A Model-Theoretic Approach to Belief Revision in Multi-Agent Belief Logic and Its Syntactic Characterizations

Abstract

Belief change studies how an agent modifies her beliefs on receiving new information. However, so far most research on belief change works on beliefs represented in propositional logic. There have been many works on integrating belief revision with reasoning about actions, and some works extending belief change from propositional logic to epistemic logics. In this paper, we study revision on beliefs of a third person represented with the multi-agent KD45 logic. Our formal technique is analogous to that of distance-based belief revision in propositional logic: to revise a KB by a formula, select from models of the formula those that are closest to models of the KB. To this end, a challenge is that in modal logics, a formula may have infinitely many Kripke models. To tackle this, we propose a variant of Moss’ canonical formulas called alternating canonical formulas, treat them as models for formulas, and define a notion of distance between them, based on the Hausdorff distance between two sets. We show that our revision satisfies all of the AGM postulates. To give syntactic characterizations of our revision, we make use of a normal form for KD45n called alternating cover disjunctive formulas (ACDFs). We give syntactic characterizations firstly on fragments of ACDFs called proper ACDFs and alternating cover conjunctive formulas (ACCFs), and finally on the whole ACDFs.