Databases Reference
In-Depth Information
Coming back to our example, it can be verified that the above
M
satisfies the
conditions of being a relaxed chase-inverse of
M
00
, thus reflecting the intuition that
M
is a good “approximation” of an inverse in our scenario.
Since
M
M
is a GLAV mapping, we can now apply the composition of
with
M
ı
M
0
to obtain an SO tgd that specifies
M
ı
M
ı
M
0
. This SO tgd is the result of
to the new schemas
S
00
and
T
0
.Weleave
adapting the original schema mapping
M
the full details to the reader.
5.2
More on Relaxed Chase-Inverses
It is fairly straightforward to see that every chase-inverse is also a relaxed chase-
inverse. This follows from a well-known property of the chase that implies that
whenever U $ I we also have that U $
M
I. Thus, the notion of relaxed
chase-inverse is a generalization of the notion of chase-inverse; in fact, it is a strict
generalization, since the schema mapping
M
in Sect.
5.1
is a relaxed chase-inverse
M
00
. However, for schema mappings that have a
chase-inverse, the notions of a chase-inverse and of a relaxed chase-inverse coin-
cide, as stated in the following theorem, which can be derived from results in
Fagin
et al. [
2009b
].
M
00
but not a chase-inverse of
of
Theorem 4.
Let
be a GLAV schema mapping from a schema
S
1
to a schema
S
2
that has a chase-inverse. Then the following statements are equivalent for every
GLAV schema mapping
M
M
from
S
2
to
S
1
:
M
is a chase-inverse of
(i)
M
.
M
is a relaxed chase-inverse of
(ii)
M
.
As an immediate application of the preceding theorem, we conclude that the
schema mapping
M
00
in Sect.
5.1
has no chase-inverse, because
M
is a relaxed
M
00
.
In Sect.
3.3
, we pointed out that chase-inverses coincide with the extended
inverses that are specified by GLAV constraints. For schema mappings that have
no extended inverses, a further relaxation of the concept of an extended inverse
has been considered, namely, the concept of a
maximum extended recovery
[
Fagin
et al.
2009b
]. It follows from results established in
Fagin et al.
[
2009b
] that relaxed
chase-inverses coincide with the maximum extended recoveries that are specified by
GLAV constraints.
M
00
but not a chase-inverse of
chase-inverse of
6
Implementations and Systems
In this section, we examine systems that implement composition and inversion and
apply them to the context of schema evolution. We do not attempt to give here a
complete survey of all the existing systems and implementations but rather focus
