Claudette Cayrol
email: ccayrol@irit.fr

Marie-Christine Lagasquie-Schiex

Bipolarity in argumentation graphs: Towards a better understanding

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Introduction

Formal models of argumentation have recently received considerable interest across different AI communities, like defeasible reasoning and multi-agent systems [START_REF] Rahwan | Argumentation in Artificial Intelligence[END_REF][START_REF] Prakken | Logics for defeasible argumentation[END_REF][START_REF] Heras | Argue to agree: a case-based argumentation approach[END_REF]. Typical applications such as for instance negotiation [START_REF] Amgoud | A formal analysis of the role of argumentation in negotiation dialogues[END_REF] and practical reasoning [START_REF] Amgoud | Generating possible intentions with constrained argumentation systems[END_REF] represent pieces of knowledge and opinions as arguments and reach some conclusion or decision on the basis of interacting arguments.

In formal argumentation, two types of approaches exist. The first one allows for the building of arguments [START_REF] Efstathiou | Algorithms for generating arguments and counterarguments in proposition logic[END_REF]. The second one corresponds to abstract argumentation frameworks that model arguments as atomic entities, ignoring their internal structure and focusing on the interactions between arguments, or sets of arguments. In this case, several semantics can be defined that formalize different intuitions about which arguments to accept from a given framework.

The first abstract framework introduced by [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF] limits the interactions to conflicts between arguments with the binary attack relation. Several specialized or extended versions of Dung's framework have been proposed (see for instance [START_REF] Amgoud | A reasoning model based on the production of acceptable arguments[END_REF][START_REF] Bench-Capon | Persuasion in practical argument using value-based argumentation frameworks[END_REF][START_REF] Kaci | Preference-based argumentation: arguments supporting multiple values[END_REF][START_REF] Dunne | Inconsistency tolerance in weighted argument systems[END_REF][START_REF] Modgil | Reasoning about preferences in argumentation frameworks[END_REF][START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF]). Among these extended versions, we are interested in the bipolar framework [START_REF] Cayrol | On the acceptability of arguments in bipolar argumentation frameworks[END_REF][START_REF] Amgoud | On bipolarity in argumentation frameworks[END_REF] which is capable of modelling a kind of positive interaction expressed by a support relation. 1 Positive interaction between arguments has been first introduced by [START_REF] Karacapilidis | Computer supported argumentation and collaborative decision making: the hermes system[END_REF][START_REF] Verheij | Deflog: on the logical interpretation of prima facie justified assumptions[END_REF]. In [START_REF] Cayrol | On the acceptability of arguments in bipolar argumentation frameworks[END_REF], the support relation is left general so that the bipolar framework keeps a high level of abstraction. The associated semantics are based on the combination of the attack relation with the support relation which results in new complex attack relations. However, introducing the notion of support between arguments within abstract frameworks has been a controversial issue and some counterintuitive results have been obtained, showing that the combination of both interactions cannot avoid a deeper analysis of the notion of support.

Moreover, there is no single interpretation of the support. Indeed, recently, a number of researchers proposed specialized variants of the support relation. Each specialization can be associated with an appropriate modelling using an appropriate complex attack. However, these proposals have been developed quite independently, based on different intuitions and with different formalizations. In this paper we do not want to discuss all the criticisms which have been advanced, our purpose is rather to show that bipolar abstract frameworks provide a convenient way to model and discuss various kinds of support. In particular, we address a comparative study of these proposals, in a common setting. Moreover, it is essential to note that our goal is not to identify an approach that would be better than another one. We rather intend to explicit the differences between various kinds of support and to propose a common framework for handling each of them.

Section 2 presents a brief review of the classical and bipolar abstract argumentation frameworks. In Sections 3 to 6 we discuss three specializations of the notion of support and propose an appropriate modelling for each of them in the bipolar framework. Related works are discussed in Section 7. In Section 8 we conclude and give some perspectives for future work.

Note that this paper is an extended version of [START_REF] Cayrol | Bipolarity in argumentation graphs: towards a better understanding[END_REF]. This extension consists in the introduction of new notions and new results (proofs are given in the appendix) and a deeper analysis of related works.

Background on abstract argumentation frameworks

Dung argumentation framework

Dung's seminal abstract framework consists of a set of arguments and one type of interaction between them, namely attack. What really means is the way arguments are in conflict.

Definition 1 (Dung AF). A Dung's argumentation framework (AF, for short) is a pair A, R where A is a finite and non-empty set of arguments and R is a binary relation over A (a subset of A × A), called the attack relation.

An argumentation framework can be represented by a directed graph, called the interaction graph, in which the nodes represent arguments and the edges are defined by the attack relation: ∀a, b ∈ A, aRb is represented by a → b.

Definition 2 (Admissibility in AF). Given A, R and S ⊆ A,

• S is conflict-free in A, R iff there are no arguments a, b ∈ S, such that aRb.

• a ∈ A is acceptable in A, R with respect to S iff ∀b ∈ A such that bRa, ∃c ∈ S such that cRb.

• S is admissible in A, R iff S is conflict-free and each argument in S is acceptable with respect to S.

Standard semantics introduced by Dung (preferred, stable, grounded) enable to characterize admissible sets of arguments that satisfy some form of optimality. Definition 3 (Extensions). Given A, R and S ⊆ A,

• S is a preferred extension of A, R iff it is a maximal (with respect to ⊆) admissible set.

• S is a stable extension of A, R iff it is conflict-free and for each a ∈ S, there is b ∈ S such that bRa.

• S is the grounded extension of A, R iff it is the least (with respect to ⊆) admissible set X such that each argument acceptable with respect to X belongs to X.

Example 1. Let AF be defined by A = {a, b, c, d, e} and R att = {(a, b), (b, a), (b, c), (c, d), (d, e), (e, c)} and represented by the following graph. There are two preferred extensions ({a} and {b, d}), one stable extension ({b, d}) and the grounded extension is the empty set.

Bipolar argumentation framework

The abstract bipolar argumentation framework presented in [START_REF] Cayrol | On the acceptability of arguments in bipolar argumentation frameworks[END_REF][START_REF] Cayrol | Coalitions of arguments: a tool for handling bipolar argumentation frameworks[END_REF] extends Dung's framework in order to take into account both negative interactions expressed by the attack relation and positive interactions expressed by a support relation (see [START_REF] Amgoud | On bipolarity in argumentation frameworks[END_REF] for a more general survey about bipolarity in argumentation). A BAF can still be represented by a directed graph G b called the bipolar interaction graph, with two kinds of edges. Let a i and a j ∈ A, a i R att a j (resp. a i R sup a j ) means that a i attacks a j (resp. a i supports a j ) and it is represented by a → b (resp. by a → b).

Definition 4 (BAF)

Example 2. For instance, in the following graph representing a BAF, there is a support from g to d and an attack from b to a New kinds of attack emerge from the interaction between the direct attacks and the supports. These new attacks together with the direct attacks will be referred to as the complex attacks of the BAF. For instance, these complex attacks can be defined using the supported attack and the secondary attack which have been introduced in [START_REF] Cayrol | Coalitions of arguments: a tool for handling bipolar argumentation frameworks[END_REF] (and previously in [START_REF] Cayrol | On the acceptability of arguments in bipolar argumentation frameworks[END_REF] with a different terminology): Definition 5 ([19] An example of complex attacks in a BAF). Let BAF = A, R att , R sup , complex attacks in BAF consist of the direct attack R att and the supported and secondary attacks defined by:

• there is a supported attack from a to b iff there is a sequence

a 1 R 1 . . . R n-1 a n , n ≥ 3, with a 1 = a, a n = b, ∀i = 1 . . . n -2, R i = R sup and R n-1 = R att .
• There is a secondary attack from a to b iff there is a sequence

a 1 R 1 . . . R n-1 a n , n ≥ 3, with a 1 = a, a n = b, R 1 = R att and ∀i = 2 . . . n -1, R i = R sup .
The set of supported (resp. secondary) attacks will be denoted R sup att (resp. R sec att ).

So, according to the above definition, new kinds of attack, from a to b, can be considered in the following cases.

Note that the above definitions combine a direct attack with a sequence of direct supports, that is a direct or indirect support.

Notation 1. In the following, a supports b means that there is a sequence of direct supports from a to b.

Example 2 (Cont'd). In this example, there is a supported attack from g (or d) to b and a secondary attack from f to a. Acceptability semantics must be redefined for taking into account complex attacks. The first step in defining acceptability is the investigation of the notion of coherence for a set of arguments. The basic requirement is to avoid conflicts. That leads to extend the notion of conflict-freeness by replacing direct attacks by complex attacks. So, in the following, given a definition of complex attacks, we will talk about conflict-freeness wrt2 these complex attacks.

