Modal logics for region-based theories of space
Résumé
The aim of this talk is to give new kinds of propositional modal logics suitable for reasoning about regions in region-based theories of space. We call them region-based propositional modal logics of space – RPMLS. The language L(?,C) of RPMLS contains Boolean variables and standard Boolean operations needed for constructing Boolean terms interpreted later on by the corresponding regions in some models of region-based theory of space. The language contains two symbols: ? for the part-of relation and C for the contact relation. Atomic formulas are of the form a ? b and aCb, where a, b are Boolean terms, and complex formulas are built from the atomic ones by means of the propositional connectives. In a sense L(?,C) is a first-order language without quantifiers, but we call it modal because many things typical of ordinary modal languages can be applied to it: relational Kripke style semantics, filtration, modal definability, p-morphisms, canonical models, etc. The relational semantics corresponds to the discrete models of the region-based theory of space, based on adjacency spaces. That is why we sometimes use the name discrete semantics. We also give a topological semantics for the language in which regions are regular closed sets in some topological space. For the logics based on the discrete semantics, we establish various modal definability results. We give axiomatization and completeness theorems with respect to both relational and topological semantics for several important RCC-like logics. We show that the relational interpretation of the language of RPMLS is also suitable for studying graphs. We study some logics suitable for describing properties of graphs like the property of colourability and connectedness.