Web Semantics 1(1) (2003)Google Scholar
  16. 16.
    Kazakov, Y., de Nivelle, H.: Subsumption of concepts in \(\mathcal{FL}_0\) for (cyclic) terminologies with respect to descriptive semantics is PSPACE-complete. In: Proc. DL 2003. CEUR Electronic Workshop Proceedings (2003),
  17. 17.
    Küsters, R.: Non-Standard Inferences in Description Logics. LNCS (LNAI), vol. 2100. Springer, Heidelberg (2001)zbMATHGoogle Scholar
  18. 18.
    Rector, A., Horrocks, I.: Experience building a large, re-usable medical ontology using a description logic with transitivity and concept inclusions. In: Proc. AAAI 1997 (1997)Google Scholar
  19. 19.
    Sofronie-Stokkermans, V.: Locality and subsumption testing in \(\mathcal{EL}\) and some of its extensions. In: Proc. AiML 2008 (2008)Google Scholar
  20. 20.
    Wolter, F., Zakharyaschev, M.: Undecidability of the unification and admissibility problems for modal and description logics. ACM Trans. Comput. Log. 9(4) (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Franz Baader
    • 1
  • Barbara Morawska
    • 1
  1. 1.Theoretical Computer ScienceTU DresdenGermany

Personalised recommendations