Database Reference
In-Depth Information
Definition 14
For a given set
S
of non-empty database schemas, we define the
graph
G
=
(V
G
,E
G
)
with
S
⊆
V
G
as follows:
A
=
(S
A
,Σ
A
)
where
Σ
A
=
Σ
tgd
A
∪
Σ
egd
1. For each database schema
is not empty,
A
we define the mapping edge
AA
={
Φ
}:
A
→
A
in
E
G
where the database
schema
A
=
(
{
r
}
,
∅
)
is defined as a new distinct vertex in
V
G
and
Φ
is defined
as follows:
•
If
Σ
egd
A
is an empty set of egds then we define
Φ
to be
TgdsToConSOtgd
Σ
tg
A
;
If
Σ
tgd
A
is an empty set of tgds then we define
Φ
to be
EgdsToSOtgd(Σ
egd
A
•
)
;
•
Otherwise we define
Φ
to be
EgdsToSOtgd
Σ
eg
A
∧
TgdsToConSOtgd
Σ
tg
A
.
2. For each inter-schema mapping defined by a non-empty set
S
of tgds, between
two database schemas
A
=
(S
A
,Σ
A
)
and
B
=
(S
B
,Σ
B
)
, we define the edge
M
AB
={
Φ
}:
A
→
B
in
E
G
where
Φ
is the SOtgd obtained from this set of tgds
by
TgdsToSOtgd(S)
.
All defined edges in
E
G
of this graph
G
will be called
atomic
mappings.
Note that a mapping graph is a necessary step in order to define a small sketch cat-
egory, as it follows from Corollary
2
, and to make the full embedding of the database
mapping system in the Categorical logic, based on functors (i.e., R-algebras) from
such a sketch category into the denotational (instance database's) category. In fact,
every directed graph can be transformed into a category as follows: the objects are
the vertices of the graph and the arrows are paths in the graph.
Thus, the first next step is to examine if for this denotational database category
we can use the standard
Set
category or, instead, we need a more appropriate one.
2.7
Review Questions
1. Can you explain the semantic difference between the ordinary tgds and
full
tgds? Why do we need a subset of the Second Order Logic? Can we deal with
incomplete knowledge with the 2-valued logics and without the Second Order
Logic?
2. Can we also deal with the inconsistent information by this kind of SOtgds? In
which way are we able to resolve the problems with the mutually-inconsistent
data in the standard GLAV semantics of an integration of a number of the source
databases into a unique global schema?
3. The many-valued logics are used for the paraconsistent logics as well. What
is gained by using the many-valued logics in order to deal with incomplete
and inconsistent information, and what are the drawbacks? Which is a minimal
many-valued logic able to also deal with incomplete and inconsistent data? It
