Star-height of an N-rational series

  • Frédérique Bassino
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1046)

Abstract

We prove a new result on N-rational series in one variable. This result gives, under an appropriate hypothesis, a necessary and sufficient condition for an N-rational series to be of star-height 1. The proof uses a theorem of Handelman on integral companion matrices.

References

  1. [BOY94]
    M. BOYLE. Symbolic dynamics and matrices. In R. Brualdi S. Friedland and V. Klee, editors, Combinatorial and Graph-Theoretic Problems in Linear Algebra, volume 50 of IMA Volumes in Mathematics and Its Applications. 1994.Google Scholar
  2. [BR88]
    J. BERSTEL and C. REUTENAUER. Rational series and their languages. Springer-Verlag, Berlin, 1988.Google Scholar
  3. [EGG63]
    L. C. EGGAN. Transition graphs and star height of regular events. Michigan Math. J., (10):385–397, 1963.CrossRefGoogle Scholar
  4. [GAN59]
    F.R. GANTMACHER. The theory of matrices. Chelsea, New-York, 1959.Google Scholar
  5. [HAN92]
    D.E. HANDELMAN. Spectral radii of primitive integral companion matrices and log-concave polynomials. In Peter Walters, editor, Symbolic Dynamic and its Applications, volume 135 of Contempory Mathemetics, pages 231–238. 1992.Google Scholar
  6. [HAS89]
    K. HASHIGUCHI. Relative star-height, star-height and finite automata with distance functions. In J.-E. Pin, editor, Formal Properties of finite automata and applications, volume 386 of Lecture Notes In Computer Sciences, pages 74–88. Springer, 1989.Google Scholar
  7. [KOE78]
    T. KATAYAMA, M. OKAMOTO, and H. ENOMOTO. Characterization of the structure-generating functions of regular sets and DOL growth functions. Inform. and Control, 36:85–101, 1978.CrossRefGoogle Scholar
  8. [LIN84]
    D. LIND. The entropies of topological Markov shifts and a related class of algebraic integers. Ergod. Th. and Dynam. Syst., 4:283–300, 1984.Google Scholar
  9. [MIN88]
    H. MINC. Nonnegative matrices. Wiley Inter-Sciences, 1988.Google Scholar
  10. [PER92]
    D. PERRIN. On positive matrices. Theoret.Comput.Sci, 94:357–366, 1992.CrossRefGoogle Scholar
  11. [SOI76]
    M. SOITTOLA. Positive rational sequences. Theoret.Comput.Sci, (2): 317–322, 1976.CrossRefGoogle Scholar
  12. [SS78]
    A. SALOMAA and M. SOITTOLA. Automata theoretic aspect of formal power series. Springer-Verlag, Berlin, 1978.Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Frédérique Bassino
    • 1
  1. 1.I.G.M., Université de Marne-la-ValléeNoisy le Grand Cedex

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