Exact and Approximate Aggregation in Constraint Query

We investigate the problem of how to extend constraint query languages with aggregate operators. We deal with standard relational aggregation, and also with aggregates specific to spatial ,data, such as volume. We study several approaches, including the addition of a new class of approximate aggregate operators which allow an error tolerance in the computation. We show how techniques based on VC-dimension can be used to give languages with approximation operators, but also show that these languages have a number of shortcomings. We then give a set of results showing that it is impossible to get constraint-based languages that admit definable aggregation operators, both for exact operators and for approximate ones. These results are quite robust, in that they show that closure under aggregation is problematic even .when the class of functions permitted in constraints is expanded. This motivates a different approach to the aggregation problem. We introduce a language FO + POLY + SUM, which permits standard discrete aggregation operators to be applied to the outputs of range-restricted constraint queries. We show that this language has a number of attractive closure and expressivity properties, and that it can compute volumes of linear-constraint databases. We also show, using techniques from machine learning, that a small extension of FO + POLY + SUM can probabilistically find approximations of volumes for polynomial-constraint databases.