NEURAL NETWORKS COMPUTING THE DUNGEAN SEMANTICS

OF ARGUMENTATION

Yoshiaki Gotou

Graduate School of Science and Technology, Niigata University, Niigata, Japan

Takeshi Hagiwara, Hajime Sawamura

Institute of Science and Technology, Niigata University, Niigata, Japan

Keywords:

Argumentation, Semantics, Neural network.

Abstract:

Argumentation is a leading principle foundationally and functionally for agent-oriented computing where rea-

soning accompanied by communication plays as essential role in agent interaction. In the work of (Makiguchi

and Sawamura, 2007a) (Makiguchi and Sawamura, 2007b), they constructed a simple but versatile neural

network for the grounded semantics (the least ﬁxed point semantics) in the Dung’s abstract argumentation

framework (Dung, 1995). This paper further develop its theory so that it can decide which argumentation se-

mantics (admissible, stable, complete semantics) a given set of arguments falls into. In doing so, we construct

a more simple but versatile neural network that can compute all extensions of the argumentation semantics.

The result leads to a neural-symbolic system for argumentation.

1 INTRODUCTION

Much attention and effort have been devoted to the

symbolic argumentation so far (Ches˜nevar et al.,

2000)(Prakken and Vreeswijk, 2002)(Besnard and

Doutre, 2004)(Rahwan and Simari, 2009), and its ap-

plication to agent-oriented computing. Argumenta-

tion can be a leading principle both foundationally

and functionally for agent-oriented computing where

reasoning accompanied by communication plays an

essential role in agent interaction. Dung’s abstract

argumentation framework and argumentation seman-

tics (Dung, 1995) have been one of the most in-

ﬂuential works in the area and community of com-

putational argumentation as well as logic program-

ming and non-monotonic reasoning. A. Garcez et al.

proposed a novel approach to argumentation, called

the neural network argumentation (Garcez et al.,

2009). In the papers (Makiguchi and Sawamura,

2007a)(Makiguchi and Sawamura, 2007b), they dra-

matically developed their initial ideas on the neu-

ral network argumentation to various directions in a

more mathematically convincing manner. In this pa-

per, we further develop its theory so that it can de-

cide which argumentation semantics (admissible, sta-

ble, complete semantics) a given set of arguments

falls into. For this purpose, we will construct a

more simple but versatile neural network that can

compute all extensions of the argumentation seman-

tics (Dung, 1995)(Caminada, 2006), and lead to a

neural-symbolic system for argumentation (Levine

and Aparicio, 1994)(Garcez et al., 2009).

The paper is organized as follows. In Section 2,

we recall what Dung’s inﬂuential work on the abstract

argumentation framework and argumentation seman-

tics is like, for the preparation of the succeeding sec-

tions. In Section 3, we introduce a 4-layer neural net-

work and a translation of a given abstract argumen-

tation framework to it, and a calculation of the argu-

mentation semantics with it. In Section 4, we describe

an implementation of our neural argumentation with

illustrating examples. Final section include previous

work,future work, and concluding remarks.

2 ARGUMENTATION

SEMANTICS

Argumentation semantics is an important subject

since it tells us what correct or justiﬁed arguments are.

However, there can be plural semantics for argumen-

Goto Y., Hagiwara T. and Sawamura H..

NEURAL NETWORKS COMPUTING THE DUNGEAN SEMANTICS OF ARGUMENTATION.

DOI: 10.5220/0003656600050014

In Proceedings of the International Conference on Neural Computation Theory and Applications (NCTA-2011), pages 5-14

ISBN: 978-989-8425-84-3

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

tation, mainly derived from the nature of argumenta-

tion. We introduce the most basic and inﬂuential deﬁ-

nitions for argumentation semantics which originated

with (Dung, 1995).

2.1 Argumentation Framework

The argumentation semantics begins with the deﬁ-

nition of (abstract) argumentation framework (Dung,

1995). We are then concerned with calculating all ex-

tensions of the semantics for an argumentation frame-

work. Then we assume the argumentation framework

to be ﬁnite.

Deﬁnition 1 (Argumentation Framework) (Dung,

1995). An argumentation framework (A F ) is a pair

hAR, attacksi where AR is a ﬁnite set of arguments,

and attacks is a binary relation over AR (in symbols,

attacks ⊆ AR ×AR).

