Tractable Iteration Mechanisms for Bag Languages

The goal of this paper is to study tractable iteration mechanisms for bags. The presence of duplicates in bags prevents iteration mechanisms developed in the context of sets to be directly applied to bags without losing tractability. We study two constructs for controlling tractability of iteration over bags. The deflationary fixpoint construct keeps removing elements from a bag until a fixpoint is reached. The bounded fixpoint construct is an inflationary iteration mechanism that never exceeds some predefined bounding bag. We study these constructs in the context of a standard (nested) bag algebra. We show that the deflationary and bounded inflationary fixpoint constructs are equally expressive and strictly more expressive than their set-based counterparts. We also show that, unlike in the set case, the bag algebra with bounded fixpoint fails to capture all PTIME queries over databases with ordered domains. We then show that adding just one construct, which can be used to assign unique tags to duplicates, captures the class of all polynomial time queries over bags when a total ordering on the domain of atomic elements is available. Finally, we compare the expressive powers of the bag algebra and the nested relational algebra with aggregate functions in the presence of these fixpoint operators.