Reasoning with Multivariant Qualitative Spaces

One of the fundamental problems in qualitative reasoning is to represent n-dimensional variable spaces, where states of a physical system are delimited by general hypersurfaces. It is impossible to handle such a situation by using conventional qualitative values, because they can take only the form of `lattices' of hyperplanes, each of which is orthogonal to a coordinate axis. We propose a framework for solving the problem by directly constructing symbolic descriptions of phase spaces which conserve the property of qualitative reasoning, i.e., we handle a class of physical models rather than a single instance. In the theory, the notions of multivariant qualitative space and qualitative region are used, which are respectively extensions of the concepts of quantity space and qualitative value into higher dimensions. A multivariant qualitative space consists of qualitative regions, topology, and geometric attributes of the regions. The geometric attributes qualitatively represent the normal vectors of relevant hypersurfaces. Behavior reasoning is carried out based on vector operations of the normal vectors and the time derivation of a state vector. The scheme has been implemented and applied to reasoning about the motion of rigid bodies constrained in a plane.