Database Reference
In-Depth Information
6. The power-view endofunctor T is a monad as well for the DB category. We
have seen that object TA is composed by the observations on a given instance
database A , obtained by the finite-length SPJRU queries. Based on this, which
kind of the monad and initial algebra semantics is in strict relationship with this
endofunctor? What is the relationship between the duality property of the DB
category and the information fluxes of the composed atomic morphisms? Why
can each morphism be represented as a composition of one epic and one monic
arrows in the DB category?
7. Two basic operations for the objects in DB category are Data Federation (iso-
morphic to union) and Data Separation (isomorphic to disjoint union). Can we
use any of them for the merging operation? If no then explain why and provide
examples.
8. The strong behavioral equivalence for the databases is obtained from the basic
observational PO relation
0 (an empty database)
and the top object Υ . This equivalence for the simple objects (databases) cor-
responds to the isomorphism of the objects in DB . Two isomorphic complex
objects are behaviorally equivalent as well. What would happen if we force that
also the converse be valid (consider the relationship between the object C and
the object, which is a coproduct in DB category, C C )?
9. C C is a coproduct of two equal databases. What does it mean from the DBMS
point of view for this complex database, and can we execute a query which
uses the relations from both databases? The same query may be executed over
only one of these two equal databases: why are C
with the bottom object
C and C observationally
equivalent? Moreover, why in this case are they not also isomorphic (so that
one can be equivalently substituted by another). How can this difference from
the isomorphism and observational equivalence express the redundancy in the
database theory?
10. In Lemma 10 , it was shown that the bottom object
0 is a zero and closed
object. Show formally that the total simple object Υ in Definition 26 is a closed
object, that is, Υ
=
T Υ . Then, based on the fact that the top complex object in
ω
Υ
( ω Υ )
=
=
···
DB category is Υ
Υ
Υ
(
Υ
) , show that it is a closed
object as well.
11. Two strongly equivalent databases are also weakly equivalent from the obser-
vational point of view. By applying the weak power-view operator T w to an
instance database A with the schema
and incomplete information, that is,
with the relational tables which can have the Skolem constants, we obtain the
subset of all views of A with only tuples without Skolem constants. In which
way a SQL query to this database A has to be modified (extended) in order to
obtain only the results without the Skolem constants? By considering the cer-
tain answers for a given query over a database A (the answers which are true in
every model of this database), are the relations in T w A and, if it is so, why?
A
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