Abstract

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.

Highlights

  • 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

  • This means that a given SO tgd can be “approximated” by GLAV schema mappings up to any fixed level of precision, even though an SO tgd is a formula of second-order logic that may not be logically equivalent to any formula of first-order logic and, in particular, to any GLAV schema mapping

  • We have embarked on a systematic study of the limiting behavior of sequences of schema mappings using concepts and tools from metric spaces

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Summary

Introduction

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 (see the surveys [11, 12] and the monograph [1]). It follows that a schema mapping M can be be viewed as a function f from the set Inst(S) of all finite S-instances to the powerset P(Inst(T)) of the set of all finite T-instances, where f (I ) = {J : (I, J ) ∈ M} This way, a sequence (Mn)n≥1 of schema mappings over a source schema S and a target schema T can be viewed as a sequence of functions from Inst(S) to the (pseudo)metric space (P(Inst(T)), dist). There are uniformly Cauchy sequences of LAV mappings that have no uniform limit. As a consequence of this, the pointwise (respectively, uniform) limits of Cauchy sequences of schema mappings can be represented by generalized schema mappings, i.e., schema mappings that allow for infinite target instances as solutions to finite source instances

Preliminaries
Metric Space of Target Instances
Limits of Sequences of GAV Mappings
Limits of Sequences of LAV Mappings
Metric Space Completion and Generalized Schema Mappings
Representing Limits of Cauchy Sequences in the Metric Completion
Connections with Representations of Structural Limits
Concluding Remarks
Full Text
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