A Decision Problem for Ultimately Periodic Sets in Non-standard Numeration Systems

Authors:
Emilie Charlier;Michel Rigo
Affiliations:
Institute of Mathematics, University of Liège, Liège, Belgium B-4000;Institute of Mathematics, University of Liège, Liège, Belgium B-4000
Venue:
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Year:
2008

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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 Xof 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 Xis 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.