Claudette Cayrol
email: ccayrol@irit.fr

Marie-Christine Lagasquie-Schiex

An axiomatic approach to support in argumentation

Keywords: abstract argumentation, bipolar argumentation, axiomatization of necessary support

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Introduction

The main feature of argumentation framework is the ability to deal with incomplete and / or contradictory information, especially for reasoning [15; 2]. Moreover, argumentation can be used to formalize dialogues between several agents by modeling the exchange of arguments in, e.g., negotiation between agents [START_REF] Amgoud | Modelling dialogues using argumentation[END_REF]. An argumentation system (AS) consists of a collection of arguments interacting with each other through a relation reflecting conflicts between them, called attack. The issue of argumentation is then to determine "acceptable" sets of arguments (i.e., sets able to defend themselves collectively while avoiding internal attacks), called "extensions", and thus to reach a coherent conclusion. Formal frameworks have greatly eased the modeling and study of AS. In particular, the framework of [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF] allows for abstracting the "concrete" meaning of the arguments and relies only on binary interactions that may exist between them.

In this paper, we are interested in bipolar AS (BAS), which handle a second kind of interaction, the support relation. This relation represents a positive interaction between arguments and has been first introduced by [18; 27]. 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. However there is no single interpretation of the support, and a number of researchers proposed specialized variants of the support relation: deductive support [START_REF] Boella | Support in abstract argumentation[END_REF], necessary support [21; 22], evidential support [23; 24], backing support [START_REF] Cohen | A survey of different approaches to support in argumentation systems[END_REF]. Each specialization can be associated with an appropriate modelling using an appropriate complex attack. These proposals have been developed quite independently, based on different intuitions and with different formalizations. [START_REF] Cayrol | Bipolarity in argumentation graphs: towards a better understanding[END_REF] presents a comparative study in order to restate these proposals in a common setting, the bipolar argumentation framework (see also [START_REF] Cohen | A survey of different approaches to support in argumentation systems[END_REF] for another survey). 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. An important result of [START_REF] Cayrol | Bipolarity in argumentation graphs: towards a better understanding[END_REF] is the highlight of a kind of duality between the deductive and the necessary specialization of support, which results in a duality in the modelling by complex attacks. In this context, new different papers have recently been written: some of them give a translation between necessary supports and evidential supports [START_REF] Polberg | Revisiting support in abstract argumentation systems[END_REF]; others propose a justification of the necessary support using the notion of subarguments [START_REF] Prakken | On support relations in abstract argumentation as abstraction of inferential relations[END_REF]; an extension of the necessary support is presented in [START_REF] Nouioua | AFs with necessities: further semantics and labelling characterization[END_REF]. From all these works it seems interesting to focus on the necessary support. However, different interpretations remain possible, leading to different ways of introducing new attacks and different ways to define acceptability of sets of arguments.

Our purpose is to propose a kind of "axiomatic approach" for studying how necessary support should be taken into account. Indeed we propose requirements (or constraints) that should be imposed to a bipolar argumentation system as "axioms" describing a desired behaviour of this system. Some of these constraints concern the new attack relations, others concern acceptability. We extend basic Dung's framework in different ways in order to propose frameworks suitable for encoding these contraints. By the way, we propose a formal study of properties of necessary support.

Some background is given in Section 2 for AS and BAS, in particular the duality identified in [START_REF] Cayrol | Bipolarity in argumentation graphs: towards a better understanding[END_REF]. Section 3 presents constraints that should be imposed for taking into account necessary support. Then different frameworks for handling these constraints are described in Section 4. Section 5 concludes and suggests perspectives of our work. The proofs are given in [START_REF] Cayrol | An axiomatic approach to support in argumentation[END_REF].

Background on abstract bipolar argumentation systems

Bipolar abstract argumentation systems extend Dung's argumentation systems. So first we recall Dung's framework for abstract argumentation systems.

Dung's framework

Dung's abstract framework consists of a set of arguments and only one type of interaction between them, namely attack. The important point is the way arguments are in conflict.

Def. 1 (Dung AS) A Dung's argumentation system (AS, 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 system can be represented by a directed graph, called the interaction graph, in which nodes represent arguments and edges are defined by the attack relation: ∀a, b ∈ A, aRb is represented by a → b.

Def. 2 (Admissibility in AS)

Given A, R and S ⊆ A. S is conflict-free in A, R if and only if (iff, for short) there are no arguments a, b ∈ S, such that (s.t., for short) aRb. a ∈ A is acceptable in A, R with respect to (wrt, for short) S iff ∀b ∈ A s.t. bRa, ∃c ∈ S s.t. cRb. S is admissible in A, R iff S is conflict-free and each argument in S is acceptable wrt S.

Standard semantics introduced by Dung (preferred, stable, grounded) enable to characterize admissible sets of arguments that satisfy some form of optimality.

Def. 3 (Extensions) Given A, R and S ⊆ A. S is a preferred extension of A, R iff it is a maximal (wrt ⊆) admissible set. S is a stable extension of A, R iff it is conflict-free and for each a ∈ S, there is b ∈ S s.t. bRa. S is the grounded extension of A, R iff it is the least (wrt ⊆) admissible set X s.t. each argument acceptable wrt X belongs to X.

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

Abstract bipolar argumentation systems

The abstract bipolar argumentation framework presented in [8; 9] 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).

Def. 4 (BAS) A bipolar argumentation system (BAS, 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.

A BAS can still be represented by a directed graph1 , 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. a → b).

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 solving this problem, some specializations of the support relation have been proposed and discussed recently. The distinction between deductive and necessary support has appeared first. Then, several interpretations have been given to the necessary support (sub-argument relation [START_REF] Prakken | On support relations in abstract argumentation as abstraction of inferential relations[END_REF], evidential support [23; 24; 25], backing support [START_REF] Cohen | A survey of different approaches to support in argumentation systems[END_REF]).

Deductive support

The deductive support has first appeared in [START_REF] Boella | Support in abstract argumentation[END_REF]. This variant 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".

In relevant literature, this interpretation is usually taken into account by adding two kinds of complex attack. The idea is to produce a new AS, containing original and new attacks, and then to use standard semantics.

The first new attack, called mediated attack in [START_REF] Boella | Support in abstract argumentation[END_REF], occurs when bR sup c and aR att c: "the acceptance of a implies the non-acceptance of c" and so "the acceptance of a implies the non-acceptance of b". Another complex attack, called supported attacks in [START_REF] Cayrol | Coalitions of arguments: a for handling bipolar argumentation frameworks[END_REF] occurs when aR sup c 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".

Def. 5 ([5] Mediated attack, [START_REF] Cayrol | Coalitions of arguments: a for handling bipolar argumentation frameworks[END_REF] Supported attack) Let BAS = 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. 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 .
So, with the deductive interpretation of the support, new kinds of attack, from a to b, can be considered in the following cases: and a is similar to the notion of subargument used in [START_REF] Prakken | On support relations in abstract argumentation as abstraction of inferential relations[END_REF].

As for deductive support, the idea is to add complex attacks in order to use standard semantics on a new AS. The first added complex attack, called extended attack in [START_REF] Nouioua | Bipolar argumentation frameworks with specialized supports[END_REF] and secondary attack in [START_REF] Cayrol | Coalitions of arguments: a for handling bipolar argumentation frameworks[END_REF] has been proposed in the following case: Suppose that aR att c and cR sup b. "The acceptance of a implies the non-acceptance of c" and so "the acceptance of a implies the non-acceptance of b". Another kind of complex attack may be considered when 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 new attack from a to b has been proposed in [START_REF] Nouioua | Argumentation frameworks with necessities[END_REF].

The formal definition of these two attacks is:

Def. 6 ([22] Extended attack) Let BAS = A, R att , R sup .
There is an extended attack from a to b iff either aR att b (direct attack), 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 (Case 1), 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 (Case 2).
So, with the necessary interpretation of the support, new kinds of attack, from a to b, can be considered in the following cases: sup . So in the following, we only focus on the necessary support since, taking advantage of the duality, all the results we obtain can be easily translated into results for deductive supports.

Axiomatic approach for handling necessary support

In relevant literature, as described in the previous section, taking into account support generally leads to add new attacks. It is the case for instance with the necessary support that leads to extended attacks. However, a deeper analysis of the original interpretation of necessary support suggests other ways to handle this support. In this section, we discuss several constraints induced by the intended meaning of necessary support, and we show that new frameworks must be proposed for encoding these constraints.

Let us come back to the original interpretation of necessary support: If cR sup b, "the acceptance of c is necessary to get the acceptance of b". Analysing this interpretation leads to at least four kinds of constraints.

Transitivity (TRA) This first requirement concerns the relation R sup alone.

