Skip to main content
SpringerLink
Log in
Book cover

International Colloquium on Automata, Languages, and Programming

ICALP 1988: Automata, Languages and Programming pp 93–104Cite as

Zeta functions of recognizable languages

  • Conference paper
  • First Online:

  • 207 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 317))

Abstract

Motivated by symbolic dynamics and algebraic geometry over finite fields, we define cyclic languages and the zeta function of a language. The main result is that the zeta function of a cyclic language which is recognizable by a finite automaton is rational.

This is a preview of subscription content, log in via an institution.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M.-P. Béal, Personal communication (1987).

    Google Scholar 

  2. J. Berstel, D. Perrin, The theory of codes, Acad. Press (1986).

    Google Scholar 

  3. J. Berstel, C. Reutenauer, Rational series and their languages, Springer-Verlag (to appear).

    Google Scholar 

  4. R. Bowen, O. Lanford, Zeta functions of restrictions of the shift transformation, Proc. Sympos. Pure Maths, 14 (1970) 43–50, A.M.S.

    Google Scholar 

  5. N. Chomsky, M.P. Schützenberger, The algebraic theory of context-free languages, in: P. Braffort and D. Hirschberg (ed.), Computer Programming and formal systems, (1963) 118–161, North Holland.

    Google Scholar 

  6. E.M. Coven, M.E. Paul, Sofic systems, Israel J. Maths 20 (1975) 165–177.

    Google Scholar 

  7. B. Dwork, On the rationality of the zeta function of an algebraic variety, Amer. J. Math. 82 (1960) 631–648.

    Google Scholar 

  8. S. Eilenberg, Automata, languages and machines, vol. A, Acad. Press (1974).

    Google Scholar 

  9. S. W. Golomb, Irreducible polynomials, synchronization codes, primitive necklaces and the cyclotomic algebra, Univ. of North Carolina Monograph Series in Proba. and Stat. 4 (1967) 358–370.

    Google Scholar 

  10. R. Hartshorne, Algebraic geometry, Springer-Verlag 1977.

    Google Scholar 

  11. K. Ireland, M. Rosen, A classical introduction to modern number theory, Springer-Verlag (1982).

    Google Scholar 

  12. N. Koblitz, P-adic numbers, p-adic analysis and zeta functions, Springer-Verlag 1984.

    Google Scholar 

  13. G. Lallement, Semigroups and combinatorial applications, John Wiley 1979.

    Google Scholar 

  14. R. Lidl, H. Niederreiter, Finite fields, Encycl. of Maths (1983), Addison Wesley.

    Google Scholar 

  15. M. Lothaire, Combinatorics on words, Encycl. of Maths., Addison Wesley (1983).

    Google Scholar 

  16. C. Reutenauer, Séries formelles et algèbres syntaxiques, J. Algebra 66 (1980) 448–483.

    Google Scholar 

  17. C. Reutenauer, Semisimplicity of the algebra associated to a biprefix code, Semigroup Forum 23 (1981) 327–342.

    Google Scholar 

  18. C. Reutenauer, Ensembles libres de chemins dans un graphe, Bull. Soc. Math. France 114 (1986) 135–152.

    Google Scholar 

  19. C. Reutenauer, Mots circulaires et polynômes irréductibles, to appear.

    Google Scholar 

  20. M.P. Schützenberger, Sur certains sous-monoïdes libres, Bull. Soc. Math. France 93 (1965) 209–223.

    Google Scholar 

  21. B. Weiss, Subshifts of finite type and sofic systems, Monatsh. Math. 77 (1973) 462–474.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LITP, 4 place Jussieu, 75005, Paris

    Jean Berstel & Christophe Reutenauer

Authors
  1. Jean Berstel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christophe Reutenauer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Timo Lepistö Arto Salomaa

Rights and permissions

Reprints and permissions

Copyright information

© 1988 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Berstel, J., Reutenauer, C. (1988). Zeta functions of recognizable languages. In: Lepistö, T., Salomaa, A. (eds) Automata, Languages and Programming. ICALP 1988. Lecture Notes in Computer Science, vol 317. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-19488-6_109

Download citation

  • DOI: https://doi.org/10.1007/3-540-19488-6_109

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-19488-0

  • Online ISBN: 978-3-540-39291-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation