Accelerating tree-automatic relations

Abstract

We study the problem of computing the transitive closure of tree-automatic (binary) relations, which are represented by tree automata. Such relations include classes of infinite systems generated by pushdown systems (PDS), ground tree rewrite systems (GTRS), PA-processes, and Turing machines, to name a few. Although this problem is unsolvable in general, we provide a semi-algorithm for the problem and prove completeness guarantee for PDS, GTRS, and PA-processes. The semi-algorithm is an extension of a known semi-algorithm for structure-preserving tree-automatic relations, for which completeness is guaranteed for several interesting parameterized systems over tree topology. Hence, there is a single generic procedure that solves reachability for PDS, GTRS, PA-processes, and several parameterized systems in a uniform way. As an application, we provide a single generic semi-algorithm for checking repeated reachability over tree-automatic relations, for which completeness is guaranteed for the aforementioned classes.