It expresses transitivity2 of the necessary support. It induces that a sequence of supports is considered as a support:

Def. 7 (Constraint TRA) ∀a, b ∈ A, if ∃n > 1 such that a = a 1 R sup . . . R sup a n = b, then a supports b.
Closure (CLO) A second constraint also concerns the relation R sup alone and expresses the fact that if cR sup b, then "the acceptance of b implies the acceptance of c". So, if cR sup b, and there exists an extension S containing b, then S also contains c. This constraint can be expressed by the property of closure of an extension under R -1 sup :3 Def. 8 (Constraint CLO) Let s be a semantics and E be an extension under s. ∀a, b ∈ A, if aR sup b and b ∈ E, then a ∈ E. Moreover, an interesting variant of this constraint could be induced by a slightly different reading of the original interpretation: "the acceptance of c is necessary to get the acceptance of b" because c is the only attacker of a particular attacker of b. This reading implies that there implicitly exists a special attack to b which can be only defeated by c. This interpretation will lead us to propose a framework with meta-arguments (see Section 4.2). Conflicting sets (CFS) Now, we consider constraints induced by the presence of both attacks and supports in a BAS. Starting from the original interpretation, if aR att c and cR sup b, "the acceptance of a implies the non-acceptance of c" and "the acceptance of b implies the acceptance of c". So, using contrapositives, "the acceptance of a implies the non-acceptance of b", and then "the acceptance of b implies the non-acceptance of a". Thus, we obtain a symmetric constraint involving a and b. However, the fact that "the acceptance of a implies the non-acceptance of b" is not equivalent to the fact that there is an attack from a to b. We have only the sufficient condition. So, the creation of a complex attack (here a secondary attack) from a to b can be viewed in some sense too strong. Hence, faced with the case when aR att c and cR sup b, we propose to assert a conflict between a and b, or in other words that the set {a, b} is a conflicting set. Similarly, if cR att b and cR sup a, "the acceptance of a implies the acceptance of c" and so "the acceptance of a implies the non-acceptance of b". nATT (resp. n+ATT) corresponds to the addition of secondary (resp. extended) attacks. In Section 4.3 we present two frameworks for handling these constraints.

Continuing the discussion one step further, if the fact that "the acceptance of a implies the non-acceptance of b" is represented by an attack from a to b, due to contrapositive, this new attack must be symmetric. However, in that case, each attack should be turned into a symmetric one. Thus, we move towards symmetric argumentation frameworks which have been studied in [START_REF] Coste-Marquis | Symmetric argumentation frameworks[END_REF]. We will not consider this case in the current paper. Some of the above constraints can he handled in a Dung's abstract framework (CLO, TRA, nATT and n+ATT) with the advantage of reusing all known Dung's results. However, as we noticed above, constraint CFS cannot be encoded in a Dung's framework. So in the next section we propose different variants of Dung's framework and of the bipolar framework in order to take into account these constraints.

New frameworks for handling necessary supports

Starting from the constraints discussed in Section 3, we propose several frameworks for handling necessary support. The first two are driven by Constraint CLO whereas the last two are driven by the constraints nATT and n+ATT. The section will end by a comparison of these frameworks.

Handling conflicting sets of arguments

We propose a generalized bipolar abstract argumentation framework consisting of a set of arguments, a binary relation representing an attack between arguments, a binary relation representing a support between arguments and a set of conflicting sets of arguments. Intuitively, knowing that a attacks b is stronger than knowing that {a, b} is a conflicting set of arguments. Knowing that a set of arguments S is conflicting will only prevent any extension from containing S. Moreover, a conflicting set may contain more than two arguments.

Def. 12 (Generalized BAS, GBAS) A generalized bipolar argumentation system is a tuple A, R att , R sup , C where A is a finite and non-empty set of arguments, R att is a binary relation over A called the attack relation, R sup is a binary relation over A called the support relation and C is a finite set of subsets of A such that ∀(a, b) ∈ R att , {a, b} ∈ C.

Conflict-freeness in a generalized bipolar argumentation system is defined as follows:

Def. 13 (Conflict-freeness in a GBAS) Let A, R att , R sup , C be a GBAS and S ⊆ A. S is conflict-free in the GBAS iff there does not exist C ∈ C such that C ⊆ S.

However, the definition of semantics depends on the interpretation of the support and also on the constraints that have to be enforced. The generalized bipolar framework can be instantiated for encoding necessary support, due to the following definition:

Def. 14 Let BAS = A, R att , R sup with R sup being a set of necessary supports. The tuple GBAS = A, R att , R sup , C with C = {{a, b}|(a, b) ∈ R att } ∪ {{a, b}|aR att c and c supports b} ∪ {{a, b}|cR att b and c supports a} is the generalized argumentation system associated with BAS.

It is easy to see that the generalized argumentation system associated with BAS enables to enforce the constraints TRA and CFS, whereas it satisfies neither Constraint nATT, nor Constraint n+ATT.

The next step is the study of acceptability in a GBAS in order to check whether Contraint CLO is taken into account. For that purpose, the first proposal is to use conflict-freeness as defined in Def. 13 and admissible, preferred and stable extensions as defined in Dung's systems. In this case, it can be proved that every stable extension is closed under R -1 sup .

Prop. 1 Let BAS = A, R att , R sup and its associated GBAS. Let S ⊆ A. If S is conflict-free in GBAS, and for each a ∈ S, there is b ∈ S s.t. bR att a, then S is closed under R -1 sup . However, this approach produces many conflicts, without adding any attacks. So in many cases, there will be no stable extension. Moreover, Constraint CLO is generally not satisfied with the preferred semantics. The following example illustrates these two drawbacks.

Ex. 3 Consider BAS represented by the following graph. Using the classical definition of semantics with conflict-freeness as defined in Def. 13, the preferred extensions of the associated GBAS are {a, x} and {a, b}, and there is no stable extension. Moreover, the preferred extension {a, b} is not closed under R -1 sup .

The preferred semantics has to be redefined in order to enforce Constraint CLO. So, our second proposal is to enforce a notion of coherence by combining conflict-freeness and closure under R -1 sup . Moreover it can be proven that:

Prop. 2 Let A, R att , R sup and its associated GBAS. Let S ⊆ A. If S is closed under R -1 sup then (S is conflict-free in GBAS iff S is conflict-free in A, R att ).
Def. 15 (Coherence in a GBAS) Let A, R att , R sup , C be a GBAS and S ⊆ A. S is coherent in the GBAS iff S is conflict-free in A, R att and S is closed under R -1 sup . Using coherence in place of conflict-freeness leads to new definitions: Def. 16 (Admissibility in a GBAS) Let A, R att , R sup , C be a GBAS and S ⊆ A. S is admissible in the GBAS iff S is coherent in the GBAS and ∀a ∈ S, ∀b ∈ A s.t. bR att a, ∃c ∈ S s.t. cR att b. S is a preferred extension of the GBAS iff it is a maximal (wrt ⊆) admissible set. S is a stable extension of the GBAS iff S is coherent4 in the GBAS and for each a ∈ S, there is b ∈ S s.t. bR att a.

Ex.3 (cont'd)

Taking into account coherence, as in Def.16, {a, x} is the unique preferred extension of the associated GBAS, and it is closed under R -1 sup . So, using Def.16 and 15, the associated GBAS enables to enforce Constraint CLO. 5 Moreover, as in Dung's framework, stable extensions are also preferred.

Prop. 3 Let A, R att , R sup , C be a GBAS and S ⊆ A. If S is a stable extension of the GBAS then S is also a preferred extension of the GBAS.

A thorough study of the generalized bipolar abstract argumentation framework would demand to define other semantics such as grounded one. However, this is not our purpose in this paper. We focus on the way to enforce different kinds of constraints related to necessary support.

A meta-framework encoding necessary support

The fact that "the acceptance of c is necessary to get the acceptance of b" can be encoded in another way. As explained in Section 3, the idea is to assume the existence of a special argument attacking b for which c is the only attacker. More precisely, if cR sup b, we create a new argument N cb and two attacks cR att N cb and N cb R att b. As c is the unique attacker of N cb , "the acceptance of b implies the acceptance of c". The meaning of N cb could be that the support from c to b is not active. A similar idea can be found in [28; 12] for the more general purpose of representing recursive and defeasible attacks and supports.

Def. 17 Let

BAS = A, R att , R sup with R sup being a set of necessary sup- ports. Let A n = {N cb |(c, b) ∈ R sup } and R n = {(c, N cb )|(c, b) ∈ R sup } ∪ {(N cb , b)|(c, b) ∈ R sup }. The tuple MAS = A ∪ A n , R att ∪ R n is the meta- argumentation system 6 associated with BAS.
Let us check whether the minimal requirements are satisfied. Let us first consider constraint TRA. From aR sup b and bR sup c, we obtain the sequence of attacks aR att N ab R att bR att N bc R att c. So, the acceptance of c implies the acceptance of b, which in turn implies the acceptance of a, as if we had directly encoded aR sup c. So, TRA is taken into account. The same result holds for CLO:

Prop. 4 Let BAS = A, R att , R sup and its associated MAS. Let S ⊆ A ∪ A n . If S is admissible in MAS, then S ∩ A is closed under R -1
sup in BAS. Constraint CFS is not enforced. We only have the following property: Note that this result is weaker than CFS since it does not imply that {a, b} is a conflicting set.

Obviously, stronger constraints such as nATT or n+ATT are not directly enforced. If aR att c and cR sup b, we obtain the sequence aR att cR att N cb R att b. No attack from a to b is added. However, we will see in Section 4.4 that the metaargumentation framework associated with BAS enables to recover the extensions obtained when enforcing Constraint nATT.

A framework with complex attacks

In this subsection we discuss two frameworks enabling to handle necessary support through the addition of complex attacks. According to the various interpretations of the necessary support, all the complex attacks are not justified. For instance, if the necessary support models a subargument relation as in [START_REF] Prakken | On support relations in abstract argumentation as abstraction of inferential relations[END_REF], only the secondary attack makes sense. Other works [START_REF] Nouioua | Argumentation frameworks with necessities[END_REF] have considered both cases of extended attack. However, to the best of our knowledge, there has been no formal study of the properties of these extended attacks, and of the consequences of these attacks on the acceptable sets of arguments.

