Extracting Argumentative Dialogues from the Neural Network that Computes the Dungean Argumentation Semantics Yoshiaki Gotou Takeshi Hagiwara Hajime Sawamura Niigata University, Japan Niigata University, Japan Niigata University, Japan gotou@cs.ie.niigata-u.ac.jp hagiwara@ie.niigata-u.ac.jp sawamura@ie.niigata-u.ac.jp Abstract 2. Can the argument status of the neural network argumenta- tion correspond to the well-known status in symbolic argu- Argumentation is a leading principle both founda- mentation framework such as in [Prakken and Vreeswijk, tionally and functionally for agent-oriented comput- 2002]? ing where reasoning accompanied by communication plays an essential role in agent interaction. We con- 3. Can the neural network argumentation compute the fixpoint structed a simple but versatile neural network for neu- semantics for argumentation? ral network argumentation, so that it can decide which argumentation semantics (admissible, stable, semi- 4. Can symbolic argumentative dialogues be extracted from stable, preferred, complete, and grounded semantics) the neural network argumentation? a given set of arguments falls into, and compute ar- gumentation semantics via checking. In this paper, The positive solutions to them helped us deeply understand we are concerned with the opposite direction from relationship between symbolic and neural network argumenta- neural network computation to symbolic argumenta- tion, and further promote the syncretic approach of symbolism tion/dialogue. We deal with the question how various and connectionism in the field of computational argumentation [Makiguchi and Sawamura, 2007a][Makiguchi and Sawamura, argumentation semantics can have dialectical proof theories, and describe a possible answer to it by ex- 2007b]. They, however, paid attention only to the grounded tracting or generating symbolic dialogues from the semantics for argumentation in examining relationship between neural network computation under various argumen- symbolic and neural network argumentation. tation semantics. Ongoingly, we constructed a simple but versatile neural net- work for neural network argumentation, so that it can decide which argumentation semantics (admissible, stable, semi-stable 1 Introduction semantics, preferred, complete, and grounded semantics) [Dung, Much attention and effort have been devoted to the symbolic 1995][Caminada, 2006] a given set of arguments falls into, argumentation so far [Rahwan and Simari, 2009][Prakken and and compute argumentation semantics via checking [Gotou, Vreeswijk, 2002][Besnard and Doutre, 2004], and its applica- 2010]. In this paper, we are concerned with the opposite direc- tion to agent-oriented computing. We think that argumenta- tion from neural network computation to symbolic argumenta- tion can be a leading principle both foundationally and func- tion/dialogue. We deal with the question how various argumen- tionally for agent-oriented computing where reasoning accom- tation semantics can have dialectical proof theories, and describe panied by communication plays an essential role in agent in- a possible answer to it by extracting or generating symbolic dia- teraction. Dung’s abstract argumentation framework and argu- logues from the neural network computation under various argu- mentation semantics [Dung, 1995] have been one of the most mentation semantics. influential works in the area and community of computational The results illustrate that there can exist an equal bidirectional argumentation as well as logic programming and non-monotonic relationship between the connectionism and symbolism in the reasoning. area of computational argumentation. And also they lead to a In 2005, A. Garcez et al. proposed a novel approach to ar- fusion or hybridization of neural network computation and sym- gumentation, called the neural network argumentation [d’Avila bolic one [d’Avila Garcez et al., 2009][Levine and Aparicio, Garcez et al., 2005]. In the papers [Makiguchi and Sawamura, 1994][Jagota et al., 1999]. 