Modularity in term rewriting revisited

We revisit modularity in term rewriting which for the last 25 years has been a very active and fruitful research field. Starting with the pioneering works of Yoshihito Toyama on the modularity of confluence and the non-modularity of termination he thus initiated an extremely productive line of research, with many non-trivial and deep results, striking counterexamples and a substantial amount of systematic theoretical foundations, methodological principles and novel proof techniques. In this focused summary we will revisit the modularity analysis for ordinary term rewriting systems, considering various confluence and termination properties and restricting ourselves mainly to the case of disjoint unions. We will summarize known results on the (non-)modularity of various confluence and termination properties, and exhibit crucial ideas, constructions and phenomena. Later on we will also briefly consider various extensions, applications, related questions and open problems, as well as recent developments.

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