Functional dependencies in fuzzy databases

Functional dependencies, which have resemblance relations and weights, are formulated as being composed of integrity constraints in a fuzzy relational database based on possibility and necessity measures. Each tuple value has a compatibility degree of necessity and possibility with a functional dependency. Whether a tuple satisfies the functional dependency is determined by the comparison of the compatibility degree with the value of the membership attribute. Our formulation does not contain any parameters. We examine inference rules under two interpretations of functional dependencies. Under the interpretation corresponding to using Godel implication, Armstrong's inference rules are sound and complete for any functional dependency with no weights, and the extended Armstrong inference rules are sound and complete for any functional dependency with weights. On the other hand, under the interpretation corresponding to using Diens implication, Armstrong's inference rules are sound and complete for functional dependencies with identity relations and no weights, and the extended Armstrong inference rules 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 and no weights and for those with resemblance relations and weights, respectively. In these cases, another set of sound inference rules holds.