# On the interaction between transitive closure and functional dependencies

## Abstract

Closure dependencies, or CDs, are introduced to capture formally transitive closure relationships between (sequences of) attributes of a relational scheme, e.g., Part and Subpart. First, CDs are studied alone. The implication problem has a simple complete axiomatization. The interaction of CDs with functional dependencies (FDs) is investigated, leading to the following main results: CDs and FDs considered together imply new CDs, but no new FDs, i.e., no other FDs than those already implied by the given FDs alone. CDs together with FDs imply only longer CDs, i.e., CDs that contain more attributes, but not shorter ones. No k-ary axiomatization can fully describe the interaction between FDs and CDs: Although simple complete axiomatizations for CDs alone and FDs alone exist, there is no complete axiomatization for CDs and FDs taken together, in which every rule is *k*-ary for some fixed *k*. Finally, we define a generalized form of CDs and state a list of interesting open problems.

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