A Transformation System for Unique Minimal Normal Forms of Conditional Knowledge Bases

Abstract

Conditional knowledge bases consisting of sets of conditionals are used in inductive nonmonotonic reasoning and can represent the defeasible background knowledge of a reasoning agent. For the comparison of the knowledge of different agents, as well as of different approaches to nonmonotonic reasoning, it is beneficial if these knowledge bases are as compact and straightforward as possible. To enable the replacement of a knowledge base \(\mathcal R\) by a simpler, but equivalent knowledge base \(\mathcal R'\), we propose to use the notions of elementwise equivalence or model equivalence for conditional knowledge bases. For elementwise equivalence, we present a terminating and confluent transformation system on conditional knowledge bases yielding a unique normal form for every \(\mathcal R\). We show that an extended version of this transformation system takes model equivalence into account. For both transformation system, we prove that the obtained normal forms are minimal with respect to subset inclusion and the corresponding notion of equivalence.