Transforming from flat algebra to nested algebra

Consider a database which at the conceptual level consists of a set of 1NF (flat) relations but which is physically stored in denormalized (prejoined) form. Denormalization reduces the number of joins required to compute a query. To reduce redundancy, each prejoined relation can be converted into a NF² (nested) relation by a series of nest operations. The authors consider the problem of converting from flat algebra to nested algebra. They provide a series of legal transformations that can be applied to a query Q_e on flat relations, to obtain an equivalent query Q_r on the nested relations. Several fundamental theorems are provided, allowing the selection conditions of the 1NF case to be applied directly to the NF² relations. This is done through changing the nesting structure internally, without unnesting, and through pushing the selection conditions down to the appropriate subrelations