Incremental Recomputation of Recursive Queries with Nested Sets and Aggregate Functions

We examine the power of incremental evaluation systems that use an SQL-like language formaintaining recursively-de ned views. We show that recursive queries such as transitive closure,and \alternating paths" can be incrementally maintained in a nested relational language, whensome auxiliary relations are allowed. In the presence of aggregate functions, even more queries canbe maintained, for example, the \same generation" query. In contrast, it is still an open problemwhether such queries are maintainable in relational calculus. We then restrict the language so thatno nested relations are involved (but we keep the aggregate functions). Such a language capturesthe capability of most practical relational database systems. We prove that this restriction doesnot reduce the incremental computational power; that is, any query that can be maintained in anested language with aggregates, is still maintainable using only at relations. We also show thatone does not need auxiliary relations of arity more than 2. In particular, this implies that therecursive queries maintainable in the nested language with aggregates, can be also maintained ina practical relational database systems using auxiliary tables of arity at most 2. This is again insharp contrast to maintenance in relational calculus, which admits a strict arity-based hierarchy.