Algebraic Dependencies

Abstract

We propose a new kind of data dependencies called algebraic dependencies, which generalize all previous known kinds. We give a complete axiomatization of algebraic dependencies in terms of simple algebraic rewriting rules. In the process we characterize exactly the expressive power of tableaux, thus solving an open problem of Aho, Sagiv and Ullman; we show that it is NP-complete to tell whether a tableau is realizable by an expression; and we give an interesting dual interpretation of the chase procedure. We also show that algebraic dependencies over a language augmented to contain union and set difference can express arbitrary domain-independent predicates of finite index over finite relations. The class of embedded implicational dependencies recently - and independently - introduced by Fagin is shown to coincide with our algebraic dependencies. Based on this, we give a simple proof of Fagin's Armstrong relation theorem.