Equivalence and normal forms for the restricted and bounded fixpoint in the Nested Algebra.

Abstract

The nested model is an extension of the traditional, "flat" relational model in which relations can also have relation-valued entries. Its "default" query language, the nested algebra, is rather weak, unfortunately, since it is only a conservative extension of the traditional, "flat" relational algebra, and thus can only express a small fraction of the polynomial-time queries. Therefore, it was proposed to extend the nested algebra with a least-fixpoint construct, but the resulting language turned out to be too powerful: many inherently exponential queries could also be expressed. Two polynomial-time restrictions of the least-fixpoint closure of the nested algebra were proposed: the restricted leastfixpoint closure (by Gyssens and Van Gucht) and the bounded fixpoint closure

(by Suciu). Here, we prove two results. First we show that that both restrictions are equivalent in expressive power. The proof technique relies on known encodings of nested relations into flat ones, and on a novel encoding method of nested relations into flat ones.