Duality for Logics of Transition Systems

  • Marcello M. Bonsangue
  • Alexander Kurz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3441)

Abstract

We present a general framework for logics of transition systems based on Stone duality. Transition systems are modelled as coalgebras for a functor T on a category χ. The propositional logic used to reason about state spaces from χ is modelled by the Stone dual \({\mathcal A}\) of χ (e.g. if χ is Stone spaces then \({\mathcal A}\) is Boolean algebras and the propositional logic is the classical one). In order to obtain a modal logic for transition systems (i.e. for T-coalgebras) we consider the functor L on \({\mathcal A}\) that is dual to T. An adequate modal logic for T-coalgebras is then obtained from the category of L-algebras which is, by construction, dual to the category of T-coalgebras. The logical meaning of the duality is that the logic is sound and complete and expressive (or fully abstract) in the sense that non-bisimilar states are distinguished by some formula.

We apply the framework to Vietoris coalgebras on topological spaces, using the duality between spaces and observation frames, to obtain adequate logics for transition systems on posets, sets, spectral spaces and Stone spaces.

Keywords

transition systems coalgebras Stone duality topological dualities modal logic 

References

  1. 1.
    Abramsky, S.: Domain theory in logical form. Annals of Pure and Applied Logic 5, 1–77 (1991)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abramsky, S.: A domain equation for bisimulation. Inf. and Comp. 92 (1991)Google Scholar
  3. 3.
    van Benthem, J., van Eijck, J., Stebletsova, V.: Modal Logic, Transition Systems and Processes. Journal of Logic and Computation 4, 811–855 (1994)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. In: CSLI (2001)Google Scholar
  5. 5.
    Bonsangue, M.M.: Topological Dualities in Semantics. ENTCS, vol. 8. Elsevier, Amsterdam (1996)Google Scholar
  6. 6.
    Bonsangue, M.M., Jacobs, B., Kok, J.N.: Duality beyond sober spaces: topological spaces and observation frames. Theor. Comp. Sci. 15(1), 79–124 (1995)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bonsangue, M.M., Kok, J.N.: Towards an infinitary logic of domains: Abramsky logic for transition systems. Inf. and Comp. 155, 170–201 (1999)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Celani, S., Jansana, R.: Priestley duality, a Sahlqvist theorem and a Goldblatt-Thomason theorem for positive modal logic. Logic Journ. of the IGPL 7, 683–715 (1999)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Font, J.M., Jansana, R., Pigozzi, D.: A Survey of Abstract Algebraic Logic. Studia Logica 74, 13–97 (2003)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Gerbrandy, J.: Bisimulations on Planet Kripke. PhD thesis, Univ. of Amsterdam (1999)Google Scholar
  11. 11.
    Goldblatt, R.I.: Metamathematics of modal logic I. Rep. on Math. Logic 6 (1976)Google Scholar
  12. 12.
    Jacobs, B.: Many-sorted coalgebraic modal logic: a model-theoretic study. Theoretical Informatics and Applications 35(1), 31–59 (2001)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1982)MATHGoogle Scholar
  14. 14.
    Johnstone, P.T.: The Vietoris monad on the category of locales. In: Continuous Lattices and Related Topics, pp. 162–179 (1982)Google Scholar
  15. 15.
    Jónsson, B., Tarski, A.: Boolean algebras with operators, part I. American Journal of Mathematics 73, 891–939 (1951)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Kracht, M.: Tools and Techniques in Modal Logic. Studies in Logic, vol. 142. Elsevier, Amsterdam (1999)MATHCrossRefGoogle Scholar
  17. 17.
    Kupke, C., Kurz, A., Venema, Y.: Stone coalgebras. Theoret. Comput. Sci. 327, 109–134 (2004)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Manes, E.G.: Algebraic Theories. Springer, Heidelberg (1976)MATHCrossRefGoogle Scholar
  19. 19.
    Michael, E.: Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71 (1951)Google Scholar
  20. 20.
    Moss, L.: Coalgebraic logic. Annals of Pure and Applied Logic 96, 277–317 (1999)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Palmigiano, A.: A coalgebraic semantics for positive modal logic. Theoret. Comput. Sci. 327, 175–195 (2004)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Rutten, J.J.M.M.: Universal coalgebra: A theory of systems. Theoret. Comput. Sci. 249, 3–80 (2000)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Vickers, S.: Topology via Logic. Cambridge University Press, Cambridge (1989)MATHGoogle Scholar
  24. 24.
    Vickers, S.: Information systems for continuous posets. Theoret. Comp. Sci. 114 (1993)Google Scholar
  25. 25.
    Worrell, J.: Terminal sequences for accessible endofunctors. In: Coalgebraic Methods in Computer Science (CMCS 1999). ENTCS, vol. 19. Elsevier, Amsterdam (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Marcello M. Bonsangue
    • 1
  • Alexander Kurz
    • 2
  1. 1.LIACSLeiden UniversityThe Netherlands
  2. 2.Department of Computer ScienceUniversity of LeicesterUK

Personalised recommendations