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M
¢
M
Schema
T
¢
Schema
S
Schema
T
†
M
M
¢¢
Mo M
¢
†
M
o M o M
¢
Schema
S
¢¢
Fig. 7.1
Application of composition and inversion in schema evolution
M
0
, where the target schema of
M
0
. The result is a
M
is the source schema of
M
ı
M
0
that has the same effect as applying first
M
0
.
schema mapping
M
and then
M
ı
M
0
combines the original transformation
For our schema evolution context,
M
0
.
Source schema evolution.
Assume now that the source schema evolves to a new
source schema
S
00
, and that we model this evolution as a schema mapping
M
with the evolution mapping
M
00
from
S
to
S
00
. Intuitively,
M
00
represents a data transformation that converts instances of
S
to instances of
S
00
. We need to obtain a new schema mapping that reflects the
original schema mapping
M
ı
M
0
after target schema evolution) but
uses
S
00
as the source schema. Note that in this case we cannot directly combine
M
00
with
M
(or rather
M
ı
M
0
are not consecutive.
To be able to apply composition, we need first to apply the inversion operator and
obtain a schema mapping
M
ı
M
0
via composition, since
M
00
and
M
that “undoes” the effect of
M
00
. Once we obtain a
M
ı
M
ı
M
0
.
The resulting schema mapping, which is now from
S
00
to
T
0
, is an adaptation of the
original schema mapping
M
, we can then apply the composition operator to produce
suitable
that works on the evolved schemas.
While the composition of two schema mappings (under a fairly natural seman-
tics [
Fagin et al. 2004
;
Melnik 2004
]) always exists and it is “only” a matter of
expressing the composition in a suitable language for schema mappings, the situ-
ation is worse for inversion. In general, a schema mapping may lose information
and, as a result, it may not be possible to revert the transformation in a way that
recovers the original data. Hence, an exact inverse [
Fagin 2007
] may not exist,
and one needs to look beyond such exact inverses. As a result, there are several
notions of “approximations” of inverses that have recently been developed: quasi-
inverses [
Fagin et al. 2008b
], maximum recoveries [
Arenas et al. 2008
], maximum
extended recoveries [
Fagin et al. 2009b
]. In this paper, motivated by the applications
to schema evolution, we take a more pragmatic approach for the treatment of the
various notions of inverse and emphasize the
operational
aspect behind them. In par-
ticular, we focus on two types of inverses, which were first introduced in
Fagin et al.
[
2009b
] and which have a clear operational semantics based on the notion of chase
that makes them attractive from a practical point of view. The first type of such oper-
ational inverses, which are called
chase-inverses
, can be used to recover the original
M
