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

top
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

top

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

top
  1. J.-P. Allouche and J. Shallit, The ring of k-regular sequences. Theoret. Comput. Sci.98 (1992) 163-197.  Zbl0774.68072
  2. G. Christol, Ensembles presque périodiques k-reconnaissables Theoret. Comput. Sci.9 (1979) 141-145.  
  3. G. Christol, T. Kamae, M. Mendès France and G. Rauzy, Suites algébriques, automates et substitutions. Bull. Soc. Math. France108 (1980) 401-419.  Zbl0472.10035
  4. A. Cobham, Uniform tag sequences. Math. Systems Theory6 (1972) 164-192.  Zbl0253.02029
  5. S. Eilenberg, Automata, Languages and Machines, Vol. A. Academic Press, New York (1974).  Zbl0317.94045
  6. C.C. Elgot and J.E. Mezei, On relations defined by generalized finite automata. IBM J. Res. Develop.9 (1965) 47-68.  Zbl0135.00704
  7. C. Frougny, Numeration Systems, in Algebraic Combinatorics on Words. CambridgeUniversity Press (to appear).  Zbl0787.68057
  8. C. Frougny and J. Sakarovitch, Synchronized rational relations of finite and infinite words. Theoret. Comput. Sci.108 (1993) 45-82.  Zbl0783.68065
  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.  Zbl0959.11015

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.