On The Semantics of Belief Revision Systems

We give semantics to belief revision operators that satisfy the Alchourron-Garden-fors-Makinson postulates by presenting an epistemic logic such that, for any such revision operator, the result of a revision can be described by a sentence in this logic. In our logic, the fact that the agent's set of beliefs is φ is represented by the sentence 0φ, where 0 is Levesque's 'only know' operator. Intuitively, 0φ is read as 'φ is all that is believed.' The fact that the agent believes ψ is represented by the sentence Bψ, read in the usual way as 'ψ is believed'. The connective d represents update as defined by Katsuno and Mendelzon. The revised beliefs are represented by the sentence 0φdBψ. We show that for every revision operator that satisfies the AGM postulates, there is a model for our epistemic logic such that the beliefs implied by the sentence 0φdBψ in this model correspond exactly to the sentences implied by the theory that results from revising φ by ψ. This means that reasoning about changes in the agent's beliefs reduces to model checking of certain epistemic sentences. The negative result in the paper is that this type of formal account of revision cannot be extended to the situation where the agent is able to reason about its beliefs. A fully introspective agent cannot use our construction to reason about the results of its own revisions, on pain of triviality.