Each element A ∈ AR is called argument and

(A,B) means argument A attacks argument B. An ar-

gumentation framework can be represented as a di-

rected graph where the arguments are represented as

nodes and the attack relation is represented as arrows.

In this paper, we don’t consider the internal structure

of each of the arguments. For this reason, we don’t

refer to the structure of attack relations.

Example 1 ( Ralph goes Fishing) (Caminada,

2008). Consider the following arguments:

Argument A. Ralph goes ﬁshing because it is Sun-

day.

Argument B. Ralph does not go ﬁshing because it is

Mother’s day, so he visits his parents.

Argument C. Ralph cannot visit his parents, because

it is a leap year, so they are on vacation.

Three arguments are represented as A, B and C. In

this case, we can see (B, A) as an attack relation be-

cause B says that Ralph does not go ﬁshing against A.

Similarly to (B, A) we can see (C, B) as an attack re-

lation. An example of an argument framework of the

argumentation is given in Figure 1.

B CA

Figure 1: A F =h {A, B,C}, {(B, A), (C, B)} i.

2.2 Dung’s Argumentation Semantics

The following deﬁnitions are basic notions and used

in deﬁning the argumentation semantics.

Deﬁnition 2. Let (AR,attacks) be an argumentation

framework,A,B∈AR and S⊆ AR.

attacks(A, B) iff (A, B) ∈ attacks.

= {A ∈ AR|attacks(S, A)} .

attacks(S, A) iff ∃C ∈ S(attacks(C, A)).

In the deﬁnition above, S

means a set of argu-

ments attacked by the arguments belonging to the S.

Deﬁnition 3 (Conﬂict-free (Dung, 1995)). Let (AR,

attacks) be an argumentation framework and let S ⊆

AR. S is said to be conﬂict-free iff S ∩ S

In example 1, according to the deﬁnition 3, {A,C}

is conﬂict-free but {A,B}is not conﬂict-free. The no-

tion of conﬂict-free means that there don’t exsit any

attack relations each other.

The notion of defend is a core of argumentation

semantics and deﬁned as follows.

Deﬁnition 4 (Defend (Dung, 1995)). Let (AR,atta-

cks) be an argumentation framework, A ∈ AR and

S ⊆ AR. S is said to defend A (in symbols, de-

fends(S,A)) iff ∀B∈AR(attacks(B,A)→attacks(S,B)).

In example 1, according to the deﬁnition 4,

defends({C}, A) holds. And note that there are at

least two attack relations when defends(S, A) holds

(except S=

0). For example, (C,B) and (B,A) exist in

example 1. For this reason, the neural network to be

proposed in section 3 needs input, ﬁrst hidden, and

second hidden layer. So we can obtain two connec-

tions from input layer to second hidden layer. At the

same time the neural net work consisted of the 3-layer

implements the notion of de fend.

The following characteristic function F is useful

for understanding the argumentation semantics and

deﬁned by the notion of defend.

Deﬁnition 5 (Characteristic Function F) (Dung,

1995). Let (AR,attacks) be an argumentation frame-

work and S ⊆ AR. We introduce a characteristic

function as follows:

• F : 2

→ 2

• F(S) = {A∈AR | defends(S,A)}

With these in mind, we give a series of deﬁnitions

for the argumentation semantics.

Deﬁnition 6 (Admissible Set (Dung, 1995)). Let

(AR,attacks) be an argumentation framework and S

⊆ AR. S is said to be admissible

iff S is conﬂict-free and ∀ A ∈ AR ( A ∈ S → de-

fends(S,A) )

iff S is conﬂict-free and S ⊆ F(S).

NCTA 2011 - International Conference on Neural Computation Theory and Applications

According to Deﬁnition 6, the empty set is surely

admissible set.

Deﬁnition 7 (Preferred Extension (Dung, 1995)).

Let (AR,attacks) be an argumentation framework and

S ⊆ AR. S is said to be a preferred extension iff S is

a maximal (w.r.t set-inclusion) admissible set.

Deﬁnition 8 (Complete Extension (Dung, 1995)).

Let (AR,attacks) be an argumentation framework and

S ⊆ AR. S is said to be a complete extension

iff S is conﬂict-free and ∀ A ∈ AR ( A ∈ S ↔

defends(S,A) )

iff S is conﬂict-free and S=F(S).