From Def. 6, new attacks called n+-attacks can be generated inductively as follows:

Def. 18 (n+-attacks) Let BAS = A, R att , R sup with R sup being a set of necessary supports. There exists a n+-attack from a to b iff either aR att b, or there is a (case 1 or case 2) extended attack from a to b, or there exists an argument c s.t. a n+-attacks c and c supports b, or there exists an argument c s.t. c supports a and c n+-attacks b. N + Rsup R att denoted the set of n+-attacks generated by R sup on R att . The AS defined by A, N + Rsup R att is denoted by AS N + . Obviously Constraints TRA, nATT and n+ATT are enforced in AS N + . Let us now consider the case when the extended attacks are restricted to secondary attacks (Case 1 of extended attacks). Following the above definition, our purpose is to define a n-attack from a to b when either aR att b, or there exists a secondary attack from a to b, or there exists an argument c s.t. a n-attacks c and c supports b. Indeed, it is easy to prove that the formal definition of this n-attack can be simplified as follows: Def. 19 (n-attack) Let BAS = A, R att , R sup . There is n-attack from a to b iff either aR att b, or there is a secondary attack from a to b. N Rsup R att denoted the set of n-attacks generated by R sup on R att . The AS defined by A, N Rsup R att is denoted by AS N .

Note that both AS N and AS N + are Dung's argumentation systems; so the classical notions given in Def. 2 and 3 can be applied without restriction, nor redefinition.

Obviously Constraints TRA and nATT are enforced in AS N , whereas Constraint n+ATT is not.

Def. 18 looks complex. However the following proposition enables to rewrite n+-attacks and n-attacks in a form which will be much easier to handle for studying their properties. anda sequence a 1 R sup . . . R sup a n with a n = a and n ≥ 1.

Prop. 6 Let BAS = A, R att , R sup . There is an n+-attack from a to b iff there is a sequence a 1 R att b 1 R sup . . . R sup b m , with b m = b and m ≥ 1,
n+-attacks as defined by Prop. 6 can be illustrated by the following figure:

a1 b1 . . . bm = b m ≥ 1 a2 . . . an = a n ≥ 1 /
Moreover, Prop. 6 can be used for identifying the following particular cases:

-The case when m = n = 1 corresponds to a direct attack from a to b.

-The case when n = 1 and m ≥ 1 corresponds to a n-attack from a to b (direct or secondary attacks, see Def. [START_REF] Martinez | An abstract argumentation framework with varied-strength attacks[END_REF]).

-The case when n = 1 and m > 1 corresponds to an extended attack -Case 1 (secondary attack) from a to b (see Def. 6).

-The case when n > 1 and m = 1 corresponds to an extended attack -Case 2 from a to b (see Def. [START_REF] Brewka | Abstract dialectical frameworks[END_REF]).

An obvious consequence of this proposition is:

Corol. 1 Let BAS = A, R att , R sup and its associated AS N and AS N + . Let

S ⊆ A. If S is conflict-free in AS N + , then S is conflict-free in AS N .
As said above, in some works necessary support can be handled by only considering n-attacks, that is by adding secondary attacks. However, although both cases of extended attacks are independent, we show that taking into account only n-attacks is already enough for inducing constraints on AS N + . Prop. 7 Let BAS = A, R att , R sup and its associated AS N . If a n+-attack from a to b can be built from BAS, there exists no admissible set in AS N containing {a, b}.

As an immediate consequence (contrapositive of Prop. 7), we have:

Corol. 2 Let BAS = A, R att , R sup and the associated AS N and AS N + . Let

S ⊆ A. If S is admissible in AS N , then S is conflict-free in AS N + .
Ex. 4 Consider BAS represented by the following graph:

c b a /
The associated AS N only contains the original attack from c to b (there is no secondary attack). If we consider only n-attacks, there is no conflict between a and b. However, it can be proved that no admissible set in AS N contains {a, b}.

The following results establish links between extensions in AS N and AS N + .

Prop. 8 Let BAS = A, R att , R sup and the associated AS N and AS N + . Let S ⊆ A. If S is admissible in AS N , then S is also admissible in AS N + . The converse of Prop 8 generally does not hold as shown by the following example.

Ex. 5 Consider BAS and its associated AS N and AS N + represented by the following graphs:

BAS AS N AS N + d c b a / / d c b a / / d c b a / / --
The set {a, b} is admissible in AS N + but is not admissible in AS N (since a does not attack c in AS N ).

However, the converse of Prop. 8 holds for maximal admissible sets:

Prop. 9 Let BAS = A, R att , R sup and its associated AS N and AS N + . Let S ⊆ A. S is maximal admissible in AS N + iff S is maximal admissible in AS N .

The same holds for stable semantics:

Prop. 10 Let BAS = A, R att , R sup and its associated AS N and AS N + . Let

S ⊆ A. S is stable in AS N + iff S is stable in AS N .
We conclude this section by providing results about the property of closure under the relation R -1 sup .

Prop. 11 Let BAS = A, R att , R sup and its associated AS N + . Let S ⊆ A and a, b ∈ A. If S is conflict-free in AS N + , a ∈ S and bR sup a, then S ∪ {b} is conflict-free in AS N + . If S is maximal (wrt ⊆) conflict-free in AS N + , then S is closed for the relation R -1 sup .

Prop. 11 does not hold when considering AS N instead of AS N + , as shown by the following example.

Ex.4 (cont'd) S = {a, b} is maximal conflict-free in AS N but it is not closed under R -1 sup . We have cR sup a but S ∪ {c} is not conflict-free in AS N . However, the property of closure under R -1 sup is recovered in AS N , if preferred (resp. stable) extensions are considered.

Prop. 12 Let BAS = A, R att , R sup and the associated AS N and AS N + . Let

S ⊆ A. If S is a preferred extension in AS N (resp. AS N + ), then S is closed for the relation R -1 sup . If S is stable in AS N (resp. AS N + ), then S is closed for the relation R -1 sup .
Due to Prop. 12, each stable (resp. preferred) extension of AS N (resp. AS N + ) is closed under R -1 sup . In that sense, Constraint CLO is enforced in AS N (resp. AS N + ).

It remains to consider Constraint CFS. This constraint is obviously satisfied by AS N + since a new attack is built for each conflict in the sense of CFS, whereas the Dung's argumentation system AS N does not capture all the conflicts induced by CFS, as illustrated by the following example.

Ex.3 (cont'd)

In the associated AS N , there is one n-attacks from x to c and one from x to b. {a, x} is the unique preferred extension of AS N . It is also stable. Note that {a, c} is conflict-free in AS N . Nevertheless {a, c} is a conflicting set in the sense of CFS.

Comparison between the different frameworks

In the previous sections, starting from a set of constraints, several frameworks (GBAS, MAS, AS N and AS N + ) have been proposed for handling necessary support. In this section, we compare these frameworks wrt two different points of view: the satisfaction of the constraints and the extensions that are produced.

First, the following table synthesizes the previous results:

GBAS MAS AS N AS N + TRA X X X X CLO X X X X CFS X - - X nATT - - X X n+ATT - - - X X (resp.
-) means that the corresponding property is (resp. not) satisfied in the corresponding framework.

Now, let us consider AS N and GBAS. We know that AS N does not satisfy CFS whereas GBAS does. However, due to Prop. 7, if S is a conflicting set of GBAS, is conflicting in AS N + and then there is no admissible set of AS N containing S. Moreover, it can be proved that each preferred extension of GBAS is (generally strictly) included in a preferred extension of AS N . This is illustrated by the following example. Prop. 13 Let BAS = A, R att , R sup and its associated GBAS and AS N . Let S ⊆ A. If S is admissible in GBAS, then S is also admissible in AS N . If S is a preferred extension in GBAS, then S is included in a preferred extension of AS N . If S is a stable extension in GBAS, then S is also a stable extension of AS N .

Ex

Note that Prop. 13 holds when considering AS N + instead of AS N , due to Prop. 8, 9 and 10.

The next issue concerns the comparison between AS N and the associated MAS of BAS. It seems that encoding a necessary support cR sup b by a metaargument N cb and the sequence aR att cR att N cb R att b is less strong than encoding n-attacks. However, there is a correspondence between the extensions which are obtained in each framework.

Prop. 14 Let BAS = A, R att , R sup and its associated MAS and AS N .

-Let S ⊆ A ∪ A n . If S is admissible in MAS, then S ∩ A is also admissible in AS N . If S is stable in MAS, then S ∩ A is also stable in AS N . -Let S ⊆ A. If S is a preferred extension in AS N , there exists S ′ admissible in MAS such that S = S ′ ∩ A. If S is a stable extension in AS N , then there exists S ′ stable in MAS such that S = S ′ ∩ A.

From Prop. 13 and 14, the following comparison between GBAS and MAS can be easily established.

Prop. 15 Let BAS = A, R att , R sup and its associated MAS and GBAS. Let S ⊆ A. If S is a preferred extension of GBAS, then there exists S ′ preferred in MAS such that S ⊆ S ′ ∩ A. If S is a stable extension of GBAS, then there exists S ′ stable in MAS such that S = S ′ ∩ A.

The following example illustrates the above propositions. In GBAS, the unique preferred (and also stable) extension is the set {a, x, e}. In AS N , the unique preferred (and also stable) extension is the set {a, x, e, d}. In MAS, the unique preferred also stable) extension is the set {a, x, e, N cb , d}.

