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Our goal is to measure the likelihood of the satisfaction of numerical constraints in the absence of prior information. We study expressive constraints, involving arithmetic and complex numerical functions, and even quantification over numbers. Such problems arise in processing incomplete data, or analyzing conditions in programs without a priori bounds on variables. We show that for constraints on n variables, the proper way to define such a measure is as the limit of the part of the n-dimensional ball that consists of points satisfying the constraints, when the radius increases. We prove that the existence of such a limit is closely related to the notion of o-minimality from model theory. Thus, for constraints definable with the usual arithmetic and exponentiation, the likelihood is well defined, but adding trigonometric functions is problematic. We look at computing and approximating such likelihoods for order and linear constraints, and prove an impossibility result for approximating with multiplicative error. However, as the likelihood is a number between 0 and 1, an approximation scheme with additive error is acceptable, and we give it for arbitrary linear constraints.

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.

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