On the Representation of Darwiche and Pearl’s Epistemic States for Iterated Belief Revision

Abstract

The seminal characterization of iterated belief revision was proposed by Darwiche and Pearl, which uses an abstract notion of epistemic states. In this work we look for a canonical representation of these epistemic states. Total preorders are not expressive enough to be used as such a canonical representation. Actually, we show that some operators can even not be represented on a countable epistemic space. Nonetheless, under a very reasonable assumption on the epistemic space, we show that OCFs (Ordinal Conditional Functions) can be considered as a canonical representation.

Iteration of Iterated Belief Revision

“…We can show that these conditions are true counterparts of the postulates (the proof is omitted for space reasons, but can be found at (Schwind, Konieczny, and Pérez 2023) and is similar to the part of the proof of Proposition 2 showing the correspondence between (CEi) and (CREi), for each i ∈ {1, 2}): Proposition 6. For each i ∈ {1, 2}, a DP revision operator • satisfies (CEiw) if and only if its corresponding DP assignment satisfies (CREiw).…”

“…Proof Sketch. We only show that • N and • R do not satisfy (CE2) and that • N does not satisfy (CE1), but the rest of the proof is available at (Schwind, Konieczny, and Pérez 2023). It is enough from Proposition 2 to show that their corresponding assignments do not satisfy the semantic counterparts of (CE1) and (CE2).…”

“…Furthermore, we will focus on TPOrepresentable revision operators, which can be viewed as transitions between TPOs, elements of the TPO-based epistemic space. For a precise definition and to see that this is just a subclass of DP iterated revision operators, please refer to (Schwind, Konieczny, and Pino Pérez 2022).…”

“…Note also that they are DP revision operators that are not TPO-representable, i.e., that require more complex structure, such as an OCF (Ordinal Conditional Function) (Spohn 1988), for being formally defined (Schwind, Konieczny, and Pino Pérez 2022).…”

Credible Models of Belief Update

“…Another distinction between revision and update is that revision involves a selection process, where we choose the most plausible models of the new evidence based on our current beliefs, while update involves a transition process, where we consider the transitions caused by the change in the new evidence for each model of our current beliefs. However, while there have been many extensions proposed and studied for belief revision operators, such as syntax dependence (Hansson 1999), non-prioritized revision (Schlechta 1997;Makinson 1998;Hansson 1998;Fermé and Hansson 1999;Hansson et al 2001;Booth et al 2012), and iteration (Darwiche and Pearl 1997;Booth and Meyer 2006;Jin and Thielscher 2007;Konieczny and Pino Pérez 2008;Schwind, Konieczny, and Pérez 2022), belief update has not received much attention since its initial characterization by Katsuno and Mendelzon (Katsuno and Mendelzon 1992). There have been some criticisms and adaptations proposed in (Herzig and Rifi 1999;Boutilier 1995;Lang 2007), but the original proposal by Katsuno and Mendelzon remains the standard one.…”