Databases Reference
In-Depth Information
The implications
1
and
2
can be simplified by replacing every occurrence of e
0
with e (according to the equality e
0
D e). In addition,
1
can be further simplified
by replacing m with f.e/. We obtain:
1
:
Emp
.e/ !
Manager
.e;f.e//
2
:
Emp
.e/ ^ .f.e/ D e/ !
SelfMgr
.e/.
At this point, the resulting implications describe a relationship between relation
symbols of
S
and relation symbols of
T
0
. The final SO tgd that describes the com-
position
M
ı
M
0
is obtained by adding all the needed universal quantifiers in front
of each implication and then by adding in all the existentially quantified functions
(at the beginning of the formula). For our example, we obtain:
9f.8e
1
^8e
2
/:
The following theorem states that SO tgds suffice for composition of GLAV map-
pings. Moreover, SO tgds are closed under composition. Thus, we do not need to go
beyond SO tgds for purposes of composition.
Theorem 3
(
Fagin et al. 2005b
).
Let
M
and
M
0
be two consecutive schema
mappings.
M
0
are GLAV, then
M
ı
M
0
canbeexpressedbyanSOtgd.
1.
If
M
and
M
0
are SO tgds, then
M
ı
M
0
can be expressed by an SO tgd.
2.
If
M
and
Moreover, it is shown in
Fagin et al.
[
2005b
] that SO tgds form a minimal lan-
guage for the composition of GLAV mappings, in the sense that every schema
mapping specified by an SO tgd is the composition of a finite number of GLAV
schema mappings.
The above theorem has an immediate consequence in the context of target
schema evolution. As long as the original schema mapping
M
is GLAV or given
M
0
by a similar type
of mapping, the new adapted mapping can be obtained by composition and can be
expressed as an SO tgd.
Additionally, the above theorem also applies in the context of source schema
evolution, provided that the source evolution mapping
by an SO tgd, and as long as we represent the target evolution
M
00
has a chase-inverse. We
summarize the applicability of Theorem
3
to the context of schema evolution as
follows.
M
0
, and
M
00
be schema mappings as in Fig.
7.1
such that
Corollary 3.
Let
M
,
M
0
are SO tgds (or, in particular GLAV mappings) and
M
00
is a GLAV
M
and
M
00
has a chase-inverse
M
, then the mapping
M
ı
M
ı
M
0
can be
mapping. If
expressed as an SO tgd.
The important remaining restriction in the above corollary is that the source
evolution mapping
M
00
must have a chase-inverse and, in particular, that
M
00
is
M
00
is lossy and, hence, a
a lossless mapping. We address next the case where
chase-inverse does not exist.
