Kernel Contraction and Base Dependence

Abstract

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 Gärdenfors’s 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 Herzig’s 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 Herzig’s formalism is that it meets other criteria for dependence, namely, Keynes’s conjunction criterion for dependence (CCD) and Gärdenfors’s 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.

Dependence in Propositional Logic: Formula-Formula Dependence and Formula Forgetting – Application to Belief Update and Conservative Extension

“…In (del Cerro and Herzig 1996), a belief state is represented as a belief set, i.e., an infinite set of formulas closed under implication. Oveisi et al (2017) pointed out that a belief base, which need not be deductively closed and is often finite, is a practical alternative for representing belief states. They also identified a similar connection between dependence and base contraction.…”

“…As mentioned in (Darwiche 1997), dependence is not only a philosophical notion but also a pragmatic notion. Over the past years, this notion has been widely used in many fields of AI, including automated reasoning (Levy, Fikes, and Sagiv 1997;Kautz, McAllester, and Selman 1997), knowledge compilation (Bryant 1992;Minato 1993;Darwiche 1997), reasoning about actions (Lin and Reiter 1997), and especially belief change (Zhang and Zhou 2009;Oveisi et al 2017). Belief update, a type of belief change, studies how an agent modifies her belief base φ in the presence of new information ψ in a dynamically changing environment.…”

Dependence in Propositional Logic: Formula-Formula Dependence and Formula Forgetting -- Application to Belief Update and Conservative Extension

“…As mentioned in (Darwiche 1997), dependence is not only a philosophical notion but also a pragmatic notion. Over the past years, this notion has been widely used in many fields of AI, including automated reasoning (Levy, Fikes, and Sagiv 1997; Kautz, McAllester, and Selman 1997), knowledge compilation (Bryant 1992;Minato 1993; Darwiche 1997), reasoning about actions (Lin and Reiter 1997), and especially belief change (Zhang and Zhou 2009;Oveisi et al 2017). Belief update, a type of belief change, studies how an agent modifies her belief base φ in the presence of new information ψ in a dynamically changing environment.…”