Euler Diagram Transformations

Abstract

Euler diagrams are a visual language which are used for purposes such as the presentation of set-based data or as the basis of visual logical languages which can be utilised for software specification and reasoning. Such Euler diagram reasoning systems tend to be defined at an abstract level, and the concrete level is simply a visualisation of an abstract model, thereby capturing some subset of the usual boolean logic. The visualisation process tends to be divorced from the data transformation process thereby affecting the user's mental map and reducing the effectiveness of the diagrammatic notation. Furthermore, geometric and topological constraints, called wellformedness conditions, are often placed on the concrete diagrams to try to reduce human comprehension errors, and the effects of these conditions are not modelled in these systems.
We view Euler diagrams as a type of graph, where the faces that are present are the key features that convey information and we provide transformations at the dual graph level that correspond to transformations of Euler diagrams, both in terms of editing moves and logical reasoning moves. This original approach gives a correspondence between manipulations of diagrams at an abstract level (such as logical reasoning steps, or simply an update of information) and the manipulation at a concrete level. Thus we facilitate the presentation of diagram changes in a manner that preserves the mental map. The approach will facilitate the realisation of reasoning systems at the concrete level; this has the potential to provide diagrammatic reasoning systems that are inherently different from symbolic logics due to natural geometric constraints. We provide a particular concrete transformation system which
preserves the important criteria of planarity and connectivity, which may form part of a framework encompassing multiple concrete systems each adhering to different sets of wellformedness conditions.