A Set-Theoretical Foundation of Qualitative Reasoning and its Application to the Modeling of Economics and Business Management Problems

Abstract

The qualitative reasoning (QR) field has developed various representation and reasoning methods for the modeling with incomplete information or incomplete knowledge. While most uncertain reasoning approaches describe uncertain or imprecisely known information as probability distribution functions, qualitative reasoning bases its model specification on qualitative descriptions that are derived from known qualitative system properties. Problems are formulated as sets of qualitative constraints and their analysis is performed by applying a qualitative calculus. This paper presents a general, unifying theory of the various existing qualitative reasoning systems that includes, as special cases, reasoning methods that use representations of qualitative differential equations and qualitative difference equations. Based on set theory, our QR framework describes fundamental concepts such as qualitative models and solutions, and relates them to the quantitative analogues of its underlying quantitative reference system. Our motivation arises from the types of models found in the management sciences. Thus we emphasize the significance of discrete, dynamic models and optimization models in the business management and economics fields, both of which have received less attention in current QR research. Finally, we extend our theoretical framework to include an approach to qualitative optimization.