Decidability questions related to abstract numeration systems

Abstract Numeration Systems and Tilings

numeration systems and tilings Valérie Berthé and Michel Rigo 1 LIRMM, CNRS-UMR 5506, Univ. Montpellier II, 161 rue Ada, 34392 Montpellier Cedex 5, France, [email protected] 2 Université de Liège, Institut de Mathématiques, Grande Traverse 12 (B 37), B-4000 Liège, Belgium, [email protected] Abstract. An abstract numeration system is a triple S = (L,Σ,<) where (Σ,<) is a totally ordered alphabet an...

Cobham’s Theorem for Abstract Numeration Systems

Abstract numeration systems generalize numeration systems whose representation map is increasing and such that the language of all the representations is regular. We show that if a set of integers is recognized by some finite automaton within two independent abstract numeration systems, then this set is a finite union of arithmetic progressions.

Multi-dimensional sets recognizable in all abstract numeration systems

numeration systems An abstract numeration system is a triple S = (L,Σ, <) where L is an infinite regular language over the totally ordered alphabet (Σ, <) By enumerating words of L in the radix order (induced by <), we define a one-to-one correspondence between N and L repS : N → L : n 7→ (n + 1)th word of L valS = rep −1 S : L → N Abstract numeration systems Example S = (a∗b∗, {a, b}, a < b) L...

Real numbers having ultimately periodic representations in abstract numeration systems

Using a genealogically ordered infinite regular language, we know how to represent an interval ofR. Numbers having an ultimately periodic representation play a special role in classical numeration systems. The aim of this paper is to characterize the numbers having an ultimately periodic representation in generalized systems built on a regular language. The syntactical properties of these words...

Numeration Systems