Reasoning by a Bipolar Argumentation
Framework for PROLEG
Tatsuki Kawasaki, Sosuke Moriguchi, and Kazuko Takahashi
School of Science&Technology, Kwansei Gakuin University,
2-1, Gakuen, Sanda, 669-1337, JAPAN
dxk96093@kwansei.ac.jp, chiguri.s@acm.org, ktaka@kwansei.ac.jp

Abstract. We develop a system allowing lawyers and law school students to analyze court judgments. We describe a transformation from thelogic programming language PROLEG to a bipolar argumentation framework (BAF) and the legal reasoning involved. Legal knowledge written ina PROLEG program is transformed into a BAF, in which the structureof argument in a judgment is clear. We describe two types of reasoning by the BAF: the entire structure and causality of arguments, andidentification of the evidence required.
Keywords: bipolar argumentation framework, PROLEG, reasoning, semantics

1

Introduction

Recently, information technology and artificial intelligence are now vigorously
applied in various fields, including those that have not yet been fully digitizedor automated. In the context of legal reasoning, although the use of artificial
intelligence has attracted a great deal of attention, higher-level and more practical support exploiting recent technological developments is required. Deriving
support for a judgment is important. When seeking to support a judgment, it
is essential to develop a system that can be easily used by lawyers who are
not computer scientists; also, the system must be highly reliable and must reason accurately and rapidly. Lawyers must be able to access the system in astraightforward manner, and the system must both describe the process leading
to judgment and the way in which the law was applied.
In terms of the former consideration, as law is logical, it is reasonable to basethe system on such logic and reason from that perspective. Several legal reasoning
systems have adopted logic programming such as Prolog as their descriptive
languages. However, it is diﬃcult for a lawyer who is not familiar with computerscience to directly write Prolog code. A PROLEG system was developed to solve
this problem [17]. The system was designed to support inferences based on theJapanese Presupposed Ultimate Facts Theory (termed “Yoken-jijitsu-ron” inJapanese) of the Japanese civil code, and it is currently applied to the Japanesepenal code. The theory deals with uncertainties that sometimes arise in court,
where a judge must give a decision even if evidence is lacking. The PROLEG

2

T. Kawasaki, S. Moriguchi, K. Takahashi

language is an extension of Prolog. Each presupposed ultimate fact is represented
using general rules written in the form of if-then statements and exceptions.
Exceptions of fact apply to all general rules and are used as court defenses.
The use of exceptions rather than negative atoms creates a structure equivalent
to that of a law, allowing lawyers to intuitively understand the program. An
interpreter is implemented in Prolog. A burden-of-proof [14] is attached to eachultimate fact to allow for decision-making even if the fact is not proven to betrue. This is achieved using the negation-as-failure inference of Prolog; thus, fora given goal, a general rule is applied and the goal is true unless there is an
exception.
In terms of legal process and application, it is appropriate to employ argumentation to describe both the judgment process and how the law was applied [1]. An argumentation system reveals both the causality of arguments (forexample, how arguments interacted to create a judgment) and the influence ofevidence. Argumentation is powerful when used to resolve conflicts, not only formalizing the structure of the process but also incorporating any uncertainties.
In the time since Dung proposed the abstract Argumentation Framework(AF) [9], many extensions and revisions of the system have been published [15].
AF represents an argumentation by a pair of a set of arguments and a set of
attacks between arguments, ignoring the contents of arguments. Several AF semantics have been defined; acceptable arguments are calculated based on thesesemantics. Visualization tools appropriate for argumentation systems have also
been developed (e.g., [16]).
Although PROLEG facilitates the representation of a law, it is diﬃcult tograsp the judgment process or argument causality from the execution trace.
On the other hand, although it is possible to create an AF representing theinteraction between a plaintiﬀ and a defendant in court, it is diﬃcult to directly
write the structure of a law per se, or the part of the law used to create anargument in an AF form. Therefore, we combined the two systems.
We developed a transformation from PROLEG to a bipolar argumentation
framework (BAF) [6], an extended AF, and showed its correctness [11]. Morespecifically, we gave a semantics for the BAF obtained as a result of the transformation, and proved that the answer set of the PROLEG program was thesame as the set of acceptable BAF arguments. Here, we describe how the BAF
reasons.
Consider the following PROLEG program representing the penal code thatdefines the “crime of murder.”1 The first clause indicates the general rule andthe second clause an exception. The text states that if the object is a human(not a dead body) and there exists both the action of murder and also theintention to murder, then the crime of murder has been committed unless there
is a legitimate defense.
crime_of_murder <= human, action_of_murder, intention_to_murder.
exception(crime_of_murder, legitimate_defense).
1

