Proximity relations in the fuzzy relational database model

The fuzzy relational model of Buckles and Petry is a rigorous scheme for incorporating non-ideal or fuzzy information in a relational database. In addition to providing a consistent scheme for representing fuzzy information in the relational structure, the model possesses two critical properties that hold for classical relational databases. These properties are that no two tuples have identical interpretations and each relational operation has a unique result. The fuzzy relational model relies on similarity relations for each scalar domain in the fuzzy database. These relations are reflexive, symmetric, and max-min transitive. In addition to introducing fuzziness into the relational model, each similarity relation induces equivalence classes in its domain. It is the existence of these equivalence classes that provides the model with the important properties possessed by classical relational databases. In this paper, we extend the fuzzy relational database model of Buckles and Petry to deal with proximity relations for scalar domains. Since reflexivity and symmetry are the only constraints placed on proximity relations, they generalize the notion of similarity relations. We show that it is possible to induce equivalence classes from proximity relations; thus, the 'nice' properties of the fuzzy relational model of Buckles and Petry are preserved. Furthermore, the removal of the max-min transitivity restriction also provides database users with more freedom to express their value structures.