Notions of Bisimulation for Heyting-Valued Modal Languages

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 HennessyMilner 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. ∗The first author is supported by the FCT grant SFRH/BPD/35000/2007. Part of the work was carried out while he was affiliated to the Institut de Matemàtica, Universitat de Barcelona, under European Commission grant MRTN-CT-2004-512234 (MODNET)