The reachability problem for ground TRS and some extensions

Abstract

The reachability problem for term rewriting systems (TRS) is the problem of deciding, for a given TRS S and two terms M and N, whether M can reduce to N by applying the rules of S. We show in this paper by some new methods based on algebraical tools of tree automata, the decidability of this problem for ground TRS's and, for every ground TRS S, we built a decision algorithm. In the order to obtain it, we compile the system S and the compiled algorithm works in a real time (as a fonction of the size of M and N). We establish too some new results for ground TRS modulo different sets of equations: modulo commutativity of an operator σ, the reachability problem is shown decidable with technics of finite tree automata; modulo associativity, the problem is undecidable; modulo commutativity and associativity, it is decidable with complexity of reachability problem for vector addition systems.