Positive/negative conditional rewriting

Abstract

This paper introduces positive/negative conditional term rewriting systems, with rules of the generic form:

$$u = v \wedge \bar u \ne \bar v \Rightarrow \lambda \to \rho$$

as often appear in algebraic specifications. We consider the algebraic semantics of such systems (viewed as sets of axioms). They do not in general have initial models; however, we show that they admit quasi-initial models, that are in some sense extremal within the class of all models. We then introduce the subclass of reducing rewrite systems, constrained by the condition:$\lambda > \rho , u, v, \bar u, \bar v$(for some reduction ordering >). For such systems, we show that an optimal rewrite relation → may be defined, and constructed as a "limit". We prove the total validity of an interpreter that computes the normal forms of terms for →. It is then shown that when → is confluent, the algebra of normal forms is a quasi-initial model. We state a general result about the converse. Lastly, we present a complete critical-pair criterion à la Knuth-Bendix to check for the confluence of reducing systems.