Je Remercie

Leila Amgoud

Daniel Leberre

Massimiliano Giacomin

Jean-Guy Mailly

Marie-Christine Lagas- Quie

Sylvie Doutre

Mélanie

Wendy Stéphie

Samantha

Élodie, Yehouda, Kalidou, Mickaël Dina Nicolas

Sébastien Jean-Michel Kamila

Priscille Audren Anaïs

Si on devait la comparer à une activité sportive, je décrirais la thèse comme étant à la fois une course de fond et une course d'orientation aux multiples arrivées ! C'est seulement après en avoir pris le départ qu'on en découvre le tracé, qu'on en réalise pleinement la dureté. Aujourd'hui, je tiens à remercier celles et ceux qui ont participé à faire de cette aventure une grande réussite ! Tout d'abord, je souhaiterais remercier mes encadrantes, Marie-Christine LAGASQUIE et Sylvie DOUTRE. En me faisant part de leur expertise, elles ont su me donner les conseils adéquats pour retrouver mon chemin quand je m'égarais, pour rebooster ma course quand je m'essoufflais à la tâche. Au delà de l'aspect technique de l'exercice, je tiens à vous remercier tout particulièrement pour votre approche humaine. J'ai traversé plusieurs moments éprouvants au cours de ces années et vous avez été très compréhensives ! J'en suis persuadé: avec d'autres encadrants, cela aurait été une toute autre histoire.

A lot of frameworks and semantics have been proposed to enhance expressivity in abstract argumentation. While a given framework specifies the way of representing and expressing an argumentation problem (types of relations between arguments, weight on attacks or arguments, higher-order relation, etc.), a semantics, defined for a specific argumentation framework, captures what is a solution of an argumentation problem, in the sense of what is acceptable.

In this thesis, I first focus on solving more efficiently argumentation problems which are expressed in the basic, seminal argumentation framework and semantics defined by Dung. Dung's semantics produce sets of jointly acceptable arguments, called extensions. A new distributed and clustering based algorithm to compute Dung's semantics is my first contribution. This algorithm has been designed for certain types of large-scale argumentation frameworks, that produce a large number of extensions. It has been implemented and tested. The results of these tests show its efficiency in the context of the large scale argumentation frameworks which are targeted.

Second, I focus on argumentation frameworks with higher order attacks, and especially Recursive Argumentation Frameworks (RAF). In this context, an attack may have as target an attack: an argument may thus be acceptable while one of its attack (receiving itself an attack) may be invalid, and so non pertinent against its target. Similarly to Dung's semantics which produce extensions, the RAF semantics produce "structures", pairs whose first element is a set of arguments and the second a set of attacks.

If algorithms already existed for Dung's framework, it was not the case for RAF. In order to address this issue, I start with studying the complexity of RAF semantics. I then extend the notion of labelling to RAF, another kind of characterization of acceptability which already existed for Dung's framework. The notion of "strongly connected component" is extended to RAF and decomposability properties of RAF semantics are studied. All these contributions pave the way for future algorithms to compute acceptability under RAF semantics.

Résumé

La théorie de l'argumentation abstraite propose des méthodes pour représenter et traiter les informations potentiellement incohérentes, et pour en tirer des conclusions ou prendre des décisions. Une telle approche est dite abstraite car elle se concentre uniquement sur la manière dont les arguments s'influencent mutuellement et pas sur la constitution des arguments. Les arguments sont donc considérés comme des entités génériques qui interagissent positivement (relation de support) ou négativement (relation d'attaque) les unes avec les autres.

Ce niveau d'abstraction permet de proposer des processus de raisonnement génériques qui peuvent être appliqués à toute définition ou formalisme concret des arguments. Le modèle de raisonnement basé sur l'argumentation est appliqué dans les systèmes multi-agents depuis des années. Le développement des techniques d'argumentation et de leur calcul est un point clé de ces applications. C'est la motivation même de mon travail : améliorer l'utilisation de l'argumentation abstraite en développant de meilleurs outils pour sa mise en oeuvre.

De nombreux cadres d'argumentation et sémantiques associées ont été proposés dans la littérature pour améliorer l'expressivité de l'argumentation abstraite. Alors qu'un cadre donné spécifie la manière de représenter et d'exprimer un problème d'argumentation (types de relations entre les arguments, poids des attaques ou des arguments, relation d'ordre supérieur, etc.), une sémantique, pour un cadre d'argumentation spécifique, capture ce qui est une solution d'un problème d'argumentation, dans le sens de ce qui est acceptable.

Dans mon travail, je me suis d'abord concentré sur la résolution efficace de certains problèmes d'argumentation qui sont exprimés dans le cadre d'argumentation classique et les sémantiques définis par Dung. Les sémantiques de Dung produisent des ensembles d'arguments conjointement acceptables, appelés extensions. Mon travail a conduit à la proposition d'un nouvel algorithme distribué et basé sur une technique de clustering pour calculer les extensions sous les sémantiques de Dung. Il a été conc ¸u pour certains types de cadres d'argumentation de "grande échelle", produisant un grand nombre d'extensions. Il a été implémenté et testé. Les résultats des tests montrent toute son efficacité pour les cadres d'argumentation à grande échelle ciblés.

Je me suis ensuite intéressé aux cadres d'argumentation d'ordre supérieur, et en particulier au cadre d'argumentation récursif (RAF). Dans ce contexte, une attaque peut avoir comme cible une autre attaque : un argument peut ainsi être acceptable alors même qu'il est attaqué parce que cette attaque (recevant elle-même une attaque) peut être invalide, et donc non pertinente contre sa cible. Là où le cadre de Dung produit des extensions, les sémantiques des RAF produisent des "structures", des paires dont le premier élément est un ensemble d'arguments et le second un ensemble d'attaques.

Si des algorithmes existaient déjà pour le cadre de Dung, il n'en était pas de même pour les RAF. J'ai donc commencé par étudier la complexité des sémantiques des RAF. J'ai ensuite étendu la notion de labelling aux RAF, une autre caractérisation de l'acceptabilité déjà existante dans le cadre de Dung. La notion de "composante fortement connexe" a été élargie aux RAF, et les propriétés de décomposabilité des sémantiques des RAF ont été étudiées. Toutes ces contributions ouvrent la voie à de futurs algorithmes pour calculer l'acceptabilité sous plusieurs sémantiques des RAF.

Part I Introduction

What is this thesis about?

Argumentation is a research field of Artificial Intelligence interested in managing contentious information. Two major sub-domains can be considered in Argumentation. The first one, called "Argument Mining", is interested in extracting arguments and their relations with each others, from natural language speeches (oral or written), in order to create a formal model to reason with (See [START_REF] Saint-Dizier | Challenges of argument mining: generating an argument synthesis based on the qualia structure[END_REF] for more information). The second one, that we will call "Argumentation Reasoning", is the one that is interested in reasoning over some argumentation model. It is useful to conclude, decide, convince, persuade or explain some issue. This way of reasoning, by considering arguments and their interactions, has proven successful in many contexts, multi-agent applications for instance (e.g. [START_REF] Carrera | A systematic review of argumentation techniques for multi-agent systems research[END_REF]).

I focus my PhD studies on Argumentation Reasoning, and more precisely, on "Abstract Argumentation", field so called because it does not focus neither on how to construct arguments nor on what the arguments are made of (their content), but rather on how arguments affect each other. Arguments are seen as generic entities that interact positively (e.g. support relation) or negatively (e.g. attack relation) with each other. This abstraction level allows to propose generic reasoning processes that could be applied to any precise definition or formalism for arguments.

There exist several approaches and formalisms to express argumentation problems. They differ on which "Argumentation Frameworks" and which "semantics" to use, to determine the argumentation solutions. These are two key notions in this research area:

• Considering the first key notion, here are some questions that have to be answered in order to "choose an Argumentation Framework" that fits with our need. Do we allow positive relations? If so, of which kind? Do we allow negative relations? If so, of which kind? Is there any notion of strength in arguments or in relations? The aim of making more complex Argumentation Frameworks is to be able to better capture human argumentation subtleties.

• Given an Argumentation Framework, the second key notion, semantics, corresponds to a formal way to say how the solution of the argumentation should be decided. It is really related with the notion of "acceptability". How to define an acceptable argumentation problem solution?

The basic, seminal Argumentation Framework and semantics have been defined by Dung in [START_REF] Minh | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF], known as Dung's Argumentation Framework (AF). Since then, a lot of propositions have been made to enhance the expressivity in Abstract Argumentation (e.g. [16,[START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF][START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF]4,[START_REF] Coste-Marquis | Weighted attacks in argumentation frameworks[END_REF]).

Future innovations in the area of Argument Mining may revolutionize the field of Artificial Intelligence, leading to the establishment of large-scale Argumentation Frameworks, built from arguments and relations collected for instance over the entire World Wide Web. Such frameworks may be large in their number of arguments, in their number of relations, in their variety of relation types, in their structure. This perspective made me choose as subject for my PhD studies the enhancement of tools for Abstract Argumentation. Carried out in the Institut de Recherche en Informatique de Toulouse (IRIT) and supervised by Marie-Christine LAGASQUIE and Sylvie DOUTRE, my thesis has thus been entitled: "Algorithms for enriched abstract argumentation frameworks for large-scale cases".

A first milestone

In the first part of my PhD, I center my research on solving more efficiently argumentation problems that are expressed in Dung's AF and semantics. This is a necessary first step before considering studying an extension of this work to other enriched Argumentation Frameworks and semantics. In Dung's setting, solutions of an argumentation problem are sets of arguments (defined under the notion of extension) which, when considered together, win the argumentation. Finding all the possible solutions of an argumentation problem, i.e. all its winning sets of arguments, can be very time consuming. Many argumentation problem instances, particularly large 1 , are too hard to be solved in an acceptable time, as shown by the results of the ICCMA argumentation solver competition 2 . This hardness is not relative to the current state of the art but rather to the intrinsic theoretical complexity of the argumentation semantics that are tackled [START_REF] Dvorak | Computational problems in formal argumentation and their complexity[END_REF].

Considering the foreseen scaling-up challenge mentioned above in addition to the complexity, there is a need for heuristics, methods and algorithms efficient enough to tackle such issues and make possible the use of automated argumentation models, even in such settings. Enhancing the computational time of enumerating the solutions of an argumentation framework has been the object of study of many works, resulting in the elaboration of several recent algorithms such as [1,[START_REF] Cerutti | Exploiting parallelism for hard problems in abstract argumentation[END_REF][START_REF] Liao | Toward incremental computation of argumentation semantics: A decomposition-based approach[END_REF][START_REF] Alviano | The pyglaf argumentation reasoner[END_REF] (see [START_REF] Charwat | Methods for solving reasoning problems in abstract argumentation-a survey[END_REF] for an overview).

To address this issue, we propose the AFDivider algorithm, a distributed and clustering-based algorithm that has for main purpose to find all the possible solutions of an argumentation problem. Those solutions are defined in terms of semantics labellings [START_REF] Caminada | On the issue of reinstatement in argumentation[END_REF][START_REF] Baroni | An introduction to argumentation semantics[END_REF], a three status based function mapping that assigns to each argument of an AF an acceptance status: accepted, rejected or undecided. An empirical analysis of the AFDivider algorithm shows that the new approach of computing Dung-like semantics is relevant and very appropriate for some types of argumentation problems. This work led to several publications: [START_REF] Lafages | Clustering and distributed computing in abstract argumentation[END_REF][START_REF] Doutre | A distributed and clustering-based algorithm for the enumeration problem in abstract argumentation[END_REF][START_REF] Castagna | Online handbook of argumentation for ai[END_REF][START_REF] Doutre | A Distributed and Clusteringbased Algorithm for the Enumeration Problem in Abstract Argumentation (JIAF 2020)[END_REF] (See Part III on page 27 for more information).

The next milestone

In the second part of my thesis, I focused on "Argumentation Frameworks with Higher-Order Attacks" (e.g. [START_REF] Barringer | Temporal dynamics of support and attack networks : From argumentation to zoology[END_REF][START_REF] Modgil | An abstract theory of argumentation that accommodates defeasible reasoning about preferences[END_REF][START_REF] Modgil | Reasoning about preferences in argumentation frameworks[END_REF]5,[START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF]). This type of Argumentation Framework is a rich extension of the classical Dung's AF: not only they consider arguments and attacks between arguments, but also attacks on attacks (see for instance [5,[START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF]). Among these frameworks, the "Recursive Argumentation Framework" (RAF) by [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF] proposes a direct approach regarding acceptability, which outputs sets of arguments and/or attacks (defined under the notion of structure), keeping the full expressiveness of higher-order attacks. A correspondence between Dung's extension-based semantics of AF and structure-based semantics of RAF without any attack on attacks has been shown in [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF], proving that RAF are a conservative generalisation of AF. This characteristic makes RAF particularly interesting to consider.

Given that the computation of RAF semantics has not been addressed so far, I dedicate the second part of my thesis to the developments of tools (new notions) for RAFs and the study of RAF properties, preparing thus the way for algorithm proposals. I first adapted the notion of AF labelling for RAF, socalled, "structure labellings". Secondly, I introduce a flattening process that transforms RAFs into AFs, ensuring interesting properties. Thirdly, relying on that flattening, I study the complexities of RAF semantics. Finally, I adapt the notion of Strongly Connected Component to RAFs and from this key notion, I study the semantics decomposability of RAF semantics (notion introduced for AFs in [START_REF] Baroni | On the input/output behavior of argumentation frameworks[END_REF]). These works led to several publications: [START_REF] Doutre | Argumentation Frameworks with Higher-Order Attacks: Labelling Semantics[END_REF][START_REF] Doutre | Argumentation Frameworks with Higher-Order Attacks: Complexity results[END_REF][START_REF] Doutre | Argumentation Frameworks with Higher-Order Attacks: Semantics and Complexity[END_REF][START_REF] Doutre | Argumentation Frameworks with Higher-Order Attacks: Labellings and Complexity[END_REF] (See Part V on page 101 for more information).

How this thesis is organized? How to read it?

The main body of this thesis contains four parts (introduction and conclusion set apart):

• Part II on page 7: Dung Argumentation Framework: Background.

• Part III on page 27: Dung Argumentation Framework: Contribution.

• Part IV on page 83: Higher-Order Attack Argumentation Frameworks: Background.

• Part V on page 101: Higher-Order Attack Argumentation Frameworks: Contribution.

Parts II and III concern the first milestone of my studies, as mentioned above, while Parts IV and V concern the second one. In each background (Parts II and IV) the notions related to argumentation required to understand the contribution parts that follow (Parts III and V) are given. Notice that, each contribution part is quite independent from the other. 3 However, for some of them, mathematical notions are required to fully understand the contributions, especially Part III. Those notions are given in appendix as explained below.

The last part of the main body, Part VI on page 165, concludes this thesis and opens perspectives for future works. This thesis has three appendices:

• Appendix 1: Mathematical Background (on page 170)

In this part are given all the mathematical background required to understand the ins and outs of this thesis. By sake of clarity, it has been separated from the main body.

• Appendix 2: Tables (on page 194)

In this part are given tables of symbols to help the reading of this thesis.

• Appendix 3: Proofs (on page 219)

In this part are given all the proofs of the propositions and theorems proposed in the different contribution chapters. 4 In order to facilitate the reading of this thesis, a bi-direction linking has been made, allowing thus to go to and from some property and its proof.

Although the different parts, chapters and sections listed below are not fully independent, here is a guide for the reading of this thesis by topic. For the reader interested in:

• Complexity, read:

-Section 16. In [START_REF] Minh | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF], Dung introduced the seminal abstract argumentation framework. It consists of a set of arguments and of a binary attack relation between them. An "Argumentation Framework" (denoted AF) can be represented as a directed graph in which nodes are arguments and directed edges are attack relations between arguments. Formally, it is defined as follows:

Definition 1 (Argumentation framework). An argumentation framework (AF) is a pair AF = ⟨A, K⟩ where A is a finite set of abstract arguments and K ⊆ A × A is a binary relation on A, called the attack relation: (a, b) ∈ K means that a attacks b. The set of all possible argumentation frameworks is denoted as Φ a f .

Example 1. Figure 1 shows an illustration of an AF. In all this document, arguments (in Latin letter) will be represented by a round box and attacks are represented by directed edges. This formalism provides a strong base to compute the "solutions" of the argumentation so represented. In Chapter 1 on the next page are presented different types of argumentation solutions, so called semantics. Chapter 2 on page 16 presents the notion of semantics semantics decomposability and gives some properties over the semantics we are interested in. Finally, in Chapter 3 on page 23 are defined AF decision problems and their complexities are given.

Chapter 1

Semantics : extensions and labellings

Basically, a semantics defines what is a "solution" of an argumentation. In this chapter we present two kinds of semantics types: extension-based ones (Section 1.1) and labelling-based ones (Section 1.2 on page 10). Then in Section 1.3 on page 12 are given the relations between these two kinds of semantics.

Extension-based semantics

In Dung-like semantics [START_REF] Minh | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF], so-called extension-based semantics, a solution of an argumentation is a group of arguments that, together, win the argumentation. The semantics thus define how to select those groups. Formally, a generic AF extension-based semantics is defined as follows:

Definition 2 (Extension-based Semantics). Let σ be a function over Φ a f . σ is said to be an AF extensionbased semantics iff the following property holds: ∀AF ∈ Φ a f , σ (AF ) ⊆ 2 A , with AF = ⟨A, K⟩ A σ -extension is defined as follows:

Definition 3 (Extension). Let σ be an AF semantics and let AF = ⟨A, K⟩ be an AF. Let S ⊆ A be a set of arguments. We say that S is a σ -extension of AF iff S ∈ σ (AF ).

Dung's semantics rely on the notion of defeat and acceptability. Definition 4 (Defeat and acceptability in Dung's framework). Let AF = ⟨A, K⟩ be an AF and S ⊆ A be a set of arguments. An argument a ∈ A is said to be: Definition 5 (Some extension-based semantics of AF). Let AF = ⟨A, K⟩ be an AF and S ⊆ A be a set of arguments. S is said to be an extension:

• defeated w.
1. Conflict-free iff S ∩ De f (S) = ∅.

2. Naive iff it is a ⊆-maximal conflict-free extension.

3. Admissible iff it is conflict-free and S ⊆ Acc(S). 4. Complete iff it is conflict-free and S = Acc(S).

5. Preferred iff it is a ⊆-maximal admissible extension. [START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF]. Grounded iff it is the ⊆-minimal complete extension.

7. Semi-stable iff it is a complete extension such that S ∪ De f (S) is maximal w.r.t. ⊆.

8. Stable iff it is conflict-free and S ∪ De f (S) = A.

In the following we will mainly focus on the complete, preferred, grounded, semi-stable and stable semantics as they are the most commonly used in the literature and so, given an AF AF , σ (AF ) is the set of extensions of AF under semantics σ , with σ being as an example the complete (co), grounded (gr), stable (st), semi-stable (sst) or preferred (pr) semantics.

Example 2. Table 1.1 on page 13 shows which extensions those semantics produce for the AF in Figure 1 on page 7. It can be noticed that on this particular AF, the stable semantics does not produce any extension.

There exists a partial order over the semantics defined in Definition 5 (see [START_REF] Caminada | On the issue of reinstatement in argumentation[END_REF] for proofs). • There is always at least one conflict-free extension.

• There is always at least one naive extension.

Conflict-free (+)

Admissible (+)

Naive (+)

Complete (+)

Preferred (+)
Semi-stable (+)

Stable (*)
Grounded (1)

Figure 1.1: AF semantics partial ordering

The cardinality of each semantics, that is, the number of extensions which can be produced by each semantics, is represented between parentheses. " * " means zero or more, "+" means one or more.

• There is always at least one admissible extension.

• There is always at least one complete extension.

• There is always a unique grounded extension.

• There is always at least one preferred extension.

• There is always at least one semi-stable extension.

• It may be the case that there is no stable extension.

Labelling-based semantics

Dung-like semantics can also be defined in terms of labellings as introduced in [START_REF] Caminada | On the issue of reinstatement in argumentation[END_REF]. A labelling maps to each argument of an AF a value representing its acceptability status. This status may be accepted (in ), rejected (out ) or in an undecided state (und ). Formally: Definition 6 (Labelling). Let AF = ⟨A, K⟩ be an AF, and S ⊆ A. A labelling of S is a total function ℓ : S → {in, out, und}. A labelling of AF is a labelling of A. The set of all labellings of AF is denoted as L (AF ). The set of all labellings of a set of arguments S is denoted as L (S).

We write in(ℓ) for a|ℓ(a) = in , out(ℓ) for a|ℓ(a) = out and und(ℓ) for a|ℓ(a) = und .

Definition 7 (legally labelled argument). Let AF = ⟨A, K⟩ be an AF, and ℓ ∈ L (AF ) be a labelling.

• An in-labelled argument is said to be legally in iff all its attackers are labelled out.

• An out-labelled argument is said to be legally out iff at least one of its attackers is labelled in.

• An und-labelled argument is said to be legally und iff it does not have an attacker that is labelled in and one of its attackers is not labelled out.

Definition 8 (Conflict-free labelling). Let AF = ⟨A, K⟩ be an AF, and ℓ ∈ L (AF ) be a labelling. ℓ is an conflict-free labelling of AF iff for each a ∈ in(ℓ), a is legally in.

Definition 9 (Naive labelling). Let AF = ⟨A, K⟩ be an AF, and ℓ ∈ L (AF ) be a labelling. ℓ is a naive labelling of AF iff it is a conflict-free labelling of AF which maximizes (w.r.t ⊆) the set of in-labelled arguments.

Definition 10 (Admissible labelling). Let AF = ⟨A, K⟩ be an AF, and ℓ ∈ L (AF ) be a labelling. ℓ is an admissible labelling of AF iff it satisfies the following conditions for any a ∈ A:

• For each a ∈ in(ℓ), a is legally in.

• For each a ∈ out(ℓ), a is legally out.

Definition 11 (Complete labelling). Let AF = ⟨A, K⟩ be an AF, and ℓ ∈ L (AF ) be a labelling. ℓ is a complete labelling of AF iff it satisfies the following conditions for any a ∈ A:

• For each a ∈ in(ℓ), a is legally in.

• For each a ∈ out(ℓ), a is legally out.

• For each a ∈ und(ℓ), a is legally und.

Definition 12 (Grounded, preferred, semi-stable and stable labelling).

Let AF = ⟨A, K⟩ be an AF, and ℓ ∈ L (AF ) be a labelling.

• ℓ is the grounded labelling of AF iff it is the complete labelling of AF which minimizes (w.r.t ⊆) the set of in-labelled arguments.

• ℓ is a preferred labelling of AF iff it is a complete labelling of AF which maximizes (w.r.t ⊆) the set of in-labelled arguments.

• ℓ is a stable labelling of AF iff it is a complete labelling of AF which has no und-labelled argument.

• ℓ is a semi-stable labelling of AF iff it is a complete labelling of AF which minimizes (w.r.t. ⊆) the set of und-labelled arguments.

It can be noticed that all complete labellings include the grounded labelling, and, as stable and preferred labellings are complete labellings, they also include the grounded labelling.

Definition 13 (Labelling-based Semantic). A labelling-based semantic σ is a total function σ : Φ a f → 2 L (AF ) which associates to an AF AF = ⟨A, K⟩ a subset of L (AF ).

Given an AF AF = ⟨A, K⟩, the set of labellings under semantics σ is denoted L σ (AF ).

Example 3. Let us consider the AF of Figure 1 on page 7. Table 1.2 on page 14 shows the labellings corresponding to the different semantics (the other possible labellings are not given). As you can see this AF has no stable labellings.

The partial ordering and cardinalities of extension-based semantics stated in Propositions 1 and 2 on page 9 and on page 10 also hold for labelling-based semantics. See • There is always at least one conflict-free labelling.

• There is always at least one naive labelling.

• There is always at least one admissible labelling.

• There is always at least one complete labelling.

• There is always a unique grounded labelling.

• There is always at least one preferred labelling.

• There is always at least one semi-stable labelling.

• It may be the case that there is no stable labelling.

Extension-based and labelling-based semantics relations

Now let consider the relation between extension-based semantics and labelling-based semantics. As proven in [START_REF] Caminada | On the issue of reinstatement in argumentation[END_REF], Proposition 5 holds.

Proposition 5 (Semantics bijection). Let σ ∈ {complete, stable, grounded, preferred, semi-stable} be a semantics. There exists a bijection between σ -extensions and σ -labellings. The correspondences stated in Table 1.3 on page 15 hold.

Given that there is a bijection between extension-based and labelling-based semantics, in the rest of this document, we will not specify extension-based or labelling-based unless it is necessary. " " means that the argument on the row belongs to the extension on the column. "•" means that the semantics on the row produces the extension on the column.

Semantics decomposability

In addition to semantics properties found for extension-based semantics, such as SCC-recursiveness (see [11] for more information), the labelling approach allowed the discovery of new ones. In this chapter we present the notion of semantics decomposability defined in [START_REF] Baroni | On the input/output behavior of argumentation frameworks[END_REF] along with the different properties that are related to it.

Several notions are required to define the semantics decomposability properties of a semantics.

Definition 14 (Labelling restriction ↓). Let ℓ be a labelling. Let S be a set of arguments. The restriction of ℓ to S denoted as ℓ ↓ S is defined as ℓ ∩ (S × {in, out, und}).

Definition 15 (Input arguments and conditioning relation). Let AF = ⟨A, K⟩ be an AF and S ⊆ A be a set. The input of S, denoted as S inp , is the set {b ∈ A \ S|∃a ∈ S, (b, a) ∈ K}. The conditioning relation of S, denoted as S K , is defined as K ∩ (S inp × S). ). An argumentation framework with input is a tuple AF , J , ℓ J , K J , including an argumentation framework AF = ⟨A, K⟩, a set of arguments J such that J ∩ A = ∅, a labelling ℓ J of the elements of J and a relation K J ⊆ J × A. Let AF , J , ℓ J , K J be an AF with input. Its standard AF is an AF that simulates the conditioning labelling of its input arguments J . To do so, some fictive arguments and interactions are added. Formally:

Definition 17 (standard AF). Given AF , J , ℓ J , K J , an argumentation framework with input, the standard AF w.r.t. AF , J , ℓ J , K J is defined as:

std-AF = A ∪ J ′ , K ∪ K ′ J
Where:

• J ′ = J ∪ a ′ |a ∈ J ∩ out(ℓ J ) • K ′ J = K J ∪ (a ′ , a)|a ∈ J ∩ out(ℓ J ) ∪ (a, a)|a ∈ J ∩ und(ℓ J ) . 1
Note: By definition, if the standard AF std-AF admits some labellings then, restricted to the input J , those labellings are exactly the labelling ℓ J . Notice that for the stable semantics it may be the case that std-AF admits no labelling.

Example 6. Let consider the AF with input AF , J , ℓ J , K J in Example 5 on the previous page. Given that the labelling of its input arguments is ℓ J = {(e, out), (d, und)}, we obtain as its corresponding standard AF the one illustrated in Given an AF with input AF , J , ℓ J , K J (with AF = ⟨A, K⟩) and its corresponding standard AF std-AF , the canonical local function is simply a function that gives the set of labellings under a certain semantics of the sub-AF we are interested in: std-AF ↓ A = AF (i.e. the input arguments and the other fictive arguments created are not in these labellings).

Definition 18 (Canonical local function). Let σ be a semantics, AF , J , ℓ J , K J be an AF with input (with AF = ⟨A, K⟩) and std-AF be its standard AF. The canonical local function F a f σ is the local function such that:

F a f σ (AF , J , ℓ J , K J ) = ℓ ↓ A |ℓ ∈ L σ (std-AF )

Example 7. Considering Example 6, we have for σ ∈ {stable, preferred, semi-stable}:

F a f
σ (AF , J , ℓ J , K J ) = {{( f , in), (g, out), (h, in), (i, out)}}

For the grounded semantics we have:

F a f
gr (AF , J , ℓ J , K J ) = {{( f , und), (g, und), (h, und), (i, und)}} And for the complete :

F a f co (AF , J , ℓ J , K J ) =    {( f ,
und), (g, und), (h, und), (i, und)} , {( f , in), (g, out), (h, in), (i, out)}    Definition 19 (Semantics fully decomposability). A semantics σ is fully decomposable (or decomposable) if and only if there is a local function F a f such that for every AF AF = ⟨A, K⟩ and every partition Ω = {ω 1 , ..., ω n } of A, we have:

L σ (AF )
= ℓ 1 ∪ ... ∪ ℓ n ∀i, ℓ i ∈ F a f (AF ↓ ω i , ω inp i , ( j∈{1,...,n} s.t. j̸ =i ℓ j ) ↓

ω inp i , ω K i )
Definition 20 (Initial argument). Let AF = ⟨A, K⟩ be an AF, and b ∈ A be an argument. b is an initial argument of AF if there is no argument in AF attacking b.

Note: In graph theory, nodes corresponding to "initial arguments" are called source nodes.

Definition 21 (Complete-compatibility). A semantics σ is complete-compatible if and only if the following conditions hold:

1. For any AF AF = ⟨A, K⟩, every labelling ℓ ∈ L σ (AF ) satisfies the following conditions:

• if a ∈ A is initial, then ℓ(a) = in • if b ∈ A and there is an initial argument in A which attacks b, then ℓ(b) = out • if c ∈ A is self-attacking, and there is no attacker of c besides c itself, then ℓ(c) = und 2. For any set of arguments J and any labelling ℓ J of J , the AF AF ′ = ⟨J ′ , K ′ ⟩, where J ′ = J ∪ {a ′ |a ∈ J ∩ out(ℓ J )} and K ′ = (a ′ , a)|a ∈ J ∩ out(ℓ J ) ∪ (a, a)|a ∈ J ∩ und(ℓ J ) , admits a unique labelling, i.e. |L σ (AF ′ )| = 1.

Proposition 6. The complete, stable, semi-stable, preferred and the grounded semantics are completecompatible.

Proposition 7. Given a complete-compatible semantics σ , if σ is fully decomposable then there is a unique local function satisfying the conditions of Definition 19, coinciding with the canonical local function F a f σ . Definition 22 (Top-down and bottom-up decomposability). Let σ be a complete-compatible semantics and F a f σ be the canonical local function corresponding to σ . σ is said to be top-down decomposable iff for any AF AF = ⟨A, K⟩ and any partition Ω = {ω 1 , ..., ω n } of A, it holds that:

L σ (AF ) ⊆    ℓ 1 ∪ ... ∪ ℓ n ℓ i ∈ F a f σ   AF ↓ ω i , ω inp i , ( j∈{1,...,n} s.t. j̸ =i ℓ j ) ↓ ω inp i , ω K i     
σ is said to be bottom-up decomposable iff for any AF AF = ⟨A, K⟩ and any partition Ω = {ω 1 , ..., ω n } of A, it holds that: Definition 24 (Decomposability w.r.t. a partition selector S ). Let S be a partition selector. A completecompatible semantics σ is top-down decomposable w.r.t. S iff for any AF AF and any partition Ω = {ω 1 , ..., ω n } ∈ S (AF ), it holds that:

L σ (AF ) ⊇    ℓ 1 ∪ ... ∪ ℓ n ℓ i ∈ F a f σ   AF ↓ ω i , ω inp i , ( j∈{1
L σ (AF ) ⊆    ℓ 1 ∪ ... ∪ ℓ n ℓ i ∈ F a f σ   AF ↓ ω i , ω inp i , ( j∈{1
,...,n} s.t. j̸ =i

ℓ j ) ↓ ω inp i , ω K i     
A complete-compatible semantics σ is bottom-up decomposable w.r.t. S iff for any argumentation framework AF and any partition Ω = {ω 1 , ..., ω n } ∈ S (AF ), it holds that: Note: For a semantics σ , to be top-down (resp. bottom-up, fully) decomposable is equivalent to be topdown (resp. bottom-up, fully) decomposable w.r.t. the partition selector that produces all possible partitions of an AF.

L σ (AF ) ⊇    ℓ 1 ∪ ... ∪ ℓ n ℓ i ∈ F a f σ   AF ↓ ω i , ω inp i , ( j∈{1
Let formally defined this partition selector: Definition 25. S D-a f is the AF partition selector defined as follows:

∀AF ∈ Φ a f , S D-a f (AF ) is the set of all possible partitions of AF Let introduce some useful notations. 2 Let AF be an AF. We denote by:

• Paths a f (AF ) the set of all paths of AF • Cycles a f (AF ) the set of all cycles of AF • PE a f the path-equivalence relation over an AF (see Definition 128 on page 174), PE a f (AF ) being the path-equivalence relation over AF

• SCC a f the abbreviation for an SCC of AF (that is, an equivalence class of arguments under PE a f . See Definition 129 on page 174)

• SCCS a f (AF ) the set of all SCCs of AF .

Example 8. Let consider the AF in Figure 2.3 on page 18. We have:

• (e ′ , e, f ) ∈ Paths a f (AF )

• ( f , g, f ) ∈ Cycles a f (AF ) 2 See also Section 16.2 on page 170 in the appendix

• SCCS a f (AF ) = {{e ′ } , {e} , {d} , { f , g} , {h} , {i}} Definition 26 (SCC a f relation: ≼). Let AF = ⟨A, K⟩ be a AF, S ∈ SCCS a f (AF ) and S ′ ∈ SCCS a f (AF ). We define the relation ≼ as a binary relation between elements of SCCS a f (AF ) as follows:

S ≼ S ′ if and only if (∃(e 1 , ..., e n ) ∈ Paths a f (AF ) s.t. e 1 ∈ S and e n ∈ S ′ ) or S = S ′ Example 9. Following Example 8, we have:

• {h} ≼ {h} • { f , g} ≼ {i} • {i} ̸ ≼ {h}
Definition 27 (USCC partition selector). The USCC partition selector (denoted S USCC ) is the partition selector such as for any AF AF = ⟨A, K⟩:

S USCC (AF ) =
Ω| Ω is a partition of A and ∀S ∈ SCCS a f (AF ), ∃ω i ∈ Ω s.t. ω i ∩ S ̸ = ∅ =⇒ S ⊆ ω i Given an AF AF = ⟨A, K⟩, we call "USCC a f " a subset S ⊆ A such that S is a union of SCC a f of AF .

Note: This partition selector has an interesting property: it does not break the SCC a f .

Example 10. Following Example 8, as an illustration, we have:

• {{e ′ , e, f , g} , {d} , {h, i}} ∈ S USCC (AF ) " " means that the semantics on the column has the property on the row. "× × × " means that the semantics on the column does not have the property on the row.

Chapter 3 AF decision problems and complexities

Dung's Argumentation Framework can be used in processes such as decision making, explanations and auctions. To do so, several decision problems can be useful. In this chapter, classical AF decision problems are defined (Section 3.1), then their complexities are given (Section 3.2 on the next page).

Definitions

Given the bijection between extension-based semantics and labelling-based semantics, and given that extensions can be transformed into labellings (and vice versa) in polynomial time, AF decision problems can be equivalently defined for the one or the other kind of semantics. We chose to define them using labellingbased semantics. Here are the classical ones: Definition 28 (Decision Problems in Abstract Argumentation).

• Credulous Acceptance Cred σ : Given an AF AF = ⟨A, K⟩ and an argument a ∈ A. Is a labelled in in some ℓ ∈ σ (AF )?

• Skeptical Acceptance Skep σ : Given an AF AF = ⟨A, K⟩ and an argument a ∈ A. Is a labelled in in each ℓ ∈ σ (AF )?

• Verification of a labelling Ver σ : Given an AF AF = ⟨A, K⟩ and a labelling ℓ. Is ℓ ∈ σ (AF )?

• Existence of a labelling Exists σ : Given an AF AF = ⟨A, K⟩. Is σ (AF ) ̸ = ∅?

• Existence of a "non-empty" labelling Exists ¬∅ σ : Given an AF AF = ⟨A, K⟩. Does there exist a labelling ℓ ∈ σ (AF ) s.t. in(ℓ) ̸ = ∅?

• Uniqueness of a solution Unique σ : Given an AF AF = ⟨A, K⟩. Is there a unique labelling ℓ ∈ σ (AF ),

i.e. σ (AF ) = {ℓ}?

Example 11. Let σ be the preferred semantics. Let AF = ⟨A, K⟩ be the AF represented in Figure 3.1 on the next page and ℓ ∈ L (AF ) be a labelling of it. Following Table 1.2 on page 14, we have two preferred 23 CHAPTER 3. AF DECISION PROBLEMS AND COMPLEXITIES labellings (ℓ 2 and ℓ 4 ):

σ (AF ) =                            ℓ 4 =         
(a, in), (b, out), (c, out), (d, in), (e, out), ( f , in), (g, out), (h, in), (i, out), ( j, und), (k, und), (l, und), (m, out), (n, in)

         , ℓ 2 =         
(a, in), (b, out), (c, out), (d, out), (e, in), ( f , out), (g, in), (h, out), (i, in), ( j, und), (k, und), (l, und), (m, out), (n, in)

                                   
As a consequence, we have:

• Cred σ (AF , l) = false • Cred σ (AF , d) = true • Skep σ (AF , d) = false • Skep σ (AF , n) = true • Ver σ (AF , ℓ) = true iff ℓ ∈ ℓ 2 , ℓ 4 • Exists σ (AF ) = true • Exists ¬∅ σ (AF ) = true • Unique σ (AF ) = false

Complexities

Table 3.1 on the following page gives the complexity class of the mentioned decision problems for the grounded, complete, preferred, stable and semi-stable semantics. This table is the result of numerous works (see [START_REF] Dvorak | Computational problems in formal argumentation and their complexity[END_REF][START_REF] Dvořák | Computational problems in formal argumentation and their complexity[END_REF] for a synthesis of these works). 1

Cred

σ Skep σ Ver σ Exists σ Exists ¬∅ σ Unique σ Grounded P-c P-c P-c trivial in L trivial Complete NP-c P-c in L trivial NP-c coNP-c Preferred NP-c Π P 2 -c coNP-c trivial NP-c coNP-c Stable NP-c coNP-c in L NP-c NP-c DP-c Semi-stable Σ P 2 -c Π P 2 -c coNP-c trivial NP-c in Θ P Part III

Dung Argumentation Framework: Contribution

Part presentation:

This part presents my contributions in the context of Dung's Argumentation Framework. The main contribution is a Distributed and Clustering-Based algorithm for the enumeration problem. This led to several publications:

• An IRIT report [START_REF] Lafages | Clustering and distributed computing in abstract argumentation[END_REF], which served as a support for subsequent works: it details the concepts used and provides the proofs that are not in some articles.

• An article in PRIMA 2019, the 22 nd edition of the international conference of Principles and Practice of Multi-Agent Systems [START_REF] Doutre | A distributed and clustering-based algorithm for the enumeration problem in abstract argumentation[END_REF], and for which I received a student grant.

• An article in OHAAI 2019, the first edition of the Online Handbook of Argumentation for Artificial Intelligence [START_REF] Castagna | Online handbook of argumentation for ai[END_REF]. OHAAI is a platform created in order that PhD students share with others their study subject and the aim of their thesis.

• An article in JIAF 2020, the 4 th edition of a French national conference so-called Journées d'Intelligence Artificielle Fondamentale [START_REF] Doutre | A Distributed and Clusteringbased Algorithm for the Enumeration Problem in Abstract Argumentation (JIAF 2020)[END_REF], which is a second publication of my PRIMA 2019 article for the French community.

The presentation of these contributions is organised as follows. Firstly, the AFDivider algorithm is presented and formally studied (Chapter 4 on the next page). Secondly, experiments with the AFDivider algorithm are presented (Chapter 5 on page 51). These results led to the idea of a so-called Compact Enumeration Representation, which is presented in a third step (Chapter 6 on page 64). To finish, related works are presented (Chapter 7 on page 70).

Chapter 4

AFDivider : presentation and formal analysis

Motivation

Finding all the possible solutions of a semantics (extension-based or equivalently labelling-based) for a given AF can be very time consuming. Many AF instances, particularly large, 1 are too hard to be solved in an acceptable amount of time, as shown by the results of the ICCMA argumentation solver competition. 2Formally, the so-called enumeration problem is a function problem defined as follows:

Definition 29 (Enumeration Problem). Given an AF AF = ⟨A, K⟩ and a semantics σ , compute the set σ (AF ) corresponding to the AF solutions.

The hardness of this problem is not relative to the current state of the art but rather to the intrinsic theoretical complexity of the semantics that are tackled. 3 Moreover, there exists a recent research field in Artificial Intelligence called "Argument Mining" whose object of study is how to extract arguments from natural language speeches, oral or written (see [START_REF] Saint-Dizier | Challenges of argument mining: generating an argument synthesis based on the qualia structure[END_REF] for more information). When major advances in this area will make available a lot of data for argumentation, this issue of solving time may become increasingly critical. There is a need for heuristics, methods and algorithms efficient enough to tackle such issues and make possible the use of automated argumentation models in the large-scale.

Enhancing the computational time of enumerating the solutions of an AF has been the object of study of many works, resulting in the elaboration of several recent algorithms such as [1,[START_REF] Cerutti | Exploiting parallelism for hard problems in abstract argumentation[END_REF][START_REF] Liao | Toward incremental computation of argumentation semantics: A decomposition-based approach[END_REF][START_REF] Alviano | The pyglaf argumentation reasoner[END_REF] (see [START_REF] Charwat | Methods for solving reasoning problems in abstract argumentation-a survey[END_REF] for an overview). During this thesis, I add my contribution to address this issue with the proposal of an algorithm, the so-called: AFDivider.

The idea that led to this algorithm is that argumentation frameworks constructed from real data should have a particular structure. Indeed, people have themes and goals while arguing. It is thus a reasonable conjecture to say that the AFs obtained from real argumentation are not random and that they have a relatively low density of relations between arguments. As a matter of fact, it is very unlikely that, in some argumentation, each presented argument attacks a large part of the other ones.

The AFDivider algorithm takes advantage of this sparsity4 of AF graphs. To do so, it uses methods that have not yet been considered for this purpose (namely clustering methods used in an original way), combined with techniques that have already been applied in other existing algorithms.

The next section presents formally the AFDivider algorithm and summarizes its main steps. Sections 4.2.1 to 4.2.4 on pages 30-42 detail each of these steps and illustrate them.

AFDivider : A Generic algorithm

First of all, let specify the kind of problems addressed. The AFDivider algorithm enumerates labelling-based semantics. It has been designed for Dung original semantics: the complete, the stable and the preferred semantics. Given that the grounded semantics can be computed in linear time and that it gives only one labelling, computing it with the AFDivider is unappropriated.

Given an argumentation framework AF = ⟨A, K⟩ and a semantics σ ∈ {complete, stable, preferred}, the AFDivider algorithm, rather than building labellings that cover the whole AF (which could be time consuming), computes the semantics labellings using a distributed and clustering-based method. Here are its four major steps graphically represented in Figure 4.1:

1. A pretreatment on AF removes "trivial" parts of it. • Any clustering method can be used to split the AF.

• Any sound and complete procedure that computes the semantics σ , can be used to compute the labellings of the different clusters.

Algorithm 1: AFDivider algorithm. Input: Let AF = ⟨A, K⟩ be an AF and σ be a semantics Result: L σ ∈ 2 L (AF ) : the set of the σ -labellings of AF Local variables:

• ℓ ′ gr : the grounded labelling restricted to the arguments labelled in and out • CCSet: the set of connected components of AF hard • ClustSet: the set of cluster structures of af i • L σ (af i ): the set of all σ -labellings of af i

1 ℓ ′ gr ← ComputeGroundedLabelling(AF ) 2 CCSet ← SplitConnectedComponents(AF , ℓ ′ gr ) 3 for all af i ∈ CCSet do in parallel 4 ClustSet ← ComputeClusters(af i ) 5 L σ (af i ) ← ComputeCompLabs(σ ,ClustSet) 6 L σ ← ∅ 7 if ∄af i ∈ CCSet s.t. L σ (af i ) = ∅ then L σ ← {ℓ ′ gr } × ∏ af i ∈CCSet L σ (af i ) 8 return L σ
Let now explain each step of the algorithm. As a running example, we will consider AF = ⟨A, K⟩, the AF represented in Figure 4.2 on page 32, that has been used in Part II, and we will illustrate the algorithm with the complete semantics.

Pretreatement: removing AF trivial parts

What we call the "trivial part" (or "fixed part") of an AF is simply a part of it that has a unique and fixed labelling that can be computed in linear time. As it will be seen in Section 4.2.3 on page 34, for each attack between clusters, several cases have to be considered and this can be very time consuming. In order to avoid this cost for attacks that are in the "trivial part", we simply cut that part from the AF and, only after that, look for clusters.

Given that we are interested in the complete, stable and preferred semantics, a good way to remove that "trivial part" is to compute the grounded labelling of the AF. Indeed, all complete, stable and preferred labellings include the grounded one, that is, the arguments labelled in or out in the grounded labelling are labelled in the same way in all σ -labellings. Furthermore, the grounded labelling is computable in linear time. This idea of preprocessing has been exploited in [START_REF] Cerutti | An SCC recursive metaalgorithm for computing preferred labellings in abstract argumentation[END_REF].

So, given an argumentation framework AF = ⟨A, K⟩, the AFDivider algorithm starts with computing the grounded labelling of AF (Algorithm 1, line 1).

Note:

The function ComputeGroundedLabelling(AF ) returns a partial labelling of AF in which the arguments are labelled in or out. In the following we are going to use ℓ gr for the grounded labelling and ℓ ′ gr for the labelling returned by ComputeGroundedLabelling(AF ). That is, und-labelled arguments according to the grounded semantics do not belong to ℓ ′ gr . We have: ℓ ′ gr = ℓ gr ↓ (in(ℓ gr )∪out(ℓ gr )) .

Algorithm 2: ComputeCompLabs algorithm. Input: Let ClustSet be a set of cluster structures for a component af , σ be a semantics Result: L σ ∈ 2 L (af ) : the set of the σ -labellings of af Local variables:

• κ j : a cluster structure • L κ j σ : the set of all σ -labellings of κ j • P κ j : the set of configurations corresponding to the σ -labellings of κ j • P: the set of all reunified labelling profiles 1 for all

κ j ∈ ClustSet do in parallel 2 L κ j σ ← ComputeClustLabs(σ , κ j ) 3 P κ j ← IdentifyConfigs(L κ j σ , κ j ) 4 L σ = ∅ 5 P = ReunifyCompConfigs( κ j ∈ClustSet P κ j ,ClustSet) 6 for all p ∈ P do 7 L σ ← L σ ∪   ∏ ξ ∈p {ℓ|ℓ ∈ ProfileLabellings(ξ , κ j ∈ClustSet L κ j σ )}   8 if σ = pr then L σ ← {ℓ|ℓ ∈ L σ s.t. ∄ℓ ′ ∈ L σ s.t. in (ℓ) ⊂ in (ℓ ′ )} 9 return L σ
Once the grounded labelling is computed, we consider a restriction AF hard of AF to those arguments that are labelled und in the grounded labelling:

AF hard = AF ↓ und (ℓ gr )
AF hard may possibly be a disconnected graph. We take advantage of that potential property in order to enhance the parallel computing as it will be explained in the following subsections. The function SplitConnectedComponents(AF , ℓ ′ gr ) thus split AF into disjoint sub-AFs obtained after removing the arguments labelled in or out in the grounded labelling (Algorithm 1, line 2). The CCSet variable is the set of the computed connected components.

Given that there are no relations between them, the identification of clusters inside them and the labelling computation of those connected components (Steps 2 and 3) can be made in a simultaneous way (Algorithm 1, line 3) according to the chosen semantics.

Note:

We refer to AF hard in order to facilitate the algorithm explanation however AF hard does not appear as a concrete data structure entity in the algorithm. When removing the trivial part, the rest of the AF is directly split following the connected components.

Identifying Clusters

For each of these connected components, a clustering is made (Algorithm 1, line 4) using any clustering method partitioning the AF (even a random partition method). In our experiments, we analyse three clustering methods (see Section 5.2).

We define a data structure, so called "cluster structure", to represent each cluster corresponding to the computed partition. These cluster structures will be particularly useful for proving the soundness and completeness of our algorithm (See Section 4.3 on page 47). Definition 30. (Cluster structure). Let AF = ⟨A, K⟩ be an AF, Ω be the partition of A, ω be an element of Ω (i.e. a set of arguments). κ = af , I, O, B is a cluster structure defined as follows:

af = AF ↓ ω I = {(a, b)|(a, b) ∈ K, b ∈ ω and a / ∈ ω} O = {(a, b)|(a, b) ∈ K, b /
∈ ω and a ∈ ω} B = {a ∈ A|(a, b) ∈ O or (b, a) ∈ I} Note: "I" means "inward attacks", "O" means "outward attacks" and "B" means "border arguments".

Example 13. Let suppose that the partition computed by the chosen clustering method produces the following partions:

• For af 1 : {{d, e, f , g} , {h, i}}.

• For af 2 : {{ j, k, l} , {m, n}}.

These partitions are illustrated in Figure 4.6 on page 36. Then a cluster structure is created in order to manipulate each cluster. Formally they are defined as follows:

• κ 1 = af 1.1 , I 1 , O 1 , B 1.
-af 1.1 = ⟨A 1 , K 1 ⟩, with A 1 = {d, e, f , g} and K 1 = {(d, e), (e, d), ( f , g), (g, f ), (e, f ), (d, g)} -I 1 = ∅ -O 1 = {(g, h)} -B 1 = {g} • κ 2 = af 1.2 , I 2 , O 2 , B 2 with: -af 1.2 = ⟨A 2 , K 2 ⟩, with A 2 = {h, i} and K 2 = {(h, i)} -I 2 = {(g, h)} -O 2 = ∅ -B 2 = {h} • κ 3 = af 1.3 , I 3 , O 3 , B 3 with: -af 1.3 = ⟨A 3 , K 3 ⟩, with A 3 = { j, k, l} and K 3 = {( j, k), (k, l), (l, j)} -I 3 = ∅ -O 3 = {(l, m)} -B 3 = {l} • κ 3 = af 1.3 , I 3 , O 3 , B 3 with: -af 1.4 = ⟨A 4 , K 4 ⟩, with A 4 = {m, n} and K 4 = {(m, n), (n, m)} -I 4 = {(l, m)} -O 4 = ∅ -B 4 = {m}

Computing the labellings

Consider now ComputeCompLabs algorithm (Algorithm 2) that computes the component labellings in a distributed way (Algorithm 1, line 5), relying on the clustering made. The σ -labellings of each cluster are computed simultaneously (Algorithm 2, line 1). Unlike the case of connected components used in Algorithm 1, there exist attacks between clusters. In order to compute all the possible σ -labellings of a given cluster, every case concerning its inward attacks (attacks whose target is in the current cluster but the source is from another cluster) has to be considered. Given that the sources of an inward attack could be labelled in , out or und in their own cluster, the σ -labellings of the current cluster have to be computed for all the labelling combinations of inward attack sources.

Note:

The number of cases to consider is 3 n , with n being the number of inter cluster attack sources. When choosing a clustering, there is thus a threshold between the size of the clusters and the number of edges cut to consider as it effects the overall solving time.

We call "context" a labelling of the cluster inward attack sources. It is formally defined as follows: Definition 31. (Context). Let κ = af , I, O, B be a cluster structure. A context µ of κ is a labelling of the inward attack sources of κ, i.e. {a|(a, b) ∈ I}.

Note: In the worst case there will be 3 |I| contexts. The exact number of contexts is 3 |{a|(a,b)∈I}| .

Each context induces an AF. Let κ = af , I, O, B be a cluster structure and µ be a context of κ. We can define af ′ , the induced AF of κ for a particular context µ, using the following ideas:

1. af ′ receives a copy of af 2. ∀s ∈ {s|(s,t) ∈ I and µ(s) = in } , ∀t ∈ {t|(s,t) ∈ I}, t is removed from af ′ with all the attacks that have t as endpoint. Formally induced AF are defined as follows:

Definition 32. (Induced AF). Let κ = af , I, O, B be a cluster structure with af = ⟨A, K⟩ and let µ be a context of κ. The induced AF af ′ of κ under the context µ is defined as following:

af ′ = A ′ , K ′
With: At this step, the computation of the labellings can be done in parallel for each induced AF, using any complete and sound procedure for the semantics σ . This is done by the function ComputeClustLabs (Algorithm 2, line 2). These labellings are so-called "Induced labellings". Once that, for all clusters, the ComputeClustLabs function has computed the σ -labellings for all the possible contexts, the σ -labellings are grouped according to their so-called "configurations" (Algorithm 2, line 3) to prepare the reunification 5 process. Each induced labelling ℓ is associated to a configuration ξ . This configuration expresses under which conditions an induced labelling, from a given cluster, can be reunified with another one from a neighbour cluster. This configuration is a 5-value labelling on the cluster border arguments (i.e. ∀a ∈ B). Configuration are formally defined as follows: Definition 34. (Configuration ξ ). Let κ = af , I, O, B be a cluster structure, with af = ⟨A, K⟩, let µ be a context of κ, and ℓ ∈ L µ(κ) σ be a computed labelling of κ under µ. Given ℓ, a configuration is a total function ξ : B → {in, out, iout, und, iund} such that:

• A ′ = A \ D • D = {a|a ∈ A
ξ : a ∈ B →                in if ℓ(a) = in out if ℓ(a) = out and ∃(b, a) ∈ K s.t. ℓ(b) = in iout if ℓ(a) = out and ∄(b, a) ∈ K s.t. ℓ(b) = in und if ℓ(a) = und and ∄(b, a) ∈ I s.t. µ(b) = und iund if ℓ(a) = und and ∃(b, a) ∈ I s.t. µ(b) = und
In words, for an argument a:

• ξ (a) = in means that a is successfully attacked neither from outside nor from inside the cluster. So it is legally in .

• ξ (a) = out means a is legally out from cluster point of view.

• ξ (a) = iout means that a is illegally out from the cluster point of view. • ξ (a) = iund means that a is illegally und from cluster point of view.

Note:

We do not need a value to represent the fact that an argument is illegaly in because if a border argument is in then all its attackers must be out. As defined in Section 4.2.4 on page 42, a simple constraint on endpoint attack labels is sufficient to ensure only such reunifications. This is not the case for the values out and und. Let illustrate that fact. Consider the cluster structure shown in Figure 4.9 on the next page. Let say that because of a certain context µ, a 1 is labelled out. From the cluster point of view a 3 could be labelled in, out or und, and the same for a 4 . Indeed, all these endpoint attack labels couples are valid. An extra constraint is needed to ensure that at least one between a 3 and a 4 is labelled in. That why we need to differentiate out and iout.

The same reasoning shows that we also need two undecided states (und and iund).

Note: When constructing a configuration ξ for a given labelling ℓ ∈ L µ(κ) σ , the distinction between the labels out and iout relies on ℓ itself. That is, for an argument a such that ℓ(a) = out, the value of ξ (a) depends on the fact that there exists or not an attacker b ∈ A of a such that ℓ(b) = in.

The distinction between the labels und and iund rather relies on the labelling of external attackers in I.

That is, for an argument a such that ℓ(a) = und, the value of ξ (a) depends on the fact that there exists or not an attack (b, a) ∈ I such that ℓ(b) = und. To explain why a different type of definition have been adopted to distinguish und from iund, let consider the cluster structure κ = af , I, O, B illustrated Figure 4.10 on the following page and its two induced AFs.

h i (a) µ 1 : g is labelled out i (b) µ 2 : g is labelled in h i (c) µ 3 : g is labelled und
We have: L µ 1 (κ) pr = {{(b, und), (c, und)}} and L µ 2 (κ) pr = {{(b, in), (c, out)} , {(b, out), (c, in)}}. Let consider the AF induced from κ under µ 1 = {(a, und)} and the labelling ℓ = {(b, und), (c, und)}. While establishing the configuration ξ associated to ℓ, we can observe that although b has an undecided internal attacker, namely c, it is not sufficient to say that b is legally labelled und from the cluster point of view, and so that ξ (b) = und. To prove this, let consider the AF induced from κ under µ 2 = {(a, out)}. This latter AF does not admit a preferred labelling such that b is labelled und. As a consequence, the fact that ℓ(b) = und is due to the fact that µ 1 (a) = und.

Following this observation, the configuration of a labelling has been defined as stated in Definition 34 on page 36.

Example 15.

Here is the result according to the complete semantics for our running example (See Figure 4.7 on the previous page):

• For κ 1 we have only one context µ κ • For κ 2 we have three contexts: µ κ 2 1 = {(g, out)}, µ κ 2 2 = {(g, in)} and µ κ 2 3 = {(g, und)}. Table 4.2 on the next page gives their corresponding labellings and induced configurations.

ℓ κ 1 1 ξ κ 1 1 ℓ κ 1 2 ξ κ 1 2 ℓ κ 1 3 ξ κ 1 3 d out in und e in out und f out in und g in in out out und und
• For κ 3 we have only one context µ κ 3 1 = ∅ (since I = ∅) that gives the labellings and their induced configurations shown in Table 4.3 on the next page.

• For κ 4 we have three contexts: µ κ 4 1 = {(l, in)}, µ κ 4 2 = {(l, out)} and µ κ 4 3 = {(l, und)}. Table 4.4 on page 41 gives their corresponding labellings and induced configurations. After that we have computed the different labellings and their corresponding configuration, we keep only the "distinct labellings" with their "merge configurations". The merge configuration as defined is the most flexible configuration of a given labelling. It ensures all the requirements for a good reunification without adding unwanted restrictions.

Note: ξ κ 4 1 (m) ̸ = iout because ℓ κ 4 1 (n) =
L κ D = {ℓ κ i |ℓ κ i ∈ L κ and ∄ℓ κ j ∈ L κ s.t. ℓ κ j = ℓ κ i and j < i} (a) Under µ κ 2 1 = {(g, out )} ℓ κ 2 1 ξ κ 2 1 h in in i out (b) Under µ κ 2 2 = {(g, in )} ℓ κ 2 2 ξ κ 2 2 h out iout i in (c) Under µ κ 2 3 = {(g, und )} ℓ κ 2 3 ξ κ 2 3 h und iund i und
ℓ κ i ∈ L κ be a labelling and C ℓ κ i = {ξ κ j |ξ κ j ∈ C κ s.t. ℓ κ j = ℓ κ i } be the set of all its possible configurations. Let ξ ∈ C ℓ κ i be a possible configuration of ℓ κ i . The merge configuration ξ ℓ κ i of ℓ κ i is defined as follows: ∀a ∈ B, ξ ℓ κ i (a) =                in if ℓ κ i (a) = in out if ℓ κ i (
Example 18. This step will affect only the cluster κ 4 as ℓ κ 4 1 = ℓ κ 4 3 = ℓ κ 4 5 and ℓ κ 4 4 = ℓ κ 4 6 . The new set of labelling/configuration of κ 4 is shown on Table 4.5 on the next page.

We can notice after this filtering and merging process that:

• One context can give several labellings.

• From a labelling is induced one and only one merge configuration.

• Several labellings can induce the same merge configuration (See Example 19 on the next page).

Reunifying the results

The labelling reunifying process is made in two steps: firstly, the reunification of the component labellings (i.e. the reunification of their cluster labellings together) and secondly, the reunification of the whole AF labellings (i.e. the reunification of its component labellings together). The first step is done in Algorithm 2 on page 31, lines 5 to 8. The second one is in Algorithm 1 on page 30, line 7. Let detail them.

Component labelling reunification

In [START_REF] Lafages | Clustering and distributed computing in abstract argumentation[END_REF][START_REF] Doutre | A distributed and clustering-based algorithm for the enumeration problem in abstract argumentation[END_REF], the first version of our algorithm, the reunification was directed made on the cluster labellings. We, later, found out that this could be enhanced if we reunify the labelling configurations instead. Indeed, this deals with much less elements as several cluster labellings could have the same configuration. We introduced so the notion of "reunified labelling profiles".

Definition 37. (Reunified labelling profiles). Let af = ⟨A, K⟩ be an AF and {κ 1 , ..., κ n } be the set of cluster structures corresponding to the clustering of af . Let {L κ 1 σ , ..., L κ n σ } be the set of distinct labelling sets of each cluster of af and {ξ κ 1 , ..., ξ κ n } be the set of their corresponding merge configuration sets. Let {I 1 , ..., I n } be the set of sets of inward attacks of the different clusters and I = n i=1 I i be the union of those sets. Let {B 1 , ..., B n } be the set of the sets of their border arguments and B = n i=1 B i be the union of those sets. Let {ξ 1 , ..., ξ n } be a set of configurations such that, for all i ∈ {1, ..., n},

ξ i ∈ ξ κ i . Let Ξ = ( n i=1 ξ i )
be the union of these configurations. p = {ξ 1 , ..., ξ n } is a reunified labelling profile (or equivalently, the configurations ξ 1 , ..., ξ n are said to be compatible together) if and only if:

∀a ∈ B,                    Ξ(a) = iout =⇒ ∃(b, a) ∈ I s.t. Ξ(b) = in Ξ(a) = iund =⇒    ∃(b, a) ∈ I s.t. Ξ(b) = und or Ξ(b) = iund and ∄(b, a) ∈ I s.t. Ξ(b) = in    Ξ(a) = in =⇒ ∀(b, a) ∈ I, Ξ(b) = out or Ξ(b) = iout Ξ(a) = und =⇒ ∀(b, a) ∈ I, Ξ(b) ̸ = in
Note: Distinct reunified labellings can have the same profile, but a labelling only has one profile. This notion defined, let continue the explanation of the AFDivider algorithm.

At this step, the ReunifyCompConfigs function is called (Algorithm 2, line 5) in order to reunify the compatible configurations of the cluster labellings together. To do that, the ReunifyCompConfigs transforms that reunifying problem into a constraint satisfaction problem (CSP).

Note:

We chose to use this method because in CSP modeling each variable can have an arbitrary value domain and constraints can have any arity and be of any nature. Those two properties make CSP modeling very straightforward and easy to automatise. Furthermore, there are a lot of CSP solvers available. This being said, here are the four steps of the transformation process:

1. For each cluster κ i , a variable V κ i is created. For each of them, the domain is the set of their computed labelling merge configurations ξ κ i .

2.

For each border argument a j , a variable V a j is created with a domain corresponding to their possible labels, i.e. {in , out , und }.

3.

For each inter-cluster attack (a, b), a constraint is added with the following set of valid tuples:

{(a = in , b = out ), (a = out , b = in ), (a = out , b = out ), (a = out , b = und ), (a = und , b = out ), (a = und , b = und )}
Let κ i = af , I, O, B be a cluster structure and ℓ be one of its induced labellings. Let ξ κ i ℓ be a value of the domain of V κ i corresponding to the merge configuration of ℓ.

For each

ξ κ i ℓ in V κ i domain:
(a) Constraints are added to map the merge configuration with the argument labels. The constraints are defined as following:

(V κ i = ξ κ i ℓ ∧ ξ κ i ℓ (a j ) = in ) =⇒ V a j = in ∀a j ∈ B, (V κ i = ξ κ i ℓ ∧ (ξ κ i ℓ (a j ) = out ∨ ξ κ i ℓ (a j ) = iout )) =⇒ V a j = out (V κ i = ξ κ i ℓ ∧ (ξ κ i ℓ (a j ) = und ∨ ξ κ i ℓ (a j ) = iund )) =⇒ V a j = und (b)
Constraints are added for all arguments labelled iout in ξ κ i ℓ :

∀a j ∈ {a|ξ κ i ℓ (a) = iout }, V κ i = ξ κ i ℓ =⇒ ∃(a k , a j ) ∈ I s.t. V a k = in (
c) Constraint are added for all arguments labelled iund in ξ κ i ℓ :

∀a j ∈ {a|ξ κ i ℓ (a) = iund }, V κ i = ξ κ i ℓ =⇒ ∃(a k , a j ) ∈ I s.t. V a k = und Note:
The constraints have to be seen as declarative rules. For example the rule:

V κ i = ξ κ i ℓ =⇒ ∃(a k , a j ) ∈ I s.t. V a k =
und as to be understand as "If the variable V κ i has the value ξ κ i ℓ , there must be a variable corresponding to one of the attackers of a j that takes the value und".

The solutions of that CSP modelling are the reunified labelling profiles (corresponding to values of the V κ i variables).

