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Basic Assumptions About True And False.
The 'student' can make mistakes, i.e. possess
wrong
knowl-
edge. From an axiomatic point of view, if an agent acts as a 'teacher' in relation to a given knowledge
set, then the 'student' will consider as true every item provided by the 'teacher'.
Conventions.
Each KB is manually initiated, however, its update will be automatic, thanks to 'learn-
ing' and reasoning abilities. In order to simplify modeling, we only use formulas such as (
P
), (
P
→
Q
) and (P↔Q). (
P
) and (
Q
) are predicates conjunctions (or their negation) of type (
p
(
A
)) or (
p
(
X
)) (or
(¬(
p
(
A
))) or (¬(
p
(
X
)))), where
= …
a set of variables. For
simplification sake, we note
P
and
Q
such predicates conjunctions. Universal quantification is implicit
for each formula having at least one variable. We consider that, to initiate learning (from the 'student'
position), the 'teacher' has to rely on the 'student's' previous knowledge. This constraint imitates hu-
mans' learning methods. Therefore, before performing a tutored learning dialog, agents must have a
part of their knowledge identical (called
basic common knowledge
). The 'teacher' will be able to teach
new knowledge by using the 'student”s already known one. However, our agents do not 'physically'
share any knowledge (their KBs are independent).
…
is a set of terms and
X x x
{ , , , }
n
x
A a a
=
{ , , , }
n
a
1
2
1
2
Connexity As A KB Fundamental Property.
During learning, each agent will attempt to make its
KB as 'connex' as possible.
Definition 1.
A KB is
connex
when its associated graph G
Γ
is connex. A graph is associated to a KB
Γ as such:
Each formula is a node. An edge is created between each couple of formulas having the same premise
or the same conclusion or when the premise of one equals the conclusion of the other. For the moment,
variables and terms are not taken into account in premise or conclusion comparison (An abductive rea-
soning mechanism is contemplated as a possible mean to compare a constant fact
q
(
a
) with a predicate
with a variable
q
(
y
). We only consider the result of a succeeding abduction.) Thus, in a connex KB,
every knowledge element is linked to every other, the path between them being more or less long. As the
dialogic situation must be as close as possible to a natural situation,
agents' KBs are not totally connex:
A human agent can often, but not always, link two items of knowledge, haphazardly taken.
Examples:
A connex KB: Γ
1
= {
t
(
z
)∧
p
(
x
)→
q
(
y
),
r
(
x
)→
q
(
y
),
s
(
x
)→
r
(
y
),
q
(
a
),
r
(
b
)}
A non connex KB: Γ
2
= {
t
(
z
)∧
p
(
x
)→
q
(
y
),
r
(
x
)→
q
(
y
),
s
(
x
)→
u
(
y
),
q
(
a
),
u
(
b
)}
Figures 1(a) and 1(b) respectively represent graphs associated to Γ
1
and Γ
2
.
Definition 2.
A connex component (or just component) is a connex subset of formulas in a KB.
Theorem 1.
Let
A
,
B
and
C
be three connex formulas sets. If
A
∪
B
and
B
∪
C
are connex then
A
∪
B
∪
C
is connex.
Proof.
Let us assume that
A
∪
B
and
B
∪
C
is connex and
G
A
,
G
B
and
G
C
are graphs respectively associated
to
A
,
B
and
C
. According to definition 1:
A
∪
B
connex is equivalent to
G
A
∪
G
B
connex. Also,
B
∪
C
con-
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