Completeness of Rewrite Rules and Rewrite Strategies for FP

Abstract

This paper treats languages whose operational semantics is given by a set of rewrite rules. For such languages, it is important to be able to determine that there are enough rules to be able to compute the correct meaning of all expressions, but not so many that the system of rules is inconsistent. A formal framework is developed in which to give a precise treatment of these completeness and soundness issues, which are then investigated in the context of an extended version of the functional programming language FP. The rewrite rules of FP are shown to be sound and complete with respect to three different notions of completeness. The latter half of the paper considers rewrite strategies. In order to implement a language based on rewrite rules, it does not suffice to know that there are “enough” rules in the language; a good strategy for determining the order in which to apply them is also needed. But what is “good”? Corresponding to each notion of completeness, there is a notion of a good rewrite strategy. These notions of goodness are examined and characterized, and examples of a number of natural good strategies are given. Although these results are presented in the context of FP, the techniques 1990 should apply well beyond the realm of FP rewriting systems. © 1990, ACM. All rights reserved.