Example 20. Let illustrate this with the CSP modelisation for the reunification of af 1 . Let Ψ af 1 = ⟨X, D,C⟩ be that modelling. Ψ af 1 is defined as following:

• X = {V κ 1 ,V κ 2 ,V g ,V h } • D =                D(V κ 1 ) = {ξ κ 1 1 , ξ κ 1 2 , ξ κ 1 3 }, D(V κ 2 ) = {ξ κ 2 1 , ξ κ 2 2 , ξ κ 2 3 }, D(V g ) = {in, out, und}, D(V h ) = {in, out, und}                • C = {c 1 , c 2 , c 3 , c 4 , c 5 } is a set of constraints, with
c 1 being the constraint that expresses the attack relation from g to h, corresponding to Step 3, c 2 being the constraint expressing the fact that the merge configurations of κ 1 impose a label on each of its border arguments (i.e. on g), corresponding to Step 4a, c 3 being the constraint expressing the fact that the merge configurations of κ 2 impose a label on each of its border arguments (i.e. on h), corresponding to Step 4a, c 4 being the constraint expressing the fact that ξ κ 2 2 can only be reunified with a merge configuration of κ 1 in which g is labelled in, corresponding to Step 4b (see Table 4.1 on page 39 and Table 4.2 on page 40), c 5 being the constraint expressing the fact that ξ κ 2 3 can only be reunified with a merge configuration of κ 1 in which g is labelled und, corresponding to Step 4c (see Table 4.1 on page 39 and Table 4.2 on page 40).

Note: c 4 and c 5 are constraints only for precise merge configurations of κ 2 . c 4 and c 5 must allow g being labelled with any label if the reunification is about another merge configuration of κ 2 . c 1 accepts only the following tuples:

-(V g = in,V h = out) -(V g = out,V h = in) -(V g = out,V h = out) -(V g = out,V h = und) -(V g = und,V h = out) -(V g = und,V h = und)
c 2 accepts only the following tuples (see Table 4.1 on page 39):

-(V κ 1 = ξ κ 1 1 ,V g = in) -(V κ 1 = ξ κ 1 2 ,V g = out) -(V κ 1 = ξ κ 1 3 ,V g = und)
c 3 accepts only the following tuples (see Table 4.2 on page 40):6

-(V κ 2 = ξ κ 2 1 ,V h = in) -(V κ 2 = ξ κ 2 2 ,V h = out) -(V κ 2 = ξ κ 2 3 ,V h = und)
c 4 accepts only the following tuples:

-(V κ 2 = ξ κ 2 2 ,V g = in) -(V κ 2 = ξ κ 2 3 ,V g = in) -(V κ 2 = ξ κ 2 3 ,V g = out) -(V κ 2 = ξ κ 2 3 ,V g = und) -(V κ 2 = ξ κ 2 1 ,V g = in) -(V κ 2 = ξ κ 2 1 ,V g = out) -(V κ 2 = ξ κ 2
1 ,V g = und) c 5 accepts only the following tuples:

-(V κ 2 = ξ κ 2 3 ,V g = und) -(V κ 2 = ξ κ 2 2 ,V g = in) -(V κ 2 = ξ κ 2 2 ,V g = out) -(V κ 2 = ξ κ 2 2 ,V g = und) -(V κ 2 = ξ κ 2 1 ,V g = in) -(V κ 2 = ξ κ 2 1 ,V g = out) -(V κ 2 = ξ κ 2 1 ,V g = und)
Note: Both c 4 and c 5 have to be respected. As a consequence, the valid tuples concerning V κ 2 and V g are the ones which are both in c 4 's valid tuples and c 5 's valid tuples. So the second and third tuples accepted by c 4 and the third and fourth tuples accepted by c 5 will be useless.

The solutions for af 1 are:

• ξ κ 1 1 = {(g, in)} , ξ κ 2 2 = {(h, iout)} • ξ κ 1 2 = {(g, out)} , ξ κ 2 1 = {(h, in)} • ξ κ 1 3 = {(g, und)} , ξ κ 2 3 = {(h, iund)}
Using the same process we obtain the following results for af 2 :

• ξ κ 3 1 = {(l, und)} , ξ ′κ 4 1 = {(m, out)} • ξ κ 3 1 = {(l, und)} , ξ ′κ 4 3 = {(m, und)}
For each of the reunified labelling profile computed (Algorithm 2, line 6 and 7), labelling parts corresponding to the configurations forming the reunified profile are reunified together.

Example 21. Following Example 20, Table 4.6 shows the complete labellings obtained for af 1 and Table 4 A special step has to be done for the preferred semantics as this reunifying process does not ensure the maximality (w.r.t. ⊆) of the set of in -labelled arguments (so not all of the labellings produced in Algorithm 2, line 7 are preferred ones). Indeed, the preferred semantics is not bottom-up decomposable (see Proposition 8 on page 21). A maximality check is done (Algorithm 2, line 8) in order to keep only the wanted labellings.

Note: This maximality check has a complexity of Θ(n 2 ), n being the number of component labelling produced by the algorithm. As a consequence, if n is large, it could be very time consuming. To enhance this check, an optimization have been made: check only the reunified labellings whose reunified profile labelling contains an und-labelled argument. Indeed, the preferred semantics is top-down decomposable. As all the cluster labellings are maximal w.r.t. their corresponding context, the only way for a reunified labelling to not be maximal is to have an und-labelled argument at the cluster border. Experimental analysis show the interest of this optimization. See Section 5.3.4 on page 58.

Note: When computing the preferred semantics with a partition selector that does not cut SCC a f , the maximality check is not necessary. Indeed, any reunified component labelling will be maximal. See the proof of Proposition 11 for more details.

Note: When computing the stable semantics, the set of labellings L σ returned by ComputeCompLabs may be empty. It happens when one of the component clusters has no stable labelling.

Whole AF labelling reunification

Now that all the component labellings are built, we can reunify the labellings of the whole AF. Indeed, given that ℓ gr is a fixed part of all σ -labellings of AF and that all the connected components are completely independent, the building of the σ -labellings of the whole AF is made with a simple Cartesian product (Algorithm 1, line 7) between the labellings of all the components and the grounded one. If one of the components has no labelling then the whole AF has no labelling (so L σ = ∅).

Example 22. To finish our illustration, following Example 21 on the previous page and following Example 12 on page 31, the complete labellings of AF are shown in Table 4.8 on the next page.

AFDivider properties: soundness and completeness

In this section we give properties ensuring that the AFDivider algorithm works well. More precisely, we ensure that it gives all the expected labellings for the complete, stable and preferred semantics; this is the notion of completeness, and that the algorithm gives only good labellings for the semantics complete, stable and preferred ; this is the notion of soundness.

In [START_REF] Baroni | On input/output argumentation frameworks[END_REF], Baroni et al. introduce several notions and proved semantics properties that are useful to prove that our proposed algorithms are sound and complete. They are presented in Chapter 2 on page 16, the most important being fully decomposable and top-down decomposable semantics properties.

A semantics will be a fully decomposable or top-down decomposable semantics if for any AF and any partition of this AF, it is possible to reconstruct all the labellings of the whole AF by combining the labellings (under the same semantic) of the partition parts.

To be more precise, the difference between a top-down decomposable semantics and a fully decomposable one is that for a top-down decomposable one, when doing this process of labelling part reunification, all the semantics labellings will be found but it is also possible to obtain non correct labellings, whereas, for a fully decomposable all and only the correct semantics labellings will be obtained.

As one may notice, in the AFDivider algorithm we do not use the notion of AF with input (introduced in [START_REF] Baroni | On input/output argumentation frameworks[END_REF], see Chapter 2). Instead, we use the notion of cluster structure. There is definitely a link between these two notions, link that we will present before proving the soundness and the completeness of our algorithm.

Relation between AFs with input and cluster structures

The aim of this comparison is to use semantics decomposability properties for cluster structures. The following example illustrates the differences between the two approaches. Given ω = {h, i}, af = AF ↓ ω is represented as follows:

h i Considering our approach, the cluster structure for ω is κ = af , I = {(g, h)}, O = ∅, B = {h} . Then three contexts exist: µ 1 = {(g, out)}, µ 2 = {(g, in)}, µ 3 = {(g, und)}. And so three induced AFs can be defined (for respectively µ 1 , µ 2 , and µ 3 ):

h i i h i
Considering the approach proposed by Baroni and co., the AF with input corresponding to ω is defined by af , {g}, µ, {(g, h)} with µ being either µ 1 , or µ 2 , or µ 3 . So three standard AFs can be defined (for respectively µ 1 , µ 2 , and µ 3 ):

g g ′ h i g h i g h i
Relying on the notion of complete-based semantics (Definition 38), Proposition 9 gives the correspondence between our induced AFs and the standard AFs.

. Let κ = ⟨af , I = ω K , O = K ∩ (ω × (A \ ω)), B = {a|(a, b) ∈ O or (b, a)
∈ I}⟩ be the cluster structure corresponding to ω. Let µ be a context of κ. The following equation holds:

L µ(κ) σ = F σ (af , ω inp , µ, ω K )
Note: As a recall, see Definition 15 on page 16 for ω K and ω inp .

□ Proof of Proposition 9: link (See page 219).

Soundness and completeness

First, we have for Algorithm 2 the two following properties: From Propositions 10 and 11, we can prove that the entire algorithm is sound and complete for the stable, preferred and complete semantics.

Regarding Algorithm 1, two similar properties can be established:

Proposition 12 (Completeness of Algorithm 1). Algorithm 1 is complete for the stable, complete and preferred semantics.

□ Proof of Proposition 12: link (See page 223).

Proposition 13 (Soundness of Algorithm 1). The following properties hold:

1. Algorithm 1 is sound for the stable and complete semantics.

2. Algorithm 1 is sound for the preferred semantics.

□ Proof of Proposition 13: link (See page 224).

Chapter 5

AFDivider : Experimental analysis

Introduction

In this chapter we present experimental results conducted with three AFDivider variants presented in Section 5.2 on the next page.

Note: A detailed technical documentation of the whole AFDivider project (user manual of the solver, software sources, project installation, experimental environment) can be found in [START_REF] Doutre | AFDivider: Manual and Documentation[END_REF].

The experiments have been made on some hard instances of the ICCMA competition for the preferred, stable and complete semantics and for the enumeration problem. 1 The AF instances studied are of Barabási-Albert (BA), Erdős-Rényi (ER), Watts-Strogatz (WS), Traffic (TR), Block world (BW) and Ferry (F2) types. The three first types have been generated by AFBenchGen2 (see [START_REF] Cerutti | Generating structured argumentation frameworks: Afbenchgen2[END_REF]), the fourth type are AF generated from real traffic data (see [START_REF] Diller | Traffic networks become argumentation frameworks[END_REF]) and the last two types are block world and ferry planning problems transformed into AF problems (see [START_REF] Cerutti | Exploiting planning problems for generating challenging abstract Arg. Frameworks[END_REF]).

To compute the labellings of a cluster given a particular labelling of its inward attack sources, we have used an already existing solver called "µ-toksia", the winner of all enumeration problem tracks of the IC-CMA 2019 session, which transforms the AF labelling problem into a SAT problem [START_REF] Niskanen | µ-toksia (version 2019-10-31): SAT-based solver for static and dynamic argumentation frameworks[END_REF]. In these experiments, we compare our algorithms (using µ-toksia) with µ-toksia itself, and with several other solvers presented in ICCMA 2017 and 2019. In the following, solver names are suffixed by the year of the ICCMA session for which they participated. For each experiment, we used 6 cores of an Intel Xeon Gold 6136 processor, each core having a frequency of 3 GHz. The RAM size was 45GB. The timeout was set to 1 hour for the real time. Analysis have been conducted on a number of solvers. The ones which are presented here are those which succeed to solve at least one of these selected hard instances. 2After the presentation of the three AFDivider variants (Section 5.2 on the next page), we analyse our result experiments (Section 5.3 on page 55), by focusing, in a first step, on the success rate of the solvers, in a second step, on their solving time, and in a third step, by comparing the clustering time with the resolution time of our AFDivider variants.

Clustering methods

Among the various clustering methods, three of them, particularly well suited for the kind of graphs that we want to address (see Chapter 5 on the previous page), are presented here. For each clustering method associated with the AFDivider algorithm, an implementation has been made.

Spectral clustering

The first clustering method that we used for the AFDivider algorithm is the so-called "spectral clustering". This method is presented and illustrated in details in the Mathematical Background, Part VI, Section 17.2 on page 188. This clustering method, originally used for data mining, relies on a similarity matrix that represents how much a data in a dataset is similar to another one. To adapt it for AF clustering, we defined the similarity between two arguments a and b as the number of attacks between them. Thus, this number may be 0, 1 or 2. Formally we defined a transformation that produces a non-directed graph from an AF. Definition 39. (Undirection transformation) Let AF = ⟨A, K⟩ be an AF. The non-directed graph G = (V, E,W ) obtained by the undirection transformation of AF (noted U (AF )) is defined as following: .

• V = A • E = {(a, b)|(a, b) ∈ K or (b, a) ∈ K} • W : E → {0, 1, 2} is defined as following: W : (a, b) →      0 if (a, b) / ∈ K and (b, a) / ∈ K, 1 if ((a,
directed weighted graphs can be represented by their adjacency matrices defined as:

M af 1 a =               d e f g h i d 0 2 0 1 0 0 e 2 0 1 0 0 0 f 0 1 0 2 0 0 g 1 0 2 0 1 0 h 0 0 0 1 0 1 i 0 0 0 0 1 0               M af 2 a =            j k l m n j 0 1 1 0 0 k 1 0 1 0 0 l 1 1 0 1 0 m 0 0 1 0 2 n 0 0 0 2 0           
Given an AF, the AF spectral clustering relies thus on a kind of adjacency matrix where the directionality of edges is omitted and where the matrix values are the number of edges between two arguments. Basically, the more an argument will be related to another, the more similar the two arguments will be considered. This similarity criterion is particularly relevant for non-dense graphs with a clustered structure. Indeed, it produces sparse matrices and as a consequence the eigenvector equation system to solve is simplified as there are many zero values. This is what motivated our choice for the spectral clustering method. Note: For a step by step illustration of how we end up with this result, see Section 17.2 on page 188.

Notice that when using the spectral clustering method for the preferred semantics computation, the optimization described in the first note of Page 47 is used for the maximality check (made in Algorithm 2, line 8).

USCC based clusterings

While the idea of the previous clustering was to find groups of arguments highly related in terms of attack relations, gathering them together regardless of their membership to common SCC structures, the two others clustering studied have been proposed to ensure that SCCs are not split into different clusters.

Given an AF, the so-called "USCC Chain" clustering forms clusters as following (each cluster being a USCC a f ):

1. First, the set of SCC is computed.

2. Then neighbour SCC singletons are joined together as chains. The first element of such a chain is thus a singleton that is not attacked by a singleton.

3. If a singleton belongs to several chains, it is kept only by the chain that has the least inward attacks (attacks coming from arguments that are not in the chain). Note that the intersection of the so built chains is empty. Ties are broken arbitrarily.

4. The last step is to join SCC and chains together so that there are not too many clusters of little size. This is done in an iterative way. The smallest group is merged to its smallest neighbour group, and that until there is no group of less than a certain number of arguments. We experimentally choose to fix this threshold to 10.

The third clustering studied is the so-called "USCC Tree" clustering. It has several common steps with the USCC Chain method, but it differs on one point: instead of forming chains that do not intersect, chains that have common arguments are merged together. This process thus produces trees.

Example 26. Both USCC based clustering methods give the same result on the previous AF example. In order to highlight the differences between the two, let consider AF = ⟨A, K⟩, the AF shown in Figure 5.4 on page 62. For the sake of brevity, the AF chosen is small. Therefore we will not illustrate the fourth step (otherwise the AF would have to be to big).

The two first steps are the same for both methods:

• Step 1: the SCC a f of AF are computed. They are graphically represented in Figure 5.5 on page 63.

• Step 2: In the second step the singletons are joined together as shown in Figure 5.6 on page 63.

At

Step 3, they will differ:

• The USCC Chain method will produce the partition: {{a, b} , {c, d} , {e, f } , {g, h, i} , { j, k}}

• The USCC Tree method will produce the partition: {{a, b} , {c, d, e, f } , {g, h, i} , { j, k}} Notice that when using these two clustering methods for the preferred semantics computation, we skip the maximality check made in Algorithm 2, line 8, as explained in the second note of Page 47.

Result presentation

Success Count Comparison

Table 5.1 on the next page shows the number of successes for each solver, by AF type selection. For each selection (columns) there are three values in each cell corresponding respectively to the preferred, the stable and the complete semantics. The following list provides references for the different solvers analysed:

• For AFDiv-spectral, see Section 5.2.1

• For AFDiv-USCC-Chain, see Section 5.2.2

• For AFDiv-USCC-Tree, see Section 5.2.2

• For ArgSemSAT2017, see [START_REF] Cerutti | ArgSemSAT-2017[END_REF] • For Argmat-dvisat2017, see [START_REF] Pu | argmat-dvisat: A division-based algorithm framework for solving argumentation problems using SAT[END_REF] • For Argmat-sat2017, see [START_REF] Pu | argmat-sat: Applying SAT solvers for argumentation problems based on boolean matrix algebra[END_REF] • For Aspartix2019, see [START_REF] Dvorák | Aspartix-v19-system description for iccma'19[END_REF] • For Cegartix2017, see [START_REF] Dvorák | Cegartix v2017-3-13: A SAT-based counter-example guided argumentation reasoning tool[END_REF] • For Coquiaas2019, see [START_REF] Lagniez | Coquiaas v3. 0 iccma 2019 solver description[END_REF] • For µ-toksia2019, see [START_REF] Niskanen | µ-toksia (version 2019-10-31): SAT-based solver for static and dynamic argumentation frameworks[END_REF] • For Pyglaf2017, see [2] • For Pyglaf2019, see [START_REF] Alviano | The pyglaf argumentation reasoner[END_REF] Note: The P-SCC-REC algorithm presented in Section 7.2.3 on page 79 has not been included in this study because no solver were available, as best as we know.

Among the first things we can observe is that the AF type, has a great impact on the aptitude of a given solver to enumerate the labellings under a given semantics. As an example, for the preferred semantics, Argmat-dvisat2017 solves nine BA instances (which is the best score among other solvers than ours) but only three WS instances. Likewise, we can also observe that for a given solver and for a given AF type, the aptitude to succeed depends on the semantics. As an example, for the ER type, Pyglaf2017 succeeds in solving six instances for the preferred semantics but four for the stable one. Such a table of experimental results is then a good tool to identify solvers specificities and capabilities.

Another observation is that the complete semantics is much harder than the two other ones. Although several decision problems under the preferred and the stable semantics are harder than under the complete one, in practice the first stumbling block to enumerate the labellings of a given semantics is (for most of the studied AF instances) the number of labellings it produces.

Let focus on our solvers results. One interesting fact is that the clustering criterion used by the AFDivider algorithm has an impact on the AF types for which it will be well suited. Indeed there are two different behaviour classes: one for the USCC clustering variants and another one for the spectral clustering one. Both behaviour classes are good on BA type and bad for BW and F2 types (whatever the semantics) but we can see that the USCC variants are good on ER and WS types which is not the case for the spectral one while the latter is better than them on TR type. These results were expected. Indeed, ER and WS type instances do not have a structure with groups of arguments such that the intra density (within group) is greater than the one inter groups (outside group). That is precisely what spectral clustering is seeking for. In contrast, some TR instances have a structure which is much less adequate to USCC variants. Given that for each cut attack three cases have to be consider when computing a cluster labellings, clusterings as the three proposed ones are not suited for BW and F2 instance types. Indeed these instances are translation of planning problems in which the notion of sequential constraints (over time or resources) is very present. This leads to AFs with particular shape for which a sequential reasoning (even though multi-threaded) is better. Nevertheless, it is worth noting that spectral clustering gives better result than USCC ones on BW type. Unlike planning AF, the BA type is completely adequate to such a cutting process. Although not presented in the following tables, other clustering methods (fully random partition, among them) have been studied for the AFDivider algorithm. All of them give pretty good results on the BA type, even better than most of the studied solvers. This shows that a clustering approach (even with a random clustering) can give interesting results.

We can see that our USCC solvers are among the best considering BA, ER and WS types (last column). The success rate over all types and for all semantics is of 43.11% for Pyglaf2017 (best rate), 42.22% for Aspartix2019 (second rate) and of 24% for both of the USCC variants. When considering only the BA, ER and WS types, we have a rate of 39.23% for Aspartix2019 (best rate), 38.52% for both of the USCC variants (second ones), 34.07% for Pyglaf2019 (third one) and 31.85% for Pyglaf2017 (forth rate).

It is also interesting to consider the reasons why some experiments failed. We observe that, for solvers other than the AFDivider variants, about 2% of the failures are due to memory overflow while 98% are due to timeout. For the AFDivider variants, about 56% are due to memory overflow and 44% are due to timeout. Knowing that in most cases, it is not because of time limitation that the AFDivider fail, a better memory management could increase its already good performances as highlighted in Section 5.3.5 on page 59.

Resolution Time Comparison

Let now consider the resolution time of those solvers. Given that the different solvers do not succeed for the same instances, making an overall average time could be misleading. Instead, in Table 5.2 on the following page, we compare the solvers two by two on instances solved by both of them (that is, same couple instance/semantics succeeded). Because of that, the average time is computed regardless of the AF type and of the semantics. To illustrate how to read this table, let consider the last cell of the first column. Over all their common instances successes, in average Pyglaf2019 takes 552 seconds more than AFDiv-spectral.

While the AFDiv-spectral does not stand out on the previous analysis, this table shows that it is faster than all other solvers (except ours). On the instances which have been successfully solved by the USCC variants and AFDiv-spectral (which are mainly of BA type), the USCC ones are faster. We can observe that no solvers are better than all others, only 3 of 12 are better than 8 other ones: AFDiv-spectral, AFDiv-USCC-Chain and Aspartix2019. The three AFDivider variants give good results compared to other solvers.

AFDiv-spectral

AFDiv-USCC-Chain AFDiv-USCC- The values in bold and large font correspond to the case where the solver given in the line is faster than the solver given in the column.

Given that several studied solvers are multi-threaded, a similar analysis has been made considering the CPU time. Even though the values may slightly differ from this table, the comparison between the solvers stays the same. As there are few changes, this table is not relevant.

Clustering Impact Comparison

Let now consider the impact of the clustering against the resolution time of our AFDivider variants (Table 5

.3 on the next page).

First of all, we can observe that the clustering is very time-efficient compare to other steps of the algorithm.

As expected, the USCC Tree clustering is faster than the USCC Chain one. Indeed merging chains is simpler than wisely separating chains sharing common arguments. We can also observe that the spectral clustering is more time consuming than the USCC ones. AFDiv-spectral takes more time to compute the labellings than the USCC variants, which means that on those common solved instances the USCC partitions are better chosen.

AFDiv-spectral

As Table 5.2 on the previous page shows, on common instances AFDiv-spectral is slower than the two other ones. Nevertheless, it is worth noting, as it has been said previously, that they form two distinct behaviour classes of solvers, that do not share so many successes. The common ones are mainly of BA type.

Table 5.3 shows another important fact. Most of the resolution time, on those instances at least, comes from the cartesian product and the result printing. Indeed, some instances may admit millions and even more of labellings which take much time to print. Even without taking the printing time into account, we can see that the cartesian product alone takes between 40% (in the spectral case) and 55% (in the other cases) of the effective calculation time. The "Compact Enumeration Representation" introduced in Chapter 6 on page 64 is a proposal to address this issue.

Maximality Check Impact Comparison

When computing the preferred semantics with the AFDivider algorithm, a maximality check of the component labellings is needed. In Page 47 two optimizations for this check have been described:

1. Checking only profiles with und -labelled arguments 2. Skipping the maximality check when using USCC clustring methods Let now consider the impact of those optimizations.

Checking only profiles with und -labelled arguments

To show its interest, we compare the labelling of components with and without the optimization, the components being clustered following the very same partition ensuring so a fair comparison.

Table 5.4 on the following page sums up the labelling step details of each experiment. We choose the first component from a TR AF type and the second one from a BW AF type. All durations are given in second. The percentages in brackets represent the duration rate of each step compared with the total component labelling time.

Without the optimization, each computed component labelling has to be checked. This step thus has a complexity of Θ(n 2 ), n being the number of computed component labellings. As shown by Table 5.4, it can be very consuming. The first component produces 3408 labellings while the second one 76366.

We can observe that with the optimization the maximality check is 100 times faster for the first component (2.251s against 0.021s), while the total component labelling is 1.5 times faster (6.631s against 4.370s). Notice that this optimization may produce an empty set of labellings to check. The value 2.5 × 10 8 for the maximality check of second component indicated that the set to check is empty.

Skipping the maximality check when using USCC clustering methods

In this experiment, we compared the use of the optimization analysed above against no maximality checking while using an USCC clustering.

None of the experiments made produced profiles with und -labelled arguments. As a consequence, in both cases the component labelling process was almost identical. The impact of skipping the maximality check was thus negligible, on those instances.

However, if we want to compare the use of this optimization against the use of no optimization at all (that is, the use of the naive maximality check in Θ(n 2 )) we can refer to the previous section to have an idea of the benefit we can get.

Memory Overflow Analysis

Table 5.5 on page 61 shows the details of the memory overflows that occurred during our experiments. Following AF types, it shows which percentage of them happened at each step of the algorithm. As an example on how to read the table, let analyse the memory overflows made by AFDiv-spectral on the TR instances: 12 of the 45 TR instances ended up with an overflow, 41.67% of them (that is, 41.67% of the 12) occurred during the clustering and cutting step, 8.33% during the cluster and component labelling step, 33.33% during the cartesian product and 16.67% during the printing step (which gives a total of 100%).

Note:

The values "NaN" are used in the table when no instance of the corresponding AF type has produced a memory overflow.

We can observe that, BA instances put aside, most of the memory overflow occurs during the clustering and cutting phase and secondarily during the cartesian product phase. For BA instances, it is during the cartesian product phase and secondarily the labelling phase.

Further investigations need to be made to understand the memory overflows that occur during the clustering and cutting phase, especially for the USCC clusterings. Indeed this could be due to implementation flaws.

If it is only an implementation flaw supposition in the clustering and cutting phase, it is a certainty for the printing phase. If stored properly, it must be possible to print all the computed labellings without using much space (as an example with the help of buffers, iterators, etc).

The memory overflows that occurred during the cartesian product and the printing phases can be avoided by using the "Compact Enumeration Representation" introduced in Chapter 6 on page 64.

Synthesis

As a synthesis, in our experimental analysis, we instantiated the generic distributed and clustering based algorithm AFDivider, that enumerates the complete, stable and preferred semantics labellings, with three different clustering methods. We compare their performances, to other solvers, according to AF types, over success rate and resolution time. It comes out from this study that the AFDiv-spectral variant is faster than most solvers (except ours) in average on common successful instances while the AFDiv-USCC-Tree and AFDiv-USCC-Chain variants succeed to solve most instances when considering Barabási-Albert (BA), Erdős-Rényi (ER) and Watts-Strogatz (WS) AF types, for the complete, stable and preferred semantics.

Based on the AF types that they solve efficiently, we identify two behaviour classes among our three solvers: one for the spectral clustering and one for the USCC based clusterings. This shows that the clustering method which is used has an important effect on the performances of the AFDivider algorithm on a particular AF type.

A major improvement of the AFDivider algorithm would then be to "know" in advance which clustering method (including ones other than those presented in Section 5.2 on page 52) should be used for a particular AF instance. Experiments could be conducted to learn, for example with a neural network, which one to use. To go further, even the cutting process could be supervised by a neural network trained to cluster AF instances following their structure. As a consequence, for any known AF type, the most appropriate clustering method would be used to solve each AF instance efficiently.

The experiments also show that a better memory management and a more compact labelling representation could enhance the efficiency of the algorithm. This is what we tried to solve introducing the notion of "compact enumeration representation" presented in Chapter 6 on page 64. AFDivider : Compact representation

AFDiv-spectral

Motivation and Definition

The last step of the AFDivider algorithm (Algorithm 1 on page 30, line 7) is a huge cartesian product between the fixed labelling part and all the component labellings. Experimental analysis shows that in some cases this enumeration construction is very time consuming. Furthermore, we realise that in order to answer common AF decision problems, this enumeration could not be necessary. These observations make us think about a "compact enumeration representation". Notice that the aim of this new algorithm is no more to enumerate the labellings/extensions under a given semantics but rather to provide a data structure from which all AF classical decision problems, and even more, can be answered. In few words, the compact enumeration representation is the set of the component labelling sets (and the fixed part).

Definition 40 (Compact Enumeration Representation). Let σ be a semantics. Let AF = ⟨A, K⟩ be an AF and Ω = {ω 0 , ..., ω n } be a partition of A such that ω 0 is the trivial part of AF (i.e. ω 0 = A∩(in(ℓ gr )∪out(ℓ gr )))

and ω 1 , ..., ω n correspond to the sets of arguments of the connected components of AF hard , following the computation made by Algorithm 1. Let af 0 , ..., af n be the set of sub-AF corresponding to Ω. The compact enumeration representation Comp σ (AF ) is the set: {L σ (af 0 ), ..., L σ (af n )}.

Example 27. Let consider one more time our running example. Let AF = ⟨A, K⟩ be the AF shown in Figure 6.1 on the following page. The compact enumeration representation corresponding to the complete semantics is the set {A, B,C} with:

A = L σ (af 0 ) = {{(a, in), (b, out), (c, out)}} B = L σ (af 1 ) =         
{(e, in), ( f , out), (g, in), (h, out)}

{(e, out), ( f , in), (g, out), (h, in)}

{(e, und), ( f , und), (g, und), (h, und)}

         C = L σ (af 2 ) =    {( j,

Decision Problems with Compact Enumeration

Given that any combination of component labelling parts produces a valid labelling of the computed semantics, all classical AF decision problems can be answered.

For the credulous (resp. skeptical) acceptance problem of an argument a, we just have to look if a is at least in one (resp. in any) labelling part in the component in which a is. For the verification problem of a labelling ℓ, we just have to look if there exists a combination of component labelling parts that produces ℓ.

For the existence problem, we just have to look if at least a combination is possible. In the case that there is no labelling the compact enumeration representation is the empty set. No product is thus possible.

For the non-empty existence problem, we just have to look if a combination produces a labelling having at least an in -labelled argument.

Finally, for the uniqueness problem, we just have to verify that one and only one combination is possible, that is, each set of the enumeration compact representation has exactly one element.

Let formally define the AF decision problems using this representation and then show that they are equivalent to the classical ones.

Definition 41 (AF decision Problems with compact enumeration representation).

Let AF = ⟨A, K⟩ be an AF decomposed into n components and Comp σ (AF ) = {L σ (af 0 ), ..., L σ (af n )} be the compact enumeration representation corresponding to σ (AF ).

• Credulous Acceptance Comp-Cred σ : Given an AF AF = ⟨A, K⟩ and an argument a ∈ A. Is it the case that: As stated by Proposition 14, the compact enumeration representation is sufficient to answer all classical decision problems. But more than that, it could be used for other type of problems such as gradual acceptability : "Given a ∈ AF , what is the portion of extensions a belongs to?". This portion is the same portion as the one in its own component.

∀L σ (af ) ∈ Comp σ (AF ), L σ (af ) ̸ = ∅ (i.

Implementation ideas

In order to enhance the time needed to answer the first three decision problems Comp-Cred σ , Comp-Skep σ and Comp-Ver σ , we can produce as output two dictionaries: one linking the component name to its labellings and the other one linking each argument to the name of the component they belong to.

Example 28. For the running example, we would have :

• Dictionary 1 :

                           A : {{(a, in), (b, out), (c, out)}} B :         
{(e, in), ( f , out), (g, in), (h, out)} {(e, out), ( f , in), (g, out), (h, in)} {(e, und), ( f , und), (g, und), (h, und)}

         C :    {( j, und), (k, und), (l, und), (m, out), (n, in)} {( j, und), (k, und), (l, und), (m, und), (n, und)}                               • Dictionary 2 :    a : A, b : A, c : A, d : B, e : B, f : B, g : B, h : B, i : B, j : C, k : C, l : C, m : C, n : C   
Given that a labelling is also a dictionary, once this representation is computed, verifying the credulous or skeptical acceptance of a given argument can be made in linear time according to the number of labellings of the argument component. 1 For the verification problem it can also be made in linear time according to the total number of component labellings and arguments.

Example 29. Let consider the answer process for these already given examples

• Is ℓ =    (a, in), (b, out), (c, out), (e, out), ( f , in), (g, out), (h, in), ( j, und), (k, und), (l, und), (m, und), (n, und)    ∈ L co (AF )?
Given that finding an element in a dictionary can be done in constant time2 , splitting ℓ following the components using Dictionnary 2 can be done in linear time according to the number of arguments. This split produces {(a, in), (b, out), (c, out)} for A, {(e, out), ( f , in), (g, out), (h, in)} for B and {( j, und), (k, und), (l, und), (m, und), (n, und)} for C. Given that labellings are also dictionaries, checking if those labellings parts belong to their corresponding component can be done in linear time according to the number of the component labellings. For A, at most one check has to be done, three for B and two for C. As a consequence, six checks at most have to be performed, which is the total number of component labellings. Let v be the number of component labellings and w be the number of arguments of the AF, the complexity of checking if a labelling is produced by a given semantics is thus in Θ(w + v),3 so in linear time according to the number of arguments and of component labellings.

• For any labelling ℓ ∈ L co (AF ), is ℓ(n) = in?
Finding the component of the argument n using Dictionary 2 is done in constant time. Then checking if n is in-labelled in any labelling of C is done in linear time according to the number of labellings of C, which is two.

Compact Enumeration Representation: Experimental Analysis

The experiments with the compact enumeration representation have been made using the very same setting as the ones presented in Chapter 5 on page 51. The only difference is that, instead of using µ-toksia2019 [START_REF] Niskanen | µ-toksia (version 2019-10-31): SAT-based solver for static and dynamic argumentation frameworks[END_REF], to compute the labellings of a cluster given a particular labelling of its inward attack sources, we used As-partix2019 [START_REF] Dvorák | Aspartix-v19-system description for iccma'19[END_REF], a solver which transforms the AF labelling problem into an ASP problem. Indeed, although µ-toksia2019 is the winner of all enumeration problem tracks of the ICCMA 2019 session, our experiments show that Aspartix2019 is better on the AF instances we selected (See Section 5.3 on page 55). Now let consider Table 6.1 on the following page. Using this representation improves the algorithm performances for the types BA, TR and BW. We do not see effect on the other ones because (at least for the studied instances) they do not lead to multiple connected components. As a consequence, doing the enumeration is strictly equivalent to not doing it.

As one can observe, avoiding the enumeration allows to resolve in some cases more instances but above all it allows to compute much faster the semantics labellings: 97.03 times much faster for BA type, 2.46 for TR and 1.15 for BW. 4It is also worth noting that much less memory is used. A lot of failures were due to memory overflow (see Section 5.3.1 on page 55). Using the compact enumeration representation allows AFDiv-spectral to solve 26 instances more (24 of them were supposed to produce a memory overflow, the two other ones a timeout). For AFDiv-USCC-Chain, 22 new instances have been solved (21 of them were supposed to produce a memory overflow, the left one a timeout). For AFDiv-USCC-Tree, 22 new instances have been solved (20 of them were supposed to produce a memory overflow, the two other ones a timeout).

Synthesis

As a synthesis for this chapter, we show that the complete enumeration of all labellings is not necessary for solving classical AF problems. This can be done with a more compact data structure. This improvement has been shown interesting for the most difficult instances (especially for BA and TR, but also for BW) of the ICCMA benchmarks (see Table 6.1 on the following page). On these instances, the new version of the algorithm (without the complete enumeration) goes faster and solves more instances that the original one. The impact of the "no-enumeration" is particularly significant for the complete semantics due to the huge number of labellings that causes memory overflow during the enumeration. Chapter 7

Related Work: Algorithms for AF

In this chapter are presented the related works associated to our algorithm proposal. We emphasize the comparison with direct approach based algorithms, as AFDivider is itself such an algorithm.

Indirect approach based algorithms

In this section, we briefly present some indirect approaches to solve AF problems, with an emphasis on SAT based algorithms, as it is the most common transformation used to solve argumentation problems.

SAT based algorithms

Logic and Abstract Argumentation are interrelated since the seminal work of Dung (see [START_REF] Besnard | Logical theories and abstract argumentation: A survey of existing works[END_REF] for an overview about these links). Moreover, given the experience and hindsight we have today, SAT solvers are really efficient (See [START_REF] Le | Sat4j 2.3.2: on the fly solver configuration[END_REF] as an example). Many problem considered as intractable are today, at least for some instances, within reach and that through the use of SAT solvers. There has been so much progress in the field that, when considering a computational problem, it is important to ask whether or not a transformation of our problem into a SAT one is advantageous or not. Indeed, this option could save us development costs and possibly, for better result in the end. Following this observation, many argumentation solvers actually transform argumentation problems into SAT problems (See [START_REF] Doutre | Preferred extensions of argumentation frameworks: Query, answering, and computation[END_REF][START_REF] Peter Wallner | Advanced SAT techniques for abstract argumentation[END_REF][START_REF] Cerutti | A SAT-based approach for computing extensions in abstract argumentation[END_REF][START_REF] Alviano | The pyglaf argumentation reasoner[END_REF] as examples). Given that they are as many approaches as there are SAT based argumentation solvers and AF semantics, we will simply recall some encoding examples for the labelling-based complete semantics, that have been presented in [START_REF] Cerutti | A SAT-based approach for computing extensions in abstract argumentation[END_REF].

Given that SAT solvers take as input a CNF formula,1 the encoding must be of that form. To do so, let introduce some notions. Let AF = ⟨A, K⟩ be an AF such that |A| = k and φ : {1, ..., k} → A be an indexing of A.

• φ (i) is thus the argument i of AF . We denote by φ (i) -the set of attackers of the argument i.

• For each argument i, three boolean variables are defined:

I i , O i , U i . For I i to be true (resp. O i , U i ), it
means that the argument i is labelled in (resp. out , und ).

• The set of boolean variables corresponding to AF , denoted as V (AF ), corresponds to:

V (AF ) = i∈{1,...,n} {I i , O i ,U i }
Following what has been said above, let define a first encoding for the complete semantics.

Definition 42. Let AF = ⟨A, K⟩ be an AF, with |A| = k and φ : {1, ..., k} → A be an indexing of A. The SAT encoding C 1 defined on V (AF ), is given by the conjunction of the formulae listed below: i∈{1,...,n}

((I i ∨ O i ∨U i ) ∧ (¬I i ∨ ¬O i ) ∧ (¬I i ∨ ¬U i ) ∧ (¬O i ∨ ¬U i )) (7.1) {i|φ (i) -=∅} (I i ∨ ¬O i ∨ ¬U i ) (7.2) {i|φ (i) -̸ =∅}   I i ∨   { j|(φ ( j),φ (i))∈K} ¬O j     (7.3) {i|φ (i) -̸ =∅}   { j|(φ ( j),φ (i))∈K} ¬I i ∨ O j   (7.4) {i|φ (i) -̸ =∅}   { j|(φ ( j),φ (i))∈K} ¬I j ∨ O i   (7.5) {i|φ (i) -̸ =∅}   ¬O i ∨   { j|(φ ( j),φ (i))∈K} I j     (7.6) {i|φ (i) -̸ =∅}   {k|(φ (k),φ (i))∈K}   U i ∨ ¬U k ∨   { j|(φ ( j),φ (i))∈K} I j       (7.7) {i|φ (i) -̸ =∅}     { j|(φ ( j),φ (i))∈K} (¬U i ∨ ¬I j )   ∧   ¬U i ∨   { j|(φ ( j),φ (i))∈K} U j       (7.8) i∈{1,...,n} I i (7.9)
Note: The last clause has been added for technical reasons, due to the algorithm introduced in [START_REF] Cerutti | A SAT-based approach for computing extensions in abstract argumentation[END_REF]. It restricts the result to "non-empty" complete labellings (that is, at least one argument is labelled in). Verifying if the "empty" labelling is a complete labellings can be done afterward trivially by adding conjunctions forcing all U i to be true.

Let describe each part of C 1 :

• Equation (7.1) states that for each argument i one and only one label has to be assigned.

• Equation (7.2) states that each unattacked argument must be labelled in .

• Equation (7.3) states that argument i is labelled in if all its attackers are labelled out .

• Equation (7.4) settles the reverse (i.e. the "only if" ) condition of the precedent point.

• Equation (7.5) corresponds to the constraint that argument i is labelled out if at least one of its attackers is labelled in .

• Equation (7.6) corresponds to the "only if" condition of the precedent point.

• Equation (7.7) states that argument i is labelled und if none of its attackers is labelled in and at least one of its attackers is labelled und .

• Equation (7.8) corresponds to the "only if" condition of the precedent point.

• Equation (7.9) ensures non-emptiness, i.e. that at least one argument is labelled in .

In [START_REF] Cerutti | A SAT-based approach for computing extensions in abstract argumentation[END_REF], it has been given the six equivalent encodings as stated by the following proposition: [START_REF] Cerutti | A SAT-based approach for computing extensions in abstract argumentation[END_REF]). Referring to the formulae listed in Definition 42 on the previous page, the following encodings are equivalent:

Proposition 15 ([
• C 1 : (7.1) ∧ (7.2) ∧ (7.3) ∧ (7.4) ∧ (7.5) ∧ (7.6) ∧ (7.7) ∧ (7.8) ∧ (7.9) • C 2 : (7.1) ∧ (7.2) ∧ (7.
3) ∧ (7.4) ∧ (7.5) ∧ (7.6) ∧ (7.7) ∧ (7.9)

• C 3 : (7.1) ∧ (7.2) ∧ (7.
3) ∧ (7.5) ∧ (7.6) ∧ (7.7) ∧ (7.8) ∧ (7.9)

• C 4 : (7.1) ∧ (7.2) ∧ (7.3) ∧ (7.4) ∧ (7.6) ∧ (7.7) ∧ (7.8) ∧ (7.9)

• C 5 : (7.1) ∧ (7.2) ∧ (7.4) ∧ (7.6) ∧ (7.8) ∧ (7.9)

• C 6 : (7.1) ∧ (7.2) ∧ (7.3) ∧ (7.5) ∧ (7.7) ∧ (7.9)

Most semantics can be encoded with such rules. Now for semantics that need some maximisation such as the preferred semantics, the SAT solver can be called iteratively until a maximal solution is found. There exists a huge amount of search strategies, each one leading to a different SAT-based algorithm. By sake of brevity, they will not be presented in this paper but the reader interested can refer to [START_REF] Doutre | Preferred extensions of argumentation frameworks: Query, answering, and computation[END_REF][START_REF] Peter Wallner | Advanced SAT techniques for abstract argumentation[END_REF][START_REF] Cerutti | A SAT-based approach for computing extensions in abstract argumentation[END_REF][START_REF] Alviano | The pyglaf argumentation reasoner[END_REF] as examples.

Other indirect approach based algorithms

Other types of transformation can be used to solve argumentation problems. Here is a non exhaustive list of approaches. References are given to go further:

• ASP-based algorithms (e.g. [START_REF] Dvorák | Aspartix-v19-system description for iccma'19[END_REF]). Notice that in ICCMA 2019 session, Aspartix, the solver presented in [START_REF] Dvorák | Aspartix-v19-system description for iccma'19[END_REF], gave very good results.

• Neural network based algorithms (e.g. [START_REF] Malmqvist | Yonas: An experimental neural argumentation solver[END_REF]). Although the ICCMA 2017 session shew that this solver is not efficient, it opens perspectives for a radical paradigm change in argumentation problem solving.

Direct or semi-direct approach based algorithms

In this section we compare the behaviour of our algorithm to other existing ones, using direct or "semi-direct" approach. By "semi-direct" we mean that the algorithm mainly and directly deals with AF and takes advantage of its structure, but in some cases, for some sub-problems as the computation of labellings/extensions of a part of the AF, uses an indirect solving method such as a transformation to SAT. Notice that several algorithms, other than those presented in this section, use solving methods similar to the ones of the algorithms presented. By sake of brevity, our algorithm is compared only to one algorithm of each method type.

In order to illustrate how these other algorithms work, we will consider the AF shown in Figure 7.1 as running example and show how the preferred labellings are computed following the different algorithms. This particular AF has been chosen because its structure let appear clusters in it, it has four SCCs and there is an interesting hierarchy between them. These two last points are very relevant for the algorithms presented in Sections 7.2.1 and 7.2.2 on the current page and on page 78.

Dynamic programming algorithm

In [START_REF] Dvořák | Towards fixed-parameter tractable algorithms for abstract argumentation[END_REF], Dvořák et al. proposed an algorithm based on a dynamic analysis of an argumentation framework. In the interest of brevity, we will just highlight the main idea of this algorithm (see [START_REF] Dvořák | Towards fixed-parameter tractable algorithms for abstract argumentation[END_REF] for a more detailed explanation).

Basically, this algorithm relies on the nice tree decomposition of a graph.

Definition 43. (Tree decomposition). Let G = (V, E) be a non directed graph. A tree decomposition of G is a pair ⟨T , X ⟩ where T = (V T , E T ) is a tree and X = (X t ) t∈V T is a set of so-called bags, which has to satisfy the following conditions: • t∈V T X t = V , i.e. X is a set covering of V .

• for each v ∈ V , T ↓ {t|v∈X t } is a connected tree.

• for each {v i , v j } ∈ E, {v i , v j } ⊆ X t for some t ∈ V T .

Definition 44. (Width of a tree decomposition). Let ⟨T , X ⟩ be a tree decomposition where T = (V T , E T ) is a tree and X = (X t ) t∈V T is a set of so-called bags. The width of such a tree decomposition is given by:

max{card(X t )|t ∈ V T } -1
Definition 45. (Tree-width of a graph). Let G = (V, E) be a non directed graph. The tree-width of G is defined by the minimum width over all its tree decompositions.

Definition 46. (Nice tree decomposition). A tree decomposition ⟨T , X ⟩ of a graph G is called nice if T is a rooted tree and if each node t ∈ T is one of the following types:

• LEAF: t is a leaf of T

• FORGET: t has only one child t ′ and X t = X t ′ \ {v} for some v ∈ X t ′

• INSERT: t has only one child t ′ and X t = X t ′ ∪ {v} for some v / ∈ X t ′

• JOIN: t has two children t " " means that the vertex v corresponding to the current line belongs to the bag X t corresponding to the current column Example 30. Figure 7.2 on the previous page shows one nice tree decomposition of the AF AF , ⟨T , X ⟩ where: [START_REF] Cerutti | An SCC recursive metaalgorithm for computing preferred labellings in abstract argumentation[END_REF] } with each bag being as summarised in Table 7.1.

• T = (V T , E T ) with: -V T = {t i |i ∈ 0, 24 } -E T = {(t i ,t i+1 )|i ∈ 0, 15 ∪ 17, 23 } ∪ {(t 8 ,t 17 )} • X = {X t i |i ∈ 0,

As node type examples, according to Definition 46:

• t 24 is a LEAF type node (no child)

• t 2 is a FORGET type node (t 3 being a child of t 2 )
• t 10 is a INSERT type node (t 11 being a child of t 10 )

• t 8 is a JOIN type node (t 9 and t 17 being the children of t 8 )

The nice tree decomposition shown in Figure 7.2 on the previous page is one among all tree decompositions of AF with the minimal width, which is 4. In other words, the tree-width of AF is 4.

There exist other possible nice tree decompositions of AF with non minimal width. As an example, the one shown in Figure 7.3 on the next page has a width of 11.

To each nice tree node is associated a sub AF defined as following: Let AF be an AF and ⟨T , X ⟩ be its tree decomposition where T = (V T , E T ) is a tree and X = (X t ) t∈V T is a set of so-called bags. We denote by X ⩾t the union of all bags X s ∈ X such that s occurs in the subtree of T rooted at t.

Let t ∈ V T be a tree node. The AF af associated with t is defined as following:

af = AF ↓ X ⩾t
Example 31. Let take as example the node t 12 in Figure 7.2 on page 74. According to Definition 47, we have:

X ⩾t 12 = X t 12 ∪ X t 13 ∪ X t 14 ∪ X t 15 ∪ X t 16 = {a, d, e, f , k}
We have so: Once the AF nice tree determined and the sub AFs associated to each tree node identified, the tree is explored from the bottom up. On each tree node, the labellings of its associated AF are computed. The node type (LEAF, INSERT, FORGET or JOIN) indicates which operations to do in order to update the computed set of labellings.

af = AF ↓ X ⩾t 12 = AF ↓ {a,d,e, f ,k}
Notice that the sub AF associated with the tree root is the whole AF. So, at the tree root, all the labellings of the AF are found. This is basically the general idea of this algorithm. This algorithm is dynamic in the sense that we are interested in the labellings of sub AF that evolve dynamically following the nice tree decomposition. To each leaf is associated an initial AF that will be transformed forgetting and inserting argument nodes in it. This approach has the advantage of breaking the SCC and eventually the hardness of the AF problem. Nevertheless it has also some disadvantages.

Indeed, each step adds or removes at most one argument. The consequence is that a lot of updates are useless and a lot of space is used for potential correct labellings. t 0 :

t 1 : t 2 :
t 3 :

t 4 :
t 5 : Although this AF admits only one preferred labelling which is {(a 1 , in), (a 2 , out), (a 3 , in), (a 4 , out)}, as we go from the leaf to the top the set of partial labellings will be updated 6 times and, at each tree node, we will have to consider all the potential partial labellings.

t 6 :
Actually, this algorithm does not work directly with labellings but with "colorings"; a coloring is a 4-state argument mapping from which are determined the semantic extensions we are interested in. Without going too deep into the details of how this coloring works, we will just highlight the fact that for each argument attacked by an argument outside the current associated AF, four colorings have to be considered, according to the four possible status of that argument. As a consequence, a lot of space is used in order to ensure that all possibilities have been explored.

This algorithm and the AFDivider algorithm have both the ability to break the SCC and hopefully the hardness the AF. However, they differ on other points and the main one is how the combinatorial effect of potential labelling number is tackled. Although the AFDivider algorithm computes all cases for a given cluster, this combinatorial effect is limited to that particular cluster and is not propagated on the whole AF. As a consequence, space and computational time are spared.

SCC decomposition based algorithms

In [START_REF] Liao | Toward incremental computation of argumentation semantics: A decomposition-based approach[END_REF], Beishui Liao proposed an algorithm that computes the labellings of an AF following its SCC decomposition.

Notice that if each SCC of a graph is considered as a super node, the resulting super graph will be acyclic.

We can thus have a hierarchical representation of this super graph: in the first layer are SCCs with no parents, in the second layer are contained all the SCCs whose parents are in the previous layers, and so on. Given that the labellings of each SCC are influenced only by the ones of its parents, it is possible to guide the research of labellings following the hierarchical representation of the SCCs of the AF. This is the main idea of the algorithm.

Example 34. In a first step, the labellings of the SCC n°1 are computed. The result is the following set of labellings:

{ℓ scc1 1 , ℓ scc1 2 , ℓ scc1 3 } with          ℓ scc1 1 = {(a, in)}, {(b, out)}, {(c, out)}, ℓ scc1 2 = {(a, out)}, {(b, in)}, {(c, out)}, ℓ scc1 3 = {(a, out)}, {(b, out)}, {(c, in)}
Then, possible labellings of the SCCs n°2 and n°3 are computed considered the labellings of the parents SCCs, in this case SCC n°1. For instance, considering ℓ scc1

1 = {(a, in)}, {(b, out)}, {(c, out)}:
• For SCC n°2 we have:

{ℓ scc2 1 , ℓ scc2 2 } with    ℓ scc2 1 = {(d, out)}, {(e, out)}, {( f , in)}, ℓ scc2 2 = {(d, out)}, {(e, in)}, {( f , out)}
• For SCC n°3 we have:

{ℓ scc3 1 , ℓ scc3 2 , ℓ scc3 3 } with          ℓ scc3 1 = {(g, out)}, {(h, out)}, {(i, in)}, ℓ scc3 2 = {(g, out)}, {(h, in)}, {(i, out)}, ℓ scc3 3 = {(g, in)}, {(h, out)}, {(i, out)}
The same thing must be done considering ℓ scc1 2 and ℓ scc1

3 .

Afterwards, the labellings of SCC n°4 are computed according to the compatible SCC parents labellings. In the interest of brevity we will not give the entire result as this AF has 47 distinct preferred labellings.

The great advantage of this approach is that no useless computation is made. When going from one layer to another, only possible labellings are considered. This reduces considerably the computational time.

Although not proposed in this paper, it is possible to parallelize the computation when there are independent branches in the acyclic super graph. But even though a distributed version of this algorithm had been proposed, it would still be very different from the AFDivider algorithm.

Indeed, this algorithm is profitable only if there are several SCCs and if the hardness of solving the AF problem is not inside the SCCs. The major difference is that the AFDivider algorithm is able to look inside SCCs and hopefully break the hardness by finding clusters in it. Another difference is that the way AFDivider parallelizes the labelling computation is not subject to any hierarchy of SCCs. As a consequence, there are no sequential constraints on the distributed computation made to construct the labellings. Finally, the used clustering method tries to balance the cluster sizes (in terms of number of arguments) so that hopefully the workload may be also balanced.

Parallel algorithms

The algorithm proposed by Cerruti et al. in [START_REF] Cerutti | Exploiting parallelism for hard problems in abstract argumentation[END_REF], named P-SCC-REC, has several common points with the AFDivider algorithm. Indeed, both algorithms are distributed and they are able to look inside SCCs. Nevertheless, the way of distributing and of "cutting" of the AF are completely different.

The P-SCC-REC algorithm is rather complex. We are going to highlight its main concepts (see [START_REF] Cerutti | Exploiting parallelism for hard problems in abstract argumentation[END_REF] for additional information).

It is a recursive algorithm. In one recursion level, the following steps are performed:

• As in the AFDivider algorithm, the grounded labelling is computed and only the hard part of the AF is considered for the next steps.

• As in the Beishui Liao's algorithm, an SCC hierarchical view of the AF is determined.

• For each SCC, a greedy labelling computing is performed, considering that all arguments attacking the given SCC is labelled out . This computation is made in a distributed way, parallelized following the SCCs.

• For each layer:

-The labelling of the SCCs are computed according to the labelling of their SCC ancestors. This computation is made in a distributed way, parallelized both following the SCCs and the SCC ancestors labellings.

* In some cases when the SCC ancestors labelling does not allow to determine quickly the labellings of the current SCC, P-SCC-REC is called recursively. The sub-AF on which it is called is that particular SCC, sligthly modified to fit with the labelling of attackers from its SCC ancestors: arguments that are attacked by in -labelled arguments from its SCC ancestors are removed. 2-Following the previous step, the set of SCC ancestors labellings of the next layer is determined.

• Some computations of SCC labelling are made using a transformation to SAT.

Example 35. Applied to AF , P-SCC-REC will behave a bit like Beishui Liao's algorithm as there is no argument labelled in or out in the grounded labelling of AF .

Notice that, given the labelling ℓ scc1 The P-SCC-REC algorithm will look inside an SCC if its SCC ancestor labelling allows it, not according to the size of this SCC and its possible hardness, whereas the AFDivider algorithm will try to found clusters similar in size whether it is necessary to break SCCs or not.

1 = {(a, in)}, {(b, out)}, {(c, out)},
There is another aspect of P-SCC-REC algorithm that may narrow its performance. If we put aside the greedy phase of the algorithm, the algorithm follows the SCC hierarchical view of the AF and parallelizes following the SCCs in one layer, and following the ancestor labellings. This later parallelization causes two problems:

1. Most of the time, it makes the number of threads explodes and so overloads the CPUs.

It leads to redundant computation as the computation cases are not based on the states of input arguments of the current SCC.

Example 36. Let consider the step to compute the SCC n°4 labellings.

• As an illustration of point 1:

-We have 21 distinct SCC ancestor labellings and so 21 threads will be created. Although AF is a small AF, the amount of threads is rather important. On a bigger one, the number of threads could quickly overload the CPUs.

• As an example of point 2:

-Even if several distinct SCC ancestor labellings are equal when restricted to the arguments f and h, the labelling computation will be made for each of them, which is highly redundant.

It is true that some of the cases computed by the AFDivider algorithm may be unused in the reunifying phase (bear in mind that it is not possible to know them in advance) but there is no waiting time due to a hierarchical view of the AF, and there is no redundant computation. Furthermore, if the AF is not too dense, the number of threads will not explode, even though the number of labellings is huge.

Part IV

Higher-Order Attack Argumentation

Frameworks: Background 82

Part presentation:

Higher-Order Attack Argumentation Frameworks are Argumentation Frameworks in which an attack can have as target an attack. In this background are presented two of such frameworks:

• The so-called "Recursive Argumentation Framework" (denoted RAF), introduced in [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF], consists of a set of arguments, a set of attacks, and mapping functions that associate to each attack a source and a target.

• The so-called "Argumentation Framework with Recursive Attack" (denoted AFRA), introduced in [5,[START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF], consists of a set of arguments and a set of named attacks.

This implies that they cannot be represented as a directed graph like AFs. However their graphical representation is as much as intuitive, as shown in Figure 7.9. Definition 48 (Recursive argumentation framework -RAF). A Recursive Argumentation Framework (RAF) RAF = ⟨A, K, s,t⟩ is a quadruple where A and K are (possibly infinite) disjoint sets respectively representing arguments and attack names, and where s : K → A and t : K → A ∪ K are functions respectively mapping each attack to its source and to its target. The set of all possible RAFs is denoted as Φ ra f . Definition 49 (AFRA). An Argumentation Framework with Recursive Attacks (AFRA) is a pair AFRA = ⟨A, K⟩ where A is a set of arguments and K is a set of named attacks, namely pairs (a, x) such that a ∈ A and x ∈ (A ∪ K). Given an attack α = (a, x), we say that a is the source of α (denoted as "src(α)") and x is the target of α (denoted as "trg(α)"). The set of all possible AFRAs is denoted as Φ a f ra .

Note: Although the definitions of AFRA and RAF are equivalent and that both frameworks can represent the same relations between arguments, substantial differences appear afterward in the way that semantics are computed.

Note: In the following, to simplify the notation and highlight the link between AFRA and RAF, while refering to AFRAs, we will use "s(α)" (resp. "t(α)") instead of "src(α)" (resp. "trg(α)") to represent the source (resp. the target) of an attack α.

Example 37. Figure 7.9 on the previous page shows a RAF/AFRA example. In all this document, arguments (in Latin letter) will be represented by a round box, while attacks (in Greek letters) will be represented by directed edges from a "circular node" to another "node" (squared or circular) through a "square node" containing the name of the attack. As one can notice, the attacks γ, κ and η have as target an attack.

In Chapter 8 are presented the background for AFRAs and in Chapter 9 the background for RAFs.

Chapter 8

Argumentation Framework with Recursive Attacks (AFRA)

In this chapter, we first present AFRA semantics (Section 8.1). Then, we present the relations between AFRA and AF (Section 8.2 on page 90).

Extension-based Semantics

What differs from AF to AFRA is that in an AFRA an attack can have an attack for target. As a consequence, an attack is not always "acceptable". In order to express this fact, an AFRA "extension-based semantic", that is, a function that defines the solutions of an AFRA, produces AFRA-extension that not only contains arguments but also attacks. As for extensions in AF, the idea behind the notion of AFRA-extension is that when presented together, the elements of the AFRA-extension (i.e. arguments plus attacks) win the argumentation.

Definitions

Formally, an AFRA extension-based semantic is defined as follows:

Definition 50 (Extension-based Semantics). Let σ be a function over Φ a f ra . σ is said to be an AFRA extension-based semantics iff the following property holds:

∀AFRA ∈ Φ a f ra , σ (AFRA) ⊆ 2 A∪K , with AFRA = ⟨A, K⟩
An AFRA-extension is thus defined as follows:

Definition 51 (AFRA-extension). A set S is said to be an AFRA-extension of some AFRA = ⟨A, K⟩ if it satisfies: S ⊆ A ∪ K.

Intuitively, any attack that does not belong to S is understood as non-acceptable and, in this sense, it cannot defeat its target. Two types of defeat relation are defined for AFRAs: Definition 52 (Direct defeat [START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF]). Let AFRA = ⟨A, K⟩ be an AFRA, α ∈ K be an attack and x ∈ (A ∪ K) be an argument or an attack. We say that α directly defeats x iff t(α) = x.

Definition 53 (Indirect defeat [START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF]). Let AFRA = ⟨A, K⟩ be an AFRA, α ∈ K be an attack and β ∈ K be an attack. We say that α indirectly defeats β iff t(α) = s(β ).

The following notion captures both defeat relations: Definition 54 (Defeat [START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF]). Let AFRA = ⟨A, K⟩ be an AFRA, α ∈ K be an attacks and x ∈ (A ∪ K) be an argument or an attack. We say that α defeats β , denoted as α → K x, iff α directly or indirectly defeats β .

From the defeat relation, we formally define the notion of acceptability as follows:

Definition 55 (Acceptability [START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF]). Let AFRA = ⟨A, K⟩ be an AFRA, S ⊆ (A ∪ K) be a subset of the elements of AFRA and x ∈ (A ∪ K) be an argument or an attack. We say that x is acceptable w.r.t. S (or defended by

S) iff ∀α ∈ K s.t. α → K x, ∃β ∈ S s.t. β → K α.
From these notion of defeat and acceptability, we can define the sets of defeated and acceptable element w.r.t. some set of elements S.

Definition 56 (Defeat and acceptable set). Let AFRA = ⟨A, K⟩ be an AFRA.

• Let S be a subset of the elements of AFRA. We denote by AFRA-De f (S) = {x|x ∈ (A ∪ K), ∃α ∈ S s.t. α → K x} the set of all the elements defeated by S.

• Let S be a subset of the elements of AFRA. We denote by AFRA-Acc(S)

= {x|x ∈ (A ∪ K), ∀α ∈ K s.t. α → K x, ∃β ∈ S s.t. β → K α}
the set of all the elements defended by S.

Note: These definitions of defeat and acceptable sets slightly differ from the ones given in [START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF], but they are equivalent. We chose these ones to be closer the corresponding definitions for RAF (once more, in order to facilitate the comparison between the two approaches).

As for Dung's Argumentation Framework, based on the notion of acceptability, semantics have been defined for AFRAs. We will focus only on semantics we are interested in although much semantics have been defined.

Definition 57 (AFRA Semantics [START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF]). Let AFRA = ⟨A, K⟩ be an AFRA and S ⊆ (A ∪ K) be a subset of its elements. S is said to be an extension:

1. AFRA-conflict-free iff S ∩ AFRA-De f (S) = ∅.
2. AFRA-admissible iff it is AFRA-conflict-free and S ⊆ AFRA-Acc(S).

3. AFRA-complete iff it is AFRA-conflict-free and S = AFRA-Acc(S).

4. AFRA-preferred iff it is a ⊆-maximal AFRA-admissible extension.

5. AFRA-grounded iff it is a ⊆-minimal AFRA-complete extension.

6. AFRA-stable iff it is AFRA-conflict-free and S ∪ AFRA-De f (S) = (A ∪ K).

AFRA-semi-stable extension iff it is an AFRA-complete extension such that S

∪ AFRA-De f (S) is max- imal w.r.t. ⊆.
Given an AFRA AFRA, we denote by co(AFRA) (resp. gr(AFRA), st(AFRA), sst(AFRA), pr(AFRA)) the set of AFRA-extensions of AFRA under the semantics AFRA-complete (resp. AFRA-grounded, AFRAstable, AFRA-semi-stable and AFRA-preferred).

Example 38. Let consider the AFRA shown in Figure 7.9 on page 83. The AFRA-extensions corresponding to the semantics mentioned are given in Table 8.1 on the next page. By sake of space the AFRA-admissible and the AFRA-conflict-free semantics are not given.

Properties

In [START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF], the following propositions and theorem have been proven. Proposition 16. The set of all AFRA-admissible extensions forms a complete partial order with respect to ⊆.

Theorem 1. The following assertions hold:

• every AFRA-admissible extension is also AFRA-conflict-free

• every AFRA-complete extension is also AFRA-admissible

• the AFRA-grounded extension is also AFRA-complete

• every AFRA-preferred extension is also AFRA-complete

• every AFRA-semi-stable extension is also AFRA-preferred

• every AFRA-stable extension is also AFRA-semi-stable

Proposition 17 (extension semantics cardinality). The following properties hold:

• There is always at least one AFRA-conflict-free extension.

• There is always at least one AFRA-admissible extension.

• There is always at least one AFRA-complete extension.

• There is always a unique AFRA-grounded extension.

• There is always at least one AFRA-preferred extension.

• There is always at least one AFRA-semi-stable extension.

• It may be the case that there is no AFRA-stable extension.

AFRA-stable

In the first part of the table, i j means that element i belongs to extension j. In the second part of the table, i j means that j is an extension of the semantics i. The cardinality of each semantics is represented between parenthesis.

" * " means zero or more, "+" means one or more.

Relation between AFRA and AF

There exists a way to express AFRAs as AFs.

