Logical Semantics for the Rewriting Calculus

The Rewriting Calculus has been proposed as a language for defining term rewriting strategies. Rules are explicitly represented as terms, and are applied explicitly to other terms to transform them. Sets of rules may be applied to (sets of) terms non-deterministically to obtain sets of results. Strategies are implemented as rules which accept other rules as arguments and apply them in certain ways. This paper describes work in progress to strengthen the Rewriting Calculus by giving it a logical semantics. Such a semantics can provide crucial guidance for studying the language and increasing its expressive power. The latter is demonstrated by adding support to the Rewriting Calculus for what we call higher-form rewriting, where rules rewrite other rules. The logical semantics used is based on ordered linear logic. The paper develops the ideas through several examples.