Complete Monotonic Semantic Path Orderings

Although theoretically it is very powerful, the semantic path ordering (SPO) is not so useful in practice, since its monotonicity has to be proved by hand for each concrete term rewrite system (TRS). In this paper we present a monotonic variation of SPO, called MSPO. It characterizes termination, i.e., a TRS is terminating if and only if its rules are included in some MSPO. Hence MSPO is a complete termination method. On the practical side, it can be easily automated using as ingredients standard interpretations and general-purpose orderings like RPO. This is shown to be a sufficiently powerful way to handle several non-trivial examples and to obtain methods like dummy elimination or dependency pairs (without the dependency graph refinement) as particular cases. Finally, we obtain some positive modularity results for termination based on MSPO.