Finer Is Better: Abstraction Refinement for Rewriting Approximations

Term rewriting systems are now commonly used as a modeling language for programs or systems. On those rewriting based models, reachability analysis, i.e. proving or disproving that a given term is reachable from a set of input terms, provides an efficient verification technique. For disproving reachability (i.e. proving non reachability of a term) on non terminating and non confluent rewriting models, KnuthBendix completion and other usual rewriting techniques do not apply. Using the tree automaton completion technique, it has been shown that the non reachability of a term t can be shown by computing an overapproximation of the set of reachable terms and prove that t is not in the over-approximation. However, when the term t is in the approximation, nothing can be said. In this paper, we improve this approach as follows: given a term t, we try to compute an over-approximation which does not contain t by using an approximation refinement that we propose. If the approximation refinement fails then t is a reachable term. This semi-algorithm has been prototyped in the Timbuk tool. We present some experiments with this prototype showing the interest of such an approach w.r.t. verification on rewriting models.