Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-11T00:13:37.276Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on constructing infinite binary words with polynomial subword complexity∗

Published online by Cambridge University Press:  25 April 2013

Francine Blanchet-Sadri ,
Bob Chen  and
Sinziana Munteanu
Francine Blanchet-Sadri
Affiliation:
Department of Computer Science, University of North Carolina, PO Box 26170, Greensboro, NC 27402–6170, USA. blanchet@uncg.edu
Bob Chen
Affiliation:
Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, Dept 0112, LaJolla, CA 92093–0112, USA
Sinziana Munteanu
Affiliation:
School of Computer Science, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213–3891, USA
Get access

Abstract

Most of the constructions of infinite words having polynomial subword complexity are quite complicated, e.g., sequences of Toeplitz, sequences defined by billiards in the cube, etc. In this paper, we describe a simple method for constructing infinite words w over a binary alphabet  { a,b }  with polynomial subword complexity pw. Assuming w contains an infinite number of a’s, our method is based on the gap function which gives the distances between consecutive b’s. It is known that if the gap function is injective, we can obtain at most quadratic subword complexity, and if the gap function is blockwise injective, we can obtain at most cubic subword complexity. Here, we construct infinite binary words w such that pw(n) = Θ(nβ) for any real number β > 1.

Keywords

Binary wordssubword complexitygap function
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 47 , Issue 2 , April 2013 , pp. 195 - 199
Copyright
© EDP Sciences 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allouche, J.-P., Sur la complexité des suites infinies. Bull. Belgian Math. Soc. 1 (1994) 133–143. Google Scholar
J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press (2003).
Arnoux, P., Mauduit, C., Shiokawa, I. and Tamura, J., Complexity of sequences defined by billiards in the cube. Bull. Soc. Math. France 122 (1994) 1–12. Google Scholar
Arnoux, P., Mauduit, C., Shiokawa, I. and Tamura, J., Rauzy’s conjecture on billiards in the cube. Tokyo J. Math. 17 (1994) 211–218. Google Scholar
Arnoux, P. and Rauzy, G., Représentation géométrique de suites de complexité 2n + 1. Bull. Soc. Math. France 119 (1991) 199–215. Google Scholar
Baryshnikov, Y., Complexity of trajectories in rectangular billiards. Communicat. Math. Phys. 174 (1995) 43–56. Google Scholar
Bedaride, N., Directional billiard complexity in rational polyhedra. Regular Chaotic Dyn. 3 (2003) 97–106. Google Scholar
Blanchet-Sadri, F., Chakarov, A., Manuelli, L., Schwartz, J. and Stich, S., Constructing partial words with subword complexities not achievable by full words. Theoret. Comput. Sci. 432 (2012) 21–27. Google Scholar
Cassaigne, J., Complexité et facteurs spéciaux. Bull. Belg. Math. Soc. 4 (1997) 67–88. Google Scholar
Cassaigne, J., Constructing infinite words of intermediate complexity. In DLT 2002, 6th International Conference on Developments in Language Theory, Kyoto, Japan. Springer-Verlag, Berlin. Lect. Notes Comput. Sci. 2450 (2003) 173–184. Google Scholar
Cassaigne, J. and Karhumäki, J., Toeplitz words, generalized periodicity and periodically iterated morphisms. European J. Combin. 18 (1997) 497–510. Google Scholar
Ferenczi, S., Complexity of sequences and dynamical systems. Discrete Math. 206 (1999) 145–154. Google Scholar
Gheorghiciuc, I., The subword complexity of a class of infinite binary words. Adv. Appl. Math. 39 (2007) 237–259. Google Scholar
Koskas, M., Complexité des suites de Toeplitz. Discrete Math. 183 (1998) 161–183. Google Scholar
M. Lothaire, Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002).
Morse, M. and Hedlund, G.A., Symbolic dynamics. Amer. J. Math. 60 (1938) 815–866. Google Scholar
Morse, M. and Hedlund, G.A., Symbolic dynamics II. Sturmian trajectories. Amer. J. Math. 62 (1940) 1–42. Google Scholar
Rote, G., Sequences with subword complexity 2n. J. Number Theory 46 (1993) 196–213. Google Scholar