Implication Problems over Positive Semirings

We study various notions of dependency in semiring team semantics. Semiring teams are essentially database relations, where each tuple is annotated with some element from a positive semiring. We consider semiring generalizations of several dependency notions from database theory and probability theory, including functional and inclusion dependencies, marginal identity, and (probabilistic) independence. We examine axiomatizations of implication problems, which are rule-based characterizations for the logical implication and inference of new dependencies from a given set of dependencies. Semiring team semantics provides a general framework, where different implication problems can be studied simultaneously for various semirings. The choice of the semiring leads to a specific semantic interpretation of the dependencies, and hence different semirings offer a way to study different semantics (e.g., relational, bag, and probabilistic semantics) in a unified framework.