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