Note that the examples shown here are simplified versions of the actual penal code;
the conditions per se are simplified and the legal terminology is not precise.

Reasoning by BAF for PROLEG

3

When evidence is provided, the facts on which that evidence bears are proved,
and it is then decided whether the crime of murder has been committed or not.
A judge should explain the judgment process to persuade those concernedwith the transparency of justice. In such a legal situation, what is required is not
only the outcome of judicial reasoning but also an explanation of the reasoning
process or the cause-and-eﬀect relationships of arguments used in reasoning.
For example, if the crime of murder was adjudged to not in fact have beencommitted; this may be because of a lack of evidence of intention to murder, or
because a legitimate defense was available.
Our transformed BAF not only shows the process and structure of judgment, but also suggests a strategy by which a user can achieve a desired goal.
If a defendant/plaintiﬀ wishes to argue that a law should or should not be applied, the BAF identifies the evidence that must be presented and any counterarguments that may arise. For example, when a prosecutor wishes to charge thecrime of murder, but finds that the lack of intention to murder is a complicating factor, s/he will look harder for evidence of intention to murder. Here, we
discuss such reasoning on our BAF.
This paper is organized as follows. In Section 2, we briefly explain PROLEG.
In Section 3, we describe the BAF that we develop and its semantics. In Section 4,
we develop the transformation rule from PROLEG to BAF. In Section 5 we
describe how the BAF reasons. In Section 6, we compare our method with thoseof others. Finally, in Section 7, we oﬀer conclusions and describe our planned
future work.

2

Legal Description Language: PROLEG

The PROLEG program P is defined as pairs of ⟨R, E⟩, a finite set of rules, anda finite set of exceptions. Each rule is a Horn clause of the form H ⇐ B1 , . . . Bn ,where H, B1 , . . . , Bn are atoms (n may be 0; we term such a rule a fact rule orsimply a fact). Each exception is in the form exception(H, B).
A fact is something given as an evidence in a court case, whereas the otherrules and exceptions describe the general case. That is, the former (the facts) are
generally given in an instantiated form whereas the other rules and exceptions
include variables. In the examples that follow, we use a proposition for simplicity.
For each rule R or exception E, we employ the functions head and bodysuch that head (R) = H and body(R) = {B1 , . . . , Bn } if R = H ⇐ B1 , . . . , Bn ;head (E) = H and body(E) = {B} if E = exception(H, B). An atom may be
defined by more than one rule. This means that there may exist R1 and R2(R1 ̸= R2 ) such that head (R1 ) = head (R2 ).
Example 1. The following is an example of a PROLEG program.
p <= q1, q2.
exception(q1, r).
q2 <=.
r <=.

4

T. Kawasaki, S. Moriguchi, K. Takahashi

The semantics of the PROLEG program P are defined as an answer set (a setof ground atoms). M is the answer set of P iﬀ M is the minimum model of the setof Horn clauses, {R ∈ R | ∀E ∈ E, if head (E) = head (R) then body(E) ̸⊆ M }.
The expressive power of PROLEG is the same as that of a normal logic programwith an answer set [10, 18].
PROLEG allows cyclic definitions. However, here, we deal with an acyclic
PROLEG program, because the Japanese civil and penal codes are usually written in an acyclic manner.