Conclusion and future works

Recent studies in argumentation have addressed the notion of support, with several interpretations (such as deductive, evidential, necessary, backing) and several approaches developed independently. In this paper we focus on necessary support and show that the intended meaning of necessary support can induce different ways to handle it. Our main contribution is to propose an axiomatic approach that is helpful for understanding and comparing the different existing proposals for handling support. First, we have proposed different kinds of constraints that should be imposed to a bipolar argumentation system using necessary supports. Then we have studied different frameworks suitable for encoding these contraints. This paper reports a preliminary work that could be pursued along different lines. First, our study must be deepened in order to give a more high-level analysis and comparison of all these frameworks. Then the axiomatic approach could be enriched by considering other constraints, such as for instance the strong requirement leading to the addition of symmetric attacks in the case of a necessary support. Moreover, it would be interesting to define such an axiomatic for other interpretations of support, or to consider other frameworks which do not explicitely define a notion of support, such as Abstract Dialectical Frameworks [START_REF] Brewka | Abstract dialectical frameworks[END_REF]. Another direction for further research would be to study how to encode necessary (or other variants) support by the addition of attacks of various strengths (see for instance [19; 7; 16; 17]). Moreover it would be interesting to see the link between our approaches and the ranking semantics proposed by

|

  Necessary supportThe necessary support has been first proposed by[21; 22] with 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". A example of this kind of support could be:Ex. 2 A dialog between three customers about the qualities of services of their hotel:-"This hotel is very well managed." (Argument a) -"Yes. In particular, the hotel staff is very competent." (Argument b) -"They are not competent! The rooms are dirty." (Argument c) Here b necessarily supports a and c attacks b (c → b → a). The link between b

/

  Duality between deductive and necessary support Deductive support and necessary support have been introduced independently. Nevertheless, they correspond to dual interpretations of the notion of support. Let us denote a D → b (resp. a N → b) when there exists a deductive (resp. necessary) support from a to b. As a D → b means that "the acceptance of a implies the acceptance of b", and a N → b means that "the acceptance of a is necessary to get the acceptance of b", it follows that a N → b is equivalent to b D → a. Following this duality, it is easy to see that the mediated attack obtained by combining the attack relation R att and the support relation R sup exactly corresponds to the secondary attack 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 }). Similarly, the supported attack obtained by combining the attack relation R att and the support relation R sup exactly corresponds to the second case of extended attack obtained by combining the attack relation R att and the support relation R -1

Def. 9 (

 9 Constraint CFS) ∀a, b, c ∈ A. If (aR att c and c supports b) or (cR att b and c supports a) then {a, b} is a conflicting set. Note that the Dung's abstract framework is not suitable for expressing such a constraint. So we will present in Section 4.1 a new framework for handling conflicting sets of arguments. Addition of new attacks (nATT and n+ATT) Beyond these properties, according to the applications and the previous works presented in literature, we may impose stronger constraints corresponding to the addition of new attacks. Two cases may be considered: Def. 10 (Constraint nATT) If aR att c and cR sup b, then there is a new attack from a to b. Def. 11 (Constraint n+ATT) If (aR att c and cR sup b) or (cR att b and cR sup a), then there is a new attack from a to b.

C

  = {{x, c}, {x, b}, {a, c}}.

Prop. 5

 5 Let BAS = A, R att , R sup and its associated MAS. Let a, b, c be arguments of A. If (aR att c and c supports b) or (cR att b and c supports a) then no admissible set in MAS contains {a, b}.

Ex. 6 (

 6 cont'd) Consider the associated MAS represented by:

  .[START_REF] Brewka | Abstract dialectical frameworks[END_REF] Consider BAS represented by:In the associated GBAS, we have C = {{x, c}, {x, b}, {c, e}, {b, d}, {a, c}, {b, e}}. The unique preferred extension of GBAS is {a, x, e}. In the associated AS N , the n-attacks from x to b is used for ensuring the acceptability of d wrt {a, x, e}. So, the unique preferred extension is {a, d, x, e}.

	x	/	c	/	e
	a		b	/	d

This is an abuse of language since, stricly speaking, this is an edge-labeled graph (with two labels) rather than a directed graph.

Irreflexivity has also been considered for instance in[21; 22].

Note that if cRsupb and cR att b, as an extension must be conflict-free, there is no extension containing both c and b, so the constraint trivially holds. Some works, as for instance[START_REF] Cayrol | Bipolarity in argumentation graphs: towards a better understanding[END_REF], exclude the case when cRsupb and cR att b.

Due to Prop.1, coherent may be replaced by conflict-free.

Note that enforcing coherence makes the set C useless due to Prop.2.

Note that it is an argumentation system in Dung's sense.


C Cayrol
email: ccayrol@irit.fr

M.-C Lagasquie-Schiex

Merging argumentation systems with weighted argumentation systems: a preliminary study

Merging argumentation systems with weighted argumentation systems: a preliminary study (SUM 2011)

Introduction

Argumentation has become an influential approach in Artificial Intelligence to model cognitive tasks such as inconsistency handling and defeasible reasoning (e.g. [START_REF] Krause | A logic of argumentation for reasoning under uncertainty[END_REF][START_REF] Prakken | Logics for defeasible argumentation[END_REF][START_REF] Amgoud | Inferring from inconsistency in preference-based argumentation frameworks[END_REF][START_REF] Verheij | Deflog: on the logical interpretation of prima facie justified assumptions[END_REF]), decision making (e.g. [START_REF] Fox | On using arguments for reasoning about values and actions[END_REF][START_REF] Gordon | The zeno argumentation framework[END_REF][START_REF] Karacapilidis | Computer supported argumentation and collaborative decision making: the hermes system[END_REF][START_REF] Kakas | Argumentation based decision making for autonomous agents[END_REF]), or negotiation between agents (e.g. [START_REF] Parsons | Agents that reason and negociate by arguing[END_REF][START_REF] Amgoud | Arguments, Dialogue and Negotiation[END_REF][START_REF] Amgoud | A unified and general framework dor argumentation-based negotiation[END_REF]). Argumentation is based on the evaluation of conflicting arguments. Most of the argumentation-based proposals are based on the abstract system proposed by Dung [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF] (a set of arguments and a binary relation capturing the conflicts between arguments). Among them we are interested in a framework proposed for the merging of argumentation systems. Indeed in [START_REF] Coste-Marquis | On the merging of Dung's argumentation systems[END_REF], a multi-agent setting has been considered in order to define argumentation systems for a group of agents from their individual argumentation systems. This amounts to make precise the set of arguments and the attack relation for the group. [START_REF] Coste-Marquis | On the merging of Dung's argumentation systems[END_REF] has proposed a two-step merging process which first expands each agent's argumentation system for taking into account all the arguments (1), then computes a set of argumentation systems which are as close as possible to these expansions [START_REF] Amgoud | A unified and general framework dor argumentation-based negotiation[END_REF]. For the first step, [START_REF] Coste-Marquis | On the merging of Dung's argumentation systems[END_REF] has introduced a new kind of interaction between arguments, the ignorance relation, reflecting that an agent cannot conclude from the other agents that there is or not an attack between two arguments. However, [START_REF] Coste-Marquis | On the merging of Dung's argumentation systems[END_REF]'s approach suffers from two important drawbacks: computing the output of the merging process is expensive and in the general case the merging process does not provide a single argumentation system, which complicates the definition of acceptability of (sets of) arguments for the group.

We propose a novel approach to the merging of argumentation systems which overcomes these limitations. Our idea is to refine the notion of ignorance and to replace it by a weighted attack. Recent works have proposed extensions of Dung's argumentation system capable of handling varied-strength attacks ( [START_REF] Martinez | On defense strength of blocking defeaters in admissible sets[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]). The idea is that attacks may have different strength and can be compared according to their relative strength. Our purpose is to take advantage of these works, in order to define a merging process such that the resulting system is a unique weighted argumentation system, easy to compute and to use.

In Section 2, we give the relevant background. Then, in Section 3, we propose a new approach for merging argumentation system into a weighted argumentation system (details and proofs are given in [START_REF] Cayrol | Merging argumentation systems with weighted argumentation systems: a preliminary study[END_REF]). Section 4 concludes the paper.

Background

We first focus on Dung's theory of argumentation [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF] in which only one interaction between arguments is taken into account.

Def 1 An abstract argumentation system AS = A, R over A is given by a finite set A of arguments and a binary relation R on A called an attack relation. Consider a i and a j ∈ A. a i Ra j means that a i attacks a j A, R defines a directed graph whose nodes are arguments and edges correspond to the attack relation. Whether a set of arguments can be accepted depends on the way arguments interact within the set but also w.r.t. the other arguments of A. Collective acceptability is based on two key notions, conflictfreeness and collective defence, and leads to several semantics in [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF].

Def 2 Let A, R be an argumentation system. Let E ⊆ A. Let a ∈ A. E is conflict-free iff (if and only if ) ∄a, b ∈ E such that aRb. E (collectively) defends a iff ∀b ∈ A, if bRa, ∃c ∈ E such that cRb. E is admissible iff E is conflict-free and ∀a ∈ E, E collectively defends a. E is a preferred extension iff E is maximal for set inclusion among the admissible sets.
Then we consider a multi-agent setting where each agent has its own argumentation system. We cannot assume that one agent knows all the arguments which are known by the other agents. Moreover, the agents may disagree on the interactions between arguments. This problem already occurs in argumentationbased dialogue. The issue addressed in [START_REF] Coste-Marquis | On the merging of Dung's argumentation systems[END_REF] concerns the combination of the different argumentation systems in order to achieve a collective decision about which arguments should be accepted. Given P = AS 1 , . . . , AS n a profile of n argumentation systems, two steps are needed:

expansion: Each AS i = A i , R i is expanded over P into PAS i (partial argumentation system) for taking into account the knowledge coming from the other agents. PAS i is built over A = ∪ i A i and contains two kinds of interaction: attack and ignorance. -fusion: The AS ′ j over A that are selected as the result of the merging process are the ones that best represent P ′ = PAS 1 , . . . , PAS n (i.e., that are the "closest" to P ′ w.r.t. a given distance). This proposal can be illustrated by the following example: For each i, the expansion PAS i of AS i over P is given by: Nevertheless, this proposal is computationally expensive and may produce several argumentation systems. Moreover, it does not enable to take into account the number of agents who agree with an attack when computing the expansion. Our proposal enables to graduate the interaction between two arguments by means of a weighted attack relation.