Moreover, the notion of coherence of a set of arguments can be still enforced by excluding sets of arguments which attack and support the same argument. This is a kind of external coherence reflected by the notion of safety [START_REF] Cayrol | On the acceptability of arguments in bipolar argumentation frameworks[END_REF]. So, as in the case of conflict-freeness, given a definition of complex attacks, we will talk about safety wrt these complex attacks. Definition 6 ([14] Safety in BAF). Let BAF = A, R att , R sup . Let R c-att be a set of complex attacks built from BAF. Consider S ⊆ A, S is safe wrt R c-att iff there are no arguments a, b ∈ S, and c ∈ A such that • b supports c or c ∈ S and • there is a complex attack from a to c belonging to R c-att .

For instance, following the example of complex attacks given by Definition 5, the set {a, b} can be considered as "incoherent" in each of the following cases: Another requirement has been considered in [START_REF] Cayrol | On the acceptability of arguments in bipolar argumentation frameworks[END_REF], which concerns only the support relation, namely the closure under R sup .

Definition 7 (Closure in BAF). Let BAF = A, R att , R sup , S ⊆ A. S is closed under R sup iff ∀a ∈ S, ∀b ∈ A, if aR sup b then b ∈ S.
So, following the same methodology as in Dung's framework, different acceptability semantics can be proposed in a bipolar argumentation framework, depending on the notion of attack (direct, supported, secondary, …) and on the notion of coherence which are used (conflict-free, safe, closed under R sup ).

Modelling various kinds of support

Handling support and attack at an abstract level has the advantage to keep genericity. An abstract bipolar framework is useful as an analytic tool for studying different notions of complex attacks, complex conflicts, and new semantics taking into account both kinds of interactions between arguments. However, the drawback is the lack of guidelines for choosing the appropriate definitions and semantics depending on the application. For instance, in Dung's framework, whatever the semantics, the acceptance of an argument which is not attacked is guaranteed. Is it always desirable in a bipolar framework? Two related questions are: Can arguments stand in an extension without being supported? Can arguments be used as attackers without being supported? It may depend on the interpretation of the support, as shown below.

In the following, we discuss three specialized variants of the support relation, which have been proposed recently: the deductive support, the necessary support and the evidential support. Let us first briefly give the underlying intuition, then some illustrative examples.

Deductive support [START_REF] Boella | Support in abstract argumentation[END_REF] is intended to capture the following intuition: If aR sup b then the acceptance of a implies the acceptance of b, and as a consequence the non-acceptance of b implies the non-acceptance of a.

Necessary support [START_REF] Nouioua | Bipolar argumentation frameworks with specialized supports[END_REF][START_REF] Nouioua | Argumentation frameworks with necessities[END_REF], is intended to capture the following intuition: If aR sup b then the acceptance of a is necessary to get the acceptance of b, or equivalently the acceptance of b implies the acceptance of a.

Evidential support [START_REF] Oren | Semantics for evidence-based argumentation[END_REF][START_REF] Oren | Moving between argumentation frameworks[END_REF] enables to distinguish between prima-facie and standard arguments. Prima-facie arguments do not require any support from other arguments to stand, while standard arguments must be supported by at least one prima-facie argument.

The following examples show that different interpretations of the support can be given strongly depending on the context, and that, according to the considered interpretation, some complex attacks need to be considered, while others are counterintuitive. It is important to note that these examples are not given here in order to express a preference over the different types of support. Their goal is only to illustrate the existing approaches. Example 3. This example has been inspired from [START_REF] Nouioua | Bipolar argumentation frameworks with specialized supports[END_REF] (and also from a variant in [START_REF] Boella | Support in abstract argumentation[END_REF]). Let us consider the following knowledge: Obtaining a Bachelor's degree with honors (bh) supports obtaining a scholarship (sch) and suppose that having at least one bad mark (bm) does not allow to obtain the honors (even if the average of marks normally allows it). One possible interpretation of the support is: obtaining a bachelor's degree is necessary for obtaining a scholarship. So, if we do not have a bh then we are sure that we do not have sch. Now let us suppose that obtaining sch may be also fulfilled if the student justifies modest incomes (mi). A more appropriate interpretation of the support is a deductive one. In that case, a secondary attack from bm to sch would be counterintuitive. Moreover, it is known that making a blank copy (bc) supports having a very bad mark. With a deductive interpretation of that support, it makes sense to add a supported attack from bc to bh. Finally, we add the knowledge: having a very good mark for each test of the examination (vg) supports obtaining a Bachelor's degree with honors.

The whole example can be formalized in a BAF represented by the following graph:

Example 4 (Example illustrating a necessary support). Let us consider the following dialogue between three agents:

• Agent 1: The room is dark, so I will light up the lamp.

• Agent 2: But the electric meter does not work.

• Agent 1: Are you sure?

• Agent 3: The electrician has detected a failure.

This dialogue shows interactions between the positions rd (the room is dark), ll (the lamp will light up), ew (the electric meter works), and fail (there is a failure in the electric meter). These interactions can be formalized in a BAF represented by the following graph:

The intuitive interpretation of the support is a necessary one since the lamp cannot light up when the electric meter does not work. In that case, it makes sense to add a secondary attack from fail to ll.

The importance of the context clearly appears in the following example inspired by an example proposed in [START_REF] Boella | Support in abstract argumentation[END_REF]:

Example 5. Let us consider the following knowledge about football matches:

• if Liverpool wins last match then Liverpool wins Premier League, • if Manchester does not win last match then Liverpool wins Premier League,

• if the best player of Liverpool is injured then Liverpool does not win last match.

The interactions between the positions wlm (Liverpool wins last match), lpl (Liverpool wins Premier League), mnw (Manchester does not win last match), and bpi (the best player of Liverpool is injured) can be formalized in a BAF represented by the following graph:

The intuitive interpretation of the support is a deductive one and not a necessary one. Indeed, Liverpool wins Premier League if Manchester does not win last match, even if Liverpool does not win last match. So adding a secondary attack from bpi to lpl is not the right modelling. We propose to restate various notions of support in the BAF framework. We will show that each specialized variant of the support can be associated with appropriate complex attacks. Then, we will be able to highlight links between these various notions of support.

We first discuss the deductive and necessary supports (Sections 4.1 and 4.2), and prove that these two specializations of the support are indeed dual. As a consequence, these two kinds of support can be handled simultaneously in a bipolar framework. Then in Section 5, we study a restricted version of evidential support and show that it can be viewed as a kind of weak necessary support.

A framework for a comparative study of deductive and necessary supports

Deductive supports

As explained above, a deductive support is intended to enforce the following constraint: If bR sup c then the acceptance of b implies the acceptance of c, and as a consequence the non-acceptance of c implies the non-acceptance of b. Suppose now that aR att c. The acceptance of a implies the non-acceptance of c and so the non-acceptance of b. This strong constraint can be taken into account by introducing a new attack, called mediated attack.

Definition 8 ([20] Mediated attack). Let BAF = A, R att , R sup .
There is a mediated attack from a to b iff there is a sequence

a 1 R sup . . . R sup a n-1 , and a n R att a n-1 , n ≥ 3, with a 1 = b, a n = a.
The set of mediated attacks will be denoted R med att .

Example 3 (Cont'd). From vgR sup bh and bmR att bh, the mediated attack bmR att vg will be added.

Moreover, the deductive interpretation of the support justifies the introduction of supported attacks (cf Definition 5 and [START_REF] Cayrol | Coalitions of arguments: a tool for handling bipolar argumentation frameworks[END_REF]). If aR sup c and cR att b, the acceptance of a implies the acceptance of c and the acceptance of c implies the nonacceptance of b. So, the acceptance of a implies the non-acceptance of b.

So, with the deductive interpretation of the support, new kinds of attack, from a to b, can be considered in the following cases: AF D has one preferred (and also stable and grounded) extension {bc, bm, sch, mi}.

The following results establish links between the different coherence requirements which can be defined, when modelling deductive support in a BAF.

Proposition 1. Consider BAF = A, R att , R sup with R sup being a set of d-supports. Given S ⊆ A, • S is safe wrt R att in BAF iff S is safe wrt R att ∪ R med att in BAF. • S is safe wrt R att in BAF iff S is conflict-free wrt R att ∪ R med att in BAF.
Proposition 2. Consider BAF = A, R att , R sup with R sup being a set of d-supports and AF D its associated Dung AF. Given S ⊆ A,

• If S is conflict-free wrt R att and closed under R sup in BAF, then S is also conflict-free in AF D (that is conflict-free wrt R D att ).

• If S is conflict-free wrt R D att and closed under R sup in BAF, then S is also safe wrt R D att in BAF.

