Defining Context-Free Power Series Coalgebraically

  • Marcello M. Bonsangue
  • Jan Rutten
  • Joost Winter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7399)

Abstract

In this paper we present a coinductive definition of context free power series in terms of behavioural differential equations. We show that our coalgebraic approach provides a unified view on many, at first sight different, existing notions of algebraicity, and we apply our behavioural differential equations to produce a new proof for a classical result by Chomsky and Schützenberger, and a simple proof that the zip-operator of two algebraic streams is algebraic.

References

  1. [AS03]
    Allouche, J.-P., Shallit, J.O.: Automatic Sequences – Theory, Applications, Generalizations. Cambridge University Press (2003)Google Scholar
  2. [BR11]
    Berstel, J., Reutenauer, C.: Noncommutative Rational Series with Applications. Cambridge University Press (2011)Google Scholar
  3. [ÉL05]
    Ésik, Z., Leiß, H.: Algebraically complete semirings and Greibach normal form. Annals of Pure and Applied Logic 133(1-3), 173–203 (2005)MathSciNetMATHCrossRefGoogle Scholar
  4. [Fli74]
    Fliess, M.: Sur divers produits de sries formelles. Bulletin de la S.M.F. 102, 181–191 (1974)MathSciNetMATHGoogle Scholar
  5. [Nij80]
    Nijholt, A.: Context-Free Grammars: Covers, Normal Forms, and Parsing. LNCS, vol. 93. Springer, Heidelberg (1980)MATHCrossRefGoogle Scholar
  6. [NR10]
    Niqui, M., Rutten, J.: Sampling, Splitting and Merging in Coinductive Stream Calculus. In: Bolduc, C., Desharnais, J., Ktari, B. (eds.) MPC 2010. LNCS, vol. 6120, pp. 310–330. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. [PS09]
    Petre, I., Salomaa, A.: Algebraic systems and pushdown automata. In: Handbook of Weighted Automata, 1st edn., pp. 257–289. Springer (2009)Google Scholar
  8. [Rut02]
    Rutten, J.: Coinductive counting: bisimulation in enumerative combinatorics. Electr. Notes Theor. Comput. Sci. 65(1), 286–304 (2002)CrossRefGoogle Scholar
  9. [Rut03]
    Rutten, J.: Behavioural differential equations: a coinductive calculus of streams, automata, and power series. Theoretical Computer Science 308(1-3), 1–53 (2003)MathSciNetMATHCrossRefGoogle Scholar
  10. [Rut05]
    Rutten, J.: A coinductive calculus of streams. Mathematical Structures in Computer Science 15(1), 93–147 (2005)MathSciNetMATHCrossRefGoogle Scholar
  11. [WBR11]
    Winter, J., Bonsangue, M.M., Rutten, J.: Context-Free Languages, Coalgebraically. In: Corradini, A., Klin, B., Cîrstea, C. (eds.) CALCO 2011. LNCS, vol. 6859, pp. 359–376. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. [SBBR10]
    Silva, A., Bonchi, F., Bonsangue, M., Rutten, J.: Generalizing the powerset construction, coalgebraically. In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2010. LIPIcs, vol. 8, pp. 272–283 (2010)Google Scholar

Copyright information

© IFIP International Federation for Information Processing 2012

Authors and Affiliations

  • Marcello M. Bonsangue
    • 2
    • 1
  • Jan Rutten
    • 1
    • 3
  • Joost Winter
    • 1
  1. 1.Centrum Wiskunde & Informatica (CWI)The Netherlands
  2. 2.LIACSLeiden UniversityThe Netherlands
  3. 3.Radboud UniversityNijmegenThe Netherlands

Personalised recommendations