On uniform belief revision

Rational belief-change policies are encoded in the so-called AGM revision functions, defined in the prominent work of Alchourrón, Gärdenfors and Makinson. The present article studies an interesting class of well-behaved AGM revision functions, called herein uniform-revision operators (or UR operators, for short). Each UR operator is uniquely defined by means of a single total preorder over all possible worlds, a fact that in turn entails a significantly lower representational cost, relative to an arbitrary AGM revision function, and an embedded solution to the iterated-revision problem, at no extra representational cost. Herein, we first demonstrate how weaker, more expressive—yet, more representationally expensive—types of uniform revision can be defined. Furthermore, we prove that UR operators, essentially, generalize a significant type of belief change, namely, parametrized-difference revision. Lastly, we show that they are (to some extent) relevance-sensitive, as well as that they respect the so-called principle of kinetic consistency.

Incompatibilities Between Iterated and Relevance-Sensitive Belief Revision

The AGM paradigm for belief change, as originally introduced by Alchourron, Gärdenfors and Makinson, lacks any guidelines for the process of iterated revision. One of the most influential work addressing this problem is Darwiche and Pearl's approach (DP approach, for short), which, despite its well-documented shortcomings, remains to this date the most dominant. In this article, we make further observations on the DP approach. In particular, we prove that the DP postulates are, in a strong sense, inconsistent with Parikh's relevance-sensitive axiom (P), extending previous initial conflicts. Immediate consequences of this result are that an entire class of intuitive revision operators, which includes Dalal's operator, violates the DP postulates, as well as that the Independence postulate and Spohn's conditionalization are inconsistent with axiom (P). The whole study, essentially, indicates that two fundamental aspects of the revision process, namely, iteration and relevance, are in deep conflict, and opens the discussion for a potential reconciliation towards a comprehensive formal framework for knowledge dynamics.

Rational Defeasible Belief Change

We present a formal framework for modelling belief change within a nonmonotonic reasoning system. Belief change and non-monotonic reasoning are two areas that are formally closely related, with recent attention being paid towards the analysis of belief change within a non-monotonic environment. In this paper we consider the classical AGM belief change operators, contraction and revision, applied to a defeasible setting in the style of Kraus, Lehmann, and Magidor. The investigation leads us to the consideration of the problem of iterated change, generalising the classical work of Darwiche and Pearl. We characterise a family of operators for iterated revision, followed by an analogous characterisation of operators for iterated contraction. We start considering belief change operators aimed at preserving logical consistency, and then characterise analogous operators aimed at the preservation of coherence—an important notion within the field of logic-based ontologies.

Non-Prioritized Iterated Revision: Improvement via Incremental Belief Merging

In this work we define iterated change operators that do not obey the primacy of update principle. This kind of change is required in applications when the recency of the input formulae is not linked with their reliability/priority/weight. This can be translated by a commutativity postulate that asks the result of a sequence of changes to be the same whatever the order of the formulae of this sequence. Technically then we end up with a sequence of formulae that we have to combine in order to obtain a meaningful belief base. Belief merging operators are then natural candidates for this task. We show that we can define improvement operators using an incremental belief merging approach. We also show that these operators can not be encoded as simple preorders transformations, contrary to most iterated revision and improvement operators.

Strong Syntax Splitting for Iterated Belief Revision

AGM theory is the most influential formal account of belief revision. Nevertheless, there are some issues with the original proposal. In particular, Parikh has pointed out that completely irrelevant information may be affected in AGM revision. To remedy this, he proposed an additional axiom (P) aiming to capture (ir)relevance by a notion of syntax splitting. In this paper we generalize syntax splitting from logical sentences to epistemic states, a step which is necessary to cover iterated revision. The generalization is based on the notion of marginalization of epistemic states. Furthermore, we study epistemic syntax splitting in the context of ordinal conditional functions. Our approach substantially generalizes the semantical treatment of (P) in terms of faithful preorders recently presented by Peppas and colleagues.

Revision by History

This article proposes a solution to the problem of obtaining plausibility information, which is necessary to perform belief revision: given a sequence of revisions, together with their results, derive a possible initial order that has generated them; this is different from the usual assumption of starting from an all-equal initial order and modifying it by a sequence of revisions. Four semantics for iterated revision are considered: natural, restrained, lexicographic and reinforcement. For each, a necessary and sufficient condition to the existence of an order generating a given history of revisions and results is proved. Complexity is proved coNP complete in all cases but one (reinforcement revision with unbounded sequence length).