Foundations of Description Logics

  • Sebastian Rudolph
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6848)


This chapter accompanies the foundational lecture on Description Logics (DLs) at the 7th Reasoning Web Summer School in Galway, Ireland, 2011. It introduces basic notions and facts about this family of logics which has significantly gained in importance over the recent years as these logics constitute the formal basis for today’s most expressive ontology languages, the OWL (Web Ontology Language) family.

We start out from some general remarks and examples demonstrating the modeling capabilities of description logics as well as their relation to first-order predicate logic. Then we begin our formal treatment by introducing the syntax of DL knowledge bases which comes in three parts: RBox, TBox and ABox. Thereafter, we provide the corresponding standard model-theoretic semantics and give a glimpse of the alternative way of defining the semantics via an embedding into first-order logic with equality.

We continue with an overview of the naming conventions for DLs before we delve into considerations about different notions of semantic alikeness (concept and knowledge base equivalence as well as emulation). These are crucial for investigating the expressivity of DLs and performing normalization. We move on by reviewing knowledge representation capabilities brought about by different DL features and their combinations as well as some model-theoretic properties associated thereto.

Subsequently, we consider typical reasoning tasks occurring in the context of DL knowledge bases. We show how some of these tasks can be reduced to each other, and have a look at different algorithmic approaches to realize automated reasoning in DLs.

Finally, we establish connections between DLs and OWL. We show how DL knowledge bases can be expressed in OWL and, conversely, how OWL modeling features can be translated into DLs.

In our considerations, we focus on the description logic \(\mathcal{SROIQ}\) which underlies the most recent and most expressive yet decidable version of OWL called OWL 2 DL. We concentrate on the logical aspects and omit data types as well as extralogical features from our treatise. Examples and exercises are provided throughout the chapter.


