Similarity, Equivalence and Decomposition of Epistemic States

Abstract

Since epistemic states will be used in this book for representing nonmonotonic inference and belief change, we will be primarily interested in the properties of epistemic states that could influence the behavior of generated inference relations and belief change operations. In order to single out the properties of epistemic states that are relevant, we will introduce in this chapter a number of notions of equivalence for epistemic states that will preserve the generated inference relations and belief sets. We will start with the strongest notion of similarity for epistemic states that will be invariant under the basic belief change operations. In other words, similar epistemic states will produce the same response under any future change made to these states; borrowing the terminology of [FUKV86], they will be equivalent forever. We will also show that any epistemic state is decomposable in this sense into a set of linear epistemic states. In addition, we will introduce the notion of a selection function which will determine the ‘inference profile’ of an epistemic state. Epistemic states generating the same selection function will be called equivalent. The notions of equivalence and similarity will supply us with powerful tools for investigating properties of epistemic states that are essential in determining their behavior in nonmonotonic inference and belief change.