Abstract
In this paper, we consider the sets F(u) of finite factors of bi-infinite words u, upon a finite alphabet A. A natural question about this notion is : are there several bi-infinite words which have the same set of finite factors. We prove that, if F(u) is rational, then, there exists a unique bi-infinite word which has F(u) as set of finite factors iff F(u) has a non-exponential complexity (the complexity of F(u) is the function n — Card(F(u) ∩ An)) ; and if this condition is realized, in fact, F(u) has a sub-linear complexity (there exists a constant C such that card(F(u) ∩ An) ≤ n+C for large enough integers n).
Furthermore, the proof gives a characterization of bi-infinite rational words : u is rational iff F(u) is a rational set of non-exponential complexity.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
BEAUQUIER D., 1984, Bilimites de langages reconnaissables, a paraitre dans Theoret. Comput. Sci.
BUCHI, J.R., 1962, On a decision method in restricted second order arithmetic, in Logic, Methodology and Philosophy of Science, (Proc. 1960 Internat. Cong.), Stanford University Press, Stanford.
EHRENFEUCHT A., LEE R.P., ROZENBERG G., 1975, Subwords complexities of various classes of deterministic developmental languages without interactions, Theoretical Computer Science 1 p. 59–76.
EILENBERG, S., 1974, Automata, Languages and Machines, Vol. A, Academic Press, New York, Vol. B, 1976.
FISCHER R., 1975, Sofic systems and graphs Monats. Math. 80 179–186.
McNAUGHTON, R., 1966, Testing and generating infinite sequences by a finite automaton, Information and Control, 9, 521–530.
NIVAT M., PERRIN D., 1982, Ensembles reconnaissables de mots bi-infinis, Proc. 14th A.C.M. Symp. on Theory of Computing, 47–59.
PANSIOT J.J, 1984, Lecture Notes in Computer Science No 166 STACS 84, Bornes inférieures sur la complexité des facteurs des mots infinis engendrés par morphismes itérés, p. 230–240.
THOMAS W., 1981, A combinatorial approach to the theory of ω-automata, Inform. Control, 48, 261–283.
THUE A., 1906, Uber unendliche Zeichenreihen, Norske Vid. Selsk. Skr. I. Mat. Nat. K1. Christiania No 7, 1–22.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1985 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Beauquier, D., Nivat, M. (1985). About rational sets of factors of a bi-infinite word. In: Brauer, W. (eds) Automata, Languages and Programming. ICALP 1985. Lecture Notes in Computer Science, vol 194. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0015728
Download citation
DOI: https://doi.org/10.1007/BFb0015728
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-15650-5
Online ISBN: 978-3-540-39557-7
eBook Packages: Springer Book Archive