Knowledge Base Resource Description Framework Description Logic Cardinality Constraint Conjunctive Query 
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.
    Andréka, H., van Benthem, J.F.A.K., Németi, I.: Modal languages and bounded fragments of predicate logic. Journal of Philosophical Logic 27(3), 217–274 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P. (eds.): The Description Logic Handbook: Theory, Implementation, and Applications, 2nd edn. Cambridge University Press, Cambridge (2007)zbMATHGoogle Scholar
  3. 3.
    Beckett, D., Berners-Lee, T.: Turtle – Terse RDF Triple Language. W3C Team Submission (January 14, 2008),
  4. 4.
    Berners-Lee, T., Hendler, J., Lassila, O.: The Semantic Web. In: Scientific American, pp. 96–101 (May 2001)Google Scholar
  5. 5.
    Blackburn, P., van Benthem, J.F.A.K., Wolter, F. (eds.): Handbook of Modal Logic. Studies in Logic and Practical Reasoning, vol. 3. Elsevier Science, Amsterdam (2006)Google Scholar
  6. 6.
    Borgida, A.: On the relative expressiveness of description logics and predicate logics. Artificial Intelligence 82(1–2), 353–367 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brachman, R.J., Levesque, H.J.: The tractability of subsumption in frame-based description languages. In: Brachman, R.J. (ed.) Proceedings of the 4th National Conference on Artificial Intelligence (AAAI 1984), pp. 34–37. AAAI Press, Menlo Park (1984)Google Scholar
  8. 8.
    Calvanese, D., Eiter, T., Ortiz, M.: Regular path queries in expressive description logics with nominals. In: Boutilier, C. (ed.) Proceedings of the 21st International Conference on Artificial Intelligence (IJCAI 2009), pp. 714–720 (2009)Google Scholar
  9. 9.
    Chandra, A.K., Merlin, P.M.: Optimal implementation of conjunctive queries in relational data bases. In: Hopcroft, J.E., Friedman, E.P., Harrison, M.A. (eds.) Proceedings of the 9th Annual ACM Symposium on Theory of Computing (STOC 1977), pp. 77–90. ACM Press, New York (1977)Google Scholar
  10. 10.
    Chang, C.C., Jerome Keisler, H.: Model Theory, 3rd edn. Studies in Logic and the Foundations of Mathematics, vol. 73. North Holland, Amsterdam (1990)Google Scholar
  11. 11.
    Ebbinghaus, H.-D., Flum, J., Thomas, W.: Mathematical Logic. Springer, Heidelberg (1996)zbMATHGoogle Scholar
  12. 12.
    Fitting, M.: First-Order Logic and Automated Theorem Proving, 2nd edn. Springer, Heidelberg (1996)CrossRefzbMATHGoogle Scholar
  13. 13.
    Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  14. 14.
    Glimm, B., Horrocks, I., Sattler, U.: Deciding \(\mathcal{SHOQ}^\cap\) knowledge base consistency using alternating automata. In: Baader, F., Lutz, C., Motik, B. (eds.) Description Logics. CEUR Workshop Proceedings, vol. 353 (2008),
  15. 15.
    Glimm, B., Horrocks, I., Sattler, U.: Unions of conjunctive queries in \(\mathcal{SHOQ}\). In: Brewka, G., Lang, J. (eds.) Proceedings of the 11th International Conference on Principles of Knowledge Representation and Reasoning (KR 2008), pp. 252–262. AAAI Press, Menlo Park (2008)Google Scholar
  16. 16.
    Glimm, B., Lutz, C., Horrocks, I., Sattler, U.: Answering conjunctive queries in the SHIQ description logic. Journal of Artificial Intelligence Research 31, 150–197 (2008)zbMATHGoogle Scholar
  17. 17.
    Golbreich, C., Zhang, S., Bodenreider, O.: The foundational model of anatomy in OWL: Experience and perspectives. Journal of Web Semantics 4(3) (2006)Google Scholar
  18. 18.
    Haarslev, V., Möller, R.: RACER System Description. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 701–705. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  19. 19.
    Hitzler, P., Krötzsch, M., Parsia, B., Patel-Schneider, P.F., Rudolph, S. (eds.): OWL 2 Web Ontology Language: Primer. W3C Recommendation (2009),
  20. 20.
    Hitzler, P., Krötzsch, M., Rudolph, S.: Foundations of Semantic Web Technologies. Chapman & Hall/CRC (2009)Google Scholar
  21. 21.
    Horridge, M., Parsia, B., Sattler, U.: Laconic and Precise Justifications in OWL. In: Sheth, A.P., Staab, S., Dean, M., Paolucci, M., Maynard, D., Finin, T., Thirunarayan, K. (eds.) ISWC 2008. LNCS, vol. 5318, pp. 323–338. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  22. 22.
