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1. ( PK ) function for 1
i
n
k , g i =
α(q i )
:
α(r i )
×
α(r i )
α(r ) is a con-
stant function ( where q i =
((_ ) 1 ( y )
(_ ) 2 ( z )
((y 1 1 =
z 1 1 )
∧···∧
(y 1 k =
z 1 k ))
((y j 1 =
z l 1 )
∨···∨
(y j n k =
z l n k )))
(_ )( 0 , 1 )
O(r i ,r i ) ), that
is , for every tuple
d 1 , d 2
α(r i )
×
α(r i )
=
r i can( I ,D) ×
r i can( I ,D) ,
.
2. ( FK ) function when n
g i (
d 1 , d 2
)
=
k
+
1
i
n , g i =
α(q i )
:
α(r i, 1 )
×
α(r i, 2 )
α(r )
( where q i =
((_ ) 1 ( x i )
(_ ) 2 ( t i ))
(_ )( 0 , 1 )
O(r i, 1 ,r i, 2 ) ) is a constant
function , that is , for every tuple
d 1 , d 2
α(r i, 1 )
×
α(r i, 2 )
=
r i, 1 can( I ,D) ×
r i, 2 can( I ,D) , g i (
d 1 , d 2
)
=
.
Proof Directly from definitions presented in this section and by Propositions 16
and 17 . From the definition of f in , we obtain that for each relational symbol r S G
(which is a relation both in
r ret( I ,D) r can( I ,D) , that
is, the ret( I ,D) is a subdatabase of the can( I ,D) , that is, we obtain the monomor-
phism f in :
G
and
G T ) we obtain that
can( I ,D) and hence f in = T ret( I ,D) .
ret( I ,D)
Remark If we were to interpret the foreign key integrity constraints in Σ tgd as
the standard inter-schema mapping based on tgds, that is, by the mapping operad
M GG =
MakeOperads(TgdsToSOtgd(Σ tgd
: G G
G ))
then we would obtain the
morphisms in DB ,
}∪ id r :
S T :
f Σ tg G ={
g 1 ,...,g k ,q
α(r)
α(r)
|
r
can(
I
,D)
can(
I
,D),
where each function g i =
α(( _ ) 1 ( x i ))
( _ )( t i )
:
α(r i, 1 )
α(r i, 2 ) ,for1
i
k , is defined by: for every tuple
d 1 ,...,d n
α(r i, 1 )
=
r i, 1 ret( I ,D) , g i (
d 1 ,...,
d n
=
{
}
=
d i 1 ,...,d i h
=
i 1 ,...,i h
)
t such that π K (
t
)
and for each i/
K
(ordered
{
}
=
sequence of indexes in key(r i, 2 ) ) such that 1
i
ar(r i, 2 ) , π i (
t
)
I T (f r 2 ,i )(v i 1 ,
...,v i h ) .
That is, the endomorphism f Σ tg G : G G
would implement the process defined
previously by the rules in (FR). But we have explained that we avoid representing
the tgd's integrity constraint as an endomorphism (i.e., a morphism with the same
source and target object,
G
in this case) with a nonempty information flux, in order
to avoid the undesirable side-effects produced by their compositions with the real
(desiderated) inter-schema mappings. We avoided it by representing the integrity
constraints by “logic” mappings with target schema
A , which have the empty in-
formation flux (as can be seen for f Σ G in this Corollary 15 ).
4.2.3.1 Query Rewriting Coalgebra Semantics
The computation of a query q over a global schema
G
requires the building of a
canonical database can(
,D) , which is generally infinite. In order to overcome this
problem, a query rewriting algorithm [ 2 ] consists of two separate phases:
1. Instead of referring explicitly to the canonical database for query answering, this
algorithm transforms the original lifted query q into a new query exp
I
(q) over
G
a global schema, called the expansion of q w.r.t. Σ tgd
G
, such that the answer to
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