[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Sturmian Trees

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

We consider Sturmian trees as a natural generalization of Sturmian words. A Sturmian tree is a tree having n+1 distinct subtrees of height n for each n. As for the case of words, Sturmian trees are irrational trees of minimal complexity.

We prove that a tree is Sturmian if and only if the minimal automaton associated to its language is slow, that is if the Moore minimization algorithm splits exactly one equivalence class at each step. We give various examples of Sturmian trees, and we introduce two parameters on Sturmian trees, called the degree and the rank. We show that there is no Sturmian tree of finite degree at least 2 and having finite rank. We characterize the family of Sturmian trees of degree 1 and having finite rank by means of a structural property of their minimal automata.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnoux, P., Rauzy, G.: Représentation géométrique de suites de complexité 2n+1. Bull. Soc. Math. Fr. 119, 199–215 (1991)

    MATH  MathSciNet  Google Scholar 

  2. Carpi, A., de Luca, A., Varricchio, S.: Special factors and uniqueness conditions in rational trees. Theor. Comput. Syst. 34, 375–395 (2001)

    MATH  Google Scholar 

  3. Courcelle, B.: Fundamental properties of infinite trees. Theor. Comput. Sci. 25, 95–165 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  4. Coven, E.M., Hedlund, G.A.: Sequences with minimal block growth. Math. Syst. Theory 7, 138–153 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  5. Lothaire, M.: Algebraic Combinatorics on Words. Encyclopedia of Mathematics, vol. 90. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  6. Pytheas-Fogg, N.: Substitutions in Dynamics, Arithmetics and Combinatorics. Lecture Notes in Mathematics, vol. 1794. Springer, Berlin (2002). Berthé, V., Ferenczi, S., Mauduit, C., Siegel, A. (eds.)

    Book  MATH  Google Scholar 

  7. Sakarovitch, J.: Élements de Théorie des Automates. Vuibert, Paris (2003)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire d’Informatique de l’Institut Gaspard-Monge (LABINFO-IGM) UMR CNRS 8049, Université Paris-Est, Marne-la-Vallée, France

    Jean Berstel & Isabelle Fagnot

  2. Laboratoire d’Informatique Algorithmique: Fondements et Applications (LIAFA), Université Denis-Diderot (Paris VII) and CNRS, Paris, France

    Luc Boasson & Olivier Carton

Authors
  1. Jean Berstel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luc Boasson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Olivier Carton
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Isabelle Fagnot
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean Berstel.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Berstel, J., Boasson, L., Carton, O. et al. Sturmian Trees. Theory Comput Syst 46, 443–478 (2010). https://doi.org/10.1007/s00224-009-9228-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-009-9228-0

Keywords

Search

Navigation