Probabilistic Query Evaluation with Bag Semantics

TL;DR

This paper investigates the complexity of evaluating conjunctive queries on probabilistic databases under bag semantics, extending known results from set semantics to more general, unbounded multiplicity models.

Contribution

It introduces a framework for probabilistic query evaluation with bag semantics, establishing complexity results and a dichotomy for self-join free conjunctive queries.

Findings

Expectations of answer tuple multiplicities are efficiently computable.

A complexity dichotomy exists: some queries are polynomial-time solvable, others are #P-hard.

The paper extends known set semantics results to bag semantics with unbounded multiplicities.

Abstract

We study the complexity of evaluating queries on probabilistic databases under bag semantics. We focus on self-join free conjunctive queries, and probabilistic databases where occurrences of different facts are independent, which is the natural generalization of tuple-independent probabilistic databases to the bag semantics setting. For set semantics, the data complexity of this problem is well understood, even for the more general class of unions of conjunctive queries: it is either in polynomial time, or #P-hard, depending on the query (Dalvi & Suciu, JACM 2012). A reasonably general model of bag probabilistic databases may have unbounded multiplicities. In this case, the probabilistic database is no longer finite, and a careful treatment of representation mechanisms is required. Moreover, the answer to a Boolean query is a probability distribution over (possibly all) non-negative integers, rather than a probability distribution over { true, false }. Therefore, we discuss two flavors of probabilistic query evaluation: computing expectations of answer tuple multiplicities, and computing the probability that a tuple is contained in the answer at most k times for some parameter k. Subject to mild technical assumptions on the representation systems, it turns out that expectations are easy to compute, even for unions of conjunctive queries. For query answer probabilities, we obtain a dichotomy between solvability in polynomial time and #P-hardness for self-join free conjunctive queries.