Advertisement

Simulations and Bisimulations for Coalgebraic Modal Logics

  • Daniel Gorín
  • Lutz Schröder
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8089)

Abstract

Simulations serve as a proof tool to compare the behaviour of reactive systems. We define a notion of Λ-simulation for coalgebraic modal logics, parametric in the choice of a set Λ of monotone predicate liftings for a functor T. That is, we obtain a generic notion of simulation that can be flexibly instantiated to a large variety of systems and logics, in particular in settings that semantically go beyond the classical relational setup, such as probabilistic, game-based, or neighbourhood-based systems. We show that this notion is adequate in several ways: i) Λ-simulations preserve truth of positive formulas, ii) for Λ a separating set of monotone predicate liftings, the associated notion of Λ-bisimulation corresponds to T-behavioural equivalence (moreover, this correspondence extends to the respective finite-lookahead counterparts), and iii) Λ-bisimulations remain sound when taken up to difunctional closure. In essence, we arrive at a modular notion of equivalence that, when used with a separating set of monotone predicate liftings, coincides with T-behavioural equivalence regardless of whether T preserves weak pullbacks. That is, for finitary set-based coalgebras, Λ-bisimulation works under strictly more general assumptions than T-bisimulation in the sense of Aczel and Mendler.

Keywords

Modal Logic Kripke Frame Positive Formula Coalgebra Structure Monotone Boolean Function 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barr, M.: Terminal coalgebras in well-founded set theory. Theoret. Comput. Sci. 114, 299–315 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Enqvist, S.: Homomorphisms of coalgebras from predicate liftings. In: Heckel, R., Milius, S. (eds.) CALCO 2013. LNCS, vol. 8089, pp. 126–140. Springer, Heidelberg (2013)Google Scholar
  3. 3.
    Hansen, H., Kupke, C.: A coalgebraic perspective on monotone modal logic. In: Coalgebraic Methods in Computer Science (CMCS 2004). ENTCS, vol. 106, pp. 121–143. Elsevier (2004)Google Scholar
  4. 4.
    Hennessy, M., Milner, R.: On observing nondeterminism and concurrency. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 299–309. Springer, Heidelberg (1980)CrossRefGoogle Scholar
  5. 5.
    Kurz, A., Leal, R.A.: Modalities in the Stone age: A comparison of coalgebraic logics. Theor. Comput. Sci. 430, 88–116 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Larsen, K., Skou, A.: Bisimulation through probabilistic testing. Inf. Comput. 94, 1–28 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Levy, P.B.: Similarity quotients as final coalgebras. In: Hofmann, M. (ed.) FOSSACS 2011. LNCS, vol. 6604, pp. 27–41. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    Marti, J., Venema, Y.: Lax extensions of coalgebra functors. In: Pattinson, D., Schröder, L. (eds.) CMCS 2012. LNCS, vol. 7399, pp. 150–169. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  9. 9.
    Pattinson, D.: Coalgebraic modal logic: Soundness, completeness and decidability of local consequence. Theoret. Comput. Sci. 309, 177–193 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Pattinson, D.: Expressive logics for coalgebras via terminal sequence induction. Notre Dame J. Formal Logic 45, 2004 (2002)MathSciNetGoogle Scholar
  11. 11.
    Pauly, M.: Bisimulation for general non-normal modal logic (1999) (unpublished Manuscript)Google Scholar
  12. 12.
    Pauly, M.: Logic for social software. Ph.D. thesis, Universiteit van Amsterdam (2001)Google Scholar
  13. 13.
    Schröder, L., Pattinson, D.: PSPACE bounds for rank-1 modal logics. ACM Trans. Comput. Log. 10, 13:1–13:33 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Schröder, L.: Expressivity of coalgebraic modal logic: The limits and beyond. Theoret. Comput. Sci. 390(2-3), 230–247 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Schröder, L., Pattinson, D.: Coalgebraic correspondence theory. In: Ong, L. (ed.) FOSSACS 2010. LNCS, vol. 6014, pp. 328–342. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  16. 16.
    Sokolova, A.: Probabilistic systems coalgebraically: A survey. Theoret. Comput. Sci. 412, 5095–5110 (2011)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Daniel Gorín
    • 1
  • Lutz Schröder
    • 1
  1. 1.Department of Computer ScienceUniversität Erlangen-NürnbergGermany

Personalised recommendations