Abelian squares are avoidable on 4 letters

  • Veikko Keränen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 623)


An abelian square is a non-empty word of the form P1P2, where P2 is a permutation of P1 that is, the number of occurrences of each letter is the same in P1 and P2. A word is called abelian 2-repetition free, or in short a-2-free, if it does not contain any abelian square as a subword. Let Σ be the four letter alphabet a,b,c,d. We give an example of a uniformly growing endo-morphism g:Σ*→Σ*; with ¦g(a)¦=85; such that the iteration of g yields an a-2-free ω-word. In fact, it is proved that the morphism g itself is a-2-free, that is, g(w) is a-2-free for every a-2-free word w in Σ*. When proving that this property holds for g, one is in practice obliged to use a computer for certain tests. We performed these tests many times by using different computing environments and methods. It is explained how rechecking can be efficiently accomplished.

The morphism g is of the form g(σ(x)) = σ(g(x)) for all x in Σ, where a performs a cyclic permutation of letters in Σ. As regards morphisms of this form, one may check by using a computer/computers that, for our example morphism g, the image word g(a) is of minimal length. This checking becomes feasible by the reason that one may restrict the study to only relatively few long prefixes and suffixes of g(a).


Cyclic Permutation Image Word Free Monoids Letter Alphabet Infinite Word 
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.


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  1. [1]
    J. Berstel, Some recent results on squarefree words, Lecture Notes in Compta. Sci. 166, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1984, 14–25.Google Scholar
  2. [2]
    R. Cori and M.R. Formisano, Partially abelian squarefree words, RAIRO Inform. Théor. et Appl. 24, 1990, 509–520.Google Scholar
  3. [3]
    M. Crochemore, Sharp characterizations of squarefree morphisms, Theoret. Compta. Sci. 18, 1982, 221–226.CrossRefGoogle Scholar
  4. [4]
    F.M. Dekking, Strongly non-repetitive sequences and progression-free sets, J. Combin. Theory Ser. A 27, 1979, 181–185.CrossRefGoogle Scholar
  5. [5]
    V. Dickert, Research topics in the theory of free partially commutative monoids, Bull Europ. Assoc. Theoret. Comput. Sci. 40, 1990, 479–491.Google Scholar
  6. [6]
    R.C. Entringer, D.E. Jackson and J.A. Schatz, On nonrepetitive sequences, J. Combin. Theory Ser. A 16, 1974, 159–164.CrossRefGoogle Scholar
  7. [7]
    P. Erdös, Some unsolved problems, Magyar Tud. Akad. Mat. Kutató Int. Közl. 6, 1961, 221–254.Google Scholar
  8. [8]
    V. Keränen, On the k-freeness of morphisms on free monoids, Ann. Acad. Sci. Fenn. Ser. AI Math. Dissertationes 61, 1986.Google Scholar
  9. [9]
    V. Keränen, On the k-freeness of morphisms on free monoids, Lecture Notes in Comput. Sci. 247, Springer-Verlag, Berlin-Heidelberg-New York-London-Paris-Tokyo, 1987, 180–188.Google Scholar
  10. [10]
    M. Leconte, A characterization of power-free morphisms, Theoret. Comput. Sci. 38, 1985, 117–122.Google Scholar
  11. [11]
    M. Leconte, Kth power-free codes, Lecture Notes in Comput. Sci. 192, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo 1985, 172–187.Google Scholar
  12. [12]
    M. Lothaire, Combinatorics on words, Addison-Wesley, Reading, Massachusetts, 1983.Google Scholar
  13. [13]
    P.A. Pleasants, Non-repetitive sequences, Troc. Cambridge Phil. Soc. 68, 1970, 267–274.Google Scholar
  14. [14]
    G. Rozenberg and A. Salomaa, The mathematical theory of L systems, Academic Press, New York-London-Toronto-Sydney-San Fransisco, 1980.Google Scholar
  15. [15]
    A. Salomaa, Jewels of formal language theory. Computer Science Press, Rockville, Maryland, 1981.Google Scholar
  16. [16]
    A. Thue, Über unendliche Zeichenreihen, Norske Vid. Selst Skr. I. Mat. Nat. Kl. Christiania 7, 1906, 1–22.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Veikko Keränen
    • 1
  1. 1.Rovaniemi Institute of TechnologyRovaniemiFinland

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