On automatic infinite permutations∗

Anna Frid; Luca Zamboni

RAIRO - Theoretical Informatics and Applications (2012)

  • Volume: 46, Issue: 1, page 77-85
  • ISSN: 0988-3754

Abstract

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An infinite permutation α is a linear ordering of N. We study properties of infinite permutations analogous to those of infinite words, and show some resemblances and some differences between permutations and words. In this paper, we try to extend to permutations the notion of automaticity. As we shall show, the standard definitions which are equivalent in the case of words are not equivalent in the context of permutations. We investigate the relationships between these definitions and prove that they constitute a chain of inclusions. We also construct and study an automaton generating the Thue-Morse permutation.

How to cite

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Frid, Anna, and Zamboni, Luca. "On automatic infinite permutations∗." RAIRO - Theoretical Informatics and Applications 46.1 (2012): 77-85. <http://eudml.org/doc/277836>.

@article{Frid2012,
abstract = {An infinite permutation α is a linear ordering of N. We study properties of infinite permutations analogous to those of infinite words, and show some resemblances and some differences between permutations and words. In this paper, we try to extend to permutations the notion of automaticity. As we shall show, the standard definitions which are equivalent in the case of words are not equivalent in the context of permutations. We investigate the relationships between these definitions and prove that they constitute a chain of inclusions. We also construct and study an automaton generating the Thue-Morse permutation.},
author = {Frid, Anna, Zamboni, Luca},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Permutation; infinite permutation; ordering; infinite word; automatic word; automatic permutation; Thue-Morse word; Thue-Morse permutation; permutation},
language = {eng},
month = {3},
number = {1},
pages = {77-85},
publisher = {EDP Sciences},
title = {On automatic infinite permutations∗},
url = {http://eudml.org/doc/277836},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Frid, Anna
AU - Zamboni, Luca
TI - On automatic infinite permutations∗
JO - RAIRO - Theoretical Informatics and Applications
DA - 2012/3//
PB - EDP Sciences
VL - 46
IS - 1
SP - 77
EP - 85
AB - An infinite permutation α is a linear ordering of N. We study properties of infinite permutations analogous to those of infinite words, and show some resemblances and some differences between permutations and words. In this paper, we try to extend to permutations the notion of automaticity. As we shall show, the standard definitions which are equivalent in the case of words are not equivalent in the context of permutations. We investigate the relationships between these definitions and prove that they constitute a chain of inclusions. We also construct and study an automaton generating the Thue-Morse permutation.
LA - eng
KW - Permutation; infinite permutation; ordering; infinite word; automatic word; automatic permutation; Thue-Morse word; Thue-Morse permutation; permutation
UR - http://eudml.org/doc/277836
ER -

References

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  1. J.-P. Allouche and J. Shallit, Automatic sequences – theory, applications, generalizations. Cambridge University Press (2003).  
  2. J.-P. Allouche, N. Rampersad and J. Shallit, Periodicity, repetitions, and orbits of an automatic sequence. Theoret. Comput. Sci.410 (2009) 2795–2803.  
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  7. J.A. Davis, R.C. Entringer, R.L. Graham and G.J. Simmons, On permutations containing no long arithmetic progressions. Acta Arith.34 (1977) 81–90.  
  8. S. Eilenberg, Automata, Languages, and MachinesA. Academic Press (1974).  
  9. D.G. Fon-Der-Flaass and A.E. Frid, On periodicity and low complexity of infinite permutations. Eur. J. Comb.28 (2007) 2106–2114.  
  10. M. Makarov, On permutations generated by infinite binary words. Sib. Èlectron. Mat. Izv.3 (2006) 304–311 (in Russian, English abstract).  
  11. M. Makarov, On an infinite permutation similar to the Thue-Morse word. Discrete Math.309 (2009) 6641–6643.  
  12. M. Makarov, On the permutations generated by Sturmian words. Sib. Math. J.50 (2009) 674–680.  
  13. M. Makarov, On the infinite permutation generated by the period doubling word. Eur. J. Comb.31 (2010) 368–378.  
  14. S. Widmer, Permutation complexity of the Thue-Morse word. Adv. Appl. Math.47 (2011) 309–329.  

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