|||Martin Spott, Using classic approximation techniques for approximate reasoning, Proc. of FUZZ-IEEE '98, p. 909-914, May 1998.
Fuzzy reasoning systems, in particular fuzzy controllers, are vague and/or uncertain descriptions of mappings from an input space into an output space. A single fuzzy rule represents partial information about this mapping; the aggregation of all rules forms a fuzzy approximation of the mapping (a fuzzy relation). From this point of view the approximation capabilities of the fuzzy system play a fundamental role for its quality. This is especially true for learning systems that require flexible adaptation mechanisms. Common fuzzy techniques are primarily invented for information processing and not for approximation purposes. Consequently, memory consumption and computation costs become infeasible, when high approximation accuracy should be achieved. For this reason an approach to fuzzy reasoning is proposed that is built on classic approximation techniques. More precisely, fuzzy relations are approximated by B-spline surfaces. The main advantages are: a well founded approximation theory, great approximation capabilities, and the possibility to interpret the B-spline basis as fuzzy rules. In addition, natural representations of positive rules (support distributions, Mamdani-inference) and negative rules (possibility distributions, GĂ¶del-inference) are presented. In this context, a very fast inference mechanism for multi-stage reasoning is proposed. In the best case, the computation costs are the square number of rules. Altogether the new method provides a memory saving representation of fuzzy rule bases, a fast inference mechanism and high approximation power. These features recommend the method not only for control purposes but also in more general settings of approximate reasoning.