Decidability of Unification in EL without Top Constructor

  • Nguyen Thanh Binh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6902)


In recent years, the description logic \(\cal{EL}\) has received a significant interest. The description logic \(\cal{EL}\) is a knowledge representation formalism used e.g in natural language processing, configuration of technical systems, databases and biomedical ontologies. Unification is used there as a tool to recognize equivalent concepts. It has been proven that unification in \(\cal{EL}\) is NP-complete. This result was based on a locality property of certain \(\cal{EL}\) unifiers. In fact, the large medical ontology SNOMED CT was built on a subset of \({\cal{EL}}\)++ formalism, however, without top-concept. It would be interesting to investigate decidability of unification in extensions of \(\cal{EL}\) without using top-concept. In this paper, we look at decidability of unification in \(\cal{EL}\) without top (\(\cal{EL}^{- \top}\)). We show that a similar locality holds for \(\cal{EL}^{- \top}\), but decidability of \(\cal{EL}^{- \top}\) unification does not follow immediately from locality as it does in the case of unification in \(\cal{EL}\). However, by restricting further the locality property, we prove that \(\cal{EL}^{- \top}\) unification is decidable and construct an NExpTime decision procedure for the problem. Moreover, the procedure allows us to compute a specific set of solutions to the unification problem.


Description Logic Dependency Path Dependency Order Biomedical Ontology 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.


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  1. 1.
    Baader, F.: Unification in commutative theories. J. of Symbolic Computation 8(5), 479–497 (1989)CrossRefzbMATHGoogle Scholar
  2. 2.
    Baader, F.: Terminological cycles in a description logic with existential restrictions. In: Gottlob, G., Walsh, T. (eds.) Proc. of the 18th Int. Joint Conf. on Artificial Intelligence (IJCAI 2003), Acapulco, Mexico, pp. 325–330. Morgan Kaufmann, Los Altos (2003)Google Scholar
  3. 3.
    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
  4. 4.
    Baader, F., Küsters, R.: Matching in description logics with existential restrictions. In: Proc. of the 7th Int. Conf. on Principles of Knowledge Representation and Reasoning (KR 2000), pp. 261–272 (2000)Google Scholar
  5. 5.
    Baader, F., Morawska, B.: Unification in the description logic \(\cal{EL}\). In: Treinen, R. (ed.) RTA 2009. LNCS, vol. 5595, pp. 350–364. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Baader, F., Narendran, P.: Unification of concepts terms in description logics. J. of Symbolic Computation 31(3), 277–305 (2001)CrossRefzbMATHGoogle Scholar
  7. 7.
    Baader, F., Nutt, W.: Basic description logics, vol. 6, pp. 43–95 (2003)Google Scholar
  8. 8.
    Baader, F., Snyder, W.: Unification theory. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. I, pp. 447–533. Elsevier Science Publishers, Springer (2001)Google Scholar
  9. 9.
    Nebel, B.: Terminological reasoning is inherently intractable. Artificial Intelligence 43, 235–249 (1990)CrossRefzbMATHGoogle Scholar
  10. 10.
  11. 11.
    Kozen, D.: Lower bounds for natural proof systems. In: Proc. 18th Ann. Symp. on Foundations of Computer Science, pp. 254–266. IEEE Computer Society, Long Beach (October 1977)Google Scholar
  12. 12.
    Küsters, R.: Non-standard Inferences in Description Logics. LNAI 2001. Springer, Heidelberg (2001)CrossRefzbMATHGoogle Scholar
  13. 13.
    McAllester, D.: Automatic Recognition of Tractability in Inference Relations. JACM 40(2) (1993)Google Scholar
  14. 14.
  15. 15.
    Rector, A., Horrocks, I.: Experience building a large, re-usable medical ontology using a description logic with transitivity and concept inclusions. In: Proceedings of the Workshop on Ontological Engineering, AAAI Spring Symposium (AAAI 1997). AAAI Press, Stanford (1997)Google Scholar
  16. 16.
  17. 17.
    Sofronie-Stokkermans, V.: Locality and subsumption testing in \(\cal{EL}\) and some of its extension. In: Proceedings of AiML (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Nguyen Thanh Binh
    • 1
  1. 1.ETH ZurichSwitzerland

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