Belief Change and Base Dependence

The AGM paradigm of belief change studies the dynamics of belief states in light of new information. For theoretical simplification, AGM idealizes a belief state as a belief set: a set of logical formulas that is closed under implication. A variant to the original AGM approach generalizes belief sets into belief bases which are not necessarily deductively closed. Many authors have argued that, compared to belief sets, belief bases are easier to represent in computers, more expressive and more inconsistency-tolerant. A strong intuition for belief change operations, Gärdenfors suggests, is that formulas that are independent of a change should remain intact. Linking belief change and dependence is significant because, for example, it can narrow the number of formulas considered during a belief change operation. Then, based on Gärdenfors’ intuition, Fariñas and Herzig axiomatize a dependence relation, and formalize the connection between dependence and belief change. The work in this thesis is also based on Gärdenfors’ intuition. We first introduce the notion of base dependence as a relation between formulas with respect to some belief base (instead of a belief set). After an axiomatization of base dependence, we present a formalization of the connection between base dependence and a particular belief base change operation, saturated kernel contraction. We also prove that base dependence is a reversible generalization of Fariñas 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 Fariñas and Herzig’s formalism is that it meets other criteria for dependence, namely, Keynes’ conjunction criterion for dependence (CCD) and Gärdenfors’ conjunction criterion for independence (CCI). We show that our base dependence formalism also meets these criteria. More interestingly, we offer a new and more specific conjunction criterion for dependence that implies both CCD and CCI, and show our base dependence formalism also meets this new criterion.