Database Reference
In-Depth Information
Based on Example 12 , for each egd (PK) for a relation r in
G
, we obtain the n
k
operators
{
q 1 ,...,q n k }⊆
k ,
q i = ( _ ) 1 ( y ) ( _ ) 2 ( z ) (y 1 1 = z 1 1 ) ∧···∧ (y 1 k = z 1 k )
(y j 1 =
T GA such that for any 1
i
n
z l n k )
z l 1 )
∨···∨
(y j n k =
( _ )( 0 , 1 ).
For a given set of k foreign-key constraints, represented by tgds (FK)
f i
x i r i, 1 ( x i )
r i, 2 ( t i ) ,i
=
1 ,...,k,
where t i is a tuple of terms with variables in x i and functional sy mbol s in f i ,we
obtain q i
O(r i, 1 ,r i, 2 ,r ) , with q i =
∧¬
( _ )( 0 , 1 ) .
Let us consider the foreign-key constraints (FK) in a global schema
(( _ ) 1 ( x i )
( _ ) 2 ( t i ))
G
given
by Example 27 (Sect. 4.2.4 ). Suppose that we have two relations r and s in
G
,
both of arity 2 and having the first attribute as a key, and the following foreign key
constraints in
G
:
1. π 2 (r)
π 1 (s) ,ofthetgd
x
y(r(x,y)
s(y,f 1 (x,y))) , with K
={
1
}
key in-
dexes of s , and K ={
2
}
.
2. π 1 (s)
π 1 (r) ,ofthetgd
x
z(s(x,z)
r(x,f 2 (x,z))) , with K
={
1
}
key in-
dexes of r , and K ={
1
}
and f 1 and f 2 introduced by Skolemization of the exis-
tential quantifications.
That is,
G =
(S G G ) with S G ={
r, s
}
, ar(r)
=
ar(s)
=
2 and
Σ tg G =
y r(x,y)
zs(y,z) ,
z s(x,z)
yr(x,y) .
x
⇒∃
x
⇒∃
Thus, in this case we obtain the schema mapping
MakeOperads {
} : G A ,
T GA =
Φ
with SOtgd Φ equal to
f 1 ,f 2
x,y r(x,y)
s y,f 1 (x,y)
r ( 0 , 1 )
∧¬
x,y s(x,z)
r x,f 2 (x,z)
r ( 0 , 1 ) .
∧∀
∧¬
α (
Consequently, we have to find an R-algebra α such that can(
) is a
model of this Data Integration system, that is, such α that satisfies the constraint
DB -morphism f Σ G =
I
,D)
=
G
α ( T GA )
={
α(q 1 ),α(q 2 ),q
}
,for
( _ ) 2 y,f 1 (x,y)
q 1 =
v 1 ·
q G, 1 =
( _ ) 1 (x,y)
∧¬
( _ )( 0 , 1 )
O(r,s,r ),
with q G, 1
O(r,s,r q 1 ) , v 1
O(r q 1 ,r
) , and for
( _ ) 2 x,f 2 (x,z)
q 2 =
v 2 ·
q G, 2 =
( _ ) 1 (x,z)
∧¬
( _ )( 0 , 1 )
O(s,r,r ),
with q G, 2 O(s,r,r q 2 ) , v 2 O(r q 2 ,r ) .
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