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, Fariñas 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 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 more specific criterion that implies both CCD and CCI, and show our base dependence formalism also meets this new criterion.

The AGM paradigm of belief change studies the dynamics of belief states in light of new information. Finding, or even approximating, those beliefs that are dependent on or relevant to a change is valuable because, for example, it can narrow the set of beliefs considered during belief change operations. A strong intuition in this area is captured by Gärdenforss preservation criterion (GPC), which suggests that formulas independent of a belief change should remain intact. GPC thus allows one to build dependence relations that are linked with belief change. Such dependence relations can in turn be used as a theoretical benchmark against which to evaluate other approximate dependence or relevance relations. Fariñas and Herzig axiomatize a dependence relation with respect to a belief set, and, based on GPC, they characterize the correspondence between AGM contraction functions and dependence relations. In this paper, we introduce base dependence as a relation between formulas with respect to a belief base, and prove a more general characterization that shows the correspondence between kernel contraction and base dependence. At this level of generalization, different types of base dependence emerge, which we show to be a result of possible redundancy in the belief base. We further show that one of these relations that emerge, strong base dependence, is parallel to saturated kernel contraction. We then prove that our latter characterization is a reversible generalization of Fariñas and Herzigs characterization. That is, in the special case when the underlying belief base is deductively closed (i.e., it is a belief set), strong base dependence reduces to dependence, and so do their respective characterizations. Finally, an intriguing feature of Fariñas and Herzigs formalism is that it meets other criteria for dependence, namely, Keyness conjunction criterion for dependence (CCD) and Gärdenforss conjunction criterion for independence (CCI). We prove that our base dependence formalism also meets these criteria. Even 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.

The AGM paradigm of belief change studies the dynamics of belief states in light of new information. Finding, or even approximating, dependent or relevant beliefs to a change is valuable because, for example, it can narrow the set of beliefs considered during belief change operations. Gärdenfors' preservation criterion (GPC) suggests that formulas independent of a belief change should remain intact. GPC allows to build dependence relations that are theoretically linked with belief change. Such dependence relations can in turn be used as a theoretical benchmark against which to evaluate other approximate dependence or relevance relations. There are already some studies, based on GPC, on the parallelism between belief change and dependence. One study offers a dependence relation parallel to AGM contraction for belief sets. Another study links base dependence relation to a more general belief base contraction, saturated kernel contraction. Here we offer yet a more general parallelism between kernel contraction and base dependence. At this level of generalization, different types of base dependence emerge. We prove that this differentiation of base dependence types is a result of possible redundancy in the base. This provides a theoretical means to distinguish between redundant and informative parts of a belief base.