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On the growth rates of complexity of threshold languages

Published online by Cambridge University Press:  11 February 2010

Arseny M. Shur
Ural State University, Ekaterinburg, Russia;
Irina A. Gorbunova
Ural State University, Ekaterinburg, Russia;
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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 $\hat{\alpha}\approx1.242$ as k tends to infinity.

Research Article
© EDP Sciences, 2010

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Brandenburg, F.-J., Uniformly growing k-th power free homomorphisms. Theoret. Comput. Sci. 23 (1983) 6982. CrossRef
Carpi, A., Dejean's, On conjecture over large alphabets. Theoret. Comput. Sci. 385 (2007) 137151. CrossRef
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.
Crochemore, M., Mignosi, F. and Restivo, A., Automata and forbidden words. Inform. Process. Lett. 67 (1998) 111117. CrossRef
Currie, J.D., Rampersad, N., Dejean's conjecture holds for n ≥ 27. RAIRO-Theor. Inf. Appl. 43 (2009) 775778. CrossRef
J.D. Currie, N. Rampersad, A proof of Dejean's conjecture,
Dejean, F., Sur un Theoreme de Thue. J. Combin. Theory Ser. A 13 (1972) 9099. CrossRef
Ehrenfeucht, A. and Rozenberg, G., On subword complexities of homomorphic images of languages. RAIRO Inform. Theor. 16 (1982) 303316. CrossRef
M. Lothaire, Combinatorics on words. Addison-Wesley (1983).
Mohammad-Noori, M. and Currie, J.D., Dejean's conjecture and Sturmian words. Eur. J. Combin. 28 (2007) 876890. CrossRef
Moulin-Ollagnier, J., Proof of Dejean's Conjecture for Alphabets with 5, 6, 7, 8, 9, 10 and 11 Letters. Theoret. Comput. Sci. 95 (1992) 187205. CrossRef
Pansiot, J.-J., À propos d'une conjecture de F. Dejean sur les répétitions dans les mots. Discrete Appl. Math. 7 (1984) 297311. CrossRef
M. Rao, Last Cases of Dejean's Conjecture, accepted to WORDS'2009.
Shur, A.M., Rational approximations of polynomial factorial languages. Int. J. Found. Comput. Sci. 18 (2007) 655665. CrossRef
Shur, A.M., Combinatorial complexity of regular languages, Proceedings of CSR'2008. Lect. Notes Comput. Sci. 5010 (2008) 289301. CrossRef
A.M. Shur, Growth rates of complexity of power-free languages. Submitted to Theoret. Comput. Sci. (2008).
Shur, A.M., Comparing complexity functions of a language and its extendable part. RAIRO-Theor. Inf. Appl. 42 (2008) 647655. CrossRef
A. Thue, Über unendliche Zeichenreihen, Kra. Vidensk. Selsk. Skrifter. I. Mat.-Nat. Kl., Christiana 7 (1906) 1–22.