From the above proposition it turns out that the notion of coherence enforced in a BAF (for instance, by using the closure under R sup ) is stronger than conflict-freeness in the corresponding AF D . Moreover, as shown by the following example, the comparison is strict, even in the case of maximal (for set-inclusion) coherent sets.

Example 7. Consider the following graph representing a BAF:

Among the sets which are conflict-free wrt R att and closed under R sup , the maximal ones are {a, x} and {b, c}.

Let us consider AF D represented by:

In AF D , there is another maximal (for set inclusion) conflict-free set, the set {a, b} which is not closed under R sup in the corresponding BAF.

The closure requirement makes the notion of coherence in a BAF very strong. However, this requirement can be justified by the deductive interpretation of the support: Assume that R sup only contains d-supports; it means that if aR sup b, the acceptance of a implies the acceptance of b; now, considering a sequence of supports a 1 R sup a 2 R sup . . . R sup a n , the acceptance of a 1 implies the acceptance of a 2 , which in turn implies the acceptance of a 3 …which implies the acceptance of a n ; so, by transitivity, the acceptance of a 1 implies the acceptance of a n . Obviously, the condition of closure under R sup enforces this property.

Going back to the interpretation of the deductive support, the following constraint hold on Example 7:

• The acceptance of a implies the acceptance of x,

• The acceptance of x implies the non-acceptance of c,

• The acceptance of b implies the acceptance of c.

It follows that the acceptance of a must imply the non-acceptance of b. However, this last constraint cannot be enforced in AF D . This problem can be easily solved by considering not only direct attacks but also supported attacks in the definition of a mediated attack. So, we propose to replace Definition 8 by the definition of a new "mediated" attack, called the super-mediated attack.

Definition 10 (Super-mediated attacks). Let BAF = A, R att , R sup with R sup being a set of d-supports. There is a super- mediated attack from a to b iff there is a direct attack or a supported attack from a to c, and a support from b to c.

The set of super-mediated attacks will be denoted R s-med att .

For instance, there is a super-mediated attack from a to b in the following case:

Then, deductive support can be better taken into account by considering the associated Dung AF consisting of the same arguments and of the relation built from the direct attacks, the supported attacks and the super-mediated attacks:

= R att ∪ R sup att ∪ R s-med att .
The following propositions show that the new AF Dc enables to recover the closure under R sup .

Proposition 3. Let BAF = A, R att , R sup with R sup being a set of d-supports. Given S ⊆ A, if S is conflict-free wrt R att and closed under R sup in BAF, then S is also conflict-free in AF Dc .

Proposition 4. Let BAF = A, R att , R sup with R sup being a set of d-supports. Given S ⊆ A, S is a ⊆-maximal conflict-free set in AF Dc iff S is ⊆-maximal among the sets which are conflict-free wrt R att and closed under R sup in BAF. We propose an inductive definition of these new attacks, called deductive complex attacks (d-attacks for short), by combining the direct, supported and mediated attacks. The set of d-attacks will be denoted R d-att .

It turns out that the set of d-attacks exactly corresponds to the attacks defined in AF Dc , the complete associated Dung AF for the deductive support.

Proposition 5. Let BAF = A, R att , R sup with R sup being a set of d-supports. R d-att = R att ∪ R sup att ∪ R s-med att . In other words, a d-attacks b iff (a, b) ∈ R att ∪ R sup att ∪ R s-med att .

Necessary supports

Necessary support corresponds to the following interpretation: If cR sup b then the acceptance of c is necessary to get the acceptance of b, or equivalently the acceptance of b implies the acceptance of c. Suppose now that aR att c. The acceptance of a implies the non-acceptance of c and so the non-acceptance of b. This constraint can be taken into account by introducing a new attack, which is exactly the secondary attack presented above (cf Definition 5 and [START_REF] Cayrol | Coalitions of arguments: a tool for handling bipolar argumentation frameworks[END_REF]).

Note that this constraint has been considered in [START_REF] Nouioua | Bipolar argumentation frameworks with specialized supports[END_REF], where it was called extended attack. Moreover, another kind of complex attack can be justified: If cR sup a and cR att b, the acceptance of a implies the acceptance of c and the acceptance of c implies the non-acceptance of b. So, the acceptance of a implies the non-acceptance of b. This constraint relating a and b should be enforced by adding a new complex attack from a to b. Note that this complex attack was not considered in [START_REF] Nouioua | Bipolar argumentation frameworks with specialized supports[END_REF] but has been added in [START_REF] Nouioua | Argumentation frameworks with necessities[END_REF].

Let us recall the definition of extended attack proposed in [START_REF] Nouioua | Argumentation frameworks with necessities[END_REF] which enables to model necessary support in a BAF. The set of the extended attacks will be denoted by R ext att .

So, with the necessary interpretation of the support, new kinds of attack, from a to b, can be considered in the following cases: Notation 5. In the following, necessary support will be called n-support and the existence of a n-support between two arguments a and b will be denoted by a n-supports b.

As in the deductive approach, it is possible to define the associated Dung AF: Notation 6. Let BAF = A, R att , R sup with R sup being a set of n-supports, the associated Dung AF for the necessary support is denoted by AF N and defined by A, R N att with R N att = R ext att .

Deductive support and necessary support have been introduced independently. However, they correspond to dual interpretations of the support in the following sense: a n-supports b is equivalent to b d-supports a. Besides, it is easy to see that the constructions of mediated attack and secondary attack are dual in the following sense: the mediated attacks obtained by combining the attack relation R att and the support relation R sup are exactly the secondary attacks obtained by combining the attack relation R att and the support relation

R -1 sup which is the symmetric relation of R sup (R -1 sup = {(b, a)|(a, b) ∈ R sup })
. Moreover, the complex attacks which are missing in [21] and added in [22] as evoked previously can be recovered by considering the supported attacks built from R att and R -1 sup . Consequently, the modelling by the addition of appropriate complex attacks satisfies this duality. It follows that {c, d} is the only preferred (and also stable and grounded) extension.

Due to the duality between necessary and deductive supports, the inductive process can be applied to the definition of extended attacks, leading to the complete associated Dung AF for the necessary support: Notation 7. Let BAF = A, R att , R sup with R sup being a set of n-supports.

• BAF sym denotes the bipolar framework defined by A, R att , R -1 sup .

• AF Dc sym denotes the complete associated Dung AF for BAF sym (obtained using the direct attacks, the supported attacks and the super-mediated attacks issued from BAF sym ).

• And the complete associated Dung AF for the necessary support is denoted by AF Nc and exactly corresponds to AF Dc sym .

The difference between AF N and AF Nc is illustrated by the following example: Example 9. This example comes from [START_REF] Nouioua | Argumentation frameworks with necessities[END_REF]. Consider BAF represented by:

The corresponding BAF sym (R -1 sup is used in place of R sup ) is represented by: Then the associated Dung AF for the deductive support AF D sym of BAF sym (which is exactly the associated Dung AF for the necessary support AF N of BAF) is represented by: Table 1 Correspondences between abstract, deductive and necessary supports.

Abstract supports of [START_REF] Cayrol | On the acceptability of arguments in bipolar argumentation frameworks[END_REF][START_REF] Cayrol | Coalitions of arguments: a tool for handling bipolar argumentation frameworks[END_REF] Deductive supports of [START_REF] Boella | Support in abstract argumentation[END_REF] Necessary supports of [START_REF] Nouioua | Bipolar argumentation frameworks with specialized supports[END_REF][START_REF] Nouioua | Argumentation frameworks with necessities[END_REF] Supported sup

AF D AF N for R -1 sup AF Dc AF Nc for R -1 sup d-attacks d-attacks for R -1
sup And the complete associated Dung AF for the deductive support AF Dc sym of BAF sym (which is exactly the complete associated Dung AF for the necessary support AF Nc of BAF) is represented by: Table 1 gives a synthetic view of the correspondences between the three approaches (abstract, deductive and necessary).

Impact on self-attacking arguments

In the literature on the argumentation domain, it is very common to find some restrictions about self-attacking arguments. As taking into account deductive or necessary supports leads to introduce new complex attacks, it is interesting to characterize the cases where these attacks correspond to self-attacks. For deductive supports, the following proposition describes these cases:

Proposition 7. Let BAF = A, R att , R sup with R

Evidential support

Evidential support [START_REF] Oren | Semantics for evidence-based argumentation[END_REF][START_REF] Oren | Moving between argumentation frameworks[END_REF] is intended to capture the notion of support by evidence: an argument cannot be accepted unless it is supported by evidence. Evidence is represented by a special argument, and the arguments which are directly supported by this special argument are called prima-facie arguments. Arguments can be accepted only if they are supported (directly or indirectly) by prima-facie arguments. Besides, only supported arguments can be used to attack other arguments.

