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

We prove that, given a double sequence w over the alphabet A (i.e. a mapping from Z Z 2 to A), if there exists a pair (n 0 ; m 0 ) 2 Z Z 2 such that p w (n 0 ; m 0 ) ! 1 100 n 0 m 0 , then w has a periodicity vector, where p w of w is the complexity function of w. 1 Introduction In combinatorics on words the notions of complexity and periodicity are of fundamental importance. The complexity function of a formal language counts, for any natural number n, the number of words in the language of length n. The complexity function of a word (finite, infinite, biinfinite) is the complexity function of the formal language whose elements are all the factors (or blocks, or also subwords) of the word. The Morse-Hedlund Theorem states that there exists an important relationship between periodicity and complexity. In particular it states that for any biinfinite word w if the number of its different factors of length n is less Dipartimento di Matematica ed Applicazioni, Universit`a degl...

We introduce the notion of periodicity for k-ary labeled trees: roughly speaking, a tree is periodic if it can be obtained by a sequence of concatenations of a smaller tree plus a "remainder". The period is the shape of such smaller tree (i.e. the corresponding unlabeled tree). This definition reduces to the classical one for string when restricted to the case of unary trees. Then, we define the greatest common divisor of two unlabeled trees and relate right congruences to unlabeled trees. This allows us to give a characterization of tree periodicity in terms of right congruences and then to prove a periodicity theorem for trees that is a generalization to trees of the Fine and Wilf's periodicity theorem for words. Keywords: Congruence, periodicity, labeled tree. Work partially supported by the ESPRIT II Basic Research Actions Program of the EC under Project ASMICS 2 (contract No. 6317) and in part by the Italian Ministry of Universities and Scientific Research MURST 40% Algoritmi, ...

. The duel paradigm appeared in the recent years as crucial to design efficient 1D or 2D pattern matching algorithms. It relies on the computation of witnesses, i.e. mismatching positions between two overlapping copies of a pattern p searched in a text t. Although this is easily done in linear time in 1D, the two previous algorithms in 2D rely on quite intricate data structures. We prove new structural properties of 2D patterns, and provide an algorithm to derive the fundamental basis of periodicities. We show how theoretical results on 2D periodicities allow to write and implement a simple algorithm that is still linear in the size of the pattern, hence optimal. 1 Introduction "Multidimensional search" or "multidimensional pattern matching" generalizes linear string searching: a pattern p, a multidimensional array that usually is connex and convex, is searched in a multidimensional array, the text, t. A strong interest appeared recently [ZT89, BYR93, ABF92, GP92, RR93, GM94]. It is i...