An Argumentation System with Indirect Attacks
Kazuko Takahashi
Kwansei Gakuin University, Sanda, Japan
Keywords:
Argumentation, Multi-agents, Semantics, Logic Programming, Persuasion.
Abstract:
We discuss argumentation frameworks with indirect attacks, such as why-questions and supports. A why-
question is regarded as a kind of attack relation, and a support is an answer to an un-presented why-question.
Based on this idea, we construct an argumentation framework with why-questions from a pair of knowledge
bases, as an instantiation of Dung’s abstract argumentation framework, and show that its extension is con-
sistent. Next, we transform this argumentation framework into an argumentation framework with supports,
and discuss its properties. The resulting framework is an instantiation of Bipolar Argumentation Framework
(BAF), defined as a triple consisting of arguments, attack relations and support relations. We define an exten-
sion of BAF, and show that the framework defined in this paper has some nice properties.
1 INTRODUCTION
Argumentation has long been an object of study in
philosophy, but recently has attracted attention in
the fields of artificial intelligence and computer sci-
ence, including multi-agent systems (Bench-Capon
and Dunne, 2007; Garc´ıa et al., 2007; Rahwan and
Simari, 2009).
Dung proposed an abstract argumentation frame-
work (AF) and expressed semantics in the form of
extensions (i.e., a set of accepted arguments) (Dung,
1995). Since then, numerous works have been
undertaken based on his framework including ex-
tended frameworks (Amgoud et al., 2008; Modgil and
Prakken, 2011; Prakken, 2010).
Dung’s abstract AF is defined as a pair consist-
ing of arguments and attack relations between argu-
ments. An attack relation is usually instantiated as a
counterargument against an opponent’s argument that
negates a statement (formula) in that argument. How-
ever, in actual argumentation, there frequently exist
indirect attacks other than counterarguments, such as
strengthening the grounds for one’s own claim or pos-
ing a query when the grounds for the opponent’s claim
are unacceptable. Such indirect attacks can be con-
sidered as a mechanism for expanding or deepening
argumentation.
Indirect attacks appear not only when contradic-
tory claims are inferred from the same fact, but also
when an agent cannot present a counterargument, and
instead questions the opponent’s conclusion. By do-
ing so, the agent may obtain new information or dis-
courage the opponent from presenting a counterargu-
ment.
Asking for grounds using a why-move is a basic
idea in argumentation systems (Walton and Krabbe,
1995), and is effective for legal reasoning. Prakken
pointed out that why-moves should be introduced in
AFs (Prakken, 2011), but neither an abstract AF with
why-moves nor its instantiation has thus far been pro-
posed.
Bipolar Argumentation System (BAF) is an ab-
stract AF in which support relations as well as attack
relations are regarded as binary relations between ar-
guments (Amgoud et al., 2008). Although the con-
cept of acceptable set obtained as a result of an argu-
mentation is defined in BAF (Cayrol and Lagasquie-
Shiex, 2010), the definition is complicated and does
not successfully relate to Dung’s semantics. A differ-
ent approach is proposed to prevent these drawbacks
by introducing support meta-arguments (Boella et al.,
2010). However, the instantiation of BAF has not
been presented, and which formulae are contained in
an acceptable set is not discussed.
In this paper, we propose a method of constructing
an AF with indirect attacks, such as why-questions or
supports, from given knowledge bases.
We regard two agents as having independent
knowledge bases, construct an AF with why-
questions AF
AS
from this pair of knowledge bases,
and show that its extension is consistent. Next, we
transform AF
AS
into an AF with supports by replac-
ing the pair consisting of a why-question and its an-
swer with a support relation. The resulting framework
551
Takahashi K..
An Argumentation System with Indirect Attacks.
DOI: 10.5220/0004323305510554
In Proceedings of the 5th International Conference on Agents and Artificial Intelligence (ICAART-2013), pages 551-554
ISBN: 978-989-8565-39-6
Copyright
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)
BAF
AS
is an instantiation of the existing BAF. We de-
fine an extension of BAF using the relationship with
AF
AS
, and show that BAF
AS
is a subset of BAF with
nice properties.
The remainder of this paper is organized as fol-
lows. In Section 2, we introduce Dung’s abstract AF
and describe basic concepts. In Section 3, we de-
fine our AF with why-questions from given knowl-
edge bases, and discuss its properties. In Section 4,
we describe the transformation of the above AF into
an AF with support relations, and discuss its proper-
ties. Finally, in Section 5 we present our conclusions.
