Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-30T21:11:50.575Z Has data issue: false hasContentIssue false
  • English
  • Français

On automatic infinite permutations∗

Published online by Cambridge University Press:  23 November 2011

Anna Frid  and
Luca Zamboni
Anna Frid
Affiliation:
Sobolev Institute of Mathematics SB RAS, Koptyug av. 4, 630090 Novosibirsk, Russia.. [email protected] Institut Camille Jordan, Université Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France.; [email protected].
Luca Zamboni
Affiliation:
Institut Camille Jordan, Université Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France.; [email protected]. Department of Mathematics and Turku Centre for Computer Science, University of Turku, 20014 Turku, Finland.
Get access

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.

Keywords

Permutationinfinite permutationorderinginfinite wordautomatic wordautomatic permutationThue-Morse wordThue-Morse permutation
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 46 , Issue 1: Special issue dedicated to the 13th "Journées Montoises d'Informatique Théorique" , January 2012 , pp. 77 - 85
Copyright
© EDP Sciences 2011

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

J.-P. Allouche and J. Shallit, Automatic sequences – theory, applications, generalizations. Cambridge University Press (2003).
Allouche, J.-P., Rampersad, N. and Shallit, J., Periodicity, repetitions, and orbits of an automatic sequence. Theoret. Comput. Sci. 410 (2009) 2795–2803. Google Scholar
Avgustinovich, S., Frid, A., Kamae, T. and Salimov, P., Infinite permutations of lowest maximal pattern complexity. Theoret. Comput. Sci. 412 (2011) 2911–2921. Google Scholar
Charlier, É., Rampersad, N. and Shallit, J., Enumeration and Decidable Properties of Automatic Sequences, Lect. Notes Comput. Sci. 6795 (2011) 165–179. Google Scholar
Christol, G., Kamae, T., France, M.M. and Rauzy, G., Suites algébriques, automates et substitutions. Bull. Soc. Math. France 108 (1980) 401–419. Google Scholar
Cobham, A., Uniform tag sequences. Math. Syst. Theor. 6 (1972) 164–192. Google Scholar
Davis, J.A., Entringer, R.C., Graham, R.L. and Simmons, G.J., On permutations containing no long arithmetic progressions. Acta Arith. 34 (1977) 81–90. Google Scholar
S. Eilenberg, Automata, Languages, and Machines A. Academic Press (1974).
D.G. Fon-Der-Flaass and Frid, A.E., On periodicity and low complexity of infinite permutations. Eur. J. Comb. 28 (2007) 2106–2114. Google Scholar
Makarov, M., On permutations generated by infinite binary words. Sib. Èlectron. Mat. Izv. 3 (2006) 304–311 (in Russian, English abstract). Google Scholar
Makarov, M., On an infinite permutation similar to the Thue-Morse word. Discrete Math. 309 (2009) 6641–6643. Google Scholar
Makarov, M., On the permutations generated by Sturmian words. Sib. Math. J. 50 (2009) 674–680. Google Scholar
Makarov, M., On the infinite permutation generated by the period doubling word. Eur. J. Comb. 31 (2010) 368–378. Google Scholar
Widmer, S., Permutation complexity of the Thue-Morse word. Adv. Appl. Math. 47 (2011) 309–329. Google Scholar