2007a][Makiguchi and Sawamura, 2007b], we dramatically de- The paper is organized as follows. In the next section, we veloped their initial ideas on the neural network argumentation to explicate our basic ideas on the neural network checking argu- various directions in a more mathematically convincing manner. mentation semantics by tracing an illustrative example. In Sec- More specifically, we illuminated the following questions which tion 3, with our new construction of neural network for argu- they overlooked in their paper but that deserve much attention mentation, we develop a dialectical proof theory induced by the since they are beneficial for understanding or characterizing the neural network argumentation for each argumentation semantics computational power and outcome of the neural network argu- by Dung [Dung, 1995]. In Section 4, we describe some related mentation from the perspective of the interplay between neural works although there is no work really related to our work except network argumentation and symbolic argumentation. for Garcez et al.’s original one and our work. The final section 1. Can the neural network argumentation algorithm deal with discusses the major contribution of the paper and some future self-defeating or other pathological arguments? works. 28 2 Basic Ideas on the neural argumentation surely converges at τ = 1. Hence, the first output vector Due to the space limitation, we will not describe the technical equals to second output vector. We judge argumentation details for constructing a neural network for argumentation and semantics by using only first input vector and converged its computing method in this paper (see [Gotou, 2010] for them). output vector. As a result we can regard a recurrent neu- Instead, we illustrate our basic ideas by using a simple argumen- ral network as a feedforward neural network except judging tation example and following a neural network computation trace grounded extension. for it. We assume readers are familiar with the Dungean seman- • The vectors of the neural network: The initial input vector tics such as admissible, stable, semi-stable, preferred, complete, for the neural network is a list consisting of 0 and a that rep- and grounded semantics [Dung, 1995][Caminada, 2006]. resent the membership of a set of arguments to be examined. Let us consider an argumentation network on the left side For example, it is [a, 0, 0] for S = Sτ =0 = {i} ⊆ AR. The of Figure 1 that is a graphic presentation of the argumen- output vectors from each layer take as the values only “-a”, tation framework AF =< AR, attacks >, where AR = “0”, “a” or “-b”.1 The intuitive meaning of them for each {i, k, j}, and attacks = {(i, k), (k, i), (j, k)}. output vector are as follows: Output layer weight is a – “a” in the output vector from the output layer repre- io ko jo weight is -b weight is -1 sents membership in Sτ′ = {X ∈ AR | def ends(Sτ , X)}2 and the argu- ih 2 kh 2 jh 2 ment is not attacked by Sτ′ . i k j – “-a” in the output vector from the output layer rep- ih 1 kh 1 jh 1 resents membership in Sτ′+ .3 – “0” in the output vector from the output layer repre- ii ki ji sents the argument belongs to neither Sτ′ nor Sτ′+ . Second hidden layer – “a” in the output vector from the second hidden Figure 1: Graphic representation of AF (left) and Neural net- layer represents membership in Sτ′ and the argument work translated from the AF (right) is not attacked by Sτ′ . – “0” in the output vector from the second hidden According to the Dungean semantics [Dung, 1995][Cami- layer represents membership not in Sτ′ or the argu- nada, 2006], the argumentation semantics for AF is determined ment is attacked by Sτ′ . as follows: Admissible set = {∅, {i}, {j}, {i, j}}, Complete ex- Fisrt hidden layer tension = {{i, j}}, Preferred extension = {{i, j}}, Semi-stable – “a” in the output vector from the first hidden layer extension = {{i, j}}, Stable extension = {{i, j}}, and Grounded represents membership in Sτ and the argument is extension = {{i, j}}. not attacked by Sτ . – “-b” in the output vector from the first hidden layer Neural network architecture for argumentation represents the membership in Sτ+ . In the Dungean semantics, the notions of ‘attack’, ‘defend (ac- – “0” in the output vector from the first hidden layer ceptable)’ and ‘conflict-free’ play the most important role in represents the others. constructing various argumentation semantics. This is true Input layer in our neural network argumentation as well. Let AF =< AR, attacks > be as above, and S be a subset of AR, to be – “a” in the output vector from the input layer repre- examined. The argumentation network on the left side of Figure sents membership in Sτ . 