Approximate reasoning in knowledge-based fuzzy sets

A fuzzy set is considered to represent deterministic uncertainty called fuzziness. In deterministic uncertainty of a fuzzy set, one may subjectively determine a membership function of a given element by one's knowledge. Different persons with different knowledge may provide different membership functions for elements in a universe with respect to a given fuzzy set. Here, knowledge plays important roles in determining or defining a fuzzy set. By adding a component of knowledge, we generalized a definition of a fuzzy set based on probability theory. In addition, by using a fuzzy conditional probability relation, granularity of knowledge is given in two frameworks, crisp granularity and fuzzy granularity. Also, two asymmetric similarity classes or subsets of knowledge are considered. When fuzzy sets represent problems or situations, a granule of knowledge might describe a class (group) of knowledge (persons) who has similar point of view in dealing the problems. In the paper, special attention is given to approximate reasoning in knowledge-based fuzzy sets representing fuzzy production rules as usually used in fuzzy expert systems.