Deﬁnition 9 (Stable Extension (Dung, 1995)). Let

(AR,attacks) be an argumentation framework and S

⊆ AR. S is said to be a stable extension

iff S is conﬂict-free and ∀ A ∈ AR ( A 6∈ S→ at-

tacks(S,A) )

iff S is conﬂict-free and S∪S

=AR.

The deﬁnition of the stable extension doesn’t need

the notion of defend. Therefore we need to construct

another layer for the neural network in order to com-

pute the stable extension. In fact, output layer is im-

plememted for it.

Deﬁnition 10 (Grounded Extension) (Dung,

1995). Let (AR,attacks) be an argumentation frame-

work. The grounded extension is the minimal ﬁxpoint

of F.

We represent Admissible Set, Preferred Extension,

Complete Extension, Stable Extension and Grounded

Extension as AS, PE, CE, SE and GE respectively.

Example 2 (Argumentation Semantics in A F ).

Let h{A,B,C,D,E},{(A,B),(B,C),(C,D),(D,C),(D,E),

(E,E)}i be an argumentation framework. The A F

represented as a directed graph is given in ﬁgure 2.

B DA C E

Figure 2: A F = h{A, B,C, D, E}, {(A,B), (B,C), (C, D), (D,

C), (D, E), (E, E)}i.

All extensions of the argumentation semantics in

A F are calculated as follows:

• AS :

0, {A}, {D} , {A,C}, { A, D}

• CE : {A}, {A,C}, {A, D}

• PE : {A,C}, {A, D}

• SE : { A, D}

• GE : {A}

In the next section, we give a neural network

which can computationally accomplish the argumen-

tation semantics described so far.

3 NEURAL NETWORK

CALCULATING DUNG’S

ARGUMENTATION

SEMANTICS

We present how to construct a neural network which

computes the argumentation semantics from A F and

determines it using the constructed neural network.

Once a neural network has been constructed for an ar-

gumentation framework, we can compute its seman-

tics automatically. This means the neural network

incorporates such an apparatus as for computing the

characteristic function (cf. Deﬁnition 5).

From this section, we represent a translated neural

network from A F as N and let (AR, attacks) be an

argumentation framework.

3.1 Translation from A F to N

3.1.1 The Number of Attacks

Before the translation algorithm from A F to N , we

introduce the function A which tell us the number of

attacks for each argument. And the values generated

from the function A deﬁne the threshold θ

of the

second hidden neuron α

Deﬁnition 11 (Function A ). We introduce a function

A as follows:

• A :AR → N

• A (X) = |{(Y, X)|Y ∈ AR∧(Y, X) ∈ attacks}|

Example 3. Examples of function A in Figure 2 are

given as follows:

• A (A) = |

0| = 0

• A (B) = |{(A, B)}| = 1

• A (C) = |{(B,C)}, {(D,C)}| = 2

• A (D) = |{(C, D)}| = 1

• A (E) = |{(D, E)}, {(E, E)}|= 2

Generally the size of attacks equals the sum

of function A for all arguments. For example 3,

attacks = A (A) + A (B) + A (C) + A (D) + A (E) =

0+1+2+1+ 2= 6. In addition the value of function

NEURAL NETWORKS COMPUTING THE DUNGEAN SEMANTICS OF ARGUMENTATION

A (X) means the degree of difﬁculty of the defend for

the argument X. Therefore if the function A (X) takes

high value, it is difﬁcult to defend the argument X.

3.1.2 Translation Algorithm

This subsubsection is a core part of this paper and

shows the translation algorithm from A F to N . It

is deﬁned as follows:

Step 1 (Input). Given an argumentation framework

hAR, attacksi where AR = {α

, α

, ..., α

Step 2 (Types of Neurons). For each argument α

∈AR (1≤k≤n), a neuron is created for each layer

(input layer, ﬁrst hidden layer, second hidden

layer and output layer) and named as α

(see Ta-

ble 1 about X).

Table 1: Representation rules of neuron.

Layer name of neuron X

Input layer i

First hidden layer h

Second hidden layer h

Output layer o

Step 3 (Weight). Let a, b be positive real numbers

and satisfy

√

b > a > 0. For each argument α

∈

AR (1≤k≤ n), do (i) ∼ (iii).

