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

Let , be two multiplicatively independent Pisot numbers, and let U , U , respectively. For every n 1, if A IN recognizable then A is de nable in hIN; +i.

We survey de nability and decidability issues related to rst-order fragments of arithmetic, with a special emphasis on Presburger and Skolem arithmetic and their (un)decidable extensions.