Exact chase inverse

Definition 2 (Exact chase-inverse). Let At be a GLAV schema mapping from a schema Si to a schema S2. We say that M is an exact chase-inverse of M if M is a GLAV schema mapping from S2 to S2 with the following property for every instance I over S 1 we have that I = chasem (chase... 

As it turns out, this candidate inverse satisfies the above requirement of being able to recover, exactly, the source instance. Indeed, it can be immediately verified that for every source instance I over S, we have that chaseMf (chasem" CD) equals I. Thus, Mt is an exact chase-inverse of M". 

The schema mapping Mt used in Sect. 3.2 is an exact chase-inverse in the sense that it can recover the original source instance I exactly. In general, however, equality with I is too strong of a requirement, and all we need is a more relaxed form of equivalence of instances, where intuitively the equivalence is modulo nulls. In this section, we start with a concrete example to show the need for such relaxation. We... 

Here, for simplicity, we focus on schema mappings on binary relations. (In particular, At" can be forced into this pattern if we ignore the major field in the two relations Takes and Takes".) The important point about this type of mappings is that they always have an exact chase-inverse. Consider now a variation on the above pattern, where Q is the same as Q. Thus, let M be the following schema mapping ... 

Thus, we recovered the two original facts of I but also the additional fact P(ni, n2) (via joining 2( i,2) and 2(2> 2))- Therefore, M is not an exact chase-inverse of M. Nevertheless, since n and n2 are nulls, the extra fact P(n, n2) does not add any new information that is not subsumed by the other two facts. Intuitively, the last instance is equivalent (although not equal) to the original source instance I. 

The existence of a chase-inverse for M implies that M has no information loss, since we can recover an instance that is the same modulo homomorphic equivalence as the original source instance. At the same time, a chase-inverse is a relaxation of the notion of an exact chase-inverse hence, it may exist even when an exact chase-inverse does not exist. 

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