The Emptiness Problem for Automata on Infinite Trees

The purpose of this paper is to give an alternative proof to the decidability of the emptiness problem for tree automata, as shown in Rabin [4]. The proof reduces the emptiness problem for automata on infinite trees to that for automata on finite trees, by showing that any automata definable set of infinite trees must contain a finitelygenerable tree. Section 1: Introduction The analysis of finite automata on infinite trees is the basis for Rabin's remarkable proof of the decidability of S2S (the monadic secondorder theory of two successors) [5]. Rabin's proof follows the now standard form of Buchi and Elgot's proof for WSlS (weak, single successor) [1, 3] and Thatcher-Wright's proof for weak S2S, and requires demonstrating effectively that the automata are closed under union, projection, and negation, and that the emptiness problem for the automata is decidable. As in the case of SIS, the main technical difficulty in the case of S2S lies in proving closure under complementation of sets accepted by nondeterministic automata on infinite trees. The problem is complicated by the fact that nondeterministic infinite tree automata are known not to be equivalent to any of the likely definitions of deterministic infinite tree automata. Curiously, the emptiness problem, which is easy for the other kinds of automata, turns out to be nontrivial for (nondeterministic) infinite tree automata. Rabin subsequently improved his original proof of the decidability of this emptiness problem, but even the second proof [4] used an involved induction and consequently does not yield a simple effective criterion for deciding emptiness. In this paper we provide such a criterion by showing that an infinite tree automaton accepts some valued tree if and only if there is a computation of the automaton containing a certain simple kind of finite subtree. Moreover, the set of finite subtrees of the kind we