Exact Model Counting of Query Expressions

Abstract

We prove exponential lower bounds on the running time of the state-of-the-art exact model counting algorithms—algorithms for exactly computing the number of satisfying assignments, or the satisfying probability, of Boolean formulas. These algorithms can be seen, either directly or indirectly, as building Decision-Decomposable Negation Normal Form (decision-DNNF) representations of the input Boolean formulas. Decision-DNNFs are a special case of d-DNNFs where d stands for deterministic. We show that any knowledge compilation representations from a class (called DLDDs in this article) that contain decision-DNNFs can be converted into equivalent Free Binary Decision Diagrams (FBDDs), also known as Read-Once Branching Programs, with only a quasi-polynomial increase in representation size. Leveraging known exponential lower bounds for FBDDs, we then obtain similar exponential lower bounds for decision-DNNFs, which imply exponential lower bounds for model-counting algorithms. We also separate the power of decision-DNNFs from d-DNNFs and a generalization of decision-DNNFs known as AND-FBDDs. We then prove new lower bounds for FBDDs that yield exponential lower bounds on the running time of these exact model counters when applied to the problem of query evaluation in tuple-independent probabilistic databases—computing the probability of an answer to a query given independent probabilities of the individual tuples in a database instance. This approach to the query evaluation problem, in which one first obtains the lineage for the query and database instance as a Boolean formula and then performs weighted model counting on the lineage, is known as grounded inference. A second approach, known as lifted inference or extensional query evaluation, exploits the high-level structure of the query as a first-order formula. Although it has been widely believed that lifted inference is strictly more powerful than grounded inference on the lineage alone, no formal separation has previously been shown for query evaluation. In this article, we show such a formal separation for the first time. In particular, we exhibit a family of database queries for which polynomial-time extensional query evaluation techniques were previously known but for which query evaluation via grounded inference using the state-of-the-art exact model counters requires exponential time.