3

Bipolar Argumentation Framework

First, we define our argumentation framework.
Definition 1 (argumentation framework). An argumentation framework is
defined as a pair ⟨AR, AT ⟩ where AR is the set of arguments and AT is a binary
relation on AR, termed an attack. If (A, A′ ) ∈ AT , we state that A attacks A′ .
A BAF is an extension of an AF in which the two relations of attack andsupport are defined over a set of arguments [6]. We define a support relation
between a power set of arguments and a set of arguments; this diﬀers from theusual BAF format, because the body of a rule generally includes more than one
atom in PROLEG.
Definition 2 (bipolar argumentation framework). A BAF is defined as a
triple ⟨AR, ATT , SUP ⟩ where AR is a finite set of arguments, ATT ⊆ AR × AR
and SUP ⊆ (2AR \ {∅}) × AR. We denote att(B, A) if (B, A) ∈ ATT , andsup(A, A) if (A, A) ∈ SUP .
Example 2. Figure 1 is a graphical representation of a bipolar argumentation
framework ⟨{a, b, c, d, e}, {(b, a), (e, d)}, {({c, d}, a)}⟩. In the figure, the straightarrow indicates an attack relation and the wavy arrow a support relation.

Fig. 1. Example of BAF.

Reasoning by BAF for PROLEG

5

We derived semantics for the BAF based on labeling [5]. Usually, labeling isa function from a set of arguments to {in, out, undec}, but undec is unnecessary
here, because the BAF is acyclic. An argument labeled in is considered to be an
accepted argument.
Definition 3 (labeling). For ⟨AR, ATT , SUP ⟩, labeling L is a function from
AR to {in, out}.
Labeling of a set of arguments is denoted as follows: L(A) = in if L(A) = infor all A ∈ A; L(A) = out, otherwise.
We use the label in to identify arguments that are neither attacked nor
supported by any other argument. When an argument is both attacked andsupported, the attack is supposed to be stronger than the support. We assign alabel out to an argument that is attacked by another argument with the label
out, and simultaneously supported by the set of arguments with the label out.
Note that an argument lacking support is labeled out, even if it is attacked byan argument labeled out.
Definition 4 (complete labeling). For ⟨AR, ATT , SUP ⟩, labeling L is complete iﬀ the following conditions are satisfied for any argument A ∈ AR.
– L(A) = in if• (∀B ∈ AR, ¬att(B, A)) ∧ (∀A ⊆ AR, ¬sup(A, A))or• (∀B ∈ AR, att(B, A) ⇒ L(B) = out) ∧ (∃A ⊆ AR, sup(A, A) ∧ L(A) =in).
– L(A) = out, otherwise.
Figure 2 shows the complete labeling of four BAFs.

Fig. 2. Examples of BAFs with complete labelings.

Example 3. For a BAF in Figure 1, L(b) = L(c) = L(e) = in and L(a) = L(d) =L({c, d}) = out.
The following theorem holds [11].
Theorem 1. For any acyclic BAF, there is exactly one complete labeling.

6

T. Kawasaki, S. Moriguchi, K. Takahashi

Note that we distinguish the case in which an argument is supported bya set of arguments from that in which it is supported by multiple arguments
separately.
Example 4. Consider two BAFs baf 1 and baf 2 shown in Figure 3. Formally,baf 1 is described as ⟨{a, b, c, d}, {(d, c)}, {({b, c}, a)}⟩ and baf 2 is described as⟨{a, b, c, d}, {(d, c)}, {({b}, a), ({c}, a)}⟩.
In baf 1 , the argument a has one support that is a set of two arguments,
whereas in baf 2 , it has two supports, both of which are singletons.
Let L1 and L2 be the complete labeling of baf 1 and baf 2 , respectively. In
baf 1 , L1 (b) = L1 (d) = in and L1 (c) = out hold. It follows that L1 ({b, c}) = out
holds. Therefore, L1 (a) = out. On the other hand, in baf 2 , L2 (b) = L2 (d) = inand L2 (c) = out hold by the similar reasoning. However, L2 ({b}) = in andL2 ({c}) = out. Therefore, L1 (a) = in.