    Horrocks, I., Sattler, U.: A tableau decision procedure for \(\mathcal{SHOIQ}\). Journal of Automated Reasoning 39(3), 249–276 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Horrocks, I., Kutz, O., Sattler, U.: The even more irresistible \(\mathcal{SROIQ}\). In: Doherty, P., Mylopoulos, J., Welty, C.A. (eds.) Proceedings of the 10th International Conference on Principles of Knowledge Representation and Reasoning (KR 2006), pp. 57–67. AAAI Press, Menlo Park (2006)Google Scholar
  24. 24.
    Kazakov, Y., Motik, B.: A resolution-based decision procedure for \(\mathcal{SHOIQ}\). Journal of Automated Reasoning 40(2-3), 89–116 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kazakov, Y.: \(\mathcal{RIQ}\) and \(\mathcal{SROIQ}\) are harder than \(\mathcal{SHOIQ}\). In: Brewka, G., Lang, J. (eds.) Proceedings of the 11th International Conference on Principles of Knowledge Representation and Reasoning (KR 2008), pp. 274–284. AAAI Press, Menlo Park (2008)Google Scholar
  26. 26.
    Kazakov, Y.: Consequence-driven reasoning for horn \(\mathcal{SHIQ}\) ontologies. In: Boutilier, C. (ed.) Proceedings of the 21st International Conference on Artificial Intelligence (IJCAI 2009), pp. 2040–2045 (2009)Google Scholar
  27. 27.
    Krötzsch, M., Rudolph, S., Hitzler, P.: Description logic rules. In: Ghallab, M., Spyropoulos, C.D., Fakotakis, N., Avouris, N. (eds.) Proceedings of the 18th European Conference on Artificial Intelligence (ECAI 2008), pp. 80–84. IOS Press, Amsterdam (2008)Google Scholar
  28. 28.
    Lehmann, J.: Dl-learner: Learning concepts in description logics. Journal of Machine Learning Research 10, 2639–2642 (2009)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Lloyd, J.W., Topor, R.W.: Making prolog more expressive. Journal of Logic Programming 1(3), 225–240 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lutz, C.: The complexity of conjunctive query answering in expressive description logics. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 179–193. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  31. 31.
    Manola, F., Miller, E. (eds.): Resource Description Framework (RDF): Primer. W3C Recommendation (2004),
  32. 32.
    Minsky, M.: A framework for representing knowledge. Artificial intelligence memo, A.I. Laboratory. Massachusetts Institute of Technology, Cambridge (1974)Google Scholar
  33. 33.
    Motik, B., Sattler, U.: A Comparison of Reasoning Techniques for Querying Large Description Logic ABoxes. In: Hermann, M., Voronkov, A. (eds.) LPAR 2006. LNCS (LNAI), vol. 4246, pp. 227–241. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  34. 34.
    Motik, B., Patel-Schneider, P.F., Grau, B.C. (eds.): OWL 2 Web Ontology Language: Direct Semantics. W3C Recommendation (2009),
  35. 35.
    Motik, B., Patel-Schneider, P.F., Parsia, B. (eds.): OWL 2 Web Ontology Language: Structural Specification and Functional-Style Syntax. W3C Recommendation (2009),
  36. 36.
    Motik, B., Shearer, R., Horrocks, I.: Hypertableau reasoning for description logics. Journal of Artificial Intelligence Research (JAIR) 36, 165–228 (2009)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Di Noia, T., Di Sciascio, E., Donini, F.M.: A tableaux-based calculus for abduction in expressive description logics: Preliminary results. In: Grau, B.C., Horrocks, I., Motik, B., Sattler, U. (eds.) Description Logics. CEUR Workshop Proceedings, vol. 477 (2009),
  38. 38.
    W3C OWL Working Group. OWL 2 Web Ontology Language: Document Overview. W3C Recommendation (2009)
  39. 39.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar
  40. 40.
    Patel-Schneider, P.F., Motik, B. (eds.): OWL 2 Web Ontology Language: Mapping to RDF Graphs. W3C Recommendation (2009),
  41. 41.
    Pratt-Hartmann, I.: Complexity of the two-variable fragment with counting quantifiers. Journal of Logic, Language and Information 14, 369–395 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Ross Quillian, M.: Semantic memory. In: Minsky, M. (ed.) Semantic Information Processing, ch. 4, pp. 227–270. MIT Press, Cambridge (1968)Google Scholar
