Bi-Rewrite Systems

In this article we propose an extension of term rewriting techniques to automate the deduction in monotone pre-order theories. To prove an inclusion a b from a given set I of them, we generate from I, using a completion procedure, a bi-rewrite system hR ; R i, that is, a pair of rewrite relations ??! R and ??! R , and seek a common term c such that a ??! R c and b ??! R c. Each component of the bi-rewrite system ??! R and ??! R is allowed to be a subset of the corresponding inclusion relation or deened by the theory of I. In order to assure the decidability and completeness of such proof procedure we study the termination and commutation of ??! R and ??! R. The proof of the commutation property is based on a critical pair lemma, using an extended deenition of critical pair. We also extend the existing techniques of rewriting modulo equalities to bi-rewriting modulo a set of inclusions. Although we center our attention on the completion process a la Knuth-Bendix, the same notion of extended critical pair is suitable of being applied to the so called unfailing completion procedures. The completion process is illustrated by means of an example corresponding to the theory of the union operator. We show that connuence of extended critical pairs may be ensured adding rule schemes. Such rule schemes contain variables denoting schemes of expressions, instead of expressions. We propose the use of the linear second-order typed-calculus to codify these expression schemes. Although the general second-order uniication problem is only semi-decidable, the second-order uniication problems we need to solve during the completion process are decidable.