(i) connect input neuron α

to ﬁrst hidden neuron

and set a as the connection weight.

(ii) connect hidden neuron α

to second hidden

neuron α

and set a as the connection weight.

(iii) connect second hidden neuron α

to output

neuron α

and set a as the connection weight.

Step 4 (Connections and Weights). Let α

∈AR. For each attack relation (α

, α

)∈

attacks, do (i) ∼ (iii).

(i) connect input neuron α

to ﬁrst hidden neuron

and set -b as the connection weight.

(ii) connect ﬁrst hidden neuron α

to second

hidden neuron α

and set -b as the connection

weight.

(iii) connect second hidden neuron α

to output

neuron α

and set -1 as the connection weight.

Step 5 (Threshold). For each argument α

∈AR

(1≤k≤n), Set A (α

)·b as the threshold θ

of the

second hidden neuron α

Step 6 (Activation Function of Input Neurons).

Set i(x) as the activation function of the input neu-

rons as follows (cf. Figure 3):

i(x) =



x (x ≥ 0)

0 (x < 0)

(1)

i(x)

Figure 3: Activation function i(x).

-b

h(x)

Figure 4: Activation function h(x).

Step 7 (Activation Function of First Hidden Neu-

rons). Set h(x) as the activation function of the

ﬁrst hidden neurons as follows (cf. Figure 4):

h(x) =







a (x ≥a

)

0 (−b < x < a

)

−b (x ≤ −b)

(2)

Step 8 (Activation Function of Second Hidden

Neurons). Set g(x) as the activation function of

the second hidden neurons as follows (cf. Figure

5):

g(x) =



a (x ≥ θ

)

0 (x < θ

)

(3)

Step 9 (Activation Function of Output Neurons).

Set f(x) as the activation function of the output

neurons as follows (cf. Figure 6):

f(x) =







a (x ≥a)

0 (−a < x < a)

−a (x ≤−a)

(4)

All A F can be translated to N by the translation al-

gorithm above. Then N must be a 4-layer neural net-

work. However, the number of neurons is not con-

stant. In detail, the number of neurons equals 4 times

the number of the arguments.

NCTA 2011 - International Conference on Neural Computation Theory and Applications

g(x)

Figure 5: Activation function g(x).

f(x)

-a

Figure 6: Activation function f(x).

The starting point of the idea of the architectute

for constructing the neural network comes from the

notion of defend. According to the deﬁnition 4, at

least two attack relations are required for it. Then we

can gain an attack relation between two neurons. Sim-

ilarly, we can gain two attack relations in the three

neurons. Threrefore 3-layer neural network can im-

plement the notion of defend. And the threshold of

the second hidden layer is speciﬁed by the number

of attack relations and deﬁnes the notion of de fend

and conflict − free. For this reason, 3-layer neu-

ral network from input layer to second hidden layer

can compute the admissible set, complete extension,

and grounded extension. However, for the purpose of

the computing the stable extension, we add the output

layer. Thus we construct the 4-layer neural network

in this idea. In addition all settings are required so

that the neural network computes argumentation se-

mantics. For this reason, the neural network includes

the characterictic function F(cf. Deﬁnition 5) and a

capability of checking conflict − free (cf. Deﬁnition

3).

And the argumentation semantics is computed via

checking an input vector converted from a set of argu-

ments (S∈AR) (cf.Deﬁnition12) and an output vector

which is ﬁnally output by N . The criteria for check-

ing the argument semantics is described in 3.3.

An example of the translation from A F to N is

seen in Example 4.

Example 4 (Translation from A F to N ). An ex-

ample of the translation from A F in Figure 2 to N

is as follow:

Step 1.

B DA C E

Step 2.

Step 3.

:weight is a

Step 4.

:weight is -b

:weight is -1

NEURAL NETWORKS COMPUTING THE DUNGEAN SEMANTICS OF ARGUMENTATION

Table 2: Threshold θ

Argument α

A (α

) Threshold θ

A 0 0

B 1 b

C 2 2b

D 1 b

E 2 2b

Step 5. Thresholds for each argument are shown in

Table 2.

Step 6 ∼ Step 9. Set the activation function to each

neuron.

