Local transformations and conjunctive-query equivalence.

Over the past several decades, the study of conjunctive queries has occupied a central place in the theory and practice of database systems. In recent years, conjunctive queries have played a prominent role in the design and use of schema mappings for data integration and data exchange tasks. In this paper, we investigate several different aspects of conjunctive-query equivalence in the context of schema mappings and data exchange. In the first part of the paper, we introduce and study a notion of a local transformation between database instances that is based on conjunctive-query equivalence. We show that the chase procedure for GLAV mappings (that is, schema mappings specified by source-to-target tuple-generating dependencies) is a local transformation with respect to conjunctive-query equivalence. This means that the chase procedure preserves bounded conjunctive-query equivalence, that is, if two source instances are indistinguishable using conjunctive queries of a sufficiently large size, then the target instances obtained by chasing these two source instances are also indistinguishable using conjunctive queries of a given size. Moreover, we obtain polynomial bounds on the level of indistinguishability between source instances needed to guarantee indistinguishability between the target instances produced by the chase. The locality of the chase extends to schema mappings specified by a second-order tuple-generating dependency (SO tgd), but does not hold for schema mappings whose specification includes target constraints. In the second part of the paper, we take a closer look at the composition of two GLAV mappings. In particular, we break GLAV mappings into a small number of well-studied classes (including LAV and GAV), and complete the picture as to when the composition of schema mappings from these various classes can be guaranteed to be a GLAV mapping, and when they can be guaranteed to be conjunctive-query equivalent to a GLAV mapping. We also show that the following problem is decidable: given a schema mapping specified by an SO tgd and a GLAV mapping, are they conjunctive-query equivalent? In contrast, the following problem is known to be undecidable: given a schema mapping specified by an SO tgd and a GLAV mapping, are they logically equivalent?