Database Reference
In-Depth Information
the autoreferential Kripke model of the Heyting algebra
F
(W)
of the closed
hereditary subsets, for any two
V
1
,V
2
∈
F
(W)
, demonstrate that
V
1
V
2
=
Γ
∪
(V
1
,V
2
)
=
W
\
{
a
∈
W
|
a
b
}
b
∈
W
\
c
∈
V
1
∪
V
2
{
b
∈
W
|
c
b
}
10. How can the database Heyting algebra
L
alg
=
F(
Υ
)
, of closed simple objects
in the
DB
category, be embedded into the
N
-modal extension of the Boolean
algebra
DB
Υ
obtained by “vectorization” of the instance databases? Explain
what is the relationship between the weak monoidal topos
DB
, its Heyting al-
gebra
L
alg
=
F(
Υ
)
and the intuitionistic logics
IPC
. What is the finite model
property (fmp) for the intermediate logics, and how is it used to define
IPC
by
the class
K
F
of all finite rooted descriptive Kripke frames?
11. WMTL is defined by the class of Heyting algebras
HA
DB
in Definition
64
,
which are the complex algebras of Kripke frames in Proposition
72
, with the
class
K
DB
of these Kripke frames (corresponding to the class of Heyting alge-
bras
K
DB
is a class of rooted descriptive frames, so that for
each (autoreferential) frame F
∈
K
DB
, its complex algebra is
F
∗
=
HA
DB
). Show that
L
alg
=
Ob
DB
sk
,
0
,
Υ
.
,
¬
F
(
Υ
)
=
⊆
,
∩
,T
∪
,
⇒
,
⊥
Explain the meaning of the autoreferential Kripke semantics for any complete
Heyting algebra
H
=
(
Υ
,
≤
,
∧
,
∨
,,
¬
,
⊥
,R
Υ
)
∈
HA
DB
by the isomorphism
(
H
)
∗
=
↓:
H
F
(
Υ
).
12. What is the strict subclass
K
of the class of rooted descriptive frames
K
DB
(corresponding to the class of Heyting algebras
)
cor-
responds to the classical propositional logic
CPC
? What is the relationship be-
tween the class of all database mapping systems without incomplete informa-
tion (which are obtained by using only the FOL tgds and egds) and this class
HA
DB
) such that
Log(
K
K
of rooted descriptive frames? Why is the intuitionistic property of WMTL de-
termined only by its subclass of
infinite
complete Heyting algebras? What are
other significative examples of the infinite complete Heyting algebras? Explain
why the “exclude-middle” axiom is not satisfied by WMTL, while it satisfies
De Morgan laws.
13. (Open problem) It is demonstrated that WMTL is a Jankov's or De Morgan in-
termediate logic. Are there other theorems of the weak monoidal database topos
logic (based on its infinitary distributive complete lattice of closed database in-
stances in
DB
category) which would produce a strictly more powerful inter-
mediate logic w.r.t. this De Morgan logic?
References
1. N.D. Belnap, A useful four-valued logic, in
Modern Uses of Multiple-Valued Logic
, ed. by
J.-M. Dunn, G. Epstein (D. Reidel, Dordrecht, 1977)