(a) baf 1 : supported by a set(b) baf 2 : supported independently

Fig. 3. Two types of support.
.

4

Transformation

4.1

Transformation rule

We here show a transformation from a PROLEG program to a BAF. The atoms,
rules, and exceptions of the PROLEG program are transformed into arguments,
supports, and attacks, respectively.
We add two types of arguments to the BAF that do not feature as explicit
atoms in PROLEG. One is an argument reflecting the absence of any rules ofinference in PROLEG. In PROLEG, an atom H that does not appear in theheader of any rule or exception is not in the answer set. On the other hand,arguments that are neither attacked nor supported are labeled in. To fill this gap,
we add the argument ab(H) that attacks H. We term this argument an absence
argument. We also add arguments showing the existence of fact rules. For a fact
rule (i.e., a rule in the form H ⇐), there are no arguments that support H in
BAF; any support is a binary relation. Therefore, we add an argument ex(H)that supports H. We term this argument an existence argument.

Reasoning by BAF for PROLEG

7

Definition 5 (transformation rule).
Transformation from a PROLEG program ⟨R, E⟩ to a BAF ⟨AR, ATT , SUP ⟩is defined as follows.
∪∪– Atom = R∈R ({head (R)} ∪ body(R)) ∪ E∈E ({head (E)} ∪ body(E))– Rule = {(body(R), head (R)) | R ∈ R ∧ body(R) ̸= ∅}– Exc = {(B, H) | exception(H, B) ∈ E}
– Existence = {H | H ⇐∈ R}– ExistenceSupport = {({ex(H)}, H) | H ∈ Existence}– Absence = Atom\({head (R) | R ∈ R} ∪ {head (E) | E ∈ E})– AbsenceAttack = {(ab(B), B) | B ∈ Absence}– AR = Atom ∪ {ex(H) | H ∈ Existence} ∪ {ab(B) | B ∈ Absence}– ATT = Exc ∪ AbsenceAttack– SUP = Rule ∪ ExistenceSupport
The following theorem indicates that the semantics is preserved during transformation [11].
Theorem 2. For PROLEG program P , let M be an answer set of P . Assume
that L is the complete labeling of the BAF transformed from P . Then, for each
atom H in P , H ∈ M iﬀ L(H) = in.
Example 5. The program in Example 1 is transformed into the following BAF:
⟨ {p, q1 , q2 , r, ex(q2 ), ex(r)}, {(r, q1 )},{({q1 , q2 }, p), ({ex(q2 )}, q2 ), ({ex(r)}, r)} ⟩.
Complete labeling of the BAF is performed in the following manner. Arguments q1 and q2 together support the argument p. The existence argumentsex(q2 ) and ex(r) are added to support q2 and r, respectively. Figure 4 shows agraphical representation of the BAF, with the complete labeling. As L(q1 ) = out
and L(q2 ) = in, the label of the set of arguments L({q1 , q2 }) = out. Also, as p
is supported by {q1 , q2 }, L(p) = out. When we ignore the existence and absence
arguments introduced during transformation, the set of arguments labeled in is{q2 , r}, which coincides with the answer set of the program of Example 1.

5

Reasoning by the BAF

We describe the two types of reasoning performed by the BAF transformed fromthe PROLEG program:
1. A portrayal of the entire structure of judgment and the causality of arguments.
2. Identification of the required evidence.

8

T. Kawasaki, S. Moriguchi, K. Takahashi

Fig. 4. BAF for the program in Example 1.

5.1

PROLEG program

Example 6. Consider the following PROLEG program. The first set of rules andexceptions states that if the object is a human (not a dead body) and there
exists both the action of murder and also the intention to murder, then thecrime of murder has been committed unless there was a legitimate defense. The
second set of rules and exceptions states that if the accused is attacked and takes
emergency, necessary, and appropriate action to defend himself/herself, then thisis a legitimate defense, unless there was no intention to harm the deceased. The
remainder of the program deals with the facts in evidence.

