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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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