1 is first translated into the neural network on the right side of – “0” in the output vector from the input layer repre- Figure 1. Then, the network architecture consists of the follow- sents the argument does not belong to S. ing constituents: • A double hidden layer network: It is a double hidden layer A trace of the neural network network and has the following four layers: input layer, first Let us examine to which semantics S = {i} belongs in AF on hidden layer, second hidden layer and output layer, which the left side of Figure 1 by tracing the neural network compu- have the ramified neurons for each argument, such as αi , tation. The overall visual computation flow is shown in Figure αh1 , αh2 and αo for the argument α. 2. • A recurrent neural network (for judging grounded exten- Stage1. Operation of input layer at τ = 0 sion): The double hidden layer network like on the right side of Figure 1 is piled up high until the input and output Sτ =0 = S = {i}. Hence, [a, 0, 0] is given to the input layer layers converge (stable state) like in Figure 2. The symbol of the neural network in Figure 1. Each input neuron computes τ represents the pile number (τ ≥ 0) which amounts to the its output value by its activation function (see the graph of the turning number of the input-output cycles of the neural net- activation function, an identity function, on the right side of the work. In the stable state, we set τ = converging. Then, input layer of Figure 2). The activation function makes the input Sτ =n stands for a set of arguments at τ = n. 1 √ Let a,b be positive real numbers and they satisfy b > a > 0. • A feedforward neural network (except judging grounded 2 Let S⊆AR and A∈AR. defends(S, A) iff ∀B ∈ AR(attacks(B, extension): When we compute argumentation semantics ex- A) → attacks(S, B)). cept grounded extension with a recurrent neural network, it 3 Let S ⊆ AR. S + = {X ∈ AR | attacks(S, X)}. 29 1st output vector 2nd output vector output value from Xo a -a a a -a a a -a + S’τ=0={ k } a -a a + S’τ=1={ k } a -a a 2 input value into Xo 0 a io ko jo io ko jo -a a2 -2a a2 a2 -2a a2 output value from Xh2 S’τ=0={ i, j } a 0 a S’τ=1={ i, j } a 0 a θi=a2+b a kh2 θk=a2+2b ih2 jh2 ih2 kh2 jh2 θi θk θj θi θk θj θj=0 input value into Xh2 a2+b -a-ab 0 a2+b -2a-ab a2 0 θX output value from Xh1 + Sτ=0={ k } a -b 0 + Sτ=1={ k } a -b a a output value ih1 kh1 jh1 ih1 kh1 jh1 -b input value into Xh1 0 2 a 2 2 2 neuron a -ab 0 a -2ab a threshold θ -b input value a 0 0 Sτ=0 = { i } Sτ=1 = { i, j } a 0 a output value from Xi ii ki ji ii ki ji weight is a a 0 0 a -a a 1 weight is -b input value into Xi a 0 0 a -a a 0 1 weight is -1 1st input vector 2nd input vector X belongs to {i, k, j} Figure 2: A trace of the neural network for argumentation with S = {i} and activation functions layer simply pass the value to the hidden layer. The input layer In summary, after the first hidden layer received the vector thus outputs the vector [a, 0, 0]. [a2 , −ab, 0], it turns out to pass the output vector [a2 + b, −a − In this computation, the input layer judges Sτ =0 = {i} and ab, 0] to the second hidden neurons. inputs a2 to ih1 through the connection between ii and ih1 whose weight is a. At the same time, the input layer inputs −ab to kh Stage 3. Operation of second hidden layer at τ = 0 through the connection between ii and kh1 whose weight is −b The second hidden layer receives a vector [a2 , −ab, 0] from first so as to make the first hidden layer know that i ∈ Sτ =0 attacks k hidden layer. Each activation function of ih2 , kh2 and jh2 is a (in symbols, attacks(i, k)). Since the output values of