Hereditary properties of words

József Balogh; Béla Bollobás

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2005)

  • Volume: 39, Issue: 1, page 49-65
  • ISSN: 0988-3754

Abstract

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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 n words of length n then, for every k n + m , it contains at most ( m + 1 ) / 2 ( m + 1 ) / 2 words of length k .

How to cite

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Balogh, József, and Bollobás, Béla. "Hereditary properties of words." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 39.1 (2005): 49-65. <http://eudml.org/doc/244882>.

@article{Balogh2005,
abstract = {Let $\{\mathcal \{P\}\}$ be a hereditary property of words, i.e., an infinite class of finite words such that every subword (block) of a word belonging to $\{\mathcal \{P\}\}$ is also in $\{\mathcal \{P\}\}$. Extending the classical Morse-Hedlund theorem, we show that either $\{\mathcal \{P\}\}$ 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 $\{\mathcal \{P\}\}$ has $m\le n$ words of length $n$ then, for every $k\ge n+m$, it contains at most $\lceil (m+1)/2\rceil \lfloor (m+1)/2 \rfloor $ words of length $k$.},
author = {Balogh, József, Bollobás, Béla},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},
keywords = {graph properties; monotone; hereditary; speed; size; graph property; dynamic symbolic},
language = {eng},
number = {1},
pages = {49-65},
publisher = {EDP-Sciences},
title = {Hereditary properties of words},
url = {http://eudml.org/doc/244882},
volume = {39},
year = {2005},
}

TY - JOUR
AU - Balogh, József
AU - Bollobás, Béla
TI - Hereditary properties of words
JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY - 2005
PB - EDP-Sciences
VL - 39
IS - 1
SP - 49
EP - 65
AB - Let ${\mathcal {P}}$ be a hereditary property of words, i.e., an infinite class of finite words such that every subword (block) of a word belonging to ${\mathcal {P}}$ is also in ${\mathcal {P}}$. Extending the classical Morse-Hedlund theorem, we show that either ${\mathcal {P}}$ 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 ${\mathcal {P}}$ has $m\le n$ words of length $n$ then, for every $k\ge n+m$, it contains at most $\lceil (m+1)/2\rceil \lfloor (m+1)/2 \rfloor $ words of length $k$.
LA - eng
KW - graph properties; monotone; hereditary; speed; size; graph property; dynamic symbolic
UR - http://eudml.org/doc/244882
ER -

References

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  2. [2] S. Ferenczi, Complexity of sequences and dynamical systems. Discrete Math. 206 (1999) 145–154. Zbl0936.37008
  3. [3] N.J. Fine and H.S. Wilf, Uniqueness theorems for periodic functions. Proc. Amer. Math. Soc. 16 (1965) 109–114. Zbl0131.30203
  4. [4] A. Heinis, The P ( n ) / n -function for bi-infinite words. Theoret. Comput. Sci. 273 (2002) 35–46. Zbl1027.68654
  5. [5] T. Kamae and L. Zamboni, Sequence entropy and the maximal pattern complexity of infinite words. Ergodic Theory Dynam. Syst. 22 (2002) 1191–1199. Zbl1014.37004
  6. [6] M. Morse and A.G. Hedlund, Symbolic dynamics. Amer. J. Math 60 (1938) 815–866. Zbl0019.33502JFM64.0798.04
  7. [7] R. Tijdeman, Periodicity and almost periodicity. Preprint, www.math.leidenuniv/ tijdeman Zbl1103.68103MR2223402

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