A General Tableau Method for Deciding Description Logics, Modal Logics and Related First-Order Fragments

  • Renate A. Schmidt
  • Dmitry Tishkovsky
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5195)


This paper presents a general method for proving termination of tableaux-based procedures for modal-type logics and related first-order fragments. The method is based on connections between filtration arguments and tableau blocking techniques. The method provides a general framework for developing tableau-based decision procedures for a large class of logics. In particular, the method can be applied to many well-known description and modal logics. The class includes traditional modal logics such as S4 and modal logics with the universal modality, as well as description logics such as \(\mathcal{ALC}\) with nominals and general TBoxes. Also contained in the class are harder and less well-studied modal logics with complex modalities and description logics with complex role operators such as Boolean modal logic, and the description logic \(\mathcal{ALBO}\). In addition, the techniques allow us to specify tableau-based decision procedures for related solvable fragments of first-order logic, including the two-variable fragment of first-order logic.


Modal Logic Decision Procedure Description Logic Closure Operator Open Branch 
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  1. 1.
    Abate, P., Goré, R.: The Tableaux Work Bench. In: Cialdea Mayer, M., Pirri, F. (eds.) TABLEAUX 2003. LNCS, vol. 2796, pp. 230–236. Springer, Heidelberg (2003)Google Scholar
  2. 2.
    Baader, F., Sattler, U.: An overview of tableau algorithms for description logics. Stud. Log. 69, 5–40 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Camb. Tracts Theor. Comput. Sci, vol. 53. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  4. 4.
    Castilho, M.A., Fariñas del Cerro, L., Gasquet, O., Herzig, A.: Modal tableaux with propagation rules and structural rules. Fundam. Inform. 3-4(32), 281–297 (1997)Google Scholar
  5. 5.
    Fariñas del Cerro, L., Gasquet, O., Herzig, A., Sahade, M.: Modal tableaux: Completeness vs. termination. In: We Will Show Them!, vol. 1, pp. 587–614. College Publ. (2005)Google Scholar
  6. 6.
    Gabbay, D.M., Kurucz, A., Wolter, F., Zakharyaschev, M.: Many-Dimensional Modal Logics: Theory and Applications. North-Holland, Amsterdam (2003)zbMATHGoogle Scholar
  7. 7.
    Gabbay, D.M., Shehtman, V.: Products of modal logics, part 1. Log. J. IGPL 6(1), 73–146 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gargov, G., Passy, S., Tinchev, T.: Modal environment for Boolean speculations. In: Proc. Gödel 1986, Plenum, pp. 253–263 (1987)Google Scholar
  9. 9.
    Gasquet, O., Herzig, A., Sahade, M.: Terminating modal tableaux with simple completeness proof. In: Proc. AiML 2006, pp. 167–186. College Publ. (2006)Google Scholar
  10. 10.
    Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. MIT Press, Cambridge (2000)zbMATHGoogle Scholar
  11. 11.
    Heuerding, A., Jäger, G., Schwendimann, S., Seyfried, M.: The Logics Workbench LWB: A snapshot. Euromath. Bull. 2(1), 177–186 (1996)Google Scholar
  12. 12.
    Horrocks, I., Sattler, U.: A description logic with transitive and inverse roles and role hierarchies. J. Log. Comput. 9(3), 385–410 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Schmidt, R.A., Tishkovsky, D.: A general tableau method for deciding description logics, modal logics and related first-order fragments,
  14. 14.
    Schmidt, R.A., Tishkovsky, D.: Using tableau to decide expressive description logics with role negation. In: Aberer, K., Choi, K.-S., Noy, N., Allemang, D., Lee, K.-I., Nixon, L., Golbeck, J., Mika, P., Maynard, D., Mizoguchi, R., Schreiber, G., Cudré-Mauroux, P. (eds.) ISWC 2007. LNCS, vol. 4825, pp. 438–451. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.

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© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Renate A. Schmidt
    • 1
  • Dmitry Tishkovsky
    • 1
  1. 1.School of Computer ScienceThe University of Manchester 

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