ki and ji step function as put on the right side of the first hidden layer in are 0, they input 0 to other first hidden neurons. Figure 2 with its threshold, θi = a2 + b, θk = a2 + 2b and In summary, after the input layer receives the input vector θj = 0 respectively. [a, 0, 0], it turns out to give the hidden layer the vector [ a · a These thresholds are defined by the ways of being attacked as + 0·(−b), a · (−b) + 0 · a + 0 · (−b), 0 · a ]= [a2 , −ab, 0]. follows: Stage 2. Operation of first hidden layer at τ = 0 • If an argument X can defend X only by itself (in Figure Now, the first hidden layer receives a vector [a , −ab, 0] from 2 1, such X is i since def ends({i}, i)), then the threshold of the input layer. Each activation function of ih1 , kh1 and jh1 is a Xh2 (θX ) is a2 +tb (t is the number of arguments bilaterally step function as put on the right side of the first hidden layer in attacking X). Figure 2. The activation function categorizes values of vectors • If an argument X can not defend X only by it- which are received from the input layer into three values as if self and is both bilaterally and unilaterally attacked the function understand each argument state. Now, the following by some other argument (in Figure 1, such X is inequalitis hold: a2 ≥ a2 , −ab ≤ −b, −b ≤ 0 ≤ a2 . Accord- k since ¬def ends({k}, k)&attacks(j, k)&attacks(i, k)), ing to the activation function, the first hidden layer outputs the then the threshold of Xh2 (θX ) is a2 + b(s + t) (s(t) is the vector [a, −b, 0]. number of arguments unilaterally(bilaterally) attacking X). Next, the first hidden layer inputs a2 + b into the second Note that l=m=1 for the argument k in Figure 1. hidden neuron ih2 through the connections between ih1 and ih2 • If an argument X is not attacked by any other arguments (in whose weight is a, kh1 and ih2 whose weight is −1, so that the Figure 1, such X is j), then the threshold of Xh (θXh ) is 0. second hidden layer can know attacks(k, i) with i ∈ Sτ =0 . At the same time, the first hidden layer inputs −a − ab into kh2 • If an argument X can not defend X only by itself and is through the connections between ih1 and kh2 whose weight is just unilaterally attacked by some other argument, then the −1, kh1 and kh2 whose weight is a, so that the second hidden threshold of Xh2 (θX ) is bs (s is the number of arguments layer can know attacks(i, k) with k ∈ Sτ+=0 and inputs 0 into unilaterally attacking X). jh2 so that the second hidden layer can know the argument j is By these thresholds and their activation functions (step func- not attacked by any arguments with j ̸∈ Sτ =0 . tions), if S defends X then Xh2 outputs a. Otherwise, Xh2 30 outputs 0 in the second hidden layer. As the result, the second Stage 7. Judging admissible set, complete extension hidden layer judges either X ∈ Sτ′ or X ̸∈ Sτ′ by two output and stable extension values (a and 0). In this way, the output vector in the second Through the above neural network computation, we have ob- hidden layer yields [a, 0, a]. This vector means that the second tained Sτ′ =0 = {i, j} and Sτ′+=0 = {k} for Sτ =0 = {i}, and hidden layer judges that the arguments i and j are defended by Sτ =0 , resulting in Sτ′ =0 = {i, j}. Sτ′ =1 = {i, j} and Sτ′+=1 = {k} for Sτ =1 = {i, j}. Moreover, we also have such a result that both the sets {i} and {i, j} are Next, the second hidden layer inputs a2 into the output neu- conflict-free. rons io and jo through the connections between ih2 and io , jh2 The condition for admissible set says that a set of arguments S and jo whose weights are a,so that the output layer can know satisfies its conflict-freeness and ∀X ∈ AR(X ∈ S → X ∈ S ′ ). i, j ∈ Sτ =0 and i, j ∈ Sτ′ =0 . At the same time, the second hid- Therefore, the neural network can know that the sets {i} and den layer inputs −2a into ko through the connections between {i, j} are admissible since it confirmed the condition at the time ih2 and ko , jh2 and ko whose weights are −1,so that output layer round τ = 0 and τ = 1 respectively. can know attacks(i, k) and attacks(j, k) with k ∈ Sτ′+=0 . The condition for complete