On synchronized sequences and their separators

Arturo Carpi; Cristiano Maggi

RAIRO - Theoretical Informatics and Applications (2010)

  • Volume: 35, Issue: 6, page 513-524
  • ISSN: 0988-3754

Abstract

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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.

How to cite

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Carpi, Arturo, and Maggi, Cristiano. "On synchronized sequences and their separators." RAIRO - Theoretical Informatics and Applications 35.6 (2010): 513-524. <http://eudml.org/doc/221953>.

@article{Carpi2010,
abstract = { 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. },
author = {Carpi, Arturo, Maggi, Cristiano},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Regular sequence; automatic sequence; separator; -synchronized sequence},
language = {eng},
month = {3},
number = {6},
pages = {513-524},
publisher = {EDP Sciences},
title = {On synchronized sequences and their separators},
url = {http://eudml.org/doc/221953},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Carpi, Arturo
AU - Maggi, Cristiano
TI - On synchronized sequences and their separators
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 6
SP - 513
EP - 524
AB - 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.
LA - eng
KW - Regular sequence; automatic sequence; separator; -synchronized sequence
UR - http://eudml.org/doc/221953
ER -

References

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  5. S. Eilenberg, Automata, Languages and Machines, Vol. A. Academic Press, New York (1974).  
  6. C.C. Elgot and J.E. Mezei, On relations defined by generalized finite automata. IBM J. Res. Develop.9 (1965) 47-68.  
  7. C. Frougny, Numeration Systems, in Algebraic Combinatorics on Words. CambridgeUniversity Press (to appear).  
  8. C. Frougny and J. Sakarovitch, Synchronized rational relations of finite and infinite words. Theoret. Comput. Sci.108 (1993) 45-82.  
  9. E. Garel, Séparateurs dans les mots infinis engendrés par morphismes. Theoret. Comput. Sci.180 (1997) 81-113.  
  10. C. Pomerance, J.M. Robson and J. Shallit, Automaticity II: Descriptional complexity in the unary case. Theoret. Comput. Sci.180 (1997) 181-201.  

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