The Least Fibred Lifting and the Expressivity of Coalgebraic Modal Logic

  • Bartek Klin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3629)


Every endofunctor B on the category Set can be lifted to a fibred functor on the category (fibred over Set) of equivalence relations and relation-preserving functions. In this paper, the least (fibre-wise) of such liftings, L(B), is characterized for essentially any B. The lifting has all the useful properties of the relation lifting due to Jacobs, without the usual assumption of weak pullback preservation; if B preserves weak pullbacks, the two liftings coincide. Equivalence relations can be viewed as Boolean algebras of subsets (predicates, tests). This correspondence relates L(B) to the least test suite lifting T(B), which is defined in the spirit of predicate lifting as used in coalgebraic modal logic. Properties of T(B) translate to a general expressivity result for a modal logic for B-coalgebras. In the resulting logic, modal operators of any arity can appear.


Equivalence Relation Modal Logic Binary Relation Test Suite Label Transition System 
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.
    Rutten, J.J.M.M.: Universal coalgebra: a theory of systems. Theoretical Computer Science 249, 3–80 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Park, D.M.: Concurrency and automata on infinite sequences. In: Nielsen, M., Schmidt, E.M. (eds.) ICALP 1982. LNCS, vol. 140, pp. 195–219. Springer, Heidelberg (1982)Google Scholar
  3. 3.
    Milner, R.: Communication and Concurrency. Prentice-Hall, Englewood Cliffs (1988)Google Scholar
  4. 4.
    Hermida, C., Jacobs, B.: Structural induction and coinduction in a fibrational setting. Information and Computation 145, 107–152 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Jacobs, B.: Exercises in coalgebraic specification. In: Blackhouse, R., Crole, R.L., Gibbons, J. (eds.) Algebraic and Coalgebraic Methods in the Mathematics of Program Construction. LNCS, vol. 2297. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. 6.
    Jacobs, B., Hughes, J.: Simulations in coalgebra. Electronic Notes in Theoretical Computer Science 82 (2003)Google Scholar
  7. 7.
    Hennessy, M., Milner, R.: Algebraic laws for nondeterminism and concurrency. Journal of the ACM 32, 137–161 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Jacobs, B.: Towards a duality result in the modal logic for coalgebras. In: Proc. CMCS 2000. Electronic Notes in Theoretical Computer Science, vol. 33 (2000)Google Scholar
  9. 9.
    Klin, B.: Abstract Coalgebraic Approach to Process Equivalence for Well-Behaved Operational Semantics. PhD thesis, BRICS, Aarhus University (2004)Google Scholar
  10. 10.
    Moss, L.: Coalgebraic logic. Annals of Pure and Applied Logic 96, 177–317 (1999)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Pattinson, D.: Expressivity Results in the Modal Logic of Coalgebras. PhD thesis, Universität München (2001)Google Scholar
  12. 12.
    Pattinson, D.: Semantical principles in the modal logic of coalgebras. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, p. 514. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  13. 13.
    Schröder, L.: Expressivity of coalgebraic modal logic: the limits and beyond. In: Sassone, V. (ed.) FOSSACS 2005. LNCS, vol. 3441, pp. 440–454. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    Jacobs, B.: Trace semantics for coalgebras. In: Proc. CMCS 2004. Electronic Notes in Theoretical Computer Science, vol. 106 (2004)Google Scholar
  15. 15.
    Jacobs, B.: Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics, vol. 141. North Holland, Amsterdam (1999)zbMATHGoogle Scholar
  16. 16.
    Aczel, P., Mendler, N.: A final coalgebra theorem. In: Dybjer, P., Pitts, A.M., Pitt, D.H., Poigné, A., Rydeheard, D.E. (eds.) Category Theory and Computer Science. LNCS, vol. 389, pp. 357–365. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  17. 17.
    Thomsen, B.: A theory of higher-order communicating systems. Information and Computation 116 (1995)Google Scholar
  18. 18.
    Cirstea, C., Pattinson, D.: Modular construction of modal logic. In: Gardner, P., Yoshida, N. (eds.) CONCUR 2004. LNCS, vol. 3170, pp. 258–275. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Bartek Klin
    • 1
  1. 1.University of Sussex, Warsaw University 

Personalised recommendations