Computing query probability with incidence algebras

Abstract

We describe an algorithm that evaluates queries over probabilistic databases using Mobius' inversion formula in incidence algebras. The queries we consider are unions of conjunctive queries (equivalently: existential, positive First Order sentences), and the probabilistic databases are tuple-independent structures. Our algorithm runs in PTIME on a subset of queries called "safe" queries, and is complete, in the sense that every unsafe query is hard for the class F P #P . The algorithm is very simple and easy to implement in practice, yet it is non-obvious. Mobius' inversion formula, which is in essence inclusion-exclusion, plays a key role for completeness, by allowing the algorithm to compute the probability of some safe queries even when they have some subqueries that are unsafe. We also apply the same lattice-theoretic techniques to analyze an algorithm based on lifted conditioning, and prove that it is incomplete.

Anytime approximation in probabilistic databases

“…[12,40]. Beyond these restrictions, there is work on tractability of positive queries [10], on queries with aggregates [17,48] and with negation [20,58]. The approximate probability computation algorithm described in this article is used by the SPROUT query engine [18] to complement its optimized exact algorithm for tractable conjunctive queries without self-joins [40].…”

Querying and Learning in Probabilistic Databases

“…From the above consideration, it follows that the lineage formulas considered in our context may take more general forms than lineage formulas resulting from (unions of) conjunctive queries (UCQs) [14,16], which produce only formulas which are restricted to positive literals.…”

“…We thus refrain from full (i.e., logical) equivalence checks, as they are known to be co-N P-complete [11]. Following Equation (14), their respective lineages are:…”

“…Recent works on efficient probability computation in PDBs have addressed this problem mostly from two ends. The first group of approaches have restricted the class of queries, i.e., by focusing on safe query plans [16,14,17], or by considering a specific class of tuple-dependencies, commonly referred to as read-once functions [59]. In particular the second group of approaches allows for applying top-k style pruning methods [51,50,8,25] at the time when the query is processed.…”

On the Connections between Relational and XML Probabilistic Data Models

“…-For UCQs over tuple-independent databases, there is a dichotomy between FP #P -hard queries and FP queries [18]; however, the only known algorithm to determine the complexity of a query is doubly exponential in the query size, and the exact complexity of this problem is unknown. -A similar dichotomy, but with a polynomial-time test, holds for CQs without self-joins over BIDs.…”