% rules regarding crime_of_murder
crime_of_murder <= human, act_of_murder, intention_to_murder.
exception(crime_of_murder, legitimate_defense).
legitimate_defense <=infringement, emergency, necessity, appropriateness,
defense_intention.
exception(legitimate_defense, aggressive_intention_to_harm).
% facts
human <=.
act_of_murder <=.
intention_to_murder <=.
infringement <=.
emergency <=.
necessity <=.
appropriateness <=.
defense_intention <=.

Reasoning by BAF for PROLEG

5.2

9

The entire structure of judgment and causality of the arguments

In this case, the entire PROLEG program is transformed into a BAF using therules shown in Section 4.
For each atom, a general rule defining both the atom and the exceptions
is transformed into the BAF using a transformation rule. If there exists a fact,
then a corresponding existence argument supporting the fact is added. If an atom
does not appear in the header of any rule or exception, then a corresponding
absence argument attacking the atom is added to the transformed BAF. We
show a graphical representation of the transformed BAF in Figure 5.

Fig. 5. Graphical representation of a transformed BAF for a murder case.

This BAF was obtained from an entire PROLEG program including facts,
and shows the structure of the entire argumentation from which we can grasp
the cause-and-eﬀect relationships of the arguments.
Using this BAF, the argumentation process is explained as follows. As thelabel of the absence argument ab(aggressive intention to harm) is in, that of theargument aggressive intention to harm is out (there was no intention to harm).
Therefore, the label of the argument legitimate defense is in (it is a legitimate
defense). The argument crime of murder has one attacking argument, the labelof which is in, and one supporting set of arguments, the label of which is in.
Hence, the label of the argument crime of murder is out (the crime of murder
was not committed).
The BAF is updated as the judgment proceeds. Counter-arguments and evidences may be incrementally added as the corresponding nodes. Then, the10T. Kawasaki, S. Moriguchi, K. Takahashi

node labels can be changed. For example, if there is an exception to a legitimate defense argument, and this is proven, a new argument is added; legitimate defense is attacked by this argument and its label is changed to out. As
another example, if evidence of aggressive intention to harm is given, then its absence argument is replaced by an existence argument, and attack by the absence
argument is replaced by support from the existence argument. As a result, thelabel of the node aggressive intention to harm is changed to in. It follows thatthe label of legitimate defense is changed to out, and that of the crime of murder
to in.
5.3

Identification of required evidence

The BAF also identifies the evidence required to apply the law or prevent its
application.
We transform a PROLEG program except for the fact part, and determine
the existence arguments required to apply or not apply the law. Unlike thefirst type of reasoning, all available rules and exceptions are assumed to be
represented, and no rules or exceptions are added.
From the definition of complete labeling, L(A) = in holds iﬀ the labels of allarguments that attack A are out and there exists an argument that supports A,
of which the label is in, or A is neither attacked nor supported.
Assume that a defendant wants to apply a law or that a plaintiﬀ wants toprevent its application. Then they seek to label the corresponding argument inand out, respectively. The BAF determines the evidence required for attainmentof either goal. This is achieved by repeatedly applying the following process:
Let A be an argument.
– Make L(A) = in.
Both of the following conditions should be satisfied.
• (attack condition) Make L(B) = out for each B such that att(B, A). If
there does not exist such an argument B, then the condition is satisfied.
• (support condition) Make L(A) = in for some A such that sup(A, A),
that is, for each A′ ∈ A, L(A′ ) = in. If there does not exist such A, thenan existence argument ex(A) and a support sup({ex(A)}, A) should be
added.
– Make L(A) = out.
Either of the following conditions should be satisfied.
• (attack condition) Make L(B) = in for some B such that att(B, A).
If there does not exist such an argument B, then this condition is notsatisfied.
• (support condition) Make L(A) = out for each A such that sup(A, A),
that is, for some A′ ∈ A, L(A′ ) = out. If there does not exist such A,
then this condition is not satisfied.
As a result, a set of existence arguments, that is, a set of evidences thatshould be provided, is found; this allows either party to attain his/her goal nomatter what evidence his/her opponent oﬀers.

