2006): ‘Admissible and Restrained Revision

Abstract

As part of the justification of their proposed framework for iterated belief revision Darwiche and Pearl advanced a convincing argument against Boutilier’s natural revision, and provided a prototypical revision operator which fits into their scheme. We show that the Darwiche-Pearl arguments lead naturally to the acceptance of a smaller class of operators which we refer to as admissible. These are characterised in terms of syntactic as well as semantic postulates. Admissible revision ensures that the penultimate input is not ignored completely, thereby eliminating natural revision, but includes the prototypical Darwiche-Pearl operator, Nayak’s well-known lexicographic revision operator, and a newly introduced operator called restrained revision. We give a syntactic and a semantic characterisation of restrained revision, and demonstrate that it satisfies desirable properties. In particular, we show that it is the most conservative of admissible revision operators, while lexicographic revision is the least conservative. This makes an interesting comparison with the Darwiche-Pearl framework in which lexicographic revision is also the least conservative, but natural revision is the most conservative. In a sense, then, restrained revision can be seen as an appropriate replacement for natural revision. Finally we show that restrained revision can also be viewed as a composite operator, consisting of natural revision preceded by an application of a “backwards revision ” operator previously studied by Papini. 1