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A New Description Logic with Set Constraints and Cardinality Constraints on Role Successors

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10483)

Abstract

We introduce a new description logic that extends the well-known logic \(\mathcal {A}\mathcal {L}\mathcal {C}\mathcal {Q}\) by allowing the statement of constraints on role successors that are more general than the qualified number restrictions of \(\mathcal {A}\mathcal {L}\mathcal {C}\mathcal {Q}\). To formulate these constraints, we use the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA), in which one can express Boolean combinations of set constraints and numerical constraints on the cardinalities of sets. Though our new logic is considerably more expressive than \(\mathcal {A}\mathcal {L}\mathcal {C}\mathcal {Q}\), we are able to show that the complexity of reasoning in it is the same as in \(\mathcal {A}\mathcal {L}\mathcal {C}\mathcal {Q}\), both without and with TBoxes.

Notes

Acknowledgment

The author thanks Viktor Kuncak for helpful discussions regarding the proof of Lemma 3.

References

  1. 1.
    Baader, F.: Concept descriptions with set constraints and cardinality constraints. LTCS-Report 17–02, Chair for Automata Theory, Institute for Theoretical Computer Science, TU Dresden, Germany, 2017. http://lat.inf.tu-dresden.de/research/reports.html
  2. 2.
    Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P.F. (eds.): The Description Logic Handbook: Theory, Implementation, and Applications. Cambridge University Press, New York (2003)zbMATHGoogle Scholar
  3. 3.
    Baader, F., Sattler, U.: Expressive number restrictions in description logics. J. Logic Comput. 9(3), 319–350 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Borgida, A., Brachman, R.J., McGuinness, D.L., Alperin Resnick, L.: CLASSIC: a structural data model for objects. In: Proceedings of the ACM SIGMOD International Conference on Management of Data, pp. 59–67 (1989)Google Scholar
  5. 5.
    Faddoul, J., Haarslev, V.: Algebraic tableau reasoning for the description logic SHOQ. J. Appl. Logic 8(4), 334–355 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Garey, M.R., Johnson, D.S.: Computers and Intractability – A guide to NP-Completeness. W.H. Freeman and Company, San Francisco (1979)zbMATHGoogle Scholar
  7. 7.
    Haarslev, V., Sebastiani, R., Vescovi, M.: Automated reasoning in \(\cal{ALCQ}\) via SMT. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS, vol. 6803, pp. 283–298. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-22438-6_22 CrossRefGoogle Scholar
  8. 8.
    Haarslev, V., Timmann, M., Möller, R.: Combining tableaux and algebraic methods for reasoning with qualified number restrictions. In: Proceedings of DL 2001, CEUR Workshop Proceedings, vol. 49. CEUR-WS.org (2001)Google Scholar
  9. 9.
    Hoehndorf, R., Schofield, P.N., Gkoutos, G.V.: The role of ontologies in biological and biomedical research: a functional perspective. Brief. Bioinform. 16(6), 1069–1080 (2015)CrossRefGoogle Scholar
  10. 10.
    Hollunder, B., Baader, F.: Qualifying number restrictions in concept languages. In: Proceedings of KR 1991, pp. 335–346 (1991)Google Scholar
  11. 11.
    Hollunder, B., Nutt, W., Schmidt-Schauß, M.: Subsumption algorithms for concept description languages. In: Proceedings of ECAI 1990, pp. 348–353. Pitman, London (United Kingdom) (1990)Google Scholar
  12. 12.
    Kuncak, V., Rinard, M.: Towards efficient satisfiability checking for boolean algebra with presburger arithmetic. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 215–230. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-73595-3_15 CrossRefGoogle Scholar
  13. 13.
    Ohlbach, H.J., Koehler, J.: Modal logics, description logics and arithmetic reasoning. Artif. Intell. 109(1–2), 1–31 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Pratt, V.R.: Models of program logic. In: Proceedings of FOCS 1979, pp. 115–122 (1979)Google Scholar
  15. 15.
    Rudolph, S., Krötzsch, M., Hitzler, P.: Type-elimination-based reasoning for the description logic \(\cal{SHIQ}\)b\(_s\) using decision diagrams and disjunctive datalog. Logical Methods Comput. Sci. 8(1), 1–38 (2012)CrossRefzbMATHGoogle Scholar
  16. 16.
    Schild, K.: A correspondence theory for terminological logics: preliminary report. In: Proceedings of IJCAI 1991, pp. 466–471 (1991)Google Scholar
  17. 17.
    Schmidt-Schauß, M., Smolka, G.: Attributive concept descriptions with complements. Artif. Intell. 48(1), 1–26 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tobies, S.: A PSpace algorithm for graded modal logic. CADE 1999. LNCS, vol. 1632, pp. 52–66. Springer, Heidelberg (1999). doi: 10.1007/3-540-48660-7_4 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Theoretical Computer ScienceTU DresdenDresdenGermany

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