Information Technology Reference
In-Depth Information
-
Ψ
=
{∼
deleted
(
f
11
)
,
∼
deleted
(
f
12
)
,
∼
deleted
(
f
13
)
∼
deleted
(
f
21
)
,
∼
deleted
(
f
22
)
,
∼
deleted
(
f
23
)
,
∼
deleted
(
f
31
)
,
∼
deleted
(
f
32
)
,
∼
deleted
(
f
33
)
,
∼
deleted
(
f
41
)
,
∼
deleted
(
f
42
)
,
∼
deleted
(
f
43
)
}
,
Υ =
{∼
deleted
(
r
31
)(
)
∼
deleted
(
r
32
)(
)
}
-
x
,
x
,
Σ
⊆{∼
deleted
(
(
))
∼
deleted
(
(
))
∼
deleted
(
(
))
-
r
11
x
,
r
12
x
,
r
21
x
,
∼
deleted
(
r
22
(
x
))
}
;
C
•
on
DF
is defined trivially. In particular,
C
(
s
(
)) =
{
s
(
)
|
=
}
on
x
y
y
x
,
for each r,
deleted
(
)
∈C
(
∼
deleted
(
))
∼
deleted
(
)
∈A
r
on
r
if
r
sm
DF
.
The possible sets
considered for the definition of the practical assumption-based argumentation
framework
pabf
DF
(
G
i
)
∈
PABFS
DF
(
G
)
Σ
≤
≤
(with
1
i
6
)aresuchthat:
•
G
1
=
{
cheap
,
good
,
fast
}
with
Σ
1
=
{∼
deleted
(
r
11
(
x
))
,
∼
deleted
(
r
12
(
x
))
,
∼
deleted
(
r
21
(
x
))
,
∼
deleted
(
r
22
(
x
))
,
∼
deleted
(
r
31
(
x
))
,
∼
deleted
(
r
32
(
x
))
}
;
•
G
2
=
{
good
,
fast
}
with
Σ
2
=
{∼
deleted
(
r
21
(
x
))
,
∼
deleted
(
r
22
(
x
))
,
∼
deleted
(
r
31
(
x
))
,
∼
deleted
(
r
32
(
x
))
}
;
=
{
good
,
fast
}
•
G
3
with
Σ
3
=
{∼
deleted
(
(
))
∼
deleted
(
(
))
r
21
x
,
r
22
x
,
∼
deleted
(
r
31
(
x
))
,
∼
deleted
(
r
32
(
x
))
}
;
•
G
4
=
{
fast
}
with
Σ
4
=
{∼
deleted
(
r
31
(
))
∼
deleted
(
r
32
(
))
}
x
,
x
.
It
is
clear
that
pabf
DF
(
G
1
)
P
pabf
DF
(
G
2
)
,
pabf
DF
(
G
1
)
P
pabf
DF
(
G
3
)
,
pabf
DF
(
G
2
)
P
pabf
DF
(
G
4
)
and
pabf
DF
(
G
3
)
P
pabf
DF
(
G
4
)
.
Having defined the PABFs, we show how a structured argument as in Definition 10
corresponds to an argument in one of the PABFs. To do this, we first define a mapping between
a structured argument and a set of assumptions.
Definition 22
(Mapping between arguments)
.
Let
DF
=
DL
be a decision framework. Let A be a structured argument in
,
P
sm
,
I
,
T
,
P
,
RV
A
(
DF
)
A
and concluding
α
∈DL
. The corresponding set of assumptions deducing
α
(denoted
(
)
)isdefined
according to the nature of A.
• fA is a hypothetical argument, then
A
.
• fA is a trivial argument built upon the fact f , then
(
)=
{
α
}
A
(
)=
{∼
deleted
(
)
}
f
.
• fA is a tree argument, then
(
A
)=
{∼
deleted
(
r
1
)
,...,
∼
deleted
(
r
n
)
}∪{
L
1
,...,
L
m
}
where:
(i) r
1
,...,
r
n
are the rules of A;
(ii) the literals L
1
,...,
L
m
are the presumptions and the decision literals of A.
The mapping is materialized through a bijection
A
(
DF
)
→A
A
:
sm
DF
where
sm
DF
is the set
of possible assumptions of one of the PABFs built upon
DF
and
A
(
DF
)
is the set of structured
arguments built upon
DF
.If
¯
Sisasetofarguments
S
A
(
DF
)
(
)
,wedenote
the corresponding
set of assumptions. Formally,
S
A
A
S
(
)=
{
(
)
|
∈
}
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