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

Robert Mercaş; Pascal Ochem; Alexey V. Samsonov; Arseny M. Shur

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2014)

- Volume: 48, Issue: 4, page 369-389
- ISSN: 0988-3754

## Access Full Article

top## Abstract

top## How to cite

topMercaş, Robert, et al. "Binary patterns in binary cube-free words: Avoidability and growth." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 48.4 (2014): 369-389. <http://eudml.org/doc/273030>.

@article{Mercaş2014,

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.},

author = {Mercaş, Robert, Ochem, Pascal, Samsonov, Alexey V., Shur, Arseny M.},

journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},

keywords = {formal languages; avoidability; avoidable pattern; cube-free word; overlap-free word; growth rate; morphism},

language = {eng},

number = {4},

pages = {369-389},

publisher = {EDP-Sciences},

title = {Binary patterns in binary cube-free words: Avoidability and growth},

url = {http://eudml.org/doc/273030},

volume = {48},

year = {2014},

}

TY - JOUR

AU - Mercaş, Robert

AU - Ochem, Pascal

AU - Samsonov, Alexey V.

AU - Shur, Arseny M.

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

JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

PY - 2014

PB - EDP-Sciences

VL - 48

IS - 4

SP - 369

EP - 389

AB - 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.

LA - eng

KW - formal languages; avoidability; avoidable pattern; cube-free word; overlap-free word; growth rate; morphism

UR - http://eudml.org/doc/273030

ER -

## References

top- [1] Growth-rate-calculator. Library for calculating growth rates of factorial formal languages (2013). Available at http://code.google.com/p/growth-rate-calculator/.
- [2] 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. Zbl1217.68164MR2862710
- [3] K.A. Baker, G.F. McNulty and W. Taylor, Growth problems for avoidable words. Theoret. Comput. Sci.69 (1989) 319–345. Zbl0693.68039MR1036470
- [4] D.A. Bean, A. Ehrenfeucht and G. McNulty, Avoidable patterns in strings of symbols. Pacific J. Math.85 (1979) 261–294. Zbl0428.05001MR574919
- [5] J.P. Bell and T.L. Goh, Exponential lower bounds for the number of words of uniform length avoiding a pattern. Inform. Comput.205 (2007) 1295–1306. Zbl1127.68073MR2334234
- [6] V.D. Blondel, J. Cassaigne and R. Jungers, On the number of α-power-free binary words for 2 <α ≤ 7/3. Theoret. Comput. Sci. 410 (2009) 2823–2833. Zbl1173.68046MR2543336
- [7] F.-J. Brandenburg, Uniformly growing k-th power-free homomorphisms. Theoret. Comput. Sci.23 (1983) 69–82. Zbl0508.68051MR693069
- [8] J. Cassaigne, Unavoidable binary patterns. Acta Informatica30 (1993) 385–395. Zbl0790.68096MR1227889
- [9] J. Cassaigne, Motifs évitables et régularités dans les mots (Thèse de Doctorat). Tech. Report. LITP-TH 94-04 (1994).
- [10] 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. Zbl0866.68057MR1469998
- [11] P. Goralcik and T. Vanicek, Binary patterns in binary words. Internat. J. Algebra Comput.1 (1991) 387–391. Zbl0759.68051MR1148238
- [12] J. Karhumäki and J. Shallit, Polynomial versus exponential growth in repetition-free binary words. J. Combin. Theory. Ser. A104 (2004) 335–347. Zbl1065.68080MR2046086
- [13] P. Ochem, A generator of morphisms for infinite words. RAIRO: ITA 40 (2006) 427–441. Zbl1110.68122MR2269202
- [14] P. Ochem, Binary words avoiding the pattern AABBCABBA. RAIRO: ITA 44 (2010) 151–158. Zbl1184.68377MR2604940
- [15] A.N. Petrov, Sequence avoiding any complete word. Mathematical Notes of the Academy of Sciences of the USSR44 (1988) 764–767. Zbl0677.20055MR975191
- [16] 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. Zbl0572.20038MR814744
- [17] 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. Zbl0959.68532MR1781724
- [18] P. Roth, Every binary pattern of length six is avoidable on the two-letter alphabet. Acta Informatica29 (1992) 95–107. Zbl0741.68083MR1154583
- [19] 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. Zbl1302.68228
- [20] A.M. Shur, Binary words avoided by the Thue–Morse sequence. Semigroup Forum53 (1996) 212–219. Zbl0859.20048MR1400647
- [21] A.M. Shur, Comparing complexity functions of a language and its extendable part. RAIRO: ITA 42 (2008) 647–655. Zbl1149.68055MR2434040
- [22] A.M. Shur, Growth rates of complexity of power-free languages. Theoret. Comput. Sci.411 (2010) 3209–3223. Zbl1196.68121MR2676864
- [23] A.M. Shur, Growth properties of power-free languages. Comput. Sci. Rev.6 (2012) 187–208. Zbl1298.68157MR2862712
- [24] A. Thue, Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske vid. Selsk. Skr. Mat. Nat. Kl. 1 (1912) 1–67. Zbl44.0462.01JFM44.0462.01
- [25] A.I. Zimin, Blocking sets of terms. Mat. Sbornik 119 (1982) 363–375; In Russian. English translation in Math. USSR Sbornik 47 (1984) 353–364. Zbl0599.20106MR678833