Generalizing Parikh's Criterion for Relevance-Sensitive Belief Revision

Parikh proposed his relevance-sensitive axiom to remedy the weakness of the classical AGM paradigm in addressing relevant change. An insufficiency of Parikh's criterion, however, is its dependency on the contingent beliefs of a belief set to be revised, since the former only constrains the revision process of splittable theories (i.e., theories that can be divided in mutually disjoint compartments). The case of arbitrary non-splittable belief sets remains out of the scope of Parikh's approach. On that premise, we generalize Parikh's criterion, introducing (both axiomatically and semantically) a new notion of relevance, which we call relevance at the sentential level. We show that the proposed notion of relevance is universal (as it is applicable to arbitrary belief sets) and acts in a more refined way as compared to Parikh's proposal; as we illustrate, this latter feature of relevance at the sentential level potentially leads to a significant drop in the computational resources required for implementing belief revision. Furthermore, we prove that Dalal's popular revision operator respects, to a certain extent, relevance at the sentential level. Last but not least, the tight relation between local and relevance-sensitive revision is pointed out.