Database Reference
In-Depth Information
2. What is the way to reduce each complex object to a
strictly
complex object and
why is this reduction invariant for the ptp arrows between two complex objects?
Explain why it is important for the both epic and monic complex arrows to have
the same number of ptp arrows in order to obtain the isomorphism? What is the
sense of the “maximality” of the principal arrows (morphisms)? What are the
canonical examples of the retraction pairs of the arrows in relationship with the
categorial symmetry of the
DB
category? What is the relationship between the
category of idempotents on a given simple object
A
and the categorial symmetry
of the
DB
category?
3. Why does the matching between two instance database represent the maximal
common information (the views) contained in both databases? Is the extension to
complex databases appropriate? (If no, explain why, and try to define it.) Is there
a natural sense for the matching operation what has to be a dual operation w.r.t.
matching? The merging endofunctor is defined as a unary operator by prefixing
with a given instance database. Is it a limitation to have a family of parameterized
unary operators instead of a unique binary endofunctor? The generalized binary
merging operator (for the objects only) is introduced later, in order to have a
maximal mathematical flexibility, in order to elaborate on a database complete
lattice. Do you have an idea how this generalization can be provided for the
arrows as well, in order to obtain a binary merging endofunctor?
4. The
DB
category is finitely (co)complete, that is, it is (co)Cartesian. The funda-
mental property is that it has all pullbacks. Why is it not a Cartesian Closed Cate-
gory (CCC)? We have demonstrated that it is not a topos, based on the properties
of its morphisms (that both an epic and monic complex arrow is not necessarily
an isomorphism), so the absence of the exponentiation is another proof. It is well
known that the CCCs are the models for the typed
λ
-calculus, with the currying
operator
Λ
. Is it reasonable to suspect that
DB
has the
λ
-calculus computation
capability, if we demonstrated that the computation power of
DB
is equivalent to
the full relational algebra
Σ
RE
with all update operations for the databases? How
can the absence of this important property of the
DB
category be explained w.r.t.
to the types of the logic formulae (i.e., SOtgds) used for the database mappings?
Can you indicate which kind of the more powerful logic formula we would need
for the database mappings in order to obtain the Cartesian Closed denotational
DB
category?
5. Explain the difference between the Universal algebra theory for a
DB
category
and the initial algebra semantics for
Σ
R
relational algebras (i.e., the syntax
monad
T
P
X
) provided in Sect.
5.1.1
. Why is it possible to substitute in this set-
ting the
DB
category with its skeletal subcategory
DB
sk
composed of only closed
objects? What is a variety
K
, and why do we deal with the
quotient
term alge-
bras? What is the meaning of the bifunctor
E
DB
OP
Set
w.r.t. the initial
algebra semantics and adjoint universal functor
U
and forgetful functor
F
, and
their composition
F
◦
U
? Show that the
DB
and
DB
sk
are concrete categories.
6. What is the reason to develop a lattice structure for the databases, from a (many)
logic theory point of view (for example, Boolean algebras are based on the par-
ticular lattice structures)? The lattice
L
DB
=
(Ob
DB
,
,
⊗
,
⊕
)
is a complete in-
finite lattice with the top and bottom objects
Υ
and
:
sk
×
K
→
0
, respectively. Why it is
⊥
