Presburger arithmetic and recognizability of sets of natural numbers by automata: New proofs of Cobham's and Semenov's theorems

Cobham's theorem for substitutions

“…Then the efforts were concentrated to simplify Cobham-Semenov's theorem and to generalize it to non-standard numeration systems given by linear recurrence relations like the Fibonacci one. Alternative and satisfactory proofs (in terms of simplification) have been proposed, among them a very interesting logical proof due to C. Michaux and R. Villemaire [MV93,MV96] (see also [Bès97,Muc03]), using p-definable sets in the formalism of first order logic in some arithmetic models like the Presburger arithmetic N, + (see [Bès01] for a survey on these methods). Another recent proof makes use of ergodic measures [Dur08] and the fact that p-recognizable subsets of N d are characterized by multidimensional substitutions [ČG86,Sal87].…”

A Generalization of Cobham's Theorem

“…There are other ways to generalize Cobham's Theorem, some of them can be found in [2], [9], [11], [17] and [21]. Related works can be found in [3], [13] or [16].…”

Words derivated from Sturmian words

“…In [2] we studied the shortest of those return words and its first occurrence. That study has permitted, in particular, to answer negatively a question posed by Michaux and Villemaire in [13]. In the next proposition we recall this result from [2] (see Fig.…”

“…In [2], the authors considered the shortest of those return words and its first occurrence. That permitted to answer negatively a question posed by Michaux and Villemaire in [13].…”