A Coalgebraic View of ε-Transitions

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8089)


In automata theory, a machine transitions from one state to the next when it reads an input symbol. It is common to also allow an automaton to transition without input, via an ε-transition. These ε-transitions are convenient, e.g., when one defines the composition of automata. However, they are not necessary, and can be eliminated. Such ε-elimination procedures have been studied separately for different types of automata, including non-deterministic and weighted automata.

It has been noted by Hasuo that it is possible to give a coalgebraic account of ε-elimination for some automata using trace semantics (as defined by Hasuo, Jacobs and Sokolova).

In this paper, we give a detailed description of the ε-elimination procedure via trace semantics (missing in the literature). We apply this framework to several types of automata, and explore its boundary.

In particular, we show that is possible (by careful choice of a monad) to define an ε-removal procedure for all weighted automata over the positive reals (and certain other semirings). Our definition extends the recent proposals by Sakarovitch and Lombardy for these semirings.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bonchi, F., Bonsangue, M., Boreale, M., Rutten, J., Silva, A.: A coalgebraic perspective on linear weighted automata. Inf. Comput. 211, 77–105 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bonchi, F., Bonsangue, M., Rutten, J., Silva, A.: Brzozowski’s algorithm (co)algebraically. In: Constable, R.L., Silva, A. (eds.) Logic and Program Semantics. LNCS, vol. 7230, pp. 12–23. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  3. 3.
    Bonchi, F., Pous, D.: Checking NFA equivalence with bisimulations up to congruence. In: POPL, pp. 457–468. ACM (2013)Google Scholar
  4. 4.
    Brengos, T.: Weak bisimulations for coalgebras over ordered functors. In: Baeten, J.C.M., Ball, T., de Boer, F.S. (eds.) TCS 2012. LNCS, vol. 7604, pp. 87–103. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  5. 5.
    Droste, M., Kuich, W., Vogler, W.: Handbook of Weighted Automata. Springer (2009)Google Scholar
  6. 6.
    Peter Gumm, H., Schröder, T.: Monoid-labeled transition systems. ENTCS 44(1), 185–204 (2001)Google Scholar
  7. 7.
    Hasuo, I., Jacobs, B., Sokolova, A.: Generic Forward and Backward Simulations. In: Proceedings of JSSST Annual Meeting (2006) (Partly in Japanese)Google Scholar
  8. 8.
    Hasuo, I., Jacobs, B., Sokolova, A.: Generic Trace Semantics via Coinduction. Logical Methods in Computer Science 3(4) (2007)Google Scholar
  9. 9.
    Jacobs, B.: From coalgebraic to monoidal traces. ENTCS 264(2), 125–140 (2010)MathSciNetGoogle Scholar
  10. 10.
    Jacobs, B., Silva, A., Sokolova, A.: Trace semantics via determinization. In: Pattinson, D., Schröder, L. (eds.) CMCS 2012. LNCS, vol. 7399, pp. 109–129. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  11. 11.
    Kerstan, H., König, B.: Coalgebraic trace semantics for probabilistic transition systems based on measure theory. In: Koutny, M., Ulidowski, I. (eds.) CONCUR 2012. LNCS, vol. 7454, pp. 410–424. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  12. 12.
    Lombardy, S., Sakarovitch, J.: The Removal of Weighted ε-Transitions. In: Moreira, N., Reis, R. (eds.) CIAA 2012. LNCS, vol. 7381, pp. 345–352. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  13. 13.
    Milius, S.: On Iteratable Endofunctors. In: CTCS. ENTCS, vol. 69, pp. 287–304. Elsevier (2002)Google Scholar
  14. 14.
    Mohri, M.: Generic ε-removal algorithm for weighted automata. In: Yu, S., Păun, A. (eds.) CIAA 2000. LNCS, vol. 2088, pp. 230–242. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  15. 15.
    Rutten, J.: Automata and coinduction (an exercise in coalgebra). In: Sangiorgi, D., de Simone, R. (eds.) CONCUR 1998. LNCS, vol. 1466, pp. 194–218. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  16. 16.
    Sangiorgi, D.: An introduction to bisimulation and coinduction. Cambridge University Press (2012)Google Scholar
  17. 17.
    Silva, A., Bonchi, F., Bonsangue, M., Rutten, J.: Generalizing the powerset construction, coalgebraically. In: FSTTCS. LIPIcs, vol. 8, pp. 272–283 (2010)Google Scholar
  18. 18.
    Silva, A., Bonchi, F., Bonsangue, M., Rutten, J.: Generalizing determinization from automata to coalgebras. LMCS 9(1) (2013)Google Scholar
  19. 19.
    Silva, A., Westerbaan, B.: A Coalgebraic View of ε-Transitions. Extended abstract, with proofs,
  20. 20.
    Sokolova, A., de Vink, E., Woracek, H.: Coalgebraic weak bisimulation for action-type systems. Sci. Ann. Comp. Sci. 19, 93–144 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.ICISRadboud University NijmegenNetherlands

Personalised recommendations