An Axiomatic Approach to Defining Approximation Measures for Functional Dependencies

Abstract

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 π _X(r)to π _Y (r). The axioms apply to measures that depend only on frequencies (i.e. the frequency of x ∈ π _X(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.

Work supported by National Science Foundation grant IIS-0082407.