Transformations and Confluence for Rewrite Systems

Abstract Many important applications of rewrite systems, e.g., automated reasoning, algebraic specifications of abstract data types, and functional/equational programming, rely either wholly or in part on rewrite systems that are constructor-based. In this paper, we study general transformations of rewrite systems that preserve confluence and normal forms. In 1985, Thatte showed that an orthogonal system can be transformed into an orthogonal constructor-based system that preserves normal forms up to a certain simple homomorphism. In 1988, Thatte claimed that this transformation works for all semiregular (confluent + nonoverlapping) systems. We show that Thatte's transformation fails to preserve confluence and normal forms for semiregular systems. We then introduce the concept of weak persistence and show that Thatte's transformation is correct for all weakly persistent confluent systems. We also give some general conditions that imply weak persistence and show that this class includes as subclasses the following: left-linear, nonoverlapping systems with confluent root overlaps (generalization of orthogonal systems), nonoverlapping noetherian systems with confluent root overlaps, and the nonlinear systems with no overlaps proved confluent by Klop. We show that our transformation scheme for convergent systems can also be applied to systems in which the innermost rewriting relation is confluent. Our results indicate that persistence plays a role in arbitrary confluent systems that in analogous to the role of the no-overlap restriction in left-linear systems; and similarly weak persistence plays a role that is analogous to that of the nonoverlapping restriction for left-linear systems.