Uncertainty That Counts

Abstract

Uncertainty is modeled by a multibase (db,μ) where db is a database with zero or more primary key violations, and μ associates a multiplicity (a positive integer) to each fact of db. In data integration, the multiplicity of a fact g can indicate the number of data sources in which g was found. In planning databases, facts with the same primary key value are alternatives for each other, and the multiplicity of a fact g can denote the number of people in favor of g.

A repair of db is obtained by selecting a maximal number of facts without ever selecting two distinct facts of the same relation that agree on their primary key. Every repair has a support count, which is the product of the multiplicities of its facts.

For a fixed Boolean query q, we define σ CERTAINTY(q) as the following counting problem: Given a multibase (db,μ), determine the weighted number of repairs of db that satisfy q. Here, every repair is weighted by its support count. We illustrate the practical significance of this problem by means of examples.

For conjunctive queries q without self-join, we provide a syntactic characterization of the class of queries q such that σ CERTAINTY(q) is in P; for queries not in this class, σ CERTAINTY(q) is $\sharp$ P-hard (and hence highly intractable).