According to Boutillier, Darwiche and Pearl and others, principles for iterated revision can be characterised in terms of changing beliefs about conditionals. For iterated contraction a similar formulation is not known. This is especially because for iterated belief change the connection between revision and contraction via the Levi and Harper identity is not straightforward, and therefore, characterisation results do not transfer easily between iterated revision and contraction. In this article, we develop an axiomatisation of iterated contraction in terms of changing conditional beliefs. We prove that the new set of postulates conforms semantically to the class of operators like the ones given by Konieczny and Pino Pérez for iterated contraction. 1 Introduction For the three main classes of theory change, revision, expansion and contraction, different characterisations are known , which are heavily supported by the correspondence between revision and contraction via the Levi and Harper identities [13, 17].
While research on iterated revision is predominant in the field of iterated belief change, the class of iterated contraction operators received more attention in recent years. In this article, we examine a non-prioritized generalisation of iterated contraction. In particular, the class of weak decrement operators is introduced, which are operators that by multiple steps achieve the same as a contraction. Inspired by Darwiche and Pearl's work on iterated revision the subclass of decrement operators is defined. For both, decrement and weak decrement operators, postulates are presented and for each of them a representation theorem in the framework of total preorders is given. Furthermore, we present two types of decrement operators which have a unique representative.
As partial justification of their framework for iterated belief revision Darwiche and Pearl convincingly argued against Boutilier's natural revision and provided a prototypical revision operator that 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. Admissible revision ensures that the penultimate input is not ignored completely, thereby eliminating natural revision, but includes the Darwiche-Pearl operator, Nayak's lexicographic revision operator, and a newly introduced operator called restrained revision. We demonstrate that restrained revision is the most conservative of admissible revision operators, effecting as few changes as possible, while lexicographic revision is the least conservative, and point out 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. Finally, we propose the establishment of a principled approach for choosing an appropriate revision operator in different contexts and discuss future work.
This paper presents an approach to belief revision in which revision is a function from a belief state and a finite set of formulas to a new belief state. In the interesting case, the set for revision S may be inconsistent but individual members of S are consistent. We argue that S will still contain interesting information regarding revision; in particular, maximum consistent subsets of S will determine candidate formulas for the revision process, and the agent's associated faithful ranking will determine the plausibility of such candidate formulas. Postulates and semantic conditions characterizing this approach are given, and representation results are provided. As a consequence of this approach, we argue that revision by a sequence of formulas, usually considered as a problem of iterated revision, is more appropriately regarded as revision by the possibly-inconsistent set of these formulas. Hence we suggest that revision by a sequence of formulas is foremost a problem of (uniterated) set revision.
This paper investigates belief revision where the underlying logic is that governing Horn clauses. It proves to be the case that classical (AGM) belief revision doesn’t immediately generalise to the Horn case. In particular, a standard construction based on a total preorder over possible worlds may violate the accepted (AGM) postulates. Conversely, Horn revision functions in the obvious extension to the AGM approach are not captured by total preorders over possible worlds. We address these difficulties by first restricting the semantic construction to "well behaved" orderings; and second, by augmenting the revision postulates by an additional postulate. This additional postulate is redundant in the AGM approach but not in the Horn case. In a representation result we show that these two approaches coincide. Arguably this work is interesting for several reasons. It extends AGM revision to inferentially-weaker Horn theories; hence it sheds light on the theoretical underpinnings of belief change, as well as generalising the AGM paradigm. Thus, this work is relevant to revision in areas that employ Horn clauses, such as deductive databases and logic programming, as well as areas in which inference is weaker than classical logic, such as in description logic.