A Framework for the Investigation of Aggregate Functions in Database Queries

In this paper we present a new approach for studying aggregations in the context of database query languages. Starting from a broad de nition of aggregate function, we address our investigation from two di erent perspectives. We rst propose a declarative notion of uniform aggregate function that refers to a family of scalar functions uniformly constructed over a vocabulary of basic operators by a bounded Turing Machine. This notion yields an e ective tool to study the e ect of the embedding of a class of built-in aggregate functions in a query language. All the aggregate functions most used in practice are included in this classi cation. We then present an operational notion of aggregate function, by considering a high-order folding constructor, based on structural recursion, devoted to compute numeric aggregations over complex values. We show that numeric folding over a given vocabulary is sometimes not able to compute, by itself, the whole class of uniform aggregate function over the same vocabulary. It turns out however that this limitation can be partially remedied by the restructuring capabilities of a query language.