On Termination and Confluence of Conditional Rewrite Systems

We consider the problem of verifying connuence and termination of conditional term rewriting systems. Recently we have obtained some interesting results for unconditional term rewriting systems (TRSs) which are non-overlapping or, more generally, locally connuent overlay systems. These results provide suucient criteria for termination plus connuence in terms of restricted termination and connuence properties (Gramlich 1994a). Here we generalize our approach to the conditional case and show how to solve the additional complications due to the presence of conditions in the rules. Our main result can be stated as follows: Any conditional TRS (CTRS) which is an innermost terminating overlay system such that all (conditional) critical pairs are joinable is complete, .e., terminating and connuent.