extension says that a set of ar- Furthermore, it should be noted that another role of the second guments S satisfies its conflict-freeness and ∀X ∈ AR(X ∈ hidden layer lies in guaranteeing that Sτ′ is conflict-free4 . It is S ↔ X ∈ S ′ ). Therefore, the neural network can know that actually true since the activation function of the second hidden the set {i, j} satisfies the condition since it has been obtained at layer makes Xh2 for the argument X attacked by Sτ output 0. τ = converging. Incidentally, the neural network knows that The conflict-freeness is important since it is another notion for the set {i} is not a complete extension since it does not appear in characterizing the Dungean semantics. the output neuron at τ = converging. In summary, after the second hidden layer received the vec- The condition for stable extension says that a set of arguments tor [a2 + b, −a − ab, 0], it turns out to pass the output vector S satisfies ∀X ∈ AR(X ̸∈ S → X ∈ S ′+ ). The neural network [a2 , −2a, a2 ] to the second hidden neurons. can know that the {i, j} is a stable extension since it confirmed the condition from the facts that Sτ =1 = {i, j}, Sτ′ =1 = {i, j} Stage 4. Operation of output layer at τ = 0 and Sτ′+=1 = {a}. The output layer now received the vector [a2 , −2a, a2 ] from the second hidden layer. Each neuron in the output layer has an ac- Stage 8. Judging preferred extension, semi-stable tivation function as put on the right side of the output layer in extension and grounded extension Figure 2. By invoking the neural network computation that was stated from This activation function makes the output layer interpret any the stages 1-7 above for every subset of AR, and AR itself as an positive sum of input values into the output neuron Xo as X ∈ input set S, it can know all admissible sets of AF, and hence Sτ′ , any negative sum as X ∈ Sτ′+ , and the value 0 as X ̸∈ Sτ′ it also can know the preferred extensions of AF by picking up and X ̸∈ Sτ′+ . As the result, the output layer outputs the vector the maximal ones w.r.t. set inclusion from it. In addition, the [a, −a, a]. neural network can know semi-stable extensions by picking up a Summarizing the computation at τ = 0, the neural network maximal S ∪ S + where S is a complete extension in AF. This received the vector [a, 0, 0] in the input layer and outputted is possible since the neural network already has computed S + . [a, −a, a] from the output layer. This output vector means that For the grounded extension, the neural network can know that the second hidden layer judged Sτ′ =0 = {i, j} and guaranteed the grounded extension of AF is Sτ′ =converging when the com- its conflict-freeness. With these information passed to the output putation stopped by starting with Sτ =0 = ∅. This is due to the layer from the hidden layer, the output layer judged Sτ′+=0 = {k}. fact that the grounded extension is obtained by the iterative com- putation of the characteristic function that starts from ∅ [Prakken Stage 5. Inputting the output vector at τ = 0 to the and Vreeswijk, 2002]. input layer at τ = 1 (shift from τ = 0 to τ = 1) Readers should refer to the paper [Gotou, 2010] for the sound- ness theorem of the neural network computation illustrated so At τ = 0, the neural network computed Sτ′ =0 = {i, j} and far. Sτ′+=0 = {k}. We continue the computation recurrently by con- necting the output layer to the input layer of the same neural network, setting first output vector to second input vector. Thus, 3 Extracting Symbolic Dialogues from the at τ = 1, the input layer starts its operation with the input vector Neural Network [a, −a, a]. We, however, omit the remaining part of the opera- In this section, we will address to such a question as if symbolic tions starting from here since they are to be done in the similar argumentative dialogues can be extracted from the neural net- manner. work argumentation. The symbolic presentation of arguments would be much better for us since it makes the neural net argu- Stage 6. Convergence to a stable state mentation process verbally understandable. The notorious criti- We stop the computation