Modal Logic for Granularity

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The aim of this project is to develop a logic which allows the representation of statements such as "At all levels of detail ..." and "At some level of detail ...". The main area of application is to reasoning about spatial information, especially in a discrete space such as a graph representing a network, or an image built of pixels with a graph modelling adjacency. The project builds on work using a non-classical modal logic, BISKT, for describing spatial relations at a single level of detail in a discrete space. The project will investigate bisimulation and the Hennessy-Milner property for the BISKT logic, as this is part of the underlying modal theory that has not been explored. In order to study granularity, the perspective of co-algebras will be applied. The existing BISKT semantics can be expressed in terms of T-coalgebras for a suitable endofunctor, T, on the category of pre-ordered sets and order-preserving functions. This setting represents a very natural space to develop the BISKT semantics and opens up a new line of research where, the objective would be to provide a coalgebraic interpretation of semantic and syntactic properties of BISKT.