2 AUGUMENTATION
FRAMEWORK
Definition 1 (Dung’s AF (Dung, 1995)). An AF is de-
fined as a pair hA , R i, where A is a set of arguments
and R ⊆ A × A is a set of attacks.
Definition 2 (conflict-free,admissible,extension). Let
hA , R i be an AF. For A, B ∈ A , and E ⊆ A ,
(1) E is conflict-free in hA ,R i iff there are no
elements A,B ∈ E such that A attacks B.
(2) E defends A in hA ,R i iff there exists an element
of E attacking each argument that attacks A.
(3) E is admissible in hA ,R i iff E is conflict-free
and defends all of its elements.
(4) E is a preferred extension of hA ,R i iff E is
maximal w.r.t. ⊆ admissible set.
Several extensions are defined as acceptable sets
of arguments within a given AF. Here we focus on
preferred extensions, and hereafter the word “exten-
sion” will mean “preferred extension.” Similar dis-
cussions are available for other extensions.
Example 1. For an AF =
h{A,B,C,D}, {(A,B),(A,C),(B,D),(C,A),(D,A)}i.
the preferred extension is {B,C} .
We instantiate AF with a logical theory.
Definition 3 (consistent,c-consistent (Modgil and
Prakken, 2011)). Let L be a set of propositional logic
formulae. If no formula ψ exists that satisfies both
ψ ∈ L and ¬ψ ∈ L, L is said to be consistent. If no
pair of φ and ψ exists that satisfies both φ ⇒ ψ ∈ L
and φ ⇒ ¬ψ ∈ L, L is said to be c-consistent, where
⇒ is a logical implication.
Let L be a set of propositional logic formulae. A
knowledge base K ⊆ L is a finite, consistent and c-
consistent set of propositional formulae. Each agent
has its own knowledge base, and uses its elements to
participate in argumentation. Note that K may not be
deductively closed; i.e., there may be a case in which
φ,φ ⇒ ψ ∈ K and ψ /∈ K hold. Also note that ¬¬ψ is
considered to be ψ. ∼ is introduced in order to make
extensions c-consistent by setting φ ⇒ ¬ψ can attack
φ ⇒ ψ. Let α be a formula φ ⇒ ψ, where φ may be
⊤. Then ∼α denotes either ¬(φ ⇒ ψ) or φ ⇒ ¬ψ.
3 AF WITH WHY-QUESTIONS
A why-question cannot occur arbitrarily, but occurs
only when an argument exists that it can attack.
Therefore, after constructing the usual arguments and
attack relations from the given pair of knowledge
bases, we construct arguments and attack relations
corresponding to why-questions.
Each agent p has its own knowledge base K
p
.
Definition 4 (argument). Let φ
1
,.. .,φ
n
and ψ be
formulae in K
p
. An argument on K
p
is a triple
(Data,Warrant,Claim), where Data = φ
1
,.. .,φ
n
,
Warrant = φ
1
∧ .. . ∧ φ
n
⇒ ψ and Claim = ψ.
For an argument P = (Data,Warrant,Claim)
on K
p
, Data,Warrant and Claim are denoted by
Dat(P),Wrr(P) and Clm(P), respectively. Fml(P)
is defined to be the set {Dat(P)} ∪ {Wrr(P)} ∪
{Clm(P)}. To simplify the problem, we consider only
the case where n = 1 in every argument; i.e., an argu-
ment is denoted by (φ,φ ⇒ ψ,ψ), where φ,φ ⇒ ψ, ψ ∈
K
p
, and denoted by (ψ) in case φ = ⊤.
Definition 5 (attack). Let A and B be arguments on
K
a
and K
b
, respectively.
If Clm(A) ⇔∼ Clm(B), then (A, B) is said to be
a rebut from A to B. If Clm(A) ⇔∼ Dat(B) or
Clm(A) ⇔∼Wrr(B), then (A,B) is said to be an un-
dercut from A to B. If (A, B) is a rebut or an undercut
from A to B, then (A,B) is an attack from A to B.
Let K
a
and K
b
be knowledge bases for agents a
and b, respectively. Let A
a
and A
b
be sets of argu-
ments on K
a
and K
b
, respectively. Also, let R
a
and
R
b
be sets of attacks from A to B and B to A, respec-
tively. Then, we introduce why-questions and their
answers.
