The application of level-2 fuzzy sets in fuzzy and uncertain data modeling

A formal framework for the uniform representation and manipulation of fuzzy and/or uncertain data is presented. This framework can be used within the context of (the formal definition of) e.g. fuzzy and uncertain databases and knowledge bases. The presented framework is based on the definition of a generalized type system, which is a generalization of a crisp type system and establishes the definition of so-called generalized types. Each generalized type is characterized by a generalized domain which contains the valid values for the type and by an implicitly defined behavior. Each generalized domain is a generalization of a crisp domain; as it is defined as a set of level-2 fuzzy sets, which in their turn are all defined over the elements of the crisp domain. This approach allows to represent fuzzy data, uncertain data and uncertainty about fuzzy data in a uniform and advantageous way. The behavior of a generalized type is defined by means of a set of operators, which are all defined over the generalized domain of the generalized type, and by a set of axioms. It is shown how these sets can be obtained as generalizations of their crisp counterparts.