PAS 1 PAS 2 PAS 3 PAS 4
[8] has proposed weighted argument systems, in which attacks are associated with a numeric weight, indicating how reluctant one would be to disregard the attack. Here, we are interested in varied-strength attacks considering that the relative strength of the attacks enables to disregard some of the defences as in [START_REF] Martinez | On defense strength of blocking defeaters in admissible sets[END_REF][START_REF] Martinez | Strong and weak forms of abstract argument defense[END_REF][START_REF] Martinez | An abstract argumentation framework with varied-strength attacks[END_REF][START_REF] Cayrol | Acceptability semantics accounting for strength of attacks in argumentation[END_REF]. Def 3 ([8] Weighted argumentation system) A weighted argumentation system over A is a triple A, R, w , denoted by WAS, where A is a finite set of arguments, R is an attack relation on A and w is a function R → R assigning real valued weights to attacks.

A weighted argumentation system can be considered as a particular case of the argumentation system with varied-strength attacks of [START_REF] Martinez | An abstract argumentation framework with varied-strength attacks[END_REF][START_REF] Cayrol | Acceptability semantics accounting for strength of attacks in argumentation[END_REF], using the natural ordering ≥ on R. Then, considering that greater is the weight w(a, b), stronger is the attack from a to b, the notions of vs-defence and vs-admissibility proposed in [START_REF] Cayrol | Acceptability semantics accounting for strength of attacks in argumentation[END_REF] Let E ⊆ A. E is conflict-free in WAS iff ∄a, b ∈ E such that bRa. E vsdefends a iff ∀b ∈ A, if bRa then ∃c ∈ E such that c vs-defends a against b. E is vs-admissible iff E is conflict-free and ∀a ∈ E, E vs-defends a.

Merging of argumentation systems into a WAS

We propose to merge a profile P = AS 1 , . . . , AS n (with AS i = A i , R i ) into a WAS such that the weight on an attack in the WAS represents the strength of the opinions of the group of agents concerning this attack. We use a global strategy considering each agent as a voting person and adding the votes for and against each interaction.

Def 5 (Merging of n ASs) The merging of P is the WAS defined by WAS = A, R, w with:

A = n i=1 A i , R = n i=1 R i and w : R →]0, +1] defined by ∀(a, b) ∈ R, w(a, b) = Σ n i=1 wi(a,b) n(a,b)
with

-w i (a, b) = 1 if (a, b) ∈ R i , w i (a, b) = 0 if (a, b) ∈ R i -and n(a, b) is the number of AS i such that a ∈ A i and b ∈ A i .
Note that the merging of a profile of ASs results in a unique WAS. And this process can be done in linear time which is not the case of the merging process proposed in [START_REF] Coste-Marquis | On the merging of Dung's argumentation systems[END_REF]. As for Dung's system, a WAS can be represented by a directed graph (nodes are arguments and weighted edges represent attacks). A missing edge has two possible interpretations: either all the agents who know both arguments agree on the non-attack, or none agent knows both arguments.

Ex 1 (cont'd) Merging the profile of ASs given in Ex. 1 produces the following WAS:

Using the notion of vs-defence, it follows that a does not vs-defend c. This conclusion is not really surprising because among the three agents who know a and b only two agree on the attack from a to b. Then, we obtain two maximal (for ⊆) vs-admissible sets {a} and {b}. Note that, in the case when there is no conflict between the agents, the WAS corresponding to the merging of the ASs is an AS (all the weights are equal to 1). A similar result has been given in [START_REF] Coste-Marquis | On the merging of Dung's argumentation systems[END_REF]. Some interesting properties enable to compare the pieces of information contained in WAS and in the initial AS i . Firstly, the arguments that are unattacked in each A i remain unattacked in the resulting WAS. Secondly, the resulting WAS reflects the unanimity between the agents about attacks: if all the agents agree about an attack, this attack is in WAS with a weight equal to 1 (a simular result exists for the non-attacks). Consequently, situations when agents disagree can also be characterized, by the existence in WAS of an attack whose weight is not equal to 1. Moreover, the pieces of information contained in WAS can be also compared with the result of the merging process as defined in [START_REF] Coste-Marquis | On the merging of Dung's argumentation systems[END_REF]: an attack weighted by 1 in WAS corresponds to an attack in each PAS i and each resulting AS ′ i (the converses do not hold). A similar result exists for the non-attacks. The behaviour in case of unanimity shows that the merging preserves the information on which the agents do not disagree. This result is not surprising since the characteristic feature of our approach is that the merging occurs at the interaction level. However, we have no such result at the output of the argumentation systems; it may happen that an argument a is accepted1 by all the agents who know it but a is not accepted in the WAS: Ex 2 Consider three agents with their respective ASs:

AS 1 AS 2 AS 3 y 1 s 1 a s 2 y 1 s 1 a y 2 s 2 s 1 a y 2 s 2
The set {a, s 1 , s 2 } is admissible in each AS i . The merging gives the following WAS:

y 1 s 1 a y 2 s 2 1
0.5 0.5 0.5 0.5 After the merging, no vs-defence exists for a against y 1 . So a cannot belong to a vs-admissible set. This result is not surprising: the attack from y 1 to a is known in AS 1 and AS 2 in which s 1 and s 2 are also known; but AS 1 and AS 2 do not agree on the attack from s 1 to y 1 and on the attack from s 2 to y 1 . So the group cannot consider that s 1 or s 2 are good vs-defenders for a against y 1 .

Note that this behaviour also appears with the merging process proposed in [START_REF] Coste-Marquis | On the merging of Dung's argumentation systems[END_REF].

Conclusion

In this paper, we have proposed a new method for merging different argumentation systems in a multi-agent setting. This approach is particularly interesting from a computational point of view because it is a linear process. It is easy to apply and exactly reflects the impact of each agent on the group. It also corrects some disadvantages of the method proposed in [START_REF] Coste-Marquis | On the merging of Dung's argumentation systems[END_REF] (this is, to our knowledge, the only existing method for merging argumentation systems) since it provides only a single output weighted argumentation system. There are several directions for future works. The first one will be to consider a reliability level for each agent and to take it into account for computing the weights of the interactions for the group. Another extension of this work will address the merging of bipolar argumentation systems, which are systems capable of handling two kinds of interaction, support and attack. For that purpose, a natural idea is to extend the weighted argumentation systems so as to handle positive and negative weights. A positive (resp. negative) weight will apply to a support (resp. an attack). That will lead to the study of bipolar weighted argumentation systems.

Ex 1

 1 Consider the profile P = AS 1 , AS 2 , AS 3 , AS 4 :

1 AS ′ 2

 12 The ignorance relation is represented by dotted arrows.AS ′Then, using a usual distance (the edit distance) and the sum aggregation function, the merging process proposed in[START_REF] Coste-Marquis | On the merging of Dung's argumentation systems[END_REF] produces the two following ASs:

become: Def 4

 4 Let WAS = A, R, w . Let a, b, c ∈ A such that cRb and bRa. c vsdefends a against b (or c is a vs-defender of a against b) iff w(c, b) ≥ w(b, a) (i.e. the attack from b to a is not strictly stronger than the one from c to b).

For instance because it belongs to all preferred extensions.


Claudette Cayrol
email: ccayrol@irit.fr

Marie-Christine Lagasquie-Schiex

Coalitions of arguments in bipolar argumentation frameworks

Bipolar argumentation frameworks enable to represent two kinds of interaction between arguments: support and conflict. In this paper, we turn a bipolar argumentation framework into a "meta-argumentation" framework where conflicts occur between sets of arguments, characterized as coalitions of supporting arguments. So, Dung's well-known semantics can be used on this metaargumentation framework in order to select the acceptable arguments.

Introduction

Argumentation has become an influential approach to model defeasible reasoning and dialogues between agents, based on the exchange of interacting arguments (see e.g. [Krause et al., 1995;[START_REF] Prakken | [END_REF]Amgoud et al., 2000;Karacapilidis and Papadias, 2001;Verheij, 2003]).

The following illustrative example presents the arguments exchanged during the meeting of the editorial board of a newspaper.

Ex. 1

Arg. a: If we have the agreement, the important information I on the person X must be published.

Arg. b: I is a private information and X does not agree for publication.

Arg. c1: Yes, I concerns X's family.

Arg. c2: Moreover, X is a private person.

Arg. d: No, X is the new prime minister.

