Gamma Acyclicity, Annotated Relations, and Consistency Witness Functions

During the early days of relational database theory it was realized that "acyclic" database schemas possess a number of desirable semantic properties. In fact, three different notions of "acyclicity" were identified and extensively investigated during the 1980s, namely, alpha-acyclicity, beta-acyclicity, and gamma-acyclicity. Much more recently, the study of alpha-acyclicity was extended to annotated relations, where the annotations are values from some positive commutative monoid. The recent results about alpha-acyclic schemas and annotated relations give rise to results about beta-acyclic schemas and annotated relations, since a schema is beta-acyclic if and only if every sub-schema of it is alpha-acyclic. Here, we study gamma-acyclic schemas and annotated relations. Our main finding is that the desirable semantic properties of gamma-acyclic schemas extend to annotated relations, provided the annotations come from a positive commutative monoid that has the transportation property. Furthermore, the results reported here shed light on the role of the join of two standard relations, Specifically, our results reveal that the only relevant property of the join of two standard relations is that it is a witness to the consistency of the two relations, provided that these two relations are consistent. For the more abstract setting of annotated relations, this property of the standard join is captured by the notion of a consistency witness function, a notion which we systematically investigate in this work.