Abstract
Two words u and v are k-abelian equivalent if they contain the same number of occurrences of each factor of length k and, moreover, start and end with a same factor of length k − 1, respectively. This leads to a hierarchy of equivalence relations on words which lie properly in between the equality and abelian equality.
The goal of this paper is to analyze Fine and Wilf’s periodicity theorem with respect to these equivalence relations. A crucial question here is to ask how far two “periodic” processes must coincide in order to guarantee a common “period”. Fine and Wilf’s theorem characterizes this for words. Recently, the same was done for abelian words. We show here that for k-abelian periods the situation resembles that of abelian words: In general, there are no bounds, but the cases when such bounds exist can be characterized. Moreover, in the cases when such bounds exist we give nontrivial upper bounds for these, as well as lower bounds for some cases. Only in quite rare cases (in particular for k = 2) we can show that our upper and lower bounds match.
Supported by the Academy of Finland under grants 137991 (FiDiPro) and 251371 and by Russian Foundation of Basic Research (grants 10-01-00424, 12-01-00448).
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
Fine, N.J., Wilf, H.S.: Uniqueness theorems for periodic functions. Proc. Amer. Math. Soc. 16, 109–114 (1965)
Castelli, M.G., Mignosi, F., Restivo, A.: Fine and Wilf’s theorem for three periods and a generalization of Sturmian words. Theoret. Comput. Sci. 218(1), 83–94 (1999)
Tijdeman, R., Zamboni, L.Q.: Fine and Wilf words for any periods II. Theoret. Comput. Sci. 410(30-32), 3027–3034 (2009)
Berstel, J., Boasson, L.: Partial words and a theorem of Fine and Wilf. Theoret. Comput. Sci. 218(1), 135–141 (1999)
Blanchet-Sadri, F.: Periodicity on partial words. Comput. Math. Appl. 47(1), 71–82 (2004)
Shur, A.M., Gamzova, Y.V.: Partial words and the period interaction property. Izv. Ross. Akad. Nauk Ser. Mat. 68(2), 191–214 (2004)
Kari, L., Seki, S.: An improved bound for an extension of Fine and Wilf’s theorem and its optimality. Fund. Inform. 101(3), 215–236 (2010)
Constantinescu, S., Ilie, L.: Fine and Wilf’s theorem for abelian periods. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS 89, 167–170 (2006)
Blanchet-Sadri, F., Tebbe, A., Veprauskas, A.: Fine and Wilf’s theorem for abelian periods in partial words. In: Proceedings of the 13th Mons Theoretical Computer Science Days (2010)
Choffrut, C., Karhumäki, J.: Combinatorics of words. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1, pp. 329–438. Springer (1997)
Huova, M., Karhumäki, J., Saarela, A.: Problems in between words and abelian words: k-abelian avoidability. Theoret. Comput. Sci. (to appear)
Karhumäki, J., Saarela, A., Zamboni, L.: On a generalization of Abelian equivalence (in preparation)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Karhumäki, J., Puzynina, S., Saarela, A. (2012). Fine and Wilf’s Theorem for k-Abelian Periods. In: Yen, HC., Ibarra, O.H. (eds) Developments in Language Theory. DLT 2012. Lecture Notes in Computer Science, vol 7410. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31653-1_27
Download citation
DOI: https://doi.org/10.1007/978-3-642-31653-1_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31652-4
Online ISBN: 978-3-642-31653-1
eBook Packages: Computer ScienceComputer Science (R0)