Abstract
Consider a non-standard numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over {0,1} without two consecutive 1. Given a set X of integers such that the language of their greedy representations in this system is accepted by a finite automaton, we consider the problem of deciding whether or not X is a finite union of arithmetic progressions. We obtain a decision procedure under some hypothesis about the considered numeration system. In a second part, we obtain an analogous decision result for a particular class of abstract numeration systems built on an infinite regular language.
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
Alexeev, B.: Minimal DFAS for testing divisibility. J. Comput. Syst. Sci. 69, 235–243 (2004)
Bertrand-Mathis, A.: Comment écrire les nombres entiers dans une base qui n’est pas entière. Acta Math. Acad. Sci. Hungar. 54, 237–241 (1989)
Cobham, A.: On the base-dependence of sets of numbers recognizable by finite automata. Math. Systems Theory 3, 186–192 (1969)
Durand, F.: A theorem of Cobham for non primitive substitution. Acta Arithmetica 104, 225–241 (2002)
Fraenkel, A.: Personal communication (February 2008)
Frougny, C., Solomyak, B.: On the representation of integers in linear numeration systems. In: Ergodic theory of Z d actions (Warwick, 1993–1994). London Math. Soc. Lecture Note Ser, vol. 228, pp. 345–368. Cambridge Univ. Press, Cambridge (1996)
Hollander, M.: Greedy numeration systems and regularity. Theory Comput. Syst. 31, 111–133 (1998)
Honkala, J.: A decision method for the recognizability of sets defined by number systems. Theoret. Inform. Appl. 20, 395–403 (1986)
Lecomte, P.B.A., Rigo, M.: Numeration systems on a regular language. Theory Comput. Syst. 34, 27–44 (2001)
Lothaire, M.: Algebraic Combinatorics on Words. Encyclopedia of Mathematics and its Applications, vol. 90. Cambridge University Press, Cambridge (2002)
Sakarovitch, J.: Élements de théorie des automates, Vuibert. English translation: Elements of Automata Theory. Cambridge University Press, Cambridge, Vuibert (2003)
Shallit, J.: Numeration systems, linear recurrences and regular sets. Inform. and Comput. 113, 331–347 (1994)
Zeckendorf, E.: Representation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas. Bull. Soc. Roy. Sci. Liege 41, 179–182 (1972)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Charlier, E., Rigo, M. (2008). A Decision Problem for Ultimately Periodic Sets in Non-standard Numeration Systems. In: Ochmański, E., Tyszkiewicz, J. (eds) Mathematical Foundations of Computer Science 2008. MFCS 2008. Lecture Notes in Computer Science, vol 5162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85238-4_19
Download citation
DOI: https://doi.org/10.1007/978-3-540-85238-4_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-85237-7
Online ISBN: 978-3-540-85238-4
eBook Packages: Computer ScienceComputer Science (R0)