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On synchronized sequences and their separators

Published online by Cambridge University Press:  15 July 2002

Arturo Carpi
Dipartimento di Matematica e Informatica, Università di Perugia, via Vanvitelli 1, 06123 Perugia, Italy; (
Cristiano Maggi
Dipartimento di Matematica, Università “La Sapienza”, Roma, Italy.
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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.

Research Article
© EDP Sciences, 2001

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Allouche, J.-P. and Shallit, J., The ring of k-regular sequences. Theoret. Comput. Sci. 98 (1992) 163-197. CrossRef
G. Christol, Ensembles presque périodiques k-reconnaissables Theoret. Comput. Sci. 9 (1979) 141-145. CrossRef
Christol, G., Kamae, T., Mendès France, M. and Rauzy, G., Suites algébriques, automates et substitutions. Bull. Soc. Math. France 108 (1980) 401-419. CrossRef
Cobham, A., Uniform tag sequences. Math. Systems Theory 6 (1972) 164-192. CrossRef
S. Eilenberg, Automata, Languages and Machines, Vol. A. Academic Press, New York (1974).
Elgot, C.C. and Mezei, J.E., On relations defined by generalized finite automata. IBM J. Res. Develop. 9 (1965) 47-68. CrossRef
C. Frougny, Numeration Systems, in Algebraic Combinatorics on Words. CambridgeUniversity Press (to appear).
Frougny, C. and Sakarovitch, J., Synchronized rational relations of finite and infinite words. Theoret. Comput. Sci. 108 (1993) 45-82. CrossRef
Garel, E., Séparateurs dans les mots infinis engendrés par morphismes. Theoret. Comput. Sci. 180 (1997) 81-113. CrossRef
Pomerance, C., Robson, J.M. and Shallit, J., Automaticity II: Descriptional complexity in the unary case. Theoret. Comput. Sci. 180 (1997) 181-201. CrossRef