  43. 43.
    Rudolph, S., Glimm, B.: Nominals, inverses, counting, and conjunctive queries or: Why infinity is your friend! Journal of Artificial Intelligence Research (JAIR) 39, 429–481 (2010)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Rudolph, S., Krötzsch, M., Hitzler, P.: All elephants are bigger than all mice. In: Baader, F., Lutz, C., Motik, B. (eds.) Proceedings of the 21st International Workshop on Description Logics (DL 2008). CEUR Workshop Proceedings, vol. 353 (2008),
  45. 45.
    Rudolph, S., Krötzsch, M., Hitzler, P.: Terminological reasoning in SHIQ with ordered binary decision diagrams. In: Pro- ceedings of the 23rd AAAI Conference on Artificial Intelligence (AAAI 2008), pp. 529–534. AAAI Press, Menlo Park (2008)Google Scholar
  46. 46.
    Rudolph, S.: Exploring relational structures via FLE. In: Wolff, K.E., Pfeiffer, H.D., Delugach, H.S. (eds.) ICCS 2004. LNCS (LNAI), vol. 3127, pp. 196–212. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  47. 47.
    Russell, S., Norvig, P.: Artificial Intelligence: A Modern Approach, 2nd edn. Prentice Hall, Englewood Cliffs (2003)zbMATHGoogle Scholar
  48. 48.
    Schild, K.: A correspondence theory for terminological logics: Preliminary report. In: Mylopoulos, J., Reiter, R. (eds.) Proceedings of the 12th International Joint Conference on Artificial Intelligence (IJCAI 1991), pp. 466–471. Morgan Kaufmann, San Francisco (1991)Google Scholar
  49. 49.
    Schmidt-Schauß, M., Smolka, G.: Attributive concept descriptions with complements. Journal of Artificial Intelligence 48, 1–26 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Schöning, U.: Logic for Computer Scientists. Birkhäuser, Basel (2008)CrossRefzbMATHGoogle Scholar
  51. 51.
    Shearer, R., Horrocks, I.: Exploiting Partial Information in Taxonomy Construction. In: Bernstein, A., Karger, D.R., Heath, T., Feigenbaum, L., Maynard, D., Motta, E., Thirunarayan, K. (eds.) ISWC 2009. LNCS, vol. 5823, pp. 569–584. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  52. 52.
    Sidhu, A., Dillon, T., Chang, E., Sidhu, B.S.: Protein ontology development using OWL. In: Proceedings of the 1st OWL Experiences and Directions Workshop (OWLED 2005). CEUR Workshop Proceedings, vol. 188 (2005),
  53. 53.
    Simancik, F., Kazakov, Y., Horrocks, I.: Consequence-based reasoning beyond horn ontologies. In: Walsh, T. (ed.) Proceedings of the 22nd International Conference on Artificial Intelligence, IJCAI 2011 (2011)Google Scholar
  54. 54.
    Sirin, E., Parsia, B., Grau, B.C., Kalyanpur, A., Katz, Y.: Pellet: A practical OWL-DL reasoner. Journal of Web Semantics 5(2), 51–53 (2007)CrossRefGoogle Scholar
  55. 55.
    Sowa, J.F.: Conceptual Structures: Information Processing in Mind and Machine. Addison-Wesley, Reading (1984)zbMATHGoogle Scholar
  56. 56.
    Staab, S., Studer, R. (eds.): Handbook on Ontologies, 2nd edn. International Handbooks on Information Systems. Springer, Heidelberg (2009)zbMATHGoogle Scholar
  57. 57.
    Stuckenschmidt, H., Parent, C., Spaccapietra, S. (eds.):Modular Ontologies: Concepts, Theories and Techniques for Knowledge Modularization. LNCS, vol. 5445. Springer, Heidelberg (2009)zbMATHGoogle Scholar
  58. 58.
    Tsarkov, D., Horrocks, I.: FaCT++ Description Logic Reasoner: System Description. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 292–297. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  59. 59.
    van Harmelen, F., Lifschitz, V., Porter, B.: Handbook of Knowledge Representation. Foundations of Artificial Intelligence. Elsevier, Amsterdam (2008)zbMATHGoogle Scholar
  60. 60.
    Wolstencroft, K., Brass, A., Horrocks, I., Lord, P., Sattler, U., Turi, D., Stevens, R.: A little semantic web goes a long way in biology. In: Gil, Y., Motta, E., Benjamins, V.R., Musen, M.A. (eds.) ISWC 2005. LNCS, vol. 3729, pp. 786–800. Springer, Heidelberg (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Sebastian Rudolph
    • 1
  1. 1.Institute AIFBKarlsruhe Institute of TechnologyGermany

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