Definition 58 (AFRA expressed as AF [START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF]). Let AFRA = ⟨A, K⟩ be an AFRA. The corresponding AF of AFRA, AFRA = A, K is defined as following:

• A = A ∪ K • K = {(a, b)|(a, b) ∈ (A ∪ K) 2 and a → K b}
In [START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF] has been shown a very important result concerning AFRAs and their corresponding AFs: there exists a one-to-one correspondence between extensions in AFRAs and their corresponding AFs for some semantics. Here we will focus on Dung's semantics and the semi-stable one but the result shown in [START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF] concerned much semantics.

Proposition 18 (Semantics correspondence: AFRA expressed as AF [START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF]). Let AFRA = ⟨A, K⟩ be an AFRA and AFRA = A, K its corresponding AF. Let S ⊆ A ∪ K.

• S is an AFRA-complete extension for AFRA iff S is a complete extension for AFRA.

• S is an AFRA-preferred extension for AFRA iff S is a preferred extension for AFRA.

• S is an AFRA-grounded extension for AFRA iff S is a grounded extension for AFRA.

• S is an AFRA-stable extension for AFRA iff S is a stable extension for AFRA.

• S is an AFRA-semi-stable extension for AFRA iff S is a semi-stable extension for AFRA.

Note: This correspondence does not correspond to a conservative generalization of AF. Indeed, if we consider for instance AFRA = ⟨A, K⟩, the non recursive AFRA illustrated in Figure 8.2(a), then the set {α, c} is an AFRA-admissible set. Notice that {α, c} is an admissible extension of AFRA as represented in Figure 8.2(b). Nevertheless, if we read AFRA as an AF (i.e. without naming its attacks) c cannot be accepted without a. This is due to the fact that the link between an attack and its source is broken in the AFRA semantics (as it can be seen in Chapter 9

Recursive Argumentation Framework (RAF)

In this chapter, we first present RAF structure-based semantics, the counterpart of AF extension-based semantics (Section 9.1). Then, we present the relations between RAF and AF that already exist in the literature (Section 9.2 on page 94). Notice that several RAF notions/ideas are similar to those defined for AFRAs. A comparison between the two approaches is given in Section 9.3 on page 99.

Structure Semantics

As in AFRA, what differs from AF to RAF is that in a RAF an attack can have an attack for target. As a consequence, an attack is not always "valid". In order to express this fact, a RAF "structure-based semantic", that is, a function that defines the solutions of a RAF produces structures: a couple whose first element is a set of arguments and the second, a set of attacks. As for extensions in AF (or in AFRA), the idea behind the notion of structure is that when presented together, the elements of the structure (i.e. arguments plus attacks) win the argumentation.

Definitions

Formally, a RAF structure-based semantic is defined as follows:

Definition 59 (Structure-based Semantics). Let σ be a function over Φ ra f . σ is said to be a RAF structurebased semantics iff the following property holds:

∀RAF ∈ Φ ra f , σ (RAF ) ⊆ 2 A × 2 K , with RAF = ⟨A, K, s,t⟩
A structure is thus defined as follows:

Definition 60 (Structure). A pair U = ⟨S, Q⟩ is said to be a structure of some RAF = ⟨A, K, s,t⟩ if it satisfies: S ⊆ A and Q ⊆ K. Notice that by x ∈ U we mean: x ∈ S ∪ Q.

91

Intuitively, the set S represents the set of "acceptable arguments" w.r.t. the structure U, while Q represents the set of "valid attacks" w.r.t. U. Any attack that does not belong to Q is understood as non-valid and, in this sense, it cannot defeat its target.

Definition 61 (Defeat and Inhibition in RAF). Let U = ⟨S, Q⟩ be a structure. The set of all arguments defeated by U, denoted RAF-De f (U), is defined as follows:

RAF-De f (U) = {a ∈ A|∃α ∈ Q s.t. s(α) ∈ S and t(α) = a}
The set of all attacks inhibited by U, denoted RAF-Inh(U), is defined as follows:

RAF-Inh(U) = {α ∈ K|∃β ∈ Q s.t. s(β ) ∈ S and t(β ) = α}
The counterpart of defeat/inhibition is the notion of acceptance: Definition 62 (RAF Acceptability). An element x ∈ (A ∪ K) is said to be acceptable w.r.t. some structure U iff every attack α ∈ K with t(α) = x satisfies one of the two following conditions:

• s(α) ∈ RAF-De f (U) • α ∈ RAF-Inh(U)
By RAF-Acc(U) we denote the set containing all acceptable arguments and attacks with respect to U.

For any pair of structures U = ⟨S, Q⟩ and

U ′ = ⟨S ′ , Q ′ ⟩, we write U ′ ⊑ U ′ iff (S ∪ Q) ⊆ (S ′ ∪ Q ′ ) and we write U ⊑ ar U ′ iff S ⊆ S ′ . As usual, we say that a structure U is ⊑-maximal (resp. ⊑ ar -maximal) iff every U ′ that satisfies U ⊑ U ′ (resp. U ⊑ ar U ′ ) also satisfies U ′ ⊑ U (resp. U ′ ⊑ ar U).
Inspired by Dung's AF semantics, the first RAF structure-based semantics, that have been defined in [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF], are the following ones: Definition 63 (RAF structure semantics). Let U = ⟨S, Q⟩ be a structure over some RAF RAF = ⟨A, K, s,t⟩. U is said to be:

1. RAF-conflict-free iff S ∩ RAF-De f (U) = ∅ and Q ∩ RAF-Inh(U) = ∅.

2. RAF-naive iff it is a ⊑-maximal RAF-conflict-free structure.

3. RAF-admissible iff it is RAF-conflict-free and (S ∪ Q) ⊆ RAF-Acc(U).

4. RAF-complete iff it is RAF-conflict-free and (S ∪ Q) = RAF-Acc(U).

5. RAF-grounded iff it is a ⊑-minimal RAF-complete structure.

6. RAF-preferred iff it is a ⊑-maximal RAF-admissible structure.

7. RAF-arg-preferred iff it is a ⊑ ar -maximal RAF-preferred structure.

RAF-stable iff S

= A \ RAF-De f (U) and Q = K \ RAF-Inh(U).
Notice that the RAF-semi-stable semantics has not been defined in [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF]. This is a contribution of this thesis. See Section 10.1 on page 102.

Example 39. Let consider the RAF shown in Figure 7.9 on page 83. The structures corresponding to the semantics mentioned in Definition 63 are given in Table 9.1 on the following page. By sake of space the RAF-admissible and the RAF-conflict-free semantics are not given.

RAF-stable

In the first part of the table, i j means that element i belongs to structure j. In the second part of the table, i j means that j is a structure of the semantics i.

Properties

In [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF], Propositions 19 and 20 and Theorem 2 have been proven. Proposition 19. The set of all RAF-admissible structures forms a complete partial order with respect to ⊑. Furthermore, for every RAF-admissible structure U, there exists a RAF-preferred (and a RAF-arg-preferred)

U ′ such that U ⊑ U ′ .
Theorem 2. The following assertions hold:

• every RAF-naive structure is also RAF-conflict-free

• every RAF-admissible structure is also RAF-conflict-free

• every RAF-complete structure is also RAF-admissible

• the RAF-grounded structure is also RAF-complete

• every RAF-preferred structure is also RAF-complete

• every RAF-arg-preferred structure is also RAF-preferred

• every RAF-stable structure is also RAF-arg-preferred

• every RAF-stable structure is also a RAF-naive Proposition 20 (Structure semantics cardinality). The following properties hold:

• There is always at least one RAF-conflict-free extension.

• There is always at least one RAF-naive extension.

• There is always at least one RAF-admissible extension.

• There is always at least one RAF-complete extension.

• There is always a unique RAF-grounded extension.

• There is always at least one RAF-preferred extension.

• There is always at least one RAF-arg-preferred extension.

• It may be the case that there is no RAF-stable extension.

Relation between RAF and AF

In this section the relation between RAFs and AFs is briefly presented. We first present semantics relation (Section 9.2.1 on page 96) then a flattening process that transforms RAF into AF (Section 9. The cardinality of each semantics is represented between parenthesis. " * " means zero or more, "+" means one or more.

Semantics correspondence

In order to establish a point of comparison, we define the notion of D-structure:

Definition 64 (D-structure). A d-structure U = ⟨S, Q⟩ is a structure that satisfies: Given the definition of the RAF-complete semantics, the following property holds:

(RAF-Acc(U) ∩ K) ⊆ Q Note:
Proposition 21. Every RAF-complete structure is a d-structure.
It is interesting to note that an AF can be viewed as a RAF without recursive attack, that is without attack whose target is an attack. Such a RAF is called a "non recursive framework" and is formally defined as follows:

Definition 66 (Non recursive framework). A framework RAF = ⟨A, K, s,t⟩ is said to be non-recursive iff : AF extension-based semantics produce sets of arguments. Unlikely, RAF structure-based semantics produce pairs of arguments and attacks. In order to make a bridge between those two frameworks and compare them, the notion of extensions w.r.t. a given RAF has to be defined: Definition 67 (Argument extensions). Given RAF = ⟨A, K, s,t⟩. Let S ⊆ A be a set of arguments. S is said to be a conflict-free extension (respectively naive, admissible, complete, preferred, grounded, stable) w.r.t. RAF iff there is some Q ⊆ K such that U = ⟨S, Q⟩ is a conflict-free (respectively naive, admissible, complete, preferred, grounded, stable) d-structure of RAF .

Example 42. Let consider the non recursive framework in Figure 9.2 on the previous page. We have: {a, c} being a complete extension w.r.t. RAF . Indeed:

U = ⟨{a, c} , {α, β , γ}⟩ is a RAF-complete d-structure of RAF .
Finally, with the notions defined above, the following theorem and its corollary can be established: Theorem 3. For each semantics σ ∈ {conflict-free, naive, admissible, complete, preferred, grounded, stable}: A set of arguments S ⊆ A is a σ -extension w.r.t. some non-recursive RAF = ⟨A, K, s,t⟩ iff it is a σ -extension w.r.t. AF = ⟨A, {(s(α),t(α))|α ∈ K}⟩.

Corollary 1. For each semantics σ ∈ {complete, preferred, grounded, stable}: Indeed there is a one-to-one correspondence between the structures of a RAF without recursive attacks and their corresponding Dung's extensions for the semantics complete, preferred, grounded and stable. Furthermore, RAFs conserve the notion conflict-freeness.

U = ⟨S, K⟩ is a σ -structure w.r.t. a non-recursive RAF = ⟨A, K, s,t⟩ iff S is σ -extension w.r.t. AF = ⟨A, {(s(α),t(α))|α ∈ K}⟩.
Note: In Section 10.1 on page 102, we prove that there is also a one-to-one correspondence between the structures of a RAF without recursive attacks and their corresponding Dung's extensions for semi-stable semantics.

RAF flattening

In [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF], a transformation has been already proposed in order to establish a link between RAF and MAF (Meta-Argumentation Framework). This flattening, inspired by the works presented in [START_REF] Gabbay | Semantics for higher level attacks in extended argumentation frames[END_REF] and [16], transforms a RAF into MAF. Given that MAFs are basically AFs, a brief presentation of this flattening process is then relevant.

Definition 68 (RAF flattening to MAF/AF). Let Raf2Af maf : Φ ra f → Φ a f be the function transforming a RAF into an AF. Raf2Af maf is defined as follows: Definition 69 (RAF Structure to MAF/AF extension). Let RAF = ⟨A, K, s,t⟩ be a RAF and U = ⟨S, Q⟩ be a structure. We denote by str2MafExt(U) the MAF/AF extension corresponding to U. It is defined as:

∀ RAF = ⟨A, K, s,t⟩ ∈ Φ ra f , Raf2Af maf : RAF → MAF = A ′ , K ′ With: A ′ = A ∪ K ∪ N N = n s(α)α |α ∈ K K ′ = (s(α), n s(α)α )|α ∈ K ∪ (n s(α)α , α)|α ∈ K ∪ {(α,t(α))|α ∈ K}
str2MafExt(U) = S ∪ {α ∈ Q|s(α) ∈ S} ∪ n s(α)α ∈ N s(α) / ∈ S and s(α) ∈ RAF-De f (U)} Theorem 4.
For each semantics σ ∈ {complete, stable, preferred, grounded} and for any RAF ∈ Φ ra f , the function str2MafExt(•) is a one-to-one correspondence between the sets of all σ -structures of RAF and the set of all σ -extensions of Raf2Af maf (RAF ).

It is important to note that even thought there is a one-to-one correspondence between the sets of all σ -structures and the set of all σ -extensions, this correspondence does not guarantee that each acceptable element w.r.t. a structure will be acceptable w.r.t. its corresponding extension in the resulting MAF/AF.

Example 45. As an illustration of that fact, we have following Example 44 on the previous page:

• U = ⟨{b} , ∅⟩ being a RAF-admissible structure of RAF . We have: RAF-Acc(U) = {α, δ , γ}. ⟨{b} , {α, δ , γ}⟩ is thus a RAF-complete structure. By the way, it is also the RAF-grounded structure.

• Raf2Af maf (U) = {b} is an admissible extension of MAF and it is also its grounded extension. We have: Acc({b}) = ∅. As a consequence, α is not acceptable w.r.t. {b}.

This remark will serve as motivation for a new flattening process. See Chapter 12 on page 114.

Relation between RAF and AFRA

In this section the relation between RAFs and AFs is briefly presented. In order to establish a point of comparison, we define a transformation from RAF structures to AFRA extensions:

Definition 70 (RAF Structure to AFRA extension [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF]). Let RAF = ⟨A, K, s,t⟩ be a RAF and U = ⟨S, Q⟩ be a structure. We denote by str2afraExt(U) the AFRA extension corresponding to U. It is defined as:

str2afraExt(U) = S ∪ {α ∈ Q|s(α) ∈ S}
In [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF], Proposition 22 and Theorem 5 have been proven.

Proposition 22. Let RAF = ⟨A, K, s,t⟩ be a RAF and U = ⟨S, Q⟩ be some RAF-conflict-free (respectively RAF-admissible ) structure. Then str2afraExt(U) is AFRA-conflict-free (respectively AFRA-admissible).

Theorem 5. For each semantics σ ∈ {complete, stable, preferred, grounded}, the function str2afraExt(•) is a one-to-one correspondence between the sets of all σ -structures and the set of all AFRA-σ -extensions.

Example 46. Let consider the RAF/AFRA shown in Figure 8.2 on page 90. Following Examples 38 and 39 on page 87 and on page 92 and as shown by the result in Table 8.1 on page 88 and Table 9.1 on page 93, there is a one-to-one correspondence between the sets of all structures and the set of all AFRA-extensions for the all the complete-based semantics mentioned. We have:

• S 1 corresponding to U 1 • S 2 corresponding to U 2 • S 3 corresponding to U 3 • S 4 corresponding to U 4 Part V
Higher-Order Attack Argumentation Frameworks: Contribution

Part presentation:

In this part is presented all the works about Recursive Argumentation Frameworks done during my thesis.

The two main contributions of this part are the adaptation of the notion of AF labelling for RAF, so-called, structure labellings, and the study of semantics decomposability property of RAF semantics. These works lead to several publications:

• An IRIT report about RAF structure labellings, [START_REF] Doutre | Argumentation Frameworks with Higher-Order Attacks: Labelling Semantics[END_REF], which serves as support for subsequent works, providing details on concepts and the properties proofs that are not in articles.

• An IRIT report about RAF Complexities, [START_REF] Doutre | Argumentation Frameworks with Higher-Order Attacks: Complexity results[END_REF], which serves to the same purpose.

• A poster in KR 2020, the 7 th International Conference on Principles of Knowledge Representation and Reasoning, [START_REF] Doutre | Argumentation Frameworks with Higher-Order Attacks: Semantics and Complexity[END_REF], about structure labellings and RAF semantics complexities.

• An article in ICTAI 2020, the 32 th International Conference on Tools with Artificial Intelligence [START_REF] Doutre | Argumentation Frameworks with Higher-Order Attacks: Labellings and Complexity[END_REF], about the structure labellings and RAF semantics complexities.

• A journal article that has been submitted and currently being reviewed in IJAIT, the International Journal of Artificial Intelligence Tools, gathering most of the works that have been done on RAFs.

Firstly, the notion of Dung labellings and the semi-stable semantics are extended to RAF. Secondly, a Flattening process that transforms RAFs into AFs, ensuring interesting properties, is introduced. Thirdly, relying on that Flattening, the complexities of RAF semantics is studied. Fourthly, the notion of Strongly Connected Component is adapted to RAFs and from this key notion, the semantics decomposability and the directionality of RAF semantics are studied. Finally, related works are presented.

Chapter 10

New semantics for RAF

In this chapter is introduced the semi-stable semantics for RAF, and a focus on RAFs with no recursive attacks is done. The notion of AF reinstatement labelling introduced in [START_REF] Caminada | On the issue of reinstatement in argumentation[END_REF] is also generalized for RAF (so called "reinstatement RAF labelling"). ) highlighted a non yet discovered semantics: the semi-stable semantics. From the semi-stable labelling semantics has been defined the semi-stable extension semantics.

Definition and some properties

As for AF, we propose that semi-stable structures be the ones that decide the most on the acceptance or the rejection of arguments and attacks. They are formally defined as follows:

Definition 71. (Semi-stable structure). Let RAF = ⟨A, K, s,t⟩ be a RAF and U = ⟨S, Q⟩ be some structure over it. U is said to be a semi-stable structure iff U is a complete structure such that: Example 47. The complete, grounded, preferred, semi-stable, arg-preferred and stable semantics corresponding to Figure 10.1 on the previous page are given in Table 10.1 on page 105. We can observe that the stable semantics produces no structure for that RAF. This example shows that a semi-stable structure is not always a stable one (see U 3 and U 4 ) and that a preferred structure is not always a semi-stable one (see U 2 ).

S ∪ Q ∪ RAF-De f (U) ∪ RAF-Inh(U)

The case of RAF with no recursive attacks

As stated in Section 9.2 on page 94, it has been proven in [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF] that in RAFs without recursive attacks there is a one-to-one correspondence between structures and Dung's extensions for the complete, grounded, preferred and stable semantics. Let now consider the case of the semi-stable semantics and show that the set of semi-stable extensions coincides with the set of semi-stable structures on RAF with no recursive attacks (i.e. RAF that happened to be simple AF). Notice that all the structures of such a RAF contain all the attacks.

Proposition 23 (Semi-stable extensions and structures). Let RAF = ⟨A, K, s,t⟩ be a RAF such that ∀α ∈ K, t(α) ∈ A. RAF can thus be considered as a simple AF. Let AF = ⟨A, K⟩ be the AF version of RAF .

U = ⟨S, K⟩ is a semi-stable structure of RAF iff S is a semi-stable extension of AF □ Proof of

Reinstatement RAF labellings

Now that relations between structure semantics and between structure and extensions semantics have been stated, we introduce the notion of labelling on RAF.

The reason why we are interested in the labelling approach to compute semantics is that labellings are more precise than structures (as there are three statuses to describe the acceptance of elements) and especially because it seems to be more practical for finding algorithms.

Definition 72. (RAF labelling). Let RAF = ⟨A, K, s,t⟩ be a recursive argumentation framework. A RAF labelling is a tuple L = ℓ A , ℓ K such that ℓ A is a total function ℓ A : A → {in, out, und} and ℓ K , a total function ℓ K : K → {in, out, und}.

We define:

• in(L) as the tuple {a ∈ A|ℓ A (a) = in}, {α ∈ K|ℓ K (α) = in} , • und(L) as the tuple {a ∈ A|ℓ A (a) = und}, {α ∈ K|ℓ K (α) = und} and • out(L) as the tuple {a ∈ A|ℓ A (a) = out}, {α ∈ K|ℓ K (α) = out} . Let x ∈ (A ∪ K).
: ∀x ∈ (A ∪ K), • (L(x) = out) ⇐⇒ (∃α ∈ K s.t. t(α) = x, ℓ K (α) = in and ℓ A (s(α)) = in) • (L(x) = in) ⇐⇒ (∀α ∈ K s.t. t(α) = x, ℓ K (α) = out or ℓ A (s(α)) = out)
An equivalent definition of reinstatement RAF labelling can be made, as for AF, using the notion of "legally labelled argument". An in -labelled element is said to be legally in iff all its attackers or their involved attacks are labelled out . An out -labelled element is said to be legally out iff at least one of its attackers and the involved attack are labelled in . An und -labelled element is said to be legally und iff it does not have any attacker and its involved attack that are labelled in and one of its attackers and the involved attack are not labelled out . Formally, "valid labellings" (notion equivalent to reinstatement RAF labellings) are defined as follows:

Definition 74 (Legally labelled elements, valid RAF labelling).

Let RAF = ⟨A, K, s,t⟩ be a recursive argumentation framework and L = ℓ A , ℓ K be a RAF labelling over RAF . Let x be an argument or an attack of RAF . x is said to be legally labelled in L if and only if the 3 following conditions hold:

• x ∈ in(L) iff (∀α ∈ K s.t. t(α) = x, ℓ K (α) = out or ℓ A (s(α)) = out) • x ∈ out(L) iff (∃α ∈ K s.t. t(α) = x, ℓ K (α) = in and ℓ A (s(α)) = in) • x ∈ und(L) iff ((∄α ∈ K s.t. t(α) = x, ℓ K (α) = in and ℓ A (s(α)) = in)
and

(∃α ∈ K s.t. t(α) = x, ℓ K (α) ̸ = out and ℓ A (s(α)) ̸ = out))
L is said to be a valid RAF labelling if all its elements are legally labelled. Example 48. As an illustration, the labelling version of Table 10.1 on the following page about Figure 10.1 on page 102 is shown in Table 10.2 on page 106. In the first part of the table, i j means that the element i belongs to the structure j.

In the second part of the table, i j means that j is a structure of the semantics i.

Structure labellings and semantics

In this chapter we show that there exists a one-to-one mapping between RAF labellings and structures. Specific semantics structures happen to be coinciding with RAF labellings under some constraints. In order to prove it, we introduce the two following functions to go from structures to labellings and vice versa:

Definition 75. (Struct2Lab and Lab2Struct). Let RAF = ⟨A, K, s,t⟩ be a RAF, U = ⟨S, Q⟩ be a structure and L = ℓ A , ℓ K be a RAF labelling. The functions Struct2Lab RAF and Lab2Struct RAF are defined as following:

• Struct2Lab RAF (U) = ℓ A , ℓ K , a RAF labelling with: -ℓ A = {(a, in)|a ∈ S} ∪ {(a, out)|a ∈ (A \ S) and a ∈ RAF-De f (U)} ∪ {(a, und)|a ∈ (A \ S) and a / ∈ RAF-De f (U)} -ℓ K = {(α, in)|α ∈ Q} ∪ {(α, out)|α ∈ (K \ Q) and α ∈ RAF-Inh(U)} ∪ {(α, und)| α ∈ (K \ Q) and α / ∈ RAF-Inh(U)}
• Lab2Struct RAF (L) = ⟨S, Q⟩, a structure with:

-S = {a|ℓ A (a) = in} -Q = {α|ℓ K (α) = in}
We write Struct2Lab and Lab2Struct instead of Struct2Lab RAF and Lab2Struct RAF when there is no ambiguity about the RAF RAF we refer to.

Complete semantics

Reinstatement RAF labellings coincide with complete structures as stated by Theorems 8 and 9 on the current page and on the next page.

Theorem 8. Let RAF = ⟨A, K, s,t⟩ be a RAF and let L = ℓ A , ℓ K be a reinstatement RAF labelling. Then Lab2Struct(L) is a complete structure.

□ Proof of Theorem 8: link (See page 232).

Theorem 9. Let RAF = ⟨A, K, s,t⟩ be a RAF and let U = ⟨S, Q⟩ be a complete structure. Then Struct2Lab(U) is a reinstatement RAF labelling.

□ Proof of Theorem 9: link (See page 233).

Preferred semantics

In this section we show that several constraints on reinstatement RAF labellings lead to the preferred semantics.

Reinstatement RAF labellings with maximal in

Reinstatement RAF labellings such that in (L) is maximal coincide with preferred structures as stated by Theorems 10 and 11.

Theorem 10. Let RAF = ⟨A, K, s,t⟩ be a RAF and let L = ℓ A , ℓ K be a reinstatement RAF labelling such that in(L) is maximal. Then Lab2Struct(L) is a preferred structure.

□ Proof of Theorem 10: link (See page 234).

Theorem 11. Let RAF = ⟨A, K, s,t⟩ be a RAF and let U = ⟨S, Q⟩ be a preferred structure. Then L = Struct2Lab(U) is a reinstatement RAF labelling such that in(L) is maximal.

□ Proof of Theorem 11: link (See page 234).

Reinstatement RAF labellings with maximal out

Reinstatement RAF labellings such that out (L) is maximal also coincide with preferred structures.

Theorem 12. Let RAF = ⟨A, K, s,t⟩ be a RAF and let L = ℓ A , ℓ K be a reinstatement RAF labelling such that out(L) is maximal. Then Lab2Struct(L) is a preferred structure.

□ Proof of Theorem 12: link (See page 236).

Theorem 13. Let RAF = ⟨A, K, s,t⟩ be a RAF and let U = ⟨S, Q⟩ be a preferred structure. Then L = Struct2Lab(U) is a reinstatement RAF labelling such that out(L) is maximal.

□ Proof of Theorem 13: link (See page 236).

Stable semantics: reinstatement RAF labellings with empty und

Reinstatement RAF labellings such that und (L) is empty coincide with stable structures as stated by Theorems 14 and 15 on the following page.

Theorem 14. Let RAF = ⟨A, K, s,t⟩ be a RAF and let L = ℓ A , ℓ K be a reinstatement RAF labelling such that und(L) = ∅. Then Lab2Struct(L) is a stable structure.

□ Proof of Theorem 14: link (See page 236).

Theorem 15. Let RAF = ⟨A, K, s,t⟩ be a RAF and let U = ⟨S, Q⟩ be a stable structure. Then L = Struct2Lab(U) is a reinstatement RAF labelling such that und(L) is empty.

□ Proof of Theorem 15: link (See page 236).

Grounded semantics

In this section we show that several constraints on reinstatement RAF labellings lead to the grounded semantics.

Reinstatement RAF labellings with maximal und

Reinstatement RAF labellings such that und (L) is maximal coincide with the grounded structure as stated by Theorems 16 and 17.

Theorem 16. Let RAF = ⟨A, K, s,t⟩ be a RAF and let L = ℓ A , ℓ K be a reinstatement RAF labelling such that und(L) is maximal. Then Lab2Struct(L) is the grounded structure.

□ Proof of Theorem 16: link (See page 237).

Theorem 17. Let RAF = ⟨A, K, s,t⟩ be a RAF and let U = ⟨S, Q⟩ be the grounded structure. Then L = Struct2Lab(U) is a reinstatement RAF labelling such that und(L) is maximal.

□ Proof of Theorem 17: link (See page 237).

Reinstatement RAF labellings with minimal in

Reinstatement RAF labellings such that in (L) is minimal coincide with the grounded structure as stated by Theorems 18 and 19.

Theorem 18. Let RAF = ⟨A, K, s,t⟩ be a RAF and let L = ℓ A , ℓ K be a reinstatement RAF labelling such that in(L) is minimal. Then Lab2Struct(L) is the grounded structure.

□ Proof of Theorem 18: link (See page 237).

Reinstatement RAF labellings with minimal out

Note that reinstatement RAF labellings such that out (L) is minimal also coincide with the grounded structure as stated by Theorems 20 and 21.

Theorem 20. Let RAF = ⟨A, K, s,t⟩ be a RAF and let L = ℓ A , ℓ K be a reinstatement RAF labelling such that out(L) is minimal. Then Lab2Struct(L) is the grounded structure.

□ Proof of Theorem 20: link (See page 237).

Theorem 21. Let RAF = ⟨A, K, s,t⟩ be a RAF and let U = ⟨S, Q⟩ be the grounded structure. Then L = Struct2Lab(U) is a reinstatement RAF labelling such that out(L) is minimal.

□ Proof of Theorem 21: link (See page 237).

Semi-stable semantics

Reinstatement RAF labellings such that und (L) is minimal coincide with semi-stable structures as stated by Theorems 22 and 23.

Theorem 22. Let RAF = ⟨A, K, s,t⟩ be a RAF and let L = ℓ A , ℓ K be a reinstatement RAF labelling such that und(L) is minimal. Then Lab2Struct(L) is a semi-stable structure.

□ Proof of Theorem 22: link (See page 238).

A one-to-one mapping

In this section are summarized the relations between labellings and structures in RAF and are also presented the links between labellings in AF and RAF with no recursive attacks. As stated in Section 10.1.2 on page 103, there exists a one-to-one mapping between structures and extensions in RAF without recursive attacks for the complete, grounded, preferred, semi-stable and stable semantics. [START_REF] Caminada | On the issue of reinstatement in argumentation[END_REF] established a one-to-one mapping between AF extensions and AF reinstatement labellings for the mentioned semantics. In this thesis, as summarized in Section 11.6.1 on the previous page, is established a one-to-one mapping between RAF structures and reinstatement RAF labellings for the same semantics.

Structures and labellings in RAF

As a consequence, for RAF with no recursive attacks, there exists obviously a one-to-one mapping between reinstatement labellings (AF notion) and structures (RAF notion) and also between reinstatement labellings (AF notion) and reinstatement RAF labellings (RAF notion).

RAF flattening

In this chapter is presented a new RAF flattening process to convert RAF into AF. We first discuss the interest of such process (Section 12.1). We, then, define it and illustrate it (Section 12.2 on the following page). Finally, some semantics properties are studied (Section 12.3 on page 116).

Motivation

What motivated the search for a flattening process that transforms RAFs into AFs is that such a tool could probably opened up perspectives for the study of RAF properties. Indeed, Dung's Argumentation Theory has been explored for decades now. A lot of properties on AFs and semantics have been defined and studied. Having a way to transform RAFs into AFs could then probably help for the extension of notions and properties defined for AF to RAF. We can think, as examples, of AF shape properties (SCC, autonomous fragments [START_REF] Baroni | Refining SCC decomposition in argumentation semantics : a first investigation[END_REF]), semantics properties (complexity, SCC-recursiveness [11], semantics decomposability [START_REF] Baroni | On the input/output behavior of argumentation frameworks[END_REF]). Furthermore, such a transformation could be used for computational purposes given that there is a wide range of AF solvers nowadays. This being said, it important to know that such a transformation has been already proposed in [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF], as seen in Section 9.2.2 on page 98. Even though this transformation proves a one-to-one semantics correspondence between RAF and MAF (and also between RAF and AF, since MAF are AF), it is not enough for proving properties as semantics complexity. Indeed, as explained in Example 45 on page 99, this correspondences do not guarantee that each acceptable element w.r.t. a structure will be acceptable w.r.t. its corresponding extension in the resulting MAF/AF. Note: Such a transformation has also been proposed in [START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF], as seen in Section 8.2 on page 90, to transform AFRA into AF. As the previous mentioned one, this transformation ensures a one-to-one semantics correspondence between AFRA and AF, but not between acceptable elements.

Moreover, in our sense, this transformation presents a counter-intuitive meaning of the link between an attack and its source. Indeed each attack is related to its source by a sequence of attacks meaning that the attack is "defended" by its source. So, every attack is always attacked in the resulting MAF even if it is not the target of an attack in the initial RAF. This has also a potential impact on properties related to the shape of the AF.

Let now introduce a new flattening process for RAF that fixes those two problems.

A new flattening process

Introduced in [START_REF] Doutre | Argumentation Frameworks with Higher-Order Attacks: Labellings and Complexity[END_REF], the new flattening process is formally defined as follows:

Definition 76. Let Raf2Af : Φ ra f → Φ a f be the function transforming a RAF into an AF. Raf2Af is defined as follows:

∀ RAF = ⟨A, K, s,t⟩ ∈ Φ ra f , Raf2Af : RAF → AF = A ′ , K ′ With: A ′ = A ∪ K ∪ Not A ∪ Not K ∪ And A,K K ′ = K ′ 1 ∪ K ′ 2 ∪ K ′ 3 ∪ K ′ 4 ∪ K ′ 5 Not A = {¬a|a ∈ A} Not K = {¬β |β ∈ K} And A,K = {a.β |β ∈ K, a = s(β )} K ′ 1 = {(a, ¬a)|a ∈ A} K ′ 2 = {(β , ¬β )|β ∈ K} K ′ 3 = {(¬a, a.β )|a ∈ A, s(β ) = a} K ′ 4 = {(¬β , a.β )|β ∈ K, s(β ) = a} K ′ 5 = {(a.β ,t(β ))|β ∈ K, s(β ) = a}
Note: "¬a", "¬β " and "a.β " are just simple argument names that represent respectively, the "negation" of argument a, the "negation" of attack β and the "conjunction" of attack β with its source a. 1This transformation represents, with AFs, the semantics of RAF defeat relation, by mean of additional arguments. Let a be an argument attacking an element β through the attack α in the RAF RAF . Given that to β be defeated by a, α must be valid (non-inhibited) and a accepted (not defeated), we represent this by creating an additional argument named "a.α" accepted in AF only when both a and α are. To do that we create two others arguments named "¬a" and "¬α". We create an attack going from a to ¬a, another going from α to ¬α, two others going from ¬a to a.α and ¬α to a.α, and finally a last one going from a.α to β . An argument (corresponding to an element of the original RAF) is thus defeated in the resulting AF if and only if there exists a valid attack in the original RAF that targets this argument and whose source is accepted.

Example 50. Let consider Figure 12.1 on the next page. Let RAF = ⟨A, K, s,t⟩ be the RAF represented in Figure 12.1(a). We have AF = Raf2Af(RAF ) being the AF represented in Figure 12.1(b).

Note: Interesting properties are formally proven in Section 12.3 and Chapters 13 and 14 on the next page, on page 119 and on page 123. Until then we can already notice that the shape structure is preserved: all and only elements that are attacked in the original RAF are attacked in the flattened version. Moreover, the correspondence between structure and extension seems intuitive. Let S = {a, b, c, d, α, δ , ¬β , ¬γ, d.δ , a.α}. S is a complete extension of AF . We have the intuition that removing the created arguments during the flattening and putting arguments and attacks apart, lead to a RAF-complete structure of RAF : U = ⟨{a, b, c, d} , {α, δ }⟩ (that is indeed RAF-complete).

Properties

In order to study the semantics properties, we defined what is an extension (of the RAF flattened version) corresponding to a structure (of the initial RAF).

Definition 77 (Extension corresponding to structure). Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be an AF (with AF = ⟨A ′ , K ′ ⟩). Let U = ⟨S, Q⟩ be a structure in RAF . We denote by "ε U " the extension in AF corresponding to a structure U, defined by:

ε U = S ∪ Q ∪ {¬a ∈ Not A |a ∈ RAF-De f (U)} ∪ {¬β ∈ Not K |β ∈ RAF-Inh(U)} ∪ {s(β ).β ∈ And A,K |β ∈ Q, s(β ) ∈ S}
The next proposition establishes the link between the RAF-De f and RAF-Inh relations of the original RAF with De f relation in the AF, and the link between the RAF-Acc relation in the RAF with the Acc relation in the AF.

Proposition 24. Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be an AF (with AF = ⟨A ′ , K ′ ⟩). Let U = ⟨S, Q⟩ be a structure in RAF . The following properties holds:

1. RAF-De f (U) ∪ RAF-Inh(U) = 12.3. PROPERTIES De f (ε U ) \      {¬a ∈ Not A |a ∈ ε U } ∪{¬β ∈ Not K |β ∈ ε U } ∪{s(β ).β ∈ And A,K |β ∈ De f (ε U ) or s(β ) ∈ De f (ε U )}      2. RAF-Acc(U) = Acc(ε U ) \      {¬a ∈ Not A |a ∈ De f (ε U )} ∪{¬β ∈ Not K |β ∈ De f (ε U )} ∪{s(β ).β ∈ And A,K |s(β ).β ∈ ε U }      □ Proof of Proposition 24: link (See page 238).
From the previous proposition, we can state the following semantics correspondence: Proposition 25. Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be an AF (with AF = ⟨A ′ , K ′ ⟩).

The following properties holds:

1. U = ⟨S, Q⟩ is a RAF-complete structure in RAF iff ε U is a complete extension in AF . 2. U = ⟨S, Q⟩ is a RAF-grounded structure in RAF iff ε U is a grounded extension in AF . 3. U = ⟨S, Q⟩ is a RAF-preferred structure in RAF iff ε U is a preferred extension in AF . 4. U = ⟨S, Q⟩ is a RAF-stable structure in RAF iff ε U is a stable extension in AF . 5. U = ⟨S, Q⟩ is a RAF-semi-stable structure in RAF iff ε U is a semi-stable extension in AF .
□ Proof of Proposition 25: link (See page 242).

The study of the semantics properties can also be done the other way around. Let define what is a structure (of the initial RAF) corresponding to an extension (of the RAF flattened version).

Definition 78 (RAF labelling and AF labelling). Let σ be a semantics. Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be an AF (with AF = ⟨A ′ , K ′ ⟩).

The function rafLab2AfLab : L σ -ra f (RAF ) → L σ (AF ), which maps to each structure labelling of RAF an AF labelling of AF , is defined as following. Let L be a structure labelling of RAF and let ℓ = rafLab2AfLab(L). We have:

• ∀x ∈ (A ∪ K): -L(x) = ℓ(x) -ℓ(¬x) = und ⇐⇒ L(x) = und -ℓ(¬x) = in ⇐⇒ L(x) = out -ℓ(¬x) = out ⇐⇒ L(x) = in • ∀α ∈ K: -ℓ(s(α).α) = in ⇐⇒ (L(s(α)) = in and L(α) = in) -ℓ(s(α).α) = out ⇐⇒ (L(s(α)) = out or L(α) = out) -ℓ(s(α).α) = und ⇐⇒ (L(s(α)) ̸ = out and L(α) ̸ = out and (L(s(α)) = und or L(α) = und)
The function afLab2RafLab : L σ (AF ) → L σ -ra f (RAF ), which maps to each AF labelling of AF a structure labelling of RAF , is defined as following. Let ℓ be a labelling of AF and let L = afLab2RafLab(ℓ).

We have:

L = ℓ ↓ A , ℓ ↓ K
The following propositions are trivially induced by Proposition 25 on the previous page.

Proposition 26. Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be an AF (with AF = ⟨A ′ , K ′ ⟩).

The following property holds: Chapter 13

L = ℓ A , ℓ K is a RAF-complete (

RAF Decision Problems and semantics complexities 13.1 RAF Decision problems

Although in [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF], results about RAF semantics complexities are given for the RAF-Cred σ problem (for the RAF-complete, RAF-preferred and RAF-stable semantics) and for the RAF-Skep σ problem (for the RAF-preferred and RAF-stable semantics), the decisions problems have not been explicitly defined nor the detailed proofs for the result given. In Definition 79, they are formally defined and in Section 13.2 on the next page their complexities are studied.

Definition 79 (Decision Problems in RAF).

• Credulous Acceptance Cred σ : Given an RAF RAF = ⟨A, K, s,t⟩ and an element x ∈ A ∪ K. Is x contained in some U ∈ σ (RAF )?

• Skeptical Acceptance Skep σ : Given an AF RAF = ⟨A, K, s,t⟩ and an element x ∈ A∪K. Is x contained in each U ∈ σ (RAF )?

• Verification of a structure Ver σ : Given an RAF RAF = ⟨A, K, s,t⟩ and a structure U. Is U ∈ σ (RAF )?

• Existence of a structure Exists σ : Given an RAF RAF = ⟨A, K, s,t⟩. Is σ (RAF ) ̸ = ∅?

• Existence of a non-empty structure Exists ¬∅ σ : Given an RAF RAF = ⟨A, K, s,t⟩. Does there exist a structure U ̸ = ∅ such that U ∈ σ (RAF )?

• Uniqueness of a solution Unique σ : Given an RAF RAF = ⟨A, K, s,t⟩. Is there a unique structure U ∈ σ (RAF ), i.e. σ (RAF ) = {U}?

Example 51. Let σ be the preferred semantics. Let RAF = ⟨A, K, s,t⟩ be the RAF represented in Figure 13.1 on the next page and U be any structure of it. Following Table 9.1 on page 93, we have three preferred structures:

σ (RAF ) =          U 2 = {a, g, α, β , γ, δ , ε, ζ , η, κ, λ } U 3 = {b, c, d, f , g, α, β , γ, ε, ζ , η, κ, λ } U 4 = {b, c, d, f , α, β , γ, ε, ζ , η, ι, κ, λ }         
As a consequence, we have so:

• RAF-Cred σ (RAF , e) = false • RAF-Cred σ (RAF , a) = true • RAF-Skep σ (RAF , b) = false • RAF-Skep σ (RAF , α) = true • RAF-Ver σ (RAF , U) = true iff U ∈ {U 2 , U 3 , U 4 } • RAF-Exists σ (RAF ) = true • RAF-Exists ¬∅ σ (RAF ) = true • RAF-Unique σ (RAF ) = false

RAF Semantics Complexities

In Section 9.2 on page 94 we saw that in terms of semantics, RAF is a generalization of Dung's AF. Knowing this, we searched for a polynomial reduction to transform RAF decision problems into AF ones and vice versa. Thereby, we could compare RAF decision problems complexities to the known AF complexities and potentially identify precisely the complexity classes of RAF decision problems.

As seen in Section 12.2 on page 115, we defined a function, Raf2Af, that transforms RAF in into AF. Let define a function, Af2Raf, that does the opposite.

Definition 80. Let Af2Raf : Φ a f → Φ ra f be the function transforming an AF into an RAF. Af2Raf is defined as following:

∀ AF = ⟨A, K⟩ ∈ Φ a f , Af2Raf : AF → RAF = A ′ , K ′ , s,t With: • A ′ = A • K ′ = {α|α = (a, b) ∈ K}. 1 • ∀α = (a, b) ∈ K, t(α) = b and s(α) = a
The definition of Af2Raf that transforms an AF into an RAF is trivial since an RAF without higher-order attacks is an AF. So it is enough to name the attacks of the AF in order to obtain a RAF. Note that, following the previous definition, no attack can be inhibited (as none of them is a target) in the RAF obtained by Af2Raf.

Both Raf2Af and Af2Raf are polynomial time and log-space functions. Let thus take advantage of these properties and prove that AF decision problems can be reduced into RAF ones and vice versa.

Proposition 28. Let RAF = ⟨A, K, s,t⟩ be an RAF and AF = Raf2Af(RAF ) be an AF (with AF = ⟨A ′ , K ′ ⟩). Let a ∈ (A ∪ K) be an element in RAF and an argument in AF , following the definition of Raf2Af. Let U = ⟨S, Q⟩ be a structure of RAF .

For each semantics σ ∈ {complete, semi-stable, stable, preferred, grounded}, we have: Given the previous property and given that Raf2Af is a polynomial time and log-space function, we can assert that: Proposition 29. The complexities of AF decision problems are at least as hard as RAF ones, for the semantics complete, semi-stable, stable, preferred, grounded.

□ Proof of Proposition 29: link (See page 252).

Let now do the other way around process.

Proposition 30. Let AF = ⟨A, K⟩ be an AF and RAF = Af2Raf(AF ) be an RAF. Let a ∈ A be an argument in AF and in RAF , following the definition of Af2Raf. For each semantics σ ∈ {complete, semi-stable, stable, preferred, grounded}, we have: Given the previous property and given that Af2Raf is a polynomial time and log-space function, we can assert that: Proposition 31. The complexities of RAF decision problems are at least as hard as AF ones, for the semantics complete, semi-stable, stable, preferred, grounded.

□ Proof of Proposition 31: link (See page 252).

Finally, we obtain the following proposition: Proposition 32. The complexities of RAF decision problems are the same as AF ones, for the semantics complete, semi-stable, stable, preferred, grounded, as stated in Table 13 Semi-stable

Σ P 2 -c Π P 2 -c coNP-c trivial NP-c in Θ P 2
Table 13.1: Complexities of RAF decision problems Proposition 32 is a very interesting one. Indeed, the complexities for the decision problems in the context of RAF, are the same as the ones in Dung's framework, despite all the additional expressivity that is brought by the higher-order attacks.

Chapter 14

Hierarchical view of RAF and semantics decomposability

In this chapter, the notion of SCC a f is extended to RAF (Section 14.1). Based on it, the semantics decomposability of RAF semantics is studied (Section 14.3 on page 133). Finally, a hierarchical view of RAF is proposed (Section 14.2 on page 131). These new properties open perspectives for future algorithms.

RAF Strongly Connected Component

In this section we build up from basic ones the notion of SCC for RAF. We also established the link between the SCCs of a given RAF and the SCCs of its AF flattened version.

Definition 81 (RAF-walk). Let RAF = ⟨A, K, s,t⟩ be a RAF and e 1 , ..., e n ∈ (A ∪ K) be elements of RAF . A RAF-walk is a sequence (e 1 , ..., e n ) with n ∈ N * such that:

• ∀i ∈ {1, ..., n}, e i ∈ (A ∪ K) • If n > 1, ∀i ∈ {1, ..., n -1}, e i ∈ A =⇒ e i+1 ∈ K and e i = s(e i+1 ) • If n > 1, ∀i ∈ {1, ..., n -1}, e i ∈ K =⇒ t(e i ) = e i+1
Definition 82 (RAF-path). A RAF-path is a RAF-walk in which all the elements are distinct. Let RAF = ⟨A, K, s,t⟩ be a RAF. We denote by "Paths ra f (RAF )" the set of RAF-paths of RAF . Note: Following Definitions 84 and 85 on the previous page and on the current page, a RAF-cycle or a RAF-closed-walk can only attack elements that belongs to it.

An equivalence relation can be defined on the notion of RAF-paths, establishing the fact that two elements x and y belong to the same equivalence class if and only if there exists a path from x to y attacking y and vice-versa (a path from y to x attacking x) in the RAF. This is an important difference with the definition of SCC in graph theory. Formally, this equivalence relation is defined, as follows:

Definition 86 (RAF Path Equivalence -PE ra f ). Given a RAF RAF = ⟨A, K, s,t⟩, a RAF Path Equivalence noted PE ra f (RAF ) is a binary relation between elements of RAF such that:

• PE ra f (RAF ) ⊆ (A ∪ K) 2 • ∀x ∈ (A ∪ K), (x, x) ∈ PE ra f (RAF )
• Given two distinct elements x, y ∈ (A ∪ K), (x, y) ∈ PE ra f (RAF ) if and only if there exist:

p ∈ Paths ra f (RAF ) such that p = (x, ..., e n-1 , y) and y = t(e n-1 ).

p ′ ∈ Paths ra f (RAF ) such that p ′ = (y, ..., e m-1 , x) and x = t(e m-1 ).

We introduce the notation x ≡ RAF y to state that (x, y) ∈ PE ra f (RAF ).

The idea behind the previous definition is the following:

• y is attacked by p (in the sense of Definition 85) so x has an effect on y.

• x is attacked by p ′ (in the sense of Definition 85) so y has an effect on x.

As both affect each other they are considered as equivalent under the PE ra f relation.

Definition 87 (RAF Strongly Connected Component -SCC ra f ). Given a RAF RAF = ⟨A, K, s,t⟩, a RAF Strongly Connected Component (SCC ra f ) is an equivalence class of elements under the relation PE ra f . The set of SCC ra f of RAF is denoted by SCCS ra f (RAF ).

An interesting property of PE ra f relation is that if two elements are equivalent then there exists a RAFclosed-walk that contains and attacks them: • {ι, g} ∈ SCCS ra f (RAF ):

Indeed, there exists a RAF-path from g to ι attacking ι which is (g, κ, ι) and another one from ι to g attacking g which is (ι, g). As a consequence, we have: ι ≡ RAF g. As there is no other element in x ∈ RAF such that x ≡ RAF ι or x ≡ RAF g then {ι, g} is a SCC ra f . Note that, although κ ∈ (g, κ, ι), κ does not belong to this SCC ra f since it is not attacked by (g, κ, ι). Let consider now the RAF RAF = ⟨A, K, s,t⟩ and the AF AF = Raf2Af(RAF ) illustrated in Figure 14.5 on page 129:

• Considering the attack β ∈ RAF , we have in Raf2Af(RAF ): (a, ¬a, a.β , a) ∈ Cycles a f (AF ) and there exists a unique path in Raf2Af(RAF ) whose first element is β and whose last is a, that is (β , ¬β , a.β , a).

• Considering the attack α ∈ RAF , we have in AF : (α, ¬α, a.α, α) ∈ Cycles a f (AF ) being the unique cycle in AF whose first and last elements are α and (a, ¬a, a.α, α) is a path in AF .

Example 58. Let consider Figure 14.4 on the next page. Let RAF be the RAF in Figure 14.4(a) on the next page and let consider the RAF-path (a, α, β ).

• In Raf2Af(RAF ), there is a path from a to β which is (a, ¬a, a.α, β ). But in a slightly different RAF it could be possible that there exists another RAF-path from a to β and so another path from a to β in Raf2Af(RAF ). See Example 59 on the next page.

• In Raf2Af(RAF ), there is a path from α to β which is (α, ¬α, a.α, β ). This path is unique as an attack has a unique target. • As (d, δ , γ, a, α, β ) is a RAF-path, there is a path in Raf2Af(RAF ) from d to β which is (d, ¬d, d.δ , γ, ¬γ, c.γ, a, ¬a, a.α, β ).

• Furthermore, as (d, ¬d, d.δ , γ, ¬γ, c.γ, a, ¬a, a.α, β ) is a path and that d ∈ RAF and β ∈ RAF then there is also a RAF-path in RAF from d to β such that the before last element is an attack. The path (d, δ , γ, a, α, β ) satisfies those conditions.

The following proposition is very important to establish the correspondence between SCC ra f and SCC a f in a RAF flattened version: Finally, the correspondence SCC ra f and SCC a f is fully established by the following proposition: Proposition 36. Let RAF = ⟨A, K, s,t⟩ be a RAF and S ⊆ A ∪ K be a subset of elements of RAF . Let S ∈ SCCS ra f (RAF ) be a SCC ra f and let S ′ ∈ SCCS a f (AF ) be an SCC a f . We say that S ′ is the SCC a f corresponding to S (and vice-versa) iff :

Proposition 35. Let RAF = ⟨A, K, s,t⟩ be a RAF and U ⊆ A ∪ K be a subset of elements of RAF . U is included in some S ∈ SCCS ra f (RAF ) iff U is included in some S ′ ∈ SCCS a f (Raf2Af(RAF )) a α β
S ∈ SCCS ra f (RAF ) iff            S ∪ {¬a ∈ Not A |a ∈ S and (|S| > 1 or (∃α ∈ K s.t. s(α) = a and t(α) = a))} ∪ {¬α ∈ Not K |α ∈ S and (|S| > 1 or t(α) = α)} ∪ {s(α).α ∈ And A,K |α ∈ S and (|S| > 1 or t(α) = α)} ∪ {s(α).α ∈ And A,K |s(α) ∈ S and t(α) ∈ S}            ∈ SCCS a f (Raf2Af(RAF )) □ Proof of
S ′ =            S ∪ {¬a ∈ Not A |a ∈ S and (|S| > 1 or (∃α ∈ K s.t. s(α) = a and t(α) = a))} ∪ {¬α ∈ Not K |α ∈ S and (|S| > 1 or t(α) = α)} ∪ {s(α).α ∈ And A,K |α ∈ S and (|S| > 1 or t(α) = α)} ∪ {s(α).α ∈ And A,K |s(α) ∈ S and t(α) ∈ S}           

SCC partial order and hierarchy

In this section, we highlight the fact that there exists a partial order over the SCC ra f of RAF according to the binary relation defined in Definition 89 on the next page. Given this partial ordering, we can define the notions of predecessors and successors of an SCC ra f :

Definition 90 (Predecessor and successor of an SCC ra f ). Let RAF = ⟨A, K, s,t⟩ be a RAF and let S 1 ∈ SCCS ra f (RAF ) and S 2 ∈ SCCS ra f (RAF ) be two distinct SCC ra f of RAF . We say that S 1 is a predecessor of S 2 (resp. S 2 is a successor of S Given two SCC ra f such that one is the predecessor of the other, the following property holds:

Proposition 38. Let RAF = ⟨A, K, s,t⟩ be a RAF and let S ∈ SCCS ra f (RAF ) and S ′ ∈ SCCS ra f (RAF ) be two distinct SCC ra f . If S is the predecessor of S ′ (resp. S ′ is the successor of S) then for any path p = (e 1 , ..., e n-1 , e n ) ∈ Paths ra f (RAF ) such that e 1 ∈ S and e n ∈ S ′ and e n-1 ∈ K then for all i ∈ {1, ..., n}, we have the following property: Here is the partial order among these SCC ra f :

e i ∈ p ∩ K =⇒ t(e i ) ∈ (S ∪ S ′ ) □ Proof of
• {b} ≼ {a, β , c} • {α} ≼ {a, β , c} • {γ} ≼ {a, β , c} • {d} ≼ {γ} and {d} ≼ {a, β , c} • {δ } ≼ {γ} and {δ } ≼ {a, β , c}
Given a RAF, if we reduce all its SCC ra f to super nodes then we obtain, as for AFs (or more generally: as for directed graphs), a "DAG" (Directed Acyclic Graph) as stated by Proposition 39. Definition 91 describes the transformation that produces the DAG corresponding to a given RAF.

Definition 91 (Dag scc transformation). Let RAF = ⟨A, K, s,t⟩ be a RAF and let associate to each S i ∈ SCCS ra f (RAF ) a super node n i . Dag scc (RAF ) is the directed graph defined as following: Let aβ c be the super node corresponding to the SCC ra f {a, β , c}. Given that the other SCC ra f of RAF are all singletons we associate to each of them a super node having as name the element it possesses. See the resulting DAG in Figure 14.9.

Dag scc (RAF ) = {n i |S i ∈ SCCS ra f (RAF )}, {(n i , n j )|S i is
This property of SCC ra f partial ordering opens perspectives for algorithms computing RAF semantics following a certain hierarchical view of a given RAF.

Decomposability of semantics

In this chapter the decomposability property of semantics defined for AF (introduced in [START_REF] Baroni | On input/output argumentation frameworks[END_REF]) is extended to RAF. We first give some formal definitions to introduce the decomposability of RAF semantics (Section 14.3.1 on the next page). Then, we give an illustration of this notion for the RAF-complete semantics (Section 14.3.2 on page 142). Finally, the decomposability properties of the RAF-complete, RAF-grounded, RAF-preferred, RAF-semi-stable and RAF-stable semantics are established and proven (Section 14.3.3 on page 153).

Definitions

In Dung's Argumentation Framework attacks are always valid. Based on the notion introduced in [START_REF] Baroni | On input/output argumentation frameworks[END_REF] and [START_REF] Baroni | On the input/output behavior of argumentation frameworks[END_REF], any AF can be splitted into several sub-frameworks by simply ignoring some attacks. Indeed the influence of an input attack on a sub-framework only depends to the acceptance state of its source in its own subframework. It is not the case for RAF. Attacks, as arguments, can be labelled in , out or und . As a consequence, we cannot just ignore attacks to split a RAF. Furthermore, as attacks can be attacked in RAF, the removal of an attack could require some other removals in cascade (see Example 67). If we want to split this RAF so that the argument g may be alone in its sub-framework, we cannot just ignore the existence of ε, because it will require the same treatment for δ , and then for γ, then β and α.

Now, if we do not suppress attacks while splitting RAFs, we will have attacks without targets or without sources. Given that, the result of such a split do not produce RAFs. A new notion is thus necessary to capture this idea of RAF splitting: "Partial RAF".

Definition 92 (Partial RAF). Let RAF = ⟨A, K, s,t⟩ be a RAF. A partial RAF RAF = Ã, K, s, t, s,t of RAF is a tuple where:

• Ã ⊆ A is a set representing arguments • K ⊆ K is a set representing attacks
• s : K → {true, false} is a boolean function that indicates whether or not an attack in K has its source in à defined as following:

∀α ∈ K, s(α) = true if s(α) ∈ Ã otherwise false
• t : K → {true, false} is a boolean function that indicates whether or not an attack in K has its target in

à ∪ K ∀α ∈ K, t(α) = true if t(α) ∈ à ∪ K otherwise false
Example 68. Figure 14.11 gives an illustration of partial RAFs. We have:

• RAF = ⟨A, K, s,t⟩ with:

-A = {a, b, c, d} and K = {α, β , γ, δ } -s(α) = a, s(β ) = b, s(γ) = c, s(δ ) = d -t(α) = β , t(β ) = c, t(γ) = δ , t(δ ) = a
• RAF 1 = Ã1 , K1 , s1 , t1 , s,t with:

-Ã1 = {a, d} and K1 = {δ } -s1 (δ ) = true -t1 (δ ) = true • RAF 2 = Ã2 , K2 , s2 , t2
, s,t with: Definition 93 (RAF partition). Let RAF = ⟨A, K, s,t⟩ be a RAF. Let Ω = {ω 1 , ..., ω n } be a partition 1 of (A ∪ K). A RAF partition of RAF is a set of partial RAFs { RAF 1 , ..., RAF n } such that:

-Ã2 = {b, c} and K2 = {α, β , γ} -s2 (α) = false, s2 (β ) = true, s2 (γ) = true -t2 (α) = true, t2 (β ) = true, t2 (
∀ω i ∈ Ω, RAF i = Ãi , Ki , si , ti , s,t with:
1 So the following property holds for Ω:

• ∀(i, j) ∈ {1, ..., n} s.t. i ̸ = j, ω i ∩ ω j = ∅ • n i=1 ω i = A ∪ K • Ãi = ω i ∩ A • Ki = ω i ∩ K
• si : Ki → {true, false} is a boolean function that indicates whether or not an attack in Ki has its source in Ãi defined as following:

∀α ∈ Ki , si (α) = true if s(α) ∈ Ãi otherwise false • ti : Ki → {true,
Ã1 ∩ Ã2 = ∅, K1 ∩ K2 = ∅ and (A ∪ K) = ( Ã1 ∪ K1 ) ∪ ( Ã2 ∪ K2 ).
Among all possible partial RAFs, we highlight to particular types: the "Well-founded partial RAF" and the "Independant partial RAF". The first type corresponds to partial RAFs in which all attacks have their sources. The second one corresponds to well founded partial RAFs that are not attacked from outside. Formally, we have the following definitions:

Definition 94 (Well founded Partial RAF). Let RAF = ⟨A, K, s,t⟩ be a RAF and RAF = Ã, K, s, t, s,t be a partial RAF of RAF . RAF is said to be "well founded" if and only if the following property holds: ∀α ∈ K, s(α) is true.

Definition 95 (Independent Partial RAF). Let RAF = ⟨A, K, s,t⟩ be a RAF and RAF = Ã, K, s, t, s,t be a partial RAF of RAF . RAF is said to be "independent" if and only if the following property holds:

RAF is well founded and ∀α ∈ K s.t. t(α) ∈ ( Ã ∪ K), α ∈ K
Example 70. Following Example 68 on the previous page: RAF 1 is well founded but not independent, RAF 2 is neither well founded nor independent.

Considering a partial RAF implies to consider also its "inputs":

Definition 96 (Partial RAF with input). Let RAF = ⟨A, K, s,t⟩ be a RAF and RAF = ⟨ Ã, K, s, t, s, t⟩ be a partial RAF of RAF . The input I of RAF is a tuple S inp , Q inp where:

• S inp is the set of arguments defined by S inp = {s(α) ∈ (A \ Ã)|α ∈ K and t(α) ∈ ( Ã ∪ K)} • Q inp is the set of attacks defined by Q inp = {α ∈ (K \ K)|t(α) ∈ ( Ã ∪ K)}
The tuple RAF , I, L inp is called a "partial RAF with input", where L inp is a structure labelling of the elements in S inp and Q inp . • S inp 1 = {c}, Q inp 1 = {γ}. As an example, we may have:

L inp 1 = ⟨{(c, out)}, {(γ, und)}⟩. • S inp 2 = {a}, Q inp 2 = ∅.
As an example, we may have: L inp 2 = ⟨{(a, in)}, ∅⟩. Then a standard RAF is the RAF that can be built from a partial RAF with inputs:

Definition 97 (Standard RAF). Let RAF = ⟨A, K, s,t⟩ be a RAF. Let RAF , I, L inp be a partial RAF with input such that RAF = Ã, K, s, t, s,t is a partial RAF of RAF . The standard RAF w.r.t. RAF , I, L inp is a RAF defined as RAF s = ⟨A s , K s , s s ,t s ⟩ where:

• A s = Ã ∪ S inp ∪ {υ, ρ, ζ } • K s = K ∪ Q inp ∪ N ∪ {θ }, with: -N = {α x |x ∈ (Und ∪ Out)} -Out = out(L inp ) -Und = und(L inp )
And where s s : K s → A s and t s : K s → (A s ∪ K s ) are functions respectively mapping each attack to its source and to its target and such that:

• ∀α ∈ ( K ∪ Q inp ), s s (α) = s(α) • ∀α ∈ Q inp ∪ ( K \ {α|α ∈ K s.t. t(α) is false}), t s (α) = t(α) • ∀α ∈ {α|α ∈ K s.t. t(α) is false}, t s (α) = ζ • ∀α x ∈ {α x ∈ N|x ∈ Out}, s s (α x ) = ρ • ∀α x ∈ {α x ∈ N|x ∈ Und}, s s (α x ) = υ • ∀α x ∈ N, t s (α x ) = x • s s (θ ) = υ • t s (θ ) = υ
The following list gives the intuition of the new elements added in the standard RAF:

• υ is the argument that will serve to label und an element of the RAF input.

• θ is the attack whose source and target is υ, making υ a self attacking argument and thus an argument that will be labelled und .

• ρ is the argument that will serve to label out an element of the RAF input.

• N is the set of attacks that will link υ and ρ to all elements of the RAF input that should be labelled out or und .

• ζ is an argument that will serve as the target of all attacks of the partial RAF whose target does not belong to the partial RAF.

Note: By definition, all RAF-complete labellings of the standard RAF2 RAF s restricted to the elements of the input I coincide with the labelling L inp .

Example 72. Figure 14.13 on the following page gives illustrations of standard RAFs. RAF s 1 is the standard RAF corresponding to the partial RAF with input:

RAF 1 , I 1 = S inp 1 = {c}, Q inp 1 = {γ} , L inp 1 = ⟨{(c, out)}, {(γ, und)}⟩
RAF s 2 is the standard RAF corresponding to the partial RAF with input:

RAF 2 , I 2 = S inp 2 = {a}, Q inp 2 = ∅ , L inp 2 = ⟨{(a, in)}, ∅⟩
Notice that ζ , υ and ρ may be disconnected from the rest of the standard RAF following the partial RAF structure and the input labelling.

Let define an operator in order to select a sub-part of a structure labelling:

Definition 98. (Structure labelling restriction ↓). Let L = ℓ A , ℓ K be a structure labelling. Let U = ⟨S, Q⟩ be a structure. The restriction of L to U denoted as L ↓ U is defined as:

ℓ A ∩ (S × {in, out, und}), ℓ K ∩ (Q × {in, out, und})
Given a RAF with input, the canonical local function is simply a function that gives the set of labellings under a certain semantics of the sub-RAF we are interested in (i.e. the input elements and the other fictive elements created are not in these labellings).

Definition 99 (RAF canonical local function). Let RAF = ⟨A, K, s,t⟩ be a RAF, RAF = Ã, K, s, t, s,t be a partial RAF of RAF , σ be a semantics, RAF , I, L inp be a RAF with input, and RAF s be its standard RAF.

F ra f σ ( RAF , I, L inp ) = {L ↓ ⟨ Ã∪ K⟩ |L ∈ L σ -ra f ( RAF s )}.
Example 73. Following Example 72 on the previous page, we have for the RAF-complete semantics in the case of RAF 1 : While considering partitions of flattened RAF, some specific partitions must be considered:

F ra f co ( RAF 1 , I 1 , L inp 1 ) = {⟨{(a,
Definition 101 (RAF-compliant partition selector). An AF partition selector S is said to be "RAF-compliant" iff for any RAF RAF = ⟨A, K, s,t⟩ and its corresponding AF AF = Raf2Af(RAF ) (with AF = ⟨A ′ , K ′ ⟩), we have the following property:

∀Ω ′ ∈ S (AF ), ∀ω ′ ∈ Ω ′ ,    x ∈ (A ∪ K) ∩ ω ′ =⇒ ¬x ∈ ω ′ α ∈ K ∩ ω ′ =⇒ s(α).α ∈ ω ′
To precise that a partition selector is RAF-compliant we use the notation: S ra f -c . Let Ω ′ ∈ S ra f -c (AF ) be a partition of AF selected by some RAF-compliant selector. We say that Ω ′ is a RAF-compliant partition of AF .

Note: Definition 101 on the previous page describe a property, the property for AF partition selector to be RAF-compliant. But it does not define any selector. The following definition defines the default RAFcompliant partition selector, which is the AF partition selector that produces, given a flattened RAF, all RAF-compliant partitions.

