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Transition Property for Cube-Free Words

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Abstract

We study cube-free words over arbitrary non-unary finite alphabets and prove the following structural property: for every pair (u, v) of d-ary cube-free words, if u can be infinitely extended to the right and v can be infinitely extended to the left respecting the cube-freeness property, then there exists a “transition” word w over the same alphabet such that uwv is cube free. The crucial case is the case of the binary alphabet, analyzed in the central part of the paper. The obtained “transition property”, together with the developed technique, allowed us to solve cube-free versions of three old open problems by Restivo and Salemi. Besides, it has some further implications for combinatorics on words; e.g., it implies the existence of infinite cube-free words of very big subword (factor) complexity.

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Notes

  1. For α > 7/3, the language of binary α-power-free words has exponential size [8]. The exponential conjecture says that for k ≥ 3 all infinite power-free languages over k letters have exponential size. This conjecture is proved for k ≤ 10 [9, 10] and odd k up to 101 [22].

References

  1. Carpi, A.: On the centers of the set of weakly square-free words on a two-letter alphabet. Inform. Process. Lett. 19, 187–190 (1984)

    Article  MathSciNet  Google Scholar 

  2. Carpi, A.: Overlap-free words and finite automata. Theoret. Comput. Sci. 115, 243–260 (1993)

    Article  MathSciNet  Google Scholar 

  3. Currie, J.D.: On the structure and extendibility of k-power free words. European J. Combinatorics 16, 111–124 (1995)

    Article  MathSciNet  Google Scholar 

  4. Currie, J.D., Shelton, R.O.: The set of k-power free words over σ is empty or perfect. European J. Combinatorics 24, 573–580 (2003)

    Article  MathSciNet  Google Scholar 

  5. Fine, N.J., Wilf, H.S.: Uniqueness theorems for periodic functions. Proc. Amer. Math. Soc. 16, 109–114 (1965)

    Article  MathSciNet  Google Scholar 

  6. Guglielmi, N., Protasov, V.: Exact computation of joint spectral characteristics of linear operators. Found. Comput. Math. 13, 37–97 (2013)

    Article  MathSciNet  Google Scholar 

  7. Jungers, R.M., Protasov, V.Y., Blondel, V.D.: Overlap-free words and spectra of matrices. Theoret. Comput. Sci. 410, 3670–3684 (2009)

    Article  MathSciNet  Google Scholar 

  8. Karhumäki, J., Shallit, J.: Polynomial versus exponential growth in repetition-free binary words. J. Combin. Theory. Ser. A 104, 335–347 (2004)

    Article  MathSciNet  Google Scholar 

  9. Kolpakov, R., Rao, M.: On the number of Dejean words over alphabets of 5, 6, 7, 8, 9 and 10 letters. Theoret. Comput. Sci. 412, 6507–6516 (2011)

    Article  MathSciNet  Google Scholar 

  10. Ochem, P.: A generator of morphisms for infinite words. RAIRO Inform. Théor App. 40, 427–441 (2006)

    Article  MathSciNet  Google Scholar 

  11. Petrova, E.A., Shur, A.M.: Constructing premaximal binary cube-free words of any level. Internat. J. Found. Comp. Sci. 23(8), 1595–1609 (2012)

    Article  MathSciNet  Google Scholar 

  12. Petrova, E.A., Shur, A.M.: Constructing premaximal ternary square-free words of any level. In: Proc. 37Th Internat. Conf. on Mathematical Foundations of Computer Science. MFCS 2012, LNCS, vol. 7464, pp 752–763 (2012)

  13. Petrova, E.A., Shur, A.M.: On the tree of ternary square-free words. In: Combinatorics on Words - 10Th Internat. Conf. WORDS 2015, Proceedings, LNCS, vol. 9304, pp 223–236. Springer, Berlin (2015)

  14. Petrova, E.A., Shur, A.M.: On the Tree of Binary Cube-Free Words Developments in Language Theory - 21St International Conference, DLT 2017, Proceedings, Lecture Notes in Computer Science, Vol. 10396, Pp. 296–307. Springer (2017)

  15. Petrova, E.A., Shur, A.M.: Transition property for cube-free words. In: Computer Science - Theory and Applications - 14Th International Computer Science Symposium in Russia, CSR 2019, Proceedings, Lecture Notes in Computer Science, vol. 11532, pp 311–324. Springer, Berlin (2019)

  16. Restivo, A., Salemi, S.: Some decision results on non-repetitive words. In: Apostolico, A., Galil, Z. (eds.) Combinatorial algorithms on words, NATO ASI series, vol. F12, pp 289–295. Springer, Berlin (1985)

  17. Shallit, J., Shur, A.: Subword complexity and pattern avoidance. Theoret. Comput. Sci. https://doi.org/10.1016/j.tcs.2018.09.010 (2018)

  18. Shur, A.M.: Two-sided bounds for the growth rates of power-free languages. In: Proc. 13Th Int. Conf. on Developments in Language Theory. DLT 2009, LNCS, vol. 5583, pp 466–477. Springer, Berlin (2009)

  19. Shur, A.M.: Deciding context equivalence of binary overlap-free words in linear time. Semigroup Forum 84, 447–471 (2012)

    Article  MathSciNet  Google Scholar 

  20. Thue, A.: Über unendliche Zeichenreihen. Norske Vid. Selsk. Skr. Mat. Nat. Kl. 7, 1–22 (1906)

    MATH  Google Scholar 

  21. Thue, A.: Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. Mat. Nat. Kl. 1, 1–67 (1912)

    MATH  Google Scholar 

  22. Tunev, I.N., Shur, A.M.: On Two stronger versions of Dejean’s conjecture. In: Proc. 37Th Internat. Conf. on Mathematical Foundations of Computer Science. MFCS 2012, LNCS, vol. 7464 , pp 801–813. Springer, Switzerland (2012)

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  1. Ural Federal University, Ekaterinburg, Russia

    Elena A. Petrova & Arseny M. Shur

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  1. Elena A. Petrova
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Correspondence to Elena A. Petrova.

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This article belongs to the Topical Collection: Special Issue on Computer Science Symposium in Russia (2019)

Guest Editor: Gregory Kucherov

E.A. Petrova — Supported by the Russian Science Foundation, grant 18-71-00043.

This paper extends the conference paper [15] with full proofs and other details omitted in [15] due to space constraints.

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Petrova, E.A., Shur, A.M. Transition Property for Cube-Free Words. Theory Comput Syst 65, 479–496 (2021). https://doi.org/10.1007/s00224-020-09979-4

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