Non - E - overlapping and weakly shallow TRSs are confluent ( Extended abstract ) Masahiko Sakai

Confluence of term rewriting systems (TRSs) is undecidable, even for flat TRSs [MOJ06] or length-two string rewrite systems [SW08]. Two decidable subclasses are known: right-linear and shallow TRSs by tree automata techniques [GT05] and terminating TRSs [KB70]. Most of sufficient conditions are for either terminating TRSs [KB70] (extended to TRSs with relative termination [HA11, KH12]) or leftlinear non-overlapping TRSs (and their extensions) [Ros73, Hue80, Toy87, Oos95, Oku98, OO97]. For non-linear TRSs, a goal is RTA open problem 58 “strongly non-overlapping and right-linear TRSs are confluent”. A best known result strengthens the right-linear assumption to simple-right-linear [TO95, OOT95], which means that each rewrite rule is right-linear and no left-non-linear variables appear in the right hand side. Other trials by depth-preserving conditions are found in [GOO98]. We have proposed a different methodology, called a reduction graph [SO10]. It has shown that “weakly non-overlapping, shallow, and non-collapsing TRSs are confluent”. An original idea comes from observation that, when non-E-overlapping, peak-elimination uses only “copies” of reductions in an original rewrite sequences. Thus, if we focus on terms appearing in peak elimination, they are finitely many. We regard a rewrite relation over these terms as a directed graph, and we construct a confluent directed acyclic graph (DAG) in a bottom-up manner, in which the shallow assumption works. The keys are, a connected convergent DAG always has a unique normal form (if it is finite), and convergence is preserved if we add an arbitrary reduction starting from that normal form. This paper briefly sketches that “non-E-overlapping and weakly-shallow TRSs are confluent” by extending reduction graph in our previous work [SO10] by introducing constructor expansion. A term is weakly shallow if each defined function symbol appears either at the root or in the ground subterms, and a TRS is weakly shallow if the both sides of rules are weakly shallow. The non-E-overlapping property is undecidable for weakly shallow TRSs [MOM12] and a decidable sufficient condition is the strongly non-overlapping condition. A Turing machine can be simulated by a weakly shallow TRS (p.27 in [Klo93]); thus the word problem is undecidable, in contrast to shallow TRSs [CHJ94].