A Constructor-Based Approach to Positive/Negative-Conditional Equational Specifications

We study algebraic specifications given by finite sets R of positive/negative-conditional equations (i.e. universally quantified first-order implications with a single equation in the succedent and a conjunction of positive and negative (i.e. negated) equations in the antecedent). The class of models of such a specification R does not contain in general a minimum model in the sense that it can be mapped to any other model by some homomorphism. We present a constructor-based approach for assigning appropriate semantics to such specifications. We introduce two restrictions: firstly, for a condition to be fulfilled we require the evaluation values of the terms of the negative equations to be in the constructor sub-universe which contains the evaluation values of all constructor ground terms; secondly, we restrict the constructor equations to have "Horn"-form and to be "constructor-preserving". A reduction relation for R is defined, which allows to generalize the fundamental results for positive-conditional rewrite systems. This reduction relation is monotonic w. r. t. consistent extension of the specification, which is of practical importance as it allows for an incremental construction process of complex specifications without destroying reduction steps which were possible before. Under the assumption of confluence, the factor algebra of the term algebra modulo the congruence of the reduction relation is a minimal model which is (beyond that) the minimum of all models that do not identify more objects of the constructor sub-universe than necessary. We define several kinds of compatibility of R with a reduction ordering for achieving decidability of reducibility, and present several criteria for the confluence of our reduction relation.