Some Classes of Term Rewriting Systems for which Termination is Decidable

Termination is one of the central properties of term rewriting systems (TRSs for short). A TRS is called terminating if it does not admit any infinite rewrite sequence. The efforts to find classes of TRSs whose termination is decidable have been made for decades and several positive results have been proposed, for example, right-ground TRSs and right-linear shallow TRSs. In this research, we study several more general classes of term rewriting systems and prove that termination is a decidable property for these classes. By showing the decidability of the existence of the cycling dependency chains, we prove that termination is decidable for semi-constructor case, the class of which is a superclass of right-ground TRSs. By analyzing the argument propagation cycling in the dependency graph and checking the reachability between two terms by tree automata techniques, we claim that termination is also decidable for left-linear shallow TRSs. Moreover, we explore the potentiality of the method for left-linear shallow case to cover right-linear shallow case and to generalize to growing TRSs. Observing that all the results we proposed so far are based on “loopingness”, we also extend our research to the field of “non-loopingness”, in which we find some new interesting non-looping examples in addition to the only one known so far given by Zantema and propose new definitions to classify these non-looping examples.