In this paper we study the problem of reducing the evaluation costs of queries on finite databases in presence of integrity constraints, by designing and materializing views. Given a database schema, a set of queries defined on the schema, a set of integrity constraints, and a storage limit, to find a solution to this problem means to find a set of views that satisfies the storage limit, provides equivalent rewritings of the queries under the constraints (this requirement is weaker than equivalence in the absence of constraints), and reduces the total costs of evaluating the queries. This problem, database reformulation, is important for many applications, including data warehousing and query optimization. We give complexity results and algorithms for database reformulation in presence of constraints, for conjunctive queries, views, and rewritings and for several types of constraints, including functional and inclusion dependencies. To obtain better complexity results, we introduce an unchase technique, which reduces the problem of query equivalence under constraints to equivalence in the absence of constraints without increasing query size.
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