Database Reference
In-Depth Information
Moreover, each R-algebra
α
of a given set of mapping-operads between a source
schema
A
B
and a target schema
determines a particular information flux from the
source into the target schema.
Definition 13
(I
NFORMATION
F
LUX
)
t
α
be
a
mapping-interpretation
(an
q
1
,...,q
n
,
1
r
∅
}=
MakeOperads(
M
AB
)
of mapping-operads, obtained from an
atomic
mapping
M
AB
:
A
→
B
R-algebra
in
Definition
11
)
of
a
given
set
M
AB
={
, and
A
=
α
∗
(S
A
)
be an instance of the schema
A
=
(S
A
,Σ
A
)
that
satisfies all constraints in
Σ
A
.
For each operation
q
i
∈
O(r
i,
1
,...,r
i,k
,r
i
)
,let
x
i
be its tuple of variables which appear at least one time free (not as an argument of
a function) in
t
i
and appear as variables in the atoms of relational symbol of the
schema
M
AB
,
q
i
=
(e
⇒
(
_
)(
t
i
))
∈
A
in the formula
e
[
(
_
)
j
/r
i,j
]
1
≤
j
≤
k
. Then, we define
=
1
≤
i
≤
n
{{
(i)
Va r (
M
AB
)
.
We define the kernel of the information flux of
M
AB
, for a given mapping-
interpretation
α
, by (we denote the image of a function
f
by '
im(f )
')
(ii)
Δ(α,
M
AB
)
={
π
x
i
(im(α(q
i
)))
|
q
i
∈
x
}|
x
∈
x
i
}
0
, f
M
AB
,
and
x
i
is not empty
}∪⊥
0
otherwise.
We define the information flux by its kernel by
(iii)
Flux(α,
M
AB
)
Va r (
M
AB
)
=∅
;
⊥
T(Δ(α,
M
AB
))
.
The flux of composition of
M
AB
and
M
BC
is defined by
(iv)
Flux(α,
M
BC
◦
=
Flux(α,
M
BC
)
.
We say that an information flux is
empty
if it is equal to
M
AB
)
=
Flux(α,
M
AB
)
∩
0
⊥
={⊥}
(and hence it is
not the empty set), analogously as for an empty instance-database.
We recall that the
kernels
of the information fluxes, defined in point (ii), will
be used as the actions for the Labeled Transition Systems (LTS) in the operational
semantics for the database mappings (in Sect.
7.3
).
The information flux of the SOtgd of the mapping
M
AB
for the instance-
level mapping
f
=
α
∗
(
M
AB
)
:
A
→
α
∗
(
B
)
composed of the set of functions
f
is denoted by
f
. Notice that
⊥∈
f
, and
α
∗
(
M
AB
)
=
={
α(q
1
),...,α(q
n
),q
⊥
}
hence the information flux
f
is an instance-database as well.
From this definition, each instance-mapping is a set of functions whose infor-
mation flux is the intersection of the information fluxes of all atomic instance-
mappings that compose this composed instance-mapping. These basic properties
of the instance-mappings will be used in order to define the database
DB
category
where the instance-mappings will be the morphisms (i.e., the arrows) of this cate-
gory, while the instance-databases (each instance-database is a set of relations of a
schema also with the empty relation
) will be its objects.
In the case of an atomic mapping, obtained from a set of egds of a given schema
⊥
A
, we have the following particular property:
