Classification of annotation semirings over query containment

We study the problem of query containment of (unions of) conjunctive queries over annotated databases. Annotations are typically attached to tuples and represent metadata such as probability, multiplicity, comments, or provenance. It is usually assumed that annotations are drawn from a commutative semiring. Such databases pose new challenges in query optimization, since many related fundamental tasks, such as query containment, have to be reconsidered in the presence of propagation of annotations. We axiomatize several classes of semirings for each of which containment of conjunctive queries is equivalent to existence of a particular type of homomorphism. For each of these types we also specify all semirings for which existence of a corresponding homomorphism is a sufficient (or necessary) condition for the containment. We exploit these techniques to develop new decision procedures for containment of unions of conjunctive queries and axiomatize corresponding classes of semirings. This generalizes previous approaches and allows us to improve known complexity bounds.