An Axiomatic Approach to Defining Approximation Measures for Functional Dependencies

We consider the problem of defining an approximation measure for functional dependencies (FDs). An approximation measure for X → Y is a function mapping relation instances, r, to non-negative real numbers. The number to which r is mapped, intuitively, describes the “degree” to which the dependency X → Y holds in r. We develop a set of axioms for measures based on the following intuition. The degree to which X → Y is approximate in r is the degree to which r determines a function from ΠX(r) to ΠY (r). The axioms apply to measures that depend only on frequencies (i.e. the frequency of x ∈ ΠX(r) is the number of tuples containing x divided by the total number of tuples). We prove that a unique measure satisfies these axioms (up to a constant multiple), namely, the information dependency measure of [5]. We do not argue that this result implies that the only reasonable, frequency-based, measure is the information dependency measure. However, if an application designer decides to use another measure, then the designer must accept that the measure used violates one of the axioms.