The Completeness Problem of Ordered Relational Databases

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

Computing Possible and Certain Answers over Order-Incomplete Data

This paper studies the complexity of query evaluation for databases whose relations are partially ordered; the problem commonly arises when combining or transforming ordered data from multiple sources. We focus on queries in a useful fragment of SQL, namely positive relational algebra with aggregates, whose bag semantics we extend to the partially ordered setting. Our semantics leads to the study of two main computational problems: the possibility and certainty of query answers. We show that these problems are respectively NP-complete and coNP-complete, but identify tractable cases depending on the query operators or input partial orders. We further introduce a duplicate elimination operator and study its effect on the complexity results.Comment: 55 pages, 56 references. Extended journal version of arXiv:1707.07222. Up to the stylesheet, page/environment numbering, and possible minor publisher-induced changes, this is the exact content of the journal paper that will appear in Theoretical Computer Scienc

Fixpoints and Bounded Fixpoints for Complex Objects

We investigate a query language for complex-object databases, which is designed to (1) express only tractable queries, and (2) be as expressive over flat relations as first order logic with fixpoints. The language is obtained by extending the nested relational algebra NRA with a bounded fixpoint operator. As in the flat case, all PTime computable queries over ordered databases are expressible in this language. The main result consists in proving that this language is a conservative extension of the first order logic with fixpoints, or of the while-queries (depending on the interpretation of the bounded fixpoint: inflationary or partial). The proof technique uses indexes, to encode complex objects into flat relations, and is strong enough to allow for the encoding of NRA with unbounded fixpoints into flat relations. We also define a logic based language with fixpoints, the nested relational calculus , and prove that its range restricted version is equivalent to NRA with bounded fixpoints

Solving equations in the relational algebra

Enumerating all solutions of a relational algebra equation is a natural and powerful operation which, when added as a query language primitive to the nested relational algebra, yields a query language for nested relational databases, equivalent to the well-known powerset algebra. We study \emph{sparse} equations, which are equations with at most polynomially many solutions. We look at their complexity, and compare their expressive power with that of similar notions in the powerset algebra.Comment: Minor revision, accepted for publication in SIAM Journal on Computin

On the Limitations of Provenance for Queries With Difference

The annotation of the results of database transformations was shown to be very effective for various applications. Until recently, most works in this context focused on positive query languages. The provenance semirings is a particular approach that was proven effective for these languages, and it was shown that when propagating provenance with semirings, the expected equivalence axioms of the corresponding query languages are satisfied. There have been several attempts to extend the framework to account for relational algebra queries with difference. We show here that these suggestions fail to satisfy some expected equivalence axioms (that in particular hold for queries on "standard" set and bag databases). Interestingly, we show that this is not a pitfall of these particular attempts, but rather every such attempt is bound to fail in satisfying these axioms, for some semirings. Finally, we show particular semirings for which an extension for supporting difference is (im)possible.Comment: TAPP 201

Deductive Optimization of Relational Data Storage

Optimizing the physical data storage and retrieval of data are two key database management problems. In this paper, we propose a language that can express a wide range of physical database layouts, going well beyond the row- and column-based methods that are widely used in database management systems. We use deductive synthesis to turn a high-level relational representation of a database query into a highly optimized low-level implementation which operates on a specialized layout of the dataset. We build a compiler for this language and conduct experiments using a popular database benchmark, which shows that the performance of these specialized queries is competitive with a state-of-the-art in memory compiled database system