About Cube-Free Morphisms

Extended Abstract
  • Gwénaël Richomme
  • Francis Wlazinski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1770)


We address the characterization of finite test-sets for cubefreeness of morphisms between free monoids, that is, the finite sets T such that a morphism f is cube-free if and only if f(T) is cube-free. We first prove that such a finite test-set does not exist for morphisms defined on an alphabet containing at least three letters. Then we prove that for binary morphisms, a set T of cube-free words is a test-set if and only if it contains twelve particular factors. Consequently, a morphism f on {a; b} is cube-free if and only if f(aabbababbabbaabaababaabb) is cube-free (length 24 is optimal). Another consequence is an unpublished result of Leconte: A binary morphism is cube-free if and only if the images of all cube-free words of length 7 are cube-free.

We also prove that, given an alphabet A containing at least two letters, the monoid of cube-free endomorphisms on A is not finitely generated.


Minimal Cover Empty Word Free Monoids Free Word Discrete Apply Mathematic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D.R. Bean, A. Ehrenfeucht and G. McNulty, Avoidable patterns in string of symbols, Pacific J. Math. 95, p261–294, 1979.MathSciNetGoogle Scholar
  2. 2.
    J. Berstel, Mots sans carré et morphismes itérés, Discrete Mathematics 29, p235–244, 1980.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J. Berstel, Axel Thue’s Work on repetitions in words, (4th Conf. on Formal Power Series and Algebraic Combinatorics, Montréal 1992) LITP Technical Report 92.70, 1992.Google Scholar
  4. 4.
    J. Berstel, Axel Thue’s papers on repetitions in words: a translation, Publications of LaCIM 20, University of Québec at Montréal.Google Scholar
  5. 5.
    J. Berstel and P. Séébold, A characterization of overlap-free morphisms, Discrete Applied Mathematics 46, p275–281, 1993.Google Scholar
  6. 6.
    F.-J. Brandenburg, Uniformly Growing k-th Power-Free Homomorphisms, Theoretical Computer Science 23, p69–82, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    C. Choffrut and J. Karhumäki, chapter: Combinatorics of Words in Handbook of Formal Languages vol.1 (G. Rozenberg and A. Salomaa Eds), Springer, 1997.Google Scholar
  8. 8.
    M. Crochemore, Sharp Characterization of Squarefree Morphisms, Theoretical Computer Science 18, p221–22, 1982.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    J. Karhumäki, On cube-free ω-words generated by binary morphisms, Discrete Applied Mathematics 5, p279–297, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    V. Keränen, On the k-freeness of morphisms on free monoids, Annales Academia Scientiarum Fennicae, 1986.Google Scholar
  11. 11.
    M. Leconte, Codes sans répétition, Thèse de 3ème cycle, LITP Université P. et M. Curie, 1985.Google Scholar
  12. 12.
    M. Leconte, A characterization of power-free morphisms, Theoretical Computer Science 38, p117–122, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    M. Lothaire, Combinatorics on words, Encyclopedia of Mathematics, Vol. 17, Addison-Wesley; reprinted in 1997 by Cambridge University Press in the Cambridge Mathematical Library.Google Scholar
  14. 14.
    M. Lothaire, Algebraic combinatorics on words, Cambridge University Press, to appear.Google Scholar
  15. 15.
    M. Morse, Recurrent geodesics on a surface of negative curvature, Transactions Amer. Math. Soc. 22, p84–100, 1921.CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    F. Mignosi and P. Séébold, If a D0L language is k-power free then it is circular, ICALP’93, LNCS 700, Springer-Verlag, 1993, p507–518.Google Scholar
  17. 17.
    G. Richomme and P. Séébold, Characterization of test-sets for overlap-free morphisms, LaRIA Internal report 9724, 1997, to appear in Discrete Applied Mathematics.Google Scholar
  18. 18.
    P. Séébold, Sequences generated by infinitely iterated morphisms, Discrete Applied Mathematics 11, p255–264, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    A. Thue, Über unendliche Zeichenreihen, Videnskapsselskapets Skrifter, I. Mat.-naturv. Klasse, Kristiania, p1–22, 1906.Google Scholar
  20. 20.
    A. Thue, Über die gegenseitige Lage gleigher Teile gewisser Zeichenreihen, Videnskapsselskapets Skrfter, I. Mat.-naturv. Klasse, Kristiania, p1–67, 1912.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Gwénaël Richomme
    • 1
  • Francis Wlazinski
    • 1
  1. 1.LaRIAUniversité de Picardie Jules VerneAmiensFrance

Personalised recommendations