Kernel Contraction and Base Dependence: Redundancy in the Base Resulting in Different Types of Dependence Mehrdad Oveisi James P. Delgrande Fred Popowich School of Computing Science Simon Fraser University Burnaby, BC, Canada {oveisi,jim,popowich}@cs.sfu.ca Francis Jeffry Pelletier Department of Philosophy University of Alberta Edmonton, AB, Canada francisp@ualberta.ca Abstract The AGM paradigm of belief change studies the dy- namics of belief states in light of new information. Finding, or even approximating, dependent or rel- evant beliefs to a change is valuable because, for example, it can narrow the set of beliefs consid- ered during belief change operations. Gärdenfors’ preservation criterion (GPC) suggests that formu- las 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 rela- tions. There are already some studies, based on GPC, on the parallelism between belief change and dependence. One study offers a dependence re- lation parallel to AGM contraction for belief sets. Another study links base dependence relation to a more general belief base contraction, saturated ker- nel contraction. Here we offer yet a more general parallelism between kernel contraction and base dependence. At this level of generalization, differ- ent 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. 1 Introduction Research into belief change provides formal means for incor- porating new and changing information. Alchourrón, Gär- denfors and Makinson [1985] provide the AGM paradigm of belief change that idealizes a belief state as a belief set K: a set of logical formulas that is closed under implication. One example of belief change operation on K is contraction which retracts α and other formulas from K as necessary to ensure α is not implied by the remaining formulas. During a belief change operation, beliefs independent of a change should remain intact. Gärdenfors [1990] states this intuition in the following preservation criterion: “If a belief state is revised by a sentence A, then all sentences in K that are independent of the validity of A should be retained in the revised state of belief” [Gärdenfors, 1990]. (GPC ) Then, based on GPC, Fariñas del Cerro and Herzig [1996] (FH) axiomatize a dependence relation, and formalize the connection between dependence and AGM contraction. A more practical and important variant of the original AGM approach uses belief bases instead of belief sets. Belief bases need not be deductively closed, and are usually finite. One very general class of base contraction is kernel contrac- tion [Hansson, 1994], which is a superclass of saturated ker- nel contraction, itself a superclass of AGM contraction. In our previous work [Oveisi et al., 2014], using belief bases instead of belief sets, we introduced base dependence as a (reversible) generalization of FH’s dependence. Based on the definitions presented in §6.1, here we will refer to this new dependence relation as strong base dependence. Thus, based on GPC, we had indeed established the correspondence be- tween strong base dependence and saturated kernel contrac- tion in that work. Indeed, as seen in both studies based on GPC mentioned above, GPC allows to build dependence relations that are the- oretically linked with belief change. Such dependence re- lations can in turn be used, for example, as a theoretical benchmark against which to evaluate other approximate de- pendence or relevance relations. Therefore, in this work, we aim to capture a yet more gen- eral base dependence relation to correspond to (full) kernel contraction, once again based on GPC. After providing the necessary background in §2, we present an in-depth motiva- tional example in §3 to show why connecting base depen- dence and kernel contraction via GPC is desirable. At this level of generalization, as discussed in §4, different types of base dependence emerge, namely strong base depen- dence and weak base dependence. In §5, weak base depen- dence is shown to be a result of redundancy in the base. In- deed, this is a second result from our study that can be used as a theoretical benchmark in other studies. The fact that weak base dependence captures redundancy may be exploited for various purposes. For example, one may use weak base de- pendence to distinguish between redundant and informative formulas in a belief base. In §6, we offer a generalization of