On synchronized sequences and their separators
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 35 (2001) no. 6, p. 513-524

We introduce the notion of a $k$-synchronized sequence, where $k$ is an integer larger than 1. Roughly speaking, a sequence of natural numbers is said to be $k$-synchronized if its graph is represented, in base $k$, by a right synchronized rational relation. This is an intermediate notion between $k$-automatic and $k$-regular sequences. Indeed, we show that the class of $k$-automatic sequences is equal to the class of bounded $k$-synchronized sequences and that the class of $k$-synchronized sequences is strictly contained in that of $k$-regular sequences. Moreover, we show that equality of factors in a $k$-synchronized sequence is represented, in base $k$, by a right synchronized rational relation. This result allows us to prove that the separator sequence of a $k$-synchronized sequence is a $k$-synchronized sequence, too. This generalizes a previous result of Garel, concerning $k$-regularity of the separator sequences of sequences generated by iterating a uniform circular morphism.

Classification:  68Q45,  68R15
Keywords: regular sequence, automatic sequence, separator
@article{ITA_2001__35_6_513_0,
author = {Carpi, Arturo and Maggi, Cristiano},
title = {On synchronized sequences and their separators},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
publisher = {EDP-Sciences},
volume = {35},
number = {6},
year = {2001},
pages = {513-524},
zbl = {1003.68064},
mrnumber = {1922292},
language = {en},
url = {http://www.numdam.org/item/ITA_2001__35_6_513_0}
}

Carpi, Arturo; Maggi, Cristiano. On synchronized sequences and their separators. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 35 (2001) no. 6, pp. 513-524. http://www.numdam.org/item/ITA_2001__35_6_513_0/

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