Database Reference
In-Depth Information
3.5
Review Questions
1. From the DBMS and from the query-answering points of view, what is the
meaning of the separation
A
B
A
⊕
B
†
and the connection
of two database
schemas
? Why is it necessary to introduce the complex mapping-
operads with the structural-operators in our setting, for a fixed source and target
databases in the case when we have a set of intermediate databases between
them? Why does the information flux of the complex mapping-operads have to
be disjoint union of the information fluxes of its simple mapping components
in Definition
15
for all structural operators different from
A
and
B
|
,
...,
_
, instead of
a simple union?
2. What is the fundamental difference between the simple atomic sketch's arrows:
View-mappings and the Integrity-constraints mappings? What is the reason that
they can be used for the database sketches and category theory? It holds from
Theorem
1
that each category
DB
is determined by a given universe
U
where
the domain
dom
can be big enough for any database mapping system but fi-
nite: why must the set of Skolem marked null values
SK
be infinite instead,
with the result that we can have the instance database with finitary relations
(a finite number of attributes) but with infinite number of tuples? Can you pro-
vide an example with a cyclic mapping between the databases with incomplete
information which requires an introduction of the Skolem constants, such that
this cyclic mapping produces the relations with an infinite number of tuples?
Which kind of tgds needs the Skolem constants, and why does the incomplete
information in a database mapping need the SOtgds? Does it mean that in all
situations, when we use the SOtgds, we must obtain the databases with some
relation composed by an infinite number of tuples?
3. How is it possible that the
n
-ary structural operators for the morphisms in Def-
initions
20
and
21
are enough to express any possible complex arrow, based
on the fact that the unique operator
for composition of complex objects
(databases) is the disjoint union (use the fact that point 1 of Definition
21
defines
the parallel composition of the arrows, points 2 and 3 define the ingoing and out-
going branching, and point 4 defines the union of different simple arrows). The
definition of the ptp arrows is dual to the structural way of the representation
of the complex arrows with the set of structural operators. Is there a possibility
to define a canonical structural expression, for a complex arrow composed by
a given set of ptp, different for that defined by Lemma
8
? Can you provide an
example?
4. The equality of the arrows in the
DB
category is not the set-theoretical equality,
but an observational one, based on the relationships of the information fluxes
of the simple arrows. Can you give an explanation why, for two given simple
instance databases
A
and
B
, the simple mapping
f
:
A
→
B
is different from
f,f
:
A
→
B
?
5. In Proposition
5
, we demonstrated that
DB
category is not a topos. Which kind
of degeneration of the
DB
category would we obtain if we force the morphism
[
the mapping
id
C
, id
C
]:
C
C
→
C
to be an isomorphism, by considering the fact that
is
the coproduct for the objects in
DB
category?