immediately after the time round τ = 1 cism for neural network as a computing machine is that connec- since the input vector to the neural network at τ = 1 coincides tionism usually does not have explanatory reasoning capability. with the output vector at τ = 1. This means that the neural We would say our attempt here is one that can turn such criticism network amounts to having computed a least fixed point of the in the area of argumentative reasoning. characteristic function that was defined with the acceptability of In our former paper [Makiguchi and Sawamura, 2007b], we arguments by Dung [Dung, 1995]. have given a method to extract symbolic dialogues from the neural network computation under the grounded semantics, and 4 showed its coincidence with the dialectical proof theory for the A set S of arguments is said to be conflict-free if there are no argu- ments A and B in S such that A attacks B. grounded semantics. In this paper, we are concerned with the 31 question how other argumentation semantics can have dialecti- Thus, we can view P(roponent, speaker)’s initial belief {i} as cal proof theories. We describe a possible answer to it by ex- justified one in the sense that it could have persuaded O(pponent, tracting or generating symbolic dialogues from the neural net- listener or audience) under an appropriate Dungean argumenta- work computation under other more complicated argumentation tion semantics. Actually, we would say it is admissibly justified semantics. We would say this is a great success that was brought under admissibly dialectical proof theory below. Formally, we by our neural network approach to argumentation since dialec- introduce the following dialectical proof theories, according to tical proof theories for various Dungean argumentation seman- the respective argumentation semantics. tics have not been known so far except only some works (e. g., [Vreeswijk and Prakken, 2000], [Dung et al., 2006]). Definition 1 (Admissibly dialectical proof theory) The admis- sibly dialectical proof theory is the dialogue extraction pro- First of all, we summarize the trace of the neural network com- cess in which the summary table generated by the neural net- putation as have seen in Section 2 as in Table 1, in order to make work computation satisfies the following condition: ∀A ∈ it easy to extract symbolic dialogues from our neural network. SP RO,τ =0 ∀k ≥ 0(A ∈ SP RO,τ =k ), where SP RO,τ =0 is the Wherein, SP RO,τ =k and SOP P,τ =k denote the followings re- input set at τ = 0. spectively: At time round τ = k(k ≥ 0) in the neural network ′ ′ computation, SP RO,τ =k = Sτ =k , and SOP P,τ =k = Sτ+=k (see Intuitively, the condition says every argument in SP RO,τ =0 is Section 2 for the notations). retained until the stable state as can be seen in Table 2. It should be noted that the condition reflects the definition of ‘admissible extension’ in [Dung, 1995]. Table 1: Summary table of the neural network computation Definition 2 (Completely dialectical proof theory) The com- SP RO,τ =k SOP P,τ =k pletely dialectical proof theory is the dialogue extraction pro- τ =0 input S {} cess in which the summary table generated by the neural network output ... ... computation satisfies the following conditions: let SP RO,τ =0 be τ =1 input ... ... the input set at τ = 0. output ... ... .. .. 1. SP RO,τ =0 satisfies the condition of Definition 1. . . ... ... 2. ∀A ̸∈ SP RO,τ =0 ∀k(A ̸∈ SP RO,τ =k ) Table 2: Summary table of the neural network computation Intuitively, the second condition says that any argument that does in Fig. 2 not belong to SP RO,τ =0 does not enter into SP RO,τ =t at any SP RO,τ =k SOP P,τ =k time round t up to a stable one k. Those conditions reflect the τ =0 input {i} {} definition of ‘complete extension’ in [Dung, 1995]. output {i, j} {k} Definition 3 (Stably dialectical proof theory) The stably di- τ =1 input {i, j} {k} alectical proof theory is the dialogue extraction process in which output {i, j} {k} the summary table generated by the neural network computation satisfies the following conditions: let SP RO,τ =0 be the input set For