THE REACHABILITY PROBLEMFOR GROUNDTRS AND SOME EXTENSIONS A. DERUYVER and R. GILLERON LIFL UA 369 CNRS Universite des sciences et techniques de LILLE FLANDRES ARTOIS U.F.R. d' I.E.E.A. Bat. M3 59655 VILLENEUVE D'ASCQ CEDEX FRANCE ~S~A~ 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 o, 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. INTRODUCTION The reachability problem for term rewriting systems (TRS) is the problem of deciding, for given TRS S and two terms M and N, whether M can reduce to N by applying the rules of S. It is well-known that this problem is undecidable for general TRS's .In a first part we study this problem for more simple systems, more specifically in the case of ground term rewriting systems. A TRS is said to be ground if its set of rewriting rules R={ li->ri I i~ I} (where I is finite) is such that li and ri are ground terms(no variable occurs in these terms). The decidability of the reachability problem for ground TRS was studied by Dauchet M. [4],[5] as a consequence of decidability of confluence for ground TRS. Oyamaguchi [15] and Togushi-Noguchi have shown this result too for ground TRS and in the same way for quasi-ground TRS.We take again this study with two innovator aspects: the modulary aspect of the decision algorithm which use all algebraical tools of tree automata, that permits to clearly describe it. the exchange between time and space aspect which have permitted to obtain some time complexities more and more reduced. Therefore we have proceeded in three steps: 1- We begin with the TRS S not modified which gives the answer to the problem with a time complexity not bounded. supported by "Greco programmation" and "PRC mathematique et informatique"