Automata Completion and Regularity Preservation

When rewriting a regular language with a left-linear term rewriting system, if the set of reachable terms is regular, we show that equational tree automata completion can compute it. This was known to be true for some known TRS classes preserving regularity, but was still an open question in the general case. The proof is not constructive: it assumes that the set of reachable terms is regular, which is undecidable. Despite being non constructive, the proof of this result has a strong practical impact: it shows how to tune completion to get the best possible precision w.r.t. sets of reachable terms. In particular, to carry out the proof, it was necessary to generalize and improve two results of completion: the termination and the precision theorems. 1998 ACM Subject Classification I.2.3 Deduction and Theorem Proving, F.4.2 Grammars and Other Rewriting Systems