During the past 15 years, schema mappings have been extensively used in formalizing and studying such critical data interoperability tasks as data exchange and data integration. Much of the work has focused on GLAV mappings, i.e., schema mappings specified by source-to-target tuple-generating dependencies (s-t tgds), and on schema mappings specified by second-order tgds (SO tgds), which constitute the closure of GLAV mappings under composition. In addition, nested GLAV mappings have also been considered, i.e., schema mappings specified by nested tgds, which have expressive power intermediate between s-t tgds and SO tgds. Even though nested GLAV mappings have been used in data exchange systems, such as IBM’s Clio, no systematic investigation of this class of schema mappings has been carried out so far. In this article, we embark on such an investigation by focusing on the basic reasoning tasks, algorithmic problems, and structural properties of nested GLAV mappings. One of our main results is the decidability of the implication problem for nested tgds. We also analyze the structure of the core of universal solutions with respect to nested GLAV mappings and develop useful tools for telling apart SO tgds from nested tgds. By discovering deeper structural properties of nested GLAV mappings, we show that also the following problem is decidable: Given a nested GLAV mapping, is it logically equivalent to a GLAV mapping?

Schema mappings have been extensively studied in the context of data exchange and data integration, where they have turned out to be the right level of abstraction for formalizing data inter-operability tasks. Up to now and for the most part, schema mappings have been studied as static objects, in the sense that each time the focus has been on a single schema mapping of interest or, in the case of composition, on a pair of schema mappings of interest. In this paper, we adopt a dynamic viewpoint and embark on a study of sequences of schema mappings and of the limiting behavior of such sequences. To this effect, we first introduce a natural notion of distance on sets of finite target instances that expresses how "close" two sets of target instances are as regards the certain answers of conjunctive que- ries on these sets. Using this notion of distance, we investigate pointwise limits and uniform limits of sequences of schema mappings, as well as the companion notions of pointwise Cauchy and uniformly Cauchy sequences of schema mappings. We obtain a number of results about the limits of sequences of GAV schema mappings and the limits of sequences of LAV schema mappings that reveal striking differences between these two classes of schema mappings. We also consider the completion of the metric space of sets of target instances and obtain concrete representations of limits of sequences of schema mappings in terms of generalized schema mappings, that is, schema mappings with infinite target instances as solutions to (finite) source instances.

Schema mappings have been extensively studied in the context of data exchange and data integration, where they have turned out to be the right level of abstraction for formalizing data inter-operability tasks. Up to now and for the most part, schema mappings have been studied as static objects, in the sense that each time the focus has been on a single schema mapping of interest or, in the case of composition, on a pair of schema mappings of interest. In this paper, we adopt a dynamic viewpoint and embark on a study of sequences of schema mappings and of the limiting behavior of such sequences. To this effect, we first introduce a natural notion of distance on sets of finite target instances that expresses how "close" two sets of target instances are as regards the certain answers of conjunctive queries on these sets. Using this notion of distance, we investigate pointwise limits and uniform limits of sequences of schema mappings, as well as the companion notions of pointwise Cauchy and uniformly Cauchy sequences of schema mappings. We obtain a number of results about the limits of sequences of GAV schema mappings and the limits of sequences of LAV schema mappings that reveal striking differences between these two classes of schema mappings. We also consider the completion of the metric space of sets of target instances and obtain concrete representations of limits of sequences of schema mappings in terms of generalized schema mappings, i.e., schema mappings with infinite target instances as solutions to (finite) source instances.

During the past decade, schema mappings have been extensively used in formalizing and studying such critical data interoperability tasks as data exchange and data integration. Much of the work has focused on GLAV mappings, i.e., schema mappings specified by source-to-target tuple-generating dependencies (s-t tgds), and on schema mappings specified by second-order tgds (SO tgds), which constitute the closure of GLAV mappings under composition. In addition, nested GLAV mappings have also been considered, i.e., schema mappings specified by nested tgds, which have expressive power intermediate between s-t tgds and SO tgds. Even though nested GLAV mappings have been used in data exchange systems, such as IBM's Clio, no systematic investigation of this class of schema mappings has been carried out so far. In this paper, we embark on such an investigation by focusing on the basic reasoning tasks, algorithmic problems, and structural properties of nested GLAV mappings. One of our main results is the decidability of the implication problem for nested tgds. We also analyze the structure of the core of universal solutions with respect to nested GLAV mappings and develop useful tools for telling apart SO tgds from nested tgds. By discovering deeper structural properties of nested GLAV mappings, we show that also the following problem is decidable: given a nested GLAV mapping, is it logically equivalent to a GLAV mapping?

Recently, two relaxed notions of equivalence of schema mappings have been introduced, which provide more potential of optimizing schema mappings than logical equivalence: data exchange (DE) equivalence and conjunctive query (CQ) equivalence. In this work, we systematically investigate these notions of equivalence for mappings consisting of s-t tgds and target egds and/or target tgds. We prove that both CQ- and DE-equivalence are undecidable and so are some important optimization tasks (like detecting if some dependency is redundant). However, we also identify an important difference between the two notions of equivalence: CQ-equivalence remains undecidable even if the schema mappings consist of s-t tgds and target dependencies in the form of key dependencies only. In contrast, DE-equivalence is decidable for schema mappings with s-t tgds and target dependencies in the form of functional and inclusion dependencies with terminating chase property.

Schema mappings are high-level specifications that describe the relationship between two database schemas. They are an important tool in several areas of database research, notably in data integration and data exchange. However, a concrete theory of schema mapping optimization including the formulation of optimality criteria and the construction of algorithms for computing optimal schema mappings is completely lacking to date. The goal of this work is to fill this gap. We start by presenting a system of rewrite rules to minimize sets of source-to-target tuple-generating dependencies (st-tgds, for short). Moreover, we show that the result of this minimization is unique up to variable renaming. Hence, our optimization also yields a schema mapping normalization. By appropriately extending our rewrite rule system, we also provide a normalization of schema mappings containing equality-generating target-dependencies (egds). An important application of such a normalization is in the area of defining the semantics of query answering in data exchange, since several definitions in this area depend on the concrete syntactic representation of the st-tgds. This is, in particular, the case for queries with negated atoms and for aggregate queries. The normalization of schema mappings allows us to eliminate the effect of the concrete syntactic representation of the st-tgds from the semantics of query answering. We discuss in detail how our results can be fruitfully applied to aggregate queries.

Minimality is an important optimization criterion for solutions of data exchange problems, well captured by the notion of the core. Though tractability of core computation has been proved, it has not yet become a part of any industrial-strength system, still being highly computationally expensive. In this demonstration, we show how core computation can be used in a data exchange modeling tool, allowing data engineers to design more robust data transfer scenarios and better understand the sources of redundancy in the target database.