Basics on Tree Automata

The theory of tree automata was established in the late sixties by Thatcher and Wright [53] and Doner [19]. They showed that the basic logical and algorithmic properties of standard automata theory can be transferred from the domain of finite words to the domain of finite trees (or terms), leading to a theory of regular tree languages. Quoting from the introduction of [53], where the theory of tree automata is called generalized finite automata theory: “...the results presented here are easily summarized by saying that conventional finite automata theory goes through for the generalization – and it goes through quite neatly!” In parallel, Rabin [44] developed a theory of automata on infinite trees, which serves as a basis for the analysis of specification logics in the context of the verification of state based systems with non-terminating behavior. The theory of automata on infinite trees is quite different from the one on finite trees and in this chapter we exclusively consider finite trees. Besides their application for showing the decidability of certain logical theories [19, 53], tree automata found their first applications in the area of term rewriting, e.g., for the automated termination analysis of certain rewriting systems, which are documented in the electronic book [16]. Nowadays, tree automata are used in various fields, e.g., in verification to model the state space of parametric systems [1] or as the underlying formalism for a logic that allows to specify properties of heap manipulating programs [38]. All these algorithmic applications of tree automata are based on the following two facts: The class of regular tree languages has strong closure properties (e.g., it is closed under boolean operations and projection), and the main algorithmic problems