A Topological Characterisation of Belief Revision over Infinite Propositional Languages

Belief revision mainly concerns how an agent updates her belief with new evidence. The AGM framework of belief revision models belief revision as revising theories by propositions. To characterise AGM-style belief revision operators, Grove proposed in 1988 a representation model using systems of spheres. This ‘spheres’ model is very influential and has been extended to characterise multiple belief revision operators. Several fundamental problems remain unsettled regarding this ‘spheres’ model. In this paper we introduce a topology on the set of all worlds of an infinite propositional language and use this topology to characterise systems of spheres. For each AGM operator ∘, we show that, among all systems of spheres deriving ∘, there is a minimal one which is contained in every other system. We give a topological characterisation of these minimal systems. Furthermore, we propose a method for extending an AGM operator to a multiple revision operator and show by an example that the extension is not unique. This negatively answers an open problem raised by Peppas.

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