Let p be agent a or b, and q its opponent.
and let WB denote either Wrr or Dat. For Q ∈
A
q
, if WB(Q) 6∈ K
p
, create a new argument called
why-argument A
whyp
= (¬WB(Q)) for p, and a
new attack called why-question (A
whyp
,Q). More-
over, if there exists an argument Q
′
∈ A
q
such that
Clm(Q
′
) ⇔ ¬WB(A
whyp
), create a new attack why-
answer (Q
′
,A
whyp
) corresponding to the answer to the
why-question.
Example 2. Figure 1 shows an example of
why-arguments and why-attacks. Assume that
ICAART2013-InternationalConferenceonAgentsandArtificialIntelligence
552
F,G,H,F ⇒ G,H ⇒ (F ⇒ G) ∈ K
q
and F ⇒ G /∈ K
p
.
First, two arguments Q,Q
′
are constructed. Then, a
why-argument A
whyp
is created, and a why-question
(A
whyp
,Q) and a why-answer (Q
′
,A
whyp
) are added.
Q
Q’
(a)
Q’
why-argument
Q
why-question
why-answer
(b)
A
whyp
F
G
F G
F
G
F G
F G
(F G)
H
H
F G
(F G)
H
H
(F
G)
Figure 1: Example of why-arguments and why-attacks.
Why-questions from p to q and why-answers from
p to q are called why-attacks from p to q.
Let A
whya
and R
whya
be a set of why-argumentsfor
a and a set of why-attacks from a to b, respectively.
Let A
whyb
and R
whyb
be a set of why-arguments for b
and a set of why-attacks from b to a, respectively.
Definition 6 (AF with why-questions on knowledge
bases (AF
AS
)). Let A be A
a
∪A
whya
∪A
b
∪A
whyb
and
R be R
a
∪ R
whya
∪ R
b
∪ R
whyb
. Then hA ,R i is said to
be an AF with why-questions on K
a
and K
b
, denoted
by AF
AS
.
In AF
AS
, an attack is either a rebut, an undercut
or a why-attack. Note that AF
AS
is an instantiation of
AF.
Proposition 1. AF
AS
does not have an odd loop; i.e.,
if (A
i−1
,A
i
) (∀i. 1 ≤ i ≤ n; A
n
= A
0
) are attacks, then
n is an even number.
Proposition 2. Let E be an extension of AF
AS
. Then,
∪
A∈E
{Clm(A)} is consistent and c-consistent, and
∪
A∈E
Fml(A) is consistent and c-consistent.
4 AN INSTANTIATION OF THE
BIPOLAR AF
4.1 Transformation from AF
AS
to
BAF
AS
A support is an argument that strengthens another
argument. It is considered as a why-answer pre-
sented without a why-question. Based on this idea,
we present a transformation T from AF
AS
to an AF
with support BAF
AS
.
Let AF
AS
be an AF with why-questions on K
a
and
K
b
. We define a set of supports for a and a set of
supports for b, denoted by S
a
and S
b
, respectively.
[Transformation from AF
AS
to BAF
AS
]
Set S
a
and S
b
equal to
/
0. Let p be agent a or b, and
q its opponent. For each why-argument Q ∈ A
whyq
and an argument P
′
such that (P
′
,Q) ∈ R
whyp
and
(Q,P) ∈ R
whyq
, (i) (P
′
,P) is added to S
p
, (ii) (P
′
,Q) is
deleted from R
whyp
for each P
′
, (iii) (Q, P) is deleted
from R
whyq
for each Q. Finally, we obtain the AF with
supports on the knowledge bases.
Definition 7 (AF with supports on the knowledge
bases (BAF
AS
)). Let A be A
a
∪ A
whya
∪ A
b
∪ A
whyb
,
R be R
a
∪R
whya
∪R
b
∪R
whyb
, and S be S
a
∪S
b
, where
each set is the end result of the transformation proce-
dure. Then hA ,R , S i is said to be an AF with supports
on K
a
and K
b
, denoted by BAF
AS
.
In this transformation T , why-questions with their
answers are replaced by supports, while the other ones
without their answers remain. Note that there is a
one-to-one relationship between AF
AS
and BAF
AS
.
