Growth Properties of Power-Free Languages

  • Arseny M. Shur
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6795)

Abstract

The aim of this paper is to give a short survey of the area formed by the intersection of two popular lines of investigation in formal language theory. The first line, originated by Thue in 1906, concerns about repetition-free words and languages. The second line is the study of growth functions for words and languages; it can be traced back to the classical papers by Morse and Hedlund on symbolic dynamics (1938, 1940). Growth functions of repetition-free languages are investigated since 1980’s. Most of the results were obtained for power-free languages, but some ideas can be applied for languages avoiding patterns and Abelian-power-free languages as well.

In this paper, we present key contributions to the area, its state-of-the-art, and conjectures that suggest answers to some natural unsolved problems. Also, we pay attention to the tools and techniques that made possible the progress in the area and suggest some technical results that would be useful to solve open problems.

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References

  1. 1.
    Berstel, J.: Growth of repetition-free words – a review. Theor. Comput. Sci. 340(2), 280–290 (2005)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Blondel, V.D., Cassaigne, J., Jungers, R.: On the number of α-power-free binary words for 2 < α ≤ 7/3. Theor. Comput. Sci. 410, 2823–2833 (2009)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brandenburg, F.-J.: Uniformly growing k-th power free homomorphisms. Theor. Comput. Sci. 23, 69–82 (1983)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Brinkhuis, J.: Non-repetitive sequences on three symbols. Quart. J. Math. Oxford 34, 145–149 (1983)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cassaigne, J.: Counting overlap-free binary words. In: STACS 1993. LNCS, vol. 665, pp. 216–225. Springer, Berlin (1993)Google Scholar
  6. 6.
    Chomsky, N., Schützenberger, M.: The algebraic theory of context-free languages. In: Computer Programming and Formal System, pp. 118–161. North-Holland, Amsterdam (1963)CrossRefGoogle Scholar
  7. 7.
    Crochemore, M., Mignosi, F., Restivo, A.: Automata and forbidden words. Inform. Processing Letters 67, 111–117 (1998)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Cvetković, D.M., Doob, M., Sachs, H.: Spectra of graphs. In: Theory and Applications, 3rd edn., p. 388. Johann Ambrosius Barth, Heidelberg (1995)Google Scholar
  9. 9.
    Dejean, F.: Sur un Theoreme de Thue. J. Comb. Theory, Ser. A 13, 90–99 (1972)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gottschalk, W.H., Hedlund, G.A.: A characterization of the Morse minimal set. Proc. of Amer. Math. Soc., 15, 70–74 (1964)Google Scholar
  11. 11.
    Goulden, I., Jackson, D.M.: An inversion theorem for cluster decompositions of sequences with distinguished subsequences. J. London Math. Soc. 20, 567–576 (1979)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Govorov, V.E.: Graded algebras. Math. Notes 12, 552–556 (1972)MathSciNetGoogle Scholar
  13. 13.
    Jungers, R.M., Protasov, V.Y., Blondel, V.D.: Overlap-free words and spectra of matrices. Theor. Comput. Sci. 410, 3670–3684 (2009)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Karhumäki, J., Shallit, J.: Polynomial versus exponential growth in repetition-free binary words. J. Combin. Theory. Ser. A 104, 335–347 (2004)CrossRefGoogle Scholar
  15. 15.
    Kobayashi, Y.: Repetition-free words. Theor. Comput. Sci. 44, 175–197 (1986)MATHCrossRefGoogle Scholar
  16. 16.
    Kobayashi, Y.: Enumeration of irreducible binary words. Discr. Appl. Math. 20, 221–232 (1988)MATHCrossRefGoogle Scholar
  17. 17.
    Kolpakov, R.M.: On the number of repetition-free words. J. Appl. Ind. Math. 1(4), 453–462 (2007)CrossRefGoogle Scholar
  18. 18.
    Kolpakov, R.: Efficient lower bounds on the number of repetition-free words. J. Int. Sequences 10, # 07.3.2 (2007)Google Scholar
