Fuzzy relational database induced by conditional probability relations
In 1982, Buckles and Petry proposed fuzzy relational database for incorporating non-ideal or fuzzy information in a relational database. The fuzzy relational database relies on the specification of similarity relation in order to distinguish each scalar domain in the fuzzy database. These relations are reflexive, symmetric, and max-min transitive. In 1989, Shenoi and Melton extended the fuzzy relational database model of Buckles and Petry to deal with proximity relation for scalar domain. Since reflexivity and symmetry are the only constraints placed on proximity relations, proximity relation is considered as a generalization of similarity relation. However, we realized that naturally relation between fuzzy information is not symmetric. Here, we consider using conditional probability relation to represent similarity between two fuzzy data. Related to the properties of conditional probability relation, we introduce an interesting mathematical relation, called weak similarity relation, as generalization of similarity relation as well as proximity relation in which conditional probability relation is regarded as a concrete example of the weak similarity relation. In this paper, we propose design of fuzzy relational database to deal with conditional probability relation for scalar domain. These relations are reflexive and not symmetric. In addition, we define a notion of asymmetric redundant tuple based on two interpretations generalizing the concept of redundancy in classical relational database. In the relation to data querying, we discuss partitioning of domains with the objective of developing similarity class. Finally, we propose a new definition of partial fuzzy functional dependency (PFFD). Fuzzy functional dependency (FFD) as an extension of functional dependency (FD), usually used in design of fuzzy relational database, can be generated by the PFFD. Inference rules that are similar to Armstrong's Axioms for the FFD are both sound and complete.
fuzzy relational database fuzzy functional dependency fuzzy conditional probability relation fuzzy set redundancy of tuples