MEASURING PROPERTIES OF FUZZY-SETS - A GENERAL TECHNIQUE AND ITS USE IN FUZZY QUERY EVALUATION

A fuzzy set can be viewed as a weighted-max combination or as a convex combination of a family of nested ordinary sets. The latter point of view allows the interpretation of a fuzzy set as a consonant random set. Then any numerical set function applied to a fuzzy set leads to a result which is a random number. The expectation attached to this random number provides a scalar evaluation of the fuzzy set with respect to the considered set-function. This approach is very general and can be used for various kinds of measurements: cardinality, confidence measures attached to the occurrence of events, diameter or perimeter of a fuzzy region, distance between two sets, etc. Also the fuzziness of the set to evaluate naturally induces a fuzzy set of possible evaluations and expected values of the lower and upper bounds of this fuzzy set of numbers can be computed. The weighted max-combination view which leads to another type of scalar evaluation is also briefly discussed. In this paper the above approach is more particularly applied to the treatment of queries asking for some global evaluation of a set of items specified by means of a fuzzy property. This kind of query can be managed by languages like SQL in case of a crisp set of items. When this set is fuzzily specified, a scalar answer can still be provided as well as a fuzzy-valued or interval-valued one if we want to keep track of the effect of the fuzziness of the query. The case where the data base contains precise information, but also the more general case of fuzzy information data bases are considered. © 1990.