On Inference Rules of Dependencies in Fuzzy Relational Data Models: Functional Dependencies

Functional dependencies, which have resemblance relations and weights, are formulated as one of constraints composing integrity constraints in a fuzzy relational database based on possibility and necessity measures. Each tuple value in a relation has a membership attribute value that expresses to what degree that tuple value belongs to the relation in the fuzzy relational database. A compatibility degree of a tuple value with a functional dependency is calculated in necessity and possibility. To what extent a tuple satisfies the functional dependency is determined by the comparison of the compatibility degree with its value of membership attribute. Our formulation does not contain any parameter. We examine inference rules under two requirements for functional dependencies. Under the requirement corresponding to using Gödel implication, Armstrong’s inference rules are sound and complete for any functional dependency with no weights, and the extended inference rules of Armstrong’s ones are sound and complete for any functional dependency with weights. On the other hand, under the requirement corresponding to using Dienes implication, Armstrong’s inference rules are sound and complete for functional dependencies with identity relations and no weights, and the extended inference rules of Armstrong’s ones are sound and complete for functional dependencies with identity relations and weights. However, Armstrong’s inference rules and their extended inference rules are not sound for functional dependencies with resemblance relations. In these cases, a different set of sound inference rules is obtained, but the completeness remains open.