A Qualitative Account of Discrete Space

Computations in geographic space are necessarily based on discrete versions of space, but much of the existing work on the foundations of GIS assumes a continuous infinitely divisible space. This is true both of quantitative approaches, using R, and qualitative approaches using systems such as the Region-Connection Calculus (RCC). This paper shows how the RCC can be modified so as to permit discrete spaces by weakening Stell’s formulation of RCC as Boolean connection algebra to what we now call a connection algebra. We show how what was previously considered a problem—with atomic regions being parts of their complements—can be resolved, but there are still obstacles to the interplay between parthood and connection when there are finitely many regions. Connection algebras allow regions that are atomic and also regions that are boundaries of other regions. The modification of the definitions of the RCC5 and RCC8 relations needed in the context of a connection algebra are discussed. Concrete examples of connection algebras are provided by abstract cell complexes. In order to place our work in context we start with a survey of previous approaches to discrete space in GIS and related areas.