Hereditary properties of words

József Balogh; Béla Bollobás

RAIRO - Theoretical Informatics and Applications (2010)

  • 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 39.1 (2010): 49-65. <http://eudml.org/doc/92765>.

@article{Balogh2010,
abstract = { 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. },
author = {Balogh, József, Bollobás, Béla},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = { Graph properties; monotone; hereditary; speed; size.; graph property; hereditary; dynamic symbolic},
language = {eng},
month = {3},
number = {1},
pages = {49-65},
publisher = {EDP Sciences},
title = {Hereditary properties of words},
url = {http://eudml.org/doc/92765},
volume = {39},
year = {2010},
}

TY - JOUR
AU - Balogh, József
AU - Bollobás, Béla
TI - Hereditary properties of words
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/3//
PB - EDP Sciences
VL - 39
IS - 1
SP - 49
EP - 65
AB - 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.
LA - eng
KW - Graph properties; monotone; hereditary; speed; size.; graph property; hereditary; dynamic symbolic
UR - http://eudml.org/doc/92765
ER -

References

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  2. S. Ferenczi, Complexity of sequences and dynamical systems. Discrete Math.206 (1999) 145–154.  
  3. N.J. Fine and H.S. Wilf, Uniqueness theorems for periodic functions. Proc. Amer. Math. Soc.16 (1965) 109–114.  
  4. A. Heinis, The P(n)/n-function for bi-infinite words. Theoret. Comput. Sci.273 (2002) 35–46.  
  5. T. Kamae and L. Zamboni, Sequence entropy and the maximal pattern complexity of infinite words. Ergodic Theory Dynam. Syst.22 (2002) 1191–1199.  
  6. M. Morse and A.G. Hedlund, Symbolic dynamics. Amer. J. Math60 (1938) 815–866.  
  7. R. Tijdeman, Periodicity and almost periodicity. Preprint, www.math.leidenuniv/~tijdeman  

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