Translating Logics for Coalgebras

  • Dirk Pattinson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2755)


This paper shows, that three different types of logics for coalgebras are institutions. The logics differ regarding the presentation of their syntax. In the first framework, abstract behavioural logic, one has a syntax-free representation of behavioural properties. We then turn to coalgebraic logic, the syntax of which is given as an initial algebra. The last framework, which we consider, is coalgebraic modal logic, the syntax of which is concretely given.


Modal Logic Natural Transformation Atomic Proposition Label Transition System Kripke Model 
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  1. 1.
    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
  2. 2.
    Adámek, J.: Free algebras and automata realizations in the language of categories. Comment. Math. Univ. Carolinae 15, 589–602 (1974)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Borceux, F.: Handbook of Categorical Algebra, vol. 2. Cambridge University Press, Cambridge (1994)CrossRefzbMATHGoogle Scholar
  4. 4.
    Carboni, A., Kelly, G., Wood, R.: A 2-categorical approach to change of base and geometric morphisms I. Cahiers de Topologie et Géometrié Différentielle Catégoriques 32(1), 47–95 (1991)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cirstea, C.: Institutionalizing coalgebraic modal logic. In: Moss, L. (ed.) Coalgebraic Methods in Computer Science (CMCS 2002). Electr. Notes in Theoret. Comp. Sci., vol. 65. Elsevier Science Publishers, Amsterdam (2002)Google Scholar
  6. 6.
    Goguen, J., Burstall, R.: Institutions: Abstract Model Theory for Specification and Programming. Journal of the Association for Computing Machinery 39(1) (1992)Google Scholar
  7. 7.
    Goldblatt, R.: A calculus of terms for coalgebras of polynomial functors. In: Corradini, M.L.A., Montanari, U. (eds.) Coalgebraic Methods in Computer Science (CMCS 2001). Electr. Notes in Theoret. Comp. Sci., vol. 44 (2001)Google Scholar
  8. 8.
    Jacobs, B.: Many-sorted coalgebraic modal logic: a model-theoretic study. Theoret. Informatics and Applications 35(1), 31–59 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jacobs, B., Hermida, C.: Structural Induction and Coinduction in a Fibrational Setting. Information and Computation 145, 107–152 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kurz, A.: A Co-Variety-Theorem for Modal Logic. In: Proceedings of Advances in Modal Logic 2, Uppsala, 1998. Center for the Study of Language and Information. Stanford University, Stanford (2000)Google Scholar
  11. 11.
    Kurz, A.: Logics for Coalgebras and Applications to Computer Science. PhD thesis, Universität München (April 2000)Google Scholar
  12. 12.
    Kurz, A., Pattinson, D.: Coalgebras and Modal Logics for Parameterised Endofunctors. Technical report, CWI (2000)Google Scholar
  13. 13.
    MacLane, S.: Categories for the Working Mathematician. Springer, Heidelberg (1971)zbMATHGoogle Scholar
  14. 14.
    Milner, R.: Communication and Concurrency. International series in computer science. Prentice-Hall, Englewood Cliffs (1989)zbMATHGoogle Scholar
  15. 15.
    Moss, L.: Coalgebraic Logic. Annals of Pure and Applied Logic 96, 277–317 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Park, D.: Concurrency and Automata on Infinite Sequences. In: Deussen, P. (ed.) GI-TCS 1981. LNCS, vol. 104. Springer, Heidelberg (1981)CrossRefGoogle Scholar
  17. 17.
    Pattinson, D.: Expressivity Results in the Modal Logic of Coalgebras. PhD thesis, Universität München (June 2001)Google Scholar
  18. 18.
    Rößiger, M.: Coalgebras and Modal Logic. In: Reichel, H. (ed.) Coalgebraic Methods in Computer Science (CMCS 2000). Electr. Notes in Theoret. Comp. Sci., vol. 33 (2000)Google Scholar
  19. 19.
    Rößiger, M.: From Modal Logic to Terminal Coalgebras. Theor. Comp. Sci. 260, 209–228 (2001)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Rutten, J.: Universal Coalgebra: A theory of systems. Theor. Comp. Sci. 249(1), 3–80 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tarlecki, A.: Institutions: An Abstract Framework for Formal Specifications. In: Astesiano, E., Kreowski, H.-J., Krieg-Brückner, B. (eds.) Algebraic Fondations of System Specification, vol. 2. Springer, Heidelberg (1999)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Dirk Pattinson
    • 1
  1. 1.Institut für InformatikLMU MünchenGermany

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