Solving equations in the relational algebra

TL;DR

This paper explores the complexity and expressive power of solving sparse equations in relational algebra, demonstrating their significance in nested relational databases and comparing with powerset algebra.

Contribution

It introduces the concept of sparse equations in relational algebra, analyzes their complexity, and compares their expressive power to powerset algebra.

Findings

Sparse equations have at most polynomially many solutions.

Solving these equations is a powerful operation in nested relational databases.

The expressive power of sparse equations is comparable to powerset algebra.

Abstract

Enumerating all solutions of a relational algebra equation is a natural and powerful operation which, when added as a query language primitive to the nested relational algebra, yields a query language for nested relational databases, equivalent to the well-known powerset algebra. We study \emph{sparse} equations, which are equations with at most polynomially many solutions. We look at their complexity, and compare their expressive power with that of similar notions in the powerset algebra.