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expensive s (
x
)
, Price (
x , high )
, reply ( accept )
,
deleted (
r 11 (
x
))
cheap s (
x
)
, Price (
x , low )
, reply ( accept )
,
deleted (
r 12 (
x
))
good s (
x
)
, Resolution (
x , high )
, reply ( accept )
,
deleted (
r 21 (
x
))
bad s (
)
, Resolution (
x , low )
, reply ( accept )
deleted (
r 22 (
))
x
,
x
fast s (
)
, DeliveryTime (
x , low )
, reply ( accept )
deleted (
r 31 (
))
x
,
x
slow s (
)
, DeliveryTime (
x , high )
, reply ( accept )
deleted (
r 32 (
))
x
,
x
Price ( a , high ) ←∼ deleted (
f 11 )
Resolution ( a , low ) ←∼ deleted (
f 12 )
DeliveryTime ( a , high ) ←∼ deleted (
f 13 )
Price ( b , high ) ←∼ deleted (
f 21 )
Resolution ( b , high ) ←∼ deleted (
)
f 22
DeliveryTime ( b , high ) ←∼ deleted (
)
f 23
Price ( c , high ) ←∼ deleted (
)
Resolution ( c , low ) ←∼ deleted (
f 31
)
f 32
DeliveryTime ( c , low ) ←∼ deleted (
f 33
)
Price ( d , low ) ←∼ deleted (
)
Resolution ( d , low ) ←∼ deleted (
f 41
f 42
)
(
d , low
) ←∼
(
f 43
)
DeliveryTime
deleted
Table 3. The rules of the PABF
There is a one-to-one mapping between arguments in our AF and arguments in some
corresponding PABFs.
Lemma 1 (Mapping between arguments) . Let
DF = DL
,
P
sm ,
I
,
T
,
P
,
RV
be a decision framework, G ∈G
a set of goals and PABFS DF ( G )
be
the set of PABFs associated with the goals G .
1. Given a structured argument built upon DF concluding
α ∈DL
, there is a corresponding argument
deducing
α
in some PABFs of PABFS ( G )
.
2. Given an atomic formula
α ∈DL
and an argument of a PABF in PABFS ( G )
deducing
α
,there
exists a corresponding structured argument in
A ( DF )
concluding
α
.
Let us consider the previous example.
Example 10 (Assumptions) . The arguments in some PABFs corresponding to the structured
arguments D 2 and C include the following set of assumptions:
D 2 )= {∼ deleted ( r 31 ( d ))
(
deleted ( f 43 )
, s ( d )
, reply ( accept ) }
,
;
C
(
)= {∼ deleted ( r 31 ( c ))
deleted ( f 33 )
, s ( c )
, reply ( accept ) }
,
;
D 2 )
(
Both of them are tree argument. The corresponding set of assumptions
considers the literals
since D 2 is built upon these rules. Moreover, the literal
deleted (
r 31 ( d ))
deleted ( f 43 )
and
s ( d )
(respectively reply ( accept )
) is a decision literal (respectively a presumption).
In order to compute our extension-based semantics, we explore the collection of PABFs
associated to our AF in order to find the PABF which deduces the strongest goals as possible.
Indeed, we have developed a mechanism to explore the collection of PABFs associated to our
AF in order to compute it. If a s-admissible set of structured arguments concludes some goals,
then there is a corresponding admissible set of assumptions in one of the corresponding PABFs
and there is no other PABF, where an admissible set of assumptions deduces stronger goals.
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