The Restricted and Bounded Fixpoint Closures of the Nested Algebra are Equivalent Marc Gyssens Dept. WNI, University of Limburg B-3590 Diepenbeek, Belgium gyssens@charlie.luc.ac.be Dan Suciu AT&T Bell Laboratories Murray Hill, NJ 07974, USA suciu@research.att.com Dirk Van Gucht Comp. Sci. Dept., Indiana University Bloomington, IN 47405-4101, USA vgucht@cs.indiana.edu 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 least-fixpoint closure (by Gyssens and Van Gucht) and the bounded fixpoint 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 “flat” relational model; thus emphasizing the inherent weakness of the nested algebra. 1 Introduction The nested model [12, 16] is an extension of the traditional, “flat” relational database model in which relations can have both “flat,” atomic entries and structured, relation-valued entries. Since the late 1980s, various query languages have been considered in the context of the nested model [1, 5, 7, 9, 11, 14, 16]. These languages can be classified according to their expressive power [2]. The nested algebra [16], which extends the traditional, “flat” relational algebra with two restructuring operators, called nest and unnest , can only express a fragment of the polynomial-time queries over nested databases. Therefore, several extensions of the nested algebra were proposed one of which is its least-fixpoint closure [1, 11].Although many more polynomial-time queries on nested databases can be expressed efficiently in this extended language, it was shown in the aforementioned papers that some intractable queries, such as computing the powerset of a relation, can also be expressed in the least-fixpoint closure of the nested algebra. Therefore, proposals were made for extensions of the nested algebra which can only express polynomial-time queries. One such proposal is the restricted least-fixpoint closure of the nested algebra introduced by Gyssens and Van Gucht [10]. In the restricted least-fixpoint closure of the nested algebra, the fixpoint construct can only be applied to expressions wherein nesting and unnesting do not occur. Another proposal to extend the expressive power of the nested algebra within is to consider the bounded-fixpoint closure of the nested algebra introduced by Suciu [15]. In the bounded-fixpoint closure of the nested algebra, the fixpoint construct can be applied to expressions in which nesting and unnesting can occur; at each iteration step, however, the intermediate result is intersected with a relation which is constant during the iteration process. Consequently, the final result of an application of the bounded-fixpoint construct is bound by that relation. It can easily be seen that the expressive power of both the restricted least-fixpoint closure and the bounded-fixpoint closure of the nested algebra is contained in , and that both extensions are strictly more powerful than the nested DBPL-5, Gubbio, Italy, 1995 1