Finding Stable Causal Interpretations of Equations

The causal ordering procedure of Iwasaki and Simon [5,61 provides a means for uncovering causal dependencies among variables constrained by a set of mathematical equations. This paper examines the procedure from a qualitative modeling viewpoint and addresses one of its limitations : context sensitivity. Causal dependencies predicted by the procedure may change depending on the context or scenario in which the underlying physical system operates . This prevents the qualitative modeler from using causal ordering to determine, a priori, a single causal interpretation of equations describing some phenomenon. We show that in some cases it is possible to find clusters of equations that possess causal stability. That is, their causal dependencies are the same in all scenarios consistent with the equations' modeling viewpoint. These unidirectional equations help the qualitative modeler by providing a stable, unambiguous causal interpretation . To identify such equations we define conditions sufficient to guarantee causal stability. In addition, we show that unidirectional equation sets are causally independent of equations outside their set . Thus, they add compositionality to the causal modeling task . Lastly, we demonstrate our ideas by uncovering the causal dependencies of Hooke's law, Gauss's law for electricity, and Bernoulli's equation .