Belief Change and Base Dependence

A strong intuition for AGM belief change operations, Gärdenfors suggests, is that formulas that are independent of a change should remain intact. Based on this intuition, Farias and Herzig axiomatize a dependence relation w.r.t. a belief set, and formalize the connection between dependence and belief change. In this paper, we introduce base dependence as a relation between formulas w.r.t. a belief base. After an axiomatization of base dependence, we formalize the connection between base dependence and a particular belief base change operation, saturated kernel contraction. Moreover, we prove that base dependence is a reversible generalization of Farias and Herzig’s dependence. That is, in the special case when the underlying belief base is deductively closed (i.e., it is a belief set), base dependence reduces to dependence. Finally, an intriguing feature of Farias and Herzig’s formalism is that it meets other criteria for dependence, namely, Keynes’ conjunction criterion for dependence (CCD) and Grdenfors’ conjunction criterion for independence (CCI). We show that our base dependence formalism also meets these criteria. More interestingly, we offer a more specific criterion that implies both CCD and CCI, and show our base dependence formalism also meets this new criterion. Introduction Belief Change The AGM paradigm of belief change studies the dynamics of belief states in light of new information (Alchourrn, GŁrdenfors, and Makinson 1985). For theoretical simplification, AGM idealizes a belief state as a belief set or a theory: a set of logical formulas that is closed under implication. An important variant of the original AGM approach uses belief bases instead of using belief sets. Belief bases are not necessarily deductively closed, and they are usually finite. Thus, they are more suitable to be represented in finite machines. Also, many authors have argued that, compared to belief sets, belief bases are more expressive (Hansson 2003), Copyright c © 2014, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. and they are more tolerant of inconsistency (Hansson and Wassermann 2002). Therefore, belief bases can be more useful in practice than belief sets. Belief Change and Dependence A long standing intuition concerning belief change is that formulas independent of a change should remain intact (GŁrdenfors 1990). In Belief Change and Dependence, Farias del Cerro and Herzig (1996) (FH) attempt to ground this intuition by axiomatizing a particular dependence relation in a close relationship to belief change. FH’s work is particularly interesting and unique in the sense that it fits the original AGM model of belief change. Their stated aim is both “to give a formal account of the notion of dependence, and to employ it in belief change.” This deep integration into the AGM model sets apart their work from other works on relevance or dependence in the context of belief change. Belief Change and Base Dependence A natural next step is to find a similar connection between dependence and belief base contraction. We call such a dependence (or relevance) relation base dependence (or base relevance). In this work, we provide an axiomatization of base dependence, and establish its relation to belief base contraction. Interestingly, base dependence turns out to be a reversible generalization of FH’s dependence. That is, in the special case that a belief base is deductively closed (i.e., it is a belief set), the base dependence relation reduces to FH’s original dependence relation. Composite Dependence One interesting aspect of FH’s work is that some of the axioms that they use to capture the concept of dependence come from intuitions put forward previously. For example, Keynes (1921) holds that there is an intuitive relationship between relevance (dependence) and logical conjunction that should stay valid for any reasonable definition of relevance. Calling it the Conjunction Criterion for Dependence, CCD, FH 151 Proceedings of the Fourteenth International Conference on Principles of Knowledge Representation and Reasoning