How to Kill Epsilons with a Dagger

A Coalgebraic Take on Systems with Algebraic Label Structure
  • Filippo Bonchi
  • Stefan Milius
  • Alexandra Silva
  • Fabio Zanasi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8446)


We propose an abstract framework for modeling state-based systems with internal behavior as e.g. given by silent or \(\epsilon \)-transitions. Our approach employs monads with a parametrized fixpoint operator \(\dagger \) to give a semantics to those systems and implement a sound procedure of abstraction of the internal transitions, whose labels are seen as the unit of a free monoid. More broadly, our approach extends the standard coalgebraic framework for state-based systems by taking into account the algebraic structure of the labels of their transitions. This allows to consider a wide range of other examples, including Mazurkiewicz traces for concurrent systems.


  1. 1.
    Adámek, J., Milius, S., Velebil, J.: Equational properties of iterative monads. Inform. Comput. 208, 1306–1348 (2010). doi:10.1016/j.ic.2009.10.006 MATHGoogle Scholar
  2. 2.
    Adámek, J., Milius, S., Velebil, J.: Elgot theories: a new perspective of the equational properties of iteration. Math. Structures Comput. Sci. 21(2), 417–480 (2011)MATHMathSciNetGoogle Scholar
  3. 3.
    Asada, K., Hidaka, S., Kato, H., Hu, Z., Nakano, K.: A parameterized graph transformation calculus for finite graphs with monadic branches. In: Peña, R., Schrijvers, T. (eds.) PPDP, pp. 73–84. ACM (2013)Google Scholar
  4. 4.
    Balan, A., Kurz, A.: On coalgebras over algebras. Theoret. Comput. Sci. 412(38), 4989–5005 (2011)MATHMathSciNetGoogle Scholar
  5. 5.
    Barr, M.: Coequalizers and free triples. Math. Z. 116, 307–322 (1970)MATHMathSciNetGoogle Scholar
  6. 6.
    Bloom, S.L., Ésik, Z.: Iteration Theories: The Equational Logic of Iterative. EATCS Monographs on Theoretical Computer Science. Springer, Heidelberg (1993)MATHGoogle Scholar
  7. 7.
    Bonsangue, M.M., Hansen, H.H., Kurz, A., Rot, J.: Presenting distributive laws. In: Heckel and Milius [15], pp. 95–109Google Scholar
  8. 8.
    Main, M.G., Melton, A.C., Mislove, M.W., Schmidt, D., Brookes, S.D. (eds.): MFPS 1993. LNCS, vol. 802. Springer, Heidelberg (1994)MATHGoogle Scholar
  9. 9.
    Freyd, P.J.: Remarks on Algebraically Compact Categories. London Mathematical Society Lecture Notes Series, vol. 177. Cambridge University Press, London (1992)Google Scholar
  10. 10.
    Gadducci, F., Montanari, U.: The tile model. In: Plotkin, G.D., Stirling, C., Tofte, M. (eds.) Proof, Language, and Interaction, pp. 133–166. MIT Press, Boston (2000)Google Scholar
  11. 11.
    Hasuo, I., Jacobs, B., Sokolova, A.: Generic forward and backward simulations. In: (Partly in Japanese) Proceedings of JSSST Annual Meeting (2006)Google Scholar
  12. 12.
    Hasuo, I., Jacobs, B., Sokolova, A.: Generic trace semantics via coinduction. Log. Methods Comput. Sci. 3(4:11), 1–36 (2007)MathSciNetGoogle Scholar
  13. 13.
    Heckel, R., Milius, S. (eds.): Algebra and Coalgebra in Computer Science. Lecture Notes in Computer Science, vol. 8089. Springer, Heidelberg (2013)MATHGoogle Scholar
  14. 14.
    Hopcroft, J., Motwani, R., Ullman, J.: Introduction to Automata Theory, Languages, and Computation, 3rd edn. Wesley, Lebanon (2006)Google Scholar
  15. 15.
    Johnstone, P.: Adjoint lifting theorems for categories of algebras. Bull. London Math. Soc. 7, 294–297 (1975)MATHMathSciNetGoogle Scholar
  16. 16.
    Kelly, G.M.: A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on. Bull. Austral. Math. Soc. 22, 1–83 (1980)MATHMathSciNetGoogle Scholar
  17. 17.
    Mazurkiewicz, A.: Concurrent program schemes and their interpretation. DAIMI PB-78, Computer Science Department, Aarhus University (1977)Google Scholar
  18. 18.
    Mac Lane, S.: Categories for the Working Mathematician. Springer, Berlin (1971)MATHGoogle Scholar
  19. 19.
    Milius, S., Palm, T., Schwencke, D.: Complete iterativity for algebras with effects. In: Kurz, A., Lenisa, M., Tarlecki, A. (eds.) CALCO 2009. LNCS, vol. 5728, pp. 34–48. Springer, Heidelberg (2009) Google Scholar
  20. 20.
    Mulry, P.S.: Lifting theorems for Kleisli categories. In: Brookes et al. [10], pp. 304–319Google Scholar
  21. 21.
    Rabinovich, A.M.: A complete axiomatisation for trace congruence of finite state behaviors. In: Brookes et al. [10], pp. 530–543Google Scholar
  22. 22.
    Silva, A., Westerbaan, B.: A coalgebraic view of \(\varepsilon \)-transitions. In: Heckel and Milius [15], pp. 267–281Google Scholar
  23. 23.
    Sobociński, P.: Relational presheaves as labelled transition systems. In: Pattinson, D., Schröder, L. (eds.) CMCS 2012. LNCS, vol. 7399, pp. 40–50. Springer, Heidelberg (2012) Google Scholar
  24. 24.
    Turi, D., Plotkin, G.: Towards a mathematical operational semantics. In: Proceedings of Logic in Computer Science (LICS’97). IEEE Computer Society (1997)Google Scholar

Copyright information

© IFIP International Federation for Information Processing 2014

Authors and Affiliations

  • Filippo Bonchi
    • 1
  • Stefan Milius
    • 2
  • Alexandra Silva
    • 3
    • 4
    • 5
  • Fabio Zanasi
    • 1
  1. 1.ENS Lyon, U. de Lyon, CNRS, INRIA, UCBLLyonFrance
  2. 2.Lehrstuhl Für Theoretische InformatikFriedrich-Alexander Universität Erlangen-NürnbergErlangenGermany
  3. 3.Institute for Computing and Information SciencesRadboud University NijmegenNijmegenThe Netherlands
  4. 4.Centrum Wiskunde & InformaticaAmsterdamThe Netherlands
  5. 5.HASLab/INESC TECUniversidade Do MinhoBragaPortugal

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