Advertisement

On Sets of Numbers Rationally Represented in a Rational Base Number System

  • Victor Marsault
  • Jacques Sakarovitch
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8080)

Abstract

In this work, it is proved that a set of numbers closed under addition and whose representations in a rational base numeration system is a rational language is not a finitely generated additive monoid.

A key to the proof is the definition of a strong combinatorial property on languages : the bounded left iteration property. It is both an unnatural property in usual formal language theory (as it contradicts any kind of pumping lemma) and an ideal fit to the languages defined through rational base number systems.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Akiyama, S., Frougny, C., Sakarovitch, J.: Powers of rationals modulo 1 and rational base number systems. Israel J. Math. 168, 53–91 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Autebert, J.M., Beauquier, J., Boasson, L., Latteux, M.: Indécidabilité de la condition IRS. ITA 16(2), 129–138 (1982)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Frougny, C., Sakarovitch, J.: Number representation and finite automata. In: Berthé, V., Rigo, M. (eds.) Combinatorics, Automata and Number Theory. Encyclopedia of Mathematics and its Applications, vol. 135, pp. 34–107. Cambridge Univ. Press (2010)Google Scholar
  4. 4.
    Greibach, S.A.: One counter languages and the IRS condition. J. Comput. Syst. Sci. 10(2), 237–247 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley (2000)Google Scholar
  6. 6.
    Lecomte, P., Rigo, M.: Abstract numeration systems. In: Berthé, V., Rigo, M. (eds.) Combinatorics, Automata and Number Theory. Encyclopedia of Mathematics and its Applications, vol. 135, pp. 108–162. Cambridge Univ. Press (2010)Google Scholar
  7. 7.
    Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press (2002)Google Scholar
  8. 8.
    Sakarovitch, J.: Elements of Automata Theory. Cambridge University Press (2009); Corrected English translation of Éléments de théorie des automates, Vuibert (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Victor Marsault
    • 1
  • Jacques Sakarovitch
    • 1
  1. 1.Telecom-ParisTech and CNRSParisFrance

Personalised recommendations