Modular Algorithms for Heterogeneous Modal Logics

  • Lutz Schröder
  • Dirk Pattinson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)


State-based systems and modal logics for reasoning about them often heterogeneously combine a number of features such as non-determinism and probabilities. Here, we show that the combination of features can be reflected algorithmically and develop modular decision procedures for heterogeneous modal logics. The modularity is achieved by formalising the underlying state-based systems as multi-sorted coalgebras and associating both a logical and an algorithmic description to a number of basic building blocks. Our main result is that logics arising as combinations of these building blocks can be decided in polynomial space provided that this is the case for the components. By instantiating the general framework to concrete cases, we obtain PSPACE decision procedures for a wide variety of structurally different logics, describing e.g. Segala systems and games with uncertain information.


Modal Operator Modal Logic Binary Feature Proof Rule Conditional Logic 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bartels, F., Sokolova, A., de Vink, E.: A hierarchy of probabilistic system types. In: Gumm, H.-P. (ed.) Coalgebraic Methods in Computer Science. ENTCS, vol. 82, Elsevier, Amsterdam (2003)Google Scholar
  2. 2.
    Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  3. 3.
    Chellas, B.: Modal Logic. Cambridge University Press, Cambridge (1980)zbMATHGoogle Scholar
  4. 4.
    Cîrstea, C., Pattinson, D.: Modular construction of modal logics. Theoret. Copmut. Sci. (to appear). Earlier version In: Gardner, P., Yoshida, N. (eds.) CONCUR 2004. LNCS, vol. 3170, pp. 258–275. Springer, Heidelberg (2004)Google Scholar
  5. 5.
    D’Agostino, G., Visser, A.: Finality regained: A coalgebraic study of Scott-sets and multisets. Arch. Math. Logic 41, 267–298 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Fine, K.: In so many possible worlds. Notre Dame J. Formal Logic 13, 516–520 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Halpern, J.Y.: Reasoning About Uncertainty. MIT Press, Cambridge (2003)zbMATHGoogle Scholar
  8. 8.
    Hansson, H., Jonsson, B.: A calculus for communicating systems with time and probabilities. In: Real-Time Systems, RTSS 90, pp. 278–287. IEEE Computer Society Press, Los Alamitos (1990)Google Scholar
  9. 9.
    Heifetz, A., Mongin, P.: Probabilistic logic for type spaces. Games and Economic Behavior 35, 31–53 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hemaspaandra, E.: Complexity transfer for modal logic. In: Abramsy, S. (ed.) LICS 1994. Logic in Computer Science, pp. 164–173. IEEE Computer Society Press, Los Alamitos (1994)Google Scholar
  11. 11.
    Jacobs, B.: Many-sorted coalgebraic modal logic: a model-theoretic study. Theor. Inform. Appl. 35, 31–59 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Jonsson, B., Yi, W., Larsen, K.G.: Probabilistic extensions of process algebras. In: Bergstra, J., Ponse, A., Smolka, S.M. (eds.) Handbook of Process Algebra, Elsevier, Amsterdam (2001)Google Scholar
  13. 13.
    Kurucz, A.: Combining modal logics. In: van Benthem, J., Blackburn, P., Wolter, F. (eds.) Handbook of Modal Logic, Elsevier, Amsterdam (2006)Google Scholar
  14. 14.
    Kutz, O., Lutz, C., Wolter, F., Zakharyaschev, M.: \(\mathcal E\)-connections of abstract description systems. Artificial Intelligence 156, 1–73 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Larsen, K., Skou, A.: Bisimulation through probabilistic testing. Inform. Comput. 94, 1–28 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Mossakowski, T., Schröder, L., Roggenbach, M., Reichel, H.: Algebraic-coalgebraic specification in CoCASL. J. Logic Algebraic Programming 67, 146–197 (2006)zbMATHCrossRefGoogle Scholar
  17. 17.
    Pattinson, D.: Expressive logics for coalgebras via terminal sequence induction. Notre Dame J. Formal Logic 45, 19–33 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Pauly, M.: A modal logic for coalitional power in games. J. Logic Comput. 12, 149–166 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Rutten, J.: Universal coalgebra: A theory of systems. Theoret. Comput. Sci. 249, 3–80 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Schröder, L.: Expressivity of coalgebraic modal logic: the limits and beyond. Theoret. Comput. Sci. Earlier version In: Sassone, V. (ed.) FOSSACS 2005. LNCS, vol. 3441, pp. 440–454. Springer, Heidelberg (2005) (to appear)Google Scholar
  21. 21.
    Schröder, L.: A semantic PSPACE criterion for the next 700 rank 0-1 modal logics, available at
  22. 22.
    Schröder, L.: A finite model construction for coalgebraic modal logic. In: Aceto, L., Ingólfsdóttir, A. (eds.) FOSSACS 2006 and ETAPS 2006. LNCS, vol. 3921, pp. 157–171. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  23. 23.
    Schröder, L., Pattinson, D.: PSPACE reasoning for rank-1 modal logics. In: Alur, R. (ed.) LICS 2006. Logic in Computer Science, pp. 231–240. IEEE Computer Society Press, Los Alamitos (2006)Google Scholar
  24. 24.
    Segala, R.: Modelling and Verification of Randomized Distributed Real-Time Systems. PhD thesis, Massachusetts Institute of Technology (1995)Google Scholar
  25. 25.
    Tobies, S.: PSPACE reasoning for graded modal logics. J. Logic Comput. 11, 85–106 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Wolter, F.: Fusions of modal logics revisited. In: Zakharyaschev, M., Segerberg, K., de Rijke, M., Wansing, H. (eds.) Advances in modal logic. CSLI Lect. Notes, vol. 1, pp. 361–379. CSLI, Stanford (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Lutz Schröder
    • 1
  • Dirk Pattinson
    • 2
  1. 1.DFKI-Lab Bremen and Department of Computer Science, Universität Bremen 
  2. 2.Department of Computing, Imperial College London 

Personalised recommendations