Talk: Kote Razmadze (Tbilisi/Göttingen)

Title:

Analysis of Multimodal Logics of Modal Maps

Abstract:

In this talk we consider mappings as structures comprised of domain, codomain and a relation between them representing a mapping, i.e. a graph of the map. Such structures can naturally be viewed as Kripke frames, and their modal logic is quite simple and well known. If, however, the domain and the codomain are not simply sets but carry some structure of their own, and the map preserves (part of) this structure, it might be possible to use a separate set of modal connectives to interpret on this additional structure. We are interested in bi-modal logics that arise in this fashion. Namely, we consider maps between Kripke frames and use one modality to reason about the Kripke structure of the domain and the codomain, while another modality is reserved for reasoning about the mapping itself. We investigate whether various properties of the map (monotonicity, p-morphicity) can be expressed in this bi-modal language and indeed axiomatize the relevant logics. We also consider maps between topological spaces and axiomatize the bi-modal logics of open, continuous and interior maps. We also conjecture that these logics have the finite model property, however at this stage the full details of the proof are not available. In this presentation we just set the stage and establish basics for the possible future research of modally reasoning about mappings between structured sets.