The Completeness Problem of Ordered Relational Databases
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Support of order in query processing is a crucial component in relational database systems, not only because the output of a query is often required to be sorted in a specific order, but also because employing order properties can significantly reduce the query execution cost. Therefore, finding an effective approach to answer queries over ordered data is important to the efficiency of query processing in relational databases. In this dissertation, an ordered relational database model is proposed, which captures both data tuples of relations and tuple ordering in relations. Based on this conceptual model, ordered relational queries are formally defined in a two-sorted first-order calculus, which serves as a yardstick to evaluate expressive power of other ordered query representations. The primary purpose of this dissertation is to investigate the expressive power of different ordered query representations. Particularly, the completeness problem of ordered relational algebras is studied with respect to the first-order calculus: does there exist an ordered algebra such that any first-order expressible ordered relational query can be expressed by a finite sequence of ordered operations? The significance of studying the completeness problem of ordered relational algebras is in that the completeness of ordered relational algebras leads to the possibility of implementing a finite set of ordered operators to express all first-order expressible ordered queries in relational databases. The dissertation then focuses on the completeness problem of ordered conjunctive queries. This investigation is performed in an incremental manner: first, the ordered conjunctive queries with data-decided order is considered; then, the ordered conjunctive queries with t-decided order is studied; finally, the completeness problem for the general ordered conjunctive queries is explored. The completeness theorem of ordered algebras is proven for all three classes of ordered conjunctive queries. Although this ordered relational database model is only conceptual, and ordered operators are not implemented in this dissertation, we do prove that a complete set of ordered operators exists to retrieve all first order expressible ordered queries in the three classes of ordered conjunctive queries. This research sheds light on the possibility of implementing a complete set of ordered operators in relational databases to solve the performance problem of order-relevant queries.