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.

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*1 An extended abstract of this paper appeared in the Proceedings of the 20th Colloquium on Trees in Algebra and Programming, Aarhus, Lecture Notes in Computer Science 915, pp. 394–408, 1995.

*2 This work is partially supported by the Grant-in-Aid for Scientific Research (C) 06680300 and the Grantin-Aid for Encouragement of Young Scientists 06780229 of the Ministry of Education, Science and Culture of Japan.