Maintaining Consistency of Imprecise Relations

Abstract

We extend functional dependencies (FDs), which are the most fundamental integrity constraints that arise in practice in relational databases, to be satisfied in an imprecise relation. The problem we tackle is the following: given an imprecise relation r over a relation schema R and a set of FDs F over R, what is the most precise approximation of r, which is also consistent with respect to F. We formalize the notion of an imprecise relation by defining tuple values to be sets of values rather than just single values as is the case when the information is precise. We interpret each value in such a set as being equally likely to be the true value. This gives rise to equivalence classes of equally likely values thus allowing us to define the merge of an imprecise relation r which replaces values in r by their equivalence class. We also define a partial order on merged imprecise relations leading to the notion of an imprecise relation being less precise than another imprecise relation. This partial order induces a lattice on the set of merged imprecise relations. An imprecise relation is consistent with respect is consistent with respect to a set of FDs F if it satisfies F. Satisfaction of an FD in an imprecise relation is defined in terms of values being equally likely rather than equal. We show that Armstrong's axiom system is sound and complete for FDs being satisfied in imprecise relations. We redefine the chase procedure for an imprecise relation r over R as a means of maintaining consistency of r with respect to F. Our main results is that the output chase (r,F) of the chase procedure is the most precise approximation of r with respect to F in the following sense. It is shown to be the join of all consistent imprecise relations s with respect to F in the following sense. It is shown to be the join of all consistent imprecise relations s such that s is a merged imprecise relation that is less precise than r. It is also shown that chase (r,F) can be computed in polynomial time in the sizes of r and F.