A Note on the Precision of the Tree Automata Completion

Tree Automata Completion is an algorithm over-approximating sets of terms reachable by rewriting. Precision of completion is conditioned by the, so-called, R/E-coherence property of the initial tree automaton. This paper shows that, in the particular case of functional TRS, this restriction can easily be removed. First, we prove that there always exists an equivalent R/E-coherent tree automaton. Second, we show how to approximate this equivalent tree automaton using completion itself and the Timbuk tool