Automated Reasoning in \(\mathcal{ALCQ}\) via SMT

  • Volker Haarslev
  • Roberto Sebastiani
  • Michele Vescovi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6803)


Reasoning techniques for qualified number restrictions (QNRs) in Description Logics (DLs) have been investigated in the past but they mostly do not make use of the arithmetic knowledge implied by QNRs. In this paper we propose and investigate a novel approach for concept satisfiability in acyclic \(\mathcal{ALCQ}\) ontologies. It is based on the idea of encoding an \(\mathcal{ALCQ}\) ontology into a formula in Satisfiability Modulo the Theory of Costs (SMT\((\ensuremath{\mathcal{C}} )\)), which is a specific and computationally much cheaper subcase of Linear Arithmetic under the Integers, and to exploit the power of modern SMT solvers to compute every concept-satisfiability query on a given ontology. We implemented and tested our approach, which includes a very effective individuals-partitioning technique, on a wide set of synthesized benchmark formulas, comparing the approach with the main state-of-the-art DL reasoners available. Our empirical evaluation confirms the potential of the approach.


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  1. 1.
    Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P.F. (eds.): The Description Logic Handbook: Theory, Implementation, and Applications (2003)Google Scholar
  2. 2.
    Bruttomesso, R., Cimatti, A., Franzén, A., Griggio, A., Hanna, Z., Nadel, A., Palti, A., Sebastiani, R.: MathSAT 4 SMT Solver. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 547–560. Springer, Heidelberg (2008)Google Scholar
  3. 3.
    Cimatti, A., Franzén, A., Griggio, A., Sebastiani, R., Stenico, C.: Satisfiability Modulo the Theory of Costs: Foundations and Applications. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 99–113. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  4. 4.
    Faddoul, J., Haarslev, V.: Algebraic Tableau Reasoning for the Description Logic \(\ensuremath{\mathcal{SHOQ}} \). Journal of Applied Logic, Special Issue on Hybrid Logics 8(4) (2010)Google Scholar
  5. 5.
    Faddoul, J., Haarslev, V.: Optimizing Algebraic Tableau Reasoning for \(\mathcal{SHOQ}\): First Experimental Results. In: Proc. of DL, Waterloo, Canada, May 4-7 (2010)Google Scholar
  6. 6.
    Farsiniamarj, N., Haarslev, V.: Practical reasoning with qualified number restrictions: A hybrid abox calculus for the description logic \(\mathcal{SHQ}\). J. AI Communications 23(2-3) (2010)Google Scholar
  7. 7.
    Haarslev, V., Möller, R.: RACER System Description. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, p. 701. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Haarslev, V., Timmann, M., Möller, R.: Combining Tableaux and Algebraic Methods for Reasoning with Qualified Number Restrictions. In: Proc. of DL 2001 (2001)Google Scholar
  9. 9.
    Hollunder, B., Baader, F.: Qualifying Number Restrictions in Concept Languages. In: Proc. of KR, Boston, USA (1991)Google Scholar
  10. 10.
    Horrocks, I., Sattler, U., Tobies, S.: Practical reasoning for very expressive description logics. Logic Journal of the IGPL 8(3) (2000)Google Scholar
  11. 11.
    Lutz, C., Areces, C., Horrocks, I., Sattler, U.: Keys, Nominals, and Concrete Domains. Journal of Artificial Intelligence Research, JAIR 23(1) (2005)Google Scholar
  12. 12.
    Sebastiani, R.: Lazy Satisfiability Modulo Theories. Journal on Satisfiability, Boolean Modeling and Computation, JSAT 3, 141–224 (2007)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Sebastiani, R., Vescovi, M.: Automated Reasoning in Modal and Description Logics via SAT Encoding: the Case Study of \(K(m)/\ensuremath{\mathcal{ALC}} \)-satisfiability. JAIR 35(1) (2009)Google Scholar
  14. 14.
    Sebastiani, R., Vescovi, M.: Axiom pinpointing in lightweight description logics via horn-SAT encoding and conflict analysis. In: Schmidt, R.A. (ed.) CADE-22. LNCS, vol. 5663, pp. 84–99. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  15. 15.
    Sirin, E., Parsia, B., Grau, B.C., Kalyanpur, A., Katz, Y.: Pellet: A practical OWL-DL reasoner. Journal of Web Semantics 5(2) (2007)Google Scholar
  16. 16.
    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
  17. 17.
    Vescovi, M., Sebastiani, R., Haarslev, V.: Automated Reasoning on TBoxes with Qualified Number Restrictions via SMT. Tech. Rep. DISI-11-001, Università di Trento (April 2011),

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Volker Haarslev
    • 1
  • Roberto Sebastiani
    • 2
  • Michele Vescovi
    • 2
  1. 1.CSEConcordia UniversityMontrealCanada
  2. 2.DISIUniversità di TrentoItaly

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