In Oren's evidential argument framework, attacks and supports may be carried out by a set of arguments (and not only by a single argument). However, for the purpose of comparing different specializations of the notion of support, we will restrict the presentation of evidential support to the case where attacks and supports are carried out by single arguments. All the definitions that we give in the following are inspired by those given in [START_REF] Oren | Semantics for evidence-based argumentation[END_REF][START_REF] Oren | Moving between argumentation frameworks[END_REF].

Given BAF = A, R att , R sup , we distinguish a subset A e ⊆ A of arguments which do not require any support to stand. These arguments will be called self-supported and correspond to the prima-facie arguments. We recall that in a BAF, a supports b means that there is a sequence of direct supports from a to b. So an evidential BAF can be defined as follows:

Definition 13 (Evidential BAF (EAF)). An evidential BAF (EAF) is a tuple A, A e , R att , R sup where A, R att , R sup is a BAF and A e ⊆ A. A e is called the set of prima-facie arguments.

So, evidential support (or e-support for short) can be defined as a particular case of the notion of (direct or indirect) support.

Definition 14 (e-Supports). Let EAF =

A, A e , R att , R sup .
• a is e-supported iff either a ∈ A e or there exists b such that b is e-supported and bR sup a.

• a is e-supported by S (or S e-supports a) iff either a ∈ A e or there is an elementary sequence b

1 R sup . . . R sup b n R sup a such that {b 1 . . . b n } ⊆ S and b 1 ∈ A e .
• S is self-supporting iff S e-supports each of its elements.

Example 8 (Cont'd).

Assume that A e = {a, c}. Then b is e-supported by {a}, d is e-supported by {c}. The sets {a, b} and {c, d} are self-supporting.

The combination of the direct attacks and the evidential support results in restrictions on the notion of attack and also on the notion of acceptability. The first idea is that only e-supported arguments may be used to make a direct attack on other arguments. This is formalized by the notion of e-supported attack.

Definition 15 (e-Supported attack). Let EAF = A, A e , R att , R sup . S carries out an e-supported attack on a iff there exists b ∈ S such that bR att a and b is e-supported by S.

The second idea concerns reinstatement: If a is attacked by b, which is e-supported, a can be reinstated either by a direct attack on b or by an attack on c such that without c, b would be no longer e-supported. In order to enforce this idea, minimal (for set-inclusion) e-supported attacks have to be considered. We have: Proposition 9. Let EAF = A, A e , R att , R sup . X is a minimal e-supported attack on the argument a iff X is the set of arguments appearing in a minimal elementary sequence b 1 R sup …R sup b n such that b 1 ∈ A e and b n R att a.

Note that a minimal e-supported attack on a given argument corresponds to a particular case of a supported attack as defined in Definition 5. In the case when b 1 R sup . . . R sup b n with b 1 ∈ A e and b n R att a, each b i carries out a supported attack on a. Now, following Oren's evidential argument framework, we propose a new definition for acceptability. There are two conditions on S, for a being acceptable wrt S. The first one is classical and concerns defence or reinstatement: S must invalidate each minimal e-supported attack on a (either by attacking the attacker of a or by rendering this attacker unsupported). The second condition requires that S e-supports a.

Definition 16 (e-Acceptability). Let EAF =

A, A e , R att , R sup . a is e-acceptable wrt S iff
• For each minimal e-supported attack X on a, there exists b ∈ S and x ∈ X such that bR att x and • a is e-supported by S.

Definition 17 (e-Admissibility). Let EAF =

A, A e , R att , R sup . S is e-admissible iff
• Each element of S is e-acceptable wrt S and • there are no arguments a, b ∈ S, such that aR att b.

Example 8 (Cont'd). Assume that A e = {a, c}.

There is only one minimal e-supported attack on d: {a, b}. As cR att a and d is e-supported by {c}, we have that d is e-acceptable wrt {c}. Then, {c, d} is e-admissible. Note that there is no e-supported attack on b. However, b does not belong to any e-admissible set, because no e-admissible set e-supports b. Assume now that A e = {a, b, c}. {b} is the only minimal e-supported attack on d. As no argument attacks b, no e-admissible set contains d. The only e-admissible set is {c, b}.

The above example enables us to highlight the relationship between the notion of evidential support and the notion of necessary support. It seems that evidential support can be viewed as a kind of weak necessary support, in the following sense: Assume that b is supported by a and c; with the necessary support interpretation, the acceptance of b implies the acceptance of a and the acceptance of c; with the evidential interpretation, if b is not self-supported, the acceptance of b implies the acceptance of a or the acceptance of c and, if b is self-supported, the acceptance of b implies no constraint on a and c.

The above comment suggests to consider the particular case when each argument is self-supported, that is A e = A. In that case, X is a minimal e-supported attack on a iff X is reduced to one argument which directly attacks a. So, classical acceptability is recovered: a is e-acceptable wrt S iff a is acceptable wrt S in Dung's sense. And as each argument is self-supported, we also recover classical admissibility. That is to say that the support relation is ignored.

Another interesting case occurs when self-supported arguments are exactly those which do not have any support, that is A e = {a ∈ A / there does not exist b such that bR sup a}. However, even in that particular case, evidential support cannot be modelled with necessary support, as shown by the following example. Taking into account these new attacks, the set is {c, b, e} is no longer admissible (there is a conflict between c and b) and {c, d} becomes admissible.

If we use the inductive definition for the complex attacks, the resulting AF Nc is the following:

In this case, {d} becomes admissible.

The above example shows that the notion of evidential support, even in the particular case of interactions between single arguments, cannot be reduced to strict necessary support (nor to deductive support). So, it is not possible to handle together in the same bipolar framework evidential support and necessary / deductive support. A fortiori, this remark is true when one considers that EAF are also able to handle attacks and supports by sets of arguments. However, the idea of considering attacks and supports between sets of arguments is related to the notion of coalitions of arguments and, in the following section, we show how coalitions can be defined using deductive and necessary supports.

Coalitions of supports

In this section, we consider only d-supports (if n-supports appear, they can be translated into d-supports without loss of generality).

Our idea is that coalitions of arguments can be used as meta arguments and our purpose is to turn a BAF into a meta Dung AF so that the usual Dung's semantics may be applied. A first attempt has been done in [START_REF] Cayrol | Coalitions of arguments: a tool for handling bipolar argumentation frameworks[END_REF]. However the proposed meta argumentation system presented some important drawbacks (the main reason is that no interpretation was given to the support).

In the current paper, we show how to fix these drawbacks using a new definition of meta Dung AF which will enable to establish a one-to-one correspondence between extensions of the meta framework and those of AF Dc .

Intuitively, each argument a gives rise to a coalition that contains all the arguments supported (directly or indirectly) by a. A coalition attacks another one if the former contains at least one argument that attacks (with R att ) an argument of the second one. As a direct consequence, we have:

Proposition 11. Let BAF = A, R att , R sup with R sup being a set of d-supports. Let S = {a 1 , a 2 , . . . , a n } ⊆ A. S is conflict-free in AF Dc iff {C(a 1 ), C(a 2 ), …, C(a n )} is conflict-free in A C , R C .
The usual Dung's semantics can then be applied on the meta framework. The main result is that there is a one-to-one correspondence between extensions of the meta framework and those of AF Dc . Proposition 12. Let BAF = A, R att , R sup with R sup being a set of d-supports. Given S ⊆ A, S = {a 1 , a 2 , . . . , a n }.

1. S is a ⊆-maximal conflict-free set in AF Dc iff {C(a 1 ), C(a 2 ), …, C(a n )} is a ⊆-maximal conflict-free set in A C , R C . 2. S is admissible in AF Dc iff {C(a 1 ), C(a 2 ), . . . , C(a n )} is admissible in A C , R C . 3. S is stable in AF Dc iff {C(a 1 ), C(a 2 ), . . . , C(a n )} is stable in A C , R C .
Applying the previous propositions on Example 11 shows that the set {e, b, c} is stable in AF Dc and ⊆-maximal among the admissible sets of AF Dc . And in Example 12, the set {e, d} is stable in AF Dc and ⊆-maximal among the admissible sets of AF Dc .

Related works

Related works can be partitioned into two parts; the first part is related to the notion of meta argumentation and the second part concerns a more general framework.

About meta argumentation

Our approach for meta argumentation is close to the approach described in [START_REF] Boudhar | Handling preferences in argumentation frameworks with necessities[END_REF] using necessary support. In this work, the meta argument associated with an argument a, called cluster, contains a and all the arguments that are directly necessary for a. Turning the necessary supports into the dual deductive supports, the cluster will contain a and all the arguments that directly d-support a. In contrast, the d-coalition contains a and all the arguments that support a, directly or not.