Definition 102 (Default RAF-compliant partition selector). The default RAF-compliant selector, denoted by S D-ra f -c , is the AF RAF-compliant partition selector defined as follows:

∀RAF = ⟨A, K, s,t⟩ ∈ Φ ra f , S D-ra f -c (AF ) =          Ω ′ Let Ω ′ be any partition of AF and (∀ω ′ i ∈ Ω ′ , ∀x ∈ (A ∪ K), x ∈ ω ′ i =⇒ ¬x ∈ ω ′ i ) and (∀ω ′ i ∈ Ω ′ , ∀α ∈ K, α ∈ ω ′ i =⇒ s(α).α ∈ ω ′ i )          , with AF = Raf2Af(RAF )
The following definition establishes the relation between RAF partition selectors and AF RAF-compliant partition selectors: Definition 103. (AF and RAF partition selector correspondance) Let S be a RAF partition selector and let S ra f -c be an AF RAF-compliant partition selector. We say that S is the RAF counterpart of S ra f -c (and vice-versa) iff :

∀RAF ∈ Φ ra f , S (RAF ) = {{ω ′ ∩ (A ∪ K)|ω ′ ∈ Ω ′ }|Ω ′ ∈ S ra f -c (Raf2Af(RAF ))}
The idea behind the previous definition is that, when we consider the partition of the flattened version of a RAF, we want that any element in And A,K belongs to the same part as the attack it is related to, any element in Not A (resp. Not K ) belongs to the same part as the argument (resp. the attack) it is related to.

Although it could be defined otherwise, we choose this definition because:

1. The acceptance of an element x ∈ (A ∪ K) is intrinsically related to the acceptance of the argument ¬x:

• x is labelled in iff ¬x is labelled out • x is labelled out iff ¬x is labelled in • x is labelled und iff ¬x is labelled und 2.
A RAF argument may be the source of several attacks, whereas an attack has only one target. It seems thus reasonable to put arguments that belong to And A,K in the same part as the argument corresponding to their attack.

The following example illustrates Definition 103.

Example 74. Let consider the frameworks in Figure 14.14 on the following page. Let S ra f -c be an AF "RAF-compliant" partition selector and S be its RAF counterpart.

Let consider Ω ∈ S such that: Definition 104. (Semantics decomposability). A semantics σ is fully decomposable (or simply decomposable) if and only if there is a local function F ra f σ such that for every RAF RAF = ⟨A, K, s,t⟩ and every partition Ω = {ω 1 , ..., ω n } of (A ∪ K) and { RAF 1 , ..., RAF n } the partition of RAF corresponding to Ω, the following property holds:

Ω = {{d},
L σ -ra f (RAF ) = {L 1 ∪ ... ∪ L n |∀i ∈ {1, ..., n}, L i ∈ F ra f σ ( RAF i , I i , L inp i )}
With RAF i = Ãi , Ki , si , ti , s,t and

I i = S inp i , Q inp i
and L inp i defined as following:

• S inp i = {s(α) / ∈ Ãi |∃α ∈ K s.t. t(α) ∈ ( Ãi ∪ Ki )} • Q inp i = {α / ∈ Ki |∃α ∈ K s.t. t(α) ∈ ( Ãi ∪ Ki )} • L inp i = ( j∈{1,...,n} s.t. j̸ =i L j ) ↓ S inp i ,Q inp i
A semantics σ is said to be top-down decomposable if and only if the following property holds:

L σ -ra f (RAF ) ⊆ {L 1 ∪ ... ∪ L n |∀i ∈ {1, ..., n}, L i ∈ F ra f σ ( RAF i , I i , L inp i )}
A semantics σ is said to be bottom-up decomposable if and only if the following property holds:

L σ -ra f (RAF ) ⊇ {L 1 ∪ ... ∪ L n |∀i ∈ {1, ..., n}, L i ∈ F ra f σ ( RAF i , I i , L inp i )}

Illustration

In this section we illustrate the decomposability property of the complete semantics, and show the link between this property in AF and in RAF. For this purpose, let consider Figure 14.15 on the following page.

Let RAF = ⟨A, K, s,t⟩ be the RAF and AF = Raf2Af(RAF ) (with AF = ⟨A ′ , K ′ ⟩) be the flattened version of RAF , as represented in Figure 14.15 on the following page. Let Ω ′ = {{d, ¬d, δ , ¬δ , d.δ , a, ¬a}, {α, ¬α, a.α, β , ¬β , b, ¬b, b.β , c, ¬c, γ, ¬γ, c.γ}} be a partition of A ′ . We have: Ω ′ ∈ S D-ra f -c (AF ) (See Definition 102 on page 140). Let Ω = Ω ′ ∩ (A ∪ K) be the partition of A ∪ K corresponding to Ω ′ . Let us split RAF along the partition Ω and AF along the partition Ω ′ . The partial RAFs RAF 1 and RAF 2 produced by the split of RAF are represented in Figure 14.16 on page 144 and the clusters κ 1 and κ 2 produced by the split of AF are represented in Figure 14.17 on page 144.

We have: Given the input elements of the partial RAFs of RAF , let define the possible labellings associated with them. For RAF 1 , we have:

• RAF 1 = Ã1 , K1 , s1
• L inp 1.1 = ⟨{(c, in )}, {(γ, in )}⟩ • L inp 1.2 = ⟨{(c, in )}, {(γ, out )}⟩ • L inp 1.3 = ⟨{(c, in )}, {(γ, und )}⟩ • L inp 1.4 = ⟨{(c, out )}, {(γ, in )}⟩ • L inp 1.5 = ⟨{(c, out )}, {(γ, out )}⟩ • L inp 1.6 = ⟨{(c, out )}, {(γ, und )}⟩ a c δ γ d ζ ρ υ θ (a) RAF s 1.1 corresponding to RAF 1 , I 1 , L inp
• µ 1.1 = {(c.γ, in )} • L inp 1.1 = ⟨{(c, in )}, {(γ, in )}⟩ • L inp 1.7 = ⟨{(c, und )}, {(γ, in )}⟩ • L inp 1.8 = ⟨{(c, und )}, {(γ, out )}⟩ • L inp 1.9 = ⟨{(c, und )}, {(γ, und )}⟩
For RAF 2 , we have:

• L inp 2.1 = ⟨{(a, in )}, ∅⟩ • L inp 2.2 = ⟨{(a, out )}, ∅⟩ • L inp 2.
3 = ⟨{(a, und )}, ∅⟩ Likewise given the border arguments of the clusters of AF , let define the possible contexts associated with them. For κ 1 , we have:

• µ 1.1 = {(c.γ, in )} • µ 1.2 = {(c.γ, out )} • µ 1.3 = {(c.γ, und )}
For κ 2 , we have: With: • •

• µ 2.1 = {(¬a, out )} • µ 2.2 = {(¬a, in )} • µ 2.3 = {(¬a, und )}
• µ 1.2 = {(c.γ, out )} • L inp 1.2 = ⟨{(c, in )}, {(γ, out )}⟩ • L inp 1.4 = ⟨{(c, out )}, {(γ, in )}⟩ • L inp 1.5 = ⟨{(c, out )}, {(γ, out )}⟩ • L inp 1.6 = ⟨{(c, out )}, {(γ, und )}⟩ • L inp 1.8 = ⟨{(c, und )}, {(γ, out )}⟩ a c δ γ d ζ ρ υ θ α γ (a) RAF s 1.3 corresponding to RAF 1 , I 1 , L inp 1.3 a c δ γ d ζ ρ υ θ α c (b) RAF s 1.7 corresponding to RAF 1 , I 1 , L inp
• µ 1.3 = {(c.γ, und )} • L inp 1.3 = ⟨{(c, in )}, {(γ, und )}⟩ • L inp 1.7 = ⟨{(c, und )}, {(γ, in )}⟩ • L inp 1.9 = ⟨{(c, und )}, {(γ, und )}⟩ a β α c b ζ γ υ θ ρ (a) RAF s 2.1 corresponding to RAF 2 , I 2 , L inp
L inp 2.1 = ⟨{(a, in )}, ∅⟩ • µ 2.1 = {(¬a, out )} • L co-ra f ( RAF s 1.1 ) =    { (a, in ) , (d, in ) , (c, in ) , (ρ, in ), (υ, und ), (ζ , in )}, { (δ , out ) , (γ, in ) , (θ , in )}    • L µ 1.1 (κ 1 ) co = { {(d,
{(δ , in ), (γ, out ), (θ , in ), (α γ , in )}    • L co-ra f ( RAF s 1.4 ) =    {(a, out ), (d, in ), (c, out ), (ρ, in ), (υ, und ), (ζ , in )}, {(δ , in ), (γ, in ), (θ , in ), (α c , in )}    • L co-ra f ( RAF s 1.5 ) =    {(a, out ), (d, in ), (c, out ), (ρ, in ), (υ, und ), (ζ , in )}, {(δ , in ), (γ, out ), (θ , in ), (α γ , in ), (α c , in )}    • L co-ra f ( RAF s 1.6 ) =    {(a, out ), (d, in ), (c, out ), (ρ, in ), (υ, und ), (ζ , in )}, {(δ , in ), (γ, und ), (θ , in ), (α γ , in ), (α c , in )}    • L co-ra f ( RAF s 1.8 ) =    {(a, out ), (d, in ), (c, und ), (ρ, in ), (υ, und ), (ζ , in )}, {(δ , in ), (γ, out ), (θ , in ), (α γ , in ), (α c , in )}    a β α c b ζ γ υ θ ρ α a (a) RAF s 2.2 corresponding to RAF 2 , I 2 , L inp
L inp 2.2 = ⟨{(a, out )}, ∅⟩ • µ 2.2 = {(¬a, in )} • L µ 1.2 (κ 1 ) co = {{(d,
• L co-ra f ( RAF s 2.1 ) =    {(a, in ), (b, in ), (c, in ), (ρ, in ), (υ, und ), (ζ , out )}, {(α, in ), (β , out ), (γ, in ), (θ , in )}    • L µ 2.1 (κ 2 ) co =       (α,
• L co-ra f ( RAF s 2.3 ) =    {(a, und ), (b, in ), (c, und ), (ρ, in ), (υ, und ), (ζ , und )}, {(α, in ), (β , und ), (γ, in ), (θ , in ), (α a , in )}    • L µ 2.3 (κ 2 ) co =       (α,
• F ra f σ ( RAF s 1.1 , I 1 , L inp 1.1 ) = ℓ ↓ Ã1 , ℓ ↓ K1 |ℓ ∈ L µ 1.1 (κ 1 ) co = {⟨{(a, in ), (d, in )}, {(δ , out )}⟩} • F ra f σ ( RAF s 1.2 , I 1 , L inp 1.2 ) = F ra f σ ( RAF s 1.4 , I 1 , L inp 1.4 ) = F ra f σ ( RAF s 1.5 , I 1 , L inp 1.5 ) = F ra f σ ( RAF s 1.6 , I 1 , L inp 1.6 ) = F ra f σ ( RAF s 1.8 , I 1 , L inp 1.8 ) = ℓ ↓ Ã1 , ℓ ↓ K1 |ℓ ∈ L µ 1.2 (κ 1 ) co = {⟨{(a, out ), (d, in )}, {(δ , in )}⟩} • F ra f σ ( RAF s 1.3 , I 1 , L inp 1.3 ) = F ra f σ ( RAF s 1.7 , I 1 , L inp 1.7 ) = F ra f σ ( RAF s 1.9 , I 1 , L inp 1.9 ) = ℓ ↓ Ã1 , ℓ ↓ K1 |ℓ ∈ L µ 1.3 (κ 1 ) co = {⟨{(a, und ), (d, in )}, {(δ , und )}⟩} • F ra f σ ( RAF s 2.1 , I 2 , L inp 2.1 ) = ℓ ↓ Ã2 , ℓ ↓ K2 |ℓ ∈ L µ 2.1 (κ 2 ) co = {⟨{(b, in ), (c, in )}, {(α, in ), (β , out ), (γ, in )}⟩} • F ra f σ ( RAF s 2.2 , I 2 , L inp 2.2 ) = ℓ ↓ Ã2 , ℓ ↓ K2 |ℓ ∈ L µ 2.2 (κ 2 ) co = {⟨{(b, in ), (c, out )}, {(α, in ), (β , in ), (γ, in )}⟩} • F ra f σ ( RAF s 2.3 , I 2 , L inp 2.3 ) = ℓ ↓ Ã2 , ℓ ↓ K2 |ℓ ∈ L µ 2.3 (κ 2 ) co = {⟨{(b, in ), (c, und )}, {(α, in ), (β , und ), (γ, in )}⟩}
Finally, we can reunify compatible structure labellings of the partial RAFs of RAF , ensuring that any produced labelling is valid:

L co-ra f (RAF ) =          ⟨{(a, in ), (d, in ), (b, in ), (c, in )}, {(δ , out ), (α, in ), (β , out ), (γ, in )}⟩ , ⟨{(a, out ), (d, in ), (b, in ), (c, out )}, {(δ , in ), (α, in ), (β , in ), (γ, in )}⟩ , ⟨{(a, und ), (d, in ), (b, in ), (c, und )}, {(δ , und ), (α, in ), (β , und ), (γ, in )}⟩         
These structure labellings coincide with the valid labellings produced by the reunification of the cluster labellings of AF :

L co (AF ) =                                                      
(a, in ), (¬a, out ), (d, in ), (¬d, out ), (b, in ), (¬b, out ), (c, in ), (¬c, out ), (δ , out ), (¬δ , in ), (α, in ), (¬α, out ), (β , out ), (¬β , in ), (γ, in ), (¬γ, out ), (d.δ , out ), (a.α, in ), (b.β , out ), (c.γ, in )

         ,         
(a, out ), (¬a, in ), (d, in ), (¬d, out ), (b, in ), (¬b, out ), (c, out ), (¬c, in ), (δ , in ), (¬δ , out ), (α, in ), (¬α, out ), (β , in ), (¬β , out ), (γ, in ), (¬γ, out ), (d.δ , in ), (a.α, out ), (b.β , in ), (c.γ, out )

         ,         
(a, und ), (¬a, und ), (d, in ), (¬d, out ), (b, in ), (¬b, out ), (c, und ), (¬c, und ), (δ , und ), (¬δ , und ), (α, in ), (¬α, out ), (β , und ), (¬β , und ), (γ, in ), (¬γ, out ), (d.δ , und ), (a.α, und ), (b.β , und ), (c.γ, und )

                                                     

Properties

In this section, we prove the decomposability properties of RAF semantics from those of AF semantics. The first steps of the demonstration consist in highlighting labellings correspondence between RAF and flattened RAF and w.r.t. some partitioning. Figure 14.24 gives an overview of those steps, leading to Proposition 41 on page 159. From this property, the second steps of the demonstration consist in establishing equivalences between RAF and AF semantics decomposability properties (See Propositions 42, 43 and 45 on pages 160-161). Figure 14.25 on the next page gives an overview of those steps, leading ultimately to Proposition 47 on page 162.

A RAF: RAF = ⟨A, K, s,t⟩ Notice that this equivalence is not sufficient to derive RAF semantics decomposability properties from AF ones. We need the propositions below to do so.

A RAF with input corresponding to ω ∈ Ω: RAF , I, L inp A standard RAF: RAF s = Ãs , Ks , s s ,t s RAF s flattened: RAF ′ s = Ã′ s , K′ s An AF: AF = ⟨A ′ , K ′ ⟩ An AF with input corresponding to ω ′ ∈ Ω ′ such that ω ′ corre- sponds to ω: AF ↓ ω ′ , J , ℓ J , K J A standard AF: std-AF = ⟨A ′ s , K ′ s ⟩ (
In order to establish a point of comparison between RAF semantics decomposability and AF semantics decomposability we introduce with the following definition the correspondence between an AF with input and a RAF with input:

Definition 105 (AF with input corresponding to RAF with input). Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be the corresponding AF of RAF (with AF = ⟨A ′ , K ′ ⟩). Let Ω be a partition of (A ∪ K) and Ω ′ ∈ S D-ra f -c (AF ) be the RAF-compliant partition of A ′ corresponding to Ω, i.e. Ω ′ = {ω ′ = ω ∪ {¬x|x ∈ ω} ∪ {s(α).α ∈ And A,K |α ∈ ω}|ω ∈ Ω}. Let ω ∈ Ω and ω ′ ∈ Ω ′ be its counterpart in AF . Let RAF = Ã, K, s, t, s,t be the partial RAF corresponding to ω. Let I = S inp , Q inp be the input elements of RAF and L inp be a structure labelling of them. Let RAF , I, L inp be a RAF with input. We define AF ↓ ω ′ , J , ℓ J , K J as the AF with input corresponding to RAF , I, L inp with:

• J = {s(α).α|α ∈ Q inp } ∪ {¬a|a ∈ S inp s.t. ∃α ∈ K and s(α) = false and s(α) = a} • ∀(s(α).α) ∈ J s.t. α ∈ Q inp , ℓ J (s(α).α) is defined as following: ℓ J (s(α).α) =          in ⇐⇒ L inp (α) = in and L inp (s(α)) = in out ⇐⇒ L inp (α) = out or L inp (s(α)) = out und ⇐⇒ L inp (α) ̸ = out and L inp (s(α)) ̸ = out and (L inp (α) = und or L inp (s(α)) = und)
• ∀¬a ∈ J s.t. a ∈ S inp , ℓ J (¬a) is defined as following: Let choose the left part of the RAF/AF for the illustration. We select thus: ω ′ = {δ , ¬δ , d.δ , a, ¬a, α, ¬α, a.α, b, ¬b}. We have so: ω = {δ , a, α, b}

ℓ J (¬a) =      in ⇐⇒ L inp (a) = out out ⇐⇒ L inp (a) = in und ⇐⇒ L inp (a) = und • K J = {(s(α).α,t(α))|α ∈ Q inp } ∪ {(
Let RAF 1 = Ã1 , K1 , s1 , t1 , s,t be a partial RAF of RAF corresponding to ω, with: Following Definition 105, AF ↓ ω ′ , J , ℓ J , K J is the AF with input corresponding to RAF 1 , I 1 , L inp

• A 1 = {a, b} and K 1 = {α, δ } • s1 (δ ) =
• ℓ J = {(c.γ, out), (¬d, und)} • K J = {(c.γ, α), (¬d, d.δ )}
Let std-AF be the standard AF corresponding to AF ↓ ω ′ , J , ℓ J , K J . Following the definition of standard AF (See Definition 17 on page 17) we have: std-AF = A ′ ∪ J ′ , K ′ ∪ K ′ J , where: An important property is the fact that, given a RAF with input RAF , I, L inp and its corresponding AF with input AF ↓ ω ′ , J , ℓ J , K J , the flattening of the standard RAF associated with RAF , I, L inp and the standard AF std-AF corresponding to AF ↓ ω ′ , J , ℓ J , K J will have a common sub-AF, attacked by same attackers (See Figure 14.29 on the next page).

• J ′ = J ∪ a ′ |a ∈ J ∩ out(ℓ J ) • K ′ J = K J ∪ (a ′ , a)|a ∈ J ∩ out(ℓ J ) ∪ (a, a)|a ∈ J ∩ und(ℓ J ) a b α c γ d δ Input elements
Example 76. Following Example 75, let consider RAF ′ s , the RAF illustrated in Figure 14.29 on the next page. There is a common sub-RAF in RAF ′ s and in std-AF (See Figure 14.28(b)). Indeed, we have: ω ′ = {δ , ¬δ , d.δ , a, ¬a, α, ¬α, a.α, b, ¬b}, J = {¬d, c.γ} and we can verify that:

• RAF ′ s ↓ ω ′ = std-AF ↓ ω ′ (
all additional arguments are not concerned) • J ⊆ K′ s and J ⊆ K ′ s (the input arguments are present in both AFs. Here: c.γ and ¬d)

• (J × ω ′ ) ∩ K′ s = (J × ω ′ ) ∩ K ′
s (the attacks from the input arguments are the same in both AFs)

Moreover, arguments attacking this sub-AF are subject to same labellings, given a "complete-based" semantics.

The following example illustrates this fact:

Example 77. Considering Examples 75 and 76, we have: As a consequence, restricted to the area of interest, both the standard AF of AF ↓ ω ′ , J , ℓ J , K J and the flattening of the standard RAF associated with RAF , I, L inp will produce the same labellings as stated by Proposition 40.

∀ℓ ∈ L σ ( RAF ′ s ), ∀ℓ ′ ∈ L σ (std-AF ), ℓ ′ (c.γ) = ℓ(c.γ) =
Proposition 40. Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be the corresponding AF of RAF (with AF = ⟨A ′ , K ′ ⟩). Let RAF , I, L inp be a RAF with input of RAF and AF ↓ ω ′ , J , ℓ J , K J be its corresponding AF with input, as defined in Definition 105 on page 155. Let RAF s = Ãs , Ks , s s ,t s be the standard RAF of RAF , I, L inp and let std-AF = ⟨A ′ s , K ′ s ⟩ be the standard AF corresponding to AF ↓ ω ′ , J , ℓ J , K J . Let RAF ′ s = Raf2Af( RAF s ) be the AF corresponding to the flattening of RAF s (with

RAF ′ s = Ã′ s , K′ s ).
Let σ be an AF complete-based semantics. The following assertion holds:

{ℓ ↓ ω ′ ∪J |ℓ ∈ L σ ( RAF ′ s )} = {ℓ ↓ ω ′ ∪J |ℓ ∈ L σ (std-AF )}
□ Proof of Proposition 40: link (See page 279).

Following Proposition 40, we can now establish the relation between the labellings of a partial RAF with input and the labellings of its corresponding AF with input for a given semantics, as made in Definition 106 and Proposition 41 on the following page.

Definition 106 (Semantics correspondence). Let σ be an AF semantics. Following the definitions of afLab2RafLab and rafLab2AfLab (See Definition 78 on page 118), we say that σ -raf is the RAF semantics corresponding to σ if and only if:

∀RAF ∈ Φ ra f , L σ -ra f (RAF ) = {afLab2RafLab(ℓ)|ℓ ∈ L σ (Raf2Af(RAF ))}
Or equivalently, if and only if:

∀RAF ∈ Φ ra f , L σ (Raf2Af(RAF )) = {rafLab2AfLab(L)|L ∈ L σ -ra f (RAF )}
Proposition 41. Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be the corresponding AF of RAF (with AF = ⟨A ′ , K ′ ⟩). Let RAF , I, L inp be a RAF with input of RAF and AF ↓ ω ′ , J , ℓ J , K J be its corresponding AF with input, as defined in Definition 105 on page 155. Let σ be an AF complete-based semantics and let σ -raf be its corresponding counterpart for RAF. The following property holds:

{L ↓ ⟨ Ã, K⟩ |L ∈ L σ -ra f ( RAF s )} = ℓ ↓ A , ℓ ↓ K ℓ ∈ F a f σ (AF ↓ ω ′ , J , ℓ J , K J )
Or equivalently that:

F ra f σ ( RAF , I, L inp ) = ℓ ↓ A , ℓ ↓ K ℓ ∈ F a f σ (AF ↓ ω ′ , J , ℓ J , K J )
□ Proof of Proposition 41: link (See page 280).

In Definition 104 on page 142 has been defined the notion of decomposibity for RAF. Definition 107 refines this notion by making it relative to a given partition selector :

Definition 107 (Top-down, bottom-up and fully decomposability w.r.t. a RAF partition selector S ). Let S be a RAF partition selector. A RAF semantics σ is top-down decomposable w.r.t. S iff for any RAF RAF and any partition Ω = {ω 1 , ..., ω n } ∈ S (RAF ), it holds that:

L σ -ra f (RAF ) ⊆ L 1 ∪ ... ∪ L n L i ∈ F ra f σ ( RAF i , I i , L inp i ) With: RAF i built from ω i and L inp i = ( j∈{1,...,n} s.t. j̸ =i L j ) ↓ I i
A RAF semantics σ is bottom-up decomposable w.r.t. S iff for any RAF and any partition Ω = {ω 1 , ..., ω n } ∈ S (RAF ), it holds that: Note: For a RAF semantics σ -raf, to be top-down (resp. bottom-up, fully) decomposable is equivalent to be top-down (resp. bottom-up, fully) decomposable w.r.t. the partition selector that produces all possible partition of a RAF.

L σ -ra f (RAF ) ⊇ L 1 ∪ ... ∪ L n L i ∈ F ra f σ ( RAF i , I i , L inp i ) With: RAF i built

Let formally defined this partition selector:

Definition 108. S D-ra f is the RAF partition selector defined as follows:

∀RAF ∈ Φ ra f , S D-ra f (RAF ) is the set of all possible partition of RAF Now, in order to take advantage of the decomposability properties already proven for AF semantics, we have to show that the decomposability of RAF semantics w.r.t. a given partition selector S is equivalent to the decomposability property of the corresponding AF semantics w.r.t. a "RAF-compliant" version of S . Following the relation stated in Proposition 41 on the previous page, we can establish the following proposition: Proposition 42. Let σ be an AF complete-based semantics and let σ -raf be the RAF semantics corresponding to σ . Let S be a RAF partition selector and let S ra f -c be the AF RAF-compliant partition selector corresponding to S . The following properties holds: Proposition 42 is not sufficient to derive RAF semantics decomposability properties from AF ones. Indeed the equivalences are established between a RAF selector (whatever it is) and its AF RAF-compliant version. To access all the wanted properties, we have to show that, for AF semantics, having a certain decomposability property w.r.t. the default AF RAF-compliant partition selector S D-ra f -c is equivalent to having the same property w.r.t. the default AF partition selector S D-a f (See Definition 25 on page 20), 4 and having a certain decomposability property w.r.t. the USCC a f RAF-compliant partition selector S ra f -c-USCC (See Definition 109 on the following page) is equivalent to having the same property w.r.t. the USCC a f partition selector S USCC .

Proposition 43 establishes, for a given AF semanctics σ (so this property concerns AF), the equivalence between the decomposability properties of σ for any partition and the decomposability properties of σ w.r.t. S D-ra f -c : Proposition 43. Let σ be an AF complete-based semantics. The following properties hold:

1. σ is top-down decomposable (equivalently w.r.t. S D-a f ) iff σ is top-down decomposable w.r.t. S D-ra f -c . " " means that the semantics on the column has the property on the row. "× × ×" means that the semantics on the column does not have the property on the row. Some AF semantics, as the preferred semantics, are not fully decomposable except w.r.t. the specific selector S USCC . To study the decomposability of such semantics for RAF, we have to define a selector equivalent to S USCC for RAF and then its "RAF-compliant" version.

Definition 109. (USCC RAF partition selector and AF correspondence) Let RAF be a RAF. Let S ra f -USCC be the RAF partition selector such that :

S ra f -USCC (RAF ) = {Ω| Ω is a partition of RAF and ∀S ∈ SCCS ra f (RAF ), ∃ω i ∈ Ω s.t. ω i ∩ S ̸ = ∅ =⇒ S ⊆ ω i }
Given a RAF RAF = ⟨A, K, s,t⟩, we call "USCC ra f " a subset S ⊆ A ∪ K such that S ∈ S ra f -USCC (RAF ). We define S ra f -c-USCC as the RAF-compliant AF partition selector corresponding to S ra f -USCC .

Those selectors defined, we can establish a property similar to Proposition 43 on the previous page, but w.r.t. S USCC and S ra f -c-USCC .

Proposition 45. Let σ be an AF complete-based semantics. The following properties hold: " " means that the semantics on the column has the property on the row. "× × ×" means that the semantics on the column does not have the property on the row.

Finally, the way is now clear to access all AF decomposability properties and know the decomposability properties of RAF semantics, w.r.t. S ra f -USCC or not, as stated by Proposition 47 (result concerning RAF).

Proposition 47. Let RAF = ⟨A, K, s,t⟩ be any RAF. The semantics properties in Table 14 " " means that the semantics on the column has the property on the row. "× × ×" means that the semantics on the column does not have the property on the row.

Chapter 15

Related Work

There are very few work related to the contributions made in this part:

• No labelling has previously been proposed for RAF.

• Before the one proposed in Chapter 12 on page 114, a flattening method to transform RAF into AF had already been introduced in [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF]. It is discussed in Sections 9.2.2 and 12.1 on page 98 and on page 114.

• In [START_REF] Cayrol | Valid attacks in Argumentation Frameworks with Recursive Attacks (long version)[END_REF], results about complexities in RAF are given on the RAF-Cred σ problem (for the complete, preferred and stable semantics) and on the RAF-Skep σ problem (for the preferred and stable semantics), but not for all RAF decision problems and semantics. Furthermore, proofs are not formally given.

• No work attempts to extend the notion of Strongly Connected Component to RAF and the decomposability of RAF semantics has not been studied before our contribution.

This thesis has been for me a very rewarding experience, of course, in terms of research methodology but also in terms of human experience. From time to time, I had to cross long deserts, littered with failures and without any potential discoveries in sight. I learned that perseverance is the key for the success. I overcame all these hard times and, today, I am glad to conclude this thesis and present below all the contributions I have made during those three years of PhD. Afterward, I present the perspectives opened by my works.

Conclusion of the first milestone

In the first contribution part of my thesis (Part III on page 27), I address the issue of the enumeration problem solving time issue of Dung's AF semantics. I proposed a generic algorithm, so-called AFDivider, that computes labellings of the complete, stable and preferred semantics in a distributed and clustering-based fashion. This algorithm is generic in the sense that it can be used with any clustering method to split the AF into clusters structures and can be associated to any sound and complete procedure that computes the labellings of the different clusters structure for the wanted semantics. It has been proven to be complete and sound for all three mentioned semantics. That is, the algorithm produces all the expected labellings of the wanted semantics and each produced labelling is indeed a correct one.

Three clustering methods have been proposed, which lead to three AFDivider solver versions. The first one, so-called AFDiv-spectral, uses a spectral clustering method to split the AF. To the best of my knowledge, it is the first algorithm that uses this methods to compute semantic labellings. The second and the third ones, so-called AFDiv-USCC-Tree and AFDiv-USCC-Chain, split the AF following a USCC partition of it (that is, a partition that does not cut the AF SCC). They differ in the way the partition is selected.

An experimental analysis of those solvers has been conducted to identify their performances, compared to other solvers, according to AF types, over success rate and resolution time. It comes out from this study that the AFDiv-spectral variant is faster than most solvers in average on common successful instances while the AFDiv-USCC-Tree and AFDiv-USCC-Chain variants succeed to solve most instances when considering Barabási-Albert (BA), Erdős-Rényi (ER) and Watts-Strogatz (WS) AF types, for the complete, stable and preferred semantics.

Based on the AF types that they solve efficiently, I identify two behaviour classes among our three solvers: one for the spectral clustering and one for the USCC based clusterings. This shows that the clustering method which is used has an important effect on the performances of the AFDivider algorithm on a particular AF type. This is an important observation that must be taken into consideration for the development of new AF solvers.

The main advantage of the AFDivider algorithm I want to highlight is the fact that cutting the AF into clusters has the great advantage of limiting the solving hardness to the clusters. Indeed, in those other approaches, the combinatorial effect due to the number of labellings is propagated to the whole AF whereas, in the AFDivider algorithm, it is limited to the clusters. This property makes it well suited for non dense AF with a clustered structure.

The Compact Enumeration Representation I proposed takes full advantage of that fact. While it has been proven sufficient to answer all classical argumentation problems, experiments show that the Compact Enumeration Representation allows to solve more hard instances of some AF types such as BA, TR (Traffic) and BW (Block World). Furthermore, it produces an output much more faster than when enumerating all the labellings, especially for BA type AF (97.03 times much faster). The impact of the Compact Enumeration Representation is particularly significant for the complete semantics due to the huge number of labellings that causes memory overflow during the enumeration.

The AFDivider algorithm, with or without Compact Enumeration Representation, opens thus a new way to approach argumentation problems and many perspectives.

Perspectives of the first milestone

Here are some ideas to go further with the AFDivider algorithm:

• A recursive clustering version of this algorithm could be made. Indeed, after the cutting process, an induced AF could still be hard to solve. It may be possible that applying recursively the same clustering process on AF parts (until a certain criterion is satisfied) could enhance the global solving time.

• Other clustering methods than those tested could be more appropriate for some AF types. This could be studied in future works.

• The Compact Enumeration Representation could be exploited to answer non classical problems such as: "What is the labelling rate in which an argument a is labelled in ?", "Is there an argument labelled in in more that 70% of the labellings ?", "Is the set of arguments S is accepted (together) in more than 60% of the labellings" and other questions of that type, and that, without explicitly enumerating all the labellings, avoiding the costly cartesian product of component labellings.

• It would be interesting to extend this work to further generalizations of AFs, using several types of relation (not only attacks but also supports), with relation and argument strength, recursive relations, and so for more complex semantics.

• Recently, a new category of AF problems appeared in ICCMA competition: Dynamic AF semantics computation. Given an AF, it consists of computing some semantics of the initial AF in a first place, then of its altered version (arguments/attacks are added/removed from the initial AF). Dynamic algorithms use the previous computed result to compute the current altered AF semantics. It would be interesting to explore the possibilities for a dynamic version of AFDivider.

• Explainability is becoming essential for reliable IA information systems. In the domain of Abstract Argumentation, having an explanation for the acceptance or the reject of some argument, more concise and informative than a whole extension/labelling, has an important interest. A work could be made to study how the AF decomposition made by our algorithm (given some clustering method) could help to explain such things.

I would like to emphasize two major improvements of the AFDivider algorithm:

• The first would be to "know" in advance which clustering method (including ones other than those presented in this paper) should be used for a particular AF instance. Experiments could be conducted to learn, for example with a neural network, which one to use.

• The second, which is the most interesting, is to go further and have even the cutting process being supervised by a neural network trained to cluster AF instances following their structure. As a consequence, for any known AF type, the most appropriate clustering method would be used to solve each AF instance efficiently. The clustering might be in a certain sense dynamic, as the rules applied to cut may be different from one AF area to another.

Conclusion of the second milestone

In the second contribution part of my thesis (Part V on page 101), I created a certain number of tools for the study of RAF structure and RAF semantics complexity and properties, with the objective of proposing a first algorithm to compute RAF semantics. In a first step, I define new semantics for RAF. The semi-stable semantics have been defined for RAF and it has been shown that RAF are still a conservative generalization of Dung's AF, even for the semistable semantics. The notion of labellings has been extended to RAF. Given a RAF, a structure labelling or RAF labelling is a tuple in which the first element is a labelling over its arguments and the second one is a labelling over its attacks. As for AF labellings, these RAF labellings are three-value based, indicating the degree of acceptance of a RAF element (i.e. accepted: in , rejected: out or undecided: und ).

In a second step, RAF semantics have been redefined in terms of RAF labelling semantics. A oneto-one mapping between structures and RAF labellings have been identified using two linking functions:

Struct2Lab, that transforms a structure into a RAF labelling and Lab2Struct, that transforms a RAF labelling into a structure. Reinstatement RAF labellings have been formally defined as coherent RAF labellings. It is shown (see Chapter 11 on page 107) that the complete, grounded, preferred, semi-stable and stable structure semantics correspond to precise types of reinstatement RAF labellings. Table 11.1 on page 111 gives the correspondence between structure semantics and reinstatement RAF labellings. Moreover, this work confirms that RAF are a conservative generalization of Dung's AF. Indeed in RAF with no recursive attacks, there is a one-to-one mapping between reinstatement labellings (AF notion) and two RAF notions (structures and reinstatement RAF labellings) for the complete, grounded, preferred, semi-stable and stable semantics. This additional precision on the acceptability status of a RAF element, from the information a simple structure can give (that is, a binary acceptability status: given a structure, whether an element belongs to it or not), opens a whole new field of research for RAF solving algorithms.

In a third step, I defined a new flattening process that transforms a RAF into an AF and this, while preserving the meaning of the RAF defeat relation, ensuring thus interesting properties such as shape or acceptability related properties that will be used in the following steps. This transformation is not only polynomial in time but also logarithmic in space. Although it has not been used in this way in this thesis, it allows the use of AF solvers to solve RAF semantics problems.

In a fourth step, decision problems for RAF have been defined and their complexity studied. Using the flattening process, an important result has been found: the complexities of RAF decision problems are the same as the ones in Dung's framework, despite all the additional expressiveness that is brought by the higher-order attacks.

In a fifth step, the notion of Strong Connected Component (SCC) has been extended to RAF. A bijection has been shown between the SCC ra f of a given RAF and the SCC a f of its flattened version. Then the decomposability properties of RAF semantics have been studied. It has been shown that the decomposability properties of RAF semantics are equivalent to the ones of their corresponding AF semantics.

Perspectives of the second milestone

As perspectives for this milestone, here is a list of interesting ideas to explore:

• More AF semantics could be extended to RAF, whether structure-based or labelling-based semantics.

• The SCC-recursiveness [11] of RAF semantics could be studied, based on the flattening process proposed.

• Other notions of graph theory, such as tree-width, could be extended to RAF, allowing thus the study of other types of properties.

• RAF generator following different structure types as well as benchmarks for RAF semantics computation could be given, as no work has been done to address those questions so far.

• Beside decision problems, other problems are of interest for argumentation frameworks, whether they have higher-order attacks or not: function problems. 1 The functional counterpart of Cred σ and Exists ¬∅ σ may turn to be particularly useful in the context of a dialogue between agents, the output being here the concerned acceptable set. Defining such problems, and investigating their complexity, would be interesting.

• All those contributions pave the way for an algorithmic investigation of the computation of RAF semantics and RAF decision problem solving. A sound and complete AFDivider-like algorithm for RAF could be proposed.

Chapter 16

Mathematical Theories

In this chapter are presented firstly, basic notions of Set Theory (Section 16.1). Secondly, some notions of Graph Theory (Section 16.2). Thirdly, definitions are given about matrices (Section 16.3 on page 175). Fourthly, the computational complexity theory is presented in Section 16.4 on page 178. Finally, basic notions of algorithm analysis are presented in Section 16.5 on page 184. Notice that the scope of this background is limited to the necessary. For an overview of set theory see [START_REF] Foreman | Handbook Set Theory[END_REF], of graph theory see [START_REF] Gross | Handbook of Graph Theory, Second Edition[END_REF], of matrix computation see [START_REF] Golub | Matrix Computations[END_REF], of complexity theory see [START_REF] Stephen | An overview of computational complexity[END_REF] and of algorithm analysis see [START_REF] Van Leeuwen | Handbook of theoretical computer science[END_REF].

Set theory

The notions of set and set partition are used throughout this document.

Definition 110 (Set). A set is a collection of distinct elements.

Definition 111 (Partition of a set). A partition Ω = {ω 1 , ..., ω n } of a set O is a set of subsets of O such that:

• ∀i, j ∈ {1, ..., n} s.t. i ̸ = j, ω i ∩ ω j = ∅ • i∈{1,...,n} ω i = O

Graph theory

In this section different types of graph are presented, and also notions related to nodes, relations, paths, subgraphs and topology.

Graph types

Definition 112. (Non-directed and directed graph). A non-directed (respectively directed) graph is an ordered pair G = (V, E) where:

• V is a set whose elements are called nodes or vertices;

• E is a set of unordered (respectively ordered) pairs of vertices called non-directed edges (respectively directed edges).

Example 78. Note: E is a set. As a consequence, in this document non-directed (respectively directed) graphs have only distinct unordered (respectively ordered) pairs in E. Non-directed (respectively directed) multigraphs, non-directed (respectively directed) graphs in which is permitted to have multiple non-directed (respectively directed) edges that have the same endpoints,1 are not considered.

Definition 113. (Weighted graph). A weighted directed (respectively non-directed) graph is an ordered pair G = (V, E,W ) where:

• (V, E) is a directed (respectively non-directed) graph.

• W : E → R is a total function that associates a weight to each directed (respectively non-directed) edge in E.

Example 79. Figure 16.2 on the next page shows examples of weighted graphs.

In the following we will implicitly consider that a non-weighted directed (respectively non-directed) graph G = (V, E) is a weighted directed (respectively non-directed) graph whose edges are weighed 1.

Node and edge relations

Definition 114 (Incidence). Let G = (V, E) be a graph (directed or not), and e = (v i , v j ) ∈ E be an edge. We say that e is incident to v i and v j , or joins v i and v j . Similarly, v i and v j are incident to e.

Definition 115 (Adjacency). Let G = (V, E) be a graph (directed or not), and v i ∈ V , v j ∈ V be two nodes of G. We say that v i and v j are adjacent if Definition 117 (Degree in a directed graph). In directed graphs three degrees are associated to each vertex. Let G = (V, E) be a directed graph and v ∈ V be a vertex. We have:

(v i , v j ) ∈ E or (v j , v i ) ∈ E.
• The inward degree of v, denoted in-deg(G, v), is its number of incident edges such that v is the second element of the edge.

• The outward degree of v, denoted out-deg(G, v), is its number of incident edges such that v is the first element of the edge.

• The degree of v, denoted deg(G, v), is its number of incident edges. Definition 118. (Weighted degree). Let G = (V, E,W ) be a weighted graph (directed or not). Let v ∈ V be a vertex and I the set of its incident edges. We define the weighted degree of v, noted deg w (G, v), as the weight sum of its incident edges:

deg w (G, v) = ∑ e∈I W (

Connectivity

Definition 119 (Walk). Let G = (V, E) be a graph (directed or not). A walk is a sequence (v 1 , ..., v n ) such that:

• ∀i ∈ {1, ..., n}, v i ∈ V • ∀i ∈ {1, ..., n -1}, (v i , v i+1 ) ∈ E
Definition 120 (Non-directed walk). Let G = (V, E) be a directed graph. A non-directed walk is a sequence (v 1 , ..., v n ) such that:

• ∀i ∈ {1, ..., n}, v i ∈ V • ∀i ∈ {1, ..., n -1}, (v i , v i+1 ) ∈ E or (v i+1 , v i ) ∈ E
In this document, we will restrict the notion of graph paths to strict simple paths (i.e. they contains only distinct elements).

Definition 121 (Path). Let G = (V, E) be a graph (directed or not). A path is a walk which contains distinct vertices.

Definition 122 (Non-directed path). Let G = (V, E) be a directed graph. A non-directed path is a nondirected walk which contains distinct vertices.

Definition 123 (Cycle). Let G = (V, E) be a graph (directed or not). A cycle is a sequence (v 1 , ..., v n ) with n ≥ 2 such that:

• (v 2 , ..., v n ) is a path • (v 1 , ..., v n-1 ) is a path • v 1 = v n
Example 82. Let consider Figure 16.1(b) on page 171. (l, j, k, l, m) is a walk, ( j, k, l) is a path and ( j, k, l, j) is a cycle.

Definition 124 (Connected and disconnected graph). Let G = (V, E) be a directed (respectively non-directed) graph. G is a connected graph if, for all distinct vertices v i ∈ V and v j ∈ V , there exists a non-directed path p (respectively a path p) in G s.t. v i is the first vertex of p and v j is the last vertex of p. Otherwise the graph is called a disconnected graph.

Definition 125 (Subgraph). Let G = (V, E) be a directed graph (respectively non-directed graph). A subgraph S = (V ′ , E ′ ) of G is a directed graph (respectively non-directed graph) such that:

• V ′ ⊆ V . • E ′ ⊆ E. • ∀(v i , v j ) ∈ E ′ , v i ∈ V ′ and v j ∈ V ′ .
Definition 126 (Graph restriction ↓). Let G = (V, E) be a graph and S ⊆ V be a set of vertices. The restriction of G to S is the subgraph of G defined as G ↓ S ≡ (S, E ∩ (S × S)).

Note: Notice that when restricting a graph G to a set S of nodes, any edge of G whose endpoints are both in S must be kept. It is not the case for the more general subgraph definition. Indeed, a subgraph of G whose set of nodes coincides with S may not keep all these edges.

Example 83. Let consider the graph G = (V, E) in Figure 16.1(b) on page 171. We have: G ↓ {m,n} = ({m, n} , {(m, n), (n, m)}). Definition 127 (Connected component). Let G be a graph (directed or not). Let H be a subgraph of G such that:

• H is connected.

• H is not contained in any connected subgraph of G which has more vertices or edges than H has.

Then H is a connected component of G.

In the following by "component" we mean "connected component".

Definition 128 (Path-equivalence relation). Let G = (V, E) be a directed graph. The binary relation of path-equivalence between nodes, denoted as PE G ⊆ (V ×V ), is defined as follows:

• ∀v i ∈ V, (v i , v i ) ∈ PE G .
• given two distinct nodes v i , v j ∈ V, (v i , v j ) ∈ PE G if and only if there is a path from v i to v j and a path from v j to v i .

Definition 129 (SCC). The strongly connected components of a directed graph G are the equivalence classes of nodes under the relation of path-equivalence. Basically, an SCC is a (directed) subgraph in which there is a path between each pair of its vertices.

Example 84. Let consider the graph in Figure 16.1(b) on page 171. {m, n} and { j, k, l} are SCCs.

Topology

Several contributions made in this thesis have been inspired by ideas from graph topologies (that is their "shape") and especially about "clusters" of nodes in graphs. Formally, clusters can be defined as follows:

Definition 130 (Cluster). Let G be a graph. A cluster of G is a connected subgraph of G.

To express the gathering, the connectivity between the nodes we use the notion of relation density:

Definition 131 (Relation density). Let G = (V, E) be a graph, S be a subset of V . The relation density R d (G) of the graph G is defined by:

R d (G) = |E| |V | The relation density R d (G ↓ S ) of the subgraph G ↓ S is defined by: R d (G ↓ S ) = |E ∩ (S × S)| |S|
In practice, given an initial graph, we will be interested by some of its connected subgraphs which have similar sizes (number of nodes) and such that their inside relation density is greater than their neighbouring relation density. The adjacency matrix M a of G is:

M a =            j k l m n j 0 1 1 0 0 k 1 0 1 0 0 l 1 1 0 1 0 m 0 0 1 0 2 n 0 0 0 2 0           
The degree matrix M d of G is:

M d =            j k l m n j 2 0 0 0 0 k 0 2 0 0 0 l 0 0 3 0 0 m 0 0 0 3 0 n 0 0 0 0 2           
And then, its laplacian matrix M l is:

M d -M a = M l =            j k l m n j 2 -1 -1 0 0 k -1 2 -1 0 0 l -1 -1 3 -1 0 m 0 0 -1 3 -2 n 0 0 0 -2 2            16.3. MATRICES Let v ∈ R 5 be a vector, with: v =            x 1 x 2 x 3 x 4 x 5           
Let u be the linear transformation mapping R5 into R 5 (i.e. u : R 5 → R 5 ), whose coefficients correspond to the matrix M l , that is, the application whose definition corresponds to matrix multiplication between M l and v ∈ R 5 .

Note: In practice we do not search for the exact expression of u, but rather directly solve the system of equations corresponding to: M l × v = λ • v, with v being a non-zero vector.

Given that:

M l × v =            2 -1 -1 0 0 -1 2 -1 0 0 -1 -1 3 -1 0 0 0 -1 3 -2 0 0 0 -2 2            ×            x 1 x 2 x 3 x 4 x 5            =            2 × x 1 -1 × x 2 -1 × x 3 + 0 × x 4 + 0 × x 5 -1 × x 1 + 2 × x 2 -1 × x 3 + 0 × x 4 + 0 × x 5 -1 × x 1 -1 × x 2 + 3 × x 3 -1 × x 4 + 0 × x 5 0 × x 1 + 0 × x 2 -1 × x 3 + 3 × x 4 -2 × x 5 0 × x 1 + 0 × x 2 + 0 × x 3 -2 × x 4 + 2 × x 5           
We have so:

u(v) =            2 × x 1 -x 2 -x 3 , -x 1 + 2 × x 2 -x 3 , -x 1 -x 2 + 3 × x 3 -x 4 , -x 3 + 3 × x 4 -2 × x 5 , -2 × x 4 + 2 × x 5           
Given the expression of u(v) and given that v is an eigenvector of M l if and only if v is a non-zero vector 2 and that there exists a value λ ∈ R such that: u(v) = λ • v, we have thus to solve the following system of equations:

                           2 × x 1 -x 2 -x 3 = λ × x 1 -x 1 + 2 × x 2 -x 3 = λ × x 2 -x 1 -x 2 + 3 × x 3 -x 4 = λ × x 3 -x 3 + 3 × x 4 -2 × x 5 = λ × x 4 -2 × x 4 + 2 × x 5 = λ × x 5 |x 1 | + |x 2 | + |x 3 | + |x 4 | + |x 5 | ̸ = 0
There are five solutions (five eigenvectors v and their corresponding eigenvalues λ ):

• The eigenvectors of M l are:

           v 1 v 2 v 3 v 4 v 5 -0.
          
• And their associated eigenvalues are:

λ 1 λ 2 λ 3 λ 4 λ 5
2.476651 × 1 × 10 -16 5.857864 × 1 × 10 -1 3.000000 3.414214 5.000000

Computational complexity theory

In this section is given a succinct overview of computational complexity theory. For a more complete view on computational complexity theory see [START_REF] Stephen | An overview of computational complexity[END_REF].

Principles

Computational complexity theory is a field of computer science whose purpose is to cluster computational problems into "complexity classes". Problems are gathered according to some criterion on the resources required to solve them. Generally the measure used to differentiate them is the time (i.e. the number of steps taken by an algorithm) needed or the space (i.e. the amount of memory) needed to solve them, but a clustering could be based on any other resource criterion. In this report, we will consider only time complexity classes. We say that a problem P belongs to the complexity class C (or P has complexity C ) if there exists an algorithm that solves P satisfying the resource requirements of C . Basically, the more a problem requires resources the more it will be considered has difficult.

In order to rank them in a fair way and so, form coherent complexity classes, problems are considered in their generic form. This means that the comparison is not made on specific "problem instances" (i.e. the problem applied to concrete data inputs). The resource requirements are expressed according to the problem "input size".

Nevertheless, considering problem's generic form is not sufficient for a proper comparison. Indeed, one can say that solving a given problem on such or such machines (that differ for example on their software or hardware architecture) would induce different resource requirements. In order to fix this issue, in computational complexity theory, we consider that algorithms are executed on some standard "model of computation", such as the so-called "(Deterministic) Turing Machine" introduced by Alan Turing in [START_REF] Mathison | On computable numbers, with an application to the entscheidungsproblem[END_REF].

For the sake of brevity, we will not explain in details how it works but simply give the intuition of it. The Turing Machine is an abstract model of computational machine. It is composed of a tape on which symbols (0 or 1) can be read and written by an head that can move the tape left and right one cell at a time. An algorithm written for a Turing Machine is simply a set of transitions going through some so-called "states". A state indicates what to do given the symbol read on the current tape cell. What is initially written on the tape corresponds to the input (i.e. an encoding of it using the Turing Machine symbols). Given an input, the number of steps made by a Turing Machine to execute an algorithm is used as a time measurement and the number of cells used on the tape is used as a space measurement.

For a particular problem the input size could be expressed with a more understandable measure than the encoding size. For example for Argumentation Framework problems, we can use the number of arguments of a given an Argumentation Framework or the number of attacks. Now, saying that "a problem P 1 has a lower time complexity than another problem P 2 " means "for inputs of size n there exists an algorithm that solves P 1 with fewer steps than any other algorithm that solves P 2 ". Notice that, is taken into account the number of steps for the worst possible case of input of size n (i.e. the input of size n that induces the most transitions to solve the problem). Finally, it is the asymptotic behaviour of P 1 and P 2 as n grows that is considered.

The comparison method being fair, problems can now be grouped in suitable complexity classes. Those correspond basically to different orders of magnitude of steps. Now let consider the complexity classes of so called "decision problems".

Decision problem theory

A decision problem is a type of computational problem that has for output a boolean. That is, given an input the solution of the problem is whether "yes" (equivalently true or 1) or "no" (equivalently false or 0). We say that the problem "accepts" or "rejects" the input.

One of the most famous decision problem, for its importance in computational complexity theory is the satisfaction problem, so-called SAT problem. It is defined as follows:

"Given a propositional formula φ , is φ satisfiable?3 " Decision problems are probably the most studied type of computational problems. Over the decades a lot of complexity classes and hierarchies between them have been established. Let consider some interesting complexity classes for our work.

Decision time complexity classes 16.4.3.1 Polynomial time: P and L

The polynomial time class P regroups computational problems for which there exists an algorithm that solves them in a number of steps that is polynomially related to the size of the input. Problems belonging to this class are considered as "easy" or "tractable". P has a subclass called "logarithmic space" denoted by L that regroups the problems of P that require an amount of space (excluded the input and the output) that is logarithmically related to the size of the input.

Non-deterministic polynomial time: NP

The non-deterministic polynomial time class NP relies on the notion of witness.

Given an input x, a witness of x can be seen as a potential proof by example that the answer of the decision problem is positive for x. Let illustrate this considering the SAT problem. Given a propositional formula φ , a witness of SAT for φ is an interpretation of φ , i.e. a value assignation of the propositional variables of φ . Given an input x, a valid witness is a valid proof by example that the decision problem accepts x.

A problem P is in NP if and only if:

1. for any instance input x, all potential witnesses of x are of polynomial size w.r.t. |x| (the size of x),

2. any witness of a given input can be verified in a polynomial number of steps w.r.t. |x|, 3. given an input x, P accepts x if and only if x has a valid witness.

As an example, the SAT problem is in NP.

NP can also be defined as the set of problems which can be solved in polynomial time on a nondeterministic Turing Machine.

The difference between a Deterministic Turing Machine and a non-deterministic one is that for each step several transitions are possible simultaneously. To illustrate this, one can imagine that at each step a new Deterministic Turing Machine could be added (a copy of the machine in its current state) for the problem solving. While a Deterministic Turing Machine follows a single computation path, a non-deterministic Turing Machine follows a computation tree. A given decision problem accepts x if there exists a non deterministic algorithm such that at least one computation branch followed by the non-deterministic Turing Machine accepts the input.

NP problems are thus computational problems for which there exists a non-deterministic algorithm that can solve them on non-deterministic Turing Machine and doing so, following a computation tree having a polynomial depth and a number of leaves relative to the input size.

NP is thus the class of computational problems for which a solution (a proposed proof by example) can be verified easily. Although there is no formal proof (at the time of writing) that P ̸ = NP, we will consider that this inequation holds in the following as it is the standard assumption.

The coNP class

The coNP class is the class regrouping the complement problems of those of NP. As for NP, coNP relies on the same notion of witness and the same witness properties, i.e. for any input x, all potential witnesses of x are of polynomial size w.r.t. |x|, and any witness of x is verifiable in an amount of steps polynomial w.r.t. |x|.

The difference between NP and coNP is that coNP regroups the decision problems for which we want all the witnesses for a certain property to be invalid.

As illustration, the coNP problem relative to the SAT problem is the following one:

"Given a propositional formula φ , is φ unsatisfiable?4 "

Here the property of interest is the satifiability of φ . UNSAT will accept φ if and only if no witness of φ (i.e. no value assignation of its propositional variables) makes φ satisfied.

The polynomial-time hierarchy

The notion of "oracle" is very important to understand what is the polynomial-time hierarchy. An oracle is a black-box abstract machine that can solve a problem of a certain complexity class in one single step. Complexity classes can be expressed via this notion.

Given a problem P, we say that P is in the complexity class C D , if there exists an algorithm solving P with a complexity C and calling an oracle that solves in one operation a sub-problem of complexity class D.

As an example, let consider the ∃ 2 QBF problem. Let φ be a propositional formula over the set of propositional variables Ω. Let v 1 ⊂ Ω and v 2 ⊂ Ω be two subsets of propositional formula such that {v 1 , v 2 } is a partition 5 of Ω. The ∃ 2 QBF is the following decision problem:

"∃v 1 such that ∀v 2 , φ is true?"

Which means:

"Does there exist a valuation of the variables of v 1 such that for all valuations of the variables of v 2 , φ is true?"

Let propose a non-deterministic algorithm to solve the ∃ 2 QBF problem. Let O be an oracle witnessing that a given propositional formula is valid. 6 O is in coNP. Indeed, to decide if φ is valid is equivalent to decide if ¬φ is unsatisfiable. The algorithm A that non-deterministically guesses a valuation of v 1 and then verifies if for all valuations of v 2 the combined valuations (of v 1 and v 2 ) are models of φ , can be viewed an NP algorithm using O as oracle. As A solves ∃ 2 QBF, we have ∃ 2 QBF belonging to the class NP coNP .

The polynomial hierarchy, denoted by PH, is the complexity class hierarchy defined as follows:

• Σ P 0 = Π P 0 = Θ P 0 = P • Σ P k+1 = NP Σ P k • Θ P k+1 = P Σ P k • Π P k+1 = coNP Σ P k
The polynomial hierarchy is the union of all these complexity classes: Notice that calling a polynomial oracle from a non-deterministic algorithm doesn't add any complexity. As a consequence, we especially have NP P = NP.

PH = ∞ k=0 Σ P k = ∞ k=0 Π P k
Notice also that using as oracle, in an algorithm, a NP based oracle or coNP based oracle of same class level (i.e. Σ P k or Π P k for a given level k) doesn't matter. Indeed the answer of one of these oracles can be switched to correspond to the one solving the complementary problem.

As a consequence, we especially have NP coNP = NP NP = Σ P 2 . And so, we have ∃ 2 QBF belonging to Σ P 2 .

The difference class: DP

The so-called "difference class" denoted by DP is a kind of conjunction of the NP and coNP classes. A problem P belongs to DP if and only if it is composed of two sub-problems, P 1 belonging to NP and P 2 belonging to coNP, and for all input x, x is accepted by P if and only if x is accepted by P 1 and P 2 .

As an illustration, the SAT-UNSAT problem belongs to DP. It is defined as follows:

"Given a couple of propositional formulas ⟨φ , Ψ⟩, is φ satisfiable and Ψ unsatisfiable?"

Following the polynomial hierarchy introduced in the previous section, the DP-hierarchy is defined as follows:

DP k = Σ P k ∧ Π P k , with k ∈ 1, +∞
Notice that "∧" means "conjunction of problems" as explained above. It is not the intersection of sets of problems.

Problem reduction, completeness and hardness 16.4.4.1 Problem reduction

Let P 1 and P 2 be two decision problems. We denote by I P 1 and I P 2 the sets of all the instances of P 1 and P 2 . Let f : I P 1 → I P 2 be an efficient7 procedure that transforms any instance of P 1 into one instance of P 2 such that for all x ∈ I P 1 , P 1 accepts x iff P 2 accepts f (x).

If such a procedure exists, it means that any algorithm solving P 2 could be used to solve P 1 by firstly converting P 1 instances into P 2 ones. Now if it holds that P 2 is in some complexity class C , it means that P 1 is also in C considering that f is an efficient problem transformer. Likewise if P 2 is not in C , then P 1 is not in C .

In computational complexity theory, polynomial reduction are considered as efficient. Polynomial reductions are thus applicable to problems in P or complexity classes above. We denote by P 1 ≤ P P 2 the relation expressing that P 1 is polynomially reducible to P 2 , and by P 1 ≤ f P P 2 that the relation holds by using f . Usually, we use polynomial reductions in P while studying problems in NP and harder complexity classes, and log-space reductions, that is procedures belonging to L (denoted by ≤ L ), while studying complexity classes within P.

Completeness and hardness

We consider that a problem is hard for a certain class C if an efficient algorithm solving it could be used to efficiently solve, by mean of reductions, all the problems in C . It is formally defined as follows: let P 1 be a problem of complexity class C . P 1 is said to be hard w.r.t. C , denoted by C -hard, if:

∀P 2 ∈ C , P 2 ≤ P P 1
A problem P is said to be complete for C , denoted by C -c, if P ∈ C and P is C -hard.

Function problems

As such type of problems has not been studied in this thesis, we do not detail the function problem Theory as it has been done for Decision Problem Theory. We rather give a definition of what a function problem is and give one example.

Formally:

Definition 137 (Function problem). A function problem P is defined as a relation R over strings of an arbitrary alphabet Σ: R ⊆ Σ * × Σ * , with: Σ * being the set of all strings over symbols in Σ, including the empty string An algorithm solves P if for every input x such that there exists a value y satisfying (x, y) ∈ R, the algorithm produces one such y.

In simple words, a function problem is a computational problem that produces an output more complex than a boolean, as it is the case for decision problems. Here is a concrete example of function problem:

Example 87. The functional version of the SAT problem presented in Section 16.4.2 on page 179, so-called FSAT, is the following: Given a boolean formula φ with variables x 1 , . . . , x n , find an assignment x i → {true, false} such that φ evaluates to true or decide that no such assignment exists.

Analysis of Algorithms 16.5.1 Presentation

As seen in the previous section the computational complexity theory classifies computational problems following their hardness into complexity classes. In this theory, the subjects of interest are thus theoretical problems. It is not about concrete implementations of theoretical problems. Instead, the study of the complexity of explicitly given algorithms is called analysis of algorithms. Both are highly related. Indeed, given a problem P, if a concrete algorithm A solves P then the complexity class of P is a lower bound for the complexity of A and the complexity of A is an upper bound for the complexity of P.

It is however important to note that, as far as we know about the current state of computational technologies, there are no such things as oracles in the real world. Even quantum machines are not non-deterministic turing machines. As a consequence, on a deterministic real world computer, algorithms solving theoretical problems whose complexity class C belongs to the polynomial hierarchy such that C / ∈ P , are thus algorithms of complexity exponential.

This being said, in this section we present basic notions to analyse the complexity of algorithms.

Asymptotic analysis

Given an algorithm A, we are interested in estimating its running time as a function of a given machineindependent parameter n ∈ N. Moreover, to have a machine-independent measurement, we identify the calculation time with the number of executed instructions. The parameter n could be for example the length of an array or the number of arguments/attacks in an AF.

To compare algorithms, we consider only their behavior for a large n (that is what we call the "asymptotic complexity") and consider their "order of magnitude" rather than the precise number of instructions executed, ensuring that the complexity measure is independent of the programming language and the machine on which the algorithm runs.

This Order of magnitude can be declined into several relation types. Here are the three most common ones:

Definition 138 (Order of magnitude relations). Let T and f be positive non-zero functions. O, Ω and Θ are order of magnitude relations defined as follows:

• "Big O": T (n) ∈ O( f ) if ∃c ∈ R * and n 0 ∈ N s.t. ∀n ≥ n 0 , T (n) ≤ c × f (n).
T is said to be asymptotically dominated by f . f is an asymptotic upper bound for T .

• "Big Ω":

T (n) ∈ Ω( f ) if ∃c ∈ R * and n 0 ∈ N s.t. ∀n ≥ n 0 , T (n) ≥ c × f (n).
f is an asymptotic lower bound for T .

• "Big Θ":

T (n) ∈ Θ( f ) if T (n) ∈ Ω( f ) and T (n) ∈ O( f ).
The "asymptotic complexity" of an algorithm is formally defined, as follows:

Definition 139 (Asymptotic complexity). Let A be an algorithm and n be its measurement parameter. The asymptotic complexity of A, denoted by T (n), is the "order of magnitude" of its execution time (in terms of number of executed instructions) when n → ∞.

Note: An asymptotic complexity can be studied for the worst (denoted T max ), the best (denoted T min ) or the average (denoted T ave ) cases for an input of size n. 8 Generally, we are interested in the complexities for the worst and average cases.

Example 88. Let A be an algorithm computing the product of two square matrices of size n, using the basic method non optimized. We have:

• T min (n) = T max (n) = T ave (n) ∈ Θ(n 2 ) • T min (n) = T max (n) = T ave (n) ∈ O(e n ) • T min (n) = T max (n) = T ave (n) ∈ Ω(n)
Notice that O(e n ) is far to be the best upper bound for T min (n

) (resp. T max (n), T ave (n)), as T min (n) ∈ O(n 2 ) (resp. T max (n) ∈ O(n 2 ), T ave (n) ∈ O(n 2 )). Likewise, Ω(n) is not the best lower bound for T min (n) (resp. T max (n), T ave (n)) as T min (n) ∈ Ω(n 2 ) (resp. T max (n) ∈ Ω(n 2 ), T ave (n) ∈ Ω(n 2 ))
Chapter 17

Mathematical Problems

This chapter presents two mathematical problems used in this thesis. Section 17.1 presents the Constraint Satisfaction Problem and Section 17.2 on page 188 a graph/data clustering method called Spectral Clustering. Notice that for the sake of space we will not go into details. The scope of this background is limited to the necessary.

Constraint Satisfaction problem

In this section is presented the formal definition of a constraint satisfaction problem (CSP). 1Given a set of changeable state objects, a Constraint Satisfaction Problem (CSP) is a mathematical problem in which we look for a configuration of object states (i.e. a mapping where each object has a particular state) that satisfies a certain number of constraints.