In most existing systems, the interaction takes the form of a conflict, usually called attack. For example, an argument can be a pair set of assumptions, conclusion , where the set of assumptions entails the conclusion according to some logical inference schema. Then, a conflict occurs between two arguments, for instance if the conclusion of one of them contradicts an assumption of the other one. In Ex. 1, b is in conflict with a. The main issue of any argumentation system is the selection of acceptable sets of arguments, based on the way arguments interact. Intuitively, an acceptable set of arguments must be in some sense coherent and strong enough (e.g. able to defend itself against all attacking arguments). The concept of acceptability has been explored through the use of argumentation frameworks, and one which is especially fruitful is Dung's argumentation framework [Dung, 1995], abstracting from the nature of the arguments. In such an abstract framework, from a set of arguments and a binary "attacks" relation, different semantics for acceptability are proposed, each one being characterized by several requirements that a set of arguments must satisfy so that the set could be selected. These selected sets of arguments are called extensions.

Recent work on argumentation [Karacapilidis and Papadias, 2001;Verheij, 2003;Cayrol and Lagasquie-Schiex, 2005a;2005b] has advocated the representation of another kind of basic interaction between arguments. Indeed, it can be the case in a dialogue that an agent advances an argument which confirms an assumption used by an argument provided by another agent. This kind of interaction is not captured by the notion of defence. It is rather a kind of support. In Example 1, we may consider that the argument c 1 given by an agent supports the argument b given by another agent. It is not only a "dialog-like speech act": a new piece of information is really given and it is given after the production of the argument b. So taking c 1 into account leads either to modify b, or to find a more intuitive solution for representing the interaction between c 1 and b1 . In [Cayrol and Lagasquie-Schiex, 2005b], Dung's framework has been extended to cope with both kinds of interaction, into a so-called bipolar abstract argumentation framework. Bipolarity refers to the existence of two independent kinds of information which represent repellent forces 2 . Semantics for acceptability have been defined, based on more complex notions of attack, called the supported and the diverted attacks. In Ex. 1, the fact that c 1 supports an attacker of a may be considered as a supported attack on a by c 1 , and the fact that d attacks a supporter of b may be considered as a diverted attack on b by d. The new semantics ensure that neither supported attacks, nor diverted attacks can occur within an extension.

However, the definitions are rather complex and choosing the corresponding extensions, then computing them seems hard. So, in this paper, our purpose is to propose more intuitive semantics, using Dung's methodology in a "metaargumentation" framework, where conflicts occur between coalitions of supporting arguments. Our motivation is the possible reusing of principles, algorithms and properties of Dung's well-known framework, now eponymously known by his name. So, our approach consists in: identifying "meta-arguments" (coalitions): conflict-free sets of arguments which are somehow related by the support relation; then defining the "meta-attack" relation: attack relation between the coalitions; finally using Dung's semantics on this "metaargumentation" framework in order to identify acceptable sets of arguments (arguments which can be chosen together without conflict) for a bipolar argumentation framework.

Our contribution is twofold: first, we provide a formal method for defining and forming coalitions on the basis of interacting components; then, we propose new semantics for characterizing acceptable sets of arguments in a bipolar framework, taking advantage of Dung's definitions.

The paper is organized as follows: Dung's approach and the bipolar framework are described in Sect. 2; in Sect. 3, coalitions of arguments are introduced and they are used in Sect. 4 as interacting components of a meta-argumentation framework; then, Sect. 5 presents some related works and perspectives.

Background Dung's framework

Let us present some basic definitions at work in Dung's theory of argumentation [Dung, 1995].

Def. 1 A finite argumentation framework is a pair A, R

where A is a finite and non-empty set of so-called arguments and R is a binary relation over A (a subset of A × A), the attacks relation.

An argumentation framework can be represented by a directed graph in which each argument is a vertex and the edges are defined by the attacks relation: ∀a, b ∈ A, aRb is represented by a → b3 .

The first important notions are the notion of acceptability and the notion of conflict. Requiring the absence of conflicts and the form of autonomy captured by the notion of self-acceptability (S acceptable w.r.t. S) leads to the notion of admissible set.

Def. 2 Let

Def. 3 S ⊆ A is admissible for A, R iff S is conflict-free for A, R and acceptable w.r.t. S.

Every extension of an argumentation framework under the standard semantics introduced by Dung (preferred, stable) is an admissible set, satisfying some form of optimality.

Def. 4 Let

S ⊆ A. S is a preferred extension of A, R iff it is maximal w.r.t. ⊆ among the admissible sets for A, R . S is a stable extension of A, R iff it is conflict-free for A, R and ∀a ∈ A \ S 5 , ∃b ∈ S s.t. bRa.
For each admissible set S, there exists a preferred extension which contains S. Each stable extension is a preferred extension, the converse is false (for instance when A = {a} and R att = {(a, a)}). The proof of these properties is given in [Dung, 1995].

Bipolar argumentation framework

As already said, arguments may be conflicting. These conflicts are captured by the attacks relation in an argumentation framework, and may be considered as negative interactions. Then, the concept of defence has been introduced in order to reinstate some of the attacked arguments, namely those whose attackers are in turn attacked. So, most logical theories of argumentation assume that if an argument a 3 defends an argument a 1 against an argument a 2 (a 3 attacks a 2 which attacks a 1 ), then a 3 is a kind of support for a 1 , so a positive interaction. It holds in Dung's framework: only negative interaction is explicitly represented by the attacks relation, and positive interaction is implicitly represented through the notion of defence. In this case, support and attack are dependent notions. It is a parsimonious strategy, but it is not a correct description of the process of argumentation in realistic examples6 : in Ex. 1, the link between the argument c 1 and the other arguments cannot be expressed with the attacks relation. So, a more complex argumentation framework is needed in order to formalize situations where two independent kinds of interaction are available: a positive and a negative one. Following [Karacapilidis and Papadias, 2001;Verheij, 2003], [Cayrol and Lagasquie-Schiex, 2005b;2005a] propose a bipolar argumentation framework. This new framework is an extension of Dung's basic framework in which a new kind of interaction between arguments is represented by the supports relation7 . The fact that c 1 supports an attacker of a may be considered as a kind of negative interaction between c 1 and a, which is however weaker than a direct attack. In the same way, the attack by d of a supporter of b may also be considered as a negative interaction between d and b. From a cautious point of view, such arguments cannot appear together in a same extension. In order to address this problem, [Cayrol and Lagasquie-Schiex, 2005b;2005a] introduce new kinds of attack.

Def. 5 A finite bipolar argumentation framework

Def. 6 Let a, b ∈ A

A supported attack for b by a is a sequence a 1 R 1 . . . R n-1 a n , n ≥ 3, with a 1 = a, a n = b, s.t. ∀i = 1 . . . n -2, R i = R sup and R n-1 = R att . A diverted attack for b by a is a sequence a 1 R 1 . . . R n-1 a n , n ≥ 3, with a 1 = a, a n = b, s.t. R 1 = R att and ∀i = 2 . . . n -1, R i = R sup .
In Ex. 1, there are a supported attack for a by c 1 and a diverted attack for b by d.

Combining different notions of conflict (one for each type of attack) with Dung's notion of acceptability, [Cayrol and Lagasquie-Schiex, 2005b] propose various bipolar semantics. However, two problems deserve further investigation: computational issues and the choice of the appropriate semantics depending on the application. For example, with a very simple example,

A = {a, b, c}, R att = {(c, b)}, R sup = {(a, b)},
the three acceptability semantics proposed in [Cayrol and Lagasquie-Schiex, 2005b] give three different results ({a, c} for the d-preferred semantics, {a} and {c} the s-preferred semantics, {c} for the c-preferred semantics).

We propose a more intuitive (and promising) approach for handling bipolarity: the support relation is first taken into account to form coalitions of arguments, which will interact with a conflict relation in a simple Dung-like framework.

Coalitions

Principles which govern the definition of a coalition are the following: each argument belongs to at least one coalition; a coalition satisfies a coherence requirement; if two arguments belong to a same coalition, they are somehow related by the support relation.

Let BAF = A, R att , R sup be a bipolar argumentation framework represented by the graph G b . G sup will denote the partial graph representing the partial framework A, R sup (see [Berge, 1973] for a background on graph theory). AF will denote the partial argumentation framework A, R att associated with BAF.

Def. 7 C ⊆ A is a coalition of BAF iff (i) The subgraph of G sup induced by C is connected; (ii) C is conflict-free for AF;
(iii) C is maximal (for ⊆) among the sets satisfying (i) and

(ii).

Note that when R att is empty, the coalitions are exactly the connected components of the partial graph G sup .

Prop. 1 Each argument which is not self-attacking belongs to at least one coalition.

Proof: Let a be an argument which is not self-attacking. Then {a} is conflict-free and the subgraph of Gsup induced by {a} is connected. So satisfies the conditions (i) and (ii) of Definition 7. Either {a} is a coalition, or there exists C a subset of A containing a and satisfying the conditions (i) and (ii). Since A is finite, there exists a coalition containing C and thus containing ✷ Ex. 2 Let BAF 2 be represented by:

b d i a c e f g h
The coalitions are:

C 1 = {b, c, d}, C 2 = {i}, C 3 = {a, b}, C 4 = {e, f, g, h}.
Definition 7 respects the desired principles but is not constructive. So an equivalent definition is proposed using the notions of maximal support path and of coalition in a set. The idea is to rely upon the connected components of the partial graph G sup .

Def. 8 Let

S ⊆ A. M = {a 1 , . . . , a n } ⊆ S is a maximal support path in S iff (i) there exists a permutation {i 1 , . . . , i n } of {1, . . . , n} s.t.
the sequence of supports a i1 R sup a i2 . . . R sup a in holds (ii) and M is maximal (for ⊆) among the subsets of S satisfying (i).

