Can Datalog Be Approximated?

Abstract

In this paper, we investigate whether recursive Datalog predicates can be approximated by finite unions of conjunctive queries. We introduce a quantitative notion of error and examine two types of approximation, namely,absoluteapproximation andrelative approximation. We also stipulate that the approximations obey certain qualitative criteria, namely we require them to beupper envelopes or lower envelopesof the Datalog predicate they approximate. We establish that absoluteapproximationby finite unions of conjunctive queries is not possible, which means that no unbounded Datalog predicate can be approximated by a finite union of conjunctive queries in such a way that the error is bounded uniformly by the same constant on all finite databases. After this, we examinerelative approximations, i.e., approximations that guarantee bounds for the error relative to the size of the Datalog predicate under consideration. Although such approximations exist in some cases, we show that for several large and well-studied classes of unbounded Datalog predicates it is not possible to find finite unions of conjunctive queries that satisfy the aforementioned qualitative criteria and have the property that the relative error of the approximation is bounded by a constant. Finally, we consider first-order approximations and obtain sharp negative results for the approximability of thetransitive closurequery by first-order queries.