Mathematical Foundations of Qualitative Reasoning

We examine different formalisms for modeling qualitatively physical systems and their associated inferential processes that allow us to derive qualitative predictions from the models. We highlight the mathematical aspects of these processes along with their potential and limitations. However, the modeling process encounters difficulties from both ends: A model must adapt to the knowledge available and the task it is built for. The possible limitations of traditional numeric methods with respect to these problems mean qualitative models can be a good alternative: (1) qualitative models cope with uncertain and incomplete knowledge, (2) a qualitative model output equals an infinity of numeric runs that are obtained at once in compact form, (3) the qualitative predictions provide the relevant qualitative distinctions in the system's behavior, and (4) the modeling primitives allow for a more intuitive interpretation. A system's evolution can be tackled in discrete terms by defining states and events that trigger transitions between states.