Canonical Conditional Rewrite Systems

Conditional equations have been studied for their use in the specification of abstract data types and as a computational paradigm that combines logic and function programming in a clean way. In this paper we examine different formulations of conditional equations as rewrite systems, compare their expressive power and give sufficient conditions for rewrite systems to have the "confluence" property. We then examine a restriction of these systems using a "decreasing" ordei~ng. With this restriction, most of the basic notions (like rewriting and computing normal forms) are decidable, the "critical pair" lemma holds, and some formulations preserve eanonicity. 1. I n t r o d u c t i o n Conditional rewriting systems arise naturally in the algebraic specification of data types and have been studied largely from this perspective [Remy-82, Kaplan-84, Bergstra-Klop-82]. See also [Brand-Darrlnger-Joyner-78]. With differing restrictions on lef t -hand sides and conditions, useful results have been obtained about the confluence of such systems. More recently, conditional rewriting systems have been shown to provide a natural computational paradigm combining logic and functional programming [Dershowitz-Plaisted-85, Fribourg-85, Goguen-Meseguer-86]. A program is a set of conditional rules and a computation is the process of finding a substi tut ion that makes two terms equal in the underlying equational theory. * This research was supported in part by the National Science Foundation under Grant DCR 85-13417. The second author is also partly supported by the Grant of the Committee on Aid to Research Activity of Faculty of Engineering and Computer Science (Concordia University), Fonds pour la Formation de Chercheurs et l'Aide a la Recherche (Quebec) and the Natural Science and Engineering Research Council (Canada).