A Modal Walk Through Space
M. Aiello, J. van Benthem

Abstract:
We investigate the major mathematical theories of space from a modal
standpoint: topology, affine geometry, metric geometry, and vector
algebra. This allows us to see new fine-structure in spatial patterns
which suggests analogies across these mathematical theories in terms
of modal, temporal, and conditional logics. Throughout the modal walk
through space, expressive power is analyzed in terms of language
design, bisimulations, and correspondence phenomena. The result is
both unification across the areas visited, and the uncovering of
interesting new questions.

Keywords: affine geometry, elementary geometry, betweenness, nearness,
mathematical morphology, modal logics of space, spatial reasoning