Definition 140 (Constraint Satisfaction Problem). A CSP is defined by a triplet Ψ = ⟨X, D,C⟩ where:

• X = {X 1 , ..., X n } is a set of variables.
• D = {D(X 1 ), ..., D(X n )} is a set of domains, where D(X i ) ⊂ Z is the finite set of values that variable X i can take (i.e. D(X i ) is the domain of X i ).

• C = {c 1 , ..., c e } is a set of constraints.

Note: In real problems, the value of the variables are not always integers but they can be converted to integers. Any value of a given variable domain can be associated with an integer. Therefore, this definition of CSP is sufficient to capture all real constraint type problems.

Definition 141 (Constraint). A constraint c i is a boolean function involving a sequence of variables X(c i ) = (X i 1 , ..., X i q ) called its scheme. The function is defined on Z q . A combination of values (or tuple)

τ ∈ Z q satisfies c i if c i (τ) = 1 (also noted τ ∈ c i ). If c i (τ) = 0 (or τ / ∈ c i ), τ violates c i .
Definition 142 (Instantiation). An instantiation of the X variables is a mapping where each X i takes a value in its domain D(X i ).

Definition 143 (CSP Solution). A solution of a CSP is an instantiation of the X variables that violates no constraint.

Note: For more details on CSP see [START_REF] Tsang | Foundations of constraint satisfaction: the classic text[END_REF].

Example 89. Let consider the CSP Ψ = ⟨X, D,C⟩ illustrated in Figure 17.1 where:

• X = {X 1 , X 2 , X 3 , X 4 } is the set of variables of Ψ • D =                D(X 1 ) = {1, 4} , D(X 2 ) = {1, 2, 3, 4} , D(X 3 ) = {1, 3} , D(X 4 ) = {1, 3}               
is a set of their domains

• C = {c 1 , c 2 , c 3 , c 4
} is a set of constraints, with c i for i ∈ {1, ..., 4}, being an inequality constraint. Their schemes, that is the set of variables on which they are applied, are: Now that the CSP problem has been properly defined, let consider some facts. We have:

-X(c 1 ) = (X 1 , X 2 ) -X(c 2 ) = (X 1 , X 3 ) -X(c 3 ) = (X 3 , X 4 ) -X(c 4 ) = (X 1 , X 4 ) 1, 4 X 1 1, 2, 3, 4 X 2 1, 3 X 3 1, 3 X 4 c 1 : ̸ = c 2 : ̸ = c 4 : ̸ = c 3 : ̸ =
• If τ = (1, 1) then c 1 (τ) = false • If τ = (1, 2) then c 1 (τ) = true • If τ = (3, 2) then c 1 (τ) = false as 3 / ∈ D(X 1 ) • {X 1 = 4, X 2 = 1, X 3 = 3, X 4 = 3} is an instanciation that does not solve Ψ as it violates c 3 • {X 1 = 4, X 2 = 2, X 3 = 3, X 4 = 1} is a solution of Ψ

Spectral Clustering

Finding clusters in graph is a subject that has been widely studied. In this section we will present an approach that we used for argumentation framework clustering: the spectral clustering. For an overview of nondirected graph clustering algorithms see [START_REF] Satu | Graph clustering[END_REF] and for directed ones see [START_REF] Fragkiskos | Clustering and community detection in directed networks: A survey[END_REF].

The spectral clustering is a clustering method which is based on the spectral analysis of a similarity laplacian matrix.

A similarity matrix is a square matrix in which the lines and the columns describe the same set of elements. The matrix coefficients (i.e. the cell values) represent how much a element is similar to another, according to given similarity measure.

In short, here is how the spectral clustering works:

• Given a similarity matrix, the laplacian of this matrix is computed.

-The lines of the laplacian matrix correspond to the coordinates of the elements in a certain similarity space.

• The eigenvectors of the laplacian matrix with their associated eigenvalues are computed.

• The eigenvalues computed are sorted increasing order. A number n of them is kept with their associated eigenvectors.

-This solving and sorting process is done in order to project the datapoints in a new space which maximizes the closeness of similar elements. This space basis is formed by the computed eigenvectors. The eigenvalue of an eigenvector represents how much the datapoints are scattered on the eigenvector corresponding axis. Given that we are interested in the dimensions that maximize the best similarity (axes on which the datapoints are closed to each other), we keep the n smallest eigenvalues and their eigenvectors.

-If there are clusters in a data set, it is reasonable to think that the number of small eigenvalues are the number of groups identified (the datapoints being rather homogeneous following that axis). An heuristic to find the appropriate number of dimensions to keep is to detect the jump in the eigenvalues sequence (sorted in increasing order).

• A matrix whose columns are the remaining eigenvectors is constructed. The lines of it represent the new elements coordinates. • Once this data treatment is done, a simple algorithm of clustering such as KMeans is applied to that new data set, seeking for a partition into n parts, based on the coordinates of the elements (see [START_REF] Lloyd | Least squares quantization in pcm[END_REF] for more information about KMeans algorithm).

Let illustrate this algorithm on a concrete example.

Example 90. Let consider the graph G = (V, E,W ) shown in Figure 17.2 and let consider the weight over the edges as the similarity between two nodes. Following Example 86 on page 178, the similarity matrix corresponding to G is the following adjacency matrix M a :

M a =            j k l m n j 0 1 1 0 0 k 1 0 1 0 0 l 1 1 0 1 0 m 0 0 1 0 2 n 0 0 0 2 0           
The degree matrix M d of G is:

M d =            j k l m n j 2 0 0 0 0 k 0 2 0 0 0 l 0 0 3 0 0 m 0 0 0 3 0 n 0 0 0 0 2           
And then, the laplacian matrix M l of G is:

M d -M a = M l =            j k l m n j 2 -1 -1 0 0 k -1 2 -1 0 0 l -1 -1 3 -1 0 m 0 0 -1 3 -2 n 0 0 0 -2 2           
From the graph similarity matrix and by means of the laplacian matrix, the datapoints are projected in a new space in which similarity is maximised. The idea is that, if a certain structure exists in the data set, we will see in that space appear some agglomerates corresponding to the node clusters.

To do that, we compute the n smallest eigenvalues2 of the laplacian matrix obtained from the similarity matrix and the vectors associated with them (this n is an arbitrary parameter).

Indeed, the eigenvectors found will correspond to the basis of that similarity space and the eigenvalues to the variance on the corresponding axes. Given that we are looking for homogeneous groups, we will consider only the axis on which the variance is low, and so the eigenvectors that have small eigenvalues. The space whose basis is the n selected eigenvectors (corresponding to the n smallest eigenvalues) is then a compression of similarity space (i.e. we keep only the dimension useful for a clustering).

For the sake of the illustration let fix the parameter n to its maximal value: n = 5. Following Example 86 on page 178, we have:

• The eigenvectors of M l are:            v 1 v 2 v 3 v 4 v 5 -0.
          
• And their associated eigenvalues are:

λ 1 λ 2 λ 3 λ 4 λ 5
2.476651 × 1 × 10 -16 5.857864 × 1 × 10 -1 3.000000 3.414214 5.000000

Now that the similarity space is found, another important step is to find how many groups we have in that space. Intuitively, the number of eigenvectors with small eigenvalues, and so, the number of axes with small variance is the number of clusters. However, within the n smallest eigenvalues determined, it is difficult to formally say what is a small eigenvalue, and so, what is the number of clusters to chose.

Sorted in ascending order, the eigenvalue sequence represents how the similarity within clusters increases as the number of clusters grows. Obviously, the more clusters, the more homogeneous they will get. The idea is to find a compromise between number of clusters and homogeneity.

As eigenvalues say, in the end, how much the corresponding clusters will be homogeneous, the heuristic we have chosen to consider is to look for the "best elbow" in that ascending order sequence. We look for the number of dimensions to keep just before the quick growth of the variance. If the topology of the graph let appear some clusters then we will indeed have elbows. We can see in Figure 17.3 that this "best elbow" in the eigenvalues sequence (blue line with squares) is in second position. In that case the number of clusters determined by that heuristic is so 2.

To compute that "best elbow" we consider the second derivative (green line with triangles) of the ascending order sequence. As the second derivative represents the concavity of the eigenvalue sequence, we can take the first value of the second derivative above a certain threshold (red line without symbol) determined experimentally (i.e. the first position where the eigenvalue sequence is enough convex).

As you can see the first point of the second derivative, corresponding to the concavity formed by the first three eigenvalues, is the first value above the threshold and then we determine that the "best elbow" is in position 2. Once the number of clusters is chosen, we remove from the similarity matrix the columns that are after this number (i.e. we remove the dimensions we are not interested in for the clustering). The lines of the resulting matrix, which columns are the kept eigenvectors, correspond to the coordinates of the nodes in that new compressed similarity space.

λ 1 λ 2 λ 3 λ 4 λ 5 -3 -2 -1 0
Finally, we just have to apply a KMeans type algorithm [START_REF] Lloyd | Least squares quantization in pcm[END_REF] to find the groups of datapoint in that space and so have the partition of nodes we wanted.

Given that the chosen number of clusters is 2, we keep only the vectors v 1 and v 2 and when binded by column the lines they form correspond to the coordinates of the nodes in a new space that maximizes the similarity.

           v 1 v 2 j -0.
          
As you can see the v 1 dimension is useless. In practice it is removed. Note: For more information on spectral clustering see [START_REF] Von | A tutorial on spectral clustering[END_REF].

Tables of symbols

AF

The abbreviation for "Argumentation Framework (Dung)"

SCC a f

The abbreviation for "AF SCC"

RAF

The abbreviation for "Recursive Argumentation Framework"

SCC ra f

The abbreviation for "RAF SCC"

USCC a f

The abbreviation for "AF USCC". See Definition 27 on page 21

USCC
(M) i, j
The cell of row i and column j of the matrix M

M a

The adjacency matrix of a certain graph. See Definition 134 on page 175

M d

The degree matrix of a certain graph. See Definition 135 on page 175

M l

The laplacian matrix of a certain graph. See Definition 136 on page 175 in (.), out (.), und (.)

The sets of arguments/elements in "." that are labelled respectively in , out and und

Ω ω

The symbols Ω and ω are used to represent respectively partitions and parts in a given partition. In the case of AFs, we have, given an AF AF = ⟨A, K⟩:

Ω = {ω 1 , ..., ω n }

S inp

Let AF = ⟨A, K⟩ be an AF and let S ⊆ A. S inp is the set of input arguments of S, that is:

S inp = {b ∈ A \ S|∃a ∈ S, (b, a) ∈ K} S K Let AF = ⟨A, K⟩ be an AF and let S ⊆ A. S K is the conditioning relation of S, that is: K ∩ (S inp × S) AF , J , ℓ J , K J
An AF with input, with:

• AF being an AF • J being a set of input arguments • ℓ J being a labelling of J

• K J being a set of conditioning relations such that K J ⊆ J × A

std-AF

Standard AF corresponding to an AF with input AF , J , ℓ J , K J is defined

as std-AF = A ∪ J ′ , K ∪ K ′ J , where J ′ = J ∪ a ′ |a ∈ J ∩ out (ℓ J ) and K ′ J = K J ∪ (a ′ , a)|a ∈ J ∩ out (ℓ J ) ∪ (a, a)|a ∈ J ∩ und (ℓ J
) . See Definition 17 on page 17

J ′ See std-AF F a f σ (AF , J , ℓ J , K J )
The canonical local function computing the σ -labellings associated with the AF with input AF , J , ℓ J , K J (see Definition 18 on page 18)

Paths a f (AF )
The set of paths of AF

Walks a f (AF )
The set of walks of AF Continued on next page ... The "default AF partition selector" is the partition selector that produces all possible partitions of an AF. See Definition 25 on page 20

S USCC

A partition selector that does not split SCC a f . See Definition 27 on page 21

AF hard

Let AF = ⟨A, K⟩ be an AF. The "hard part" of AF is the sub AF defined as: AF hard = AF ↓ {a|a∈A,ℓ gr (a)=und } . See Section 4.2.1 on page 30 κ = af , I, O, B Let AF = ⟨A, K⟩ be an AF κ is a cluster structure where: The set of configurations corresponding to the cluster structure κ. See Algorithm 2 on page 31

• af is a sub AF of AF such that for some ω ⊆ A, af = AF ↓ ω • I = {(a, b)|(a, b) ∈ K, b ∈ ω and a / ∈ ω} • O = {(a, b)|(a, b) ∈ K, b / ∈ ω and a ∈ ω} • B = {a|(
P A set of reunified labelling profiles U (AF )
The undirection transformation that transforms an AF into a weighted non-directed graph. See Definition 39 on page 52

Comp σ (AF )

The compact enumeration representation of AF according to a semantics σ . See Definition 40 on page 64

MAF

A "Meta-Argumentation Framework" (MAF). They are basically Argumentation Framework. See [START_REF] Gabbay | Semantics for higher level attacks in extended argumentation frames[END_REF] for more information

Cred σ , Skep σ , Ver σ , Exists σ , Exists ¬∅ σ , Unique σ
Respectively the Credulous Acceptance, Skeptical Acceptance, Verification of a labelling, Existence of an extension/labelling/structure, Existence of a "non-empty" extension/labelling/structure and the Uniqueness of a solution AF decision problems. See Definition 28 on page 23 • A is a set of arguments • K is a set of attacks • s is a mapping function that associates an attack with its source (i.e. s : K → A) • t is a mapping function that associates an attack with its target (i.e.

t :

K → A ∪ K) U = ⟨S, Q⟩ A structure. S is a set of arguments and Q is a set of attacks RAF-De f (U)
Let RAF = ⟨A, K, s,t⟩ be a RAF and U = ⟨S, Q⟩ be a structure of RAF . RAF-De f (U) is the set of arguments defeated by U, that is:

RAF-De f (U) = {a ∈ A|∃α ∈ Q s.t. s(α) ∈ S and t(α) = a}
See Definition 61 on page 92 RAF-Inh(U) Let RAF = ⟨A, K, s,t⟩ be a RAF and U = ⟨S, Q⟩ be a structure of RAF . RAF-Inh(U) is the set of attacks inhibited by U, that is: ⊑ ⊑ ar Let U = ⟨S, Q⟩ and U ′ = ⟨S ′ , Q ′ ⟩ be any pair of structures. We write means "(x, y) ∈ PE ra f (RAF )" and "x ̸ ≡ RAF y" means "(x, y) / ∈ PE ra f (RAF )" RAF = Ã, K, s, t, s,t Let RAF = ⟨A, K, s,t⟩ be a RAF. RAF is a partial RAF of RAF where:

RAF-Inh(U) = {α ∈ K|∃β ∈ Q s.t. s(β ) ∈ S
U ′ ⊑ U ′ iff (S ∪ Q) ⊆ (S ′ ∪ Q ′ ) and U ⊑ ar U ′ iff S ⊆ S ′ . L = ℓ A , ℓ K A structure labelling L σ -ra f (RAF ) L σ -ra f (RAF )
• à ⊆ A is a set representing arguments • K ⊆ K is a set representing attacks • s : K → {true, false} is a boolean function that indicates whether or not an attack in K has its source in à defined as following:

∀α ∈ K, s(α) = true if s(α) ∈ Ã otherwise false
• t : K → {true, false} is a boolean function that indicates whether or not an attack in K has its target in Ã

∀α ∈ K, t(α) = true if t(α) ∈ Ã ∪ K otherwise false
See Definition 92 on page 134

I = S inp , Q inp Let RAF = ⟨A, K, s,

S (RAF )

A partition selector S is a function receiving as input a RAF RAF = ⟨A, K, s,t⟩ and returning a set of partitions of A ∪ K

S D-ra f

The "default RAF partition selector" is the partition selector that produces all possible partitions of a RAF. See Definition 108 on page 159 S ra f -USCC A RAF partition selector that does not split SCC ra f . See Definition 109 on page 161

S ra f -c-USCC
The AF partition selector that corresponds to S ra f -USCC . See Definition 109 on page 161

RAF-Cred σ , RAF-Skep σ , RAF-Ver σ , RAF-Exists σ , RAF-Exists ¬∅ σ , RAF-Unique σ
Respectively the Credulous Acceptance, Skeptical Acceptance, Verification of a labelling, Existence of an extension/labelling/structure, Existence of a "non-empty" extension/labelling/structure and the Uniqueness of a solution RAF decision problems. See Definition 79 on page 119 s(α).α Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be the flattened version of RAF . Let α ∈ K be an attack of RAF . s(α).α is the created argument that represents the "conjunction" of attack α with its source s(α).

See Definition 76 on page 115 (Notice that s(α).α ∈ And A,K ) An SCC decomposition hierarchy of an AF. This layered schema is used to illustrate the algorithms proposed in [START_REF] Liao | Toward incremental computation of argumentation semantics: A decomposition-based approach[END_REF] and [START_REF] Cerutti | Exploiting parallelism for hard problems in abstract argumentation[END_REF] Figure 7.8 on page 80 An illustration for the recursiveness of the P-SCC-REC algorithm (cite [START_REF] Cerutti | Exploiting parallelism for hard problems in abstract argumentation[END_REF]) decision problem A decision problem is a type of computational problem that has for output a boolean. That is, given an input the solution of the problem is whether "yes" (equivalently true or 1) or "no" (equivalently false or 0). We say that the problem "accepts" or "rejects" the input. See Section heuristic Any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, short-term goal or approximation. vector space Also called a linear space, a vector space is a set of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. In all the following proofs, by L σ () we mean "the set of labellings under the semantics σ according to the mathematical definition of σ " whereas by L * σ () we mean "the set of labellings under the semantics σ computed with our algorithm". Thus, proving completeness is proving that L σ () ⊆ L * σ () and proving soundness is proving L * σ () ⊆ L σ (). We assume, in the following proofs, that the external existing solver used to compute the labellings of the induced AFs from the different cluster structures is sound and complete for the grounded, complete, stable and preferred semantics.

ε U Let RAF = ⟨A, K, s,

Table of tables

Proofs of Section 4.3.1: Relation between AFs with input and cluster structures

Proof of Proposition 9 on page 49. Let af ′ be the induced AF of κ under the context µ. Let af , ω inp , µ, ω K be an AF with input (See Definition 16 on page 17) and std-AF be its standard argumentation framework (See Definition 17 on page 17). Let prove that:

L µ(κ) σ = F σ (af , ω inp , µ, ω K )
By definition of the induced AF (Definition 32 on page 35), we have:

af ′ = ω ′ , K ′
Where:

• D = {a|a ∈ ω and (s, a) ∈ ω K and s ∈ in (µ)} being the set of arguments attacked by an in -labelled argument in µ.

• ω ′ = ω \ D • K ′ = (K ∩ (ω ′ × ω ′ )) ∪ {(a, a)|(s, a) ∈ ω K and s ∈ und (µ)}
af ′ is so the AF obtained from af after the removal of the arguments attacked by an in -labelled argument of the context and after the adding of self-attacks on each argument attacked by an und -labelled argument of the context.

By definition of the standard argumentation framework (Definition 17 on page 17), we have:

std-AF = ω ∪ J ′ , (K ∩ (ω × ω)) ∪ K ′ J
Where:

• J ′ = ω inp ∪ {a ′ |a ∈ ω inp ∩ out (µ)}. See footnote. 1 • K ′ J = ω K ∪ {(a ′ , a)|a ∈ ω inp ∩ out (µ)} ∪ {(a, a)|a ∈ ω inp ∩ und (
µ)} Let std-AF 1 be the AF corresponding to std-AF ↓ ω∪{a|a∈in (µ)}∪{a|a∈und (µ)} . Given that to obtain std-AF 1 from std-AF we just have to remove the arguments labelled out in µ and those attacking them, 2 we have then:

{ℓ ↓ ω |ℓ ∈ L σ (std-AF 1 )} = {ℓ ↓ ω |ℓ ∈ L σ (std-AF )} (18.1)
Let ω ′ be the set of arguments such that ω ′ = ω \ D (as defined above). Let std-AF 2 be the AF corresponding to std-AF 1 ↓ ω ′ ∪{a|a∈und (µ)} . Given that to obtain std-AF 2 from std-AF 1 we just have to remove the arguments labelled in in µ and those they attack, 3 we have then:

{ℓ ↓ ω ′ |ℓ ∈ L σ (std-AF 2 )} = {ℓ ↓ ω ′ |ℓ ∈ L σ (std-AF 1 )} (18.2)
Considering the AF std-AF 2 , let U = {a|a ∈ ω ′ and (b, a) ∈ ω K and b ∈ ω inp ∩ und (µ)} be the set of arguments of ω ′ attacked by an argument labelled und in µ. Let u ∈ U be one of these arguments.

Given that u is attacked by an und -labelled argument, u must be labelled und or out . Nevertheless having an argument labelled und cannot have as consequence an argument labelled in or out . And so, if u is labelled out in some labelling of std-AF 2 , it is not due to the set of arguments labelled und in µ.

Knowing this, we have:

L σ (af ′ ) = {ℓ ↓ ω ′ |ℓ ∈ L σ (std-AF 2 )} (18.3)
From Equation (18.3) and Equation (18.2), we have:

L σ (af ′ ) = {ℓ ↓ ω ′ |ℓ ∈ L σ (std-AF 1 )} (18.4)
Let ℓ D be the labelling of the set of arguments D defined as following: ℓ D = {(a, out )|a ∈ D}.

From Equation (18.4) and Equation (18.1), we have:

{ℓ ∪ ℓ D |ℓ ∈ L σ (af ′ )} = {ℓ ↓ ω |ℓ ∈ L σ (std-AF )} (18.5)
By definition of an induced labelling set (Definition 33 on page 35), we have:

L µ(κ) σ = {ℓ ∪ ℓ D |ℓ ∈ L σ (af ′ )} (18.6)
By definition of a canonical local function (Definition 18 on page 18), we have:

F σ (af , ω inp , µ, ω K ) = {ℓ ↓ ω |ℓ ∈ L σ (std-AF )} (18.7)
From Equations (18.5) to (18.7) on the previous page, we prove thus that:

L µ(κ) σ = F σ (af , ω inp , µ, ω K ) ■
Proofs of Section 4.3.2: Soundness and completeness

Proof of Proposition 10 on page 50. Let af = ⟨A, K⟩ be an AF, Ω = {ω 1 , ..., ω n } be a partition of A and {κ 1 , ..., κ n } be the set of cluster structures corresponding to Ω, with each κ i being defined as:

κ i = af ↓ ω i , I = ω K i , O = K ∩ (ω i × (A \ ω i )), B = {a|(a, b) ∈ O or (b,
L σ (af ) ⊆ {ℓ ω 1 ∪ ... ∪ ℓ ω n |ℓ ω i ∈ F σ (af ↓ ω i , ω inp i , ( j∈{1,...,n} s.t. j̸ =i ℓ ω j ) ↓ ω inp i , ω K i )} (18.8)
Given that the labellings of all cluster structures are computed for every possible context, we have, by definition of the context and of the input arguments:

∀i, ∀ℓ inp = ( j∈{1,...,n} s.t. j̸ =i ℓ ω j ) ↓ ω inp i , ∃µ κ i s.t. µ κ i = ℓ inp (18.9)
Given that the external solver that computes the labellings of af ↓ ω i according to the semantics σ is sound and complete, and considering std-AF being the standard AF w.r.t to the AF with input af ↓ ω i , ω inp i , µ κ i , ω K i , we have:

∀i, ∀µ κ i , ∀ℓ std-AF ∈ L σ (std-AF ), ∃ℓ ∈ L * µ(κ i ) σ s.t. ℓ = ℓ std-AF ↓ ω i (18.10)
So we have:

∀i, ∀µ κ i , ∀ℓ std-AF ∈ L σ (std-AF ), ℓ std-AF ↓ ω i ∈ L κ i D (18.11)
And so (following Definition 18 on page 18):

∀ω i , F σ (af ↓ ω i , ω inp i , ( j∈{1,...,n} s.t. j̸ =i ℓ ω j ) ↓ ω inp i , ω K i ) ⊆ L κ i D (18.12)
As a consequence and because of Equation (18.8) we have (∏ denoting the cartesian product):

L σ (af ) ⊆ ∏ κ i L κ i D (18.13) Let χ = {ℓ|ℓ ∈ ∏ κ i L κ i
D and ∃a ∈ A s.t. a is illegally labelled in ℓ} be the set of all possible incorrect labellings (i.e. the set of labellings in which there exists an argument that is not legally labelled).

We have, by definition of σ :

L σ (af ) ⊆ ( ∏ κ i L κ i D ) \ χ (18.14)
Given that, for all computed labellings, we keep only the merged configuration, that is the most flexible possible configuration, our CSP modelisation does not add extra constraints. The proposed reunification removes, thus, only the labellings belonging to χ.

As a consequence, we have:

L σ (af ) ⊆ L * σ (af ) (18.15)
We prove so that for any top-down decomposable semantics σ our algorithm is complete, and so for the complete, stable and preferred semantics following Proposition 8 on page 21.

■

Proof of Proposition 11 on page 50.

• Assertion 1: Let af = ⟨A, K⟩ be an AF and Ω = {ω 1 , ..., ω n } be a partition of A corresponding to the clustering of af . Let σ be a fully decomposable and complete-based semantics and let ℓ * be a labelling of af according to σ obtained by Algorithm 2.

Let suppose that ℓ * / ∈ L σ (af ). We will prove that it is impossible with a reductio ad absurdum. As σ is a complete-based and fully decomposable semantics we can say that (Definition 19 on page 19):

ℓ * / ∈ {ℓ ω 1 ∪ ... ∪ ℓ ω n |ℓ ω i ∈ F σ (af ↓ ω i , ω inp i , ( j∈{1,...,n} s.t. j̸ =i ℓ ω j ) ↓ ω inp i , ω K i )} (18.16)
And so:

∃ω i ∈ Ω s.t. ℓ * ↓ ω i / ∈ F σ (af ↓ ω i , ω inp i , ( j∈{1
ℓ ω j ) ↓ ω inp i
, ω K i ) (18.17)

In the following we denote by ω the particular ω i for which Equation (18.17) holds in order to simplify the notation.

Let κ = af ↓ ω , I = ω K , O = K ∩ (ω × (A \ ω)), B = {a|(a, b) ∈ O or (b, a)
∈ I} be the cluster structure corresponding to ω.

Let µ be a context of κ such that µ = ( j∈{1,...,n} s.t. ω j ̸ =ω ℓ ω j ) ↓ ω inp .

Let L * µ(κ) σ be the set of labellings of κ under the context µ produced by Algorithm 2.

Let ℓ ′ * ∈ L * µ(κ) σ be the labelling coinciding with ℓ * ↓ ω (i.e. ℓ ′ * = ℓ * ↓ ω ).

We have so:

ℓ ′ * ∈ L * µ(κ) σ (18.18)
Whereas:

ℓ ′ * / ∈ F σ (af ↓ ω , ω inp , µ, ω K ) (18.19)
And so:

L * µ(κ) σ ̸ = F σ (af ↓ ω , ω inp , µ, ω K ) (18.20)
Nevertheless, according to Proposition 9 on page 49 we must have:

L * µ(κ) σ = F σ (af ↓ ω , ω inp , µ, ω K ) (18.21)
Thus, there is a contradiction between Equation (18.20) on the previous page and Equation (18.21).

From this contradiction we can conclude that:

L * σ (af ) ⊆ L σ (af ) (18.22)
We prove so that for any fully decomposable and complete-based semantics σ our algorithm is sound, and so for the complete and stable semantics, following Proposition 8 on page 21.

• Assertion 2: Let af = ⟨A, K⟩ be an AF. Given that Algorithm 2 is complete for the preferred semantics (see Proposition 10 on page 50), L * σ , the set of all labellings reunified from the different clusters obtained in Algorithm 2 line 7, contains all the preferred labellings of af .

In Algorithm 2 line 8, we keep from L * σ only the maximal (w.r.t ⊆ of in -labelled arguments) labellings, that are by definition the preferred labellings. As a consequence, L pr contains only and all the preferred labellings of af . Algorithm 2 is, thus, sound and complete for the preferred semantics.

■

Proof of Proposition 12 on page 50. (Completeness of Algorithm 1 + Algorithm 2). Let AF = ⟨A, K⟩ be an AF, ℓ gr be its grounded labelling, AF hard = AF ↓ {a|a∈A,ℓ gr (a)=und } be the hard part of AF and {af 1 = ⟨A 1 , K 1 ⟩ , ..., af n = ⟨A n , K n ⟩} be the set of AFs obtained from AF hard components.

Let σ be the complete, stable or preferred semantics. Let L * σ (AF ) be the set of labellings obtained from Algorithm 1. Let L * σ (af i ) be the set of labellings obtained from Algorithm 2 for the component af i . Let L σ (AF ) be the set of labellings of AF .

Let Ω = {ω gr , A 1 , ..., A n } be a partition of A with ω gr = {a|a ∈ in (ℓ gr ) or a ∈ out (ℓ gr )}.

Let ℓ ∈ L σ (AF ) be a labelling of AF according to σ .

Given that (following Definition 18 on page 18):

F σ (AF ↓ ω gr , ω inp gr , ( i∈{1,...,n} ℓ A i ) ↓ ω inp gr , ω K gr ) = {ℓ gr } (18.23)
We have by definition of top-down decomposable semantics (following Definition 22 on page 19):

L σ (AF ) ⊆ {ℓ gr ∪ A i ℓ A i } with ℓ A i ∈ F σ (AF ↓ A i , A inp i , ( j∈{1,...,n} s.t. j̸ =i ℓ A j ) ↓ A inp i , A K i ) (18.24)
Given that Algorithm 2 is complete for top-down decomposable semantics (i.e. ∀p ∈ Ω,

L σ (AF ↓ p ) ⊆ L * σ (AF ↓ p )), ∀A i , ℓ A i ∈ L * σ (af i ) (18.25)
Furthermore:

∀ℓ * ∈ L * σ (AF ), ℓ * = ℓ gr ∪ ℓ * i , with ℓ * i ∈ L * σ (af i ) (18.26)
We have so:

{ℓ gr ∪ A i ℓ A i } = L * σ (AF ) (18.27)
Finally, we have:

L σ (AF ) ⊆ L * σ (AF ) (18.28)
We prove so that our algorithm is complete for the complete, stable and preferred semantics. ■ Proof of Proposition 13 on page 50.

• Assertion 1: Algorithm 1 is sound for the stable and complete semantics.

Let AF = ⟨A, K⟩ be an AF, ℓ gr be its grounded labelling, AF hard = AF ↓ {a|a∈A,ℓ gr (a)=und } be the hard part of AF and {af 1 = ⟨A 1 , K 1 ⟩ , ..., af n = ⟨A n , K n ⟩} be the set of AFs obtained from AF hard components.

Let σ be the complete or stable semantics.

Let L * σ (AF ) be the set of labellings of AF obtained from Algorithm 1. Let L σ (AF ) be the set of labellings of AF .

Let ℓ * ∈ L * σ (AF ) be a labelling of AF computed by Algorithm 1. Let L * σ (af i ) be the set of labellings of af i obtained from Algorithm 2. Following Algorithm 1, we have:

ℓ * = ℓ gr ∪ ℓ * i , with ℓ * i ∈ L * σ (af i ) (18.29)
Let Ω = {ω gr , A 1 , ..., A n } be a partition of A with ω gr = {a|a ∈ in (ℓ gr ) or a ∈ out (ℓ gr )}.

We have (following Definition 18 on page 18):

F σ (
L σ (AF ) = {ℓ gr ∪ A i ℓ A i } with ℓ A i ∈ F σ (af ↓ A i , A inp i , ( j∈{1,...,n} s.t. j̸ =i ℓ A j ) ↓ A inp i , A K i ) (18.31)
Given that Equation (18.31) holds and that Algorithm 2 is sound for fully decomposable semantics (i.e. ∀p ∈ Ω, L * σ (AF ↓ p ) ⊆ L σ (AF ↓ p )), we have:

ℓ * ∈ L σ (AF ) (18.32)
And thus:

L * σ (AF ) ⊆ L σ (AF ) (18.33)
We prove so that for the complete and stable semantics our algorithm is sound.

• Assertion 2: Algorithm 1 is sound for the preferred semantics.

Let AF = ⟨A, K⟩ be an AF, ℓ gr be its grounded labelling, AF hard = AF ↓ {a|a∈A,ℓ gr (a)=und } be the hard part of AF and {af 1 = ⟨A 1 , K 1 ⟩ , ..., af n = ⟨A n , K n ⟩} be the set of AFs obtained from AF hard components.

Let L * pr (AF ) be the set of labellings of AF obtained from Algorithm 1. Let L pr (AF ) be the set of labellings of AF .

Let L * pr (af i ) be the set of labellings of af i obtained from Algorithm 2. Following Algorithm 1, we have: Let AF ′ = ⟨A, K \ {(a ′ , a)|(a ′ ∈ A 0 and a / ∈ A 0 ) or (a ′ / ∈ A 0 and a ∈ A 0 )}⟩ be the AF constructed by removing from AF the attacks between its fixed part and its non fixed part. As in AF all arguments in the fixed part attacking arguments outside the fixed part is labelled out (Equation (18.38)) their attacks have no effect. The consequence is the following:

L * pr (AF ) = {ℓ gr ∪ ℓ A 1 ∪ ... ∪ ℓ A n |ℓ A i ∈ L * pr (af i )} (18.
L pr (AF ′ ) = L pr (AF ) (18.39)
Notice that AF ′ has n + 1 connected components corresponding to the partition Ω. Given that there is no connection (attack) between those connected components, each A i ∈ Ω is an USCC a f (see Definition 27 on page 21). As a consequence, following the definition of S USCC (Definition 27 on page 21), we have:

Ω ∈ S USCC (AF ′ ) (18.40)
As the preferred semantics is fully decomposable w.r.t. S USCC (Definition 27 on page 21), we have (following Definition 24 on page 20):

L σ (AF ′ ) = {ℓ A 0 ∪ ... ∪ ℓ A n |ℓ A i ∈ F pr (AF ↓ A i , A inp i , ( j∈{0,...,n} s.t. j̸ =i ℓ A j ) ↓ A inp i , A K i )} (18.41)
Notice that:

F pr (AF ′ ↓ A 0 , A inp 0 , ( i∈{1,...,n} ℓ A i ) ↓ A inp 0 , A K 0 ) =
ℓ A j ) ↓ A inp i , A K i ) = L * pr (af i ) (18.43)
From the Equations (18.41) to (18.43) on pages 225-226, we have:

L pr (AF ′ ) = {ℓ gr ∪ ℓ A 1 ∪ ... ∪ ℓ A n |ℓ A i ∈ L * pr (af i )} (18.44)
From Equations (18.39) and (18.44) on the previous page and on this page, we have:

L pr (AF ) = {ℓ gr ∪ ℓ A 1 ∪ ... ∪ ℓ A n |ℓ A i ∈ L * pr (af i )} (18.45)
Finally, from Equations (18.34) and (18.45) on the previous page and on this page we have:

L * pr (AF ) = L pr (AF ) (18.46)
We prove so that Algorithm 1, when using Algorithm 2 to compute the component labellings, is sound and complete for the preferred semantics.

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Proofs of Chapter 6: Compact representation

Proof of Proposition 14 on page 66. Following Algorithm 1, let AF be decomposed into n components and let Comp σ (AF ) = {L σ (af 0 ), ..., L σ (af n )} be the compact enumeration representation corresponding to σ (AF ).

Assertion 1: Cred σ (AF , a) ≡ Comp-Cred σ (AF , a).

• Case 1: If Cred σ (AF , a) is true, then Comp-Cred σ (AF , a) is true.

If Cred σ (AF , a) is true, then there exists a labelling ℓ ∈ σ (AF ) such that a ∈ in (ℓ).

Given that σ (AF ) ̸ = ∅ and that Algorithm 1 is complete for σ then ∀L σ (af i ) ∈ Comp σ (AF ), we have:

L σ (af i ) ̸ = ∅.
Moreover, as Algorithm 1 is complete for σ , there exists a combination of component labellings ℓ 0 , ..., ℓ n , with ℓ i ∈ L σ (af i ) and L σ (af i ) ∈ Comp σ (AF ), such that: ℓ = n i=0 ℓ i . As a consequence, there exists i ∈ {0, ..., n} such that a ∈ in (ℓ i ).

We have so: Comp-Cred σ (AF , a) being true.

• Case 2: If Comp-Cred σ (AF , a) is true, then Cred σ (AF , a) is true.

If Comp-Cred σ (AF , a) is true, then ∀L σ (af i ) ∈ Comp σ (AF ), L σ (af i ) ̸ = ∅, and there exists a set L σ (af ) ∈ Comp σ (AF ) such that ∃ℓ j ∈ L σ (af ) s.t. ℓ j (a) = in . As no component labelling set is empty, following Algorithm 1, there is thus a combination of component labellings ℓ 0 , ..., ℓ n , with ℓ i ∈ L σ (af i ) and L σ (af i ) ∈ Comp σ (AF ), including that particular ℓ j . Let ℓ = n i=0 ℓ i . Given that Algorithm 1 is sound for σ , then we have: ℓ ∈ σ (AF ). As a ∈ in (ℓ), then we have: Cred σ (AF , a) being true.

We prove so that: Cred σ (AF , a) ≡ Comp-Cred σ (AF , a). Given that σ (AF ) ̸ = ∅ and that Algorithm 1 is complete for σ then ∀L σ (af i ) ∈ Comp σ (AF ), we have:

L σ (af i ) ̸ = ∅.
Let L σ (af j ) be the labelling set of the particular component to which a belongs and let, for i ∈ ({0, ..., n} \ { j}), L σ (af i ) ∈ Comp σ (AF )} be a set of labellings of the compact enumeration representation different from L σ (af j ). As Algorithm 1 is sound for σ , then ∀ℓ j ∈ L σ (af j ), ∀ℓ 0 ∈ L σ (af 0 ), ..., ∀L σ (ℓ n ) ∈ af n , (ℓ j n i=0 ℓ i ) ∈ σ (AF ). As ∀ℓ ∈ σ (AF ), a ∈ in (ℓ), then a ∈ in (ℓ j n i=0 ℓ i ). As a consequence: a ∈ in (ℓ j ).

We have so: Comp-Skep σ (AF , a) being true.

• Case 2: If Comp-Skep σ (AF , a) is true, then Skep σ (AF , a) is true. If Comp-Skep σ (AF , a) is true, then ∀L σ (af i ) ∈ Comp σ (AF ), L σ (af i ) ̸ = ∅,
and there exists a set L σ (af j ) ∈ Comp σ (AF ) such that ∀ℓ j ∈ L σ (af j ), ℓ j (a) = in .

As no component labelling set is empty and as Algorithm 1 is sound for σ , we have, with L σ (af i ) ∈

Comp σ (AF ) for i ∈ ({0, ..., n} \ { j}), the following assertions: ∀ℓ j ∈ L σ (af j ), ∀ℓ 0 ∈ L σ (af 0 ), ..., ∀L σ (ℓ n ) ∈ af n , (ℓ j n i=0 ℓ i ) ∈ σ (AF ) and a ∈ in (ℓ j n i=0 ℓ i ). Given that Algorithm 1 is complete for σ and that ∀ℓ = ℓ j n i=0 ℓ i , we have a ∈ in (ℓ), then we have:

Skep σ (AF , a) being true.

We prove so that: Skep σ (AF , a) ≡ Comp-Skep σ (AF , a).

Assertion 3: Ver σ (AF , ℓ) ≡ Comp-Ver σ (AF , ℓ)

• Case 1: If Ver σ (AF , ℓ) is true, then Comp-Ver σ (AF , a) is true.

If Ver σ (AF , ℓ) is true, then ℓ ∈ σ (AF ). Given that Algorithm 1 is complete for σ , then there exists a combination of component labellings ℓ 0 , ..., ℓ n , with ℓ i ∈ L σ (af i ) and L σ (af i ) ∈ Comp σ (AF ), such that: ℓ = n i=0 ℓ i . As a consequence we have:

Comp-Ver σ (AF , a) is true. • Case 2: If Comp-Ver σ (AF , a) is true, then Ver σ (AF , ℓ) is true.
If Comp-Ver σ (AF , ℓ) is true, there exists a combination of component labellings ℓ 0 , ..., ℓ n , with ℓ i ∈ L σ (af i ) and L σ (af i ) ∈ Comp σ (AF ), such that: ℓ = n i=0 ℓ i . Given that Algorithm 1 is sound for σ then ℓ ∈ σ (AF ). As a consequence we have: Ver σ (AF , a) is true.

We prove so that: Ver σ (AF , ℓ) ≡ Comp-Ver σ (AF , ℓ). If Exists σ (AF ) is true, then σ (AF ) ̸ = ∅. Let ℓ ∈ σ (AF ) be a labelling. As Algorithm 1 is complete for σ , there exists thus a combination of component labellings ℓ 0 , ..., ℓ n , with ℓ i ∈ L σ (af i ) and L σ (af i ) ∈ Comp σ (AF ) such that ℓ = n i=0 ℓ i . As a consequence, we have:

∀L σ (af i ) ∈ Comp σ (AF ), L σ (af i ) ̸ = ∅.
We prove so that if Exists σ (AF ) is true, then Comp-Exists σ (AF ) is true.

• Case 2: If Comp-Exists σ (AF ) is true, then Exists σ (AF ) is true.

If Comp-Exists σ (AF ) is true, then ∀L σ (af i ) ∈ Comp σ (AF ), L σ (af i ) ̸ = ∅. Let ℓ = n i=0
ℓ i be a labelling with ℓ i∈{0,...,n} ∈ L σ (af i ). As Algorithm 1 is sound for σ , then ℓ ∈ σ (AF ).

We prove so that if Comp-Exists σ (AF ) is true, then Exists σ (AF ) is true.

We prove so that: Exists σ (AF ) ≡ Comp-Exists σ (AF ). σ (AF ) is true, then ∀L σ (af i ) ∈ Comp σ (AF ), L σ (af i ) ̸ = ∅ and ∃L σ (af i ) s.t. ℓ i ∈ L σ (af i ) and in (ℓ i ) ̸ = ∅. As Algorithm 1 is sound for σ then there exists a labelling ℓ = n i=0 ℓ i , with ℓ i ∈ L σ (af i ), such that ℓ ∈ σ (AF ) and in (ℓ) ̸ = ∅.

We prove so that if Comp-Exists ¬∅ σ (AF ) is true, then Exists ¬∅ σ (AF ) is true. We prove so that: Exists ¬∅ σ (AF ) ≡ Comp-Exists ¬∅ σ (AF ).

Assertion 6: Unique σ (AF ) ≡ Comp-Unique σ (AF )

• Case 1: If Unique σ (AF ) is true, then Comp-Unique σ (AF ) is true.

If Unique σ (AF ) is true, then σ (AF ) = {ℓ}.

As Algorithm 1 is complete for σ , there exists thus a combination of component labellings ℓ 0 , ..., ℓ n , with

ℓ i ∈ L σ (af i ) and L σ (af i ) ∈ Comp σ (AF ) such that ℓ = n i=0 ℓ i . As a consequence, ∀L σ (af i ) ∈ Comp σ (AF ), L σ (af i ) ̸ = ∅.
As Algorithm 1 is sound for σ and as |σ

(AF )| = 1, then ∀L σ (af i ) ∈ Comp σ (AF ), |L σ (af i )| = 1.
We prove so that if Unique σ (AF ) is true, then Comp-Unique σ (AF ) is true.

S = A \ RAF-De f (U) and Q = K \ RAF-Inh(U)
For any x ∈ (A ∪ K), x is either in U or is defeated or inhibited by U. As a consequence,

(S ∪ Q ∪ RAF-De f (U) ∪ RAF-Inh(U)) is maximal w.r.t. to inclusion.
We prove so that every stable structure is a semi-stable structure.

2. (Semi-stable structures are preferred ones). Let RAF = ⟨A, K, s,t⟩ be a RAF and U = ⟨S, Q⟩ be a semi-stable structure. Let suppose that U is not a preferred structure. U being by definition a complete structure (Definition 71 on page 103), there exists thus a preferred structure U ′ = ⟨S ′ , Q ′ ⟩ such that U ⊏ U ′ . We have thus by definition of ⊑-inclusion:

S ⊆ S ′ and Q ⊆ Q ′ (18.47)
From the strict inclusion, we also have:

(S ∪ Q) ⊂ (S ′ ∪ Q ′ ) (18.48)
It follows, from Equations (18.47) and (18.48) and from Definition 61 on page 92 that:

(RAF-De f (U) ∪ RAF-Inh(U)) ⊂ (RAF-De f (U ′ ) ∪ RAF-Inh(U ′ )) (18.49)
Combining Equations (18.48) and (18.49), we have:

(S ∪ Q ∪ RAF-De f (U) ∪ RAF-Inh(U)) ⊂ (S ′ ∪ Q ′ ∪ RAF-De f (U ′ ) ∪ RAF-Inh(U ′ )) (18.50)
Given that U ′ is also a complete structure, the consequence of Equation (18.50) is that U is not a semi-stable structure, as (S ∪ Q ∪ RAF-De f (U) ∪ RAF-Inh(U)) is not maximal. There is thus a contradiction.

230

We prove so that every semi-stable structure is also a preferred structure. According to Theorem 6 on page 103, U is also a semi-stable structure. As any semi-stable structure

■

U ′ = ⟨S ′ , Q ′ ⟩ maximizes the set (S ′ ∪ RAF-De f (U ′ ) ∪ Q ′ ∪ RAF-Inh(U ′ ))
and as there exists U, a structure such that (S ∪ RAF-De f (U) ∪ Q ∪ RAF-Inh(U)) is maximized to point that it includes all the arguments and attacks of RAF , then for U ′ to be maximal we necessarily have

(S ′ ∪ RAF-De f (U ′ ) ∪ Q ′ ∪ RAF-Inh(U ′ ))
also including all the arguments and attacks of RAF . U ′ is then a stable structure.

We prove thus that if there exists a stable structure, then the semi-stable structures coincide with the stable structures.

■

Proof of Proposition 23 on page 103. Let RAF = ⟨A, K, s,t⟩ be a RAF such that ∀α ∈ K, t(α) ∈ A. RAF can thus be considered as a simple AF. Let AF = ⟨A, K⟩ be the AF version of RAF .

Step 1: Let prove in a first place that if U = ⟨S, K⟩ is a semi-stable structure of RAF then S is a semistable extension of AF .

Let U = ⟨S, K⟩ be a semi-stable structure over RAF . Notice that the set of attacks of U is K as attacks are always valid in RAF , and so that RAF-Inh(U) = ∅. Let suppose that S is not a semi-stable extension of AF . There exists thus an extension S ′ of AF such that:

(S ∪ RAF-De f (S)) ⊂ (S ′ ∪ RAF-De f (S ′ )) (18.51)
We have thus :

(S ∪ RAF-De f (S) ∪ K) ⊂ (S ′ ∪ RAF-De f (S ′ ) ∪ K) (18.52)
Let U ′ = ⟨S ′ , K⟩ be the structure over RAF whose set of arguments is the extension S ′ . For the same reason as U, the set of attacks of U ′ is K and RAF-Inh(U ′ ) = ∅.

As RAF-Inh(U) = ∅ and RAF-Inh(U ′ ) = ∅, we can thus say from Equation (18.52) that:

(S ∪ RAF-De f (S) ∪ K ∪ RAF-Inh(U)) ⊂ (S ′ ∪ RAF-De f (S ′ ) ∪ K ∪ RAF-Inh(U ′ )) (18.53)
Given that all attacks are valid in RAF , we have: RAF-De f (S) = RAF-De f (U) and RAF-De f (S ′ ) = RAF-De f (U ′ ). We have thus from Equation (18.53):

(S ∪ RAF-De f (U) ∪ K ∪ RAF-Inh(U)) ⊂ (S ′ ∪ RAF-De f (U ′ ) ∪ K ∪ RAF-Inh(U ′ )) (18.54)
As stated by Equation (18.54), (S ∪ RAF-De f (S) ∪ K ∪ RAF-Inh(U)) is not maximal. It follows that U is not a semi-stable structure, which is a contradiction.

We prove so that if U = ⟨S, K⟩ is a semi-stable structure of RAF then S is a semi-stable extension of AF .

Step 2: Let now prove that if S is a semi-stable extension of AF then U = ⟨S, K⟩ is a semi-stable structure of RAF .

Let S be a semi-stable extension of AF and let U = ⟨S, K⟩ be a structure over RAF whose set of arguments is S. Notice that the set of attacks of U is K as attacks are always valid in RAF .

Let suppose that U is not a semi-stable structure. There exists thus a semi-stable structure U ′ = ⟨S ′ , K⟩ such that:

(S ∪ RAF-De f (U) ∪ K ∪ RAF-Inh(U)) ⊂ (S ′ ∪ RAF-De f (U ′ ) ∪ K ∪ RAF-Inh(U ′ )) (18.55)
Given that all attacks are valid in RAF , we have: RAF-Inh(U) = ∅ and RAF-Inh(U ′ ) = ∅. We have thus from Equation (18.55):

(S ∪ RAF-De f (U)) ⊂ (S ′ ∪ RAF-De f (U ′ )) (18.56)
Furthermore, as all attacks are valid in RAF , we have: RAF-De f (S) = RAF-De f (U) and RAF-De f (S ′ ) = RAF-De f (U ′ ). We have thus from Equation (18.56):

(S ∪ RAF-De f (S)) ⊂ (S ′ ∪ RAF-De f (S ′ )) (18.57)
As stated by Equation (18.57), (S ∪ RAF-De f (S)) is not maximal. It follows that S is not a semi-stable extension, which is a contradiction.

We prove so that if S is a semi-stable extension of AF then U = ⟨S, K⟩ is a semi-stable structure of RAF .

With steps 1 and 2, we have thus proven that: Step 1:

U = ⟨S, K⟩ is a semi-stable structure of RAF iff S is a semi-stable
(S ∪ Q) ⊆ Acc(U) Let x ∈ (S ∪ Q).
By definition of Lab2Struct(L), we have L(x) = in . Given that L is a reinstatement RAF labelling, we have: As a consequence and following Definition 62 on page 92, we have: x ∈ Acc(U).

∀α ∈ K s.t. t(α) = x, ℓ K (α) = out or ℓ A (s(α)) = out
We prove so that:

(S ∪ Q) ⊆ Acc(U) (18.59)
Step 2:

(S ∪ Q) ⊇ Acc(U)
Let x ∈ Acc(U), x being an argument or an attack. According to Definition 62 on page 92, for all α ∈ K such that t(α) = x, we have: s(α) ∈ RAF-De f (U) or α ∈ RAF-Inh(U).

Let y be s(α) or α. Given that y ∈ (RAF-De f (U) ∪ RAF-Inh(U)), there exists an attack β such that s(β ) ∈ S, β ∈ Q and t(β ) = y. By definition of Lab2Struct(L), we have ℓ A (s(β )) = in and ℓ K (β ) = in .

Given L is a reinstatement RAF labelling, we have L(y) = out . As a consequence, we have:

∀α ∈ K s.t. t(α) = x, ℓ A (s(α)) = out or ℓ K (α) = out (18.60)
Then, given that L is a reinstatement RAF labelling, we have : L(x) = in . By definition of Lab2Struct(L) we have so x ∈ (S ∪ Q).

We prove so that :

(S ∪ Q) ⊇ Acc(U) (18.61)
Finally, because of Equations (18.59) and ( 18.61) we have:

(S ∪ Q) = Acc(U) (18.62)
We prove thus that Lab2Struct(L) is a complete structure. ■ Proof of Theorem 9 on page 108. Let L = Struct2Lab(U). In order to prove that L is a reinstatement RAF labelling (with L = ℓ A , ℓ K ) we have to prove that, for all x ∈ (A ∪ K): We prove so that for all x ∈ (A ∪ K):

(L(x) = out ) =⇒ (∃α ∈ K s.t. t(α) = x, ℓ K (α) = in and ℓ A (s(α)) = in ) (18.63)
Step 2:

(L(x) = out ) ⇐= (∃α ∈ K s.t. t(α) = x, ℓ K (α) = in and ℓ A (s(α)) = in ) Let x ∈ (A ∪ K)
be an argument or an attack. If there exists an attack α ∈ K such that t(α) = x, ℓ K (α) = in and ℓ A (s(α)) = in , then according to the definition of Struct2Lab(U), we have α ∈ Q and s(α) ∈ S.

As a consequence, we have x ∈ (RAF-De f (U) ∪ RAF-Inh(U)). We have thus, according to the definition of Struct2Lab(U): L(x) = out .

We prove so that for all x ∈ (A ∪ K):

(L(x) = out ) ⇐= (∃α ∈ K s.t. t(α) = x, ℓ K (α) = in and ℓ A (s(α)) = in ) (18.64)
Step 3:

(L(x) = in ) =⇒ (∀α ∈ K s.t. t(α) = x, ℓ K (α) = out or ℓ A (s(α)) = out )
Let x ∈ (A ∪ K) be an argument or an attack such that L(x) = in . According to the definition of Struct2Lab(U), we have then x ∈ U and as U is a complete structure we have x ∈ Acc(U). As a consequence, for all α ∈ K such that t(α) = x, we have: s(α) ∈ RAF-De f (U) or α ∈ RAF-Inh(U). According to the definition of Struct2Lab(U), we have then: ℓ A (s(α)) = out or ℓ K (α) = out .

We prove so that for all x ∈ (A ∪ K):

(L(x) = in ) =⇒ (∀α ∈ K s.t. t(α) = x, ℓ K (α) = out or ℓ A (s(α)) = out ) (18.65)
Step 4:

(L(x) = in ) ⇐= (∀α ∈ K s.t. t(α) = x, ℓ K (α) = out or ℓ A (s(α)) = out ) Let x ∈ (A ∪ K)
be an argument or an attack such that for all attacks α ∈ K s.t. t(α) = x, ℓ K (α) = out or ℓ A (s(α)) = out . For any such attack α, we have then, according to the definition of Struct2Lab(U): α ∈ RAF-Inh(U) or s(α) ∈ RAF-De f (U). As a consequence, we have x ∈ Acc(U) and so x ∈ (S ∪ Q), U being a complete structure. According to the definition of Struct2Lab(U), we have then: L(x) = in .

We prove so that for all x ∈ (A ∪ K):

(L(x) = in ) ⇐= (∀α ∈ K s.t. t(α) = x, ℓ K (α) = out or ℓ A (s(α)) = out ) (18.66)
Equations (18.63) Let prove that out (L) ⊂ out (L ′ ), and so that :

1. ∀y ∈ out (L), y ∈ out (L ′ ) 2. ∃z ∈ out (L ′ ), z / ∈ out (L)
Step 1: ∀y ∈ out (L), y ∈ out (L ′ )

Let y be an attack or an argument such that y ∈ out (L). Given L is a reinstatement RAF labelling, we have by definition:

(L(y) = out ) =⇒ (∃α ∈ K s.t. t(α) = y, ℓ K (α) = in and ℓ A (s(α)) = in )
Then according to Equation (18.67) on the previous page, α ∈ in (L ′ ) and s(α) ∈ in (L ′ ). As L ′ is also a reinstatement RAF labelling, we have so y ∈ out (L ′ ).

Step 2: ∃z ∈ out (L ′ ), z / ∈ out (L) Let x be an attack or an argument such that x ∈ in (L ′ ) and x / ∈ in (L). Given L and L ′ are reinstatement RAF labellings, we have by definition:

(L ′ (x) = in ) ⇐⇒ (∀α ∈ K s.t. t(α) = x, ℓ K ′ (α) = out or ℓ A ′ (s(α)) = out ) (18.69) (L(x) ̸ = in ) ⇐⇒ (∃α ∈ K s.t. t(α) = x, ℓ K (α) ̸ = out and ℓ A (s(α)) ̸ = out ) (18.70)
Let α be such an attack with t(α) = x, ℓ K (α) ̸ = out and ℓ A (s(α)) ̸ = out .

By definition of α we have, α / ∈ out (L) and s(α) / ∈ out (L). Furthermore, given that L ′ (x) = in , we have following Equation (18.69), α ∈ out (L ′ ) or s(α) ∈ out (L ′ ).

We prove thus that there exists z such that, z ∈ out (L ′ ) and z / ∈ out (L). ■ Lemma 2. Let L and L ′ be two reinstatement RAF labellings. If out(L) ⊂ out(L ′ ) then in(L) ⊂ in(L ′ ).

Proof of Lemma 2. Let L and L ′ be two reinstatement RAF labellings such that out (L) ⊂ out (L ′ ), meaning that:

∀w ∈ out (L), w ∈ out (L ′ ) (18.71) and ∃x ∈ out (L ′ ), x / ∈ out (L) (18.72)
Let prove that in (L) ⊂ in (L ′ ), and so that :

1. ∀y ∈ in (L), y ∈ in (L ′ ) 2. ∃z ∈ in (L ′ ), z / ∈ in (L)
Step 1: ∀y ∈ in (L), y ∈ in (L ′ )

Let y be an attack or an argument such that y ∈ in (L). Given L is a reinstatement RAF labelling, we have by definition:

(L(y) = in ) =⇒ (∀α ∈ K s.t. t(α) = y, ℓ K (α) = out or ℓ A (s(α)) = out )
Then according to Equation (18.71) on the previous page, we have:

(L(y) = in ) =⇒ (∀α ∈ K s.t. t(α) = y, ℓ K ′ (α) = out or ℓ A ′ (s(α)) = out )
As L ′ is also a reinstatement RAF labelling, we have then y ∈ in (L ′ ).

Step 2: ∃z ∈ in (L ′ ), z / ∈ in (L) Let x be an attack or an argument such that x ∈ out (L ′ ) and x / ∈ out (L). Given L and L ′ are reinstatement RAF labellings, we have by definition:

(L ′ (x) = out ) ⇐⇒ (∃α ∈ K s.t. t(α) = x, ℓ K ′ (α) = in and ℓ A ′ (s(α)) = in ) (18.73) (L(x) ̸ = out ) ⇐⇒ (∀α ∈ K s.t. t(α) = x, ℓ K (α) ̸ = in or ℓ A (s(α)) ̸ = in ) (18.74)
According to Equation (18.74), for any attack α such that t(α) = x, we have: α / ∈ in (L) or s(α) / ∈ in (L). However, we have following Equation (18.73), there exists at least one attack α ∈ K s.t. α ∈ in (L ′ ) and s(α) ∈ in (L ′ ).

We prove thus that there exists z such that, z ∈ in (L ′ ) and z / ∈ in (L). ■

Proof of Theorem 12 on page 108. Let L be a reinstatement RAF labelling such that out (L) is maximal. Let suppose that Lab2Struct(L) is not a preferred structure. Then according to Theorem 10 on page 108, in (L) is not maximal. There exists thus a reinstatement RAF labelling L ′ such that in (L) ⊂ in (L ′ ). We have then, following Lemma 1 on page 234, out (L) ⊂ out (L ′ ), which is a contradiction. ■ Proof of Theorem 13 on page 108. Let U be a preferred structure. According to Theorem 11 on page 108, L = Struct2Lab(U) is a reinstatement RAF labelling such that in (L) is maximal. Let suppose that out (L) is not maximal. There exist thus a reinstatement RAF labelling L ′ such that out (L) ⊂ out (L ′ ).

We have then, following Lemma 2 on the previous page, in (L) ⊂ in (L ′ ), which is a contradiction. ■

Proofs of Section 11.3: Stable semantics

Proof of Theorem 14 on page 109. Let L = ℓ A , ℓ K be a reinstatement RAF labelling such that und (L) = ∅. Let U = Lab2Struct(L). Let x be any attack or argument such that x / ∈ U. Given that und (L) = ∅, we have according to Definition 75 on page 107: L(x) = out . There exists then an attack α such that: ℓ A (s(α)) = in ∩ ℓ K (α) = in . We have then s(α) ∈ U and α ∈ U.

Therefore, according to Definition 61 on page 92, we have: x ∈ RAF-De f (U) or x ∈ RAF-Inh(U), following the nature of x. This means that U defeats or inhibits any argument and attack which is not in it.

We prove so that U is, thus, a stable structure. ■

Proof of Theorem 15 on page 109. Let U be a stable structure and let x be an argument or an attack. If x ∈ U then L(x) = in . Let consider the case when x / ∈ U. Given U is a stable structure then there exists an attack α in U that defeats or inhibits x. We have then, according to Definition 75 on page 107: L(x) = out .

In both cases L(x) ̸ = und . We prove so that und (L) = ∅. ■

Proofs of Section 11.5: Semi-stable semantics

Proof of Theorem 22 on page 110. Let RAF = ⟨A, K, s,t⟩ be a RAF. Let L = ℓ A , ℓ K be a reinstatement RAF labelling such that und (L) is minimal. Let suppose that U = Lab2Struct(L) is not a semi-stable structure (with U = ⟨S, Q⟩). There exists thus a semi-stable structure U ′ = ⟨S ′ , Q ′ ⟩ such that:

(S ∪ Q ∪ RAF-De f (U) ∪ RAF-Inh(U)) ⊂ (S ′ ∪ Q ′ ∪ RAF-De f (U ′ ) ∪ RAF-Inh(U ′ )) (18.75)
As a consequence, we have:

(A ∪ K) \ (S ∪ Q ∪ RAF-De f (U) ∪ RAF-Inh(U)) ⊃ (A ∪ K) \ (S ′ ∪ Q ′ ∪ RAF-De f (U ′ ) ∪ RAF-Inh(U ′ )) (18.76) Let L ′ = Struct2Lab(U ′ ).
Following Equation (18.76), we have according to the definition of Struct2Lab:

und (L) ⊃ und (L ′ )
Then und (L) is not minimal, which is a contradiction. We prove so that U = Lab2Struct(L) is a semi-stable structure.

■

Proof of Theorem 23 on page 110. Let U = ⟨S, Q⟩ be a semi-stable structure. By definition, we have thus:

(S ∪ Q ∪ RAF-De f (U) ∪ RAF-Inh(U)) being maximal As a consequence, (A ∪ K) \ (S ∪ Q ∪ RAF-De f (U) ∪ RAF-Inh(U)) is minimal. Let L = Struct2Lab(U).
According to the definition of Struct2Lab and following the previous statement, we have thus und (L) being minimal. We prove so that L = Struct2Lab(U) is a reinstatement RAF labelling such that und (L) is minimal.

■

Proofs of Chapter 12: Flattening

Proof of Proposition 24 on page 117. Assertion 1:

RAF-De f (U) ∪ RAF-Inh(U) = De f (ε U ) \         {¬a ∈ Not A |a ∈ ε U } ∪{¬β ∈ Not K |β ∈ ε U } ∪{s(β ).β ∈ And A,K |β ∈ De f (ε U ) or s(β ) ∈ De f (ε U )}         • Step 1: RAF-De f (U) ∪ RAF-Inh(U) ⊆ De f (ε U ) \         {¬a ∈ Not A |a ∈ ε U } ∪{¬β ∈ Not K |β ∈ ε U } ∪{s(β ).β ∈ And A,K |β ∈ De f (ε U ) or s(β ) ∈ De f (ε U )}         Let x ∈ RAF-De f (U) ∪ RAF-Inh(U).
There exists thus an attack α ∈ Q such that s(α) ∈ S and t(α) = x. We have thus:

α ∈ ε U and s(α) ∈ ε U
As a consequence, following the definition of Raf2Af, we have:

¬α ∈ De f (ε U ) and ¬s(α) ∈ De f (ε U )
And so: s(α).α ∈ ε U Given that s(α).α ∈ ε U , we have so:

x ∈ De f (ε U )

Given that x ∈ A ∪ K, we have, following the definition of Raf2Af: x / ∈ (Not A ∪ Not K ∪ And A,K ). As a consequence, we have:

x

∈ De f (ε U ) \ (Not A ∪ Not K ∪ And A,K )
We prove so that: RAF

-De f (U) ∪ RAF-Inh(U) ⊆ De f (ε U ) \         {¬a ∈ Not A |a ∈ ε U } ∪{¬β ∈ Not K |β ∈ ε U } ∪{s(β ).β ∈ And A,K |β ∈ De f (ε U ) or s(β ) ∈ De f (ε U )}         • Step 2: RAF-De f (U) ∪ RAF-Inh(U) ⊇ De f (ε U ) \         {¬a ∈ Not A |a ∈ ε U } ∪{¬β ∈ Not K |β ∈ ε U } ∪{s(β ).β ∈ And A,K |β ∈ De f (ε U ) or s(β ) ∈ De f (ε U )}         Let x ∈ De f (ε U ) \         {¬a ∈ Not A |a ∈ ε U } ∪{¬β ∈ Not K |β ∈ ε U } ∪{s(β ).β ∈ And A,K |β ∈ De f (ε U ) or s(β ) ∈ De f (ε U )}         .
Let consider four cases: x ∈ Not A , x ∈ Not K , x ∈ And A,K and x ∈ A ∪ K. Let show that the three first cases are impossible.