Nevertheless, in [START_REF] Boudhar | Handling preferences in argumentation frameworks with necessities[END_REF], the binary relation which encodes the necessary support is assumed irreflexive and transitive. The transitive nature of the necessary support enables to recover indirect support in the clusters. So, clusters are similar to d-coalitions. However, the irreflexive nature of the necessary support excludes some of the d-coalitions. In the particular case when the support relation is irreflexive and transitive, using the results presented in Section 6 and the duality between deductive and necessary support, it can be proved that the meta framework based on clusters enables to encode the attacks in AF Nc , and not only the extended attacks defined in Definition 12.

Example 14. Consider BAF using n-supports and represented by:

The clusters are C(a) = {a, x}, C(b) = {b, c}, C(c) = {c}, C(x) = {x}.
Note that there is a meta attack from C(a) to C(b) whereas there is no extended attack from a to b.

Using deductive support, [START_REF] Boella | Support in abstract argumentation[END_REF] described a meta argumentation framework in which meta arguments are auxiliary arguments representing pairs of interacting arguments. More precisely,

• A direct attack from x to c is encoded by a path of length 3 in the meta framework:

acc(x)R att X xc R att Y xc R att acc(c); • A support from b to c is encoded by a path of length 2 in the meta framework: acc(c)R att Z bc R att acc(b);
where X xc (resp. Y xc , Z bc ) is read as "the attack from x to c is not active" (resp. "the attack from x to c is active", "b does not support c") and acc(x) is read as "x is acceptable".

The above approach enables to encode a mediated attack, but does not enable to encode a supported attack (see the following example):

Example 7 (Cont'd). In this example the meta argumentation framework proposed by [START_REF] Boella | Support in abstract argumentation[END_REF] is represented by: And the Dung meta argumentation framework obtained with our proposition (see Section 6) is represented by: With both approaches, we obtain the same ⊆-maximal admissible set {a, x} in the original framework. Nevertheless, an essential difference between both approaches concerns the conflict-free sets: following the meta argumentation framework proposed by [START_REF] Boella | Support in abstract argumentation[END_REF], the set {a, c} is conflict-free that is not the case in our approach.

Moreover, the reading of the auxiliary arguments is not intuitive. From the reading giving by Boella, there should be a symmetric attack between X xc and Y xc . As for the encoding of a deductive support from an argument b to an argument c, it seems strange to create an attack from "c is accepted" to "b does not support c".

The last approach proposing a meta argumentation framework is given by [START_REF] Oren | Moving between argumentation frameworks[END_REF]. In order to compare with our approach, we still consider the particular case where attacks and supports are carried out by single arguments. The meta arguments represent groups of arguments and are built from the notion of self-supporting path, as follows:

• Each ⊆-maximal self supporting path of A, R sup is a meta argument. • For each x ∈ A such that x is directly attacked, each ⊆-maximal self supporting path of A \ {x}, R sup is a meta argument.
There is a meta attack from S 1 to S 2 iff there exist a ∈ S 1 and b ∈ S 2 such that aR att b. The meta argumentation framework of [START_REF] Oren | Moving between argumentation frameworks[END_REF] captures the e-admissibility in the sense that {S 1 , S 2 , . . . , S p } is admissible in the meta framework iff the set of arguments S 1 ∪ S 2 . . . ∪ S p is e-admissible in the evidential argumentation framework.

Let us consider again Example 10.

Example 10 (Cont'd).

Assuming that A e = {a, c, e}, the meta argumentation framework proposed by [START_REF] Oren | Moving between argumentation frameworks[END_REF] is represented by: It follows that {{e, b}, {c}} is admissible in the meta framework. Note that there are two distinct meta arguments that contain b, reflecting the notion of weak necessary support: e is necessary for b or a is necessary for b. So the Dung meta argumentation framework obtained with our approach is represented by: It follows that {C(c), C(d), C(e)} is admissible in the meta framework, and so {c, d, e} is admissible in AF Dc . In our approach, b belongs to only one d-coalition. So, there is only one attack against the meta argument C(d) = {c, d}, for which there exists a counter attack (C(d) attacks C(b)). In contrast, in the meta argumentation framework proposed by [START_REF] Oren | Moving between argumentation frameworks[END_REF], there are two attacks against the meta argument {c, d}: one attack is by the meta argument {a, b}, for which there exists a counter attack; the other attack is by the meta argument {b, e}, for which there is no counter attack.

Other works

Another interesting related work has been proposed in the more general setting of Abstract Dialectical Framework (ADF for short) [START_REF] Brewka | Abstract dialectical frameworks[END_REF].

This framework allows to represent a variety of dependencies between nodes in an interaction graph. In a BAF, there are two kinds of edges, one for the support and one for the attack. In contrast, there is only one kind of edge in an ADF. An edge between a and b represents a dependency between a and b. The kind of dependency is specified by associating an acceptance condition with each node of the graph. The acceptance condition of s specifies how the status of s depends on the status of the parents of s, and gives the exact conditions under which s is accepted.

Acceptance conditions are much more flexible than the conditions described above for deductive, necessary or evidential support. For instance, if c depends on a and b, the following constraint can be taken into account: c is accepted if and only if exactly one of {a, b} is accepted.

Formally, an ADF is a directed graph whose nodes represent arguments which can be accepted or not. For each node s, the set of its parents in the graph is denoted by par(s). An acceptance condition of s, denoted by C s , is a function that assigns to each subset R of par(s) one of the values in, out. C s (R) = in means that if the nodes in R are accepted and those in par(s) \ R are not accepted, then s is accepted. So, the exact conditions under which s is accepted are given by the subsets R ⊆ par(s) such that C s (R) = in.

Note that if s has no parent in the graph, then s is accepted if and only if C s (∅) = in. Moreover, as explained in [START_REF] Brewka | Abstract dialectical frameworks[END_REF], if each edge represents an attack, C s (R) = in iff R = ∅.

In the following, we show that the ADF model does not always enable to capture exactly the notion of deductive support. We consider three different examples. In each case, starting from a BAF considered as a dependency graph, we write the possible acceptance conditions and try to determine whether some of them may correspond to deductive or necessary support. • With C 2 b , b is accepted iff c is not accepted, whatever the status of a. So none notion of support is captured.

This example shows that a deductive support cannot always be captured in the ADF model.

({a}) = in, C 1 b ({c}) = in, C 1 b ({a, c}) = out, C 1 b (∅) = out; C 2 b ({a}) = in, C 2 b ({c}) = in, C 2 b ({a, c}) = in, C 2 b (∅) = out; C 3 b ({a, c}) = in, C 3 b ({a}) = out, C 3 b ({c}) = out, C 3 b (∅) = out.
So, we have:

• With C 1 b , b is accepted if and only if exactly one of {a, c} is accepted.

• With C 2 b , b is accepted iff a is accepted or c is accepted. So C 2
b enables to model a support which is both deductive ("if" part) and weak necessary ("only if" part).

• With C 3 b , b is accepted if and only if a is accepted and b is accepted. So C 3 b enables to model a support which is both necessary ("only if" part) and weak deductive ("if" part).

This example shows that a purely deductive (resp. purely necessary) support cannot always be captured in the ADF model.

Conclusions and future works

In this paper, we have considered three recent proposals for specializing the support relation in abstract argumentation: the deductive support, the necessary support and the evidential support. These proposals have been developed independently within different frameworks and with appropriate modellings, based on different intuitions.

We have restated these proposals in a common setting, the bipolar argumentation framework. Basically, the idea is to keep the original arguments, to add complex attacks defined by the combination of the original attack and the support, and to modify the classical notions of acceptability. We have proposed a comparative study of the modellings obtained for the considered variants of the support, which has enabled us to highlight relationships and differences between these variants. Namely, we have shown a kind of duality between the deductive and the necessary interpretations of support, which results in a duality in the modelling by complex attacks. In contrast, the evidential interpretation is quite different and cannot be captured with deductive or necessary supports.

So, the abstract bipolar argumentation framework is a suitable tool for handling applications where deductive as well as necessary supports are expressed. By cons, it is no longer the case as soon as evidential supports also appear in the same applications.

Evidential support has been captured by a meta argumentation framework, which instantiates Dung's framework with meta arguments. Following the same line, we have also proposed a meta argumentation framework taking into account the deductive/necessary supports and preserving some semantics. This proposition allows for new understandings of the differences between the variants of support.

This paper addresses how various notions of support can be handled in abstract argumentation and so it is a first step towards a better understanding of the notion of support in argumentation.

The next step is to discuss how these different notions of support can be built from the internal structure of the arguments (see [START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF]). In particular, it would be interesting to study how these three types of support (deductive, necessary, evidential) can capture different types of reasoning when arguments are built from pieces of knowledge.

