Narrowing Directed by a Graph of Terms

Narrowing provides a complete procedure to solve equations modulo confluent and terminating rewriting systems. But it seldom terminates. This paper presents a method to improve the termination. The idea consists in using a finite graph of terms built from the rewriting system and the equation to be solved, which helps one to know the narrowing derivations possibly leading to solutions. Thus, the other derivations are not computed. This method is proved complete. An example is given and some improvements are proposed. 1 I n t r o d u c t i o n Solving equations (unification) modulo term rewriting systems is an important problem, which is necessary to combine functional and logic programming, for example as in EQLOG langage [Goguen-Meseguer-86]. In this framework, completeness and termination of unification methods are desirable. Indeed, during a clause superposition attempt, it is interesting to detect the unsatisflability of the equation to be solved, which implies that the superpositlon is not possible. Conditional completion procedures also need methods that solve the equations appearing in the conditions. If such an equation is unsatisfiable, the conditional rewrite rule containing it can be deleted, because this rule can't be applied. And to detect that, a complete and terminating procedure is necessary. This paper presents a method that improves the termination of basic narrowing. Basic narrowing gives a complete set of solutions modulo confluent and terminating rewriting systems, by computing all the narrowing derivations issued from the equation to be solved [Hullot-80]. Actually, only the derivations that lead to a syntactically unifiable equation give solutions. Thus, the others are useless. Our idea consists in using this fact to prune useless paths from the search tree. Let's look at a simple example. Consider the following confluent and terminating rewriting system : r l : h(f(g(Y))) -+ h(/(y)) r2 : h ( 0 ) ~ 0