-Let suppose that x ∈ Not A with x = ¬b. Given that ¬b ∈ De f (ε U ), according to the definition of Raf2Af, we have b ∈ ε U . As a consequence, we have:

x ∈ {¬a ∈ Not A |a ∈ ε U }, which is a contradiction.
-Let suppose that x ∈ Not K with x = ¬α. Given that ¬α ∈ De f (ε U ), according to the definition of Raf2Af, we have α ∈ ε U . As a consequence, we have:

x ∈ {¬β ∈ Not K |β ∈ ε U }, which is a contradiction.
-Let suppose that x ∈ And A,K with x = s(α).α. Given that s(α).α ∈ De f (ε U ), according to the definition of Raf2Af, we have thus: ¬s(α) ∈ ε U or ¬α ∈ ε U . And so we have:

s(α) ∈ De f (ε U ) or α ∈ De f (ε U ). As a consequence, we have: x ∈ {s(β ).β ∈ And A,K |β ∈ De f (ε U ) or s(β ) ∈ De f (ε U )}, which is a contradiction.
As a consequence, following the definition of Raf2Af, we have: ¬α ∈ ε U or ¬s(α) ∈ ε U Furthermore, since s(α).α ∈ And A,K , we have:

s(α).α ∈ De f (ε U ) ∩ And A,K
As a consequence, as it is the case of any attack α attacking x, we have:

x ∈ Acc(ε U )

Given that x ∈ A ∪ K, we have, following the definition of Raf2Af: x / ∈ (Not A ∪ Not K ∪ And A,K ). As a consequence, we have:

x

∈ Acc(ε U ) \ (Not A ∪ Not K ∪ And A,K )
We prove so that:

RAF-Acc(U) ⊆ Acc(ε U ) \         {¬a ∈ Not A |a ∈ De f (ε U )} ∪{¬β ∈ Not K |β ∈ De f (ε U )} ∪{s(β ).β ∈ And A,K |s(β ).β ∈ ε U }         • Step 2: RAF-Acc(U) ⊇ Acc(ε U ) \         {¬a ∈ Not A |a ∈ De f (ε U )} ∪{¬β ∈ Not K |β ∈ De f (ε U )} ∪{s(β ).β ∈ And A,K |s(β ).β ∈ ε U }         Let x ∈ Acc(ε U ) \         {¬a ∈ Not A |a ∈ De f (ε U )} ∪{¬β ∈ Not K |β ∈ De f (ε U )} ∪{s(β ).β ∈ And A,K |s(β ).β ∈ ε U }        
Let consider four cases: x ∈ Not A , x ∈ Not K , x ∈ And A,K and x ∈ A ∪ K. Let show that the three first cases are impossible.

-Let suppose that x ∈ Not A with x = ¬b. Given that ¬b ∈ Acc(ε U ), according to the definition of Raf2Af, we have b ∈ De f (ε U ). As a consequence, we have:

x ∈ {¬a ∈ Not A |a ∈ De f (ε U )},
which is a contradiction.

-Let suppose that x ∈ Not K with x = ¬α. Given that ¬α ∈ Acc(ε U ), according to the definition of Raf2Af, we have α ∈ De f (ε U ). As a consequence, we have:

x ∈ {¬β ∈ Not K |β ∈ De f (ε U )},
which is a contradiction.

-Let suppose that x ∈ And A,K with x = s(α).α. Given that s(α).α ∈ Acc(ε U ), according to the definition of Raf2Af, we have thus: ¬s(α) ∈ De f (ε U ) and ¬α ∈ De f (ε U ). And so we have: s(α) ∈ ε U and α ∈ ε U . As a consequence, we have: x ∈ {s(β ).β ∈ And A,K |s(β ).β ∈ ε U }, which is a contradiction.

We prove so that: x ∈ A ∪ K.

Given that x ∈ Acc(ε U ), then following the definition of Raf2Af, for any argument s(α).α attacking x, we have: s(α).α ∈ De f (ε U ). As a consequence, following the definition of Raf2Af, we have:

¬s(α) ∈ ε U or ¬α ∈ ε U
And so:

s(α) ∈ De f (ε U ) or α ∈ De f (ε U )
As Assertion 1 holds and as s(α) ∈ A and α ∈ K, we have:

s(α) ∈ RAF-De f (U) or α ∈ RAF-Inh(U)
as it is the case of any attack α attacking x, we have:

x ∈ RAF-Acc(U)
We prove so that:

RAF-Acc(U) ⊇ Acc(ε U ) \         {¬a ∈ Not A |a ∈ De f (ε U )} ∪{¬β ∈ Not K |β ∈ De f (ε U )} ∪{s(β ).β ∈ And A,K |s(β ).β ∈ ε U }         ■ Proof of Proposition 25 on page 117. Assertion 1: U = ⟨S, Q⟩ is a RAF-complete structure in RAF iff ε U is a complete extension in AF . U = ⟨S, Q⟩ is a RAF-complete structure in RAF iff (S ∪ Q) = RAF-Acc(U).
Following Proposition 24 on page 117, we have thus:

U = ⟨S, Q⟩ is a RAF-complete structure in RAF iff (S ∪ Q) = Acc(ε U ) \         {¬a ∈ Not A |a ∈ De f (ε U )} ∪{¬β ∈ Not K |β ∈ De f (ε U )} ∪{s(β ).β ∈ And A,K |s(β ).β ∈ ε U }        
And so:

U = ⟨S, Q⟩ is a RAF-complete structure in RAF iff (S ∪ Q) ∪         {¬a ∈ Not A |a ∈ De f (ε U )} ∪{¬β ∈ Not K |β ∈ De f (ε U )} ∪{s(β ).β ∈ And A,K |s(β ).β ∈ ε U }         = Acc(ε U ) (18.77)
Given that, for all s(β ).β ∈ And A,K such that s(β ).β ∈ ε U we have following the definition of ε U : β ∈ Q, s(β ) ∈ S, from Equation (18.77), we have then:

U = ⟨S, Q⟩ is a RAF-complete structure in RAF iff (S ∪ Q) ∪         {¬a ∈ Not A |a ∈ De f (ε U )} ∪{¬β ∈ Not K |β ∈ De f (ε U )} ∪{s(β ).β ∈ And A,K |β ∈ Q, s(β ) ∈ S}         = Acc(ε U ) (18.78)
Following the definition of Raf2Af:

¬a ∈ Not A iff a ∈ A and ¬β ∈ Not K iff β ∈ K (18.79)
Furthermore, following Proposition 24 on page 117, we have:

De f (ε U ) =         RAF-De f (U) ∪ RAF-Inh(U) ∪{¬a ∈ Not A |a ∈ ε U } ∪ {¬β ∈ Not K |β ∈ ε U } ∪{s(β ).β ∈ And A,K |β ∈ De f (ε U ) or s(β ) ∈ De f (ε U )}         (18.80)
From Equations (18.79) and ( 18.80) we have so:

a ∈ De f (ε U ) iff a ∈ RAF-De f (U) and β ∈ De f (ε U ) iff β ∈ RAF-Inh(U) (18.81)
As a consequence, we have from Equations (18.78) and (18.81) on the previous page:

U = ⟨S, Q⟩ is a RAF-complete structure in RAF iff (S ∪ Q) ∪         {¬a ∈ Not A |a ∈ RAF-De f (U)} ∪{¬β ∈ Not K |β ∈ RAF-Inh(U)} ∪{s(β ).β ∈ And A,K |β ∈ Q, s(β ) ∈ S}         = Acc(ε U )
Following the definition of ε U , we have thus:

U = ⟨S, Q⟩ is a RAF-complete structure in RAF iff ε U = Acc(ε U ) We prove so that: U = ⟨S, Q⟩ is a RAF-complete structure in RAF iff ε U is a complete extension in AF Assertion 2: U = ⟨S, Q⟩ is a RAF-grounded structure in RAF iff ε U is a grounded extension in AF . ε U is a grounded extension in AF iff there is no complete extension ε U ′ in AF (with U ′ = ⟨S ′ , Q ′ ⟩) such that: ε U ′ ⊂ ε U .
We have so:

ε U ∈ σ gr (AF ) iff ∄ε U ′ ∈ σ co (AF ) s.t. Acc(ε U ′ ) ⊂ Acc(ε U ) thus: ε U ∈ σ gr (AF ) iff ∄ε U ′ ∈ σ co (AF ) s.t. U ′ ⊆ U Given that ε U ′ ̸ = ε U iff U ′ ̸ = U, we have thus: ε U ∈ σ gr (AF ) iff ∄ε U ′ ∈ σ co (AF ) s.t. U ′ ⊂ U
We prove so that ε U is a grounded extension in AF iff U = ⟨S, Q⟩ is a RAF-grounded structure in RAF .

Assertion 3: U = ⟨S, Q⟩ is a RAF-preferred structure in RAF iff ε U is a preferred extension in AF . ε U is a preferred extension in AF iff there is no complete extension ε U ′ in AF (with U ′ = ⟨S ′ , Q ′ ⟩) such that: ε U ⊂ ε U ′ .
We have so:

ε U ∈ σ pr (AF ) iff ∄ε U ′ ∈ σ co (AF ) s.t. Acc(ε U ) ⊂ Acc(ε U ′ )
Following Proposition 24 on page 117, we have:

ε U ∈ σ pr (AF ) iff ∄ε U ′ ∈ σ co (AF ) s.t. RAF-Acc(U) ∪         {¬a ∈ Not A |a ∈ De f (ε U )} ∪{¬β ∈ Not K |β ∈ De f (ε U )} ∪{s(β ).β ∈ And A,K |s(β ).β ∈ ε U }         ⊂ RAF-Acc(U ′ ) ∪         {¬a ∈ Not A |a ∈ De f (ε U ′ )} ∪{¬β ∈ Not K |β ∈ De f (ε U ′ )} ∪{s(β ).β ∈ And A,K |s(β ).β ∈ ε U ′ }        
We have thus:

ε U ∈ σ pr (AF ) iff ∄ε U ′ ∈ σ co (AF ) s.t. RAF-Acc(U) ⊆ RAF-Acc(U ′ ) Given that, following Assertion 1, ε U ′ and ε U are complete iff U ′ and U are RAF-complete, we have thus: ε U ∈ σ pr (AF ) iff ∄ε U ′ ∈ σ co (AF ) s.t. U ⊆ U ′ Given that ε U ′ ̸ = ε U iff U ′ ̸ = U, we have thus: ε U ∈ σ pr (AF ) iff ∄ε U ′ ∈ σ co (AF ) s.t. U ⊂ U ′ We prove so that ε U is a preferred extension in AF iff U = ⟨S, Q⟩ is a RAF-preferred structure in RAF . Assertion 4: U = ⟨S, Q⟩ is a RAF-stable structure in RAF iff ε U is a stable extension in AF . • Step 1: If U = ⟨S, Q⟩ is a RAF-stable structure in RAF then ε U is a stable extension in AF . If U = ⟨S, Q⟩ is a RAF-stable structure in RAF then ∄x ∈ (A∪K) such that x / ∈ U and x / ∈ (RAF-De f (U)∪ RAF-Inh(U)). If U is a RAF-stable structure in RAF then U is also RAF-complete. Following As- sertion 1, ε U is thus a complete extension in AF . Let suppose that ε U is not stable. There exists thus x ∈ (A ′ ∪ K ′ ) such that x / ∈ ε U and x / ∈ De f (ε U ).
Let consider two cases: x ∈ A ∪ K and x / ∈ (A ∪ K).

Case 1: Given that U is RAF-stable, if x ∈ A ∪ K, we have so:

x ∈ (S ∪ Q ∪ RAF-De f (U) ∪ RAF-Inh(U)).
As shown in Proof of Assertion 1 (Equation (18.81) on page 243):

a ∈ A ∩ De f (ε U ) iff a ∈ RAF-De f (U) and β ∈ K ∩ De f (ε U ) iff β ∈ RAF-Inh(U) (18.82) As a consequence if x ∈ A ∪ K then: x ∈ (S ∪ Q ∪ (A ∩ De f (ε U )) ∪ (K ∩ De f (ε U ))) As x / ∈ ε U and x / ∈ De f (ε U ) there is a contradiction. Case 2: If x / ∈ (A ∪ K) then: x ∈ (Not A ∪ Not K ∪ And A,K ). Given that: x / ∈ ε U and x / ∈ De f (ε U )
We have thus three possible cases:

-x ∈ Not A \ {¬a ∈ Not A |a ∈ (De f (ε U ) ∪ ε U )} -x ∈ Not K \ {¬β ∈ Not K |β ∈ (De f (ε U ) ∪ ε U )} -x ∈ And A,K \ {s(β ).β ∈ And A,K |β ∈ (De f (ε U ) ∪ ε U )}
Given that following the definition of Raf2Af:

¬a ∈ Not A iff a ∈ A and ¬β ∈ Not K iff β ∈ K and s(β ).β ∈ And A,K iff β ∈ K (18.83)
We have thus:

-x ∈ Not A \ {¬a ∈ Not A |a ∈ (A ∩ De f (ε U )) ∪ (A ∩ ε U )} -x ∈ Not K \ {¬β ∈ Not K |β ∈ (K ∩ De f (ε U )) ∪ (K ∩ ε U )} -x ∈ And A,K \ {s(β ).β ∈ And A,K |β ∈ (K ∩ De f (ε U )) ∪ (K ∩ ε U )}
As shown in Proof of Assertion 1 (Equation (18.81) on page 243):

A ∩ De f (ε U ) = RAF-De f (U) and K ∩ De f (ε U ) = RAF-Inh(U) (18.84)
We have thus:

thus:

ε U ∈ σ sst (AF ) iff ∄ε U ′ ∈ σ co (AF ) s.t. U ∪     RAF-De f (U) ∪RAF-Inh(U)     ⊆ U ′ ∪     RAF-De f (U ′ ) ∪RAF-Inh(U ′ )     Given that ε U ′ ̸ = ε U iff U ′ ̸ = U, we have thus: ε U ∈ σ sst (AF ) iff ∄ε U ′ ∈ σ co (AF ) s.t. U ∪     RAF-De f (U) ∪RAF-Inh(U)     ⊂ U ′ ∪     RAF-De f (U ′ ) ∪RAF-Inh(U ′ )    
We prove so that ε U is a semi-stable extension in AF iff U = ⟨S, Q⟩ is a RAF-semi-stable structure in RAF . ■

Proofs of Chapter 13: Complexity

Proof of Proposition 28 on page 121.

RAF-Ver

σ accepts (RAF , U) iff U ∈ σ (RAF ) iff ε U ∈ σ (AF ) (
• RAF-Cred σ ≤ Raf2Af L Cred σ • RAF-Skep σ ≤ Raf2Af L Skep σ • RAF-Ver σ ≤ Raf2Af L Ver σ • RAF-Exists σ ≤ Raf2Af L Exists σ • RAF-Exists ¬∅ σ ≤ Raf2Af L Exists ¬∅ σ • RAF-Unique σ ≤ Raf2Af L Unique σ ■ Proof of
• Cred σ ≤ Af2Raf L RAF-Cred σ • Skep σ ≤ Af2Raf L RAF-Skep σ • Ver σ ≤ Af2Raf L RAF-Ver σ • Exists σ ≤ Af2Raf L RAF-Exists σ • Exists ¬∅ σ ≤ Af2Raf L RAF-Exists ¬∅ σ • Unique σ ≤ Af2Raf L RAF-Unique σ ■
Proof of Proposition 32 on page 122. Given that Raf2Af and Af2Raf are polynomial time procedures and that Propositions 29 and 31 on page 121 and on page 122 holds, then all the complexities are the same. ■

The following lemma states that the source and the target of an attack (whose source is different from its attack) belong to the same SCC ra f iff there exists a RAF-path from the target to the source and whose before last element is an attack: Lemma 3. Let RAF = ⟨A, K, s,t⟩ be a RAF and let α ∈ K. The following proposition holds: If there exists a RAF-path p = (e 1 = t(α), ..., e n = s(α)) ∈ Paths ra f (RAF ) such that e n-1 ∈ K, then (by Definition 82 on page 123) t(α) ̸ = s(α). Furthermore, c = (s(α), α, e 1 = t(α), ..., e n = s(α)) ∈ Cycles ra f (RAF ) and c attacks s(α) and t(α). Following Proposition 34 on page 126, we have so: t(α) ≡ RAF s(α). • According to the definition of Raf2Af (Definition 76 on page 115, see rules Not K , And A,K , K ′ 2 , K ′ 4 and K ′ 5 ), there is a walk in Raf2Af(AF ) from α to t(α) which corresponds to the sequence of arguments: (α, ¬α, s(α).α, t(α)). As t(α) = α then we have: (α, ¬α, s(α).α, α) ∈ Cycles a f (AF ). Furthermore, there exists an unique cycle whose first element is α. Indeed, by definition, an attack only has one target.

• Likewise (see rules Not A , And A,K , K ′ 1 , K ′ 3 and K ′ 5 ), there is a walk in Raf2Af(AF ) from s(α) to t(α) which corresponds the sequence of arguments: (s(α), ¬s(α), s(α).α, t(α)). Given that t(α) ̸ = s(α), this walk is a path. Notice that this path may not be the unique going from s(α) to t(α). Indeed it could exists another attack whose source is s(α) and from which there exists a path to t(α).

Assertion 2: e i ∈ K s.t. t(e i ) = s(e i ) =⇒ ∃c = (s(e i ), ..., s(e i )) ∈ Cycles a f (AF ) and ∃!(e i , ...,t(e i )) ∈ Paths a f (AF )

Let α ∈ K be an attack of RAF such that t(α) = s(α).

• According to the definition of Raf2Af (Definition 76 on page 115, see rules Not K , And A,K , K ′ 2 , K ′ 4 and K ′ 5 ), there is a walk in Raf2Af(AF ) from α to t(α) which corresponds to the sequence of arguments: (α, ¬α, s(α).α, t(α)). Given that t(e i ) ̸ = e i , this walk is a path. Furthermore, this path is the unique going from α to t(α) as, by definition, an attack only has one target.

• Likewise (see rules Not A , And A,K , K ′ 1 , K ′ 3 and K ′ 5 ), there is a walk in Raf2Af(AF ) from s(α) to t(α) which corresponds the sequence of arguments: (s(α), ¬s(α), s(α).α, t(α)). Given that t(α) = s(α), this walk is a cycle. Notice that this cycle may not be the unique cycle whose first element is s(α). Indeed it could exists another attack whose source is s(α) and from which there exists a cycle to s(α).

Assertion 3:

e i ∈ K s.t. t(e i ) ̸ = e i =⇒        
∃!(e i , ...,t(e i )) ∈ Paths a f (AF ) and (∃(s(e i ), ...,t(e i )) ∈ Paths a f (AF ) or ∃(s(e i ), ...,t(e i )) ∈ Cycles a f (AF ))

        Let α ∈ K such that t(α) ̸ = α.
• According to the definition of Raf2Af (Definition 76 on page 115, see rules Not K , And A,K , K ′ 2 , K ′ 4 and K ′ 5 ), there is a walk in Raf2Af(AF ) from α to t(α) which corresponds to the sequence of arguments: (α, ¬α, s(α).α, t(α)). Given that t(α) ̸ = α, this walk is a path. Furthermore, this path is the unique path going from α to t(α) as, by definition, an attack only has one target.

We have thus:

α ∈ K s.t. t(α) ̸ = α =⇒ ∃!(α, ...,t(α)) ∈ Paths a f (AF ) (18.85)
• Likewise (see rules Not A , And A,K , K ′ 1 , K ′ 3 and K ′ 5 ), there is a walk in Raf2Af(AF ) from s(α) to t(α) which corresponds the sequence of arguments: (s(α), ¬s(α), s(α).α, t(α)). If t(α) ̸ = s(α), then this walk is a path. Otherwise it is a cycle. Notice that, in both cases, it may not be the unique path or cycle going from s(α) to t(α). Indeed it could exists another attack whose source is s(α) and from which there exists a path to t(α).

We have thus:

α ∈ K s.t. t(α) ̸ = α =⇒ ∃(s(α), ...,t(α)) ∈ Paths a f (AF ) or ∃(s(α), ...,t(α)) ∈ Cycles a f (AF ) ( 18
        Let α ∈ K such that t(α) ̸ = s(α).
• According to the definition of Raf2Af (Definition 76 on page 115, see rules Not K , And A,K , K ′ 2 , K ′ 4 and K ′ 5 ), there is a walk in Raf2Af(AF ) from α to t(α) which corresponds to the sequence of arguments: (α, ¬α, s(α).α, t(α)). If t(α) ̸ = α, then this walk is a path. Otherwise, it is a cycle. Furthermore, this path or cycle is the unique one going from α to t(α) as, by definition, an attack only has one target.

We have thus:

α ∈ K s.t. t(α) ̸ = α =⇒ ∃!(α, ...,t(α)) ∈ Paths a f (AF ) or ∃!(α, ...,t(α)) ∈ Cycles a f (AF ) (18.87) • Likewise (see rules Not A , And A,K , K ′ 1 , K ′ 3 and K ′ 5 )
, there is a walk in Raf2Af(AF ) from s(α) to t(α) which corresponds the sequence of arguments: (s(α), ¬s(α), s(α).α, t(α)). Given that t(α) ̸ = s(α), this walk is a path. Notice that it may not be the unique path going from s(α) to t(α). Indeed it could exists another attack whose source is s(α) and from which there exists a path to t(α).

We have thus: The following lemma establishes the elementary link between RAF-paths and paths in the AF version of a RAF: Lemma 5. Let (e 1 , ..., e n ) ∈ Paths ra f (RAF ).

α ∈ K s.t. t(α) ̸ = α =⇒ ∃(s(α), ...,t(α)) ∈ Paths a f (AF ) ( 18
1. For i ∈ {1, ..., n -2} (so n ≥ 3), if e i ∈ A then e i+1 ∈ K and then there is a path in Raf2Af(AF ) from e i to e i+2 .

2. For i ∈ {1, ..., n -1} (so n ≥ 2), if e i ∈ K then there is a unique path in Raf2Af(AF ) from e i to e i+1 .

Proof of Lemma 5. Assertion 1: For i ∈ {1, ..., n -2}, if e i ∈ A then e i+1 ∈ K and then there is a path in Raf2Af(AF ) from e i to e i+2 . Given that (e 1 , ..., e n ) ∈ Paths ra f (RAF ) then ∀(i, j) ∈ {1, ..., n} 2 s.t. i ̸ = j, e i ̸ = e j . Furthermore, we have: e i ∈ A =⇒ e i+1 ∈ K. As s(e i+1 ) = e i , we have: t(e i+1 ) ̸ = s(e i+1 ). We also have: t(e i+1 ) ̸ = e i+1 . According to Assertion 5 of Lemma 4 on page 255, we have then: ∃!(e i+1 , ...,t(e i+1 )) ∈ Paths a f (AF ) and ∃(s(e i+1 ), ..., t(e i+1 )) ∈ Paths a f (AF ). As s(e i+1 ) = e i and t(e i+1 ) = e i+2 , we have thus: (e i , ..., e i+2 ) ∈ Paths a f (AF ).

Assertion 2: For i ∈ {1, ..., n -1}, if e i ∈ K then there is a unique path in Raf2Af(AF ) from e i to e i+1 .

Given that (e 1 , ..., e n ) ∈ Paths ra f (RAF ) then ∀(i, j) ∈ {1, ..., n} 2 s.t. i ̸ = j, e i ̸ = e j . Furthermore, we have: e i ∈ K =⇒ t(e i ) = e i+1 . As t(e i ) ̸ = e i , we have following Assertion 3 of Lemma 4 on page 255: ∃!(e i , ..., e i+1 ) ∈ Paths a f (AF ). ■ Following Lemma 5, Lemma 6 deepens the link between RAF-paths and paths in the AF version of a RAF: Lemma 6. Let {e 1 , ..., e n } ∈ Paths ra f (RAF ) with n > 1. If e n-1 ∈ K then there is a path in Raf2Af(RAF ) from e 1 to e n .

Proof of Lemma 6. If e n-1 ∈ K, then according to Lemma 5 there is a path in Raf2Af(RAF ) from e n-1 to e n . Let prove this property for n > 2. Let e j ∈ A be the last argument of the RAF-path such that j ≤ n -2. There are three cases to consider:

• Case 1: If there is no such e j then for i ∈ {1, ..., n -2}, all e i are attacks. According to Lemma 5 (Item 2) there is thus a walk in Raf2Af(RAF ) from e 1 to e n (i.e. the concatenation of the unique RAF-paths from each e i to e i+1 , for i = {1, . . . , n -1}). Furthermore, given that ∀(l, k) ∈ {1, ..., n} 2 s.t. l ̸ = k, e l ̸ = e k , this walk is thus a path.

• Case 2: Else if n = 3 and e j = e 1 , then according to Lemma 5 on the previous page (Item 1) there is thus a path in Raf2Af(RAF ) from e 1 to e 3 .

• Case 3: Else (i.e. n > 3 and there exists such e j ), then according to Lemma 5 on the previous page (Item 1) there is a path in Raf2Af(RAF ) from e j to e j+2 . Furthermore given that for i ∈ { j + 1, ..., n -1}, all e i are attacks and given that ∀(l, k) ∈ {1, ..., n} 2 s.t. l ̸ = k, e l ̸ = e k , there is thus a path in Raf2Af(RAF ) from e j to e n .

If we are in Case 3, then e j-1 ∈ K and according to Lemma 5 on the previous page (Item 2) there is a path in Raf2Af(RAF ) from e j-1 to e j . By replacing n by j, by applying iteratively Case 1, 2 or 3, and by considering that ∀(l, k) ∈ {1, ..., n} 2 s.t. l ̸ = k, e l ̸ = e k , we end up with the conclusion that: if e n-1 ∈ K then there is a path in Raf2Af(RAF ) from e 1 to e n . ■ Note: We cannot apply this lemma for RAF-path of length 2 with an argument as first element (a, α) due to the condition "if e n-1 ∈ K".

Following Lemmas 5 and 6 on the previous page, Lemma 7 establishes the general relation between RAF-paths and paths in the AF version of a RAF with the following equivalence: Lemma 7. Let RAF = ⟨A, K, s,t⟩ be a RAF and let AF = Raf2Af(RAF ) be its corresponding AF. The following property holds: • Case 1: If o 1 ∈ A then according to the definition of Raf2Af, we have:

∃p ′ = (x, ..., y) ∈ Paths a f (AF ) s.t. (x, y) ∈ (A ∪ K) 2
• o 2 ∈ Not A and o 2 = ¬o 1 • o 3 ∈ And A,K and o 3 = o 1 .α with α ∈ K and s(α) = o 1 • o 4 ∈ (A ∪ K) and o 4 = t(α) • (α, ¬α) ∈ K ′ • (¬α, o 1 .α) ∈ K ′ As a consequence, in RAF , we have: α ∈ K s.t. s(α) = o 1 and t(α) = o 4 .
• Case 2: Now, if o 1 ∈ K then according to the definition of Raf2Af, we have:

• o 2 ∈ Not K and o 2 = ¬o 1 • o 3 ∈ And A,K and o 3 = s(o 1 ).o 1 • o 4 ∈ (A ∪ K) and o 4 = t(o 1 ) • (s(o 1 ), ¬s(o 1 )) ∈ K ′ • (¬s(o 1 ), s(o 1 ).o 1 ) ∈ K ′ As a consequence, in RAF , we have: o 1 ∈ K s.t. t(o 1 ) = o 4 .
In both cases o 4 ∈ (A ∪ K) and o 1 ̸ = o 4 and there is a RAF-path in RAF from o 1 to o 4 , that is whether {o 1 , α, o 4 } or {o 1 , o 4 }. Let l be the length of the one or the other path, that is 3 in the first case and 2 in the second one. We have so e l-1 = α or e l-1 = o 1 . In both cases we have: e l-1 ∈ K.

If o 4 = e n then the property holds: there exists a RAF-Path from x to y attacking y. That is, the before last element of this RAF-Path is an attack whose target is y (i.e. e l-1 ).

Otherwise, by replacing o 1 by o 4 in the two cases studied above and by applying them iteratively until o 4 = e n , we can obtain a well formed RAF-path4 p = {x = e 1 , ..., y = e n } from x to y. Furthermore, e n-1 ∈ K.

We prove so that: If |S| > 1 and x ∈ S then there exists x ′ ∈ S such that x ̸ = x ′ . There is thus a path from x to x ′ and a path from x ′ to x. It follows that there is a cycle from x to x. Let c ∈ Cycles a f (AF ) be such a cycle. According to the definition of Raf2Af, for any x ∈ A ∪ K, the only argument attacked by x is ¬x. As a consequence, we have: ¬x ∈ c and so: ¬x ∈ S. x and there exists a cycle c = (x, ..., x) ∈ Cycles a f (AF ) such that s(α).α ∈ c. Given that s(α).α is attacked by only two arguments: ¬α and ¬s(α), and so, defended by: α and s(α), we have: Proof of Proposition 35 on page 129. In two steps:

∃p ′ = (x, ..., y) ∈ Paths a f (AF ) s.t. (x, y) ∈ (A ∪ K) 2 =⇒ ∃p = (x =
• Step 1: If U is included is some S ∈ SCCS ra f (RAF ) then U is included in some S ′ ∈ SCCS a f (Raf2Af(RAF ))
Let S ∈ SCCS ra f (RAF ) and U ⊆ S. If |S| = 1 then trivially x ∈ S belongs to a unique SCC of Raf2Af(RAF ). Otherwise (i.e. if |S| > 1), according to the definition of SCC ra f (Definition 87 on page 125), for any couple (x, y) ∈ S 2 s.t. x ̸ = y there is a RAF-path from x to y in which y is attacked and a RAF-path from y to x in which x is attacked.

Following Lemma 6 on page 257, there is thus a path from x to y and a path from y to x in Raf2Af(RAF ).

As a consequence x and y are in the same SCC of Raf2Af(RAF ). It follows that any element of S belongs to the same SCC.

We prove so that if U is included is some S ∈ SCCS ra f (RAF ) then U is included in some S ′ ∈ SCCS a f (Raf2Af(RAF )). We prove so that U ⊆ S with S being some SCC ra f of RAF .

• Step 2: If U is included is some S ′ ∈ SCCS a f (Raf2Af(RAF )) then U is included in some S ∈ SCCS ra f (RAF ) Let S ′ ∈ SCCS a f (Raf2Af(RAF )) and U ⊆ S ′ . If |S ′ | = 1 then
We prove from Case 1 and Case 2 that Proposition 35 on page 129 holds. ■ Proof of Proposition 36 on page 129. Let RAF = ⟨A, K, s,t⟩ be a RAF, S ⊆ A ∪ K be a subset of elements of RAF and AF = Raf2Af(RAF ) be the corresponding AF of RAF .

Step 1: Let prove that:

S ∈ SCCS ra f (RAF ) =⇒                 S ∪ {¬a ∈ Not A |a ∈ S and (|S| > 1 or (∃α ∈ K s.t. s(α) = a and t(α) = a))} ∪ {¬α ∈ Not K |α ∈ S and (|S| > 1 or t(α) = α)} ∪ {s(α).α ∈ And A,K |α ∈ S and (|S| > 1 or t(α) = α)} ∪ {s(α).α ∈ And A,K |s(α) ∈ S and t(α) ∈ S}                 ∈ SCCS a f (Raf2Af(RAF ))
If S ∈ SCCS ra f (RAF ) then, according to Proposition 35 on page 129, S is included in some S ′ ∈ SCCS a f (AF ). If S ′ ∈ SCCS a f (AF ) then, according to Proposition 35 on page 129, S ′ ∩ (A ∪ K) ⊆ S. We have so: S ′ \ S ⊆ (Not A ∪ Not K ∪ And A,K ).

Let S ∈ SCCS ra f (RAF ) and let S ′ ∈ SCCS a f (AF ) be the SCC such that S ⊆ S ′ and S ′ \ S ⊆ (Not A ∪ Not K ∪ And A,K ).

Let consider the case where

|S| = 1. Let x ∈ S. • Let x ∈ A.
-Let suppose that there exists a RAF-closed-walk p = (x, ..., e n-1 , x) ∈ ClosedWalk ra f (RAF ).

Given that x ∈ A then we have: e n-1 ∈ K and t(e n-1 ) = x, e 2 ∈ K and s(e 2 ) = x. If the length of p is greater than three this means that there exists e i ∈ p for some i ∈ 3, n -1 such that e i-1 ∈ K and so t(e i-1 ) = e i . Given that p is a RAF-closed-walk and that both x and e i are attacked in p then following Proposition 34 on page 126, we have: e i ∈ S, which contradicts: |S| = 1. As a consequence, if such RAF-closed-walk exists it must be the case that its length equals three, as x ∈ A. We have so : p = (x, α, x). There exists thus α ∈ K such that s(α) = x and t(α) = x. As s(α) ∈ S and t(α) ∈ S, we have according to Lemma 9 on page 260: ¬x ∈ S ′ and x.α ∈ S ′ . Furthermore, given that |S| = 1, there is no y ∈ A ∪ K such that x ̸ = y and x ≡ RAF y. As a consequence, we have: S ′ = {x, ¬x, x.α}. We finally have:

S ′ =                 S ∪ {¬a ∈ Not A |a ∈ S and (|S| > 1 or (∃α ∈ K s.t. s(α) = a and t(α) = a))} ∪ {¬α ∈ Not K |α ∈ S and (|S| > 1 or t(α) = α)} ∪ {s(α).α ∈ And A,K |α ∈ S and (|S| > 1 or t(α) = α)} ∪ {s(α).α ∈ And A,K |s(α) ∈ S and t(α) ∈ S}                
(18.90) -If there is no RAF-closed-walk p ∈ ClosedWalk ra f (RAF ) from x to x then, there is no attack α ∈ K such that s(α) = x and t(α) = x. We have thus: S ′ = S = {x} and Equation (18.90) holds.

• Let x ∈ K.

-Let suppose that there exists a RAF-closed-walk p = (x, ..., e n-1 , x) ∈ ClosedWalk ra f (RAF ) from x to x such that e n-1 ∈ K. As x in K, e 2 and x are attacked by p. As a consequence, we have x ≡ RAF e 2 . This equivalence contradicts |S| = 1, except if n = 2 and x = e 2 . As a consequence, we have: p = (x, x). We have so: t(x) = x. According to the definition of Raf2Af, there exists a cycle in AF from x to x which is (x, ¬x, s(x).x, x). We have thus: ¬x ∈ S ′ and s(x).x ∈ S ′ . As there is no y ∈ A ∪ K such that x ̸ = y and x ≡ RAF y (i.e. |S| = 1) and as x ∈ S and t(x) = x, we finally have: S ′ = {x, ¬x, s(x).x} and Equation (18.90) holds. -If there is no RAF-closed-walk p ∈ ClosedWalk ra f (RAF ) from x to x then: t(x) ̸ = x. We have thus: S ′ = S = {x} and Equation (18.90) holds.

2. Let consider the case where |S| > 1.

• Let a ∈ A be an argument in RAF .

Given that S ⊆ S ′ , we have, according Lemma 8 on page 259, the following property:

|S| > 1 and a ∈ S ⇐⇒ |S| > 1 and ¬a ∈ S ′ (18.91) The negation of this equivalence is the following:

|S ′ | ≤ 1 or t(α) / ∈ S ′ ⇐⇒ |S ′ | ≤ 1 or (∄β ∈ K s.t. t(β ) = t(α) and s(β ).β ∈ S ′ )
Given that |S ′ | > 1, we have:

t(α) / ∈ S ′ ⇐⇒ ∄β ∈ K s.t. t(β ) = t(α) and s(β ).β ∈ S ′
Given that t(α) / ∈ S ′ , we have thus:

∄β ∈ K s.t. t(β ) = t(α) and s(β ).β ∈ S ′
That particularly holds for β = α and so, as a consequence, we have:

s(α).α / ∈ S ′
The following property holds then:

|S| > 1 and α / ∈ S and t(α) / ∈ S =⇒ s(α).α / ∈ S ′ (18.96)

• Let s(α) / ∈ S ′ . We have: |S ′ | > 1, α / ∈ S ′ and s(α) / ∈ S ′ . Given that the case where t(α) / ∈ S has already been treated with Equation (18.96), let consider the case where t(α) ∈ S and so t(α) ∈ S ′ . If t(α) is not in the same SCC a f as s(α), then there is no cycle c ∈ Cycles a f (AF ) going from t(α) to t(α) such that s(α) ∈ c. Likewise, if t(α) is not in the same SCC a f as α means that there is no cycle c ∈ Cycles a f (AF ) going from t(α) to t(α) such that α ∈ c. Following the definition of Raf2Af, there exist a walk from s(α) to t(α) which is (s(α), ¬s(α), s(α).α, t(α)) and a walk from α to t(α) which is (α, ¬α, s(α).α, t(α)). As a consequence, there is no cycle c = (t(α), ...,t(α)) ∈ Cycles a f (AF ) such that s(α).α ∈ c. We have thus: s(α).α / ∈ S ′ . The following property holds then: From Cases 1 and 2, we prove so that: We prove so that: (S 1 ≼ S 2 and S 2 ≼ S 3 ) =⇒ S 1 ≼ S 3

|S| > 1
S ∈ SCCS ra f (RAF ) =⇒                 S ∪
From Steps 1, 2 and 3, we prove so that ≼ is a partial order. ■

Proof of Proposition 38 on page 132. Let p = (e 1 , ..., e n-1 , e n ) ∈ Paths ra f (RAF ) such that e 1 ∈ S and e n ∈ S ′ and e n-1 ∈ K. Let suppose that there exists e i ∈ p ∩ K such that t(e i ) / ∈ (S ∪ S ′ ). If t(e i ) / ∈ (S ∪ S ′ ), we have thus t(e i ) ̸ = e n . Let V ∈ SCCS ra f (RAF ) be the SCC ra f such that t(e i ) ∈ V . Given that (e 1 , ..., e i , e i+1 = t(e i )) ∈ Paths ra f (RAF ), e 1 ∈ S and t(e i ) ∈ V , we have then: S ≼ V . Furthermore, given that t(e i ) ̸ = e n we have: (e i+1 = t(e i ), ..., e n ) ∈ Paths ra f (RAF ). As t(e i ) ∈ V , e n ∈ S ′ , e n-1 ∈ K then we have: V ≼ S ′ . Given that S ≼ V and V ≼ S ′ then S is not a predecessor of S ′ , which is a contradiction. We prove so that: ∀i ∈ {1, ..., n}, e i ∈ p ∩ K =⇒ t(e i ) ∈ (S ∪ S ′ ) ■ Proof of Proposition 39 on page 133. Given that the notion of predecessor, as defined, proceed from the relation ≼, which is a partial order, then trivially Dag scc (RAF ) is acyclic. ■

Proofs of Section 14.3: RAF semantics decomposability Lemma 11. Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be the corresponding AF of RAF (with AF = ⟨A ′ , K ′ ⟩). Let Ω be a partition of (A ∪ K) and Ω ′ ∈ S D-ra f -c (AF ) be the RAF-compliant partition of A ′ corresponding to Ω, i.e. Ω ′ = {ω ′ = ω ∪ {¬x|x ∈ ω} ∪ {s(α).α ∈ And A,K |α ∈ ω}|ω ∈ Ω}. Let ω ∈ Ω and ω ′ ∈ Ω ′ be its counterpart in AF . Let RAF = Ã, K, s, t, s,t be the partial RAF corresponding to ω. Let I = S inp , Q inp be the input elements of RAF and L inp be a structure labelling of them. Let RAF , I, L inp be a RAF with input and AF ↓ ω ′ , J , ℓ J , K J be its corresponding AF with input, as defined in Definition 105 on page 155. Let RAF s = Ãs , Ks , s s ,t s be the standard of RAF , I, L inp and let std-AF = ⟨A ′ s , K ′ s ⟩ be the standard AF corresponding to AF ↓ ω ′ , J , ℓ Note: In order to follow easily the following proofs, here is a reminder of some set relations:

• Considering the definition of RAF = Ã, K, s, t, s,t , RAF s = Ãs , Ks , s s ,t s , RAF ′ s = Ã′ s , K′ s and the definition of Raf2Af, we have:

-Ã = A ∩ ω = Ãs ∩ ω -K = K ∩ ω = Ks ∩ ω
-Ãs ∪ Ks ∪ Not Ãs ∪ Not Ks ∪ And Ãs , Ks = Ã′ s • Considering the definitions of AF = ⟨A ′ , K ′ ⟩, AF ↓ ω ′ , J , ℓ J , K J , std-AF = ⟨A ′ s , K ′ s ⟩ and the definition of Raf2Af, we have:

-A ∪ K ∪ Not A ∪ Not K ∪ And A,K = A ′ -ω ′ = A ′ ∩ ω ′ = A ′ s ∩ ω ′
Proof of Lemma 11 on the previous page. Assertion 1: J ⊆ (Not A ∪ And A,K ) Trivial considering Definition 105 on page 155.

Assertion 2: (¬s(α), s(α).α) ∈ K′ s ∪ K ′ s s.t. s(α).α ∈ ω ′ and ¬s(α) / ∈ ω ′ =⇒ s(α) ∈ S inp Let consider two cases: (¬s(α), s(α).α) ∈ K′ s and (¬s(α), s(α).α) ∈ K ′ s .

• Case 1: Let consider RAF ′ s and let (¬s(α), s(α).α) ∈ K′ s s.t. s(α).α ∈ ω ′ and ¬s(α) / ∈ ω ′ . Given that s(α).α ∈ Ã′ , we have following the definition of Raf2Af: α ∈ Ks . α is an attack of the standard RAF RAF ′ s . Furthermore, given that s(α).α ∈ ω ′ , that Ω ′ ∈ S D-ra f -c (RAF ) and that ω ′ ∈ Ω ′ , we have so: α ∈ ω ′ and ¬α ∈ ω ′ . As α ∈ Ks ∩ ω ′ , then we have: α ∈ K. α is thus an element of RAF . Furthermore, as ¬s(α) / ∈ ω ′ , we have: s(α) / ∈ ω ′ . As a consequence, s(α) is not an element of RAF . Given that s(α) / ∈ RAF and α ∈ RAF , we have so: s(α) ∈ S inp .

• Case 2: Let consider std-AF and let (¬s(α), s(α).α) ∈ K ′ s s.t. s(α).α ∈ ω ′ and ¬s(α) / ∈ ω ′ . Given that s(α).α ∈ A ′ s , we have following the definition of Raf2Af: α ∈ K. α is an attack of the original RAF RAF . Given that s(α).α ∈ ω ′ , that Ω ′ ∈ S D-ra f -c (RAF ) and that ω ′ ∈ Ω ′ , we have so: α ∈ ω ′ and ¬α ∈ ω ′ . As α ∈ K ∩ ω ′ , we have: α ∈ RAF . Furthermore, as ¬s(α) / ∈ ω ′ , we have: s(α) / ∈ ω ′ . As a consequence, s(α) is not an element of RAF . Given that s(α) / ∈ RAF and α ∈ RAF , we have so: s(α) ∈ S inp .

From Cases 1 and 2, we prove so that Assertion 2 holds. Assertion 3: (s(α).α,t(α)) ∈ K′ s ∪ K ′ s s.t. s(α).α / ∈ ω ′ and t(α) ∈ ω ′ =⇒ α ∈ Q inp

• Case 1: Let consider RAF ′ s and let (s(α).α,t(α)) ∈ K′ s s.t. s(α).α / ∈ ω ′ and t(α) ∈ ω ′ . Given that s(α).α ∈ Ã′ , we have following the definition of Raf2Af: α ∈ Ks . α is an attack of the standard RAF RAF ′ s . Given that t(α) ∈ ω ′ and that α ∈ Ks , we have so: t(α) ∈ Ãs ∪ Ks . As a consequence and as t(α) ∈ ω ′ , we have: t(α) ∈ Ã ∪ K. Given that s(α).α / ∈ ω ′ , that Ω ′ ∈ S D-ra f -c (RAF ) and that ω ′ ∈ Ω ′ , we have so: α / ∈ ω ′ and ¬α / ∈ ω ′ . As a consequence, we have α / ∈ K. As t(α) belongs to RAF and α does not belong to RAF , We have: α ∈ Q inp .

• Case 2: Let consider std-AF and let (s(α).α,t(α)) ∈ K ′ s s.t. s(α).α / ∈ ω ′ and t(α) ∈ ω ′ . Given that s(α).α ∈ A ′ s , we have following the definition of Raf2Af: α ∈ K. α is an attack of the original RAF RAF . As a consequence, we have: t(α) ∈ A ∪ K. Given that t(α) ∈ ω ′ and that t(α) ∈ A ∪ K, we have: t(α) ∈ Ã ∪ K. Given that s(α).α / ∈ ω ′ , that Ω ′ ∈ S D-ra f -c (RAF ) and that ω ′ ∈ Ω ′ , we have so: α / ∈ ω ′ and ¬α / ∈ ω ′ . As a consequence, we have α / ∈ K. As t(α) belongs to RAF and α does not belong to RAF , We have: α ∈ Q inp .

From Cases 1 and 2, we prove so that Assertion 3 holds. Assertion 4: (x, y) ∈ K ′ s ∪ K′ s s.t. y ∈ ω ′ =⇒ x ∈ ω ′ ∪ J . Let consider two cases: (x, y) ∈ K′ s and (x, y) ∈ K ′ s .

• Case 1: (x, y) ∈ K ′ s Let (x, y) ∈ K ′ s s.t. y ∈ ω ′ . Let suppose that x / ∈ (ω ′ ∪ J ). Following the definition of standard AF, we have thus: x ∈ J ′ , J ′ being the set of added arguments to fit with the labelling of the input arguments (see Definition 17 on page 17). Also, following the definition of standard AF: ∄(x, y) ∈ K ′ s s.t. x ∈ J ′ and y ∈ ω ′ . We have a contradiction. We prove so that:

(x, y) ∈ K ′ s s.t. y ∈ ω ′ =⇒ x ∈ ω ′ ∪ J
• Case 2: (x, y) ∈ K′ s Let (x, y) ∈ K′ s s.t. y ∈ ω ′ . Trivially, x may belong to ω ′ . Let thus suppose that x / ∈ ω ′ and prove that x ∈ J . Let consider three cases: y ∈ A ∪ K, y ∈ Not A ∪ Not K and y ∈ And A,K .

-Case 2.1: y ∈ A ∪ K Following the definition of Raf2Af, we have: x ∈ And Ãs , Ks . Let assume that x = s(α).α with α ∈ Ks and t(α) = y. If α ∈ Ks ∩ K then we have: α ∈ ω. Given that ω ⊆ ω ′ , that ω ′ ∈ Ω ′ , that Ω ′ ∈ S D-ra f -c (AF ), we have: s(α).α ∈ ω ′ , which contradicts: x / ∈ ω ′ . We have so: α ∈ Ks \ K. Thus α does not belong to RAF . As t(α) ∈ A ∪ K and t(α) ∈ ω ′ , t(α) belongs to RAF . Given that α / ∈ K and t(α) ∈ Ã ∪ K, we have: α ∈ Q inp and so α ∈ K. In the flattening process of RAF the RAF-walk (s(α), α,t(α)) will produce the following walks: (s(α), ¬s(α), s(α).α,t(α)) and (α, ¬α, s(α).α,t(α)). Given that (s(α).α,t(α)) ∈ K ′ , that s(α).α / ∈ ω ′ and that t(α) ∈ ω ′ , we have: (s(α).α,t(α)) ∈ K J . As a consequence we have: x ∈ J .

-Case 2.2: y ∈ Not A ∪ Not K Following the definition of Raf2Af, we have: x ∈ Ãs ∪ Ks . Let assume that y = ¬x. If x ∈ ( Ãs ∪ Ks ) ∩ ( Ã ∪ K), then we have: x ∈ ω. Given that ω ⊆ ω ′ , we have: x ∈ ω ′ , which contradicts:

x / ∈ ω ′ . We have so: x ∈ ( Ãs ∪ Ks ) \ ( Ã ∪ K). We have so: x ∈ S inp ∪ Q inp . x is thus an element of RAF and we have: x ∈ A ∪ K. Given that Ω ′ ∈ S D-ra f -c (RAF ) and that ω ′ ∈ Ω ′ , as y = ¬x ∈ ω ′ , we have: x ∈ ω ′ , which contradicts x / ∈ ω ′ . As a consequence:

∄(x, y) ∈ K′ s s.t. x / ∈ ω ′ and y ∈ Not A ∪ Not K (18.99) -Case 2.3: y ∈ And A,K Let assume that y = s(α).α with α ∈ Ks . Following the definition of Raf2Af, we have so: x ∈ Not Ãs ∪ Not Ks . Let assume that x = ¬z with z ∈ Ãs ∪ Ks . We have so: z = s(α) or z = α. If x ∈ Not Ks then: z = α. Given that s(α).α ∈ ω ′ , that Ω ′ ∈ S D-ra f -c (RAF ) and that ω ′ ∈ Ω ′ , we have so: α ∈ ω ′ and ¬α ∈ ω ′ , which contradicts x / ∈ ω ′ . We have so: x ∈ Not Ãs and so: z = s(α) and z ∈ Ãs . Given that α ∈ ω ′ and α ∈ Ks , we have so: α ∈ K. α is thus an element of RAF and so of RAF . Given that ¬s(α) / ∈ ω ′ , that Ω ′ ∈ S D-ra f -c (RAF ) and that ω ′ ∈ Ω ′ , we have so: s(α) / ∈ ω ′ . As a consequence, z = s(α) ∈ Ãs \ Ã. s(α) is not an element of RAF . As α ∈ K, we have so: s(α) ∈ S inp and so s(α) ∈ A.

In the flattening process of RAF the RAF-walk (s(α), α,t(α)) will produce the following walks: (s(α), ¬s(α), s(α).α,t(α)) and (α, ¬α, s(α).α,t(α)). Given that (¬s(α), s(α).α) ∈ K ′ , that ¬s(α) /

∈ ω ′ and that s(α).α ∈ ω ′ , we have: (¬s(α), s(α).α) ∈ K J . As a consequence we have: x ∈ J . From Cases 2.1, 2.2, 2.3, we prove that:

(x, y) ∈ K′ s s.t. y ∈ ω ′ =⇒ x ∈ ω ′ ∪ J
From Cases 1 and 2, we prove so that Assertion 4 holds. ■ Lemma 12. Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) (with AF = ⟨A ′ , K ′ ⟩) be the corresponding AF of RAF (with AF = ⟨A ′ , K ′ ⟩). Let Ω be a partition of (A ∪ K) and Ω ′ ∈ S D-ra f -c (AF ) be the RAF-compliant partition of A ′ corresponding to Ω, i.e. Ω ′ = {ω ′ = ω ∪ {¬x|x ∈ ω} ∪ {s(α).α ∈ And A,K |α ∈ ω}|ω ∈ Ω}. Let ω ∈ Ω and ω ′ ∈ Ω ′ be its counterpart in AF . Let RAF = Ã, K, s, t, s,t be the partial RAF corresponding to ω. Let I = S inp , Q inp be the input elements of RAF and L inp be a structure labelling of them. Let RAF , I, L inp be a RAF with input of RAF and AF ↓ ω ′ , J , ℓ J , K J be its corresponding AF with input, as defined in Definition 105 on page 155. Let RAF s = Ãs , Ks , s s ,t s be the standard RAF of RAF , I, L inp and let std-AF = ⟨A ′ s , K ′ s ⟩ be the standard AF corresponding to AF ↓ ω ′ , J , ℓ J , K J . Let RAF ′ s = Raf2Af( RAF s ) be the corresponding AF of RAF s (with RAF ′ s = Ã′ s , K′ s ). The following assertions hold:

1. A ′ s ∩ Ã′ s = ω ′ ∪ J 2. K ′ s ∩ (ω ′ × ω ′ ) = K′ s ∩ (ω ′ × ω ′ ) 3. K ′ s ∩ (J × ω ′ ) = K′ s ∩ (J × ω ′ )
Proof of Lemma 12. Assertion 1: A ′ s ∩ Ã′ s = ω ′ ∪ J

• Step 1: Let prove that ω ′ ⊆ A ′ s ∩ Ã′ s . First of all, notice that if x ∈ ω ′ then x ∈ AF ↓ ω ′ , and so by definition of the standard AF, we have x ∈ std-AF . More precisely we have: x ∈ A ′ s . The following property holds then: x ∈ ω ′ =⇒ x ∈ A ′ s (18.100) Secondly, notice that if x ∈ ω then x belongs to RAF . We have thus, by definition of the standard RAF: x ∈ Ãs ∪ Ks . As a consequence, x and ¬x belong to RAF ′ s and we have: x ∈ Ã′ s and ¬x ∈ Ã′ s . The following property holds then:

x ∈ ω =⇒ x ∈ Ã′ s and ¬x ∈ Ã′ s (18.101)

Let prove that x ∈ ω ′ \ ω belongs to Ã′ s . We have two cases to consider: x ∈ (ω ′ ∩ (Not A ∪ Not K )) and x ∈ (ω ′ ∩ And A,K ).

-Case 1: x ∈ (ω ′ ∩ (Not A ∪ Not K ))

Let assume that x = ¬y as it belongs whether to Not A or Not K . Given that ¬y ∈ ω ′ , that ω ′ ∈ Ω ′ and that Ω ′ ∈ S D-ra f -c (AF ), we have then: y ∈ ω ′ ∩ (A ∪ K) and so: y ∈ ω. According to Equation (18.101), we have thus: y ∈ Ã′ s and ¬y ∈ Ã′ s . As a consequence, we have: x ∈ Ã′ s . The following property holds then:

x ∈ (ω ′ ∩ (Not A ∪ Not K )) =⇒ x ∈ Ã′ s (18.102) -Case 2: x ∈ (ω ′ ∩ And A,K )

Let assume that x = s(α).α as it belongs to And A,K . Given that s(α).α ∈ ω ′ , that ω ′ ∈ Ω ′ and that Ω ′ ∈ S D-ra f -c (AF ), we have then: ¬α ∈ ω ′ and α ∈ ω ′ . Furthermore, given that α ∈ K, we have: α ∈ ω. As ω ∩ K = K, we have: α ∈ K (i.e. α is an attack belonging to RAF ). By definition of the standard RAF, we have thus: α ∈ Ãs . Now, given that α ∈ Ãs , by definition of Raf2Af, we have thus: s(α) ∈ Ã′ s , α ∈ Ã′ s and s(α).α ∈ Ã′ s . As a consequence we have: x ∈ Ã′ s . The following property holds then:

x ∈ (ω ′ ∩ And A,K ) =⇒ x ∈ Ã′ -Case 1: x ∈ Not A .

Let assume that x = ¬y with y ∈ A. Following the definition of Raf2Af, we have:

(¬y, z) ∈ K ′ =⇒ z ∈ And A,K

We know that such an attack exists as ¬y ∈ J . Let s(α).α be an argument such that (¬y, s(α).α) ∈ K ′ . Following the definition of Raf2Af, we have: s(α) = y. We have so: s(α) / ∈ ω ′ and ¬s(α) / ∈ ω ′ . Furthermore, we have: s(α).α ∈ ω ′ .

Following Assertion 2 of Lemma 11 on page 267, we have thus: y ∈ S inp and so y ∈ Ãs . Then, following the definition of Raf2Af, we have thus: y ∈ Ã′ s and ¬y ∈ Ã′ s , and so x ∈ Ã′ s . The following equation holds then:

x ∈ J ∩ Not A =⇒ x ∈ Ã′ s (18.106) -Case 2: x ∈ And A,K .

Let assume that x = s(α).α with s(α) ∈ A and α ∈ K. Following the definition of Raf2Af, we have:

(s(α).α, z) ∈ K ′ =⇒ z ∈ A ∪ K
Let z be an element such that (s(α).α, z) ∈ K ′ s.t. z ∈ ω ′ . We know that such an attack exists as s(α).α ∈ J . Following Assertion 3 of Lemma 11 on page 267, we have thus: α ∈ Q inp . Following the definition of standard RAF, we have so: α ∈ Ks . Then, following the definition of Raf2Af, we have thus: α ∈ Ã′ s , ¬α ∈ Ã′ s and s(α).α ∈ Ã′ s , and so x ∈ Ã′ s . The following equation holds then:

x ∈ J ∩ And A,K =⇒ x ∈ Ã′ s (18.107) From Equations (18.105) to (18.107) on pages 271-272 we prove so that:

J ⊆ A ′ s ∩ Ã′ s • Step 3: Let prove that A ′ s ∩ Ã′ s ⊆ ω ′ ∪ J . Let x ∈ A ′ s ∩ Ã′ s .
Notice that following the definition of standard AF (Definition 17 on page 17 used to create std-AF ), we have: Ã′ s ∩ J ′ = ∅, J ′ being the set of added arguments to fit with the labelling of the input arguments in std-AF . 5 Given that x ∈ Ã′ s , we have so: x / ∈ J ′ . As a consequence, we have: x ∈ ω ′ ∪ J . We prove so that:

A ′ s ∩ Ã′ s ⊆ ω ′ ∪ J
From Steps 1, 2 and 3, we prove that Assertion 1 holds.

Assertion 2: K ′ s ∩ (ω ′ × ω ′ ) = K′ s ∩ (ω ′ × ω ′ ) • Step 1: Let prove that K ′ s ∩ (ω ′ × ω ′ ) ⊆ K′ s ∩ (ω ′ × ω ′ ) Let (x, y) ∈ K ′ s ∩ (ω ′ × ω ′ ).
Given that (x, y) ∈ (ω ′ × ω ′ ) implies that: x / ∈ J ′ and y / ∈ J ′ , we have thus, following Assertion 1: x ∈ A ′ s ∩ Ã′ s and y ∈ A ′ s ∩ Ã′ s . Following the definition of Raf2Af, we have the following four cases to consider: (x ∈ A ∪ K and y ∈ Not A ∪Not K ), (x ∈ Not A and y ∈ And A,K ), (x ∈ Not K and y ∈ And A,K ) and (x ∈ And A,K and y ∈ A∪K) -Case 1: x ∈ A ∪ K and y ∈ Not A ∪ Not K According to the definition of Raf2Af, we have: y = ¬x. If x ∈ A ∪ K, then we have x ∈ ω. As a consequence we have: x ∈ Ã ∪ K and then x ∈ Ãs ∪ Ks . The flattening of RAF s will thus produced an attack (x, ¬x) ∈ K′ s , with x ∈ Ã′ s and ¬x ∈ Ã′ s . The following property holds then: Given that Ω ′ ∈ S D-ra f -c (RAF ), that ω ′ ∈ Ω ′ and that ¬z ∈ ω ′ , we have: z ∈ ω ′ . Moreover, as z ∈ A, we have: z ∈ ω. z is thus an element of RAF . Let assume that y = s(α).α with α ∈ K. We have thus: s(α) = z.

Given that s(α).α ∈ ω ′ , that Ω ′ ∈ S D-ra f -c (RAF ) and that ω ′ ∈ Ω ′ , we have so: α ∈ ω ′ . As α ∈ K, we have: α ∈ ω. α is thus an element of RAF . As a consequence, we have: α ∈ Ks and s(α) ∈ Ãs . Following the definition of Raf2Af, we have so: {s(α), ¬s(α), s(α).α} ∈ Walks a f ( RAF ′ s ). The following property then holds: s(α) ∈ ω ′ and s(α).α ∈ ω ′ =⇒ {s(α), ¬s(α), s(α).α} ∈ Walks a f ( RAF As a consequence we have: (¬α, s(α).α) ∈ K′ s and so: (x, y) ∈ K′ s .

The following property then holds: From Cases 1, 2, 3 and 4 we prove so that:

(x, y) ∈ K ′ s ∩ (ω ′ × ω ′ )
(x, y) ∈ K ′ s ∩ (ω ′ × ω ′ ) =⇒ (x, y) ∈ K′ s ∩ (ω ′ × ω ′ ) (18.114) • Step 2: Let prove that K′ s ∩ (ω ′ × ω ′ ) ⊆ K ′ s ∩ (ω ′ × ω ′ ) Let (x, y) ∈ K′ s ∩ (ω ′ × ω ′ ).
According to the definition of Raf2Af, we have whether: (x ∈ A ∪ K and y ∈ Not A ∪ Not K s.t. y = ¬x), (x ∈ Not A and y ∈ And A,K ), (x ∈ Not K and y ∈ And A,K ) or (x ∈ And A,K and y ∈ A ∪ K).

Let consider those cases.

-Case 1: x ∈ A ∪ K and y ∈ Not A ∪ Not K s.t. y = ¬x Given that x ∈ A ∪ K then, following the definition of Raf2Af, we have: (x, ¬x) in K ′ . As x ∈ ω ′ , Ω ′ ∈ S D-ra f -c (RAF ) and ω ′ ∈ Ω ′ , we have so: ¬x ∈ ω ′ . Both x and ¬x are thus arguments of AF ↓ ω ′ . As x ∈ ω ′ ∩ (A ∪ K), we have following the definition of Raf2Af and the definition of standard AF the following property:

x ∈ ω ′ ∩ (A ∪ K) =⇒ (x, ¬x) ∈ K ′ s (18.115)

We have so: (x, y) ∈ K ′ s . -Case 2: x ∈ Not A and y ∈ And A,K As x ∈ Not A , let assume that x = ¬z, with z ∈ A. As y ∈ And A,K , let assume that y = s(α).α with α ∈ K such that s(α) = z.

Given that s(α).α ∈ ω ′ , that Ω ′ ∈ S D-ra f -c (RAF ) and that ω ′ ∈ Ω ′ , we have: α ∈ ω ′ . α is thus an argument of AF ↓ ω ′ . Following the definition of Raf2Af, and the definition of standard AF, we have thus the following property:

α ∈ ω ′ ∩ K =⇒ {α, ¬α, s(α).α} ∈ Walks a f (std-AF ) (18.116) Furthermore, we have also the following property:

α ∈ ω ′ ∩ K and s(α) ∈ ω ′ =⇒ {s(α), ¬s(α), s(α).α} ∈ Walks a f (std-AF ) (18.117)

As a consequence, we have: (¬s(α), s(α).α) ∈ K ′ s , and so: (x, y) ∈ K ′ s . -Case 3: x ∈ Not K and y ∈ And A,K As x ∈ Not K , let assume that x = ¬α with α ∈ K and y = s(α).α. Given that s(α).α ∈ ω ′ , that Ω ′ ∈ S D-ra f -c (RAF ) and that ω ′ ∈ Ω ′ , we have: α ∈ ω ′ . Following Equation (18.116), we have thus: {α, ¬α, s(α).α} ∈ Walks a f (std-AF ).

The following property holds then: s(α).α ∈ ω ′ =⇒ {α, ¬α, s(α).α} ∈ Walks a f (std-AF ) (18.118)

As a consequence, we have: (¬α, s(α).α) ∈ K ′ s and so: (x, y) ∈ K ′ s -Case 4: x ∈ And A,K and y ∈ A ∪ K As x ∈ And A,K , let assume that x = s(α).α with α ∈ K and y = t(α). Following Equation (18.118) and the fact that y ∈ ω ′ , we have: {α, ¬α, s(α).α,t(α)} ∈ Walks a f (std-AF ).

As a consequence, we have: (s(α).α,t(α)) ∈ K ′ s and so: (x, y) ∈ K ′ s .

From Cases 1, 2, 3 and 4, the following property then holds:

(x, y) ∈ K′ s s.t. x ∈ ω ′ and y ∈ ω ′ =⇒ (x, y) ∈ K ′ s (18.119)

From Steps 1 and 2, we prove that Assertion 2 holds.

Assertion 3: K ′ s ∩ (J × ω ′ ) = K′ s ∩ (J × ω ′ )

• Step 1: K ′ s ∩ (J × ω ′ ) ⊆ K′ s ∩ (J × ω ′ ) Let (x, y) ∈ K ′ s ∩(J ×ω ′ ). Following Assertion 1 of Lemma 11 on page 267, we have whether: x ∈ Not A or x ∈ And A,K . Let consider those two cases.

-Case 1: x ∈ J ∩ Not A and y ∈ ω ′ .

