Query Containment Research Articles

Containment of conjunctive queries

Conjwzctiue queries are queries over a relational database and are at the core of relational query languages such as SQL. Testing for containment (and equivalence) of such queries arises as part of many advanced features of query optimization, for example, using rnateriahzed views, processing correlated nested queries, semantic query optimization, and global query optimization. Earlier formal work on the topic has examined conjunctive queries over sets of tuples, where each query can be viewed as a function from sets to sets. Containment (and equivalence) of conjunctive queries has been naturally defined based on set mcluslon and has been shown to be an NP-complete problem. Even in SQL, however, queries over multzsets of tuples may be posed. In fact, relations are treated as multisets by default, with duplicates being ehmmated only after explicit requests Thus, in order to reason about containment/equivalence of a large class of SQL queries, it is necessary to consider a generalization of conjunctive queries, in which relations are interpreted as multmets of tuples: The view of a relation as a set of tuples must be generahzed. In this paper we study conjunctive queries over databases in which each tuple has an associated label. This generalized notion of a database allows us to consider relations that are mcsltzsets and relatlons that are fuzzy sets. As a special case, we can also model traditional set-relatlons by making the label associated with a tuple be either “true” (meaning that the tuple is in the relation) or “false” (meaning that the tuple is not in the relation). In order to keep our results general, we consider a variety of label systems, where each label system is essentially a set of conditions on the labels that can be associated with tuples. Once a result is established for a label system, it holds for all mterpretations of relatlons that satisfy these conditions. For example, we present a necessary and sufficient condition for containment of conjunctive queries for label systems of a type that abstracts both the traditional set-relations and fuzzy sets. We also present a different necessary and sufficient condition for containment of a restricted class of conjunctive queries for a label system that abstracts relations as multlsets. Finally, we show that containment of unions of conjunctwe queries is decidable for label systems of the first type and undecidable for label systems of the second type This result underscores the fundamental difference between viewing relations as sets and as multmets, and motivates a closer examination of relatlons as multisets, given them importance in SQL.

On the Efficiency of Transforming Database Logic Programs

We investigate the complexity of various optimization techniques for deductive databases. In particular, we provide polynomial-time algorithms for restricted versions of common program transformations, and show that a minor relaxation of these restrictions leads to N P -hardness. To this end, we define the k-containment problem on conjunctive queries and show that while the two-containment problem is in P (and in NLOGSPACE for function-free queries), the three-containment problem is N P -complete. These results provide a complete description of the complexity of conjunctive query containment. We extend this description to provide a natural characterization of certain optimization problems in deductive databases, such as the detection of serializability and commutativity among pairs of linear rules, the detection of one-boundedness in sirups, and the detection of basis-linearizability in simple nonlinear recursions.

CONSTRUCTING PARTIAL AND COMPLETE INTELLIGENT ANSWERS FOR RECURSIVE QUERIES

In this paper we study the issue of providing intelligent answers to recursive queries initially proposed in [Imie88]. In contrast to conventional query answers, which are sets of tuples, intelligent answers are rules describing the characteristics of the results. This way, an intelligent answer facilitates users to better capture the meaning of the query. To this aim, we first extend the notion of intelligent answers by allowing them to be partial (or incomplete). Complete and partial intelligent answers are both useful, in that they provide (possibly partial) characterizations of queries. We will show that complete intelligent answers also give a more efficient way to evaluate queries. For constructing intelligent answers, we propose a general approach that enables us to combine several cases studied in [Imie88]. Moreover, we will show that semantic constraints can be systematically utilized to provide complete intelligent answers to a wider range of queries. We consider two commonly used classes of semantic constraints called implication and referential constraints, and prove that the completeness of intelligent answers with respect to semantic constraints can be reduced to the semantics-based query containment problem. Finally, a necessary and sufficient condition is given to solve this containment problem.

Some results on the containment and minimization of (in) equality queries

The containment and minimization of conjunctive queries are two basic problems of query optimization in relational and deductive databases. For equality queries, the problems are NP-complete due to the “homomorphism property”. For (in) equality queries, however, the containment problem has recently been proved to be π p 2-complete, while the minimization is virtually unknown. In this paper, we identify subclasses of (in) equality queries which have the homomorphism property and a similar minimization procedure to the one for equality queries. We also give interesting examples to show that queries outside these classes do not have the property.

Equivalence of Datalog queries is undecidable

Datalog is a powerful query language for relational databases [10]. We consider the problems of determining containment, equivalence, and satisfiability of Datalog queries. We show that containment and equivalence are recursively unsolvable. This should be contrasted with the work of Aho, Sagiv, and Ullman on relational queries [1]. Satisfiability is easily decidable for Datalog quer ies. We also consider Datalog f which allows function symbols. Here, satisfiability is recursively unsolvable.

Testing containment of conjunctive queries under functional and inclusion dependencies

Much of the work to date on the optimization of queries for relational databases has focussed on the case where the only dependencies allowed are functional dependencies. We extend this work to the case where inclusion dependencies are also allowed. We show that there are substantial special cases where the presence of inclusion dependencies does not make the basic problems of optimization any harder than they are when there are no dependencies at all. In particular, we show that the problems of query containment, equivalence, and nonminimality remain in NP when either (a) all dependencies are inclusion dependencies or (b) the set of dependencies is what we call “key-based.” These results assume that infinite databases are allowed. If only finite databases are allowed, new containments and equivalences may arise, as we illustrate by an example, and the problems may be substantialy more difficult. We can, however, prove a “finite controllability” theorem that shows that no such examples exist for case (b), or for (a) when the only inclusion dependencies allowed are those having “width” equal to one.