Lazy narrowing: Strong completeness and eager variable elimination

Abstract

Narrowing is an important method for solving unification problems in equational theories that are presented by confluent term rewriting systems. Because narrowing is a rather complicated operation, several authors studied calculi in which narrowing is replaced by more simple inference rules. This paper is concerned with one such calculus. Contrary to what has been stated in the literature, we show that the calculus lacks strong completeness, so selection functions to cut down the search space are not applicable. We prove completeness of the calculus and we establish an interesting connection between its strong completeness and the completeness of basic narrowing. We also address the eager variable elimination problem. It is known that many redundant derivations can be avoided if the variable elimination rule, one of the inference rules of our calculus, is given precedence over the other inference rules. We prove the completeness of a restricted variant of eager variable elimination in the case of orthogonal term rewriting systems.