Reasoning by BAF for PROLEG

11

Example 7. Figure 6 shows a PROLEG program excluding the fact part of Example 6. For convenience, each node is named a, b, . . . , k, respectively.

Fig. 6. Reasoning about required evidences.

– In this BAF, consider the conditions required to make L(a) = in.
• By attack condition for a, L(e) = out should be satisfied. To achievethis, attack condition for e or support condition for e should be satisfied.
By attack condition for e, L(k) = in should be satisfied, and since k has
no support, ex(k) is required. By support condition for e, at least oneof L(f ) = out, L(g) = out, L(h) = out, L(i) = out or L(j) = out holds.
However, this is impossible since f, g, h, i and j are neither attacked nor
supported.
• By support condition for a, L(b) = L(c) = L(d) = in should be satisfied.
Therefore, ex(b), ex(c) and ex(d) are required.
As a result, the defendant should provide the four evidences ex(k), ex(b), ex(c)and ex(d) to apply the law.
– On the other hand, consider the conditions required to make L(a) = out.
• By attack condition for a, L(e) = in should be satisfied. To achieve this,L(k) = out should be satisfied, but this is impossible since k is neither
attacked nor supported.
• By support condition for a, either L(b), L(c) or L(d) should be out, but
this is impossible since b, c and d are neither attacked nor supported.
Therefore, the plaintiﬀ never prevents application of the law.

12

T. Kawasaki, S. Moriguchi, K. Takahashi

In this example, only one set of existence arguments is found to make L(a) =in, and no argument is found L(a) = out. However, in general, we may findmultiple sets in both cases. For example, assume that a defendant wishes tomake L(a) = in in the BAF shown in Figure 7. The support required to makeL(a) = in is one of ex(b) or ex(c). The support required to make L(f ) = in is
one of ex(g) or ex(h). Thus, we find four sets of required evidences.

Fig. 7. Multiple sets of evidences are obtained.

6

Related Works

We compare our BAF semantics to the semantics derived by others.
Several works on BAF semantics have been undertaken. In almost all of them,the BAFs are given in advance or obtained by a translation from artificial logic
programs. Such programs principally discuss argumentation structures that are
seldom seen in actual judgments. We sought to apply real-world legal reasoning.
A significant issue during transformation is to create BAF semantics preserving
legal reasoning; no previous BAF semantics met this criterion.
Cayrol et al. investigated BAF semantics, defining several types of indirect
attacks by combining attacks with supports. They also defined several types ofextension [6]. Next, the concept of “coalition” (a set of arguments) was introduced and used to define a meta-AF [7, 8]. The idea was to reduce a BAF to an
AF by deleting the support relations between arguments in the same coalition.
An argument in BAF is accepted if it is included in an accepted coalition of
the meta-AF. Boella et al. pointed out that this approach does not allow use of
the Dung semantics, and revised the semantics by introducing diﬀerent metaarguments and meta-supports [2]. However, if we adopt these semantics, thesemantics of PROLEG and BAF do not coincide [11]. It follows that we cannot

Reasoning by BAF for PROLEG

13

combine arguments to form a single support without considering their original
relationships in PROLEG.
Noueioua et al. proposed a BAF that considered a support relation to bea “necessity” relation [12]. In this approach, each atom corresponds to eachargument, similar to our approach. They proved the correspondence between anormal logic program and their BAF. The main drawback of the method is thatit does not discriminate support by a set of arguments from support given by
separate multiple arguments. They do not reflect the case in which a set of bodygoals support its head goal in a logic program.
Oren and Norman developed an evidential argumentation by introducing
a special argument, corresponding to an environment, into a BAF [13]. This
concept is similar to the existence argument of our method. The diﬀerence isthat they introduced a single argument from which both attack and support
relations arise. In contrast, we add an existence or absence argument for eachfact. The structure of their evidential argumentation is more compact than ours,
but the evidences bearing upon individual facts are unclear.
Unlike the works cited above, Brewka et al. developed an abstract dialecticalframework (ADF) as a generalization of the Dung AF [3, 4]. In the ADF, eachnode is associated with an acceptance condition depending on the parent nodes,
and each link exhibits an individual strength. A bipolar ADF is a subclass of
ADF in which a link is either attacked or supported depending on the polarityof its strength. A BAF transformed from PROLEG may be considered to be an
instantiation of an ADF. It would be interesting to explore whether an ADF
semantics could be simply applied to a BAF transformed from PROLEG.