Therefore, we can define T
−1
.
Example 3. Figure 2(a) shows AF
AS
. In this fig-
ure, X and Y are why-arguments and the edges con-
nected from/to them are why-attacks. Figure 2(b)
shows BAF
AS
obtained via T . In the figure, → rep-
resents an attack, and ⇒ represents a support.
X
Y
A
B
C
D
A
B
C
Y
D
(a) AF
AS
(b) BAF
AS
Figure 2: A transformation from AF
AS
to BAF
AS
.
4.2 Bipolar AF
An abstract bipolar AF (BAF) includes a support re-
lationship. We transform this framework to AF via
T
−1
, and define an extension of BAF using the corre-
sponding AF.
Definition 8 (BAF (Amgoud et al., 2008)). A BAF
is defined as a triple hA ,R ,S i, where A is a set of
arguments, R ⊆ A × A is a set of attack relations,
and S ⊆ A × A is a set of support relations.
Example 4. Figure 2(b) shows BAF =
h{A,B,C,D,Y}, {(A, B),(B,A),(C,A),(Y,D)}, {(D,B)}i.
Proposition 3. An abstract BAF can be transformed
into an abstract AF via T
−1
.
An extension of the resulting AF E
AF
is defined
according to Definition 2. An extension of BAF is
defined using E
AF
.
AnArgumentationSystemwithIndirectAttacks
553
Definition 9 (extension of BAF). Let hA ,R ,S i be a
BAF, and let AF be the corresponding Dung’s AF. Let
E
AF
be an extension of AF. Then E
AF
∩ A is an ex-
tension of BAF.
From this definition, the following property holds.
Proposition 4. Let E
AF
be an extension of AF
AS
on
K
a
and K
b
. Then there exists E
BAF
of BAF
AS
, ob-
tained from AF
AS
by T , such that E
BAF
⊆ E
AF
holds.
Example 5. In Figure 2, there is only one extension
in both AF
AS
and BAF
AS
. The extension of AF
AS
is
{C, X,Y}, while the extension of BAF
AS
is {C,Y}.
4.3 Properties of BAF
AS
BAF itself is defined as an abstract framework. It
can include cyclic arguments, and a pair of arguments
may be an attack and a support at the same time.
Moreover, it is immaterial which agent presents an
argument, and the order in which arguments are pre-
sented is also irrelevant. Therefore, the internal mean-
ing of an extension of BAF is unclear. On the other
hand, BAF
AS
obtained from AF
AS
via the transforma-
tion T
−1
is a subset of BAF that satisfies several nice
properties.
First, Propositions 1 and 2 in AF
AS
are preserved
in BAF
AS
.
In addition, the following properties hold.
Proposition 5. Let hA ,R ,S i be BAF
AS
. If (A
′
,A) is
in S , then Clm(A
′
) ⇔ Wrr(A) or Clm(A
′
) ⇔ Dat(A)
holds, and Clm(A
′
) /∈ K
a
∩ K
b
.
This proposition follows from the definition of
a why-question, and shows that we can construct
BAF
AS
directly from K
a
and K
b
.
Proposition 6. Let hA ,R ,S i be BAF
AS
. R ∩ S =
/
0.
This proposition follows from the definition of an
attack and a support.
Proposition 7. Let hA ,R , S i be BAF
AS
. Let A,A
′
,B
be arguments in A . If (A
′
,A) is in S and (B,A) is in
R , then (B,A
′
) is not in S , and (A
′
,B) is not in S .
This proposition shows that we need not consider
a case against our intuition in which B and A
′
are in
the support relation when B attacks A and A
′
supports
A.
We proved all these properties, although the
proofs are not shown here because of the space limit.
5 CONCLUSIONS
We proposed the construction of an AF with why-
questions from a pair of knowledge bases, as an in-
stantiation of an abstract AF, and showed that its ex-
tension is consistent. Moreover, we transformed this
framework into an AF with supports, and discussed
its properties.
Our main contributions are as follows. (1) Agents
argue using different knowledge bases, whereas a sin-
gle knowledge base is used in most systems. (2) An
AF with why-questions is constructed. (3) A new,
simple definition of BAF extension is given, and a
subset with some nice properties is presented. The
former two points are advantageous for handing ac-
tual argumentation.
As future research, we are considering the con-
struction of a system with changing knowledge bases.
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