A FFD Inference System
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.
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