Terminating Minimal Model Generation Procedures for Propositional Modal Logics

  • Fabio Papacchini
  • Renate A. Schmidt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8562)


Model generation and minimal model generation are useful for tasks such as model checking and for debugging of logical specifications. This paper presents terminating procedures for the generation of models minimal modulo subset-simulation for the modal logic K and all combinations of extensions with the axioms T, B, D, 4 and 5. Our procedures are minimal model sound and complete. Compared with other minimal model generation procedures, they are designed to have smaller search space and return fewer models. In order to make the models more effective for users, our minimal model criterion is aimed to be semantically meaningful, intuitive and contain a minimal amount of information. Depending on the logic, termination is ensured by a variation of equality blocking.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baumgartner, P., Fürbach, U., Niemelä, I.: Hyper tableaux. In: Orłowska, E., Alferes, J.J., Moniz Pereira, L. (eds.) JELIA 1996. LNCS, vol. 1126, pp. 1–17. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  2. 2.
    Bry, F., Yahya, A.: Positive unit hyperresolution tableaux and their application to minimal model generation. J. Automat. Reason. 25(1), 35–82 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Cialdea Mayer, M.: A proof procedure for hybrid logic with binders, transitivity and relation hierarchies. In: Bonacina, M.P. (ed.) CADE 2013. LNCS (LNAI), vol. 7898, pp. 76–90. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  4. 4.
    Clarke, E.M., Schlingloff, B.: Model checking. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 1635–1790. Elsevier (2001)Google Scholar
  5. 5.
    Denecker, M., De Schreye, D.: On the duality of abduction and model generation in a framework for model generation with equality. Theoret. Computer Sci. 122(1&2), 225–262 (1994)CrossRefzbMATHGoogle Scholar
  6. 6.
    Henzinger, M.R., Henzinger, T.A., Kopke, P.W.: Computing simulations on finite and infinite graphs. In: Proc. FCS-36, pp. 453–462. IEEE Comput. Soc. (1995)Google Scholar
  7. 7.
    Hintikka, J.: Model minimization—An alternative to circumscription. J. Automat. Reason. 4(1), 1–13 (1988)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Horridge, M., Parsia, B., Sattler, U.: Extracting justifications from bioportal ontologies. In: Cudré-Mauroux, P., Heflin, J., Sirin, E., Tudorache, T., Euzenat, J., Hauswirth, M., Parreira, J.X., Hendler, J., Schreiber, G., Bernstein, A., Blomqvist, E. (eds.) ISWC 2012, Part II. LNCS, vol. 7650, pp. 287–299. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  9. 9.
    Horrocks, I., Hustadt, U., Sattler, U., Schmidt, R.A.: Computational modal logic. In: Blackburn, P., van Benthem, J., Wolter, F. (eds.) Handbook of Modal Logic, pp. 181–245. Elsevier (2007)Google Scholar
  10. 10.
    Horrocks, I., Sattler, U.: A description logic with transitive and inverse roles and role hierarchies. J. Logic Comput. 9(3), 385–410 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Lorenz, S.: A tableaux prover for domain minimization. J. Automat. Reason. 13(3), 375–390 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Massacci, F.: Single step tableaux for modal logics. J. Automat. Reason. 24(3), 319–364 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Nguyen, L.A.: Constructing finite least Kripke models for positive logic programs in serial regular grammar logics. Logic J. IGPL 16(2), 175–193 (2008)CrossRefzbMATHGoogle Scholar
  14. 14.
    Niemelä, I.: Implementing circumscription using a tableau method. In: Proc. ECAI 1996, pp. 80–84. Wiley (1996)Google Scholar
  15. 15.
    Niemelä, I.: A tableau calculus for minimal model reasoning. In: Miglioli, P., Moscato, U., Ornaghi, M., Mundici, D. (eds.) TABLEAUX 1996. LNCS, vol. 1071, pp. 278–294. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  16. 16.
    Papacchini, F., Schmidt, R.A.: A tableau calculus for minimal modal model generation. Electr. Notes Theoret. Computer Sci. 278(3), 159–172 (2011)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Papacchini, F., Schmidt, R.A.: Computing minimal models modulo subset-simulation for propositional modal logics. In: Fontaine, P., Ringeissen, C., Schmidt, R.A. (eds.) FroCoS 2013. LNCS (LNAI), vol. 8152, pp. 279–294. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  18. 18.
    Reiter, R.: A theory of diagnosis from first principles. Artificial Intelligence 32(1), 57–95 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Schlobach, S., Huang, Z., Cornet, R., van Harmelen, F.: Debugging incoherent terminologies. J. Automat. Reason. 39(3), 317–349 (2007)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Fabio Papacchini
    • 1
  • Renate A. Schmidt
    • 1
  1. 1.The University of ManchesterUK

Personalised recommendations