Abstract
For a set H of natural numbers, the H-power of a language L is the set of all words p k where p ∈, L and k ∈, H. The root of L is the set of all primitive words p such that p n belongs to L for some n ≥ 1. There is a strong connection between the root and the powers of a regular language L namely, the H-power of L for an arbitrary finite set H with 0, 1, 2 ∉, H is regular if and only if the root of L is finite. If the root is infinite then the H-power for most regular sets H is context-sensitive but not context-free. The stated property is decidable.
The research of this author was supported by a collaboration between the HAS, Budapest, Hungary, and the JSPS, Tokyo, Japan, and also by the German-Hungarian project No. WTZ HUN 00/040.
After finishing his diploma at the Friedrich Schiller University Jena he started studies in Tarragona at the 1st International PhD School in Formal Languages and Applications.
Corresponding author.
In the French literature the terminus rational is used instead of regular. We follow the more common usage and shall speak of regular languages.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Y. Bar-Hillel, M. Perles, E. Shamir, On formal properties of simple phrase structure grammars, Z. Phonetik Sprachwiss. Kommunikationsforsch. 14 (1961), 143–172.
J.R. Büchi, Weak second-order arithmetic and finite automata, Z. f. Math. Logiku. Grundl. d. Math. 6 (1960), 66–92.
T. Cachat, The power of one-letter rational languages, in [12], 145–154.
H. Calbrix, M. Nivat, Prefix and period languages of rational ω-languages, in Proc. Developments in Language Theory, Magdeburg 1995, World Scientific, 1996, 341–349.
C. Choffrut, J. Karhumäki, Combinatorics of words, in [15], 329–438.
S. Eilenberg, M.P. Schützenberger, Rational sets in commutative monoids, J. of Algebra 13 (1969), 173–191.
J.E. Hopcroft, J.D. Ullman, Introduction to automata theory, languages, and computation, Addison-Wesley, Reading (Mass.), 1979.
S. Horváth, M. Ito, Decidable and undecidable problems of primitive words, regular and context-free languages, J. Universal Computer Science 5 (1999), 532–541.
S. Horváth, M. Kudlek, On classification and decidability problems of primitive words, PU.M.A. 6 (1995), 171–189.
H. Jürgensen, S. Konstantinidis, Codes, in [15], 511–607.
S.C. Kleene, Representation of events in nerve nets and finite automata, Automata Studies, Princeton Univ. Press, Princeton (N.J.), 1956, 2–42.
W. Kuich, G. Rozenberg, A. Salomaa (Eds.), Developments in Language Theory, 5th International Conference, Wien 2001, Lecture Notes in Computer Science 2295, Springer-Verlag, Berlin-Heidelberg, 2002.
G. Lischke, The root of a language and its complexity, in [12], 272–280.
R.C. Lyndon, M.P. Schützenberger, On the equation a M = bNcP in a free group, Michigan Math. Journ. 9 (1962), 289–298.
G. Rozenberg, A. Salomaa (Eds.), Handbook of formal languages, Vol. 1, Springer-Verlag, Berlin-Heidelberg, 1997.
H.J. Shyr, Free monoids and languages, Hon Min Book Company, Taichung, 1991.
S. Yu, Regular languages, in [15], 41–110.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Horváth, S., Leupold, P., Lischke, G. (2003). Roots and Powers of Regular Languages. In: Ito, M., Toyama, M. (eds) Developments in Language Theory. DLT 2002. Lecture Notes in Computer Science, vol 2450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45005-X_19
Download citation
DOI: https://doi.org/10.1007/3-540-45005-X_19
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40431-6
Online ISBN: 978-3-540-45005-4
eBook Packages: Springer Book Archive