We examine the notion of bisimulation and its ramifications, in the context of the family of Heyting-valued modal languages introduced by M. Fitting. Each modal language in this family is built on an underlying space of truth values, a Heyting algebra H. All the truth values are directly represented in the language, which is interpreted on relational frames with an H-valued accessibility relation. We define two notions of bisimulation that allow us to obtain truth invariance results. We provide game semantics and, for the more interesting and complicated notion, we are able to provide characteristic formulae and prove a Hennessy-Milner type theorem. If the underlying algebra H is finite, Heyting-valued modal models can be equivalently reformulated to a form relevant to epistemic situations with many interrelated experts. Our definitions and results draw inspiration from this formulation, which is of independent interest to Knowledge Representation applications.
Multiple Belief Change extends the classical AGM framework for Belief Revision introduced by Alchourron, Gardenfors, and Makinson in the early '80s. The extended framework includes epistemic input represented as a (possibly infinite) set of sentences, as opposed to a single sentence assumed in the original framework. The transition from single to multiple epistemic input worked out well for the operation of belief revision. The AGM postulates and the system-of-spheres model were adequately generalized and so was the representation result connecting the two. In the case of belief contraction however, the transition was not as smooth. The generalized postulates for contraction, which were shown to correspond precisely to the generalized partial meet model, failed to match up to the generalized epistemic entrenchment model. The mismatch was fixed with the addition of an extra postulate, called the limit postulate, that relates contraction by multiple epistemic input to a series of contractions by single epistemic input. The new postulate however creates problems on other fronts. First, the limit postulate needs to be mapped into appropriate constraints in the partial meet model. Second, via the Levi and Harper Identities, the new postulate translates into an extra postulate for multiple revision, which in turn needs to be characterized in terms of systems of spheres. Both these open problems are addressed in this article. In addition, the limit postulate is compared with a similar condition in the literature, called (K*F), and is shown to be strictly weaker than it. An interesting aspect of our results is that they reveal a profound connection between rationality in multiple belief change and the notion of an elementary set of possible worlds (closely related to the notion of an elementary class of models from classical logic).
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.