A Logic for Rewriting Strategies

As term-rewriting has matured, with applications in areas such as automatic program transformation, the necessity of programmed control for applying rules has become apparent. Early term-rewriting systems used a xed strategy, such as innermost-leftmost matching, choosing among possibly overlapping rules nondeterministically. This mode of control is su cient for a con uent, terminating system of rules. When a rule system is not con uent or its termination cannot be guaranteed, the selection and order in which rules are applied can be critical. Hence, modern rewriting systems have evolved programmable strategies to control rewriting. The introduction of programmed strategies into rewriting systems is fairly recent. A primitive form of control is provided by conditional rewrite systems, in which boolean-valued guards on rules allow the ring condition for a rule to depend upon state variables, as well as critical pair matching between a pattern and a term. However, conditional rewrite rules are not su cient for programming large applications. A programming language is needed to express strategies. In this paper we advocate that strategies are best understood as programs over a domain of terms and that the control of these programs involves recursion and nondeterministic choice. A natural formalism for reasoning about such programs is a weakest-precondition logic in a modality in which predicates are interpreted over a domain of nitely-branching trees. As a modal logic for terms, we have adopted the -calculus [7], enriched with modalities that express path quanti cation in terms. We develop rules in this logic for the constructions of Stratego [12, 13, 15, 14], a domain-speci c language designed speci cally for programming strategies for term rewriting. Stratego provides a compositional semantics with explicit recursion, which allows strategies to be applied at sites deep within terms. Patterns are rst-class constructs of Stratego. As in a logic programming language, conditional control in Stratego is based upon the success or nite failure of strategies, rather than if-then-else expressions that test conditions coded as boolean values. Although essentially a rst-order language, Stratego also employs a restricted form of higher-order strategies, namely term congruences, which lift term constructors to strategy constructors.