A schema mapping is a specification that describes how data structured under one schema (the source schema) is to be transformed into data structured under a different schema (the target schema). Although the notion of an inverse of a schema mapping is important, the exact definition of an inverse mapping is somewhat elusive. This is because a schema mapping may associate many target instances with each source instance, and many source instances with each target instance. Based on the notion that the composition of a mapping and its inverse is the identity, we give a formal definition for what it means for a schema mapping M′ to be an inverse of a schema mapping M for a class S of source instances. We call such an inverse an S-inverse. A particular case of interest arises when S is the class of all source instances, in which case an S-inverse is a global inverse. We focus on the important and practical case of schema mappings specified by source-to-target tuple-generating dependencies, and uncover a rich theory. When S is specified by a set of dependencies with a finite chase, we show how to construct an S-inverse when one exists. In particular, we show how to construct a global inverse when one exists. Given M and M′, we show how to define the largest class S such that M′ is an S-inverse of M.
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