As x ∈ Not A , let assume that x = ¬z with z ∈ A. Following the definition of Raf2Af, we have: y ∈ And A,K . Let thus assume that y = s(α).α with α ∈ K such that s(α) = z. As s(α).α ∈ ω ′ , following Equation (18.111) on page 273, we have:

{α, ¬α, s(α).α} ∈ Walks a f ( RAF ′ s ) As a consequence we have: (¬α, s(α).α) ∈ K′ s and so: (x, y) ∈ K′ s . The following property then holds: (x, y) ∈ K ′ s ∩ (J × ω ′ ) s.t. x ∈ Not A =⇒ (x, y) ∈ K′ s ∩ (J × ω ′ ) (18.120) -Case 2: x ∈ J ∩ And A,K and y ∈ ω ′ . As x ∈ And A,K , let assume that x = s(α).α with α ∈ K such that t(α) = y with y ∈ A ∪ K. Following Assertion 3 of Lemma 11 on page 267, we have: α ∈ Q inp . Following the definition of the standard RAF, we have so: α ∈ Ks . Following the definition of Raf2Af and the fact that y ∈ ω ′ , we have then: {α, ¬α, s(α).α,t(α)} ∈ Walks a f ( RAF ′ s )}. The following property then holds: α ∈ J =⇒ {s(α), ¬s(α), s(α).α,t(α)} ∈ Walks a f ( RAF ′ s ) (18.121)

As a consequence: (s(α).α,t(α)) ∈ K′ s and so: (x, y) ∈ K′ s . The following property then holds:

(x, y) ∈ K ′ s ∩ (J × ω ′ ) s.t.
x ∈ And A,K =⇒ (x, y) ∈ K′ s ∩ (J × ω ′ ) (18.122)

From Cases 1 and 2, we prove so that:

(x, y) ∈ K ′ s ∩ (J × ω ′ ) =⇒ (x, y) ∈ K′ s ∩ (J × ω ′ ) (18.123)

•

Step 2: K′ s ∩ (J × ω ′ ) ⊆ K ′ s ∩ (J × ω ′ ) Let (x, y) ∈ K′ s ∩(J ×ω ′ ). Following Assertion 1 of Lemma 11 on page 267, we have whether: x ∈ Not A or x ∈ And A,K . Let consider those two cases.

-Case 1: x ∈ Not A \ ω ′ and y ∈ ω ′ As x ∈ Not A , let assume that x = ¬z with z ∈ A. We have so: y ∈ And A,K . Let assume that y = s(α).α with α ∈ K such that s(α) = z. In the flattening process of RAF the RAF-walk (s(α), α,t(α)) will produce the following walks: (s(α), ¬s(α), s(α).α,t(α)) and (α, ¬α, s(α).α,t(α)). Given that (¬s(α), s(α).α) ∈ K ′ , that ¬s(α) / ∈ ω ′ and that s(α).α ∈ ω ′ , we have: (¬s(α), s(α).α) ∈ K J . Following the definition of standard AF, we have thus: (¬s(α), s(α).α) ∈ K ′ s , and so: (x, y) ∈ K ′ s . -Case 2: x ∈ And A,K \ ω ′ and y ∈ ω ′ As x ∈ And A,K , let assume that x = s(α).α with α ∈ K. We have so: y = t(α) and y ∈ (A∪K), following the definition of Raf2Af. In the flattening process of RAF the RAF-walk (s(α), α,t(α)) will produce the following walks: (s(α), ¬s(α), s(α).α,t(α)) and (α, ¬α, s(α).α,t(α)). Given that (s(α).α,t(α)) ∈ K ′ , that ¬s(α).α / ∈ ω ′ and that t(α) ∈ ω ′ , we have: (s(α).α,t(α)) ∈ K J . Following the definition of standard AF, we have thus: (s(α).α,t(α)) ∈ K ′ s , and so: (x, y) ∈ K ′ s .

From Cases 1 and 2, the following property then holds:

(x, y) ∈ K′ s s.t. x ∈ J and y ∈ ω ′ =⇒ (x, y) ∈ K ′ s (18.124)

From Steps 1 and 2, we prove that Assertion 3 holds. ■ Lemma 13. Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be the corresponding AF of RAF (with AF = ⟨A ′ , K ′ ⟩). Let Ω be a partition of (A ∪ K) and Ω ′ ∈ S D-ra f -c (AF ) be the RAF-compliant partition of A ′ corresponding to Ω, i.e. Ω ′ = {ω ′ = ω ∪ {¬x|x ∈ ω} ∪ {s(α).α ∈ And A,K |α ∈ ω}|ω ∈ Ω}. Let ω ∈ Ω and ω ′ ∈ Ω ′ be its counterpart in AF . Let RAF = Ã, K, s, t, s,t be the partial RAF corresponding to ω. Let I = S inp , Q inp be the input elements of RAF and L inp be a structure labelling of them. Let RAF , I, L inp be a RAF with input of RAF and AF ↓ ω ′ , J , ℓ J , K J be its corresponding AF with input, as defined in Definition 105 on page 155. Let RAF s = Ãs , Ks , s s ,t s be the standard of RAF , I, L inp and let std-AF = ⟨A ′ s , K ′ s ⟩ be the standard AF corresponding to AF ↓ ω ′ , J , ℓ J , K J . Let RAF ′ s = Raf2Af( RAF s ) be the corresponding AF of RAF s (with RAF ′ s = Ã′ s , K′ s ). Let σ be a complete-based AF semantics (See Definition 38 on page 49) and let σ -raf be the RAF semantics corresponding to σ . 6 The following assertions hold:

1. ∀¬a ∈ J ∩ Not A , ∀ℓ ∈ L σ ( RAF ) and s(α).α ∈ J ∩ And A,K . Given that s(α).α ∈ J ∩ And A,K , we have following Definition 105 on page 155: α ∈ Q inp and s(α) ∈ S inp . As a consequence, following the definition of standard RAF we have: α ∈ Ks and s(α) ∈ Ãs . The flattening process of RAF s will thus produce the following walks:

(s(α), ¬s(α), s(α).α,t(α)) and (α, ¬α, s(α).α,t(α)). As a consequence ℓ(s(α)), ℓ(¬s(α)), ℓ(α) and ℓ(¬α) must be defined.

Let L = afLab2RafLab(ℓ). Given that σ -raf is the RAF semantics corresponding to σ , we have: L ∈ L σ -ra f ( RAF s ). Moreover, according to Definition 78 on page 118 we have: According to Assertion 3 of Lemma 12 on page 270, we have: Given that à ⊆ Ãs and K ⊆ Ks , we have following Equation (18.133):

K ′ s ∩ (J × ω ′ ) = K′ s ∩ (J × ω ′ ) ( 18
{L ↓ ⟨ Ã, K⟩ |L ∈ L σ -ra f ( RAF s )} = { ℓ ↓ Ã, ℓ ↓ K |ℓ ∈ L σ ( RAF ′ s )}
Given that à ∪ K ⊆ ω ′ , we have following Equation (18.132):

{L ↓ ⟨ Ã, K⟩ |L ∈ L σ -ra f ( RAF s )} = { ℓ ↓ Ã, ℓ ↓ K |ℓ ∈ F a f σ (AF ↓ ω ′ , J , ℓ J , K J )} (18.134) Given that ( Ã ∪ K) ⊆ (A ∪ K), that ω ′ ∩ ((A ∪ K) \ ( Ã ∪ K)) ∩ (A ∪ K) = ∅ and that F a f σ (AF ↓ ω ′ , J , ℓ J , K J )
produces labellings of arguments that are in ω ′ , we have following Equation (18.134): Thus, from Equations (18.135) and (18.136) on the previous page, we prove that:

{L ↓ ⟨ Ã, K⟩ |L ∈ L σ -ra f ( RAF s )} = { ℓ ↓ A , ℓ ↓ K |ℓ ∈ F a f σ (AF ↓ ω ′ , J , ℓ J , K J )} (
F ra f σ ( RAF , I, L inp ) = ℓ ↓ A , ℓ ↓ K ℓ ∈ F a f σ (AF ↓ ω ′ , J , ℓ J , K J )

■

Proof of Proposition 42 on page 160. Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be the corresponding AF of RAF (with AF = ⟨A ′ , K ′ ⟩). Let Ω ′ ∈ S D-ra f -c (AF ) be any RAF-compliant partition of A ′ , with Ω ′ = {ω ′ 1 , ..., ω ′ n }. Let Ω ∈ S (RAF ) be the partition of RAF corresponding to Ω ′ , with As σ -raf is the RAF semantics corresponding to σ , we have:

Ω = {ω i = ω ′ i ∩ (A ∪ K)|ω ′ i ∈ Ω ′ }.
L σ -ra f (RAF ) = { ℓ ↓ A , ℓ ↓ K |ℓ ∈ L σ (AF )} (18.137)
σ is top-down decomposable w.r.t. S ra f -c iff :

∀Ω ′ ∈ S ra f -c (AF ), L σ (AF ) ⊆ ℓ 1 ∪ ... ∪ ℓ n ℓ i ∈ F a f σ (AF ↓ ω ′ i , J i , ℓ J i , K J i ) With: ℓ J i = ( j∈{1,...,n} s.t. j̸ =i ℓ j ) ↓ J i (18.138) We have so, from Equations (18.137) and (18.138), σ being top-down decomposable w.r.t. S ra f -c iff : ∀Ω ′ ∈ S ra f -c (AF ), L σ -ra f (RAF ) ⊆ ℓ 1 ↓ A , ℓ 1 ↓ K ∪ ... ∪ ℓ n ↓ A , ℓ n ↓ K ℓ i ∈ F a f σ (AF ↓ ω ′ i , J i , ℓ J i , K J i ) With: ℓ J i = ( j∈{1,...,n} s.t. j̸ =i ℓ j ) ↓ J i (18.139) Following Proposition 41 on page 159, for all i ∈ {1, ..., n}, we have : As σ -raf is the RAF semantics corresponding to σ , we have:

F ra f σ ( RAF i , I i , L inp i ) = ℓ ↓ A , ℓ ↓ K ℓ ∈ F a f σ (AF ↓ ω ′ i , J i , ℓ J i , K J i ) (18
L σ -ra f (RAF ) = { ℓ ↓ A , ℓ ↓ K |ℓ ∈ L σ (AF )} (18.143)
σ is bottom-up decomposable w.r.t. S ra f -c iff : ∀Ω ′ ∈ S ra f -c (AF ), L σ (AF ) ⊇ ℓ 1 ∪ ... ∪ ℓ n ℓ i ∈ F a f σ (AF ↓ ω ′ i , J i , ℓ J i , K J i ) With: ℓ J i = ( j∈{1,...,n} s.t. j̸ =i ℓ j ) ↓ J i (18.144) We have so, from Equations (18.143) and (18.144) on the next page, σ being bottom-up decomposable w.r.t. S ra f -c iff : ∀Ω ′ ∈ S ra f -c (AF ), L σ -ra f (RAF ) ⊇ ℓ 1 ↓ A , ℓ 1 ↓ K ∪ ... ∪ ℓ n ↓ A , ℓ n ↓ K ℓ i ∈ F a f σ (AF ↓ ω ′ i , J i , ℓ J i , K J i ) With: ℓ J i = ( j∈{1,...,n} s.t. j̸ =i ℓ j ) ↓ J i (18.145) Following Proposition 41 on page 159, for all i ∈ {1, ..., n}, we have : The hard demonstration part of Proposition 43 on page 160 is to show that if an AF semantics σ is not fully (resp. top-down, bottom-up) decomposable then σ is not fully (resp. top-down, bottom-up) decomposable w.r.t. S D-ra f -c . We prove this property by choosing an AF and a partition for which σ is not fully (resp. top-down, bottom-up) decomposable. Then, we transform this AF into a RAF by naming its attacks. Next, we flatten this RAF into a new AF. Finally, we show with this new AF and the partition corresponding that σ is not fully (resp. top-down, bottom-up) decomposable w.r.t. S D-ra f -c .

F ra f σ ( RAF i , I i , L inp i ) = ℓ ↓ A , ℓ ↓ K ℓ ∈ F a f σ (AF ↓ ω ′ i , J i , ℓ J i , K J i ) (18
Likewise, the hard demonstration part of Proposition 45 on page 161 is to show that if an AF semantics σ is not fully (resp. top-down, bottom-up) decomposable w.r.t. S USCC then σ is not fully (resp. top-down, bottom-up) decomposable w.r.t. S ra f -c-USCC . We prove this property by the very same process, except that this time it is w.r. Let consider the semantics preferred , the bottom-up decomposability property and the following partition of AF 1 : Ω 1 = ω 1 1 = {a} , ω 1 2 = {b} . We have:

L pr (AF 1 ) ⊉ ℓ 1 ∪ ... ∪ ℓ n ℓ j ∈ F a f pr (AF ↓ ω 1 j , J 1 j , ℓ J 1 j , K J 1 j ) Indeed: L pr (AF 1 ) = {{(a, in), (b, out)} , {(b, in), (a, out)}}

Let AF 1 = ⟨A 1 , K 1 ⟩ be an AF and Ω 1 = ω 1 1 , ..., ω 1 n be any partition of AF 1 such that they satisfy Equation (18.156) on the previous page. Let RAF 2 = Af2Raf(AF 1 ) be the non recursive RAF corresponding to AF 1 , with RAF 2 = ⟨A 2 , K 2 , s 2 ,t 2 ⟩ (see Definition 80 on page 121). Let Ω 2 = ω 2 1 , ..., ω 2 n be the partition of RAF 2 such that: ∀ω 2 j ∈ Ω 2 , ω 2 j = ω 1 j ∪ α|α ∈ K 2 s.t. s(α) ∈ ω 1 j . Let AF 3 = Raf2Af(RAF 2 ) (with AF 3 = ⟨A 3 , K 3 ⟩) be the AF corresponding to the flattening of RAF 2 . Let Ω 3 = ω 3 1 , ..., ω 3 n be the partition of AF 3 such that: ∀ω 3 j ∈ Ω 3 , ω 3 j = ω 2 j ∪ ¬x|x ∈ (A 2 ∪ K 2 ) ∩ ω 2 j ∪ s(α).α|α ∈ K 2 ∩ ω 2 j . Following Lemma 15 on page 287, we have:

L σ (AF 3 ) R ℓ 1 ∪ ... ∪ ℓ n ℓ i ∈ F a f σ (AF 3 ↓ ω 3 j , J 3 j , ℓ J 3 j , K J 3 j ) (18.157)
Notice that Ω 3 is a RAF-compliant partition of AF 3 (i.e. Ω 3 ∈ S D-ra f -c (AF 3 )). As a consequence, if σ is not top-down (resp. bottom-up, fully) decomposable then the following statement holds: ∃AF ∈ {Raf2Af(RAF )|RAF ∈ Φ ra f } and Ω ∈ S D-ra f -c (AF ) s.t.

L σ (AF ) R ℓ 1 ∪ ... ∪ ℓ n ℓ i ∈ F a f σ (AF ↓ ω i , J i , ℓ J i , K J i ) (18.158)
Considering the contrapositive of the previous implication, we prove so that:

σ

5 --

 5 4 on page 178 (in the appendix): Computational complexity theory -Chapter 3 on page 23: AF decision problems and complexities -Section 9.1 on page 91: Structure semantics -Chapter 12 on page 114: RAF flattening Chapter 13 on page 119: RAF decision problems and semantics complexities • algorithms, read: -Section 16.2 on page 170 (in the appendix): Graph theory -Section 16.3 on page 175 (in the appendix): Matrices -Chapter 17 on page 186 (in the appendix): Mathematical problems -Chapter 1 on page 8: Semantics : extensions and labellings -Part III on page 27: Dung Argumentation Framework: Contribution • Semantics Decomposability: -Section 16.1 on page 170 (in the appendix): Set theory -Chapter 2 on page 16: AF semantics decomposability -Section 9.1 on page 91: Structure semantics -Chapter 11 on page 107: Structure labellings and semantics -Chapter 12 on page 114: RAF flattening 6 Chapter 14 on page 123: Hierarchical view of RAF and semantics decomposability

Figure 1 :

 1 Figure 1: Example of an AF

  r.t. S iff ∃b ∈ S s.t. (b, a) ∈ K. • accepted w.r.t. S iff ∀(b, a) ∈ K, ∃c ∈ S s.t. (c, b) ∈ K. We define the sets of defeated and accepted arguments w.r.t. S as follows: De f (S) = {a ∈ A|∃b ∈ S s.t. (b, a) ∈ K} Acc(S) = {a ∈ A|∀(b, a) ∈ K, ∃c ∈ S s.t. (c, b) ∈ K} Let now give the formal definition of some AF semantics :

Proposition 1 (

 1 Extension-based semantics partial ordering). The following properties hold: • Stable extensions are also semi-stable extensions • Stable extensions are also naive extensions • Semi-stable extensions are also preferred extensions • Preferred extensions are also complete extensions • The Grounded extension is also a complete extension • Complete extensions are also admissible extensions • Admissible extensions are also conflict-free extensions • Naive extensions are also conflict-free extensions Proposition 2 (Extension-based semantics cardinality). The following properties hold:

Figure 1 .

 1 Figure 1.1 illustrates Propositions 1 and 2.

Figure 1 .1 on page 10 .

 110 Proposition 3 (Labelling-based semantics partial ordering). The following properties hold: • Stable labellings are also semi-stable labellings • Stable labellings are also naive labellings • Semi-stable labellings are also preferred labellings • Preferred labellings are also complete labellings • The grounded labelling is also a complete labelling • Complete labellings are also admissible labellings • Admissible labellings are also conflict-free labellings • Naive labellings are also conflict-free labellings Proposition 4 (Labelling-based semantics cardinality). The following properties hold:

S 1 S 2 S 3 S 4 S 5 1 :

 123451 Semantic extensions of the AF in Figure 1 on page 7

Example 4 .Figure 2 . 1 : An AF 16 Definition 16 (

 4211616 Figure 2.1: An AF

Example 5 .Figure 2 . 2 :

 522 Figure 2.2: An illustration of the AF with input (Example 5)

Figure 2 Figure 2 . 3 :

 223 Figure 2.3: The standard AF corresponding to Example 5 on the previous page

Definition 23 (

 23 ,...,n} s.t. j̸ =i ℓ j ) ↓ Partition selector). A partition selector S is a function receiving as input an AF AF = ⟨A, K⟩ and returning a set of partitions of A.

Figure 3 . 1 :

 31 Figure 3.1: The AF of Example 1 on page 7

2 .Figure 4 . 1 :

 241 Figure 4.1: AFDivider operating diagram Algorithms 1 and 2 on the next page and on page 31 give the formal definition of the AFDivider algorithm. They are said to be generic algorithms in the sense that:

Example 12 .

 12 Let consider AF = ⟨A, K⟩, the AF represented in Figure 4.2 on the next page. Its grounded labelling is as follows: ℓ gr (a) = in, ℓ gr (b) = ℓ gr (c) = out and ∀x ∈ A \ {a, b, c}, ℓ gr (x) = und.

Figure 4 . 3 onFigure 4 . 2 :

 4342 Figure 4.2: AF for the running example

Figure 4 . 3 :

 43 Figure 4.3: Grounded labelling

Figure 4 Figure 4 . 4 :

 444 Figure 4.7 on page 37 illustrates them.

3 .af 2 Figure 4 . 5 :

 3245 Figure 4.5: The connected components of AF hard

  and (s, a) ∈ I and s ∈ in(µ)} • K ′ = (K \ {(s,t)|s ∈ D or t ∈ D}) ∪ {(a, a)|a ∈ A and (s, a) ∈ I and s ∈ und(µ)} Example 14. Following Example 13 on page 33, κ 2 has three possible contexts: µ 1 = {(g, out)}, µ 2 = {(g, in)} and µ 3 = {(g, und)}.

Figure 4 .

 4 8 on page 38 represents the three AFs induced from κ 2 under those contexts.

Definition 33 .af 2 Figure 4 . 6 :

 33246 Figure 4.6: Cluster partitions

κ 4 : second cluster of af 2 Figure 4 . 7 :

 4247 Figure 4.7: Cluster structures of the components

Figure 4 . 8 : 2 Figure 4 . 9 :

 48249 Figure 4.8: AFs induced from κ 2 a 3 a 4

1 1 =Figure 4 . 10 :

 1410 Figure 4.10: Example for the distinction between und and iund

κ 4 4

 4 in and the attack (n, m) exists in κ 4 . ξ κ 4 4 (m) ̸ = iund because ℓ (n) = und and the attack (n, m) exists in κ 4 .

Definition 35 .

 35 (Distinct labelling set). Let κ = af , I, O, B be a cluster, let L κ = {ℓ κ 1 , ..., ℓ κ n } be the set of labellings computed from κ, and L κ D be the distinct labelling set of κ. L κ D is defined as following:

Example 16 .= {ℓ κ 4 1 , ℓ κ 4 2 , ℓ κ 4 4Example 17 .Figure 4 . 11 :

 1642417411 Figure 4.11: Illustration of merge configurationGiven that it is possible for a labelling to have several and different configurations, we introduce the notion of "merge configuration".

Table 4 . 4 :

 44 κ 4 labellings and configurations (three contexts). Definition 36. (Merge configuration). Let κ = af , I, O, B be a cluster, let L κ = {ℓ κ 1 , ..., ℓ κ n } be the set of labellings and C κ = {ξ κ 1 , ..., ξ κ n } the set of their corresponding configurations computed from κ, let

  a) = out and ∃(b, a) ∈ af s.t. ℓ(b) = in iout if ℓ κ i (a) = out and ∄(b, a) ∈ af s.t. ℓ(b) = in und if ℓ κ i (a) = und and ∃ξ ∈ C ℓ κ i s.t. ξ (a) = und iund otherwise

Figure 4 . 12 :Example 19 .

 41219 Figure 4.12: Same merge configuration for different labellings

Example 23 .

 23 Consider the following AF, AF = ⟨A, K⟩: g h i

af 2 Figure 5 . 1 :af 2 Figure 5 . 2 :

 251252 Figure 5.1: The connected components of AF hard

Example 25 .

 25 Following Example 24, the spectral clustering produces the partitions represented in Figure 5.3 on page 62.

  6 6 4 6 0 3 5 4 8 9 0 5 5 0 32 39 12 19 25 12 15 19 12

af 2 Figure 5

 25 Figure 5.3: Cluster partition

Figure 6 . 1 :

 61 Figure 6.1: The AF of the running example

Figure 7 . 1 :

 71 Figure 7.1: AF example AF

Figure 7 . 2 :

 72 Figure 7.2: Nice tree decomposition of AF

Figure 7 . 3 :

 73 Figure 7.3: Nice tree decomposition of AF

Figure 7 .

 7 Figure 7.4 on the following page shows the AF af associated with the node t 12 .

Figure 7 . 4 :

 74 Figure 7.4: af , the AF associated with the node t 12

a 1 a 2 a 3 a 4 Figure 7 . 5 : AF example ∅ a 1 a 1 , a 2 a 2 a 2 , a 3 a 3 a 3 , a 4

 4751122334 Figure 7.5: AF example

Figure 7 . 6 :

 76 Figure 7.6: Nice tree decomposition

Figure 7 . 7 :

 77 Figure 7.7: The SCC decomposition for AF (the first layer being Layer 0) Example 33.Figure 7.7 is the SCC hierarchical view of AF .

Figure 7 .

 7 7 is the SCC hierarchical view of AF .

Figure 7 . 8 :

 78 Figure 7.8: SCC n°2 under ℓ scc1 1

Figure 7 . 9 :

 79 Figure 7.9: Example of a RAF/AFRA Formally, RAFs and AFRAs are defined as follows:

Figure 8 . 1 S 2 S 3

 8123 Figure 8.1 on page 89 illustrates Theorem 1 and Proposition 17. Notice that we have a partial ordering similar to the one for AF (see Figure 1.1 on page 10).

Figure 8 . 1 :

 81 Figure 8.1: AFRA semantics partial ordering

Figure 8 Figure 8 . 2 :

 882 Figure 8.2: An example of AFRA expressed as AF

Structures U 1 U 2 U 3

 123

Figure 9 .

 9 Figure 9.1 on the following page illustrates Theorem 2 and Proposition 20. Notice that we have a partial ordering similar to the one for AF (see Figure 1.1 on page 10).

Figure 9 . 1 :

 91 Figure 9.1: RAF semantics partial ordering

Figure 9 . 2 :

 92 Figure 9.2: A non recursive RAF

Example 43 .Figure 9 . 3 :

 4393 Figure 9.3: The AF corresponding to the non recursive framework of Figure 9.2 on the previous page

Figure 9 . 4 :

 94 Figure 9.4: Example of RAF flattening with Raf2Af maf

Example 44 .

 44 Let consider the RAF RAF illustrated in Figure 9.4(a).

  Figure 9.4(b) illustrates AF = Raf2Af maf (RAF ), the MAF/AF corresponding to RAF .

Figure 10 . 1 :

 101 Figure 10.1: Running example

  Proposition 23: link (See page 231).

  Given a certain L, we use the notation L(x) to indicate the labelling of x in L. It could mean ℓ A (x) or ℓ K (x), following the nature of x. Definition 73. (Reinstatement RAF labelling). Let RAF = ⟨A, K, s,t⟩ be a recursive argumentation framework and L = ℓ A , ℓ K be a RAF labelling. L is a reinstatement RAF labelling iff it satisfies the following conditions

U 1 U 2 U 3

 123

Theorem 19 .

 19 Let RAF = ⟨A, K, s,t⟩ be a RAF and let U = ⟨S, Q⟩ be the grounded structure. Then L = Struct2Lab(U) is a reinstatement RAF labelling such that in(L) is minimal. □ Proof of Theorem 19: link (See page 237).

Theorem 23 .

 23 Let RAF = ⟨A, K, s,t⟩ be a RAF and let U = ⟨S, Q⟩ be a semi-stable structure. Then L = Struct2Lab(U) is a reinstatement RAF labelling such that und(L) is minimal. □ Proof of Theorem 23: link (See page 238).

Figure 11 . 1 :

 111 Figure 11.1: Running example

  Figure 12.1: RAF flattening illustration

Figure 13 . 1 :

 131 Figure 13.1: Same RAF as Figure 10.1 on page 102

1 .

 1 Cred σ accepts (AF , a) iff RAF-Cred σ accepts (RAF , a). 2. Skep σ accepts (AF , a) iff RAF-Skep σ accepts (RAF , a).3. Ver σ accepts (AF , S) iff RAF-Ver σ accepts (RAF , U = ⟨S, K⟩). 4. Exists σ accepts AF iff RAF-Exists σ accepts RAF . 5. Exists ¬∅ σ accepts AF iff RAF-Exists ¬∅ σ accepts RAF . 6. Unique σ accepts AF iff RAF-Unique σ accepts RAF . □ Proof of Proposition 30: link (See page 252).

. 1 .□

 1 Proof of Proposition 32: link (See page 253).

Example 52 .Figure 14 . 1 :Figure 14 . 2 :Example 55 .

 5214114255 Figure 14.1: Example of RAF-paths: valid in green, invalid in red

Proposition 33 .

 33 Let RAF = ⟨A, K, s,t⟩ be a RAF and let x and y be two distinct elements of RAF . The following property holds:x ≡ RAF y iff there exists a RAF-closed-walk c = (e 1 , ..., e n ) ∈ ClosedWalk ra f (RAF ) such that:• x ∈ c. So there exists i ∈ {2, ..., n} such that: x = e i • y ∈ c. So there exists j ∈ {2, ..., n} such that: y = e j• e i-1 ∈ K • e j-1 ∈ K □ Proof ofProposition 33: link (See page 253). Note: The two last conditions of the previous proposition are related to the fact that x (respectively y) must be attacked by the RAF-closed-walk. Example 56. Let consider Figure 14.3 on the following page.

•□Figure 14 . 3 :

 143 Figure 14.3: Example of two SCC ra f : one in green and one in blue

Example 59 .Figure 14 . 4 :

 59144 Figure 14.4: RAF flattening illustration (Same as Figure 12.1)

Figure 14 . 5 :Example 62 .

 14562 Figure 14.5: RAF flattening illustration

Figure 14 . 6 :Example 63 .Figure 14 . 7 :

 14663147 Figure 14.6: RAF flattening illustration n°2

Figure 14 . 8 :Proposition 37 .

 14837 Figure 14.8: SCC ra f to SCC a f 2nd exampleOne SCC ra f and its corresponding SCC a f in green One SCC ra f and its corresponding SCC a f in blue (light blue used for elements not in the initial RAF)

Example 65 .Figure 14 . 9 :

 65149 Figure 14.9: Dag scc (RAF ) corresponding to the RAF in Figure 14.7(a) on page 131

  a predecessor of S j } Proposition 39. Let RAF = ⟨A, K, s,t⟩ be a RAF. Dag scc (RAF ) is acyclic. □ Proof of Proposition 39: link (See page 267). Example 66. Let consider the RAF in Figure 14.7(a) on page 131 and let illustrate the notion of Dag scc .

Figure 14 . 10 :

 1410 Figure 14.10: Example of attacks in cascade

2 Figure 14 . 11 :

 21411 Figure 14.11: Illustration of partial RAFs

Example 71 .Q inp 2 be the input of RAF 2 . Let L inp 1 (respectively L inp 2 ) 2 Figure 14 . 12 :

 71221221412 Figure 14.12: Illustration of partial RAFs with inputs Note that input elements do not belong to the partial RAFs.

RAF 2 ,Figure 14 . 13 :

 21413 Figure 14.13: Standard RAFs example

  out), (d, in)}, {(δ , in)}⟩} The next step consists in establishing a link between a RAF partition and a partition of its flattened version. Let first define a RAF partition selector: Definition 100. (RAF Partition selector). A RAF partition selector S is a function receiving as input a RAF RAF = ⟨A, K, s,t⟩ and returning a set of partitions of A ∪ K.

  Figure 14.14: RAF-compliant partition example

Figure 14 . 16 :κ 2 Figure 14 . 17 :

 141621417 Figure 14.15: Running example for semantics decomposability illustration

  af ′ 1.1 the induced AF of κ 1 under µ 1.1 (Notice that δ has disappeared since its attacker is labelled in ).

Figure 14 . 18 :

 1418 Figure 14.18: When c and γ are accepted

Figures 14 . 2 Figure 14 . 19 :

 1421419 Figures 14.18 to 14.23 on pages 145-150 show the standard RAFs produced from RAF 1 and RAF 2 associated with the induced AF produced from κ 1 and κ 2 . It is worth to notice that several standard RAFs can be associated to a same induced AF (see for instance Figures 14.19 and 14.20 on the next page and on page 147).Let consider Figure14.18. Notice that, only for this figure, the elements belonging to the partial RAF are highlighted in green and those being the RAF inputs in blue, we have:3

3 Figure 14 . 20 :

 31420 Figure 14.20: When neither c nor γ is out and one of them is und

1 Figure 14 . 21 :

 11421 Figure 14.21: When a is accepted With:

  in ) , (¬d, out ), (δ , out ) , (¬δ , in ), (d.δ , out ), (a, in ) , (¬a, out )}} Let consider Figure 14.19 on page 146. We have: • L co-ra f ( RAF s 1.2 ) =    {(a, out ), (d, in ), (c, in ), (ρ, in ), (υ, und ), (ζ , in )},

2 Figure 14 . 22 :

 21422 Figure 14.22: When a is rejected With:

=

  in ), (¬d, out ), (δ , in ), (¬δ , out ), (d.δ , in ), (a, out ), (¬a, in )}} Let consider Figure14.20 on page 147. We have:• L co-ra f ( RAF s 1.3 ) =    {(a, und ), (d, in ), (c, in ), (ρ, in ), (υ, und ), (ζ , in )}, {(δ , und ), (γ, und ), (θ , in ), (α γ , in )}    • L co-ra f ( RAF s 1.7 ) =    {(a, und ), (d, in ), (c, und ), (ρ, in ), (υ, und ), (ζ , in )}, {(δ , und ), (γ, in ), (θ , in ), (α c , in )}    • L co-ra f ( RAF s 1.9 ) =    {(a, und ), (d, in ), (c, und ), (ρ, in ), (υ, und ), (ζ , in )},{(δ , und ), (γ, und ), (θ , in ), (α γ , in ), (α c , in )} {{(d, in ), (¬d, out ), (δ , und ), (¬δ , und ), (d.δ , und ), (a, und ), (¬a, und )}} Let consider Figure14.21 on the previous page. We have:

3 Figure 14 . 23 :• L inp 2 . 3 =

 3142323 Figure 14.23: When a is undecided With:• L inp 2.3 = ⟨{(a, und )}, ∅⟩ • µ 2.3 = {(¬a, und )}



  in ), (¬α, out ), (a.α, und ), (β , und ), (¬β , und ), (b.β , und ), (b, in ), (¬b, out ), (c, und ), (¬c, und ), (γ, in ), (¬γ, out ), (c.γ, und ) Notice that we have the following equalities:

Figure 14 . 24 :

 1424 Figure 14.24: Demonstration overview: schema n • 1

  ¬s(α), s(α).α)|α ∈ K and s(α) = false} Example 75. Let RAF = ⟨A, K, s,t⟩ and AF = Raf2Af(RAF ) be respectively the RAF and the AF represented in Figure14.26 on the next page. Let Ω ′ = {{d, ¬d, c, ¬c, γ, ¬γ, c.γ}, {δ , ¬δ , d.δ , a, ¬a, α, ¬α, a.α, b, ¬b}} be a partition of A ′ . We have: Ω ′ ∈ S D-ra f -c (AF ). Let Ω = Ω ′ ∩ (A ∪ K) be the partition of A ∪ K corresponding to Ω ′ . Let us split RAF along the partition Ω and AF along the partition Ω ′ .

  false and s1 (α) = true • t1 (δ ) = true and t1 (α) = true Let I 1 = ⟨{c, d}, {γ}⟩ be the input corresponding to RAF 1 , as represented in Figure 14.27 on the next page. Let as an example L inp 1 = ⟨{(c, in), (d, und)} , {(γ, out)}⟩ be a labelling of I 1 .

  Figure 14.26: Decomposability illustration

Figure 14 Figure 14 . 28 : 1 (

 1414281 Figure 14.27: RAF 1

Figure 14 .

 14 Figure 14.29: RAF ′ s : the flattened version of RAF s

Figure 16 .Figure 16 . 1 :

 16161 Figure 16.1: Example of graphs

Figure 16 . 2 :

 162 Figure 16.2: Example of weighted graphs

  That is: deg(G, v) = out-deg(G, v)+ in-deg(G, v). Example 80. Let consider Figure 16.1 on the previous page. In both graphs: • m and n are incident to e = (m, n) and vice versa • m and n are adjacent In the non-directed graph, we have: deg(G, m) = 2. In the directed graph, we have: deg(G, m) = 3, in-deg(G, m) = 2 and out-deg(G, m) = 1.

  e) Note: In order to simplify the notation, deg(G, v) (resp. deg w (G, v)) will be noted deg(v) (resp. deg w (v)) when there is no ambiguity about the graph in which the degree is measured. Example 81. Let consider Figure 16.2. In both graphs we have: deg(m) = 11.

Example 85 .Figure 16 . 3 :

 85163 Figure 16.3: A weighted non-directed graph

Figure 16 .

 16 Figure 16.4 on the next page illustrates this hierarchy.

Figure 16 . 4 :

 164 Figure 16.4: Polynomial hierarchy

Figure 17 . 1 :

 171 Figure 17.1: CSP illustration

Figure 17 . 2 :

 172 Figure 17.2: The weighted non-directed graph of Example 86

Figure 17 . 3 :

 173 Figure 17.3: eigenvalues sorted by ascending order

2 Figure 17 . 4 :

 2174 Figure 17.4: Node datapoints projected in similarity space Figure 17.4 shows two clusters as expected: G ↓ {m,n} and G ↓ { j,k,l} .

Table 18 . 2 :

 182 ra f The abbreviation for "RAF USCC". See Definition 109 on page 161 BA, ER, WS, TR, F2,BW The abbreviations for respectively Barabási-Albert, Erdős-Rényi, Watts-Strogatz, Traffic, Ferry and Block world graph types. 194 Graph and matrix symbols Symbol Meaning G = (V, E) A graph G with V being a set of nodes and E being a set of edges G = (V, E,W ) A graph G with V being a set of nodes, E being a set of edges and W being a function that associates a weight to any e ∈ E deg(G, v) Let G = (V, E) be a non-weighted graph and v ∈ V . deg(G, v) is the number of incident edges to v in G. See Definitions 116 and 117 on page 172 deg w (G, v) Let G = (V, E,W ) be a weighted graph and v ∈ V . deg(G, v) is the weight sum of the incident edges to v in G. See Definition 118 on page 172 R d (G) Let G be a graph. R d (G) is the relation density in G with: R d (G) = |E| |V | See Definition 131 on page 174 M A matrix. See Definition 132 on page 175

  being a partition of A. In the case of RAFs, we have, given a RAF RAF = ⟨A, K, s,t⟩: Ω = {ω 1 , ..., ω n } being a partition of A ∪ K Table 18.4: AF related Symbols Symbol Meaning Φ a f The set of all possible AF AF = ⟨A, K⟩ An AF where: • A is a set of arguments • K is a set of attacks (i.e. K ⊆ A × A) af = ⟨A, K⟩ Same definition as the previous one. This notation is used occasionally to represent a sub-AF and especially connected component sub-AFs of an AF named AF . See for example Definition 40 on page 64. Continued on next page ... Table 18.4: AF related Symbols (continued) Symbol Meaning De f (S) The set of arguments defeated w.r.t. the set of arguments S Acc(S) The set of arguments acceptable w.r.t. the set of arguments S L (AF ) Let AF be an AF. L (AF ) is the set of all possible labellings of AF L (S) Let S be a set of arguments. L (S) is the set of all possible labellings of S L σ (AF ) Let AF be an AF. L σ (AF ) is the set of AF labellings under the semantics σ ℓ A labelling

32 µ 35 ξ 35 DD 43 Ξ

 32353543 a, b) ∈ O or (b, a) ∈ I} See Definition 30 on page Let κ = af , I, O, B be a cluster structure. A context µ of κ is a labelling of the inward attack sources of κ, i.e. {a|(a, b) ∈ I}. See Definition 31 on page 34 L µ(κ) σ The set of induced labelling produced by the cluster structure κ under the context µ. See Definition 33 on page Let κ = af , I, O, B be a cluster structure, L µ(κ) σ a set of induced labellings and ℓ ∈ L µ(κ) σ be a labelling. The configuration ξ corresponding to ℓ is a five value-based labelling of the border arguments of κ. See Definition 34 on page 36 af ′ Let κ = af , I, O, B be a cluster structure and µ be a context of κ. We denote by af ′ the induced AF from κ under µ. See Definition 32 on page The set of deleted arguments from the an induced AF. See Definition 32 on page 35 Continued on next page ...Table 18.4: AF related Symbols (continued) Symbol Meaning L κ The set of distinct labellings of a cluster structure κ. See Definition 35 on page 39 ξ ℓ κ i The merge configuration corresponding to the labelling ℓ i of the cluster structure κ. See Definition 36 on page 41 p = {ξ 1 , ..., ξ n } A reunified labelling profile. See Definition 37 on page Let p = {ξ 1 , ..., ξ n } be a reunified labelling profile. Ξ is the union of those configurations: Ξ = ( n i=1 ξ i ) P κ

  and t(β ) = α} See Definition 61 on page 92 RAF-Acc(U) Let RAF = ⟨A, K, s,t⟩ be a RAF and U = ⟨S, Q⟩ be a structure of RAF . RAF-Acc(U) is the set of elements acceptable w.r.t. U, that is: {e ∈ (A ∪ K)|(∃α ∈ K s.t. t(α) = e) =⇒ (s(α) ∈ RAF-De f (U) or α ∈ RAF-Inh(U))}. See Definition 62 on page 92

  is the set of structure labellings of RAF under the semantics σ Lab2Struct(ℓ) Let RAF = ⟨A, K, s,t⟩ be a RAF. The function Lab2Struct transforms a labelling ℓ of RAF into its corresponding structure U of RAF . See Definition 75 on page 107 Continued on next page ...Table 18.5: RAF related Symbols (continued) Symbol Meaning Struct2Lab(U) Let RAF = ⟨A, K, s,t⟩ be a RAF. The function Struct2Lab transforms a structure U of RAF into its corresponding labelling ℓ of RAF . See Definition 75 on page 107 SCCS ra f (RAF ) The set of SCC ra f of RAF . See Definition 87 on page 125 Paths ra f (RAF ) The set of RAF-paths of RAF . See Definition 82 on page 123 Cycles ra f (RAF ) The set of RAF-cycles of RAF . See Definition 83 on page 124 ClosedWalk ra f (RAF ) The set of RAF-closed-walks of RAF . See Definition 84 on page 124 PE ra f (RAF ) The path-equivalence relation over RAF . See Definition 87 on page 125 ≡ RAF ̸ ≡ RAF Other notations for the path-equivalence relation over given RAF . "x ≡ RAF y"

  t⟩ be a RAF and AF = Raf2Af(RAF ) be the flattened version of RAF . Let U be a structure of RAF . ε U is the extension of AF corresponding to U. See Definition 77 on page 116 rafLab2AfLab(L) Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be the flattened version of RAF . rafLab2AfLab is a function that transforms a structure labelling L of RAF into the labelling of AF corresponding to it. See Definition 78 on page 118 afLab2RafLab(ℓ) Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be the flattened version of RAF . afLab2RafLab is a function that transforms a labelling ℓ of AF into the structure labelling of AF corresponding to it. See Definition 78 on page 118 Raf2Af maf (RAF ) Let RAF = ⟨A, K, s,t⟩ be a RAF. The function Raf2Af maf flattens a RAF into a MAF/AF. See Definition 68 on page 98 str2MafExt(U) Let RAF = ⟨A, K, s,t⟩ be a RAF and MAF = Raf2Af maf (RAF ). Let U be a structure labelling of RAF . str2MafExt(U) is the extension of MAF corresponding to U. See Definition 69 on page 98 Continued on next page ... Table 18.6: Transformation related symbols: RAF and AF (continued) Symbol Meaning S ra f -c (AF ) Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be the flattened version of RAF . S ra f -c is a partition selector that selects partitions of AF that are "compliant" with RAF . See Definition 101 on page 139 S D-ra f -c (AF ) Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be the flattened version of RAF . S D-ra f -c , so-called "default RAF-compliant partition selector", is the partition selector that produces all the RAF-compliant partitions of AF . See Definition 102 on page 140

Figure 1 Figure 1 . 1

 111 Figure 1 on page 7 An AF example Figure 1.1 on page 10 AF semantics partial ordering and cardinality

Figures 2 .

 2 Figures 2.1 to 2.3 on pages 16-18 Given an initial AF, these figures illustrate what is an AF with input and its corresponding standard AF Figure 3.1 on page 24 A recall of Example 1 on page 7 to illustrate AF decision problems

Figures 4 .Figure 4 . 9 Figure 4 .Figure 4 .Figure 6 . 1

 4494461 Figures 4.2 to 4.8 on pages 32-38 An illustration of the different steps of the AFDivider algorithm Figure 4.9 on page 38 An cluster example showing the interest of the 5-value labelling used for the reunification made by the the AFDivider algorithm Figure 4.11 on page 40 An illustration of merge configuration Figure 4.12 on page 42 An illustration of same merge configuration for different labellings Figures 5.1 to 5.3 on pages 52-62 An illustration of the spectral clustering partition method Figures 5.4 to 5.6 on pages 62-63 An illustration of the USCC-based partition methods Figure 6.1 on page 65 A recall of Example 1 on page 7 to illustrate the Compact Enumeration Representation Continued on next page ...

Figure 7 .Figure 7 . 7

 777 Figure 7.1 on page 73An AF example to illustrate the other direct-approach algorithms Figures 7.2 to 7.6 on pages 74-77 There figures illustrate the algorithm proposed in [43] Figure 7.7 on page 78An SCC decomposition hierarchy of an AF. This layered schema is used to illustrate the algorithms proposed in[START_REF] Liao | Toward incremental computation of argumentation semantics: A decomposition-based approach[END_REF] and[START_REF] Cerutti | Exploiting parallelism for hard problems in abstract argumentation[END_REF]

Figure 7 .Figure 8 . 2 Figure 9 . 1 Figure 9 . 2 Figure 9 . 3 Figure 9 . 4 Figure 12 Figure 13 . 1 Figure 14 . 1

 7829192939412131141 Figure 7.9 on page 83 An example of a RAF/AFRA Figure 8.1 on page 89 AFRA semantics partial ordering and cardinality Figure 8.2 on page 90 An illustration of the fact that AFRA does not correspond to a conservative generalization of AF Figure 9.1 on page 95 RAF semantics partial ordering and cardinality Figure 9.2 on page 96 An example of non recursive RAF Figure 9.3 on page 97 The AF corresponding to the non recursive framework of Figure 9.2 on page 96 Figure 9.4 on page 98 An example of RAF flattening with the Raf2Af maf function Figure 10.1 on page 102 A RAF example Figure 11.1 on page 112 A recall of Figure 10.1 on page 102 to illustrate RAF semantics labellings Figure 12.1 on page 116 A RAF flattening illustration Figure 13.1 on page 120 A recall of Figure 10.1 on page 102 to illustrate RAF decision problems Figure 14.1 on page 124 An example of RAF-paths Figure 14.3 on page 127 An example of SCC ra f

Figure 14 .Figure 14 .Figure 14 .Figure 14 .Figure 16 Figure 16 . 4

 1414141416164 Figure 14.9 on page 133 Dag scc (RAF ) corresponding to the RAF in Figure 14.7(a) on page 131 Figure 14.10 on page 134 A RAF example having attacks in cascade Figure 14.11 on page 135 Illustration of partial RAFs Figure 14.12 on page 137 Illustration of partial RAFs with inputs Figure 14.13 on page 139 An example of standard RAFs Figure 14.14 on page 141 An example of RAF-compliant partitionFigures 14.15 to 14.23 on pages 143-150 An illustration of the decomposability RAF semantics Figures 14.24 and 14.25 on page 153 and on page 154 Schemas showing the demonstration overview of RAF semantics decomposability properties Figures 14.26 to 14.29 on pages 156-158 These figures illustrate that given a RAF and its flattened version, there is a correspondence between its RAFs with input and AFs with input, between their respective standard RAFs and standard AFs, and finally, between the flattened version of a standard RAF and its corresponding standard AF Figure 16.1 on page 171 Examples of graph Figure 16.2 on page 172 Examples of weighted graph Figure 16.3 on page 176 A weighted non-directed graph Figure 16.4 on page 182 A schema of the Polynomial Hierarchy Continued on next page ...

Figure 17 .Figure 17 . 3

 17173 Figure 17.1 on page 187 An illustration of CSP modeling Figure 17.2 on page 189 The weighted non-directed graph of Example 86 on page 178 Figure 17.3 on page 191 Following Example 90 on page 192 illustrating the Spectral Clustering method, this graphics shows eigenvalues sorted by ascending order Figure 18.1 on page 261 A counter example that shows that Assertion 2 of Lemma 10 on page 260 is only an implication and not an equivalence Figure 18.2 on page 283 Example giving an intuition for the proof of Propositions 43 and 45

34 )

 34 Let A 0 = {a|a ∈ in (ℓ gr ) or a ∈ out (ℓ gr )} be the fixed part of AF . The set of argument sets Ω = {A 0 , A 1 , ..., A n } is then a partition of A. By definition of the grounded labelling, we have:∃a ∈ A s.t. ℓ gr (a) = und =⇒ (∀a ′ ∈ A s.t. (a ′ , a) ∈ K, ℓ gr (a) ̸ = in ) (18.35)Given that:und (ℓ gr ) ∩ A 0 = ∅ (18.36)And that by construction of A 0 :∀i ∈ {1, ..., n}, ∀a ∈ A i , ℓ gr (a) = und(18.37)The consequence of Equation(18.35) is:∀i ∈ {1, ..., n}, ∀(a ′ , a) ∈ K s.t. a ′ ∈ A 0 and a ∈ A i , ℓ gr (a ′ ) = out(18.38)

Assertion 2 :

 2 Skep σ (AF , a) ≡ Comp-Skep σ (AF , a). • Case 1: If Skep σ (AF , a) is true, then Comp-Skep σ (AF , a) is true. If Skep σ (AF , a) is true, then σ (AF ) ̸ = ∅ and ∀ℓ ∈ σ (AF ), we have: a ∈ in (ℓ).

Assertion 4 :

 4 Exists σ (AF ) ≡ Comp-Exists σ (AF )• Case 1: If Exists σ (AF ) is true, then Comp-Exists σ (AF ) is true.

Proof of Theorem 7

 7 on page 103. Let suppose that there exists a stable structure U = ⟨S, Q⟩. Following the definition of stable structures (Definition 63 on page 92), we have: S = A \ RAF-De f (U) and Q = K \ RAF-Inh(U). As a consequence, we have (S ∪ RAF-De f (U) ∪ Q ∪ RAF-Inh(U)) including all the arguments and attacks of RAF .

  extension of AF ■ Proofs of Chapter 11: Semantics and Labellings Proofs of Section 11.1: Complete semantics Proof of Theorem 8 on page 107. Let U = Lab2Struct(L). According to Definition 63 on page 92, U being a complete structure (with U = ⟨S, Q⟩) means that (S ∪ Q) = Acc(U). In a first step, let us prove that (S ∪ Q) ⊆ Acc(U) and then that (S ∪ Q) ⊇ Acc(U).

( 18 . 58 )

 1858 So two cases must be considered:ℓ K (α) = out or ℓ A (s(α)) = out . 1. ℓ K (α) = out .Given L is a reinstatement RAF labelling there exists an attack β such that t(β ) = α, ℓ K (β ) = in and ℓ A (s(β )) = in . As a consequence, β ∈ Q and s(β ) ∈ S. According to Definition 61 on page 92, we have so: α ∈ RAF-Inh(U).

2 .

 2 ℓ A (s(α)) = out . Given L is a reinstatement RAF labelling there exists an attack γ such that t(γ) = s(α), ℓ K (γ) = in and ℓ A (s(γ)) = in . As a consequence, γ ∈ Q and s(γ) ∈ S. According to Definition 61 on page 92, we have so: s(α) ∈ RAF-De f (U).

1 .

 1 (L(x) = out ) =⇒ (∃α ∈ K s.t. t(α) = x, ℓ K (α) = in and ℓ A (s(α)) = in ) 2. (L(x) = out ) ⇐= (∃α ∈ K s.t. t(α) = x, ℓ K (α) = in and ℓ A (s(α)) = in ) 3. (L(x) = in ) =⇒ (∀α ∈ K s.t. t(α) = x, ℓ K (α) = out or ℓ A (s(α)) = out ) 4. (L(x) = in ) ⇐= (∀α ∈ K s.t. t(α) = x, ℓ K (α) = out or ℓ A (s(α)) = out ) Step 1: (L(x) = out ) =⇒ (∃α ∈ K s.t. t(α) = x, ℓ K (α) = in and ℓ A (s(α)) = in )Let x ∈ (A ∪ K) be an argument or an attack such that L(x) = out . According to the definition of Struct2Lab(U), we have x ∈ (RAF-De f (U) ∪ RAF-Inh(U)). Following the definitions of RAF-De f (U) and RAF-Inh(U), we can state that there exists an attack α such that α ∈ Q, s(α) ∈ S and t(α) = x. According to the definition of Struct2Lab(U), we have so ℓ K (α) = in and ℓ A (s(α)) = in .

Assertion 1 :Assertion 3 :

 13 RAF-Cred σ accepts (RAF , a) iff Cred σ accepts (AF , a). RAF-Cred σ accepts (RAF , a) iff ∃U ∈ σ (RAF ) s.t. a ∈ U iff ∃ε U ∈ σ (AF ) s.t. a ∈ ε U (following Proposition 25 on page 117) iff Cred σ accepts (AF , a) Assertion 2: RAF-Skep σ accepts (RAF , a) iff Skep σ accepts (AF , a). RAF-Skep σ accepts (RAF , a) iff ∀U ∈ σ (RAF ), a ∈ U iff ∀ε U ∈ σ (AF ), a ∈ ε U (following Proposition 25 on page 117) iff Skep σ accepts (AF , a) RAF-Ver σ accepts (RAF , U) iff Ver σ accepts (AF , ε U ).

∃(e 1

 1 = t(α), ..., e n = s(α)) ∈ Paths ra f (RAF ) s.t. e n-1 ∈ K ⇐⇒ t(α) ̸ = s(α) and (t(α) and s(α) are in the same S ∈ SCCS ra f (RAF )) Proof of Lemma 3. In two steps: • Step 1: ∃(e 1 = t(α), ..., e n = s(α)) ∈ Paths ra f (RAF ) s.t. e n-1 ∈ K =⇒ t(α) ̸ = s(α) and (t(α) and s(α) are in the same S ∈ SCCS ra f (RAF ))

• Step 2 :

 2 t(α) ̸ = s(α) and (t(α) and s(α) are in the same S ∈ SCCS ra f (RAF )) =⇒ ∃(e 1 = t(α), ..., e n = s(α)) ∈ Paths ra f (RAF ) s.t. e n-1 ∈ K By definition, t(α) ≡ RAF s(α) =⇒ ∃(e 1 = t(α), ..., e n = s(α)) ∈ Paths ra f (RAF ) s.t. e n-1 ∈ K.

  ⇐⇒ ∃p = (x = e 1 , ..., y = e n ) ∈ Paths ra f (RAF ) s.t. e n-1 ∈ K Proof of Lemma 7. In two steps: Step 1: Let prove that: ∃p ′ = (x, ..., y) ∈ Paths a f (AF ) s.t. (x, y) ∈ (A ∪ K) 2 =⇒ ∃p = (x = e 1 , ..., y = e n ) ∈ Paths ra f (RAF ) s.t. e n-1 ∈ K Let p ′ = (x = o 1 , ..., y = o m ) ∈ Paths a f (AF ) s.t. (x, y) ∈ (A ∪ K) 2 . Let consider two cases :

Step 2 :Lemma 8 .• Step 1 :

 281 e 1 , ..., y = e n ) ∈ Paths ra f (RAF ) s.t. e n-1 ∈ K Let prove that: ∃p = (x = e 1 , ..., y = e n ) ∈ Paths ra f (RAF ) s.t. e n-1 ∈ K =⇒ ∃p ′ = (x, ..., y) ∈ Paths a f (AF ) s.t. (x, y) ∈ (A ∪ K) 2 Let p = (x = e 1 , ..., y = e n ) ∈ Paths ra f (RAF ) s.t. e n-1 ∈ K. According to Lemma 6 on page 257: ∃p ′ = (x, ..., y) ∈ Paths a f (AF ). Furthermore, (x, y) ∈ (A ∪ K) 2 . We prove from Step 1 and Step 2 that Lemma 7 on the previous page holds. ■ Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be the AF corresponding to RAF . Let S ∈ SCCS a f (AF ) and x ∈ A ∪ K. The following property holds: |S| > 1 and x ∈ S ⇐⇒ |S| > 1 and ¬x ∈ S Proof of Lemma 8. In two steps: |S| > 1 and x ∈ S =⇒ |S| > 1 and ¬x ∈ S

Figure 18 . 1 :

 181 Figure 18.1: Counter example illustration

∈

  {¬a ∈ Not A |a ∈ S and (|S| > 1 or (∃α ∈ K s.t. s(α) = a and t(α) = a))} ∪ {¬α ∈ Not K |α ∈ S and (|S| > 1 or t(α) = α)} ∪ {s(α).α ∈ And A,K |α ∈ S and (|S| > 1 or t(α) = α)} ∪ {s(α).α ∈ And A,K |s(α) ∈ S and t(α) ∈ S} SCCS a f (Raf2Af(RAF )) • If e n = o 1 then we have: (e 1 , ..., e n-1 , e n = o 1 , ..., o m-1 , o m ) ∈ Paths ra f (RAF ). As e 1 ∈ S 1 and o m ∈ S 3 , we have: S 1 ≼ S 3 . • Otherwise, if e n ̸ = o 1 then we have: |S 2 | > 1. Given that (e n , o 1 ) ∈ PE ra f (RAF ) then: ∃(e n = x 1 , ..., x p-1 , x p = o 1 ) ∈ Paths ra f (RAF ) s.t. x p-1 ∈ K and x p = o 1 . As a consequence: (e 1 , ..., e n-1 , e n = x 1 , ..., x p-1 , x p = o 1 , ..., o m-1 , o m ) ∈ Paths ra f (RAF ). We have thus: S 1 ≼ S 3 .

• Step 2 :

 2 [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF].101) to(18.103), we prove the following property:x ∈ ω ′ =⇒ x ∈ Ã′ s (18.104)From Equations (18.100) and (18.104) on the previous page and on this page we prove that:ω ′ ⊆ A ′ s ∩ Ã′ s Let prove that J ⊆ A ′ s ∩ Ã′s . By definition of the standard AF, we have:J ⊆ A ′ s (18.105)Let prove that: J ⊆ Ã′ s . Let x ∈ J . Following Assertion 1 of Lemma 11 on page 267, we have: x ∈ Not A or x ∈ And A,K . Let consider two cases.

  (x, y) ∈ K ′ s ∩ (ω ′ × ω ′ ) s.t. x ∈ A ∪ K and y ∈ Not A ∪ Not K =⇒ (x, y) ∈ K′ s ∩ (ω ′ × ω ′ ) (18.108)-Case 2: x ∈ Not A and y ∈ And A,KAs x ∈ Not A , let assume that x = ¬z with z ∈ A.

  we have: (¬z, z.α) ∈ K′ s and so: (x, y) ∈ K′ s . The following property then holds:(x, y) ∈ K ′ s ∩ (ω ′ × ω ′ ) s.t. x ∈ Not A and y ∈ And A,K =⇒ (x, y) ∈ K′ s ∩ (ω ′ × ω ′ ) (18.110)-Case 3: x ∈ Not K and y ∈ And A,KAs x ∈ Not K , let assume that x = ¬α with α ∈ K and that y = s(α).α.Given that Ω ′ ∈ S D-ra f -c (RAF ), that ω ′ ∈ Ω ′ and that ¬α ∈ ω ′ , we have: α ∈ ω ′ . Moreover, as α ∈ K, we have: α ∈ ω. α is thus an element of RAF . As a consequence, we have: α ∈ Ks . Following the definition of Raf2Af, we have so:{α, ¬α, s(α).α} ∈ Walks a f ( RAF ′ s ). The following property then holds: α ∈ ω ′ =⇒ {α, ¬α, s(α).α} ∈ Walks a f ( RAF ′ s )(18.111)

  ⇐⇒ L inp (a) = out out ⇐⇒ L inp (a) = in und ⇐⇒ L inp (a) = und 2. ∀s(α).α ∈ J ∩ And A,K , ∀ℓ ∈ L σ ( RAF′ ⇐⇒ L inp (α) = in and L inp (s(α)) = in out ⇐⇒ L inp (α) = out or L inp (s(α)) = out und ⇐⇒ L inp (α) ̸ = out and L inp (s(α)) ̸ = out and (L inp (α) = und or L inp (s(α)) = und) 3. ∀a ∈ J , ∀ℓ 1 ∈ L σ ( RAF ′ s ), ∀ℓ 2 ∈ L σ (std-AF ), ℓ 1 (a) = ℓ 2 (a)Proof of Lemma 13. Assertion 1:∀¬a ∈ J ∩ Not A , ∀ℓ ∈ L σ ( RAF ′ ⇐⇒ L inp (a) = out out ⇐⇒ L inp (a) = in und ⇐⇒ L inp (a) = und Let ℓ ∈ L σ ( RAF ′ s) and ¬a ∈ J ∩ Not A . Given that ¬a ∈ J ∩ Not A we have following Definition 105 on page 155: a ∈ S inp . As a consequence, following the definition of standard RAF we have: a ∈ Ãs . Following the definition of Raf2Af, we have so: a ∈ Ã′ s . As a consequence ℓ(a) must be defined. Let L = afLab2RafLab(ℓ). Given that σ -raf is the RAF semantics corresponding to σ , we have: L ∈ L σ -ra f ( RAF s ). Moreover, according to Definition 78 on page 118 we have:∀x ∈ ( Ãs ∪ Ks ), ℓ(x) = L(x)Given that L ∈ L σ -ra f ( RAF ′ s ), we have following the definition of standard RAF:∀x ∈ S inp , L(x) = L inp (x)As a consequence, we have:∀x ∈ S inp , ℓ(x) = L inp (x) (18.125)As σ is complete-based, we have: ℓ(a) = in ⇐⇒ ℓ(¬a) = out , ℓ(a) = out ⇐⇒ ℓ(¬a) = in and ℓ(a) = und ⇐⇒ ℓ(¬a) = und . From those equivalences and from Equation (18.125), we prove so that Assertion 1 holds. Assertion 2: ∀s(α).α ∈ J ∩ And A,K , ∀ℓ ∈ L σ ( RAF ′ ⇐⇒ L inp (α) = in and L inp (s(α)) = in out ⇐⇒ L inp (α) = out or L inp (s(α)) = out und ⇐⇒ L inp (α) ̸ = out and L inp (s(α)) ̸ = out and (L inp (α) = und or L inp (s(α)) = und ) Let ℓ ∈ L σ ( RAF ′ s

  ∀x ∈ ( Ãs ∪ Ks ), ℓ(x) = L(x)Given that L ∈ L σ -ra f ( RAF ′ s ), we have following the definition of standard RAF:∀x ∈ (S inp ∪ Q inp ), L(x) = L inp (x)As a consequence, we have:∀x ∈ (S inp ∪ Q inp ), ℓ(x) = L inp (x) (18.126)As for all s(α).α ∈ And Ãs , Ks (σ being complete-based), we have:ℓ(s(α).α) ⇐⇒ ℓ(α) = in and ℓ(s(α)) = in out ⇐⇒ ℓ(α) = out or ℓ(s(α)) = out und ⇐⇒ ℓ(α) ̸ = out and ℓ(s(α)) ̸ = out and (ℓ(α) = und or ℓ(s(α)) = und )And as Equation (18.126) holds, we prove so that Assertion 2 holds.Assertion 3: ∀a ∈ J , ∀ℓ 1 ∈ L σ ( RAF ′ s ), ∀ℓ 2 ∈ L σ (std-AF ), ℓ 1 (a) = ℓ 2 (a) Let ℓ 1 ∈ L σ ( RAF ′ s ) be any σ -labelling of RAF ′ s . Let ℓ 2 ∈ L σ (std-AF ) be any σ -labelling of std-AF .By Definition of the standard AF we have:∀ℓ ∈ L σ (std-AF ), ∀x ∈ J , ℓ(x) = ℓ J (x)We have so:∀x ∈ J , ℓ 2 (x) = ℓ J (x) (18.127)Following Definition 105 on page 155 (σ being complete-based), we have:• ∀(s(α).α) ∈ J s.t. α ∈ Q inp , ℓ J (s(α).α) = ⇐⇒ L inp (α) = in and L inp (s(α)) = in out ⇐⇒ L inp (α) = out or L inp (s(α)) = out und ⇐⇒ L inp (α) ̸ = out and L inp (s(α)) ̸ = out and (L inp (α) = und or L inp (s(α)) = und ) • ∀¬a ∈ J s.t. a ∈ S inp , ⇐⇒ L inp (a) = out out ⇐⇒ L inp (a) = in und ⇐⇒ L inp (a) = undFollowing Equation (18.127), we have so:• ∀(s(α).α) ∈ J s.t. α ∈ Q inp , ℓ 2 (s(α).α) = ⇐⇒ L inp (α) = in and L inp (s(α)) = in out ⇐⇒ L inp (α) = out or L inp (s(α)) = out und ⇐⇒ L inp (α) ̸ = outand L inp (s(α)) ̸ = out and (L inp (α) = und or L inp (s(α)) = und ) • ∀¬a ∈ J s.t. a ∈ S inp , ⇐⇒ L inp (a) = out out ⇐⇒ L inp (a) = in und ⇐⇒ L inp (a) = und Given that by definition J = {s(α).α|α ∈ Q inp } ∪ {¬a|a ∈ S inp }, we prove, following Assertions 1 and 2, that: ∀a ∈ J , ∀ℓ 1 ∈ L σ ( RAF ′ s ), ∀ℓ 2 ∈ L σ (std-AF ), ℓ 1 (a) = ℓ 2 (a). ■ Proof of Proposition 40 on page 158. According to Assertions 1 and 2 of Lemma 12 on page 270, we have: std-AF ↓ ω ′ = RAF ′ s ↓ ω ′ (18.128)

1 .

