Equivalence and Normal Forms for the Restricted and Bounded Fixpoint in the Nested Algebra

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 xpoint 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 xpoint closure of the nested algebra were proposed: the restricted xpoint closure (by Gyssens and Van Gucht) and the bounded xpoint closure (by Su-ciu). Here, we prove two results. First we show that that both restrictions are equivalent in expressive power. The proof technique relies on known en-codings of nested relations into at ones, and on a novel technique, called type substitution, by which we reduce the equivalence of the two restrictions to its obvious counterpart in the \\at" relational model. Second we prove that both the bounded xpoint queries and the restricted xpoint queries admit normal forms, in which the xpoint occurs exactly once. The proof technique relies on a novel encoding method of nested relations into at ones.