[10:07 13/5/2013 exs047.tex] LogCom: Journal of Logic and Computation Page: 545 541–562
Partial semantics of argumentation 545
Based on Definition 3.4, some basic properties of the set of relevant arguments of a given subset
are formulated in the following theorem.
Theorem 3.1
Let AF =A,R be an abstract argumentation framework, and B,C ⊆ A sets of arguments. It holds
that:
(1) rlvt
AF
(B) ⊇B;
(2) rlvt
AF
(B)
−
=∅, i.e., rlvt
AF
(B) is an unattacked set;
(3) if B ⊇ C, then rlvt
AF
(B) ⊇rlvt
AF
(C); and
(4) rlvt
AF
(B∪C) =rlvt
AF
(B)∪rlvt
AF
(C).
The proof of Theorem 3.1 is presented in Appendix A.1.
Since rlvt
AF
(B) is an unattacked set, we may use rlvt
AF
(B) to induce a sub-framework of
AF: AF↓
rlvt
AF
(B)
. Under a semantics S ∈{CO,PR,GR,ID} satisfying the directionality criterion,
the extensions of AF↓
rlvt
AF
(B)
can be independently computed (according to Definition 2.2).
E
S
(AF↓
rlvt
AF
(B)
) is called a partial semantics of AF w.r.t. B, in which S ∈{CO,PR,GR,ID}.
Definition 3.5
Let AF =A,R be an abstract argumentation framework, B ⊆A a set of arguments, and rlvt
AF
(B) the
set of relevant arguments of B. Under a semantics S ∈{CO,PR,GR,ID} satisfying the directionality
criterion, the partial semantics of AF w.r.t. B is defined as E
S
AF↓
rlvt
AF
(B)
.
Theorem 3.2
Let AF =A,R be an abstract argumentation framework, and B ⊆ A a set of arguments. Under a
semantics S ∈{CO,PR,GR,ID} satisfying the directionality criterion, for every argument α ∈ B, α
is sceptically justified (respectively, credulously justified and indefensible) w.r.t. E
S
(AF), if and only if
α is sceptically justified (respectively, credulously justified and indefensible) w.r.t. E
S
AF↓
rlvt
AF
(B)
.
The proof of Theorem 3.2 is presented in Appendix A.2.
According to this theorem, when querying the justification status of some arguments in an AF, we
only need to compute the partial semantics of the AF w.r.t. these arguments.
Example 3.4
Continue Example 3.3. Let us consider the following two approaches.
Firstly, use the partial semantics of AF
1
to evaluate the status of arguments in B
1
, B
2
and B
3
,
respectively.
Under preferred semantics,
E
PR
AF
1
↓
rlvt
AF
1
(B
1
)
={E
1,1
,E
1,2
,E
1,3
},in which
E
1,1
={α
1
,α
9
},E
1,2
={α
1
,α
10
},E
1,3
={α
2
,α
10
};
E
PR
AF
1
↓
rlvt
AF
1
(B
2
)
={E
2,1
},in which E
2,1
={α
13
};
E
PR
AF
1
↓
rlvt
AF
1
(B
3
)
={E
3,1
},in which E
3,1
=∅.
According to these partial semantics, in terms of Definition 2.3, it holds that α
9
and
α
10
are credulously justified w.r.t. E
PR
AF
1
↓
rlvt
AF
1
(B
1
)
, α
13
is sceptically justified w.r.t.
E
PR
AF
1
↓
rlvt
AF
1
(B
2
)
, and α
14
is indefensible w.r.t. E
PR
AF
1
↓
rlvt
AF
1
(B
3
)
.
at Zhejiang University on May 17, 2013http://logcom.oxfordjournals.org/Downloaded from
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