From Semi-Heuristic Fuzzy Techniques To Optimal Fuzzy Methods: Mathematical Foundations and Applications

Fuzzy techniques have been successfully used in various application areas ranging from control to image processing to decision making. In all these applications, there is usually: a general idea, and then there are several possible implementations of this idea; e.g., we can use: { diierent membership functions, { diierent \and" and \or" operations, { diierent defuzziications, etc. In the rst approximation, the results are usually reasonably robust and independent on this choice, so any heuristic or semi-heuristic choice works OK. However: if we want to further improve the semi-heuristic \good enough" control or image processing techniques, we must actually make the selection that would lead to an optimal fuzzy method. Even if we know the exact optimality criterion, the resulting optimization problem is so complicatedly non-linear that, often, no known optimization techniques can be used. In many real-life situations, we do not even know the exact criterion: the relative quality of diierent controls or image processing techniques may be not by exact numerical criteria, but rather by expert opinions which are, themselves, fuzzy. 1 In this paper, we describe a general mathematical technique for solving such optimization problems. This technique is based on the group-theoretic (symmetry) methodology, a methodology which is one of the main successful tools of modern physics. This methodology has lead us to the justiication of many heuristic formulas in fuzzy logic, fuzzy control, neural networks, etc. (many examples are presented in our 1997 Kluwer book Applications of continuous mathematics to computer science). We also present new applications of this techniques, including applications: to optimal fuzzy control, and to optimal fuzzy image processing.