example, Table 2 is the table for S = {i} summarized at τ = 0. from the neural network computation in Fig. 2 1. SP RO,τ =0 satisfies the conditions of Definition 2. We assume dialogue games are performed by proponents 2. AR = SP RO,τ =n ∪ SOP P,τ =n , where AF =< (PRO) and opponents (OPP) who have their own sets of argu- AR, attacks > and n denotes a stable time round. ments that are to be updated in the dialogue process. In advance of the dialogue, proponents have S(= Sτ =0 ) as an initial set Intuitively, the second condition says that PRO and OPP cover SP RO,τ =0 , and opponents have an empty set {} as an initial set AR exclusively and exhaustively. Those conditions reflect the SOP P,τ =0 . definition of ‘stable extension’ in [Dung, 1995]. We illustrate how to extract dialogues from the summary table For the dialectical proof theories for preferred [Dung, 1995] by showing a concrete extraction process of dialogue moves in and semi-stable semantics [Caminada, 2006], we can similarly Table 2: define them taking into account maximality condition. So we 1. P(roponent, speaker): PRO declares a topic as a set of be- omit them in this paper. liefs by saying {i} at τ = 0. OPP just hears it with no As a whole, the type of the dialogues in any dialectical proof response {} for the moment. (dialogue extraction from the theories above would be better classified as a persuasive dialogue first row of Table 2) since it is closer to persuasive dialogue in the dialogue classifi- cation by Walton [Walton, 1998]. 2. P(roponent, or speaker): PRO further asserts the incre- mented belief {i, j} because the former beliefs defend j, 4 Related Work and at the same time states the belief {i, j} conflicts with {k} at τ = 0. (dialogue extraction from the second row of Garcez et al. initiated a novel approach to argumentation, called Table 2) the neural network argumentation [d’Avila Garcez et al., 2005]. However, the semantic analysis for it is missing there. That is, 3. O(pponent, listener or audience): OPP knows that its belief it is not clear what they calculate by their neural network argu- {k} conflicts with PRO’s belief {i, j} at τ = 0. (dialogue mentation. Besnard et al. proposed three symbolic approaches extraction from the second row of Table 2) to checking the acceptability of a set of arguments [Besnard and 4. No further dialogue moves can be promoted at τ = 1, re- Doutre, 2004], in which not all of the Dungean semantics can be sulting in a stable state. (dialogue termination by the third dealt with. So it may be fair to say that our approach with the and fourth rows of Table 2) neural network is more powerful than Besnard et al.’s methods. 32 Vreeswijk and Prakken proposed a dialectical proof theory for [d’Avila Garcez et al., 2009] Artur S. d’Avila Garcez, Luı́s C. the preferred semantics [Vreeswijk and Prakken, 2000]. It is Lamb, and Dov M. Gabbay. Neural-Symbolic Cognitive Rea- similar to that for the grounded semantics [Prakken and Sartor, soning. Springer, 2009. 1997], and hence can be simulated in our neural network as well. [Dung et al., 2006] P. M. Dung, R. A. Kowalski, and F. Toni. In relation to the neural network construction and computa- Dialectic proof procedures for assumption-based, admissible tion for the neural-symbolic systems, the structure of the neural argumentation. Artificial Intelligence, 170:114–159, 2006. network is a similar 3-layer recurrent network, but our neural [Dung, 1995] P.M. Dung. On the acceptability of arguments network computes not only the least fixed point (grounded se- mantics) but also the fixed points (complete extension). This is a and its fundamental role in nonmonotonic reasoning, logics most different aspect from Hölldobler and his colleagues’ work programming and n-person games. Artificial Intelligence, [Hölldobler and Kalinke, 1994]. 77:321–357, 1995. [Gotou, 2010] Yoshiaki Gotou. Neural Networks calcu- lating Dung’s Argumentation Semantics. Master’s 5 Concluding Remarks thesis, Graduate School of Science and Technology, Niigata University, Niigata, Japan, December 2010. 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