Database Reference
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Example 10 Let us show how we construct the set S and the compacting of tuples
given by Definition 11 above:
Let us consider an operad q i
MakeOperads(
M AB ) , obtained from a nor-
.
=
malized implication φ Ai ( x )
r B ( t ) in
M AB , ((y
f 1 (x,z))
r 1 (x,y,z)
(f r 3 (y,z,w ,w) .
r 2 (v,x,w)
=
1 ))
r B (x,z,w,f 2 (v,z)) , so that q i is equal to
x,y,z,v,w,w
the expression (e
(the
ordering of variables in the atoms (with database relational symbols) from left
to right), t
( _ )( t ))
O(r 1 ,r 2 ,r 3 ,r B ) , where x
=
.
=
=
x,z,w,f 2 (v,z)
, and the expression e equal to (y
f 1 (x,z))
( _ ) 1 ( t 1 )
( _ ) 2 ( t 2 )
( _ ) 3 ( t 3 ) , with t 1 =
x,y,z
, t 2 =
v,x,w
and t 3 =
y,z,w ,w
.
Consequently, we obtain
= ( 1 , 1 ),( 2 , 2 ) , ( 2 , 1 ),( 1 , 3 ) , ( 3 , 1 ),( 2 , 3 ) , ( 3 , 2 ),( 4 , 3 )
S
that are the positions of duplicates (or joined variables) of x,y,z , and w , respec-
tively.
Thus, for given tuples d 1 =
a 1 ,a 2 ,a 3
α(r 1 ) , d 2 =
b 1 ,b 2 ,b 3
α(r 2 ) and
d 3 =
c 1 ,c 2 ,c 3 ,c 4
α(r 3 ) , the statement
π j h ( d j )
| (j h ,j),(n h ,n)
S
=
π n h ( d n )
is equal to
π 1 ( d 1 )
π 2 ( d 2 ) π 2 ( d 1 )
π 1 ( d 3 ) π 3 ( d 1 )
π 2 ( d 3 ) π 3 ( d 2 )
π 4 ( d 3 ) ,
=
=
=
=
which is true when a 1 = b 2 , a 2 = c 1 , a 3 = c 2 , and b 3 = c 4 .
The compacting of these tuples is equal to
Cmp S,
d 1 , d 2 , d 3 =
d
=
a 1 ,a 2 ,a 3 ,b 1 ,b 3 ,c 3
,
w /c 3 ]
with the assignment to variables
[
x/a 1 ]
,
[
y/a 2 ]
,
[
z/a 3 ]
,
[
v,b 1 ]
,
[
w/b 3 ]
, and
[
.
x/a 1 ,y/a 2 ,z/a 3 ,v/b 1 ,w/b 3 ,w /c 3 ]
is obtained by this assign-
ment g to the tuple of variables x , so that the sentence e
That is, d
=
x
[
[
( _ ) n /r n ] 1 n k /g is well
defined and equal to
a 2 =
I T (f 1 )(a 1 ,a 3 )
r 1 (a 1 ,a 2 ,a 3 )
r 2 (b 1 ,a 1 ,b 3 )
r 3 (a 2 ,a 3 ,c 3 ,b 3 ),
that is, to
a 2 = I T (f 1 )(a 1 ,a 3 ) r 1 ( d 1 ) r 2 ( d 2 ) r 3 ( d 3 ),
and if this formula is satisfied by such an assignment g , i.e., I T (e
[
( _ ) n /r n ] 1 n k /g)
=
1, then
= g(x),g(z),g(w),g f 2 (v,z)
= a 1 ,a 3 ,b 3 ,I T (f 2 )(b 1 ,a 3 ) ,
g ( t )
f(
d 1 , d 2 , d 3
)
=
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