In Ex. 1, {c 1 , b} and {c 2 , b} are maximal support paths in A 1 but not coalitions (they do not respect the condition (iii)). In Ex. 2, {a, b, c, d} {e, f, g, h} and {e, i} are maximal support paths in A 2 but not coalitions (the first one and the third one are not conflict-free for AF 2 ).

Def. 9 C ⊆ S is a coalition in S iff there exists {M 1 , . . . , M p } a maximal (for ⊆) set of non-disjoint maximal support paths in S s.t.

C = M 1 ∪ M 2 . . . ∪ M p .
It can be proved that: ⇐ Consider that C is a connected component of the subgraph of Gsup induced by S. Let a be an element of C. C is also the set of all the elements b in S such that there exists a chain of supports between a and b. Let M1 be a maximal support path containing a, and C ′ be the union of all the maximal support paths which are non-disjoint from M1. From the first part of the proof, C ′ is a maximal connected subset of S in the subgraph of Gsup induced by S, and

Prop. 2 C is a coalition in S iff C is a connected component 9 of
C ′ is included in C. Since C is maximal connected, we have C = C ′ . So, C is a coalition in S. ✷ Prop. 3 C ⊆ A is a coalition of BAF iff (i) there exists S ⊆ A maximal (for ⊆) conflict-free for AF s.t. C is a coalition in S and
(ii) C is maximal (for ⊆) among the subsets of A satisfying (i).

9 C is a connected component of G iff (i) ∀v1, v2 ∈ C, if v1 = v2, ∃ a chain from v1 to v2 in G and (ii) C is maximal in G w.r.t. ⊆ for (i).
Proof: ⇒ Let C be a coalition of BAF. C is conflict-free for AF, so there exists S maximal conflict-free for AF containing C. If the subgraph of Gsup induced by S is connected, C = S. If it is not the case, it is sufficient to prove that C a connected component of the subgraph of Gsup induced by S (due to Proposition 2). Since C is a coalition, C is connected in this subgraph. If C is not a connected component, there exists C ′ connected in S, which strictly contains C. So, we obtain C ′ conflict-free whose induced subgraph is connected. That is in contradiction with the maximality of C. So, C is a coalition in S, which proves the first item. The second item follows from the definition of a coalition.

⇐ Assume that C is a coalition in S, maximal conflictfree for AF, and that there does not exist S ′ maximal conflict-free for AF and C ′ coalition in S ′ such that C ′ strictly contains C. Obviously, C is conflict-free for AF and the subgraph of Gsup induced by C is connected. It remains to prove that C is maximal (condition (iii) of Definition 7).

that C is strictly included in D connected and conflict-free. There exists S ′ maximal conflict-free for AF containing D. Since D is connected, there exists C ′ a connected component of S ′ containing D. So, C is strictly included in C ′ , coalition of S ′ . That is in contradiction with our assumption on C. So, C is a coalition of BAF. ✷

Prop. 2 and 3 suggest a procedure for computing the coalitions of BAF:

Step 1: Consider AF and determine the maximal conflict-free sets for AF.

Step 2: For each set of arguments Si obtained at Step 1, determine the connected components of the subgraph of Gsup induced by Si.

Step 3: Keep the maximal (for ⊆) sets obtained at

Step 2.

It can be proved that: It can be proved that:

Prop. 5 Let C 1 and C 2 be two distinct coalitions of BAF. If C 1 and C 2 are non-disjoint then C 1 c-attacks C 2 or C 2 c-attacks C 1 .
Proof: The subgraph of Gsup induced by C1 (resp. C2) is connected. Since C1 and C2 are non-disjoint, the subgraph induced by their union is connected. C1 and C2 are distinct, so their union strictly contains C1 (and also C2).

As a coalition is maximal connected conflict-free, C1 ∪C2 cannot be conflict-free. But C1 and C2 are conflict-free. So there exists a1 in C1 and a2 in C2 such that a1Ratta2 (C1 c-attacks C2) or a2Ratta1 (C2 c-attacks C1). ✷

So we define a new argumentation framework CAF = C(A), c-attacks , referred to as the coalition framework associated with BAF. Dung's definitions apply to CAF, and it can be proved that:

Prop. 6 Let {C 1 , . . . , C p } be a finite set of distinct coalitions. {C 1 . . . , C p } is conflict-free for CAF iff C 1 ∪ . . . ∪ C p is conflict-free for AF.
Proof: Assume that {C1, . . . , Cp} is not conflict-free for CAF . There exists 1 ≤ i ≤ p and 1 ≤ j ≤ p such that Ci c-attacks Cj. So there exists ai in Ci aj in Cj such that aiRattaj. Then, C1 ∪ . . . ∪ Cp is not conflict-free for AF. Conversely, assume that C1 ∪ . . . ∪ Cp is not conflict-free for AF. There exists a and b in C1 ∪ . . . ∪ Cp such that aRattb. Each Ci is conflict-free (it is a coalition), so a and b do not belong to a same Ci. There exist i = j with a belongs to Ci and b belongs to Cj. Hence, we have Ci c-attacks Cj and the set {C1, . . . , Cp} is not conflict-free for CAF. ✷ However, even if the set of coalitions {C 1 , . . . , C p is conflict-free for CAF, there may exist a supported or a diverted attack in C 1 ∪ . . . ∪ C p .

Ex. 2 (cont'd)

For CAF 2 , {C 1 , C 2 } is conflict-free. However, there is a diverted attack for i by c in BAF 2 . {C 3 , C 4 } is conflict-free for CAF 2 . However, there is a supported attack for e by b in BAF 2 . So, we have a "meta-argumentation" framework (CAF) with a set of "meta-arguments" (the set of coalitions C(A)) and a "meta-attack" relation on these coalitions (the c-attacks relation). A coalition gathers arguments which are close in some sense and can be advanced together. However, as coalitions may conflict, following Dung's methodology, we can compute preferred and stable extensions of CAF. Such an extension contains coalitions which are collectively acceptable. The last step consists in gathering the elements of the different coalitions of an extension. By this way, we are able to select the best groups of arguments (w.r.t. the given interaction relations).

Def. 11 Let

S ⊆ A. S is a cp-extension 10 of BAF iff there exists {C 1 , . . . , C p } a preferred extension of CAF s.t. S = C 1 ∪ . . . ∪ C p . S is a cs-extension 11 of BAF iff there exists {C 1 , . . . , C p } a stable extension of CAF s.t. S = C 1 ∪. . .∪C p .
10 cp means coalition-preferred. 11 cs means coalition-stable.

When the only preferred extension of CAF is the empty set, we define the empty set as the unique cp-extension of BAF.

Ex. 2 (cont'd)

There is only one preferred extension of CAF 2 , which is also stable: {C 1 , C 2 }. So, S = {b, c, d, i} is the cpextension (and also the cs-extension) of BAF 2 . Some nice properties of Dung's classical framework are preserved:

A BAF has always a (at least one) cp-extension. It is a consequence of Def. 11.

In contrast, there does not always exist a cs-extension of BAF. The reason is that there may be no stable extension of CAF.

Each cs-extension is also a cp-extension. The converse is false.

There cannot exist two cp-extensions s.t. one strictly contains the other one. It follows from Def. 7 and 11.

However, other properties are lost. The following example shows that a cp-extension is not always admissible for AF, and a cs-extension is not always a stable extension of AF.

Ex. 3 Let BAF 3 be represented by:

a b c d e
The coalitions are:

C 1 = {a}, C 2 = {b, c, d}, C 3 = {e}.
There is only one preferred extension of CAF 3 , which is also stable: {C 1 , C 3 }. So, S = {a, e} is the cp-extension (and also the cs-extension) of BAF 3 . We have dR att e, but a does not defend e against d (neither by a direct attack, nor by a diverted or a supported attack). Indeed, a attacks an element of the coalition which attacks e. So, S is not admissible for AF 3 . S does not contain c, but there is no attack (no supported attack and no diverted attack) of an element of S against c. So, S is not a stable extension of AF 3 .

A coalition is considered as a whole and its members cannot be used separately in an attack process.

Ex. 3 suggests that admissibility is lost due to the size of the coalition {b, c, d}, and that it would be more fruitful to consider two independent coalitions {c, b} and {c, d}. However, a new formalization of coalitions in terms of conflictfree maximal support paths does not enable to recover Dung's properties as shown below.

Def. 12 An elementary coalition of BAF is a subset EC = {a 1 , . . . , a n } of A s.t.:

(i) there exists a permutation {i 1 , . . . , i n } of {1, . . . , n} s.t.

the sequence of supports a i1 R sup a i2 . . . R sup a in holds;

(ii) EC is conflict-free for AF;

(iii) EC is maximal (for ⊆) among the subsets of A satisfying (i) and (ii).

EC(A) denotes the set of elementary coalitions of BAF. Consider ECAF = EC(A), c-attacks the elementary coalition framework associated with BAF. Note that the lost of admissibility in Dung's sense is neither surprising, nor problematic for us: admissibility is lost because it takes into account "individual" attack and defence (d is not defended against b); whereas, with meta-argumentation and coalitions, we want to consider "collective" attack and defence (d is "reinstated" because the coalition {a, b, c} which attacks d is attacked by another coalition {e}).

Def. 13 Let

Discussion

We have proposed in this paper a "meta-argumentation framework" which takes into account two opposite kinds of interaction: the support relation is used in order to identify "coalitions" (sets of arguments which can be used together without conflict and which are related by the support relation); then the attack relation is used in order to identify conflicts between coalitions and to define new acceptability semantics as in Dung's framework.

