Functional dependencies in Horn theories

Abstract

This paper studies functional dependencies in Horn theories, both when the theory is represented by its clausal form and when it is defined as the Horn envelope of a set of models. We provide polynomial algorithms for the recognition of whether a given functional dependency holds in a given Horn theory, as well as polynomial algorithms for the generation of some representative sets of functional dependencies. We show that some problems of inferring functional dependencies (e.g., constructing an irredundant FD-cover) are computationally difficult. We also study the structure of functional dependencies that hold in a Horn theory, showing that every such functional dependency is in fact a single positive term Boolean function, and prove that for any Horn theory the set of its minimal functional dependencies is quasi-acyclic. Finally, we consider the problem of condensing a Horn theory, prove that any Horn theory has a unique condensation, and develop an efficient polynomial algorithm for condensing Horn theories.