A Coalgebraic Perspective on Minimization and Determinization

  • Jiří Adámek
  • Filippo Bonchi
  • Mathias Hülsbusch
  • Barbara König
  • Stefan Milius
  • Alexandra Silva
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7213)


Coalgebra offers a unified theory of state based systems, including infinite streams, labelled transition systems and deterministic automata. In this paper, we use the coalgebraic view on systems to derive, in a uniform way, abstract procedures for checking behavioural equivalence in coalgebras, which perform (a combination of) minimization and determinization. First, we show that for coalgebras in categories equipped with factorization structures, there exists an abstract procedure for equivalence checking. Then, we consider coalgebras in categories without suitable factorization structures: under certain conditions, it is possible to apply the above procedure after transforming coalgebras with reflections. This transformation can be thought of as some kind of determinization. We will apply our theory to the following examples: conditional transition systems and (non-deterministic) automata.


Transition System Label Transition System Unique Morphism Input Alphabet Concrete Category 
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.


  1. 1.
    Adámek, J.: Free algebras and automata realizations in the language of categories. Comment. Math. Univ. Carolin. 15, 589–602 (1974)MATHGoogle Scholar
  2. 2.
    Adámek, J., Bonchi, F., Hülsbusch, M., König, B., Milius, S., Silva, A.: A coalgebraic perspective on minimization and determinization (extended version), http://alexandrasilva.org/files/fossacs12-extended.pdf
  3. 3.
    Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories – The Joy of Cats. Wiley (1990)Google Scholar
  4. 4.
    Adámek, J., Koubek, V.: On the greatest fixed point of a set functor. TCS 150, 57–75 (1995)MATHCrossRefGoogle Scholar
  5. 5.
    Adámek, J., Milius, S., Moss, L.S., Sousa, L.: Well-pointed coalgebras. In: Birkedal, L. (ed.) FOSSACS 2012. LNCS, vol. 7213, Springer, Heidelberg (2012)Google Scholar
  6. 6.
    Boreale, M.: Weighted Bisimulation in Linear Algebraic Form. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 163–177. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Brzozowski, J.A.: Canonical regular expressions and minimal state graphs for definite events. Mathematical Theory of Automata 12(6), 529–561 (1962)Google Scholar
  8. 8.
    Brzozowski, J., Tamm, H.: Theory of Átomata. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 105–116. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Ferrari, G.L., Montanari, U., Tuosto, E.: Coalgebraic minimization of HD-automata for the pi-calculus using polymorphic types. TCS 331(2-3), 325–365 (2005)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Fitting, M.: Bisimulations and boolean vectors. In: Advances in Modal Logic, vol. 4, pp. 1–29. World Scientific Publishing (2002)Google Scholar
  11. 11.
    Gumm, H.P.: From T-Coalgebras to Filter Structures and Transition Systems. In: Fiadeiro, J.L., Harman, N.A., Roggenbach, M., Rutten, J. (eds.) CALCO 2005. LNCS, vol. 3629, pp. 194–212. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  12. 12.
    Gumm, H.P.: On minimal coalgebras. Applied Categorical Structures 16, 313–332 (2008)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Hasuo, I., Jacobs, B., Sokolova, A.: Generic trace semantics via coinduction. LMCS 3(4:11), 1–36 (2007)MathSciNetGoogle Scholar
  14. 14.
    Hennessy, M., Lin, H.: Symbolic bisimulations. TCS 138(2), 353–389 (1995)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Hermida, C., Jacobs, B.: Structural induction and coinduction in a fibrational setting. Information and Computation 145, 107–152 (1998)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation, 3rd edn. Wesley (2006)Google Scholar
  17. 17.
    Kurz, A.: Logics for Coalgebras and Applications to Computer Science. PhD thesis, Ludwigs-Maximilians-Universität München (2000)Google Scholar
  18. 18.
    Mac Lane, S.: Categories for the Working Mathematician. Springer, Heidelberg (1971)MATHGoogle Scholar
  19. 19.
    Mulry, P.S.: Lifting Theorems for Kleisli Categories. In: Main, M.G., Melton, A.C., Mislove, M.W., Schmidt, D., Brookes, S.D. (eds.) MFPS 1993. LNCS, vol. 802, pp. 304–319. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  20. 20.
    Panangaden, P.: Duality in probabilistic automata Slides (May 19, 2011), http://www.cs.mcgill.ca/~prakash/Talks/duality_talk.pdf
  21. 21.
    Power, J., Turi, D.: A coalgebraic foundation for linear time semantics. In: Proc. of CTCS 1999. ENTCS, vol. 29, pp. 259–274 (1999)Google Scholar
  22. 22.
    Rutten, J.J.M.M.: Universal coalgebra: a theory of systems. TCS 249, 3–80 (2000)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Staton, S.: Relating coalgebraic notions of bisimulation. LMCS 7(1) (2011)Google Scholar
  24. 24.
    Worrell, J.: On the final sequence of a finitary set functor. TCS 338(1-3), 184–199 (2005)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jiří Adámek
    • 1
  • Filippo Bonchi
    • 2
  • Mathias Hülsbusch
    • 3
  • Barbara König
    • 3
  • Stefan Milius
    • 1
  • Alexandra Silva
    • 4
  1. 1.Technische Universität BraunschweigGermany
  2. 2.CNRS, ENS Lyon, Université de Lyon LIP (UMR 5668)France
  3. 3.Universität Duisburg-EssenGermany
  4. 4.Radboud UniversityNijmegenNetherlands

Personalised recommendations