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i.e
r
with
head
(
r
)
∈{
g
,
¬
g
}
and
g
∈RV
, by having the assumption
∼
deleted
(
r
)
in
its set of assumptions. However, each framework in
PABFS
DF
(
G
)
can or cannot adopt the
rules concluding the goals (or their negation) which are not in the reservation value, i.e.
r
with
head
(
in its
set of assumptions. Referring to the running example and considering the goal
cheap
the
strongest structured arguments concluding
cheap
,requires
r
12
(
r
)
∈{
g
,
¬
g
}
and
g
∈
G
−RV
by having the assumption
∼
deleted
(
r
)
x
)
to be built within the PABF
if
sm
DF
.
Case
(iv)
defines the contrary relation of a PABFwhich trivially comes from the incompatibility
relation and which comes from the contradiction of
deleted
(
∼
deleted
(
r
12
(
x
))
∈A
r
)
with
∼
deleted
(
r
)
whatever the rule
r
is.
Arguments will be built upon rules, the candidate decisions, and by making suppositions
within the presumable beliefs. Formally, given a decision framework
DF
and a practical
assumption-based framework
pabf
DF
(
G
)=
L
DF
,
R
DF
,
A
C
on
DF
sm
DF
,
,wedefine
Σ
=
{∼
deleted
(
)
∈A
sm
DF
|
∈T
and
head
(
)
∈{
¬
}
∈
G
−RV}
r
r
r
g
,
g
and
g
as the set of goal rules considered in this PABF.
The practical assumption-based argumentation frameworks built upon a decision framework
and associated with some goals include (or not) the rules concluding these goals which are
more or less prior. This allows us to associate the set
PABFS
DF
(
G
)
with a priority relation,
denoted
, modeling the intuition that, in solving a decision problem, high-ranked goals are
preferred to low-ranked goals.
Definition 21
(Priority over PABF)
.
Let
DF
=
DL
P
P
I
T
P
RV
,
sm
,
,
,
,
be a decision framework,
G
∈G
a set of goals such that
G
⊇RV
and
PABFS
DF
(
G
)
be the set of PABFs associated with the goals
G
.
∀
G
1
,
G
2
such that
RV ⊆
G
1
,
G
2
⊆
G
∀
pabf
DF
(
G
1
)
,
pabf
DF
(
G
2
)
∈
PABFS
DF
(
G
)
,
pabf
DF
(
G
1
)
P
pabf
DF
(
G
2
)
iff:
•
G
1
⊃
G
2
,and
•
∀
g
1
∈
G
1
\
G
2
there is no g
2
∈
G
2
such that g
2
P
g
1
.
Due to the properties of set inclusion, the priority relation
P
is transitive, irreflexive and
asymmetric over
PABFS
DF
(
G
)
.
In order to illustrate the previous notions, let us go back to our example.
Example 9
(PABF)
.
Given the decision framework (cf example 3) capturing the decision problem
of the buyer (
RV
=
{
fast
}
{
fast
,
cheap
,
good
}
. We
denote this set
G
. We will consider the collection of practical assumption-based argumentation
frameworks
PABFS
DF
(
G
)
). We consider the set of goals
.
Let
pabf
DF
(
G
)=
L
DF
,
R
DF
,
A
C
on
DF
sm
DF
,
be a practical assumption-based argumentation
framework in
PABFS
DF
(
G
)
.ThisPABFisdefinedasfollows:
•
L
DF
=
DL ∪ {
deleted
}
,where
DL
is defined as in the previous example and
deleted
specifies if a rule does not hold;
•R
DF
is defined by the rules in Table 3;
•
A
sm
DF
=
Δ
∪
Γ
∪
Υ
∪
Σ
where:
-
Δ
=
{
s
(
x
)
|
x
∈{
a
,
b
,
c
,
d
}}
,
Φ =
{
reply
(
)
|
∈{
accept
,
reject
}}
-
y
y
,
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