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not know the preferences of its interlocutors. We say that a decision is a minimal concession
whenever there is no other preferred decisions.
Definition 19
(Minimal concession)
.
Let
DF
=
DL
,
P
sm
,
I
,
T
,
P
,
RV
be a decision framework.
is a
concession
with respect to
dec
∈D
The decision
dec
∈D
iff there exists a set of decisions
D
such that
dec
∈
D
and for all
D
⊆D
with
dec
∈
D
, it is not the case that
D
P
D
. The decision
dec
is a
minimal concession
wrt
dec
iff it is a concession wrt
dec
and there is no
dec
∈D
such that
•
dec
is a concession wrt
dec
,and
•th es
D
⊆D
with
dec
∈
D
with
D
P
D
.
The minimal concessions are computed by the computational counterpart of our
argumentation framework.
{
s
(
c
)
}
Example 8
(Minimal concession)
.
According to the buyer,
is a minimal concession with
{
s
(
d
)
}
respect to
.
3.6 Computational counterpart
Having defined our argumentation framework for decision making, we need to find a
computational counterpart for it. For this purpose, we move our AF to an ABF (cf. Section 2)
which can be computed by the dialectical proof procedure of (Dung et al., 2006) extended
in (Gartner & Toni, 2007). So that, we can compute the suggestions for reaching a goal.
Additionally, we provide the mechanism for solving a decision problem, modeling the
intuition that high-ranked goals are preferred to low-ranked goals which can be withdrawn.
The idea is to map our argumentation framework built upon a decision framework
into a collection of assumption-based argumentation frameworks, that we call
practical
assumption-based argumentation frameworks
(PABFs for short). Basically, for each rule
r
in the
theory we consider the assumption
in the set of possible assumptions. By
means of this new predicate, we distinguish in a PABF the several distinct arguments that give
rise to the same conclusion. Considering a set of goals, we allow each PABF in the collection
to include (or not) the rules whose heads are these goals (or their strong negations). Indeed,
two practical assumption-based frameworks in this collection may differ in the set of rules
that they adopt. In this way, the mechanism consists of a search in the collection of PABFs.
∼
deleted
(
r
)
Definition 20
(PABF)
.
Let
DF
=
DL
,
P
sm
,
I
,
T
,
P
,
RV
be a decision framework and
G
∈G
a set
of goals such that
G
⊇RV
.A
practical assumption-based argumentation framework built upon
DF
associated with the goals
G
is a tuple
pabf
DF
(
G
)=
L
DF
,
R
DF
,
A
sm
DF
,
C
on
DF
where:
4
;
(i)
L
DF
=
DL∪{
deleted
}
R
DF
, the set of inference rules, is defined as follows:
-
For each rule r
(ii)
∈T
∈R
DF
such that
head
(
)=
head
(
)
, there exists an inference rule R
R
r
and
body
(
)=
body
(
)
∪{∼
deleted
(
)
}
R
r
r
;
-
If
r
1
,
r
2
∈T
with
head
(
r
1
)
I
head
(
r
2
)
and it is not the case that
head
(
r
2
)
P
head
(
r
1
)
,
then the inference rule
deleted
(
r
2
)
←∼
deleted
(
r
1
)
R
DF
.
is in
A
A
sm
DF
= Δ
∪
Φ
∪
Ψ
∪
Υ
∪
Σ
(iii)
sm
DF
, the set of assumptions, is defined such that
where:
Δ =
{
(
)
∈L|
(
)
}
-
D
a
D
a
is a decision literal
,
4
We assume
deleted
∈L
.
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