An Automata Theoretic Approach to the Theory of Rational Tree Relations

We investigate rational relations over trees. Our starting point is the definition of rational tree relations via rational expressions by Raoult (Bull. Belg.Math. Soc. 1997). We develop a new class of automata, called asynchronous tree automata, which recognize exactly these relations. The automata theoretic approach is convenient for the solution of algorithmic problems (like the emptiness problem). The second contribution of this paper is a new subclass of the rational tree relations, called separate-rational tree relations, defined via a natural restriction on asynchronous tree automata. These relations are closed under composition, preserve regular tree languages, and generate precisely the regular sets in the unary case (all these properties fail for the general model), and they are still more powerful than, for instance, the automatic tree relations.