Syntactic Confluence Criteria for Positive/Negative-Conditional Term Rewriting Systems

We study the combination of the following already known ideas for showing confluence of unconditional or conditional term rewriting systems into practically more useful confluence criteria for conditional systems: Our syntactic separation into constructor and non-constructor symbols, Huet’s introduction and Toyama’s generalization of parallel closedness for non-terminating unconditional systems, the use of shallow confluence for proving confluence of terminating and non-terminating conditional systems, the idea that certain kinds of limited confluence can be assumed for checking the fulfilledness or infeasibility of the conditions of conditional critical pairs, and the idea that (when termination is given) only prime superpositions have to be considered and certain normalization restrictions can be applied for the substitutions fulfilling the conditions of conditional critical pairs. Besides combining and improving already known methods, we present the following new ideas and results: We strengthen the criterion for overlay joinable terminating systems, and, by using the expressiveness of our syntactic separation into constructor and non-constructor symbols, we are able to present criteria for level confluence that are not criteria for shallow confluence actually and also able to weaken the severe requirement of normality (stiffened with left-linearity) in the criteria for shallow confluence of terminating and non-terminating conditional systems to the easily satisfied requirement of quasi-normality. Finally, the whole paper also gives a practically useful overview of the syntactic means for showing confluence of conditional term rewriting systems. This research was supported by the Deutsche Forschungsgemeinschaft, SFB 314 (D4-Projekt)