Database Reference
In-Depth Information
This function is monotonic w.r.t.
from the fact that
J
i
J
k
iff
i
≤
k
and thus
Ψ(J
i
)
)
.
This chain is represented in
DB
by the following composition of monomor-
phisms:
Ψ(J
k
)
, with least fixed point
Ψ(J
n
)
=
J
n
=
can
F
(
I
,
D
G
∅
=
J
0
→
J
1
→
J
2
→···
→
J
n
=
can
F
(
I
,
D
),
so that
G
∅
=
J
0
J
1
J
2
···
J
n
=
I
D
can
F
(
,
)
.
Notice that each infinite canonical database of a global database schema
G
is
weakly equivalent to the finite instance-database
can
F
(
I
,
D
)
that
is not a model
of
G
but is obtained as the least fixpoint of the operator
Ψ
.
Thus,
can(
I
,
D
)
≈
w
can
F
(
I
,
D
)
, where
can(
I
,
D
)
is an infinite model of
G
, and
can
F
(
I
,
D
)
is a finite weakly equivalent object to it in the
DB
category.
4.3
Review Questions
1. What is the relationship between the functorial semantics for database schemas
and Tarski's semantics? What is the relationship between the functorial seman-
tics for the database mappings and Tarski semantics and Second-order Logic
semantics? What is the meaning of the homomorphism
α
∗
in Proposition
14
?
Can we define a similar homomorphism between schema database mappings and
the morphisms in
DB
category and, if so, formalize such a homomorphism and
explain what is its meaning in the database theory.
2. What is the fundamental difference between the graph of a given database map-
ping system and the sketch category obtained from it? Why are we using the FOL
with identity? Is the identity binary predicate of the FOL used in this database
logic theory different for any database schema and the sketch obtained from this
schema in Proposition
15
? If not, in which FOL theory is it a binary predicate,
and what is exactly its interpretation? Can we have two different interpretations
of this predicate for a given
DB
category?
3. For the homomorphism
α
∗
in Proposition
14
, which represents a functorial inter-
pretation of the database schemas, is it possible to determine if such an interpre-
tation is a model of a schema
))
in Proposition
15
? For which kind of the schemas is every homomorphism
α
∗
in Proposition
14
also a model of such schemas? How can we use the morphism
f
Σ
A
in Proposition
15
in order to differentiate the models from non-models?
4. What are the main differences between the Data Exchange and Data integra-
tion? How is the Data Exchange generalized in this new framework? What is
the relationship between DATALOG and Data Integration? Explain the possi-
ble solutions for the data-inconsistency problems of information integrated from
the different sources? In which way is the data-inconsistency, obtained from a
database mapping system, checked by using the functorial semantics? Why is
the
DB
category able to support also the data-inconsistency without the prob-
lems of the explosive inconsistency that such situation makes to the formalism
based on the FOL?
A
without the definition of the sketch
Sch
(G(
A
