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, Knuth-Bendix 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.

Abstract. This paper investigates the use of tree automata with global equalities and disequalities (TAGED for short) in reachability analysis over term rewriting systems (TRSs). The reachability problem being in general undecidable on non terminating TRSs, we provide TAGEDbased construction, and then design approximation-based semi-decision procedures to model-check useful temporal patterns on in nite state rewriting graphs. To show that the above TAGED-based construction can be e ectively carried out, complexity analysis for rewriting TAGEDde nable languages is given. Recently, reachability analysis turned out to be a very e cient veri cation technique for proving properties on in nite systems modeled by term rewriting systems (TRSs for short). In the rewriting theory, the reachability problem is the following: given a TRS R and two terms s and t, can we decide whether s → * R t or not? This problem, which can easily be solved on strongly terminating TRSs, is undecidable on non terminating TRSs. However, on the one hand, there exist several syntactic classes of TRSs for which this problem becomes decidable [16,20,34]. On the other hand, in addition to classical proof tools of rewriting, given a set E ⊆ T (F) of initial terms, provided that s ∈ E, one can prove s → * R t by using over-approximations of R * (E) [21,16] and proving that t does not belong to these approximations. Recently, the veri cation of temporal properties of systems modeled by TRSs has been investigated [15,28,27]. To apply these very interesting and promising theoretical results to applications This work has been funded by the French ANR-06-SETI-014 RAVAJ project.

This paper addresses the following general problem of tree regular model-checking: decide whether R * (L) ∩ Lp = ∅ where R * is the reflexive and transitive closure of a successor relation induced by a term rewriting system R, and L and Lp are both regular tree languages. We develop an automatic approximation-based technique to handle thisundecidable in general -problem in the case when term rewriting system rules are non left-linear.