On an Information Theoretic Approximation Measure for Functional Dependencies

We investigate the problem of de ning an approximation measure for functional dependencies (FDs). For xed sets of attributes, X and Y , an approximation measure is a function which maps relation instances to real numbers. The number to which an instance is mapped, intuitively, describes the strength of the dependency, X ! Y , in that instance. We de ne an approximation measure for FDs based on a connection between Shannon's information theory and relational database theory. Our measure is normalized to lie between zero and one (inclusive), and maps a relation instance to zero if and only if X ! Y holds in the instance. Hence, the smaller the number to which an instance is mapped, the \closer" X ! Y is to being an FD in the instance. To put our measure in context, we compare it to a slight variation of a measure previously de ned by Kivinen and Mannila, g3. We denote the variation as ĝ3, although, our results, essentially, apply unchanged to g3. The purpose of comparing our measure with ĝ3 is to develop a deeper understanding of not only our measure, but also, ĝ3. Moreover, we gain a deeper understanding of the natural intuitive notion of an approximate FD. We observe that our measure and ĝ3 agree at their extremes but are quite di erent in-between. As a result, we conclude that our measure and ĝ3 are signi cantly di erent. An interesting question emerges from this conclusion: is there a rigorous way to determine when one measure better captures the meaning of the degree to which an FD is approximate?