Spatial Aggregate: Theory and Application to Qualitative Physics

EEective reasoning about a physical system requires an appropriate mapping from the system characteristics to abstractions that match the impedance of the task at hand. In Qualitative Physics, three ontological abstractions are widely used: device, process, and constraint. We present a framework and a new ontological abstraction, the eld ontology, to unify many reasoning tasks involving image-like analogue representations such as the velocity eld for uid motion, phase space for dynamical systems, and connguration space for mechanism analysis. A eld is deened as a mapping from one continuum (say R m) to another (say R n). A eld is typically information-rich. To transform the information-rich representation into symbolic descriptions well suited for explaining the structure and behavior of the system, we propose a theory of multi-layer spatial abstractions. Abstraction in each layer is represented by a neighborhood graph whose nodes are objects and edges are adjacency relations. The multi-layer theory has two advantages: (1) A nonlocal property of a lower layer can be redescribed as a local property of a higher layer, and (2) On each layer the neighborhood graph provides a common interface to support identical modular computations. To illustrate our theory, we examine the computational structure of three implemented programs { KAM, MAPS, and J&S { that integrate symbolic, numerical, and visual reasoning. We show a small set of generic operators that construct, transform, lter, classify, and search neighborhood graphs capture the commonalities of these programs. We develop a language, a way of organizing programs around neighborhood graphs, to make programs written in this style clear.