Spatial Logic and the Complexity of Diagrammatic Reasoning

Researchers have sought to explain the observed \eecacy" of diagrammatic reasoning (DR) via the notions of \limited abstraction" and inexpressivity 17, 20]. We argue that application of the concepts of computational complexity to systems of dia-grammatic representation is necessary for the evaluation of precise claims about their eecacy. We show here how to give such an analysis. Centrally, we claim that recent formal analyses of diagrammatic representations (DRs) (eg: 14]) fail to account for the ways in which they employ spatial relations in their representational work. This focus raises some problems for the expressive power of graphical systems, related to the topological and geometrical constraints of the medium. A further idea is that some diagrammatic reasoning may be analysed as a variety of topological inference 15]. In particular, we show how reasoning in some diagrammatic systems is of polynomial complexity , while reasoning in others is NP hard. A simple case study is carried out, which pinpoints the inexpressiveness and complexity of versions of Euler's Circles. We also consider the expressivity and complexity of cases where a limited number of regions is used, as well as Venn diagrams, \GIS-like" representations, and some three dimensional cases.