7

Conclusion

We have described the transformation from a PROLEG description to a BAF,
and the legal reasoning using the BAF. We gave semantics to the BAF preserving
the features of a PROLEG program. The BAF reflects the structure of thejudgment process and causality among arguments. We have developed reasoning
on the BAF, that is diﬃcult to emulate or understand using a PROLEG program
or trace of its execution. Our system enables lawyers and law school students toanalyze judgments.
In future, we will improve reasoning by the BAF and create a graphicalinterface.

Acknowledgment
This work was supported by JSPS KAKENHI Grant Number JP17H06103.

References
1. Bench-Capon, T., Prakken H. and Sartor, G.: Argumentation in legal reasoning.
Argumentation in Artificial Intelligence, 363-382 (2009).

14

T. Kawasaki, S. Moriguchi, K. Takahashi

2. Boella, G., Gabbay, D. M., Torre, L. van der and Villata, S.: Support in abstract
argumentation In Proc. of COMMA2010, 40-51 (2010).
3. Brewka, G. and Woltran, S.: Abstract dialectical frameworks. In Proc. ofKR2010,102-111 (2010).
4. Brewka, G., Ellmauthaler, S., Strass, H., Wallner, J.P. and Woltran, S.: Abstract
dialectical frameworks revisited. In Proc. of IJCAI2013, 803-809 (2013).
5. Caminada, M.: On the issue of reinstatement in argumentation. In Proc. of
JELIA2006, 111-123 (2006).
6. Cayrol, C. and Lagasquie-Schiex, M.: On the acceptability of arguments in bipolarargumentation frameworks. In Proc. of ECSQARU2005, 378-389 (2005).
7. Cayrol, C. and Lagasquie-Schiex, M.: Coalitions of arguments: A tool for handling
bipolar argumentation frameworks. In International Journal of Intelligent Systems,
Volume 25, 83-109 (2010).
8. Cayrol, C. and Lagasquie-Schiex, M.: Bipolarity in argumentation graphs: Towards
a better understanding. International Journal of Approximate Reasoning, Volume
54, 876-899 (2013).
9. Dung, P. M.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence, Volume 77, 321-357 (1995).
10. Gelfond, M. and Lifschitz, V.: The stable model semantics for logic programming.
In Proc. of ICLP, 1070-1080 (1988).
11. Kawasaki, T., Moriguchi, S. and Takahashi, K.: Transformation from PROLEG toa bipolar argumentation framework. In Proc. of SAFA2018, 36-47 (2018).
12. Nouioua, F. and Risch, V.: Argumentation framework with necessities. In Proc. of
SUM2011, 163-176 (2011).
13. Oren, N. and Norman, T. J.: Semantics for evidence-based argumentation. In Proc.
of COMMA2008, 276-284 (2008).
14. Prakken, H., Reed, C. and Walton, D.: Dialogues about the burden of proof. In
Proc. of ICAIL2005, 115-124 (2005).
15. Rahwan, I. and Simari, G.(eds.): Argumentation in Artificial Intelligence, Springer(2009).
16. Reed, C. and Rowe, G.: Araucaria: Software for argument analysis, diagramming
and representation. In International Journal of AI Tools, Volume 13, 961-980(2004).
17. Satoh, K. et al.: PROLEG: An Implementation of the Presupposed Ultimate Fact
Theory of Japanese Civil Code by PROLOG Technology. In JSAI-isAI 2010: New
Frontiers in Artificial Intelligence, 153-164 (2010).
18. Satoh, K. et al.: On generality of PROLEG knowledge representation. In Proc. of
JURISIN2012, 115-128 (2012).