A FFD Inference System

Abstract

Fuzzy functional dependencies (FFDs) express the relationships among the attributes of objects. If we have a relation scheme R composed of a set of attributes U, and a FFD X→_θY for R, then X→_θY is meant to all the relations of R. That is, X→_θY holds in R means that X→_θY holds in all the relations of R. Since X→_θY is an integrity constraint, the enforcement of X→_θY is to guarantee that X→_θY is satisfied by all the relations of R. Moreover, given a FFD set F, the enforcement of FFDs in F is not only to guarantee that these FFDs in F are satisfied by all the relations of R, but also to guarantee that each FFD logically implied by F is satisfied by all the relations of R. However, to tell all FFDs which are logically implied by F or to tell whether a given FFD is logically implied by F does not turn out to be a trivial task. Intuitively, if we know that X→_θY holds in R, we may expect X→_αY holds in R as well for α < θ. If X→_θY holds in R, then we may expect XZ functionally determines YZ to a degree at least θ for all the relations of R. Additionally, if X→_αY holds in R and Y→_βZ holds in R, then we may expect to have X functionally determines Z to a certain degree x. An intuitive guess could be X = min(α,β) because (1) if X functionally determines Y to a degree θ and Y functionally determines Z to a degree θ then we may expect that X functionally determines Z to a same degree θ; (2) if X functionally determines Y to a degree α ≥ θ and Y functionally determines Z to β ≥ θ then at least we expect that X functionally determines Z to a degree θ. This intuition conforms to what the extended Armstrong’s axioms represent. Therefore, like the classical model, we may think of a number of inference rules that could be applied, based on F, to deduce other FFDs (if any). The basic requirement for being an inference rule is that the rule itself should be correct: that is, any FFD deduced using the rule from F must be really a FFD that is logically implied by F. A system composed of such inference rules is called correct if and only if each rule is correct.