Qualitative Subdivision Algebra: Moving Towards the Quantitative

Qualitative Reasoning has achieved significant successes in providing abstractions for processes and functionality. However many important questions (e.g modeling shape) are not resolvable at purely qualitative levels, and traditional hybrid models have resorted to quantitative data when it came to these issues. This requires that different sets of data be maintained and updated continuously, and does not provide a mechanism for obtaining the "adequate" level of approximation. An alternative approach for hybrid qualitative-quantitative reasoning is that of subdividing the qualitative regions, where the desired discretization is determined by the needs of the application. We define a class of such subdivision algebras, called `k-proper'', which limit the uncertainty involved in a transitive inference to at most k regions. We construct such algebras for linear and circular domains (-,0,+ and Front/Left/Behind/Right), and construct models for 2D shapes and of 3D positional information with Qualitative Vector Algebra. We use this algebra to build a 3D spatial reasoning system, and a 2D shape modeler.