# Derivation of constraints and database relations

## Abstract

In this paper we investigate which constraints may be derived from a given set of constraints. We show that, given a set of relations \(\mathcal{R}\), all of which are invariant under some set of permutations *P*, it is possible to derive any other relation which is invariant under *P*, using only the projection, Cartesian product, and selection operators, together with the effective domain of \(\mathcal{R}\), provided that the effective domain contains at least three elements. Furthermore, we show that the condition imposed on the effective domain cannot be removed. This result sharpens an earlier result of Paredaens [13], in that the union operator turns out to be superfluous. In the context of constraint satisfaction problems, this result shows that a constraint may be derived from a given set of constraints containing the binary disequality constraint if and only if it is closed under the same permutations as the given set of constraints.

## Keywords

Relational database relational algebra constraint derivation## Preview

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