More generally, it would be interesting to extend the discussion to the case when attacks and supports can be carried out by sets of arguments, as in the evidential argumentation framework.

Another interesting topic for further research is the representation of defeasible support in bipolar frameworks. A promising proposal has been given in [START_REF] Boella | Support in abstract argumentation[END_REF]. As interactions between arguments are represented by auxiliary arguments in the meta argumentation framework, an attack from an argument to a support can be easily represented by an attack in the meta framework. A direct representation of defeasible support in a BAF or in the meta framework based on coalitions must be investigated.

As regards meta argumentation, the study of coalitions could be continued in connection with recent works about weighted argumentation frameworks (see [START_REF] Cayrol | Graduality in argumentation[END_REF][START_REF] Kaci | Preference-based argumentation: arguments supporting multiple values[END_REF][START_REF] Dunne | Inconsistency tolerance in weighted argument systems[END_REF][START_REF] Cayrol | Acceptability semantics accounting for strength of attacks in argumentation[END_REF][START_REF] Martinez | Strong and weak forms of abstract argument defense[END_REF]). The following questions appear to be relevant: How can weights on the arguments and/or weights on the supports be combined into weights on coalitions? Can the cardinality be taken into account for weighting a coalition? How can weighted coalitions be handled in the meta framework? Proposition 4. Let BAF = A, R att , R sup with R sup being a set of d-supports. Given S ⊆ A, S is a ⊆-maximal conflict-free set in AF Dc iff S is ⊆-maximal among the sets which are conflict-free wrt R att and closed under R sup in BAF.

Proof.

• (⇒)-Part: Let S be a ⊆-maximal conflict-free set in AF Dc . So, S is conflict-free wrt R att in BAF. Assume that S is not closed under R sup in BAF. Then, there exist a ∈ S and b / ∈ S with aR sup b. As S is ⊆-maximal conflict-free in AF Dc , S ∪ {b} is not conflict-free in AF Dc . So there exists c ∈ S such that either bR Dc att c or cR Dc att b. In each case, it is possible to build an attack between a and c, which contradicts the fact that S is conflict-free set in AF Dc . Let us enumerate the different cases which may be encountered.

• If bR att c, as aR sup b, we obtain a supported attack from a to c.

• If bR sup att c, as aR sup b, we also obtain a supported attack from a to c.

• If bR s-med att c, as aR sup b, we obtain a super-mediated attack from a to c.

• If cR att b, as aR sup b, we obtain a mediated attack from c to a.

• If cR sup att b, as aR sup b, we obtain a super-mediated attack from c to a.

• If cR s-med att b, as aR sup b, we also obtain a super-mediated attack from c to a.

So, we have proved that S is conflict-free wrt R att and closed under R sup in BAF. It remains to prove that S is ⊆-maximal. Assume that it is not the case. Then, there exists S ′ ⊆ A, such that S is strictly included in S ′ and S ′ is conflict-free wrt R att and closed under R sup in BAF. Due to Proposition 2, it holds that S ′ is conflict-free in AF D (that is conflict-free wrt R D att ). As S is strictly included in S ′ , there exists b ∈ S ′ such that b / ∈ S. So, since S is conflict-free wrt R Dc att and S ′ is conflict-free wrt R D att , there is an attack between b and an element a of S and this attack must be a super-mediated one (as S ′ is conflict-free wrt R D att , this attack can be neither a direct attack, nor a supported attack, nor a simple mediated attack). Two cases may occur.

• If the attack is from a to b, there exists a supported attack from a to an argument d and a support from b to d. The supported attack from a to d is composed of a support from a to an argument e and an attack from e to d. As S is closed under R sup in BAF, e belongs to S and so to S ′ . So, there is in fact a mediated attack from e, element of S ′ , to b, which contradicts the fact that S ′ is conflict-free wrt R D att .

• If the attack is from b to a, there exists a supported attack from b to an argument d and a support from a to d.

As S is closed under R sup in BAF, d belongs to S and so to S ′ . So, there is a supported attack from b to an element of S ′ , which contradicts the fact that S ′ is conflict-free wrt R D att .

• (⇐)-Part: Let S be a subset of A, ⊆-maximal among the sets which are conflict-free wrt R att and closed under R sup in BAF. Due to Proposition 3, we know that S is conflict-free in AF Dc . It remains to prove that S is ⊆-maximal among the conflict-free sets in AF Dc . Assume that it is not the case. Then, there exists S ′ ⊆ A, such that S is strictly included in S ′ and S ′ is conflict-free in AF Dc . As A is finite, we can assume that S ′ is ⊆-maximal. So, from the (⇒)-part of the proof, we know that S ′ is conflict-free wrt R att and closed under R sup in BAF. That is in contradiction with S being ⊆-maximal among the sets which are conflict-free wrt R att and closed under R sup in BAF. Proof.

• It is easy to prove that R att ∪ R • either aR att a,

• or ∃b ∈ A, such that a supports b and bR att a,

• or ∃b ∈ A, such that a supports b and aR att b,

• or ∃b and c ∈ A, such that a supports c, cR att b and a supports b.

Proof. The proposition is an obvious consequence of definitions and propositions concerning the deductive support.

Proposition 8. Let BAF = A, R att , R sup with R sup being a set of n-supports. Consider a ∈ A. a is a self-attacking argument of AF Nc iff

• either aR att a,

• or ∃b ∈ A, such that b supports a and bR att a,

• or ∃b ∈ A, such that b supports a and aR att b,

• or ∃b and c ∈ A, such that c supports a, cR att b and b supports a.

Proof. The proposition is an obvious consequence of Proposition 7 following the duality between deductive and necessary supports.

Proposition 9. Let EAF = A, A e , R att , R sup . X is a minimal e-supported attack on the argument a iff X is the set of arguments appearing in a minimal elementary sequence b 1 R sup …R sup b n such that b 1 ∈ A e and b n R att a.

Proof. We first notice that if b 1 R sup . . . R sup b n is an elementary sequence such that b 1 ∈ A e and b n R att a, the set X = {b 1 , . . . , b n-1 , b n } is an e-supported attack on a (this follows from the definition of e-supported attack). Furthermore, assuming that the sequence b 1 R sup . . . R sup b n is minimal exactly means that there is no elementary sequence c 1 R sup …R sup c p such that c 1 ∈ A e , c p R att a and {c 1 . . . Obviously, Y e-supports b, so Y is an e-supported attack on a. Moreover Y ⊆ X. As X be a minimal e-supported attack on the argument a, we conclude that Y = X.

It remains to prove the minimality of the sequence. Assume that there exists an elementary sequence c 1 R sup . . . R sup c p such that c 1 ∈ A e , c p R att a and {c 1 . . . c p } ⊂ X. Due to the preliminary remarks given in the proof, Z = {c 1 . . . c p } is an e-supported attack of a. As Z ⊂ X, there is a contradiction with the fact that X be a minimal e-supported attack on the argument a.

• (⇐)-Part: Let X be the set of arguments appearing in a minimal elementary sequence b 1 R sup . . . R sup b n such that b 1 ∈ A e and b n R att a. From the preliminary remarks given in the proof, X is an e-supported attack on a. It remains to prove the minimality. Assume that there exists Z such that Z ⊂ X and Z is an e-supported attack on a. From the (⇒)-part of the proof, we can build an elementary sequence c 1 R sup . . . R sup c p such that c 1 ∈ A e , c p R att a and {c 1 . . . c p } ⊆ Z.

As Z ⊂ X, we have {c 1 . . . c p } ⊂ X, which contradicts the fact that X is the set of arguments appearing in a minimal elementary sequence.

  A bipolar argumentation framework (BAF, for short) is a tuple A, R att , R sup where A is a finite and non-empty set of arguments, R att is a binary relation over A called the attack relation and R sup is a binary relation over A called the support relation.

Example 6 (

 6 Example for an evidential support). Let us consider the BAF represented by the graph: Assume first that the only prima-facie argument is c. So, d may stand, but neither a nor b is grounded in prima-facie arguments. As a consequence, the attack on d cannot be taken into account. So, c and d will be accepted. Assume now that the prima-facie arguments are a and c. So, b and d may stand and the attack on d must be considered. In that case the accepted arguments are a, b and c. In order to reinstate d, an attack could be added either from c to b or from c to a. Indeed, an attack from c to a invalidates the attack on d by rendering b unsupported. Finally, assume that the prima-facie arguments are a, b and c. The attack from b to d holds without the support by a. So an attack from c to a does not enable to reinstate d. There must be an attack from c to b for "saving" d.

Notation 2 .

