On Strengthening the Logic of Iterated Belief Revision: Proper Ordinal Interval Operators
This work addresses foundational issues in belief revision logic for AI and knowledge representation, offering a more nuanced framework that avoids reductionism while maintaining modesty, though it appears incremental in building on prior principles.
The paper tackles the problem of strengthening the logic of iterated belief revision by introducing novel conditions that supplement existing DP postulates and principle (P), resulting in operators representable with proper ordinal interval assignments, which are more fine-grained than simple world orderings. It shows these operators satisfy iterated versions of AGM-era postulates like Superexpansion, which are not generally sound for other operators.
Darwiche and Pearl's seminal 1997 article outlined a number of baseline principles for a logic of iterated belief revision. These principles, the DP postulates, have been supplemented in a number of alternative ways. Most of the suggestions made have resulted in a form of `reductionism' that identifies belief states with orderings of worlds. However, this position has recently been criticised as being unacceptably strong. Other proposals, such as the popular principle (P), aka `Independence', characteristic of `admissible' revision operators, remain commendably more modest. In this paper, we supplement both the DP postulates and (P) with a number of novel conditions. While the DP postulates constrain the relation between a prior and a posterior conditional belief set, our new principles notably govern the relation between two posterior conditional belief sets obtained from a common prior by different revisions. We show that operators from the resulting family, which subsumes both lexicographic and restrained revision, can be represented as relating belief states that are associated with a `proper ordinal interval' (POI) assignment, a structure more fine-grained than a simple ordering of worlds. We close the paper by noting that these operators satisfy iterated versions of a large number of AGM era postulates, including Superexpansion, that are not sound for admissible operators in general.