Confluence of Conditional Rewriting Modulo

Sets of equations E play an important computational role in rewriting-based systems R by defining an equivalence relation =E inducing a partition of terms into E-equivalence classes on which rewriting computations, denoted ->R/E and called *rewriting modulo E*, are issued. This paper investigates *confluence of ->R/E*, usually called *E-confluence*, for *conditional* rewriting-based systems, where rewriting steps are determined by conditional rules. We rely on Jouannaud and Kirchner's framework to investigate confluence of an abstract relation R modulo an abstract equivalence relation E on a set A. We show how to particularize the framework to be used with conditional systems. Then, we show how to define appropriate finite sets of *conditional pairs* to prove and disprove E-confluence. In particular, we introduce *Logic-based Conditional Critical Pairs* which do not require the use of (often infinitely many) E-unifiers to provide a finite representation of the *local peaks* considered in the abstract framework. We also introduce *parametric Conditional Variable Pairs* which are essential to deal with conditional rules in the analysis of E-confluence. Our results apply to well-known classes of rewriting-based systems. In particular, to *Equational (Conditional) Term Rewriting Systems*. In this realm, our E-confluence results strictly subsume previous approaches by Huet and Jouannaud and Kirchner.