Probabilistic Query Evaluation with Bag Semantics

Typically, probabilistic databases (PDBs) are probability distributions over the subsets of a finite set of facts. However, many practical implementations of relational databases use a bag semantics that allows multiple copies of a fact. As facts may appear arbitrarily often, extending PDBs to a bag semantics naturally leads to infinite PDBs, the mathematical framework of which as only been introduced recently by Grohe and Lindner (PODS 2019, ICDT 2020). In this paper, we study the problem of query evaluation over PDBs with bag semantics (bag PDBs). We focus on tuple-independent bag PDBs, which means that the multiplicities of different facts are independent. In the set-based setting, the complexity of probabilistic query evaluation is well-understood for unions of conjunctive queries (UCQs): it is either in polynomial time, or #P-hard (Dalvi and Suciu, JACM 2012). The setting with bag semantics differs substantially. As PDBs are no longer finite, we need feasible representations. Moreover, the answer to a Boolean query is a probability distribution over numbers, rather than a single probability. This yields different reasonable definitions of the query evaluation problem. First, we discuss computing the expectation and variance of query answers. Surprisingly, under mild assumptions on the representation, both problems are solvable in polynomial time for all UCQs. Second, we investigate the problem of computing the probability that the answer is at most k, where k is a parameter. While key arguments from prior work do not carry over to our setting, we show that the original dichotomy for Boolean self-join free CQs persists for most representations. Still, there are representation systems where the problem is always solvable in polynomial time.