On the growth rates of complexity of threshold languages
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 44 (2010) no. 1, pp. 175-192.

Threshold languages, which are the (k/(k-1))+-free languages over k-letter alphabets with k ≥ 5, are the minimal infinite power-free languages according to Dejean's conjecture, which is now proved for all alphabets. We study the growth properties of these languages. On the base of obtained structural properties and computer-assisted studies we conjecture that the growth rate of complexity of the threshold language over k letters tends to a constant α ^≈1.242 as k tends to infinity.

DOI : 10.1051/ita/2010012
Classification : 68Q70, 68R15
Mots-clés : power-free languages, Dejean's conjecture, threshold languages, combinatorial complexity, growth rate 
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Shur, Arseny M.; Gorbunova, Irina A. On the growth rates of complexity of threshold languages. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 44 (2010) no. 1, pp. 175-192. doi : 10.1051/ita/2010012. http://archive.numdam.org/articles/10.1051/ita/2010012/

[1] F.-J. Brandenburg, Uniformly growing k-th power free homomorphisms. Theoret. Comput. Sci. 23 (1983) 69-82. | Zbl

[2] A. Carpi, On Dejean's conjecture over large alphabets. Theoret. Comput. Sci. 385 (2007) 137-151. | Zbl

[3] C. Choffrut, J. Karhumäki, Combinatorics of words, edited by G. Rosenberg and A. Salomaa. Handbook of formal languages, Vol. 1, Chap. 6. Springer, Berlin (1997) 329-438.

[4] M. Crochemore, F. Mignosi and A. Restivo, Automata and forbidden words. Inform. Process. Lett. 67 (1998) 111-117.

[5] J.D. Currie, N. Rampersad, Dejean's conjecture holds for n ≥ 27. RAIRO-Theor. Inf. Appl. 43 (2009) 775-778. | Zbl

[6] J.D. Currie, N. Rampersad, A proof of Dejean's conjecture, http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.1129v3.pdf

[7] F. Dejean, Sur un Theoreme de Thue. J. Combin. Theory Ser. A 13 (1972) 90-99. | Zbl

[8] A. Ehrenfeucht and G. Rozenberg, On subword complexities of homomorphic images of languages. RAIRO Inform. Theor. 16 (1982) 303-316. | Numdam | Zbl

[9] M. Lothaire, Combinatorics on words. Addison-Wesley (1983). | Zbl

[10] M. Mohammad-Noori and J.D. Currie, Dejean's conjecture and Sturmian words. Eur. J. Combin. 28 (2007) 876-890. | Zbl

[11] J. Moulin-Ollagnier, Proof of Dejean's Conjecture for Alphabets with 5, 6, 7, 8, 9, 10 and 11 Letters. Theoret. Comput. Sci. 95 (1992) 187-205. | Zbl

[12] J.-J. Pansiot, À propos d'une conjecture de F. Dejean sur les répétitions dans les mots. Discrete Appl. Math. 7 (1984) 297-311. | Zbl

[13] M. Rao, Last Cases of Dejean's Conjecture, accepted to WORDS'2009.

[14] A.M. Shur, Rational approximations of polynomial factorial languages. Int. J. Found. Comput. Sci. 18 (2007) 655-665. | Zbl

[15] A.M. Shur, Combinatorial complexity of regular languages, Proceedings of CSR'2008. Lect. Notes Comput. Sci. 5010 (2008) 289-301. | Zbl

[16] A.M. Shur, Growth rates of complexity of power-free languages. Submitted to Theoret. Comput. Sci. (2008). | Zbl

[17] A.M. Shur, Comparing complexity functions of a language and its extendable part. RAIRO-Theor. Inf. Appl. 42 (2008) 647-655. | EuDML | Numdam | Zbl

[18] A. Thue, Über unendliche Zeichenreihen, Kra. Vidensk. Selsk. Skrifter. I. Mat.-Nat. Kl., Christiana 7 (1906) 1-22. | JFM

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