Advertisement

Pushing the Boundaries of Reasoning About Qualified Cardinality Restrictions

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

Abstract

We present a novel hybrid architecture for reasoning about description logics supporting role hierarchies and qualified cardinality restrictions (QCRs). Our reasoning architecture is based on saturation rules and integrates integer linear programming. Deciding the numerical satisfiability of a set of QCRs is reduced to solving a corresponding system of linear inequalities. If such a system is infeasible then the QCRs are unsatisfiable. Otherwise the numerical restrictions of the QCRs are satisfied but unknown entailments between qualifications can still lead to unsatisfiability. Our integer linear programming (ILP) approach is highly scalable due to integrating learned knowledge about concept subsumption and disjointness into a column generation model and a decomposition algorithm to solve it. Our experiments indicate that this hybrid architecture offers a better scalability for reasoning about QCRs than approaches combining both tableaux and ILP or applying traditional (hyper)tableau methods.

Keywords

Qualified Number Restrictions Column Generation Role Hierarchy Saturation Rule Subroles 
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.

References

  1. 1.
    Baader, F., Brandt, S., Lutz, C.: Pushing the \(\cal{EL}\) envelope. In: Proceeding of IJCAI, pp. 364–369 (2005)Google Scholar
  2. 2.
    Baader, F., Sattler, U.: An overview of tableau algorithms for description logics. Stud. Logica. 69(1), 5–40 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barnhart, C., Johnson, E.L., Nemhauser, G.L., Savelsbergh, M.W.P., Vance, P.H.: Branch-and-price: column generation for solving huge integer programs. Oper. Res. 46(3), 316–329 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bate, A., Motik, B., Cuenca Grau, B., Simančík, F., Horrocks, I.: Extending consequence-based reasoning to \(\cal{SRIQ}\). In: Proceeding of KR, pp. 187–196 (2016)Google Scholar
  5. 5.
  6. 6.
    Chvatal, V.: Linear Programming. Freeman, New York (1983)zbMATHGoogle Scholar
  7. 7.
    Dantzig, G.B., Wolfe, P.: Decomposition principle for linear programs. Oper. Res. 8(1), 101–111 (1960)CrossRefzbMATHGoogle Scholar
  8. 8.
  9. 9.
    Faddoul, J., Haarslev, V.: Algebraic tableau reasoning for the description logic \(\cal{SHOQ}\). J. Appl. Logic 8(4), 334–355 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Faddoul, J., Haarslev, V.: Optimizing algebraic tableau reasoning for \(\cal{SHOQ}\): First experimental results. In: Proceeding of DL, pp. 161–172 (2010)Google Scholar
  11. 11.
    Farsiniamarj, N., Haarslev, V.: Practical reasoning with qualified number restrictions: a hybrid Abox calculus for the description logic \(\cal{SHQ}\). AI Commun. 23(2–3), 334–355 (2010)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Freund, R., Mizuno, S.: Interior point methods: current status and future directions. Optima 51, 1–9 (1996)zbMATHGoogle Scholar
  13. 13.
    Gilmore, P.C., Gomory, R.E.: A linear programming approach to the cutting-stock problem. Oper. Res. 9(6), 849–859 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Haarslev, V., Hidde, K., Möller, R., Wessel, M.: The RacerPro knowledge representation and reasoning system. Semant. Web 3(3), 267–277 (2012)Google Scholar
  15. 15.
    Haarslev, V., Möller, R.: RACER system description. In: Goré, R., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS, vol. 2083, pp. 701–705. Springer, Heidelberg (2001). doi: 10.1007/3-540-45744-5_59 CrossRefGoogle Scholar
  16. 16.
    Haarslev, V., Sebastiani, R., Vescovi, M.: Automated reasoning in \(\cal{ALCQ}\) via SMT. In: Proceeding of CADE, pp. 283–298 (2011)Google Scholar
  17. 17.
    Hansen, P., Jaumard, B., de Aragão, M.P., Chauny, F., Perron, S.: Probabilistic satisfiability with imprecise probability. Int. J. Approximate Reasoning 24(2–3), 171–189 (2000)CrossRefzbMATHGoogle Scholar
  18. 18.
  19. 19.
    Hollunder, B., Baader, F.: Qualifying number restrictions in concept languages. In: Proceeding of KR, pp. 335–346 (1991)Google Scholar
  20. 20.
    Jaumard, B., Hansen, P., de Aragão, M.P.: Column generation methods for probabilistic logic. ORSA J. Comput. 3(2), 135–148 (1991)CrossRefzbMATHGoogle Scholar
  21. 21.
    Klinov, P., Parsia, B.: Pronto: a practical probabilistic description logic reasoner. In: Bobillo, F., Costa, P.C.G., d’Amato, C., Fanizzi, N., Laskey, K.B., Laskey, K.J., Lukasiewicz, T., Nickles, M., Pool, M. (eds.) UniDL/URSW 2008-2010. LNCS, vol. 7123, pp. 59–79. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-35975-0_4 CrossRefGoogle Scholar
  22. 22.
  23. 23.
    Lübbecke, M., Desrosiers, J.: Selected topics in column generation. Oper. Res. 53, 1007–1023 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Megiddo, N.: On the complexity of linear programming. In: Advances in Economic Theory, pp. 225–268. Cambridge University Press (1987)Google Scholar
  25. 25.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1988)CrossRefzbMATHGoogle Scholar
  26. 26.
    Ohlbach, H., Köhler, J.: Modal logics, description logics and arithmetic reasoning. Artif. Intell. 109(1–2), 1–31 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
  28. 28.
    Roosta Pour, L., Haarslev, V.: Algebraic reasoning for \(\cal{SHIQ}\). In: Proceeding of DL, pp. 530–540 (2012)Google Scholar
  29. 29.
    Samwald, M.: Genomic CDS: an example of a complex ontology for pharmacogenetics and clinical decision support. In: 2nd OWL Reasoner Evaluation Workshop, pp. 128–133 (2013)Google Scholar
  30. 30.
    Simančík, F., Motik, B., Horrocks, I.: Consequence-based and fixed-parameter tractable reasoning in description logics. Artif. Intell. 209, 29–77 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Vanderbeck, F.: Branching in branch-and-price: a generic scheme. Math. Program. 130(2), 249–294 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Vlasenko, J., Daryalal, M., Haarslev, V., Jaumard, B.: A saturation-based algebraic reasoner for \(\cal{ELQ}\). In: PAAR@IJCAR, Coimbra, Portugal, pp. 110–124 (2016)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Concordia UniversityMontrealCanada

Personalised recommendations