Queries with Arithmetic on Incomplete Databases

Abstract

The standard notion of query answering over incomplete database is that of certain answers, guaranteeing correctness regardless of how incomplete data is interpreted. In majority of real-life databases, relations have numerical columns and queries use arithmetic and comparisons. Even though the notion of certain answers still applies, we explain that it becomes much more problematic in situations when missing data occurs in numerical columns. We propose a new general framework that allows us to assign a measure of certainty to query answers. We test it in the agnostic scenario where we do not have prior information about values of numerical attributes, similarly to the predominant approach in handling incomplete data which assumes that each null can be interpreted as an arbitrary value of the domain. The key technical challenge is the lack of a uniform distribution over the entire domain of numerical attributes, such as real numbers. We overcome this by associating the measure of certainty with the asymptotic behavior of volumes of some subsets of the Euclidean space. We show that this measure is well-defined, and describe approaches to computing and approximating it. While it can be computationally hard, or result in an irrational number, even for simple constraints, we produce polynomial-time randomized approximation schemes with multiplicative guarantees for conjunctive queries, and with additive guarantees for arbitrary first-order queries. We also describe a set of experimental results to confirm the feasibility of this approach.

Coping with Incomplete Data: Recent Advances

“…In some cases there are simple extensions: for example, the approximation schemes of Section 4.2 can be used by treating type-specific comparisons similarly to disequalities. On the other hand, the probabilistic approximations of Section 4.3 are more complicated, though they can provide more refined information about query answers [23]. Bridging the gap between theoretical models and typed real-life schemas is essential for applying these results in practice.…”

Tractability Frontiers in Probabilistic Team Semantics and Existential Second-Order Logic over the Reals

“…By symmetry, it suffices to show that the assignments in X mapping x 2 to b also have a positive total weight in X. By assumption there is an extension Z of X[ b/ u] satisfying the quantifier-free part of (10). It follows that the total weight of assignments in Z that map v 1 to c is positive.…”

“…Finally, let us return to the universal quantification of z 0 , which precedes the existential quantification of z in (10).…”

“…See e.g. [24,44] for declarative frameworks for machine learning utilizing logic, [10,42] for very recent works on logical query languages with arithmetic, and [31] for applications of descriptive complexity in machine learning.…”

Tractability frontiers in probabilistic team semantics and existential second-order logic over the reals

“…See e.g. [15,32] for declarative frameworks for machine learning utilizing logic, [4,30] for very recent works on logical query languages with arithmetic, and [21] for applications of descriptive complexity in machine learning.…”