System Z FO : Default reasoning with system Z-like ranking functions for unary first-order conditional knowledge bases

Abstract A propositional conditional of the form ( B | A ) , representing the default rule “If A , then usually B ”, goes beyond the limits of classical logic, and the semantics of a knowledge base consisting of such conditionals must take into account the three-valued nature of conditionals. Ordinal conditional functions (OCF), also called ranking functions, assign a degree of implausibility to possible worlds and provide a convenient semantic framework for conditionals. To each consistent knowledge base R , system Z associates a unique minimal OCF accepting R , enabling inductive reasoning from this model. In this article, we present an approach of transforming the ideas of the popular system Z to the first-order case. We develop system Z FO that considers first-order conditionals of the form ( B ( x ) | A ( x ) ) . The notion of tolerance used in system Z is extended to the notion of tolerance pairs, taking also a partition of the domain elements into account. We show that each tolerance pair induces a system Z-like ranking function, and we provide an algorithm systematically generating all tolerance pairs. We introduce the notion of minimal tolerance pairs and a corresponding refinement of the algorithm computing all minimal tolerance pairs. An inference relation based on minimal tolerance pairs is presented, and it is shown that system Z FO properly generalizes the basic ideas underlying system Z.