3.2 Calculating Argumentation

Semantics by using N

In this subsection, we introduce notions and rules for

calculating the argumentation semantics by N .

Notation (Notation of Input-output Values of Neu-

rons). Let α

∈ AR. We represent the input values

of neurons as I

and similarly the output as O

(see Table 1 for X). The Figure 7 represents the input-

output values of neurons.

weight is a

weight is -b

weight is -1

...

Figure 7: Representation of the input-output values of neu-

rons

An input vector is given to N at ﬁrst and used for

judging the argumentation semantics. The vector is

deﬁned in the following Deﬁnition 12.

Deﬁnition 12 (First Input Vector I ). Let |AR| = n

and S ⊆ AR. A function I from a set of arguments to

input vector is deﬁned as follws:

• I : 2

→ {a, 0}

• I (S) = [I

, I

, ..., I

], I



a (α

∈S)

0 (α

6∈S)

Example 5 (First Input Vector). Let Figure 2 be an

argumentation framework and S={A,C, E} . The ﬁrst

input vector I (S) = [a, 0, a, 0, a] by Deﬁnition 12.

Only when calculating the grounded extension, an

output vector is fed back as an input vector until N

gets to be a converging state (cf. Deﬁnition 14). The

following notions are necessary for the feedbacks.

Deﬁnition 13 (Time Round τ). In the computation

with a neural network N , the passage of time till the

output-vector is read off at the output-neurons since

the input-neurons are given an input-vector is called

time round, symbolically denoted τ. It has 0 as initial

value, and is incremented by 1 every time on input-

vector is recurrently given to N .

Deﬁnition 14 (Converging State of N ). If the

input-vector given to N is identical with the output-

vector read off at the same τ, the computation of N is

said to be in a converging state.

3.3 Argumentation Semantics in

The following deﬁnitions show how to interpret a re-

sult of computation of N and determine the argu-

mentation semantics of N . The argumentation se-

mantics in N can be deﬁned only using the input-

output vectors except the grounded extension. Let

k be a natural number, | AR |= n, the ﬁrst input

vector be [i

, i

, ..., i

] and the output vector be

, o

, ..., o

Deﬁnition 15 (Admissible Set in N ). A set S⊆AR

is an admissible set in N iff ∀k(i

= a →o

= a).

Example 6. Let S = {D}. The input-output vector in

the translated N from Figure 2 (cf. Example 4) is the

following:

First Input Vector. [0, 0, 0, a, 0]

Output Vector. [a, 0, 0, a, 0]

According to Deﬁnition 15, {D} is an admissible set

in N .

Deﬁnition 16 (Complete Extension in N ). A set

S⊆AR is a complete extension in N iff ∀k(i

= a ↔

= a).

Example 7. Let S = {A, D}. The input-output vector

in the translated N from Figure 2 (cf. Example 4) is

the following:

NCTA 2011 - International Conference on Neural Computation Theory and Applications

First Input Vector. [a, 0, 0, a, 0]

Output Vector. [a, 0, 0, a, 0]

According to Deﬁnition 16, {A, D} is a complete ex-

tension in N .

Deﬁnition 17 (Stable Extension in N ). A

set S⊆AR is a stable extension in N iff

∀k(i

= a ↔o

= a) ∧ ∀l(i

= 0 →o

= −a).

Example 8. Let S = {A,C, E}. The input-output vec-

tor in the translated N from Figure 2 (cf. Example 4)

is the following:

First Input Vector. [a, 0, a, 0, a]

Output Vector. [a, −a, a, −a, a]

According to Deﬁnition 17, {A,C, E} is a stable ex-

tension in N .

Deﬁnition 18 (Grounded Extension in N ). Let S =

0. S

= {α

∈ AR | O

=a in the converging state of

N } is the grounded extension.

Example 9. The input-output vector in the translated

N from Figure 2 (cf. Example 4) is the following:

When calculating the grounded extension, the ﬁrst in-

put vector must be I (

0) = [0, 0, 0, 0, 0]. And the out-

put vector in the converging state of N is used for

checking the grounded extension.

First Input Vector. [0, 0, 0, 0, 0]

Output Vector. [a, −a, 0, 0, 0]

According to Deﬁnition 18, {A} is a grounded exten-

sion in N .

