We survey the properties of sets of integers recognizable by automata when they are written in p-ary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given by Muchnik for the theorem of Cobham-Semenov, the original proof being published in Russian. 1

This paper extends some results of Allouche and Shallit for q-regular sequences to numeration systems in algebraic number fields and to linear numeration systems. We also construct automata that perform addition and multiplication by a fixed number. 1 Introduction A sequence is called q-automatic if its n-th term can be generated by a finite state machine from the q-ary digits of n. The concept of automatic sequences was introduced in 1969 and 1972 by Cobham [8, 9]. In 1979 Christol [6] (see also Christol, Kamae, Mend`es France and Rauzy [7]) discovered a nice arithmetic property of automatic sequences: a sequence with values in a finite field of characteristic p is p-automatic if and only if the corresponding power series is algebraic over the field of rational functions over this finite field. A brief survey on this subject is given in [2], see also [10]. Some generalizations of this concept were studied in [27, 23, 24, 3], see also the survey [1]. An automatic sequence has to t...

. There exist various well-known characterizations of sets of numbers recognizable by a finite automaton, when they are represented in some integer base p 2. We show how to modify these characterizations, when integer bases p are replaced by linear numeration systems whose characteristic polynomial is the minimal polynomial of a Pisot number. We also prove some related interesting properties. 1 Introduction Since the work of [9], sets of integers recognizable by finite automata have been studied in numerous papers. One of the jewels in this topic is the famous Cobham's theorem [11]: the only sets of numbers recognizable by finite automata, independently of the base of representation, are those which are ultimately periodic. Other studies are concerned with computation models equivalent to finite automata in the recognition of sets of integers. The proposed models use first-order logical formulae [9], uniform substitutions [12], algebraic formal series [10]. We refer the reader to the...

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