This paper investigates the problem of approximating conjunctive queries without self-joins on probabilistic databases by lower and upper bounds that can be computed more efficiently. We study this problem via an indirection: Given a propositional formula Φ, find formulas in a more restricted language that are greatest lower bound and least upper bound, respectively, of Φ. We study bounds in the languages of read-once formulas, where every variable occurs at most once, and of read-once formulas in disjunctive normal form.We show equivalences of syntactic and model-theoretic characterisations of optimal bounds for unate formulas, and present algorithms that can enumerate them with polynomial delay. Such bounds can be computed by queries expressed using first-order queries extended with transitive closure and a special choice construct.Besides probabilistic databases, these results can also benefit the problem of approximate query evaluation in relational databases, since the bounds expressed by queries can be computed in polynomial combined complexity.

This paper presents a query evaluation technique for positive relational algebra queries with aggregates on a representation system for probabilistic data based on the algebraic structures of semiring and semimodule. The core of our evaluation technique is a procedure that compiles semimodule and semiring expressions into so-called decomposition trees, for which the computation of the probability distribution can be done in polynomial time in the size of the tree and of the distributions represented by its nodes. We give syntactic characterisations of tractable queries with aggregates by exploiting the connection between query tractability and polynomial-time decomposition trees.A prototype of the technique is incorporated in the probabilistic database engine SPROUT. We report on performance experiments with custom datasets and TPC-H data.

This article describes an approximation algorithm for computing the probability of propositional formulas over discrete random variables. It incrementally refines lower and upper bounds on the probability of the formulas until the desired absolute or relative error guarantee is reached. This algorithm is used by the SPROUT query engine to approximate the probabilities of results to relational algebra queries on expressive probabilistic databases.

This article charts the tractability frontier of two classes of relational algebra queries in tuple-independent probabilistic databases. The first class consists of queries with join, projection, selection, and negation, but without repeating relation symbols and union. The second class consists of quantified queries that express the following binary relationships amongst sets of entities: set division, set inclusion, set equivalence, and set incomparability. Quantified queries are expressible in relational algebra using join, projection, nested negation, and repeating relation symbols. Each query in the two classes has either polynomial-time or #P-hard data complexity and the tractable queries can be recognised efficiently. Our result for the first query class extends a known dichotomy for conjunctive queries without self-joins to such queries with negation. For quantified queries, their tractability is sensitive to their outermost projection operator: They are tractable if no attribute representing set identifiers is projected away and #P-hard otherwise.

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