We study the following problem: how to test if Q(2) is contained in Q(1), where Q(1) and Q(2) are conjunctive queries with arithmetic comparisons? This problem is fundamental in a large variety of database applications. Existing algorithms first normalize the queries, then test a logical implication using multiple containment mappings from Q(1) to Q(2). We are interested in cases where the containment can be tested more efficiently. This work aims to (a) reduce the problem complexity from Pi(2)(P)-completeness to NP-completeness in these cases; (b) utilize the advantages of the homomorphism property (i.e., the containment test is based on a single containment mapping) in applications such as those of answering queries using views; and (c) observing that many real queries have the homomorphism property. The following are our results. (1) We show several cases where the normalization step is not needed, thus reducing the size of the queries and the number of containment mappings. (2) We develop an algorithm for checking various syntactic conditions on queries, under which the homomorphism property holds. (3) We further reduce the conditions of these classes using practical domain knowledge that is easily obtainable. (4) We conducted experiments on real queries, and show that most of the queries pass this test.

We consider the problem of answering queries using views, where queries and views are conjunctive queries with arithmetic comparisons over dense orders. Previous work only considered limited variants of this problem, without giving a complete solution. We first show that obtaining equivalent rewritings for conjunctive queries with arithmetic comparisons is decidable. Then, we consider the problem of finding maximally contained rewritings (MCRs) where the decidability proof does not carry over. We investigate two special cases of this problem where the query uses only semi-interval comparisons. In both cases decidability of finding MCRs depends on the query containment test. First, we address the case where the homomorphism property holds in testing query containment. In this case decidability is easy to prove but developing an efficient algorithm is not trivial. We develop such an algorithm and prove that it is sound and complete. This algorithm applies in many cases where the query uses only left (or right) semi-interval comparisons. Then, we develop a new query containment test for the case where the containing query uses both left and right semi-interval comparisons but with only one left (or right) semi-interval subgoal. Based on this test, we show how to produce an MCR which is a Datalog query with arithmetic comparisons. The containment test that we develop obtains a result of independent interest. It finds another special case where query containment in the presence of arithmetic comparisons can be tested in nondeterministic polynomial time. (c) 2006 Elsevier B.V. All rights reserved.