Interpretability, Complexity, and Modular Structure of Fuzzy Systems

Zadeh’s original motivation for fuzzy logic and the Fuzzy Rule-Based System (FRBS) was linguistic and hence possessed highly interpretable components. But as the complexity of a typical FRBS increases, it often becomes more like an uninterpretable neural network, and the Principle of Incompatibility predicts a degradation in interpretability for the same accuracy. This is particularly true of the so-called Takagi-Sugeno-Kang (TSK), or simply the Sugeno, approximator. We argue that imposing additional structure on the TSK system can significantly improve the tradeoff inherent in the Principle of Incompatibility. A promising structure was proposed recently in which the membership functions are local and sufficiently differentiable, and the consequent polynomials are rule-centered. This structure leads to the general interpretation that the consequent polynomials are Taylor series expansions. On this interpretation, a foundation for an algebra and a calculus of FRBSs can be built. We will illustrate these aspects of the proposed structure and discuss issues of modularity, functionality, and scalability of FRBSs.