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the tgd of the constraint. This intuitively means that the association is expanded
to include the referenced attributes of the constraint. The procedure is repeated
until no more schema constraints can be applied, in which case the association
has become maximal. This maximal association is a logical association. Maier
et al. [
Maier et al. 1979
] have shown that for the relational model, two differ-
ent chase sequences with the same set of dependencies, i.e., constraints, generate
identical results. Popa [
Popa 2000
] has shown a similar result for the case of the
nested relational model. These two results mean that the the result of the chase of a
user or a structural association with a set of constraints is unique. To illustrate how
the logical relations are computed, consider again the example on Fig.
5.4
b. Let
A
represent the structural association
Public
Grant
.gi;am;co;in;ma;as/
.The
tgd expressing the foreign key constraint from the attribute
company
to
name
is
Public
co
.Its
left-hand side is the same as
A
, thus, the question on whether it is logically implied
by
A
is yes, which means that a chase step can be applied on
A
using the specific
constraint. This will enhance
A
with the contents of the right-hand side of the tgd of
the constraint, bringing the association into the form:
Grant
.gi;am;co;in;ma;as/
!
Public
Company
.na;sy/
,
na
D
Public
Grant
.
gi
;
am
;
co
;
in
;
ma
;
as
/;
Public
Company
.
na
;
sy
/;
na
D
co
Further chase steps on the association using the foreign key constraints on the
attributes
manager
and
assistant
will further expand the association into the form:
Public
Company
.
na
;
sy
/;
Contact
.
cim
;
phm
/;
Contact
.
cia
;
pha
/;
na
Grant
.
gi
;
am
;
co
;
in
;
ma
;
as
/;
Public
D
co
^
cim
D
ma
^
cia
D
as
:
Since no other constraint can be further applied to it
A
,
A
in its last form is a logical
association.
Associations form the basis for understanding how the correspondences can be
combined together to form groups that will produce semantically meaningful map-
pings. The technique presented here forms the basis of the Clio [
Fagin et al. 2009
]
mapping tool. Given a set of correspondences, Clio generates a mapping scenario
with nested tgds. Similar technique has also been adapted by other tools, such as
Spicy [
Bonifati et al. 2008a
]orHePToX[
Bonifati et al. 2005
]. This is done by con-
sidering pairs of source and target logical associations. For each such pair, the set of
correspondences covered by the pair is discovered. A correspondence is said to be
covered by the pair
A;B
of a source and a target association, if its left and right part
(apart from the equality condition) are logically implied by
A
and
B
, respectively.
A mapping is formed by creating a tgd whose left-hand side consists of association
A
, and whose right-hand side is the association
B
enhanced with the conditions
of the covered correspondences. Note that the covering of a correspondence is
based on the notion of homomorphism. If there are multiple homomorphisms, then
there are multiple alternative mappings. Consider, for instance, the source-target
logical association pair
Public
Company
.na;sy/
and
Company
.na2;id;
Grand
/
.
