Congruences, Automata and Periodicities

We show the deep connection between two apparently unrelated topics in automata theory and combinatorics on words: the equivalence problem of nite automata and the Fine and Wilf's periodicity theorem. In fact they both refer to the problem of checking whether two congruences agree on a nite set. This also allows us to introduce a notion of periodicity for labelled trees and to derive a periodicity theorem for trees that generalizes the Fine and Wilf's theorem.