 2 In the following, deductive support will be called d-support and the existence of a d-support between two arguments a and b will be denoted by a d-supports b. Definition 9 (Modelling deductive support). Let BAF = A, R att , R sup with R sup being a set of d-supports. The combi- nation of the direct attacks and the d-supports results in the addition of supported attacks and mediated attacks. As explained above, modelling deductive support in a BAF can be done in considering the associated Dung AF consisting of the same arguments and of the relation built from the direct attacks, the supported attacks and the mediated attacks: Notation 3. Let BAF = A, R att , R sup with R sup being a set of d-supports, the associated Dung AF for the deductive support is denoted by AF D and defined by A, R D att with R D att = R att ∪ R sup att ∪ R med att . Example 3 (Cont'd). The following attacks are added: a supported attack from bc to bh and a mediated attack from bm to vg. Then support can be ignored, and we obtain the following AF D :

Notation 4 .

 4 Let BAF = A, R att , R sup with R sup being a set of d-supports, the complete associated Dung AF for the deductive support is denoted by AF Dc and defined by A, R Dc att with R Dc att

Example 7 (

 7 Cont'd). In AF Dc , the ⊆-maximal conflict-free sets are {a, x} and {b, c}. An alternative method for modelling all the attacks induced by the deductive support is to give an inductive definition for these new attacks. Let us first illustrate this method on Example 7: Example 7 (Cont'd). The new attacks could be obtained with two steps: Note that the new attack from a to b can also be obtained as a new kind of "mediated" attack from the (supported) attack (a, c) and the support (b, c).

  Definition 11 (d-Attacks). Let BAF = A, R att , R sup with R sup being a set of d-supports. There exists a d-attack from a to b iff • either aR att b, or aR sup att b, or aR med att b (Basic case), • or there exists an argument c such that a supports c and c d-attacks b (Case 1), • or there exists an argument c such that a d-attacks c and b supports c (Case 2).

Proposition 6 (

 6 Modelling necessary support). Let BAF = A, R att , R sup with R sup being a set of n-supports. The combination of the direct attacks and the n-supports can be handled by turning the n-supports into the dual d-supports and then adding the supported attacks and mediated attacks. Example 8. Consider BAF represented by: Assume that the support relation has been given a necessary interpretation. That is a is necessary for b and c is necessary for d. It is equivalent to consider that there is a deductive support from b to a and also from d to c. Then, we add a supported attack from d to a and a mediated attack from c to b. The resulting AF N is represented by:

Proposition 8 .

 8 sup being a set of d-supports. Consider a ∈ A. a is a self-attacking argument of AF Dc iff • either aR att a, • or ∃b ∈ A, such that a supports b and b R att a, • or ∃b ∈ A, such that a supports b and a R att b, • or ∃b and c ∈ A, such that a supports c, c R att b and a supports b. So if we need Dung AF without self attacking arguments, we have to restrict BAF. A similar property is obtained for the necessary support using the duality between necessary and deductive supports: Let BAF = A, R att , R sup with R sup being a set of n-supports. Consider a ∈ A. a is a self-attacking argument of AF Nc iff • either aR att a, • or ∃b ∈ A, such that b supports a and b R att a, • or ∃b ∈ A, such that b supports a and a R att b, • or ∃b and c ∈ A, such that c supports a, c R att b and b supports a.

Example 10 .

 10 We complete Ex 8 by adding an argument e and a support from e to b: Assume that A e = {a, c, e} (this is represented by a double box around the elements of A e ). The only ⊆-maximal eadmissible set is {c, e, b}. Indeed, d is not e-acceptable wrt {c} since {e, b} is a minimal e-supported attack on d and neither b nor e is attacked. Now, if we handle the same graph with necessary supports, we first take R -1 sup and then add supported and mediated attacks. This results in adding an attack from d to a and an attack from c to b:

Definition 18 (

 18 d-Coalition). Let BAF = A, R att , R sup with R sup being a set of d-supports. Let a ∈ A, the d-coalition3 associated with the argument a is defined by: C(a) = {a} ∪ {b s.t. a supports b}.Note that C(a) corresponds to the set of nodes that are reachable from a by support edges in the directed graph G b . Formally, we define a meta argumentation framework corresponding to a BAF in the following way: For each argument a ∈ A, the d-coalition C(a) is the meta argument associated with a. Definition 19 (Meta framework). Let BAF = A, R att , R sup with R sup being a set of d-supports. The Dung meta argu- mentation framework corresponding to BAF is A C , R C , where • A C denotes the set of all the meta arguments obtained from A (A C = {C(a), a ∈ A}) • R C is an attack relation defined by: C(a)R C C(b) iff there exists x ∈ C(a) and y ∈ C(b) such that xR att y. The following examples illustrate this definition. Example 11. Consider BAF represented by: We have C(a) = {a, b, c}, C(b) = {b, c}, C(c) = {c}, C(d) = {d}, C(e) = {e}. So the corresponding Dung meta argumentation framework is represented by: Example 12. Consider BAF represented by: We have C(a) = {a, b, c}, C(b) = {a, b, c}, C(c) = {a, b, c}, C(d) = {d}, C(e) = {e}. So the corresponding Dung meta argumentation framework is represented by: 3 d-Coalition means "deductive coalition". Note that three distinct meta arguments correspond to the same d-coalition of arguments. Example 13. Consider BAF represented by: We have C(a) = {a, b, c}, C(b) = {b, c}, C(c) = {c}. So the corresponding Dung meta argumentation is represented by: Note that we obtain self-attacking arguments. The attacks in the meta framework can be characterized in terms of attacks in AF Dc . Proposition 10. Let BAF = A, R att , R sup with R sup being a set of d-supports. Let a, b ∈ A. C(a)R C C(b) iff there is an attack from a to b in AF Dc .

  In contrast, with the necessary support, turned into deductive supports, we obtain the d-coalitions C(a) = {a}, C(b) = {b, a, e}, C(c) = {c}, C(d) = {d, c}, C(e) = {e}. Note that there is only one d-coalition containing b.

Example 15 . 1 b 2 b 1 b 2 bExample 16 . 1 b 1 b

 1512121611 Consider the BAF represented by: We are interested in the acceptance condition C b . Considering par(b) = {a}, there exist four possible cases for defining C b : Sets R s.t. Conditions under which b is accepted C b (R) = in None No R s.t. C b (R) = in; so there is no condition under which b is accepted (b cannot be accepted whatever the status of a) C 0 b ∅ ∃ one R s.t. C b (R) = in; so there is only one condition under which b is accepted C {a} ∃ one R s.t. C b (R) = in; so there is only one condition under which b is accepted C ∅, {a} ∃ two R s.t. C b (R) = in; so there are two different conditions under which b is accepted The second case exactly corresponds to an attack from a to b (C b (R) = in iff R = ∅). Since, in our example, there is no attack from a to b, we do not use it. Thus, in order to characterize the support from a to b, it only remains two possible acceptance conditions for b, C 1 b and C 2 b . • C 1 b ({a}) = in means b is accepted if a is accepted. And as C specifies only one condition under which b is accepted, we also have the equivalence b is accepted if and only if a is accepted. So, C 1 b models a support which is both deductive and necessary. • C 2 b (∅) = in means b is accepted if a is not accepted. C ({a}) = in means b is accepted if a is accepted. So b will be accepted whatever the status of a. Thus the notion of support is not captured. We complete the above example by adding an attack from c to b. So, we consider the BAF represented by: We are interested in the acceptance condition C b . As there is an attack from c to b, if c is accepted, then b cannot be accepted. This constraint can be expressed by the following constraint on C b : If C b (R) = in then c / ∈ R. So only two subsets of par(b) may be in: ∅ and {a}. And we obtain the same discussion as in the above example. As there is no attack from a to b, we cannot have C b (R) = in iff R = ∅. So, there are two possible acceptance conditions for b, C 1 b and C 2 b defined by: C (R) = in iff R = {a} and C 2 b (∅) = in, C 2 b ({a}) = in. • C 1 b ({a}) = in, means b is accepted if a is accepted and c is not accepted. And as C specifies only one condition under which b is accepted, we also have the equivalence b is accepted if and only if a is accepted and c is not accepted. It follows that if b is accepted, then a is accepted. So, C 1 b enables to model a necessary support from a to b (and not the deductive support since the condition if a is accepted then b is accepted does not hold).

Example 17 .

 17 Consider the BAF represented by: We are interested in the acceptance condition C b , and among all the possible functions we discuss C 1 b , C 2 b , C 3 b defined by: C 1 b

Proposition 5 .