3.4 Soundness of Neural Network

Argumentation

The follwing theorem states that the computation of

the neural network provides a sound computation in

the sense that it yields the same extension as Dung’s

semantics.

Theorem 1. Let S ∈ AR. S is an extension of argu-

mentation semantics calculated in N iff S is an exten-

sion of the argumentation semantics in A F .

Due to the space limitation, we will not describe

the details of the proof (see Chapter 5 in (Gotou,

2010)). The idea of the proof is showing the sound-

ness of the deﬁnition of the threshold θ

. Be-

cause the threshold deﬁnes the notion of defend

and conflict − free and argumentation semantics are

mostly deﬁned by them.

Figure 8: A program behavior from Step 1 to Step 4.

4 IMPLEMENTATION

In this section, we brieﬂy describe an implementa-

tion of the neural network argumentation by using its

program results. When the program receives A F , it

outputs all extensions for each semantics(Admissible,

Complete, Stable, Grounded). The following is the

ﬂow of the program.

Step 1. Input the number of arguments.

Step 2. Arguments named in alphabetical order are

generated.

Step 3. Input the name of an argument which is at-

tacked for each argument, otherwise type’0’.

Step 4. After inputting every counterargument,all at-

tack relations are output.

Step 5. Input-output values of neurons are output for

every calculation process.

Step 6. All extensions for each semantics are output.

Here, we will show an example of the

program behavior, assuming the pro-

gram receives A F =h{A, B,C, D, E},

{(A, B), (B,C), (C, D), (D,C), (D, E), (E, E)}i which

is given in Figure 2. An example of the program

behavior from Step 1 to Step 4 is given in Figure 8.

Now we calculate some semantics for A F . There

are 5 arguments and therefore we input 5 (Step 1).

Thereupon 5 arguments (named A, B, C, D, E) are

generated (Step 2). Then we input the name of an ar-

gument which is attacked. For the argument A in this

example, A attacks B and has no more arguments it at-

tacks. Accordingly, we input B at ﬁrst and next, input

NEURAL NETWORKS COMPUTING THE DUNGEAN SEMANTICS OF ARGUMENTATION

Figure 9: Computation process (1).

0 (Step 3). After inputting every counterargument, all

attack relations are outputted (Step 4).

We set 1 as a and 2 as b in the implementa-

tion. The following shows the computation process

for checking the grounded extension (input S=

0). See

Figure 9. The vectors in lines represent the input-

output values of each neuron and are listed in alpha-

betical order about arguments and the top is the input

and the bottom is the output. Calculating only the

grounded semantics, the output vector is fed back as

a new input vector. In the time round 1, the input vec-

tor equals to the output vector. We can regard it as

a converging state of N . Thus {A} is the grounded

extension in N by Deﬁnition 18.

The computation process for input S = {A, D} is

shown in Figure 10. According to Deﬁnition 15, 16

Figure 10: Computation process (2).

and 17, {A, D} are admissible, complete and stable.

The results of all extensions of the argumentation

semantics is shown as Figure 11. We need the power-

of-AR computations in N in order to get all exten-

sions of the argumentation semantics.

Figure 11: Results of argumentation semantics.

5 PREVIOUS WORK

Previously, Makiguchi and Sawamura proposed the

neural network that computes the Dungean argu-

mentation semantics (Makiguchi and Sawamura,

2007a)(Makiguchi and Sawamura, 2007b). How-

ever, it turned out that the neural network can-

not compute the Dungean argumentation semantics

in some abstract argumentation framework. For

example, let be h {A, B,C}, {(B, A), (C, B)} i an

argumentation framework shown in Figure 1 and

S = {A,C}. According to the study (Makiguchi

and Sawamura, 2007a)(Makiguchi and Sawamura,

2007b), ﬁrst input vector =[1, 0, 1] and input vector

=[0, −1, 1].This means that {A,C} is not justiﬁed in

admissible set by the work (Makiguchi and Sawa-

mura, 2007a)(Makiguchi and Sawamura, 2007b).

However, according to the deﬁnition 6, {A,C} should

be justiﬁed in admissible set. Thus the neural network

cannot compute some extensions correctly. On the

other hand the neural network we proposed can com-

pute all extensions. In the same example, ﬁrst input

vector =[a, 0, a] and input vector =[a, −a, a]. Accord-

ing to the deﬁnition 15, {A,C} is justiﬁed in admissi-

ble set as well as Dungean argumentation semantics.

