On decomposition of relational databases

A central issue in relational database theory is that of decomposition. It has been agreed that decompositions should be injective, so as not to lose information, and surjective, so they decompose a relation into independent components. Injectiveness and surjectiveness are in general second-order notions. We show here how to express these notions in a first-order manner, assuming that we are dealing only with first-order constraints. As a consequence we get that the reconstruction map, which is the inverse to the decomposition map, is also first-order, but is not necessarily the natural join. This result is derived by applying Beth's Definability Theorem from model theory. For the case that the constraints used are implicational dependencies, we derive the exact syntactic form of the reconstruction map, and show that if the decomposition map is both injective and surjective then the reconstruction map is the natural join.