Hereditary properties of words

Balogh, József; Bollobás, Béla

doi:10.1051/ita:2005003

Balogh, József ; Bollobás, Béla

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 39 (2005) no. 1, pp. 49-65

Résumé

Let $𝒫$ be a hereditary property of words, i.e., an infinite class of finite words such that every subword (block) of a word belonging to $𝒫$ is also in $𝒫$ . Extending the classical Morse-Hedlund theorem, we show that either $𝒫$ contains at least $n + 1$ words of length $n$ for every $n$ or, for some $N$ , it contains at most $N$ words of length $n$ for every $n$ . More importantly, we prove the following quantitative extension of this result: if $𝒫$ has $m \leq n$ words of length $n$ then, for every $k \geq n + m$ , it contains at most $⌈ (m + 1) / 2 ⌉ ⌊ (m + 1) / 2 ⌋$ words of length $k$ .

EuDML MR Zbl

DOI : 10.1051/ita:2005003

Classification : 05C
Keywords: graph properties, monotone, hereditary, speed, size

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     title = {Hereditary properties of words},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {49--65},
     year = {2005},
     publisher = {EDP Sciences},
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Balogh, József; Bollobás, Béla. Hereditary properties of words. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 39 (2005) no. 1, pp. 49-65. doi: 10.1051/ita:2005003

Bibliographie
Cité par

[1] S. Ferenczi, Rank and symbolic complexity. Ergodic Theory Dyn. Syst. 16 (1996) 663-682. | Zbl

[2] S. Ferenczi, Complexity of sequences and dynamical systems. Discrete Math. 206 (1999) 145-154. | Zbl

[3] N.J. Fine and H.S. Wilf, Uniqueness theorems for periodic functions. Proc. Amer. Math. Soc. 16 (1965) 109-114. | Zbl

[4] A. Heinis, The $P (n) / n$ -function for bi-infinite words. Theoret. Comput. Sci. 273 (2002) 35-46. | Zbl

[5] T. Kamae and L. Zamboni, Sequence entropy and the maximal pattern complexity of infinite words. Ergodic Theory Dynam. Syst. 22 (2002) 1191-1199. | Zbl

[6] M. Morse and A.G. Hedlund, Symbolic dynamics. Amer. J. Math 60 (1938) 815-866. | JFM

[7] R. Tijdeman, Periodicity and almost periodicity | Zbl | MR

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