For this reason, there are several differences between

our work and the work (Makiguchi and Sawamura,

2007a)(Makiguchi and Sawamura, 2007b). And the

main differences are shown as follws:

• The former architecture is 3-layer neural network

but the latter is 4-layer.

• The former neural network is completely deﬁned

by invariable but the latter is mostly deﬁned by

variable.

• The former neural network needs the deﬁnition of

types of attack relation but the latter doesn’t.

• The former is the recurrent neural network but the

latter is the feedforward neural network (except

the computation of the grounded extension).

• The former neural network cannot compute some

NCTA 2011 - International Conference on Neural Computation Theory and Applications

extensions but the latter can compute all exten-

sions.

The neural network we proposed is constructed so

that it can compute the Dungean argumentation se-

mantics correctly. And the Dungean argumentation

semantics is deﬁned by the notions of defend and

conﬂict-free mainly. For this reason, all parameters

are set up so that the notions are implememted into

the neural network.

6 RELATED WORK

D’Avila Garcez et al. initiated a novel approach to

argumentation, called the neural network argumen-

tation (Garcez et al., 2005). However, the semantic

analysis for it is missing there. That is, it is not clear

what they calculate by their neural network argumen-

tation.

Besnard and Doutre proposed three symbolic ap-

proaches to checking the acceptability of a set of ar-

guments (Besnard and Doutre, 2004), in which not all

of the Dungean semantics can be dealt with. So it

may be fair to say that our approach with the neural

network is more powerful than Besnard et al.’s meth-

ods.

Vreeswijk and Prakken proposed a dialectical

proof theory for the preferred semantics (Vreeswijk

and Prakken, 2000). It is similar to that for the

grounded semantics (Prakken and Sartor, 1997), and

hence can be simulated in our neural network as well.

H¨olldobler and his colleagues gave a method to

encode a logic program to a 3-layer recurrent neu-

ral network and compute the least ﬁxed point of

it (H¨olldobler and Kalinke, 1994) in the seman-

tics of logic programming. We, on the other hand,

constructed a 4-layer neural network, but our neu-

ral network computes not only the least ﬁxed point

(grounded semantics) but also the ﬁxed points (com-

plete extension) in the argumentation semantics.

7 CONCLUDING REMARKS

It is a long time since connectionism appeared as an

alternative movement in cognitive science or com-

puting science which hopes to explain human intel-

ligence or soft information processing. It has been a

matter of hot debate how and to what extent the con-

nectionism paradigm constitutes a challenge to classi-

cism or symbolic AI. On the other hand, much effort

has been devoted to a fusion or hybridization of neu-

ral net computation and symbolic one (Jagota et al.,

1999).

In this paper, we proposed a neural network which

computes all extensions of an argumentation seman-

tics exactly for every argumentation framework and

showed the soundness of neural network argumenta-

tion. The results yield a strong evidence to show that

such a symbolic cognitive phenomenon as human ar-

gumentation can be captured within an artiﬁcial neu-

ral network.

With the neural argumentation framework pre-

sented in the paper, we showed that symbolic dialec-

tical proof theories can be obtained from the neural

network computing various argumentation semantics,

which allow to extract or generate symbolic dialogues

from the neural network computation under various

argumentation semantics. The results illustrate that

there can exist an equal bidirectional relationship be-

tween the connectionism and symbolism in the area

of computational argumentation (Gotou et al., 2011).

The simplicity and efﬁciency of our neural net-

work may be favorable to our future plan such as

introducing learning mechanism into the neural net-

work argumentation, implementing the neural net-

work engine for argumentation, and so on. Specif-

ically, it might be possible to take into account the

so-called core method developed in (H¨olldobler and

Kalinke, 1994) and CLIP in (Garcez et al., 2009) al-

though our neural-symbolic system for argumentation

is much more complicated due to the complexitiesand

varieties of the argumentation semantics.

The neural argumentation framework allows to

build an Integrated Argumentation Environment

(IAE) based on the logic of multiple-valued argumen-

tation (Takahashi and Sawamura, 2004), powered by

the neural network. We are planning to incorporate

the neural argumentation framework to IAE.

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