Expressing discrete spatial relations under granularity

The logic UBiSKt is a bi-intuitionistic modal logic with universal modalities having a semantics in which formulae are interpreted as subgraphs. We use the logic to formalize the idea of looking at subgraphs at different levels of detail. We show how the logic can be used to define various spatial relations between subgraphs. These relations are considered with respect to change in level of detail. We use an existing formulation of generalized partitions on graphs and hypergraphs in terms of relations to give a novel generalization of the classical modal logic S5 to a bi-intuitionistic modal logic we call UBiSKtS5.