Cardinality Constraints for Uncertain Data

Modern applications require advanced techniques and tools to process large volumes of uncertain data. For that purpose we introduce cardinality constraints as a principled tool to control the occurrences of uncertain data. Uncertainty is modeled qualitatively by assigning to each object a degree of possibility by which the object occurs in an uncertain instance. Cardinality constraints are assigned a degree of certainty that stipulates on which objects they hold. Our framework empowers users to model uncertainty in an intuitive way, without the requirement to put a precise value on it. Our class of cardinality constraints enjoys a natural possible world semantics, which is exploited to establish several tools to reason about them. We characterize the associated implication problem axiomatically and algorithmically in linear input time. Furthermore, we show how to visualize any given set of our cardinality constraints in the form of an Armstrong instance, whenever possible. Even though the problem of finding an Armstrong instance is precisely exponential, our algorithm computes an Armstrong instance with conservative use of time and space. Data engineers and domain experts can jointly inspect Armstrong instances in order to consolidate the certainty by which a cardinality constraint shall hold in the underlying application domain.