On the optimal approximation of queries using tractable propositional languages

Abstract

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.

Anytime approximation in probabilistic databases

“…It considers relational algebra queries and pc-tables that go beyond these restrictions. This article unifies work reported previously on decomposition-based approximation for positive queries [41] and for queries with negation [20] and on model-based approximation [19]. The approximation algorithms used by the probabilistic database management systems MystiQ [46], MayBMS [27], and MCDB [29] are variations of the Monte Carlo algorithm presented above.…”

“…Within this framework, recent work characterizes lower bounds for conjunctive queries where the bounds themselves are expressed as conjunctive queries [4,19].…”

“…This article describes the approximate probability computation algorithm used by the SPROUT query engine [18,27] and thereby unifies aspects of this algorithm that have previously been reported separately [19,20,41]. The algorithm works for arbitrary propositional formulas and thus for any relational algebra query and pc-table, it is anytime, and admits absolute and relative error guarantees.…”

Negative Knowledge for Certain Query Answers

“…), relate them to other approximate query answering notions, both in databases [16,23,15,34] and in AI [27,32], and to existing approaches that explain why tuples do not appear in query answers [36,21]. As for quality of approximations of certain answers, these are best confirmed experimentally, as was demonstrated recently [19].…”

Querying and Learning in Probabilistic Databases

“…Building upon results of [28,49,55], we start by considering bounds for propositional formulas, from which we extend to the more general case of first-order lineage. Then, we show that these bounds converge monotonically to the probabilities P (φ i ) of each query answer φ i , as we continue to ground Φ.…”