Implication Problems over Positive Semirings
Minna Hirvonen
TL;DR
The paper investigates implication problems for dependencies in semiring-based team semantics, unifying relational, bag, and probabilistic interpretations via $K$-teams, where each tuple carries a weight from a semiring $K$. It introduces and analyzes axiomatizations that are sound and complete for several dependency classes, including functional dependencies, inclusion dependencies, marginal identities, and conditional independences, under additively cancellative positive semirings. A key contribution is the introduction of weighted inclusion dependencies, which collapse to standard INDs or MI on suitable semirings, and the development of unary variants that enable complete axiomatizations in non-additively cancellative settings. The results illuminate how semiring structure governs the behavior and interrelations of dependencies and establish tractable decision procedures in many cases, with Armstrong relations and cycle rules extending classical database theory to a general semiring setting. Overall, the work provides a unified, algebraic framework for dependency reasoning across domains, with potential impact on data provenance, probabilistic modeling, and semantics-aware query optimization.
Abstract
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.