The Restricted and Bounded Fixpoint Closures of the Nested Relational Algebra are Equivalent

The nested model is an extension of the traditional, \ at" 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, \ at" 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 leastxpoint 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 leastxpoint closure of the nested algebra were proposed: the restricted leastxpoint closure (by Gyssens and Van Gucht) and the bounded xpoint closure (by Suciu). Here, we prove that both restrictions are equivalent in expressive power. We also exhibit a proof technique, called type substitution, by which we reduce our result to its obvious counterpart in the \ at" relational model; thus emphasizing the inherent weakness of the nested algebra.