Quantification of fuzzy mappings: a relevance of rule-based architectures

Fuzzy mappings are fundamental constructs of granular computing and rule-based systems, in particular. The notion of relevance of the fuzzy mappings is instrumental in the quantification of their quality prior to any detailed construction. For the purposes of such quantification, we introduce shadowed sets and discuss their role therein. It is revealed that shadowed sets deliver an interesting three-valued quantification of the relevance of the fuzzy mappings (labeled as acceptable mapping, marginal mapping, and a lack of mapping). The paper includes a number of detailed calculations concerning two commonly exploited classes of triangular and Gaussian fuzzy sets.