  19. 19.
    Kolpakov, R., Kucherov, G., Tarannikov, Y.: On repetition-free binary words of minimal density. Theor. Comput. Sci. 218, 161–175 (1999)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Noonan, J., Zeilberger, D.: The Goulden-Jackson Cluster Method: Extensions, Applications, and Implementations. J. Difference Eq. Appl. 5, 355–377 (1999)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Ochem, P.: A generator of morphisms for infinite words. RAIRO Theor. Inform. Appl. 40, 427–441 (2006)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Ochem, P., Reix, T.: Upper bound on the number of ternary square-free words. In: Proc. Workshop on Words and Automata (WOWA 2006), S.-Petersburg, #8 (2006) (electronic)Google Scholar
  23. 23.
    Rampersad, N.: Words avoiding (7/3)-powers and the Thue–Morse morphism. Int. J. Foundat. Comput. Sci. 16, 755–766 (2005)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Richard, C., Grimm, U.: On the entropy and letter frequencies of ternary square-free words. Electronic J. Combinatorics 11, # R14 (2004)MathSciNetGoogle Scholar
  25. 25.
    Restivo, A., Salemi, S.: Overlap-free words on two symbols. In: Perrin, D., Nivat, M. (eds.) Automata on Infinite Words. LNCS, vol. 192, pp. 196–206. Springer, Heidelberg (1985)Google Scholar
  26. 26.
    Samsonov, A.V., Shur, A.M.: On Abelian repetition threshold. In: Samsonov, A.V., Shur, A.M. (eds.) Proc. 13th Mons Days of Theoretical Computer Science, pp. 1–11. Univ. de Picardie Jules Verne, Amiens (2010)Google Scholar
  27. 27.
    Séébold, P.: Overlap-free sequences. In: Perrin, D., Nivat, M. (eds.) Automata on Infinite Words. LNCS, vol. 192, pp. 196–206. Springer, Heidelberg (1985)Google Scholar
  28. 28.
    Shur, A.M.: The structure of the set of cube-free Z-words over a two-letter alphabet. Izvestiya Math. 64(4), 847–871 (2000)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Shur, A.M.: Factorial languages of low combinatorial complexity. In: Ibarra, O.H., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 397–407. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  30. 30.
    Shur, A.M.: Comparing complexity functions of a language and its extendable part. RAIRO Inform. Theor. Appl. 42, 647–655 (2008)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Shur, A.M.: Combinatorial complexity of regular languages. In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds.) Computer Science – Theory and Applications. LNCS, vol. 5010, pp. 289–301. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  32. 32.
    Shur, A.M.: Two-sided bounds for the growth rates of power-free languages. In: Ibarra, O.H., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 466–477. Springer, Heidelberg (2006)Google Scholar
  33. 33.
    Shur, A.M.: Growth rates of complexity of power-free languages. Theor. Comput. Sci. 411, 3209–3223 (2010)MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Shur, A.M.: Growth of power-free languages over large alphabets. In: Ablayev, F., Mayr, E.W. (eds.) CSR 2010. LNCS, vol. 6072, pp. 350–361. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  35. 35.
    Shur, A.M.: On the existence of minimal β-powers. In: Gao, Y., Lu, H., Seki, S., Yu, S. (eds.) DLT 2010. LNCS, vol. 6224, pp. 411–422. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  36. 36.
    Shur, A.M.: Numerical values of the growth rates of power-free languages, arXiv:1009.4415v1 (cs.FL) (2010) Google Scholar
  37. 37.
    Shur, A.M., Gorbunova, I.A.: On the growth rates of complexity of threshold languages. RAIRO Inform. Theor. Appl. 44, 175–192 (2010)MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Thue, A.: Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. I, Mat. Nat. Kl. 1, 1–67 (1912)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Arseny M. Shur
    • 1
  1. 1.Ural State UniversityEkaterinburgRussia

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