Sofic and Almost of Finite Type Tree-Shifts

We introduce the notion of sofic tree-shifts which corresponds to symbolic dynamical systems of infinite trees accepted by finite tree automata. We show that, contrary to shifts of infinite sequences, there is no unique minimal deterministic irreducible tree automaton accepting an irreducible sofic tree-shift, but that there is a unique synchronized one, called the Shannon cover of the tree-shift. We define the notion of almost finite type tree-shift which is a meaningful intermediate dynamical class in between irreducible finite type tree-shifts and irreducible sofic tree-shifts. We characterize the Shannon cover of an almost finite type tree-shift and we design an algorithm to check whether a sofic tree-shift is almost of finite type.