Natural rewriting and narrowing for general term rewriting systems

We address the problem of efficient rewriting and narrowing strategies for general term rewriting systems. Several strategies have been proposed over the last two decades, the most efficient of all being the natural rewriting and narrowing strategies of Escobar. All the strategies so far, including natural rewriting and narrowing, assume that the given term rewriting system is left-linear and constructor-based. Although these restrictions are reasonable for some functional programming languages, they limit the expressive power of equational languages, and they preclude certain applications of narrowing to equational theorem proving and to languages combining equational and logic programming. In this paper, we propose a conservative generalization of natural rewriting and narrowing that does not require the rules to be left-linear and constructor-based. We also establish the soundness and completeness of these generalizations.