In this meta AF, called "Coalition AF" (CAF), some nice properties of Dung's framework are preserved (link between new stable extensions -cs-extensions -and new preferred extensions -cp-extensions -, existence and maximality for set-inclusion of cp-extensions), but other properties are lost (the cp-extensions and the cs-extensions are not always admissible).

The notion of bipolar argumentation system is a recent notion in the field of argumentation, and, in our opinion, the use of coalitions in this framework is the first. So, we think it is important to compare our approach with other works on that key notion. Indeed, the concept of coalition has already been related to argumentation.

Collective argumentation framework [Bochman, 2003] [Nielsen and Parsons, 2006] A collective argumentation framework is an abstract framework where the initial data are a set of arguments and a binary "attack" relation between sets of arguments. The key idea is the following: a set of arguments can produce an attack against other arguments, which is not reducible to attacks between particular arguments. That is in agreement with our notion of coalition, since in our work, a coalition is considered as a whole and its members cannot be used separately in an attack process.

The proposal by Nielsen and Parsons is similar to Bochman's proposal. Both proposals take the attacks between sets of arguments as initial data, and define semantics by properties on subsets of arguments. However, Nielsen and Parsons propose an abstract framework which allows sets of arguments to attack single arguments only, and they stick as close as possible to the semantics provided by Dung. In contrast, Bochman departs from Dung's methodology and give new specific definitions for stable and admissible sets of arguments. Our proposal essentially differs from collective argumentation in two points. First, we keep exactly Dung's construction for defining semantics, but we apply this construction in a meta-argumentation framework (the coalition framework). The second main difference lies in the meaning of a coalition: we intend to gather as many arguments as possible in a coalition, and a coalition cannot be broken in a defence process. Generation of coalition structures in multi-agent systems [START_REF] Dang | [END_REF]Amgoud, 2005] In multi-agent systems, the coalition formation is a process in which independent and autonomous agents come together to act as a collective. A coalition structure (CS) is a partition of the set of agents into coalitions. Each coalition has a value (the utility that the agents in the coalition can jointly get minus the cost which this coalition induces for each agent). So the value of a CS is obtained by aggregating the values of the different coalitions in the structure. One of the main problems is to generate a preferred CS, that is a structure which maximizes the global value. Recently, [Amgoud, 2005] has proposed an abstract framework where the initial data are a set of coalitions equipped with a conflict relation.

A preferred CS is a subset of coalitions which is conflictfree and defends itself against attacks. Coalitions may conflict for instance if they are non-disjoint or if they achieve a same task. However, the generation of the coalitions is not studied in [Amgoud, 2005]. So, one perspective is to apply our work to the formation of coalitions taking into account interactions between the agents. Arguments represent agents in that case. Indeed, it is very important to put together agents which want to cooperate ("supports" relation) and to avoid gathering agents who do not want to cooperate ("attacks" relation). Then, the concept of cpextension provides a tool for selecting the best groups of agents (w.r.t. the given interaction relations).

More generally, the work reported here is generic and takes place in abstract frameworks, since no assumption is made on the nature of the arguments. Arguments may have a logical structure such as a pair explanation, conclusion , may just be positions advanced in a discussion, or may be agents interacting in a multi-agent system. All that we need is the bipolar interaction graph describing how the arguments under consideration are interrelated. We think that this generic work should stimulate discussion across boundaries.

A first perspective is to propose new benchmarks, with new kinds of real-world problems that can be modelled by our approach and not modelled by previous proposals. Another perspective from a computational point of view will be to evaluate the complexity of our approach.

  A, R att , R sup consists of a finite and non-empty set A of arguments, a binary relation R att on A called the attacks relation and another binary relation R sup on A called the supports relation. Consider a finite bipolar argumentation framework BAF = A, R att , R sup . Note also that BAF can still be represented by a directed graph G b called the bipolar interaction graph with two kinds of edges 8 , one for the attacks relation and another one for the supports relation. Consider a, b ∈ A, aR att b is represented by a → b and aR sup b is represented by a → b. Ex. 1 (cont'd)The whole discussion during the editorial board meeting can now be formalized by the bipolar framework BAF 1 represented by:

  the subgraph of G sup induced by S. Proof: ⇒ Let C be a coalition in S. We have to prove that C is a maximal connected subset in the subgraph of Gsup induced by S. C is the union of non-disjoint maximal support paths in S, namely M1, . . . , Mp. Let a and b be two distinct elements of C. There exists Mi (resp. Mj) a maximal support path in S containing a (resp. b). If Mi and Mj are identical, there exists a sequence of supports from a to b (or from b to a). If Mi and Mj are distinct, they are non-disjoint. There exists c which belongs to Mi and Mj. So, there is a sequence of supports between a and c (in Mi) and also a sequence of supports between b and c (in Mj). So there exists a chain of supports between a and b. So, C is a connected subset of S in the subgraph of Gsup induced by S. If C is not maximal connected, there exists C ′ connected subset of S strictly contains C. C ′ an element d which does not belong to C. Since C is not empty and ′ connected, there exists a chain of supports between d and an element e of C. That chain is the concatenation of non-disjoint support paths, each one being part of a maximal support path. of these support paths is included in C since the set {M1, M2, . . . , Mp} is maximal. So, d belongs to C. So, C is maximal connected: C is a connected component of the subgraph of Gsup induced by S.

Prop. 4

 4 Let a, b ∈ A s.t. aR sup b and {a, b} is conflict-free in AF. Then, there exists a coalition of BAF containing both a and b.Proof: Since {a, b} is conflict-free, there exists S maximal conflict-free subset of A containing a and b. Consider the subgraph of Gsup induced by S. This subgraph contains the edge (a, b), since a and b belong to S and aRsupb. So {a, b} is included in a connected component of this subgraph, and due to Proposition 2 and Proposition 3, {a, b} is included in a coalition. ✷4 A meta-argumentation frameworkLet C(A) denote the set of coalitions of BAF. We define a conflict relation on C(A) as follows.Def. 10 Let C 1 and C 2 be two coalitions of BAF. C 1 c-attacks C 2 iff there exists an argument a 1 in C 1 and an argument a 2 in C 2 s.t. a 1 R att a 2 .

  The coalitions are C 1 = {a, b, d}, C 2 = {a, b, c}. The unique cp-extension is {a, b, d}. The elementary coalitions are EC 1 = {a, d}, EC 2 = {a, b, c}. So the unique ecpextension is {a, d}. It seems difficult to justify the elimination of b.And admissibility can still be lost:Ex. 5 Let BAF 5 be represented by:The elementary coalitions are EC 1 = {a, b, c}, EC 2 = {d}, EC 3 = {e}. The unique ecp-extension is S = {e, d}. S is not admissible for AF 5 : we have bR att d, but e does not defend d against b (neither by a direct attack, nor by a diverted or a supported attack).

  a ∈ A and S ⊆ A. a is acceptable w.r.t. 4 S iff ∀b ∈ A s.t. bRa, ∃c ∈ S s.t. cRb. A set of arguments is acceptable w.r.t. S when each of its elements is acceptable w.r.t. S. S is conflict-free for A, R iff ∄a, b ∈ S, s.t. aRb.

  S ⊆ A. S is a ecp-extension of BAF iff there exists {EC 1 , . . . , EC p } a preferred extension of ECAF s.t. S = EC 1 ∪ . . . ∪ EC p . In BAF 3 , there are 4 elementary coalitions {a}, {c, b}, {c, d}, {e}. The unique ecp-extension {a, c, d} is admissible for AF 3 . Let BAF 4 be represented by:

	Ex. 3 (cont'd) However, some counter-intuitive results are obtained with
	ecp-extensions:
	Ex. 4

We adopt an incremental point of view, considering that pieces of information given by different agents enable them to provide more and more arguments. We do not want to revise already advanced arguments. Contrastedly, we intend to represent as much as possible all the kinds of interaction between these arguments. A comparison between both approaches is a topic for future work, for example from the point of view of computational complexity.

Note that bipolarity already appears in argumentation during the definition of the arguments or the selection of the "best" arguments (even if we only consider one kind of interaction)[START_REF] Amgoud | [END_REF].

If a does not attack b then, in the directed graph, there is no edge from a to b.

We abbreviate "with respect to" as "w.r.t.", and "such that" as "s.t.".

i.e. a belongs to A and does not belong to S.

For example, when the arguments are logical explanation arguments, i.e. under the form of a pair Σ, Φ with Σ being a consistent set of formulae and Φ being a formula implied by Σ, the attacks relation can be classically defined by: let a = Σa, Φa , b = Σ b , Φ b be two arguments, a attacks b iff ¬Φa ∈ Σ b . In the same way, the supports relation can be defined by: a supports b iff Φa ∈ Σ b . In that case, support and defence are different notions: a defends b iff there exists another argument c = Σc, Φc such that ¬Φa ∈ Σc and ¬Φc ∈ Σ b .

If the supports relation is removed, Dung's framework is retrieved.

 8 Positive and negative interactions were both envisaged in Wigmore Charts[START_REF] Wigmore | The science of judicial proof as given by logic, psychology, and general experience, and illustrated judicial trials[END_REF], which use a complex graphical notation for legal argument structuring. More recently, Yoshimi[Yoshimi, 2004] has published an interesting work on the theory of the structure of debate, where debates and positions are represented by sets of arguments equipped with two inter-argument relations : dispute and support.

Acknowledgements

We would like to thank the referees for their comments which helped us to improve this paper.