Bounded fixpoints for complex objects

Abstract

We study a query language for complex-object databases, which is designed to (1) express only tractable queries, and (2) be as expressive over flat relations as first-order logic with fixpoints. The language is obtained by extending the nested relational algebra, NRA , with a “bounded fixpoint” operator. Similar to results for flat relations, all tractable queries over ordered databases are expressible in this language. The main result consists in proving that this language is a conservative extension of the first-order logic with fixpoints, or of the while-queries, (depending on the interpretation of the bounded fixpoint: inflationary or partial). That is, a query from flat relations to flat relations is expressible in our language if and only if it is expressible in first-order logic with fixpoints, or in the while-queries, respectively. The proof technique for this theorem uses indexes to encode complex objects into flat relations. It can serve as basis for an implementation method of complex objects databases in terms of relational databases, which works well for queries expressed both with fixpoints and with bounded fixpoint. We also define a complex object logical calculus with fixpoints and prove that its range-restricted fragment is equivalent to NRA with bounded fixpoints.