Equivalence , Containment and Rewriting of Aggregate Queries

The primary goal of this thesis is to lay the theoretical foundations for a formal study of aggregate query optimization. This requires gaining a coherent understanding of equivalences and containments between aggregate queries of varied forms. A secondary goal of this thesis is to solve the view usability problem for varied types of aggregate queries. The view usability problem is that of determining whether it is possible to find a reformulation of a query that uses other queries, instead of base relations. This is of importance in query optimization, since it enables reuse of previously computed results. It is also a necessity when integrating information from heterogeneous sources, when each source is described as a view over a global schema. In order to achieve the primary goal, the general problem of determining equivalences among aggregate queries is considered. Our aggregate queries may contain disjunctions, comparisons and negated subgoals. First, the bounded equivalence problem, i.e., the problem of determining equivalence of queries with respect to databases whose size is bound by a constant, is studied. A complete characterization of when bounded equivalence is decidable is presented. It is shown how to determine equivalence by a reduction to local equivalence (a special case of bounded equivalence). Characterizations of equivalence for the special case of quasilinear queries are presented. Equivalence of quasilinear queries can be determined in polynomial time. Containment of aggregate queries is reduced to equivalence of aggregate queries. All the characterizations are couched in terms of abstract characterizations of aggregation functions. Thus, the results presented are easily extendible to additional aggregation functions. These results also solve additional open problems, including