Advertisement

Equations and Coequations for Weighted Automata

  • Julian SalamancaEmail author
  • Marcello  Bonsangue
  • Jan Rutten
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9234)

Abstract

We study weighted automata from both an algebraic and a coalgebraic perspective. In particular, we consider equations and coequations for weighted automata. We prove a duality result that relates sets of equations (congruences) with (certain) subsets of coequations. As a consequence, we obtain two equivalent but complementary ways to define classes of weighted automata. We show that this duality cannot be generalized to linear congruences in general but we obtain partial results when weights are from a field.

References

  1. 1.
    Adámek, J., Milius, S., Myers, R.S.R., Urbat, H.: Generalized eilenberg theorem i: local varieties of languages. In: Muscholl, A. (ed.) FOSSACS 2014 (ETAPS). LNCS, vol. 8412, pp. 366–380. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  2. 2.
    Arbib, M.A., Manes, E.G.: Foundations of system theory: the hankel matrix. J. Comput. Syst. Sci. 20(3), 330–378 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ballester-Bolinches, A., Cosme-Llópez, E., Rutten, J.J.M.M:. The dual equivalence of equations and coequations for automata. CWI Technical report FM-1403, pp. 1–41 (2014, To appear inInformation and Computation)Google Scholar
  4. 4.
    Bezhanishvili, N., Kupke, C., Panangaden, P.: Minimization via duality. In: Ong, L., de Queiroz, R. (eds.) WoLLIC 2012. LNCS, vol. 7456, pp. 191–205. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  5. 5.
    Bonchi, F., Bonsangue, M., Boreale, M., Rutten, J., Silva, A.: A coalgebraic perspective on linear weighted automata. Inf. Comp. 211, 77–105 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bonchi, F., Bonsangue, M.M., Hansen, H.H., Panangaden, P., Rutten, J., Silva, A.: Algebra-coalgebra duality in brzozowski’s minimization algorithm. ACM Trans. Comput. Logic 15(1), 3 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Burri, S.N., Sankappanava, H.P.: A course in universal algebra. Graduate Texts in Mathematic, vol. 78. Springer, New York (1981) CrossRefGoogle Scholar
  8. 8.
    Eilenberg, S.: Automata, languages, and machines, vol. B. Academy Press, New York (1976) zbMATHGoogle Scholar
  9. 9.
    Gehrke, M., Grigorieff, S., Pin, J.É.: Duality and equational theory of regular languages. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 246–257. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  10. 10.
    Hungerford, T.W.: Algebra. Graduate Texts in Mathematics, vol. 73. Springer, New York (1974) Google Scholar
  11. 11.
    Kuich, W.: Semirings and formal power series. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, Word, Language, Grammar, vol. 1, pp. 609–677. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  12. 12.
    Petković, T.: Varieties of fuzzy languages. In: Proceedings of International Conference on Algebraic Informatics. Aristotle University of Thessaloniki (2005)Google Scholar
  13. 13.
    Rutten, J.J.M.M.: Universal coalgebra: a theory of systems. Theor. Comput. Sci. 249(1), 3–80 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Schützenberger, M.P.: On the definition of a family of automata. Inf. Control 4, 245–270 (1961)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Julian Salamanca
    • 1
    Email author
  • Marcello  Bonsangue
    • 1
    • 2
  • Jan Rutten
    • 1
    • 3
  1. 1.CWI AmsterdamAmsterdamThe Netherlands
  2. 2.LIACS - Leiden UniversityLeidenThe Netherlands
  3. 3.Radboud University NijmegenNijmegenThe Netherlands

Personalised recommendations