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Binary patterns in binary cube-free words: Avoidability and growth∗

Published online by Cambridge University Press:  17 July 2014

Robert Mercaş ,
Pascal Ochem ,
Alexey V. Samsonov  and
Arseny M. Shur
Robert Mercaş
Affiliation:
Christian-Albrechts-Universität zu Kiel, Institut für Informatik, Germany.. rgm@informatik.uni-kiel.de
Pascal Ochem
Affiliation:
CNRS, LIRMM, France.; Pascal.Ochem@lirmm.fr
Alexey V. Samsonov
Affiliation:
Ural Federal University, Ekaterinburg, Russia.; vonosmas@gmail.com ; Arseny.Shur@usu.ru ;
Arseny M. Shur
Affiliation:
Ural Federal University, Ekaterinburg, Russia.; vonosmas@gmail.com ; Arseny.Shur@usu.ru ;
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Abstract

The avoidability of binary patterns by binary cube-free words is investigated and the exact bound between unavoidable and avoidable patterns is found. All avoidable patterns are shown to be D0L-avoidable. For avoidable patterns, the growth rates of the avoiding languages are studied. All such languages, except for the overlap-free language, are proved to have exponential growth. The exact growth rates of languages avoiding minimal avoidable patterns are approximated through computer-assisted upper bounds. Finally, a new example of a pattern-avoiding language of polynomial growth is given.

Keywords

Formal languagesavoidabilityavoidable patterncube-free wordoverlap-free wordgrowth ratemorphism

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References

Growth-rate-calculator. Library for calculating growth rates of factorial formal languages (2013). Available at http://code.google.com/p/growth-rate-calculator/.
G. Badkobeh, S. Chairungsee and M. Crochemore, Hunting redundancies in strings. In Proc. 15th Developments in Language Theory. DLT 2011, Vol.6795 of Lect. Notes Sci. Springer, Berlin (2011) 1–14.
Baker, K.A., McNulty, G.F. and Taylor, W., Growth problems for avoidable words. Theoret. Comput. Sci. 69 (1989) 319–345. Google Scholar
Bean, D.A., Ehrenfeucht, A. and McNulty, G., Avoidable patterns in strings of symbols. Pacific J. Math. 85 (1979) 261–294. Google Scholar
Bell, J.P. and Goh, T.L., Exponential lower bounds for the number of words of uniform length avoiding a pattern. Inform. Comput. 205 (2007) 1295–1306. Google Scholar
Blondel, V.D., Cassaigne, J. and Jungers, R., On the number of α-power-free binary words for 2 <α ≤ 7/3. Theoret. Comput. Sci. 410 (2009) 2823–2833. Google Scholar
Brandenburg, F.-J., Uniformly growing k-th power-free homomorphisms. Theoret. Comput. Sci. 23 (1983) 69–82. Google Scholar
Cassaigne, J., Unavoidable binary patterns. Acta Informatica 30 (1993) 385–395. Google Scholar
J. Cassaigne, Motifs évitables et régularités dans les mots (Thèse de Doctorat). Tech. Report. LITP-TH 94-04 (1994).
C. Choffrut and J. Karhumäki, Combinatorics of words. In vol. 1 of Handbook of Formal Languages, edited by G. Rozenberg and A. Salomaa. Springer-Verlag (1997) 329–438.
Goralcik, P. and Vanicek, T., Binary patterns in binary words. Internat. J. Algebra Comput. 1 (1991) 387–391. Google Scholar
Karhumäki, J. and Shallit, J., Polynomial versus exponential growth in repetition-free binary words. J. Combin. Theory. Ser. A 104 (2004) 335–347. Google Scholar
Ochem, P., A generator of morphisms for infinite words. RAIRO: ITA 40 (2006) 427–441. Google Scholar
Ochem, P., Binary words avoiding the pattern AABBCABBA. RAIRO: ITA 44 (2010) 151–158. Google Scholar
Petrov, A.N., Sequence avoiding any complete word. Mathematical Notes of the Academy of Sciences of the USSR 44 (1988) 764–767. Google Scholar
A. Restivo and S. Salemi, Overlap free words on two symbols.In Automata on Infinite Words, edited by M. Nivat and D. Perrin, Ecole de Printemps d’Informatique Théorique, Le Mont Dore, 1984, vol. 192 of Lect. Notes Sci. Springer-Verlag (1985) 198–206.
G. Richomme and F. Wlazinski, About cube-free morphisms. In STACS 2000, Proc. 17th Symp. Theoretical Aspects of Comp. Sci., vol. 1770 of Lect. Notes Sci. Edited by H. Reichel and S. Tison. Springer-Verlag (2000) 99–109.
Roth, P., Every binary pattern of length six is avoidable on the two-letter alphabet. Acta Informatica 29 (1992) 95–107. Google Scholar
A.V. Samsonov and A.M. Shur, Binary patterns in binary cube-free words: Avoidability and growth. In Proc. 14th Mons Days of Theoretical Computer Science. Univ. catholique de Louvain, Louvain-la-Neuve (2012) 1–7. electronic.
Shur, A.M., Binary words avoided by the Thue–Morse sequence. Semigroup Forum 53 (1996) 212–219. Google Scholar
Shur, A.M., Comparing complexity functions of a language and its extendable part. RAIRO: ITA 42 (2008) 647–655. Google Scholar
Shur, A.M., Growth rates of complexity of power-free languages. Theoret. Comput. Sci. 411 (2010) 3209–3223. Google Scholar
Shur, A.M., Growth properties of power-free languages. Comput. Sci. Rev. 6 (2012) 187–208. Google Scholar
Thue, A., Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske vid. Selsk. Skr. Mat. Nat. Kl. 1 (1912) 1–67. Google Scholar
Zimin, A.I., Blocking sets of terms. Mat. Sbornik 119 (1982) 363–375; In Russian. English translation in Math. USSR Sbornik 47 (1984) 353–364. Google Scholar