 5 Let BAF = A, R att , R sup with R sup being a set of d-supports. R d-att = R att ∪ R sup att ∪ R s-med att . In other words, ad-attacksb iff (a, b) ∈ R att ∪ R sup att ∪ R s-med att .

  sup att ∪ R s-med att ⊆ R d-att . If aR att b, or aR sup att b, or aR med att b, then by definition (Basic case), a d-attacks b. If there is a super-mediated attack from a to b, composed of a supported attack from a to c (so, a d-attacks c -Basic case), and a support from b to c, then, by definition (Case 2), there is a d-attack from a to b. • Conversely, we have to prove that R d-att ⊆ R att ∪ R sup att ∪ R s-med att . We give a proof by structural induction. Let (a, b) such that a d-attacks b. Basic case: either aR att b, or aR sup att b, or aR med att b. So, aR Dc att b. Case 1: there exists an argument c such that a supports c and c d-attacks b. Assuming that cR Dc att b, we have to prove that aR Dc att b. As cR Dc att b, we have either cR att b, or cR sup att b, or cR s-med att b. If cR att b or cR sup att b, as a supports c, we obtain a supported attack from a to b. If cR s-med att b, as a supports c, we obtain a super-mediated attack from a to b. So, in each case, aR Dc att b.

Case 2 :Proposition 6 .Proposition 7 .

 267 there exists an argument c such that a d-attacks c and b supports c. Assuming that aR Dc att c, we have to prove that aR Dc att b. As aR Dc att c, we have either aR att c, or aR sup att c, or aR s-med att c. If aR att c, as b supports c, we obtain a mediated attack from a to b. If aR sup att c, or aR s-med att c, as b supports c, we obtain a super-mediated attack from a to b. So, in each case, aR Dc att b. Let BAF = A, R att , R sup with R sup being a set of n-supports. The combination of the direct attacks and the n-supports can be handled by turning the n-supports into the dual d-supports and then adding the supported attacks and mediated attacks. Proof. Given BAF = A, R att , R sup , let us consider BAF sym = A, R att , R -1 sup 4 and its associated Dung AF for the deductive support AF D sym . So, using the duality between n-supports and d-supports, we have AF N = AF D sym . Let BAF = A, R att , R sup with R sup being a set of d-supports. Consider a ∈ A. a is a self-attacking argument of AF Dc iff

  c p } ⊆ {b 1 . . . b n }. • (⇒)-Part: Let X be a minimal e-supported attack on the argument a. By definition, there exists b ∈ X such that bR att a and b is e-supported by X. So, either b ∈ A e or there is an elementary sequence b 1 R sup . . . R sup b n-1 R sup b such that {b 1 . . . b n-1 } ⊆ X and b 1 ∈ A e . Let Y be a subset of A defined as follows: If b ∈ A e , then Y = {b}, else Y = {b 1 . . . b n-1 , b}.

  Definition 12([22] Extended attack). Let BAF = A, R att , R sup . There is an extended attack from a to b iff 1. either aR att b, 2. or there is a sequence a 1 R att a 2 R sup . . . R sup a n , n ≥ 3, with a 1 = a, a n = b, 3. or there is a sequence a 1 R sup . . . R sup a n , and a 1 R att a p , n ≥ 2, with a n = a, a p = b.

wrt: with respect to.

Acknowledgements

We wish to thank the reviewers for their very constructive work. Thanks to their suggestions, this paper has been greatly improved.

Appendix A. Proofs Proposition 1. Consider BAF = A, R att , R sup with R sup being a set of d-supports. Given S ⊆ A,

• S is safe wrt R att in BAF iff S is safe wrt R att ∪ R med att in BAF. • S is safe wrt R att in BAF iff S is conflict-free wrt R att ∪ R med att in BAF.

Proof.

• For the first result, it is sufficient to prove that if S is safe wrt R att in BAF, then S is also safe wrt R med att in BAF. Assume that S is not safe wrt R med att in BAF. Then there exists a, b ∈ S, and c ∈ A such that (b supports c or c ∈ S) and there is a mediated attack from a to c. So, there is a sequence c 1 (= c)R sup . . . R sup c p , and aR att c p , p ≥ 2, and either c ∈ S or b supports c with b ∈ S. If c ∈ S, we obtain a contradiction with the assumption that S is safe wrt R att in BAF.

By concatenating the two sequences, we obtain a sequence of supports from b to c p , and so a contradiction with the fact that S is safe wrt R att in BAF.

• Due to the above result, if S is safe wrt R att in BAF, then S is safe wrt R att ∪ R med att in BAF, and so by definition of safety, S is conflict-free wrt R att ∪ R med att in BAF. Conversely, let S be conflict-free wrt R att ∪ R med att . Assume that S is not safe wrt R att . Then there exists a, b ∈ S, and c ∈ A such that (b supports c or c ∈ S) and there is a direct attack from a to c.

If c ∈ S, we obtain a contradiction with the assumption that S is conflict-free wrt R att in BAF.

If b ∈ S and b supports c, we obtain exactly a mediated attack from a to b and so a contradiction with the fact that S is conflict-free wrt R med att in BAF.

Proposition 2. Consider BAF = A, R att , R sup with R sup being a set of d-supports and AF D its associated Dung AF. Given S ⊆ A,

• If S is conflict-free wrt R att and closed under R sup in BAF, then S is also conflict-free in AF D (that is conflict-free wrt R D att ).

• If S is conflict-free wrt R D att and closed under R sup in BAF, then S is also safe wrt R D att in BAF.

Proof.

• For the first result, it is sufficient to prove that "if S is conflict-free wrt R att and closed under R sup , then S is conflict-free wrt R sup att ∪ R med att in BAF". This proof is made by a reduction ad absurdum. Assume that there are arguments a, b ∈ S with a supported attack from a to b. There is a sequence a 1 (= a)R sup . . . R sup a n-1 R att b, with n ≥ 3. As S is closed under R sup , we have a n-1 ∈ S. So there is a direct attack between two elements of S, which contradicts the assumption that S is conflict-free wrt R att . Now assume that there are arguments a, b ∈ S with a mediated attack from a to b. There is a sequence b 1 (= b)R sup . . . R sup b p , and aR att b p , p ≥ 2. As S is closed under R sup , we have b p ∈ S. So there is a direct attack between two elements of S, which contradicts the assumption that S is conflict-free wrt R att .

• The second result follows from the following remark: In Definition 6, if S is closed under R sup , the condition (b supports c or c ∈ S) reduces to c ∈ S. So, if S is closed under R sup , S is safe wrt a complex attack is equivalent to S is conflict-free wrt that complex attack.

Proof. Let S be conflict-free wrt R att and closed under R sup in BAF. Due to Proposition 2, we know that S is conflict-free in AF D (that is conflict-free wrt R D att ). So, we have to prove that there is no super-mediated attack between two elements of S. Assume that it is not the case, and that there exists a super-mediated attack from a to b in S. Then, there exists c ∈ A such that aR • x = a, y = b and xR att y: there is a direct attack from a to b.

• x = a, there is a sequence of supports from b to y and xR att y: there is a mediated attack from a to b • y = b, there is a sequence of supports from a to x and xR att y: there is a supported attack from a to b • there is a sequence of supports from a to x, a sequence of supports from b to y and xR att y: there is a super-mediated attack from a to b

In each case, we find an attack from a to b in AF Dc . • (⇐)-Part: Assume that there is an attack from a to b in AF Dc . Due to the definition of AF Dc , three cases may be encountered.

• there is a direct attack from a to b: Proposition 12. Let BAF = A, R att , R sup with R sup being a set of d-supports. Given S ⊆ A, S = {a 1 , a 2 , . . . , a n }.

Proof.

1. Follows directly from the above proposition, since each argument corresponds to exactly one meta argument 2. Assume that S = {a 1 , a 2 , . . . , a n } is admissible in AF Dc . So S is conflict-free in AF Dc . We know that {C(a 1 ), C(a 2 ), . . . , C(a n )} is also conflict-free in A C , R C . Now assume that C(a i ) is attacked by a meta argument C(b). Due to the previous results, we know that b attacks a i in AF Dc . As S is admissible in AF Dc , there exists a j ∈ S such that a j attacks b in AF Dc . So, C(a j ) attacks C(b) in A C , R C . We have proved that {C(a 1 ), C(a 2 ), . . . , C(a n )} is admissible in A C , R C .

The proof for the converse is analogous.

3. Assume that S = {a 1 , a 2 , . . . , a n } is stable in AF Dc . So S is conflict-free in AF Dc . We know that {C(a 1 ), C(a 2 ), . . . , C(a n )} is also conflict-free in A C , R C . Assume that C(b) is a meta argument not belonging to {C(a 1 ), C(a 2 ), . . . , C(a n )}. So b / ∈ S. As S is stable, there exists a i ∈ S such that a i attacks b in AF Dc . So, C(a i ) attacks C(b) in A C , R C . We have proved that {C(a 1 ), C(a 2 ), . . . , C(a n )} is stable in A C , R C . The proof for the converse is analogous.