Graph Granularity through Bi-intuitionistic Modal Logic

This thesis concerns the use of a bi-intuitionistic modal logic, UBiSKt, in the field of Knowledge Representation and Reasoning. The logic is shown to be able to represent qualitative spatial relations between subgraphs at different levels of detail, or granularity. The level of detail is provided by the modal accessibility relation R defined on the set of nodes and edges. The connection between modal logic and mathematical morphology is exploited to study notions of granulation on subgraphs, namely the process of changing granularity, and to define qualitative spatial relations between these “granular” regions. In addition, a special case of graph and hypergraph granularity is analysed, namely when the accessibility relation gives rise to a partition of the underlying set of nodes and edges. Different S5 extensions of intuitionistic modal logic are considered and compared in the thesis. It is shown that these logics, and their associated semantics, provide different ways of partitioning a graph, a hypergraph, or, more generally, a partially ordered set.