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-
Φ
=
{
B
∈B|
B
∈P
sm
}
,
-
Ψ
=
{∼
(
r
)
|
r
∈T
and
head
(
r
)
∈{
L
,
¬
L
}
s.t. L
∈G}
,
deleted
-
Υ
=
{∼
deleted
(
r
)
|
r
∈T
and
head
(
r
)
∈{
g
,
¬
g
}
s.t. g
∈RV}
;
-
Σ
=
{∼
deleted
(
r
)
|
r
∈T
and
head
(
r
)
∈{
g
,
¬
g
}
s.t. g
∈
G
−RV}
;
(iv)
.
The set of practical assumption-based argumentation frameworks built upon
DF
and associated with
the goals
G
with
C
on
DF
the set of contraries is defined such that for all
α
∈A
sm
DF
,y
∈C
on
(
α
)
iff y
I
α
RV ⊆
G
⊆
G
will be denoted
PABFS
DF
(
G
)
.
Case
(i)
defines the language. In order to capture the decision problem within an
assumption-based argumentation framework, we have extended the decision language to
include a predicate symbol
deleted
, which is used to specify whether or not a rule is
adopted within the PABF. It is worth noticing that the definition of arguments in the ABF (cf
Definition 6) focuses attention on the candidate assumptions and ignores the internal structure
of arguments. In order to distinguish in a PABF the several distinct arguments that give rise
to the same conclusion, we have named the rules used to deduce it. Therefore, an argument
in a PABF contains a set of assumptions of the following schemata
∼
deleted
(
)
r
,forallrule
r
used by the argument.
Case
(ii)
defines the inference rules. Firstly, there is an inference rule for each rule of the theory.
Forthispurpose,thebodyofeachrule
r
is extended by adding the assumption
∼
deleted
(
)
r
.
Referring to Example 3, the rule
r
11
(
x
)
becomes
expensive
←
s
(
)
,
Price
(
x
,
high
)
,
reply
(
accept
)
∼
deleted
(
(
))
x
,
r
11
x
.
In this way, the assumption
allows an argument to use this rule.
Secondly, the inference rules include not only the original deduction rules but also the conflicts
amongst the rules having incompatible heads. It is worth noticing that the attack relation
between arguments in the ABF (cf Def. 7) ignores the possible conflicts amongst the heads of
rules which are not assumptions. In order to capture these conflicts, we have introduced rules
which allow the defeasibility of rules. Referring to the example, we introduce, e.g.,
deleted
(
r
12
(
∼
deleted
(
r
11
(
x
))
))
modeling the given incompatibility
cheap
I
expensive
. Obviously, we also introduce,
deleted
(
r
11
(
x
))
←∼
deleted
(
r
11
(
x
))
modeling the given incompatibility
expensive
I
cheap
. Our treatment of conflicting rules
requires not to interfere with our treatment of priorities which is inspired by (Kowalski &
Toni, 1996). Referring to the example, we introduce, e.g.,
deleted
(
r
12
(
x
))
←∼
deleted
(
r
12
(
x
))
modeling the given priority
cheap
P
fast
. In this way, the corresponding literal
∼
deleted
(
r
31
(
x
))
←∼
deleted
(
r
31
(
x
))
x
must be assumed in order to handle this priority. Obviously, we do not
introduce,
deleted
(
r
31
(
))
←∼
deleted
(
r
12
(
))
.
Case
(iii)
defines the assumptions. The decisions are obviously possible assumptions. In the
same way, a PABF adopts an presumable belief if this is a presumption of the corresponding
AF
. Referring to the example, an argument, which assumes that the reply is an acceptance,
can be built since
reply
(
accept
)
∈A
x
x
sm
DF
. Each framework adopt the epistemic rules, i.e
r
with
head
(
)
∈{
¬
}
∈G
∼
deleted
(
)
r
L
,
L
and
L
, by having the assumption
r
in its set of
assumptions.
We want to go through the set of goals such that high-ranked goals are preferred to low-ranked
goals and the reservation value is the minimal set of goals we want to reach. For this
purpose, we adopt the rules concluding the goals (or their negation) in the reservation value,
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