Bag-set Semantics Research Articles

Combined-semantics equivalence of conjunctive queries: Decidability and tractability results

Query containment and equivalence are fundamental problems in the context of query processing and optimization. In this paper, we consider combined-semantics equivalence of conjunctive (CQ) queries. The combined-semantics formalism [10,11] generalizes the well-known and practically useful notions of set, bag, and bag-set semantics for CQ queries. It also provides tools for studying practical SQL queries, specifically important types of queries arising in on-line analytical processing. In our study, we introduce a containment-based algorithm for deciding combined-semantics equivalence for pairs of CQ queries that belong to a large well-behaved class of “explicit-wave” queries. Our algorithm, as well as our general sufficient condition for containment of combined-semantics CQ queries, generalizes in a uniform way the tests reported in [3,6,11]. Moreover, we single out a subclass of explicit-wave CQ queries for which it is tractable to determine combined-semantics equivalence.

On solving efficiently the view selection problem under bag and bag-set semantics

In this paper, we investigate the problem of view selection for workloads of conjunctive queries under bag and bag-set semantics. In particular, for both semantics we aim to limit the search space of candidate viewsets. We also start delineating boundaries between query workloads for which certain even more restricted search spaces suffice. They suffice in the sense that they do not compromise optimality in that they contain at least one of the optimal solutions. We start with the general case for both bag and bag-set semantics, where we give a tight condition that candidate views can satisfy and still the search space (thus limited) does contain at least one optimal solution. We show that these results, for both semantics, reduce the size of the search space significantly. Further on, due to this analysis for both semantics, a delineation of the space of viewsets and the space of the corresponding equivalent rewritings for a certain conjunctive query workload is given. We show that for chain query workloads under both bag and bag-set semantics, taking only chain views may miss optimal solutions, whereas, if we further limit the queries to be path-queries (i.e., chain queries over a single binary relation), then, under bag semantics, path-views suffice. Concentrating to bag-set semantics, we show that the path-viewsets do not suffice for every path-query workload.

Query containment under bag and bag-set semantics

Conjunctive queries (CQs) are at the core of query languages encountered in many logic-based research fields such as AI, or database systems. The majority of existing work assumes set semantics but often in real applications the manipulation of duplicate tuples is required. One of the major problems that arises as part of advanced features of query optimization, data integration, query reformulation and many other research topics is testing for containment of such queries. In this work, we investigate the complexity of query containment problem for CQs under bag semantics (i.e. duplicate tuples are allowed in both the database and the results of queries) and under bag-set semantics (i.e. duplicates are allowed in the result of the queries but not in the database). We derive complexity results for these problems for five major subclasses of CQs; and we also find necessary conditions for CQ query containment. The general case of these problems remains open.

Equivalence of queries that are sensitive to multiplicities

The query equivalence problem has been studied extensively for set-semantics and, more recently, for bag and bag-set semantics. However, SQL queries often combine set, bag and bag-set semantics. For example, an SQL query that returns a multiset of elements may call a subquery or view that returns a set of elements. Queries may access both relations that do not contain duplicates, as well as relations with duplicates. As another example, in SQL one can compute a multiset-union of queries, each of which returns a set of answers. This paper presents combined semantics, which formally models query evaluation combining set, bag and bag-set semantics. The equivalence problem for queries evaluated under combined semantics is studied. A sufficient condition for equivalence is presented. For several important common classes of queries necessary and sufficient conditions for equivalence are presented.

View selection for real conjunctive queries

Given a query workload, a database and a set of constraints, the view-selection problem is to select views to materialize so that the constraints are satisfied and the views can be used to compute the queries in the workload efficiently. A typical constraint, which we consider in the present work, is to require that the views can be stored in a given amount of disk space. Depending on features of SQL queries (e.g., the DISTINCT keyword) and on whether the database relations on which the queries are applied are sets or bags, the queries may be computed under set semantics, bag-set semantics, or bag semantics. In this paper we study the complexity of the view-selection problem for conjunctive queries and views under these semantics. We show that bag semantics is the “easiest to handle” (we show that in this case the decision version of view selection is in NP), whereas under set and bag-set semantics we assume further restrictions on the query workload (we only allow queries without self-joins in the workload) to achieve the same complexity. Moreover, while under bag and bag-set semantics filtering views (i.e., subgoals that can be dropped from the rewriting without impacting equivalence to the query) are practically not needed, under set semantics filtering views can reduce significantly the query-evaluation costs. We show that under set semantics the decision version of the view-selection problem remains in NP only if filtering views are not allowed in the rewritings. Finally, we investigate whether the cgalg algorithm for view selection introduced in Chirkova and Genesereth (Linearly bounded reformulations of conjunctive databases, pp. 987–1001, 2000) is suitable in our setting. We prove that this algorithm is sound for all cases we examine here, and that it is complete under bag semantics for workloads of arbitrary conjunctive queries and under bag-set semantics for workloads of conjunctive queries without self-joins.

Deciding equivalences among conjunctive aggregate queries

Equivalence of aggregate queries is investigated for the class of conjunctive queries with comparisons and the aggregate operators count, count-distinct, min, max, and sum. Essentially, this class contains unnested SQL queries with the above aggregate operators, with a where clause consisting of a conjunction of comparisons, and without a having clause. The comparisons are either interpreted over a domain with a dense order (like the rationals) or with a discrete order (like the integers). Characterizations of equivalence differ for the two cases. For queries with either max or min, equivalence is characterized in terms of dominance mappings, which can be viewed as a generalization of containment mappings. For queries with the count-distinct operator, a sufficient condition for equivalence is given in terms of equivalence of conjunctive queries under set semantics. For some special cases, it is shown that this condition is also necessary. For conjunctive queries with comparisons but without aggregation, equivalence under bag-set semantics is characterized in terms of isomorphism. This characterization essentially remains the same also for queries with the count operator. Moreover, this characterization also applies to queries with the sum operator if the queries have either constants or comparisons, but not both. In the general case (i.e., both comparisons and constants), the characterization of the equivalence of queries with the sum operator is more elaborate. All the characterizations given in the paper are decidable in polynomial space.

Equivalences among aggregate queries with negation

Query equivalence is investigated for disjunctive aggregate queries with negated subgoals, constants and comparisons. A full characterization of equivalence is given for the aggregation functions count, max, sum, prod, top2 and parity . A related problem is that of determining, for a given natural number N , whether two given queries are equivalent over all databases with at most N constants. This problem is called bounded equivalence . A complete characterization of decidability of bounded equivalence is given. In particular, it is shown that this problem is decidable for all the above aggregation functions as well as for cntd (count distinct) and avg . For quasilinear queries (i.e., queries in which predicates that occur positively are not repeated), it is shown that equivalence can be decided in polynomial time for the aggregation functions count, max, sum, prty, prod, top2 and avg . A similar result holds for cntd provided that a few additional conditions hold. The results are couched in terms of abstract characteristics of aggregation functions, and new proof techniques are used. Finally, the results above also imply that equivalence, under bag-set semantics, is decidable for nonaggregate queries with negation.