New Abstract Criteria for Termination and Confluence of Conditional Rewrite Systems

Abstract

We consider the problem of verifying confluence and termination of conditional term rewriting systems (TRSs). For unconditional TRSs the critical pair lemma holds which enables a finite test for confluence of (finite) terminating systems. And for ensuring termination of unconditional TRSs a couple of methods for constructing appropiate well-founded term orderings are known. If however termination is not guaranteed then proving confluence is much more difficult. Recently we have obtained some interesting results for unconditional TRSs which provide sufficient criteria for termination plus confluence in terms of restricted termination and confluence properties. In particular, we have shown that any innermost terminating and locally confluent overlay system is complete, i.e. terminating and confluent. 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 which is an innermost terminating semantical overlay system such that all (conditional) critical pairs are joinable is complete.