 1 .129) According to Assertion 4 of Lemma 11 on page 267, we have:(x, y) ∈ K′ s ∪ K ′ s s.t. y ∈ ω ′ =⇒ x ∈ ω ′ ∪ J (18.130)According to Assertion 3 of Lemma 13 on page 277, we have:∀a ∈ J , ∀ℓ 1 ∈ L σ ( RAF ′ s ), ∀ℓ 2 ∈ L σ (std-AF ), ℓ 1 (a) = ℓ 2 (a)Following Equation (18.128) on the previous page, both AFs are identical restricted to ω ′ 2. Following Equations (18.129) and (18.130) on the previous page, this common subAF is attacked in both AFs identically (by same arguments and attack relations) 3. Following Equation (18.131) on the previous page, all arguments attacking this common subAF is labelled identically in both AFs and for all possible labellings 4. The semantics σ is complete-basedWe have so:{ℓ ↓ ω ′ ∪J |ℓ ∈ L σ ( RAF ′ s )} = {ℓ ↓ ω ′ ∪J |ℓ ∈ L σ (std-AF )} ■Proof of Proposition 41 on page 159. Given that ω ′ ⊆ (ω ′ ∪ J ), we have, following Equations (18.128) and (18.129) on the previous page given in the proof of Proposition 40 on page 158:{ℓ ↓ ω ′ |ℓ ∈ L σ ( RAF ′ s )} = {ℓ ↓ ω ′ |ℓ ∈ L σ (std-AF )} Equivalently, we have so: {ℓ ↓ ω ′ |ℓ ∈ L σ ( RAF ′ s )} = F a f σ (AF ↓ ω ′ , J , ℓ J , K J ) (18.132)Now, given that σ -raf is the RAF semantics corresponding to σ , we have:L σ -ra f ( RAF s ) = {afLab2RafLab(ℓ)|ℓ ∈ L σ ( RAF′ s )} Or equivalently: L σ -ra f ( RAF s ) = { ℓ ↓ Ãs , ℓ ↓ Ks |ℓ ∈ L σ ( RAF ′ s )} (18.133)

  [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF].135) By Definition we have:F ra f σ ( RAF , I, L inp ) = {L ↓ ⟨ Ã, K⟩ |L ∈ L σ -ra f ( RAF s )}(18.136)

3 Figure 18 . 2 :

 3182 Figure 18.2: Example giving an intuition for the proof of Propositions 43 and 45 Example 92. Let consider Figure 18.2. Let AF 1= ⟨A 1 , K 1 ⟩ be the AF illustrated in Figure 18.2(a), RAF 2 = ⟨A 2 , K 2 , s 2 ,t 2 ⟩ be the RAF illustrated in Figure 18.2(b) such that RAF 2 = Af2Raf(AF 1 ) and AF 3 = ⟨A 3 , K 3 ⟩ be the AF illustrated in Figure 18.2(c) such that AF 3 = Raf2Af(RAF 2 ).Let consider the semantics preferred , the bottom-up decomposability property and the following partition of AF 1 : Ω 1 = ω 1 1 = {a} , ω 1 2 = {b} . We have:

  is top-down (resp. bottom-up, fully) decomposable w.r.t. S D-ra f -c =⇒ σ is top-down (resp. bottom-up, fully) decomposable (18.159) ■ Proof of Proposition 44 on page 160. Trivial considering Proposition 43 on page 160 and the decomposability properties of AF semantics shown in Table 2.1 on page 22. ■ Proof of Proposition 45 on page 161. Following the definitions of S USCC (See Definition 27 on page 21) and of S ra f -c-USCC (See Definition 109 on page 161), we have:∀RAF ∈ Φ ra f , S ra f -c-USCC (Raf2Af(RAF )) ⊆ S USCC (Raf2Af(RAF )) (18.160)That is, for any RAF RAF = ⟨A, K, s,t⟩, each partition of Raf2Af(RAF ) produced by S ra f -c-USCC is also a partition of Raf2Af(RAF ) produced by S USCC .It follows that:σ is top-down (resp. bottom-up, fully) decomposable w.r.t. S USCC =⇒ σ is top-down (resp. bottom-up, fully) decomposable w.r.t. S ra f -c-USCC(18.161) Let prove that the reciprocal proposition is also true.

Table 1 .

 1 2: Semantic labellings "•" means that the semantics on the row produces the labelling on the column.

		ℓ 1	ℓ 2	ℓ 3	ℓ 4	ℓ 5	ℓ 6
	a	in	in	in	in	in	in
	b	out out out out out out
	c	out out out out out out
	d	out out	in	in	und und
	e	in	in	out out und und
	f	out out	in	in	und und
	g	in	in	out out und und
	h	out out	in	in	und und
	i	in	in	out out und und
	j	und und und und und und
	k	und und und und und und
	l	und und und und und und
	m	und out und out und out
	n	und	in	und	in	und	in
	grounded					•
	complete	•	•	•	•	•	•
	preferred		•		•
	stable

Table 1 .

 1

3: Reinstatement labelling and extension based semantics correspondence

Chapter 2

  The semantics properties in Table2.1 on the next page hold.

		Complete Grounded Preferred Semi-stable Stable
	Full decomposability	× × ×	× × ×	× × ×
	Top-down decomposability			× × ×
	Bottom-up decomposability	× × ×	× × ×	× × ×

• {{e ′ , e, f } , {d, g} , {h, i}} / ∈ S USCC (AF ) Proposition 8. Full decomposability w.r.t. S USCC × × × Top-down decomposability w.r.t. S USCC × × × Bottom-up decomposability w.r.t. S USCC × × × Table 2.1: AF Semantics decomposability properties

Table 4 .

 4

1: κ 1 labellings and configurations under µ κ 1 1 = ∅.

Table 4 .

 4 2: Labellings and configurations of κ 2 (three contexts).

Table 4 .

 4 .7 shows the ones for af 2 .

		ℓ 1.1 ℓ 1.2 ℓ 1.3
	d out	in	und
	e	in	out und
	f out	in	und
	g	in	out und
	h out	in	und
	i	in	out und
	Table 4.6: Complete labellings
			ℓ 2.1 ℓ 2.2
		j und und
		k und und
		l und und
		m und out
		n und	in

7: Complete labellings

Table 4 . 8 :

 48 Complete labellings of AF

		ℓ 1	ℓ 2	ℓ 3	ℓ 4	ℓ 5	ℓ 6
	a	in	in	in	in	in	in
	b out out out out out out
	c out out out out out out
	d out out	in	in	und und
	e	in	in	out out und und
	f out out	in	in	und und
	g	in	in	out out und und
	h out out	in	in	und und
	i	in	in	out out und und
	j und und und und und und
	k und und und und und und
	l und und und und und und
	m und out und out und out
	n und	in	und	in	und	in

  Let σ be a complete-based semantics. Let AF = ⟨A, K⟩ be an AF and ω ⊆ A be a set of arguments. Let af = ⟨ω, K ∩ (ω × ω)⟩ be the restricted AF corresponding to AF ↓ ω

	Definition 38 (Complete-based semantics). A semantics σ is complete-based if and only if the following condition holds:
	∀AF ∈ Φ a f , L σ (AF ) ⊆ L co (AF )
	Note: By definition complete-based semantics are also complete-compatible (See Definition 21 on page 19).
	Proposition 9.

Table 5 .

 5 1: Success count for preferred -stable -complete semantics (best values in bold and large font).

  AFDiv-USCC-Chain AFDiv-USCC-Tree

	Clustering and cutting time	0.37	0.16	0.15
	Cluster and component labelling time	57.09	31.10	30.52
	Cartesian product time	40.63	40.66	40.65
	Printing time	24.88	24.46	25.89
	Total resolution time	122.97	96.39	97.20
	Table 5.3: Average real time comparison (in seconds) of AFDivider variants over 26 instances.

Table 5 .

 5

	4: Component labelling time details (in second)
	For the second component, with the optimization the maximality check is 2.5 × 10 8 times faster (1961.143s against 8×10 -6 s) while the total component labelling is 8.4 times faster (2228.455s against 266.525s).

Table 5 . 5

 55

: Memory overflow analysis: rate of instances passing algorithm steps

  • Verification of an extension Comp-Ver σ : Given an AF AF = ⟨A, K⟩ and a labelling ℓ. Is there a combination of component labellings ℓ 1 , ..., ℓ n with ℓ i ∈ L σ (af i ) and L σ (af i ) ∈ Comp σ (AF ) such that: ℓ = n i=1 ℓ i ?• Existence of an extension Comp-Exists σ : Given an AF AF = ⟨A, K⟩. Is it the case that: ∀L σ (af ) ∈ , ..., ℓ n , with ℓ i ∈ L σ (af i ) and L σ (af i ) ∈ Comp σ (AF ), such that: ∃i ∈ {1, ..., n}, in(ℓ i ) ̸ = ∅?• Uniqueness of a solution Comp-Unique σ : Given an AF AF = ⟨A, K⟩. Is it the case that: ∀L σ (af i ) ∈

	Comp σ (AF ), L σ (af ) ̸ = ∅?
	• Existence of a non-empty extension Comp-Exists ¬∅ σ : Given an AF AF = ⟨A, K⟩. Does there exist a combination of component labellings ℓ 1 Comp σ (AF ), |L σ (af i )| = 1?
	Proposition 14. Let σ ∈ {complete, stable, preferred} be a semantics. Let AF = ⟨A, K⟩ be any AF, a ∈ A be an argument and ℓ be any labelling of AF . We have the following equivalence (that is, in any case both decision problems give the same answer):
	1. Cred σ (AF , a) ≡ Comp-Cred σ (AF , a)
	2. Skep σ (AF , a) ≡ Comp-Skep σ (AF , a)
	3. Ver σ (AF , ℓ) ≡ Comp-Ver σ (AF , ℓ)
	4. Exists σ (AF ) ≡ Comp-Exists σ (AF )
	5. Exists ¬∅ σ (AF ) ≡ Comp-Exists ¬∅ σ (AF )
	6. Unique σ (AF ) ≡ Comp-Unique σ (AF )
	□ Proof of Proposition 14: link (See page 226).
	e. Comp-Exists σ (AF ) is true) and that there exists a set L σ (af ) ∈ Comp σ (AF ) such that ∃ℓ ∈ L σ (af ) and ℓ(a) = in?
	• Skeptical Acceptance Comp-Skep σ : Given an AF AF = ⟨A, K⟩ and an argument a ∈ A. Is it the case that: ∀L σ (af ) ∈ Comp σ (AF ), L σ (af ) ̸ = ∅ (i.e. Comp-Exists σ (AF ) is true) and that there exists a set L σ (af ) ∈ Comp σ (AF ) such that ∀ℓ ∈ L σ (af ), ℓ(a) = in?

Table 6 .

 6

			Enum.	AFDiv-spectral	AFDiv-USCC-Chain	AFDiv-USCC-Tree
	BA	Nb Suc. Time avg.	yes no yes 59.34 59.62 167.80 61.84 61.66 169.74 60.25 59.57 167.63 10 10 2 10 10 2 10 10 2 14 14 13 14 14 13 14 14 13 no 4.82 0.80 0.56 0.49 0.52 0.40 0.50 0.50 0.35
	ER	Nb Suc. Time avg.	yes no yes no	0 0 NaN 2133.79 NaN 1737.72 1311.29 1917.24 1695.76 1471.12 1919.99 2 0 6 6 6 5 6 4 2 0 6 6 6 5 6 4 NaN 2133.79 NaN 1737.72 1311.29 1917.24 1695.76 1471.12 1919.99
	TR	Nb Suc. Time avg.	yes no yes 154.89 76.64 4 5 5 6 no 101.47 45.82 21.69 106.40 71.57 0 1 1 4 2 2 NaN 288.41 209.99 NaN 244.01 209.89 NaN 0 1 1 0 1 2 2 1 0.64 83.66 70.91 0.68
	WS	Nb Suc. Time avg.	yes no yes no	0 0 NaN NaN	0 0 NaN NaN	0 0 NaN 1304.46 849.96 1385.25 1524.22 875.35 1473.26 4 5 4 4 5 4 4 5 4 4 5 4 NaN 1304.47 849.96 1385.25 1524.22 875.35 1473.26
	BW	Nb Suc. Time avg.	yes no yes 72.90 90.41 4 5 4 5 no 69.21 72.77 2867.53 NaN 0 0 1 0 NaN NaN	0 0 NaN NaN	0 0 NaN NaN	0 0 NaN NaN	0 0 NaN NaN	0 0 NaN NaN

1: AFDivider solvers success with and without enumeration for preferred -stable -complete semantics.

Table 7 .

 7 ′ , t ′′ and X t = X t ′ = X t ′′

	v\Xt	Xt 0	Xt 1	Xt 2	Xt 3	Xt 4	Xt 5	Xt 6	Xt 7	Xt 8	Xt 9	Xt 10	Xt 11	Xt 12	Xt 13	Xt 14	Xt 15	Xt 16	Xt 17	Xt 18	Xt 19	Xt 20	Xt 21	Xt 22	Xt 23	Xt 24
	a
	b
	c
	d
	e
	f
	g
	h
	i
	j
	k
	l

1: Bags of X

Table 9 .

 9 1: RAF semantics with structures

  Following the previous definition, all valid attacks w.r.t. a d-structure U belong to U.Example 40. Let consider RAF = ⟨A, K, s,t⟩, the RAF illustrated in Figure7.9 on page 83 and let consider U 1 , U 2 , U 3 and U 4 , the structures shown in Table9.1 on page 93. We have: U 1 , U 2 , U 3 and U 4 being d-structures.

RAF semantics for d-structures can be defined as follows: Definition 65. A conflict-free (respectively naive, admissible, complete, preferred, grounded, stable) dstructure is a RAF-conflict-free (respectively RAF-naive, RAF-admissible, RAF-complete, RAF-preferred, RAF-grounded, RAF-stable) structure which is also a d-structure.

  is maximal w.r.t. to inclusion.

	Theorem 6. The following assertions hold:

1. Every stable structure is a semi-stable structure 2. Every semi-stable structure is a preferred structure □ Proof of Theorem 6: link (See page 230).

Theorem 7. Let RAF = ⟨A, K, s,t⟩ be a RAF. If there exists a stable structure, then the semi-stable structures coincide with the stable structures.

□ Proof of Theorem 7: link (See page 231).

Table 10

 10

	.1: Semantics structures

Table 11 .

 11 1 on the following page sums up the whole previous sections of Chapter 11. It shows the correspondence between structure semantics and reinstatement RAF labellings. Example 49. Following Example 48 on page 104 and the different properties proven in this chapter, the semantics labellings of the RAF illustrated in Figure 11.1 on page 112 are given in Table 11.2 on page 113.

	Restriction on Reinstatement RAF labelling	Semantics	Theorems
	no restrictions	complete semantics	Theorems 8 and 9
	empty und	stable semantics	Theorems 14 and 15
	maximal in	preferred semantics	Theorems 10 and 11
	maximal out	preferred semantics	Theorems 12 and 13
	maximal und	grounded semantics	Theorems 16 and 17
	minimal in	grounded semantics	Theorems 18 and 19
	minimal out	grounded semantics	Theorems 20 and 21
	minimal und	semi-stable semantics Theorems 22 and 23

Table 11 .

 11

1: Reinstatement RAF labellings and structures semantics 11.6.2 AF labellings and RAF labellings when no recursive attack exists

  resp. RAF-grounded, RAF-preferred, RAFstable and RAF-semi-stable) structure labelling of RAF iff ℓ = rafLab2AfLab(L) is a complete (resp.

grounded, preferred, stable and semi-stable) labelling of AF . Proposition 27. Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be an AF. The following property holds: ℓ is a complete (resp. grounded, preferred, stable and semi-stable) labelling of AF iff L = afLab2RafLab(ℓ) is a RAF-complete (resp. RAF-grounded, RAF-preferred, RAF-stable and RAF-semistable) structure labelling of RAF .

  false} is a boolean function that indicates whether or not an attack in Ki has its target in Ãi ∪ Ki ∀α ∈ Ki , ti (α) = true if t(α) ∈ Ãi ∪ Ki otherwise false Example 69. Following Example 68 on the previous page, RAF 1 and RAF 2 form a RAF partition of RAF as

  1. σ -raf is top-down decomposable w.r.t. S iff σ is top-down decomposable w.r.t. S ra f -c . 2. σ -raf is bottom-up decomposable w.r.t. S iff σ is bottom-up decomposable w.r.t. S ra f -c . 3. σ -raf is fully decomposable w.r.t. S iff σ is fully decomposable w.r.t. S ra f -c .

□ Proof of Proposition 42: link (See page 281).

Table 14 .

 14 2. σ is bottom-up decomposable (equivalently w.r.t. S D-a f ) iff σ is bottom-up decomposable w.r.t. Proposition 43, new results concerning AF semantics are induced. Indeed, we can now have the decomposability properties w.r.t. S D-ra f -c of AF semantics, and state the following proposition: Proposition 44. Let RAF = ⟨A, K, s,t⟩ be any RAF and AF = Raf2Af(RAF ) be the corresponding AF of RAF . The semantics properties in Table14.1 on the following page hold for the flattened RAF.

	From □ Proof of Proposition 44: link (See page 289).

S D-ra f -c . 3. σ is fully decomposable (equivalently w.r.t. S D-a f ) iff σ is fully decomposable w.r.t. S D-ra f -c . □ Proof of Proposition 43: link (See page 288). 1: AF Semantics decomposability properties w.r.t. S D-ra f -c

Table 14 . 2

 142 1. σ is top-down decomposable w.r.t. S USCC iff σ is top-down decomposable w.r.t. S ra f -c-USCC . 2. σ is bottom-up decomposable w.r.t. S USCC iff σ is bottom-up decomposable w.r.t. S ra f -c-USCC . 3. σ is fully decomposable w.r.t. S USCC iff σ is fully decomposable w.r.t. S ra f -c-USCC . □ Proof of Proposition 45: link (See page 289).From Proposition 45, new results concerning AF semantics are induced. Indeed, we can now have the decomposability properties w.r.t. S ra f -c-USCC of AF semantics, and state the following proposition: Proposition 46. Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be the corresponding AF of RAF . The semantics properties in Table14.2 on the next page hold for the flattened RAF.

	□ Proof of Proposition 46: link (See page 290).

: AF Semantics decomposability properties w.r.t. S ra f -c-USCC

  .3 hold.

	□ Proof of Proposition 47: link (See page 290).
	RAF-co RAF-gr RAF-pr RAF-sst RAF-st
	Full decomposability	× × ×	× × ×	× × ×
	Top-down decomposability			× × ×
	Bottom-up decomposability	× × ×	× × ×	× × ×

Full decomposability w.r.t. S ra f -USCC × × × Top-down decomposability w.r.t. S ra f -USCC × × × Bottom-up decomposability w.r.t. S ra f -USCC × × × Table 14.3: RAF Semantics decomposability properties

Table 18

 18

	.1: Shortcut symbols

The abbreviation for the Latin phrase "id est", meaning "that is" e.g.

The abbreviation for the Latin phrase "exempli gratia", meaning "for example" w.r.t.

The abbreviation for "with regard to"

s.t.

The abbreviation for "such that" iff The abbreviation for "if and only if"

Table 18 .

 18

		3: AF and RAF related symbols
	Symbol	Meaning
	σ	A semantics
	S	A set of arguments/elements
		Continued on next page ...

Table 18 .

 18

		4: AF related Symbols (continued)
	Symbol	Meaning
	Cycles a f (AF )	The set of cycles of AF
	PE a f (AF )	The path-equivalence relation over AF
	SCCS a f (AF )	The set of SCC a f of AF
	S (AF )	A partition selector S is a function receiving as input an AF AF = ⟨A, K⟩ and returning a set of partitions of A
	S D-a f

Table 18 .

 18

		5: RAF related Symbols
	Symbol	Meaning
	Φ ra f	The set of all possible RAF
		Continued on next page ...

  t⟩ be a RAF and RAF is a partial RAF of RAF . We denote by I the input of RAF . See Definition 96 on page 136RAF , I, L inpLet RAF be a partial RAF and I be an the input of RAF . The tuple RAF , I, L inp is called a "partial RAF with input", where L inp is a structure labelling of the elements in S inp and Q inp . See Definition 96 on page 136 Continued on next page ... ⟨A s , K s , s s ,t s ⟩ Let RAF , I, L inp be a partial RAF with input. RAF s is the standard RAF corresponding to it. See Definition 97 on page 138 ζ υ ρ θ And Ãs , Ks Not Ãs Not Ks Let RAF , I, L inp be a partial RAF with input. In the process of flattening created the standard RAF RAF s corresponding to it, several elements and sets of elements are created. See Definition 97 on page 138 F ra f ( RAF , I, L inp ) Let RAF , I, L inp be a partial RAF with input, the canonical function F ra f ( RAF , I, L inp ) computes the structure labellings corresponding to it. See Definition 99 on page 138 ≼ A partial order relation between SCC ra f . See Definition 89 on page 132 Dag scc (RAF ) A function that creates a the directed graph corresponding to the ≼ relation over RAF . See Definition 91 on page 133

		Table 18.5: RAF related Symbols (continued)
	Symbol	Meaning
	RAF s =

Table 18 .

 18 6: Transformation related symbols: RAF and AF Let AF = ⟨A, K⟩ be an AF. The function Af2Raf transforms an AF into a RAF by naming its attacks. See Definition 80 on page 121 Raf2Af(RAF ) Let RAF = ⟨A, K, s,t⟩ be a RAF. The function Raf2Af transforms a RAF into an AF. See Definition 76 on page 115 Not A Not K And A,K Let RAF = ⟨A, K, s,t⟩ be a RAF. In the process of flattening RAF with Raf2Af, several sets of arguments are created. Among those are Not A , Not K and And A,K . See Definition 76 on page 115 ¬x Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be the flattened version of RAF . Let x ∈ (A ∪ K) be an element of RAF . ¬x is the created argument that represents the "negation" of element x. See Definition 76 on page 115 (Notice that ¬x ∈ (Not A ∪ Not K ))

	Symbol	Meaning
	Af2Raf(AF )

Table 18 .

 18 NP NP , P NP , NP coNP Several time complexity classes. See Section 16.4.3 on page 180

		7: Complexity Symbols
	Symbol	Meaning
	P	A decision problem. See Sections 16.4.1 and 16.4.2 on page 178 and on page 179
	C	A complexity class. See Section 16.4.1 on page 178
	≤ P , ≤ L	Respectively polynomial reduction and logarithmic in space reduction. See Section 16.4.4.1 on page 182
	L, P, NP, DP, DP 2 , coNP, Σ P 2 , Π P 2 , Θ P 2 ,

C -hard

Let P be a problem and C be a complexity classes. P ∈ C -hard if and only if P is at least as hard as the other problems of the class C . See Section 16.4.4.2 on page 183 C -c Let P be a problem and C be a complexity classes. P ∈ C -c if an only if P ∈ C -hard and P ∈ C . See Section 16.4.4.2 on page 183

Table of figures

 of Table 18.8: Table of figures

Table 18 .

 18 8: Table of figures (continued)

Table 18 .

 18 8: Table of figures (continued)

	Figure	Description
	Figures 14.4 to 14.5 on pages 128-129	Different cases of RAF flattening that illustrate some interesting properties leading to Propositions 35 and 36 on page 129
	Figures 14.7 to 14.8 on pages 131-132	Illustrations of the correspondance between SCC a f and SCC ra f

Table 18 .

 18 8: Table of figures (continued)

Table 18

 18

		.9: Table of tables
	Table	Description
	Tables 1.1 and 1.2 on page 13 and on page 14	Semantic extensions and labellings of the AF in Figure 1 on page 7
	Table 1.3 on page 15	Labelling and extension based semantics correspondence
	Table 2.1 on page 22	AF Semantics decomposability properties
	Tables 4.1 to 4.8 on pages 39-48	Labelling tables illustrating the different steps of the AFDivider algorithm on the AF represented in Figure 4.2 on page 32
	Table 5.1 on page 56	AF solver success count analysis
	Table 5.2 on page 57	AF solver resolution time analysis
	Table 5.3 on page 58	A comparison of the average real time of the AFDivider variants
	Table 6.1 on page 69	AFDivider success count and resolution time analysis when using the Compact Enumeration Representation
	Table 7.1 on page 75	Bags produced by the algorithm proposed in [43]
	Table 8.1 on page 88	AFRA semantics extension for the AFRA illustrated in Figure 7.9 on page 83
	Table 9.1 on page 93	RAF semantics extension for the RAF illustrated in Figure 7.9 on page 83
		Continued on next page ...
		209

Table 18 .

 18 9: Table of tables (continued)

Table Description Table 10 .

 Description10

	1 on page 105	A recall of Table 9.1 on page 93 with the addition of the semi-stable semantics
	Table 10.2 on page 106	RAF labellings corresponding to Table 10.1 on page 105
	Table 11.1 on page 111	Correspondence between reinstatement RAF labellings and structures semantics
	Table 11.2 on page 113	RAF semantics labellings for the RAF illustrated in Figure 11.1 on page 112
	Table 13.1 on page 122	Complexities of RAF decision problems

Table 14 .

 14 1 on page 161 AF Semantics decomposability properties w.r.t. S D-ra f -c Table 14.2 on page 162 AF Semantics decomposability properties w.r.t. S ra f -c-USCC

Table 14

 14

	.3 on page 162	RAF Semantics decomposability properties
	Table 18.2 on page 195	Graph and matrix symbols
	Table 18.3 on page 195	AF and RAF related symbols
	Table 18.4 on page 196	AF related Symbols
	Table 18.5 on page 199	RAF related Symbols
	Table 18.6 on page 203	Transformation related symbols: RAF and AF
	Table 18.7 on page 204	Complexity Symbols
	Table 18.8 on page 205	Tables of figures

Table 18 .

 18 9 on the previous page Tables of tablesGlossaryAbstract Argumentation Abstract argumentation theory proposes methods to represent and deal with contentious information, and to draw conclusions or take decision from it. Such an abstract approach focuses on how arguments affect each other. Arguments are seen as generic entities which interact positively (support relation) or negatively (attack relation) with each other. 2, 70, 166, 210 adjacency See Definition 115 on page 171. 210 adjacency matrix See Definition 134. 53, 175, 176, 189, 210 AF trivial part Part of an AF that has a unique and fixed labelling that can be computed in linear time. See Section 4.2.1. i, 30, 210 AF with input See Definition 16. 17, 18, 48, 49, 153, 155, 157-159, 210 AF-extension A set of arguments of an AF. 210, 213 AFRA Argumentation Framework with Recursive Attacks. See Definition 49. 83, 210 AFRA-extension A set of arguments and attacks of an AFRA. 210, 213 algorithm A finite sequence of well-defined, computer-implementable instructions, typically to solve a class of specific problems or to perform a computation. 3, 5, 27, 70, 104, 123, 165, 210 application A mathematical application is a relationship between two sets in which each element of the former is related to a single element of the latter. 175, 177, 210 Argument Mining A sub research field of Argumentation interested in extracting arguments and their relations with each others, from natural language speeches (oral or written), in order to create a formal model to reason with. 2, 28, 210 argumentation A research field of Artificial Intelligence interested in managing contentious information. 2, 210 Argumentation Framework An argumentation framework, in the general sense, is a particular formalism to express argumentation problems. The expression has also come to mean an argumentation problem instance modeled after a particular formalism (by the way, a particular argumentation framework). 2, 3, 83, 210 Argumentation Reasoning A sub research field of Argumentation interested in reasoning over some argumentation model. It is useful to conclude, decide, convince, persuade or explain some issue. 2, 210 211 conjecture In mathematics, a conjecture is an assertion for which a proof is not yet known, but which is strongly believed to be true, in the absence of a counterexample. More generally speaking, its an opinion based on probabilities, appearances. 29, 210 connected component See Definition 127. 30-32, 34, 47, 64, 68, 174, 210 CPU time Amount of time for which a CPU was used for processing instructions of a computer program, as opposed to, for example, waiting for input/output operations. The CPU time is often measured in clock ticks or as a percentage of the CPU capacity. 57, 210, 216 CSP A Contraint Satisfaction Problem is a mathematical problem that looks for a configuration of object states (i.e. a mapping where each object has a particular state) that satisfies a certain number of constraints. See Section 17.1. 43-45, 186, 187, 210 datapoint Discrete unit of information. In a statistical or analytical context, a datapoint is usually derived from a measurement or research and can be represented numerically and/or graphically. 188, 190, 192, 210

  Functional problem consisting in finding all the possible solutions of a computational problem. See Definition 29 for the enumeration problem in Argumentation.[START_REF] Castagna | Online handbook of argumentation for ai[END_REF][START_REF] Stephen | An overview of computational complexity[END_REF][START_REF] Le | Sat4j 2.3.2: on the fly solver configuration[END_REF][START_REF] Peter Wallner | Advanced SAT techniques for abstract argumentation[END_REF] 165, 210 extension See glossary: AF-extension and AFRA-extension. 3, 8, 9, 85, 102, 103, 111, 166, 210 extension-based See glossary: extension-based semantics. 8, 12, 28, 210 extension-based semantics A semantics that produces extensions. 3, 8, 12, 16, 23, 85, 91, 97, 210, 213 field A field is one of the fundamental algebraic structures of general algebra. It is a set with two binary operations making possible addition, multiplication and the calculation of opposites and inverses, allowing the definition of subtraction and division operators. 210, 215, 217 Glossary full decomposability See glossary: semantics decomposability. 210 fully decomposable See glossary: semantics decomposability. 18-20, 47, 48, 142, 159-161, 210 function problem Type of computational problem that has for output a more complex output than decision problems. See Definition 137. 28, 168, 183, 210 graph See glossary: non-directed graph, directed graph, weighted graph. 210 hard We consider that a computational problem is hard for a certain complexity class if an efficient algorithm solving it could be used to efficiently solve, by mean of reductions, all the problems in that given class. See Section 16.4.4.2. 121, 122, 183, 210

	7, 23, 24, 55, 64-66, 120-122, 163, 167, 168, 210	16.4.2.
	degree matrix See Definition 135 on page 175. 176, 189, 210
	directed graph See Definition 112. 210, 214
	Dung's Argumentation Framework Dung introduced in [39] the seminal abstract argumentation frame-work. See Definition 1. 2, 23, 27, 86, 210
	eigenvalue See Definition 133. 175, 178, 188, 190, 191, 210
	eigenvector See Definition 133. 53, 175, 177, 178, 188, 190, 192, 210
	enumeration problem flattening Process by which an argumentation framework is transformed into another one (often using a less complex formalism) while keeping some properties of interest. 3, 94, 98, 99, 101, 114, 115, 157, 158, 163, 167, 210

  [START_REF] Alviano | The pyglaf argumentation reasoner[END_REF][START_REF] Stephen | An overview of computational complexity[END_REF] 188, 191, 210 Higher-Order Attack Argumentation Framework A argumentation framework that allow attacks to have as target an attack. See Parts IV and V. ii, 82, 83, 100, 210 Attack going into a cluster. SeeDefinition 30. 32,[START_REF] Doutre | Argumentation Frameworks with Higher-Order Attacks: Labelling Semantics[END_REF][START_REF] Dvorák | Cegartix v2017-3-13: A SAT-based counter-example guided argumentation reasoning tool[END_REF][START_REF] Le | Sat4j 2.3.2: on the fly solver configuration[END_REF][START_REF] Fragkiskos | Clustering and community detection in directed networks: A survey[END_REF][START_REF] Peter Wallner | Advanced SAT techniques for abstract argumentation[END_REF] 143, 210 labelling Generally speaking, a labelling is an acceptance value mapping of a set of elements. See Definition 6 for AF, Definition 72 for RAF. 3, 11, 29, 101, 104, 165, 167, 210 labelling-based See glossary: labelling-based semantics. 8, 12, 28, 210 labelling-based semantics A semantics that produces labellings. 12, 23, 29, 167, 210, 214 laplacian matrix See Definition 136. 175, 176, 188, 190, 210 legally labelled See Definition 7 for AF and Definition 74 for RAF. 11, 104, 210, 216 linear application See glossary: linear transformation. 210 linear combination expression constructed from a set of terms by multiplying each term by a constant and adding the results. 210, 215 linear transformation A linear application (also called linear transformation) is an application between two vector spaces over a field which respects vector addition and scalar multiplication, and thus more generally preserves linear combinations. See glossary: field, linear combination, vector space. 175, 177, 210, 214 log-space function A polynomial time function that can be executed using at most a memory space logarithmic w.r.t. to the size of the input. Section 16.4.3.1. 121, 122, 210 matrix See Definition 132. 52, 53, 170, 175, 177, 188-192, 210 merge configuration See Definition 36. 39-44, 210 multi-threaded See glossary: multi-threading. 55, 57, 210 multi-threading a form of parallelization or division of work to enable simultaneous processing. Instead of giving a large workload to a single CPU core, threaded programs divide the work into several software tasks (threads). These tasks are processed in parallel by different CPU cores to save time. 210, 215 non-directed graph See Definition 112. 210, 214 order of magnitude See Definition 138. 184, 210 outward attack Attack coming out of a cluster. See Definition 30. 32, 143, 210 Glossary real time As opposed to CPU time (See glossary: CPU time), Real Time is the actual, real world, time that a process takes to run. 51, 210 reinstatement labelling A labelling is in which all elements are legally labelled. 102, 111, 167, 210 reunified labelling profile See Definition 37. 31, 42-44, 46, 210 SAT The decision problem consisting in deciding if a given propositional formula is satisfiable or not. 51, 70-73, 80, 179-181, 183, 210 SCC Given an AF (resp. a RAF), an SCC is a set of arguments (resp. elements) that are equivalent w.r.t. the PE a f (resp. w.r.t. the PE ra f ) relation. See glossary: path-equivalence relation. 20, 54, 73, 78-80, 123, 174, 210 SCC decomposition See Section 7.2.2. ii, 78, 210 semantics Given an argumentation framework, a semantics corresponds to a formal way to say how the solution of the argumentation should be decided. 2-4, 8, 28, 83, 85, 101, 165, 210 semantics decomposability Properties of a semantics stating if the latter is computable in a distributed way, that is by considering sub-parts of an argumentation framework (whatever the formalism and for all instances). The top-down (resp. bottom-up) decomposability property ensures that the distributed computation made is complete (resp. sound). The fully decomposability property is the intersection of both properties. 3, 7, 16, 48, 101, 114, 123, 210, 212, 214, 217 set A set is a collection of distinct elements. 210 similarity criterion Criterion used in the spectral clustering method. See Section 17.2. 53, 210 sound An algorithm is said to be sound for a given problem if and only if it produces only valid solutions for the wanted problem. See Section 4.3 as an example. 50, 165, 168, 210, 216 soundness See glossary: sound. 32, 47, 48, 210 space In mathematics, a space is a set with additional structures, allowing to define objects analogous to those of usual geometry. The elements can be called points, vectors, functions, etc., depending on the context. See for an example glossary: vector space. 188, 190, 192, 210 sparse A graph is said to be sparse when its density is low. See Definition 131. 29, 53, 210, 216 See glossary: structure-based semantics. 167, 210 structure-based semantics A semantics that produces RAF structures. 3, 91, 92, 97, 210, 217 top-down decomposability See glossary: semantics decomposability. 210 top-down decomposable See glossary: semantics decomposability. 19, 20, 47, 142, 159-161, 210 USCC Union of SCCs. See Definition 27 for AF and Definition 109 for RAF. 21, 54-58, 60, 210 USCC Chain A clustering method. See Section 5.2.2. 54, 57, 210 USCC Tree A clustering method. See Section 5.2.2. 54, 57, 210 valid attack Given a RAF and a structure U, an attack α is said to be valid if U does not inhibit α. See glossary: inhibited. 92, 210 variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. 190, 191, 210

	structure-based
	partial RAF See Definition 92. 134, 210
	partial RAF with input See Definition 96. 136, 210
	partition See Definition 111. An AF partition is a partition of its arguments. A RAF partition is a partition of its elements (arguments and attacks). 18-20, 32, 47, 53, 54, 56, 64, 135, 139, 140, 210
	partition selector An application that produces some set of partitions from a given argumentation frame-work (whether AF or RAF). See Definition 23 for AF and Definition 100 for RAF. 20, 21, 139, 140, 159, 161, 210
	path See Definition 121. 210
	path-equivalence relation See Definition 128 for AF and Definition 86 for RAF. 20, 210, 216 sparsity See glossary: sparse. 210 polynomial reduction A problem reduction that is polynomial in time. See glossary: problem reduction. 120, 183, 210 spectral clustering A clustering method. See Section 17.2. 52, 53, 55, 57, 60, 165, 188, 192, 210
	standard AF See Definition 17. 17, 18, 49, 153, 156-158, 210 problem reduction Basically, a procedure that transforms a given computational problem into another one. See Section 16.4.4.1 for details. 210, 215 standard RAF See Definition 97. 210
	RAF Recursive Argumentation Framework. See Definition 48. 83, 210 structure Given a RAF, a structure is a pair whose first element is a set of arguments and the second a set of attacks. 3, 91, 103, 167, 210 RAF path See Definition 82. 210 RAF-compliant See Definition 101. 139, 140, 155, 160, 161, 210 structure labelling Given a RAF, a structure labelling is a pair whose first element is a labelling of argu-ments and the second a labelling of attacks. 3, 101, 117, 118, 136, 138, 151, 155, 167, 210

incidence See Definition 114 on page 171. 210 independant partial RAF See Definition 95. 136, 210 induced AF See Definition 32. 34, 35, 49, 145, 210 inhibited Basically, the "inhibition" is the notion of defeat but for attacks. Given a structure U, an attack α is said to be inhibited by U if there exists β ∈ U such that β has its source in U and its target is α. See Definition 61. 92, 121, 210, 217 input argument See Definition 15. 16-18, 80, 157, 210 inward attack

  a) ∈ I} Let σ be a top-down decomposable semantics. Let L κ i D be the set of distinct labellings of κ i according to the semantics σ . Let L * σ (af ) be the set of labellings of af according to σ obtained by Algorithm 2. Let L * µ(κ i ) σ be the set of labellings of κ i under the context µ. By definition we have (Definition 22 on page 19):

  ,...,n} s.t. j̸ =i

  AF ↓ ω gr , ω inp gr , (

	i∈{1,...,n}	ℓ A i ) ↓ ω	inp gr	, ω K gr ) = {ℓ gr }	(18.30)
	Because σ is a fully decomposable semantics we have so (Definition 19 on page 19):

Assertion 5 :

 5 Exists ¬∅ σ (AF ) ≡ Comp-Exists ¬∅ σ (AF ) • Case 1: If Exists ¬∅ σ (AF ) is true, then Comp-Exists ¬∅ σ (AF ) is true. If Exists ¬∅ σ (AF ) then ∃ℓ ∈ σ (AF ) s.t. in (ℓ) ̸ = ∅. As Algorithm 1 is complete for σ , there exists thus a combination of component labellings ℓ 0 , ..., ℓ n , with ℓ i ∈ L σ (af i ) and L σ (af i ) ∈ Comp σ (AF ) such that ℓ = n i=0 ℓ i . As a consequence, we have: ∀L σ (af i ) ∈ Comp σ (AF ), L σ (af i ) ̸ = ∅ and ∃i ∈ {0, ..., n} s.t. in (ℓ i ) ̸ = ∅. We prove so that if Exists ¬∅ σ (AF ) is true, then Comp-Exists ¬∅ σ (AF ) is true. • Case 2: If Comp-Exists ¬∅ σ (AF ) is true, then Exists ¬∅ σ (AF ) is true. If Comp-Exists ¬∅

•

  Case 2: If Comp-Unique σ (AF ) is true, then Unique σ (AF ) is true. If Comp-Unique σ (AF ) is true, then ∀L σ (af i ) ∈ Comp σ (AF ), |L σ (af i )| = 1. For i ∈ {0, ..., n}, let ℓ ibe the unique labelling of the component af i . Stable structures are semi-stable ones). Let RAF = ⟨A, K, s,t⟩ be a RAF and U = ⟨S, Q⟩ be a stable structure. According to the definition of a stable structure (Definition 63 on page 92), we have:

	Proofs of Part V: Contributions about RAF
	Proofs of Chapter 10: New RAF semantics
	Proof of Theorem 6 on page 103.
	1. (

  Proof of Theorem 10 on page 108. Let L be a reinstatement RAF labelling such that in (L) is maximal. Let suppose that U = Lab2Struct(L) is not a preferred structure. According to Definition 63 on page 92, Proposition 19 on page 94 and Theorem 2 on page 94, there exists then a complete structureU ′ such that U ⊏ U ′ (strict inclusion). Let L ′ = Struct2Lab(U ′ ). Then in (L ′ ) ⊂ in (L). As a consequence L is not a reinstatement RAF labelling such that in (L) is maximal, which is a contradiction. ■Proof of Theorem 11 on page 108. Let U be a preferred structure and L = Struct2Lab(U). Let us suppose that L is not a reinstatement RAF labelling such that in (L) is maximal. Then there exists a reinstatement RAF labellingL ′ such that in (L) ⊂ in (L ′ ). Let U ′ = Lab2Struct(L ′ ).Then U ′ is a complete structure such that U ⊏ U ′ (strict inclusion). As a consequence, U is not a preferred structure, which is a contradiction. ■ Lemma 1. Let L and L ′ be two reinstatement RAF labellings. If in(L) ⊂ in(L ′ ) then out(L) ⊂ out(L ′ ).

		and
		∃x ∈ in (L ′ ), x / ∈ in (L)	(18.68)
	RAF labelling.	to (18.66) on pages 233-234 being stated, we prove thus that L is a reinstatement ■
	Proofs of Section 11.2: Preferred semantics
	Proof of Lemma 1. Let L and L ′ be two reinstatement RAF labellings such that in (L) ⊂ in (L ′ ), meaning that:
		∀w ∈ in (L), w ∈ in (L ′ )	(18.67)

  following Proposition 25 on page 117) iff Ver σ accepts (AF , ε U ) Assertion 4: RAF-Exists σ accepts RAF iff Exists σ accepts AF . RAF-Exists σ accepts RAF iff ∃U ∈ σ (RAF ) iff ∃ε U ∈ σ (AF ) (following Proposition 25 on page 117) iff Exists σ accepts AF Assertion 5: RAF-Exists ¬∅ σ accepts RAF iff Exists ¬∅ σ accepts AF . RAF-Exists ¬∅ σ accepts RAF iff ∃(U = ⟨S, Q⟩) ∈ σ (RAF ) s.t. (S ∪ Q) ̸ = ∅ iff ∃ε U ∈ σ (AF ) s.t. ε U ̸ = ∅ (following Proposition 25 on page 117) RAF-Unique σ accepts RAF iff Unique σ accepts AF .

	iff Exists ¬∅ σ accepts AF
	Assertion 6:

RAF-Unique

σ accepts RAF iff ∃!U ∈ σ (RAF ) iff ∃!ε U ∈ σ (AF ) (following

Proposition 25 on page 117) iff Unique σ accepts AF ■ Proof of Proposition 29 on page 121. Given that Raf2Af is a polynomial time, log-space function (See Sections 16.4.3.1 and 16.4.4.1 on page 180 and on page 182), then according to Proposition 28 on page 121, for each semantics σ ∈ {complete, semi-stable, stable, preferred, grounded} we have:

  Proposition 30 on page 122. This proof is trivial considering Theorem 3 on page 97 and Proposition 23 on page 103. ■ Proof of Proposition 31 on page 122. Given that Af2Raf is a polynomial time, log-space function, then according to Proposition 30 on page 122, for each semantics σ ∈ {complete, semi-stable, stable, preferred, grounded} we have:

  .88) From Equations(18.87) and (18.88) on the previous page and on this page we prove Assertion 4.Assertion 5: e i ∈ K s.t. t(e i ) ̸ = s(e i ) and t(e i ) ̸ = e i ⇐⇒ ∃!(e i , ...,t(e i )) ∈ Paths a f (AF ) and ∃(s(e i ), ..., t(e i )) ∈ Paths a f (AF )Trivially Assertion 5 is deduced from Assertions 3 and 4. ■

•

  Step 2: |S| > 1 and ¬x ∈ S =⇒ |S| > 1 and x ∈ S If |S| > 1 and ¬x ∈ S then there exists x ′ ∈ S such that ¬x ̸ = x ′ . There is thus a path from ¬x to x ′ and a path from x ′ to ¬x. It follows that there is a cycle from ¬x to ¬x. Let c ∈ Cycles a f (AF ) be such a cycle. According to the definition of Raf2Af, for any ¬x ∈ Not A ∪ Not K , the only argument attacking ¬x is x. As a consequence, we have: x ∈ c and so: x ∈ S.From Steps 1 and 2, we prove that: |S| > 1 and x ∈ S ⇐⇒ |S| > 1 and ¬x ∈ S ■ Lemma 9. Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be the AF corresponding to RAF . Let S ∈ SCCS a f (AF ) and α ∈ K. The following property holds: s(α) ∈ S and t(α) ∈ S ⇐⇒ ¬s(α) ∈ S and s(α).α ∈ S and ¬t(α) ∈ SIf |S| > 1 and x ∈ S then there exists x ′ ∈ S such that x ̸ = x ′ . There is thus a path from x to x ′ and a path from x ′ to x. It follows that there is a cycle from x to x. Let c ∈ Cycles a f (AF ) be such a cycle. According to the definition of Raf2Af, x can only be attacked by arguments in And A,K . There exists thus an attack α ∈ K such that t(α) = x and s(α).α ∈ c. As a consequence, we have: s(α).α ∈ S.If |S| > 1 and s(α).α ∈ S then there exists x ′ ∈ S such that s(α).α ̸ = x ′ . There is thus a path from s(α).α to x ′ and a path from x ′ to s(α).α. It follows that there is a cycle from s(α).α to s(α).α. Let c ∈ Cycles a f (AF ) be such a cycle. According to the definition of Raf2Af, the only argument attacked by s(α).α is t(α) = x. As a consequence, we have: x ∈ S. Assertion 2: |S| > 1 and x ∈ S =⇒ |S| > 1 and ∃α ∈ K s.t. t(α) = x and (s(α) ∈ S or α ∈ S) According to the proof of Assertion 1, given that |S| > 1 and x ∈ S, ∃α ∈ K s.t. t(α) =

	Proof of Lemma 9. In two steps:

• Step 1: s(α) ∈ S and t(α) ∈ S =⇒ ¬s(α) ∈ S and s(α).α ∈ S and ¬t(α) ∈ S If s(α) ∈ S and t(α) ∈ S then according to the definition of Raf2Af, there exists a path from t(α) to s(α) whose second element is ¬t(α) and a path from s(α) to t(α) which is (s(α), ¬s(α), s(α).α,t(α)). Let c ∈ Cycles a f (AF ) be the cycle formed by merging both paths. Given that: ¬s(α) ∈ c and s(α).α ∈ c and ¬t(α) ∈ c, we also have: ¬s(α) ∈ S and s(α).α ∈ S and ¬t(α) ∈ S.

• Step 2: ¬s(α) ∈ S and s(α).α ∈ S and ¬t(α) ∈ S =⇒ s(α) ∈ S and t(α) ∈ S Given that |S| > 1, then according to Lemma 8 on the previous page, we have:

¬t(α) ∈ S =⇒ t(α) ∈ S

and ¬s(α) ∈ S =⇒ s(α) ∈ S From Steps 1 and 2, we prove that Lemma 9 holds. ■ Lemma 10. Let RAF = ⟨A, K, s,t⟩ be a RAF and AF = Raf2Af(RAF ) be the AF corresponding to RAF . Let S ∈ SCCS a f (AF ) and x ∈ A ∪ K. The following properties holds: 1. |S| > 1 and x ∈ S ⇐⇒ |S| > 1 and ∃α ∈ K s.t. t(α) = x and s(α).α ∈ S 2. |S| > 1 and x ∈ S =⇒ |S| > 1 and ∃α ∈ K s.t. t(α) = x and (s(α) ∈ S or α ∈ S) Proof of Lemma 10. Assertion 1: |S| > 1 and x ∈ S ⇐⇒ |S| > 1 and ∃α ∈ K s.t. t(α) = x and s(α).α ∈ S • Step 1: |S| > 1 and x ∈ S =⇒ |S| > 1 and ∃α ∈ K s.t. t(α) = x and s(α).α ∈ S • Step 2: |S| > 1 and ∃α ∈ K s.t. t(α) = x and s(α).α ∈ S =⇒ |S| > 1 and x ∈ S

  trivially x ∈ S ′ belongs to an unique SCC ra f of RAF . Otherwise (i.e. if |S ′ | > 1), according to the definition of SCC a f , for any couple (x, y) ∈ S ′2 s.t. x ̸ = y there is a path from x to y and a path from y to x. It is particularly the case for any couple (x, y) ∈ U 2 .

Following Lemma 7 on page 258, there is thus a RAF-path: (x = e 1 , ..., y = e n ) ∈ Paths ra f (RAF ) s.t. e n-1 ∈ K and another one: (y = o 1 , ..., x = o m ) ∈ Paths ra f (RAF ) s.t. o m-1 ∈ K. As a consequence: (x, y) ∈ PE ra f .

•

  Let α ∈ K be an attack in RAF . We have to cases to consider: α ∈ S and α / ∈ S.-Let α ∈ S. As |S| > 1, then α is equivalent to another element of RAF w.r.t. PE ra f , which is also in S. There is thus, following Proposition 33 on page 126, a RAF-closed-walk c = (e 1 = α, ..., e n = α) ∈ ClosedWalk ra f (RAF ) such that n > 2. Following Proposition 34 on page 126, all elements of U = {e i ∈ c|i ∈ {2, ..., n} s.t. e i-1 ∈ K} belong to the same SCC ra f . We have thus: e 2 = t(α) ∈ S. Given that α ∈ S and t(α) ∈ S, we also have: α ∈ S ′ and t(α) ∈ S ′ . Furthermore, according to the definition of Raf2Af, we have: (α, ¬α, s(α).α,t(α)) ∈ Walks a f (AF ). Notice that if t(α) = α then it must be the case that: c = (α, α) according to the definition of RAF-closed-walk. This contradicts n > 2. As a consequence, we have: t(α) ̸ = α and so: (α, ¬α, s(α).α,t(α)) ∈ Paths a f (AF ). α and t(α) being in the same SCC a f of AF , there is also a path from t(α) to α in AF . Given that there is a path from

t(α) to α in AF and that (α, ¬α, s(α).α,t(α)) ∈ Paths a f (AF ), we have: ¬α, s(α).α and α being equivalent w.r.t. PE a f . As a consequence we have: ¬α ∈ S ′ and s(α).α ∈ S ′ . The following property holds then:

|S| > 1 and α ∈ S ∩ K =⇒ ¬α ∈ S ′ and s(α).α ∈ S ′ (18.92)

-Let α / ∈ S.

Let consider two cases: (s(α) ∈ S and t(α) ∈ S), (s(α) / ∈ S or t(α) / ∈ S). * Let suppose that s(α) ∈ S and t(α) ∈ S. We have thus s(α) ∈ S ′ and t(α) ∈ S ′ . Following to Lemma 9 on page 260, the following property then holds:

α ∈ K \ S s.t

. s(α) ∈ S and t(α) ∈ S =⇒ ¬s(α) ∈ S ′ and s(α).α ∈ S ′ (18.93) * Let suppose that s(α) / ∈ S or t(α) / ∈ S. We have thus: s(α) / ∈ S ′ or t(α) / ∈ S ′ . Let x be whether s(α) or t(α) and let suppose that x / ∈ S ′ . As |S| > 1 and S ⊆ S ′ then we have: |S ′ | > 1. As x / ∈ S ′ , we also have according to Lemma 8 on page 259: ¬x / ∈ S ′ . As consequence, the two following properties hold: |S| > 1 and α ∈ K \ S s.t. s(α) / ∈ S =⇒ ¬s(α) / ∈ S ′ (18.94) |S| > 1 and α ∈ K \ S s.t. t(α) / ∈ S =⇒ ¬t(α) / ∈ S ′ (18.95) Let consider two cases: s(α) / ∈ S ′ and t(α) / ∈ S ′ . • Let t(α) / ∈ S ′ . According to Lemma 10 on page 260 (Assertion 1), we have: |S ′ | > 1 and t(α) ∈ S ′ ⇐⇒ |S ′ | > 1 and ∃β ∈ K s.t. t(β ) = t(α) and s(β ).β ∈ S ′

  Raf2Af( RAF s ) be the corresponding AF of RAF s (with RAF ′ s s.t. s(α).α ∈ ω ′ and ¬s(α) / ∈ ω ′ =⇒ s(α) ∈ S inp 3. (s(α).α,t(α)) ∈ K′ s ∪ K ′ s s.t. s(α).α / ∈ ω ′ and t(α) ∈ ω ′ =⇒ α ∈ Q inp 4. (x, y) ∈ K′ s ∪ K ′ s s.t. y ∈ ω ′ =⇒ x ∈ ω ′ ∪ J

J , K J . Let RAF ′ s = ′ s = Ã′ s , K′ s ).

The following assertions hold:

1. J ⊆ (Not A ∪ And A,K ) 2. (¬s(α), s(α).α) ∈ K′ s ∪ K

  s.t. x ∈ Not K and y ∈ And A,K -Case 4: x ∈ And A,K and y ∈ A ∪ K As x ∈ And A,K , let assume that x = s(α).α such that y = t(α). As s(α).α ∈ ω ′ and as y ∈ ω, we have following Equation (18.111) on the previous page: {α, ¬α, s(α).α,t(α)} ∈ Walks a f ( RAF ′ s ). As a consequence we have: (s(α).α,t(α)) ∈ K′ s and so: (x, y) ∈ K′ s . The following property then holds:(x, y) ∈ K ′ s ∩ (ω ′ × ω ′ ) s.t.x ∈ And A,K and y ∈ A ∪ K

	=⇒	(18.112)
	(x, y) ∈ K′ s ∩ (ω ′ × ω ′ )

=⇒ (x, y) ∈ K′ s ∩ (ω ′ × ω ′ ) (18.113)

  Let { RAF 1 , ..., RAF n } be the partition of RAF corresponding to Ω, with RAF i = Ãi , Ki , si , ti , s,t being the partial RAF corresponding to ω i ∈ Ω. Let I i = S inp i , Q inp i be the input elements of RAF i and L inp i be a structure labelling of them. For i ∈ {1, ..., n}, let RAF i , I i , L inp i be a RAF with input and AF ↓ ω ′ i , J i , ℓ J i , K J i be its corresponding AF with input, as defined in Definition 105 on page 155. Assertion 1: σ -raf is top-down decomposable w.r.t. S iff σ is top-down decomposable w.r.t. S ra f -c .

  .140) Let denote ℓ i ↓ A , ℓ i ↓ K by L i , for i ∈ {1, ..., n}. Then, following Equations (18.139) and (18.140), we have σ being top-down decomposable w.r.t. S ra f -c iff :∀Ω ′ ∈ S ra f -c (AF ), L σ -ra f (RAF ) ⊆ L 1 ∪ ... ∪ L n L i ∈ F ra f σ ( RAF i , I i , L inp i ) With: L inp i = ( j∈{1,...,n} s.t. j̸ =i L j ) ↓ I i ∀Ω ∈ S (RAF ), L σ -ra f (RAF ) ⊆ L 1 ∪ ... ∪ L n L i ∈ F ra f σ ( RAF i , I i , L inp i )With:L inp i = ( j∈{1,...,n} s.t. j̸ =i L j ) ↓ I i (18.142) As a consequence, we have σ being top-down decomposable w.r.t. S ra f -c iff σ -raf is top-down decomposable w.r.t. S . Assertion 2: σ -raf is bottom-up decomposable w.r.t. S iff σ is bottom-up decomposable w.r.t. S ra f -c .

	And so:
	(18.141)

  .146) Let denote ℓ i ↓ A , ℓ i ↓ K by L i , for i ∈ {1, ..., n}. Then, following Equations (18.145) and (18.146), we have σ being bottom-up decomposable w.r.t. S ra f -c iff :∀Ω ′ ∈ S ra f -c (AF ), L σ -ra f (RAF ) ⊇ L 1 ∪ ... ∪ L n L i ∈ F ra f σ ( RAF i , I i , L inp i ) With: L inp i = ( j∈{1,...,n} s.t. j̸ =i L j ) ↓ I i ∀Ω ∈ S (RAF ), L σ -ra f (RAF ) ⊇ L 1 ∪ ... ∪ L n L i ∈ F ra f σ ( RAF i , I i , L inp i )With:L inp i = ( j∈{1,...,n} s.t. j̸ =i L j ) ↓ I i (18.148) As a consequence, we have σ being bottom-up decomposable w.r.t. S ra f -c iff σ -raf is bottom-up decomposable w.r.t. S . Assertion 3: σ -raf is fully decomposable w.r.t. S iff σ is fully decomposable w.r.t. S ra f -c . Trivial considering Assertions 1 and 2. ■ Lemmas 14 and 15 on page 285 and on page 287 pave the road for the demonstration of Propositions 43 and 45 on page 160 and on page 161.

	(18.147)
	And so:

This notion of largeness of an argumentation framework is not so simple to define. It is related to the fact that the computation of the solutions is complex either because of the number of arguments, or of the number of interactions, or because of the structure of the argumentation framework.

http://argumentationcompetition.org

Although there are some definitions of Parts II and III that are used in Part V, this does not require to read these parts first. Those definitions can be read when needed.

Some additional lemmas with their proof are also given in this appendix.

This part is necessary to understand how the complexity of RAF decision problems has been proven.

This part is necessary to understand how the decomposability of RAF semantics has been proven.

The fictive arguments are denoted by a ′ in the definition of J ′ and the fictive interactions are the pairs (a ′ , a) or (a, a) appearing in the definition of K ′ J .

See Section 16.4 on page 178, in the appendix, for the mathematical notions related to the complexity classes.

Table

3.1: Complexities of Dung's Abstract Framework

This notion of largeness of an AF is not so simple to define. It is related to the fact that the computation of the solutions is complex either because of the number of arguments, or of the number of interactions, or because of the structure of the AF.

http://argumentationcompetition.org

Notice that in the literature it is the decision problem versions of function problems that are studied rather than the actual function problems. This is due to the convenience of Decision Problem Theory. The complexity of their decision versions is sufficient to give a good idea of their hardness. See[START_REF] Dvorak | Computational problems in formal argumentation and their complexity[END_REF] for an overview.

A graph is said to be sparse when its density is low.

This process is later described in details. For the moment, we will simply say that it produces valid labellings.

Notice that in the CSP modeling the border arguments can only be labelled in , out or und . The potential illegal aspect of labellings (that is, labels iout and iund , as it can be seen in Table4.2 on page 40) is captured by the constraints added in Steps 4b and 4c. In this example, it corresponds to constraints c 4 and c 5 .

In ICCMA Competition the enumeration problems for the preferred, stable and complete semantics are named respectively "EE-PR", "EE-ST" and "EE-CO"

 2 In addition to the solvers presented in Table5.1 on page 56, Eqargsolver2019, Taas-dredd2019 and Yonas2019 have also been analysed.[START_REF] Le | Sat4j 2.3.2: on the fly solver configuration[END_REF]

This is due to the fact that finding an element in a dictionnary is done in constant time (i.e. Θ(1)) in the average case as explained by the following footnote.

Good implementations of dictionaries use some hash function to create the keys of the stored elements. If the hash function is well chosen w.r.t. the size of the dictionary itself and the definition domain of the stored elements, then the verification of the membership of an element to the dictionary can be done in average in constant time (i.e. Θ(1)).

See Section 16.5 on page 184 for an explanation on Θ(w + v).

These comparisons do not take into account average solving time of the compact enumeration representation mode when standard enumeration mode gives no result.

A CNF is a propositional formula which is a conjunction of clauses. A clause is a propositional formula which is a disjunction of literals. A literal is a propositional variable or the complement of a propositional variable (e.g. v or ¬v).

P-SCC-REC is called recursively with a parameter which corresponds to the set of arguments attacked by und -labelled arguments from the SCC ancestors.

The words "negation" and "conjunction" are used only by abuse of language and not in a logic meaning (even if there is a link, see[START_REF] Cayrol | Logical Encoding of Argumentation Frameworks with Higher-order Attacks and Evidential Supports[END_REF]).

This means that K ′ received the names of the attacks that are in K.

A standard RAF being a RAF, standard RAF labellings are simply RAF labellings.

The use of these colors illustrates more easily the common points between the RAF side and the AF side and also the fact that the inputs can still appear on the RAF side but not on the AF side.

As a reminder, for an AF semantics σ , to be top-down (resp. bottom-up, fully) decomposable is equivalent to be top-down (resp. bottom-up, fully) decomposable w.r.t. S D-a f .

In computational complexity theory, a function problem is a computational problem where a single output is expected for every input, but the output is more complex than that of a decision problem: it is not simply "yes" or "no".

The endpoints of an edge are the vertices incident to it. See Definition 114.

That is, whetherx 1 ̸ = 0, x 2 ̸ = 0, x

̸ = 0, x

̸ = 0 or x

̸ = 0

A propositional formula is said to be satisfiable if there exists a model of it, that is a value (true or false) assignation of its propositional variables for which φ is true.

A propositional formula is said to be unsatisfiable iff there exists no model of it.

Ω = v 1 ∪ v 2 and v 1 ∩ v 2 = ∅.

A formula φ is said to be valid if all valuations of its propositional variables are models of φ , i.e. φ is always true.

The complexity of f should be "easy" compared to the complexity of solving P 1 or P 2 .

A CSP modelling is used for in AFDivider algorithm. See Section 4.2.4 on page 42.

There exist algorithms, such as Krylov-Schur method, able to compute eigenvectors from smallest to greatest eigenvalue and stop at any wanted step (e.g. number of vectors found). With such an algorithm it is not necessary to find all the solutions as we are interested only in the small eigenvalues.

For each a ∈ ω inp ∩ out (µ) an argument a ′ is created.

All these arguments are not in ω.

All these arguments are not in ω ′ .

The RAF-path p is indeed well formed because at each iteration we obtain whether:{o 1 , α, o 4 } s.t. o 1 ∈ A, α ∈ K and o 4 ∈ (A ∪ K) or: {o 1 , o 4 } s.t. o 1 ∈ K and o 4 ∈ (A ∪ K).This implies so that there cannot be two consecutive arguments in p.

Remind that J ∩ J ′ = ∅ and that the arguments that are in J ′ are different from those created in RAF s to fit input labelling of RAF , I, L inp .

See Definition 106 on page 159 for RAF and AF semantics correspondence.

Remerciements

We prove so that: x ∈ A ∪ K.

Given that x ∈ De f (ε U ), following the definition of Raf2Af, there exists thus an argument s(α).α ∈

Part VI

Conclusion and Perspectives

Matrices

In this section are presented notions on matrices, the key notions being the eigenvectors and values, and laplacian matrices.

Definition 132 (Matrix). A matrix M with m lines and n columns, or a m × n matrix, with values in some field of scalars K is an application of {1, 2, ..., m -1, m} × {1, 2, ..., n -1, n} in K . M i, j ∈ K is the image of the couple (i, j). i is called the line index and j the column index.

Definition 133 (Eigenvector and eigenvalue). Let O be a vector space over some field K of scalars, let u be a linear transformation mapping O into O (i.e. u : O → O), and let v ∈ O be a non-zero vector.

v is an eigenvector of u if and only if there exists a scalar λ ∈ K such that:

In this case λ is called eigenvalue (associated with the eigenvector v).

For more details on eigenvectors and eigenvalues see [START_REF] Robert | Elementary linear algebra[END_REF], Chapter 6.

Definition 134 (Adjacency matrix). Let G = (V, E,W ) be a weighted non-directed graph. The adjacency matrix M a of G is an n × n matrix (with n = |V |) defined as:

If the weights of a graph G represent similarity measures then adjacency matrix is called the similarity matrix of G.

Definition 135 (Degree matrix). Given a weighted non-directed graph G = (V, E,W ), the degree matrix M d for G is an n × n matrix (with n = |V |) defined as:

Note: M d is a diagonal matrix.

Definition 136 (Laplacian matrix). Given a weighted non-directed graph G = (V, E,W ), the laplacian matrix M l for G is an n × n matrix (with n = |V |) defined as: Glossary asymptotic Given a mathematical function f , the "asymptotic" behavior of f is its limiting behavior, that is for a large input (w.r.

Following Lemma 1 on page 234, we thus also have:

As a consequence, we have: und (L ′ ) ⊂ und (L). We prove so that L = Struct2Lab(U) is a reinstatement RAF labelling such that und (L) is maximal. ■ Proof of Theorem 18 on page 109. L be a reinstatement RAF labelling such that in (L) is minimal. Let suppose that U = Lab2Struct(L) is not the grounded structure. By definition of the grounded structure (Definition 63 on page 92), we can thus say that there exists a structure U ′ that is the grounded structure and such that U ′ ⊏ U (strict inclusion). Let L ′ = Struct2Lab(U ′ ) be the reinstatement RAF labelling corresponding with the grounded structure. As U ′ ⊏ U we have, by definition of Struct2Lab: in (L ′ ) ⊂ in (L). We have then a contradiction.

We prove so that U = Lab2Struct(L) is the grounded structure. ■

Proof of Theorem 19 on page 109. Let U be the grounded structure and L = Struct2Lab(U). Let suppose that in (L) is not minimal. There exists then a reinstatement RAF labelling

From the definition of Lab2Struct, we can say that: U ′ ⊏ U. This contradicts the definition of the grounded structure (Definition 63 on page 92). We prove so that L is thus a reinstatement RAF labelling such that out (L) is minimal.

Given that U is RAF-stable, we have thus:

There is thus a contradiction.

We prove so that if U = ⟨S, Q⟩ is a RAF-stable structure in RAF then ε U is a stable extension in AF .

As shown in Proof of Assertion 1 (Equation (18.81) on page 243):

We have thus:

We prove so that if ε U is a stable extension in AF then U = ⟨S, Q⟩ is a RAF-stable structure in RAF .

We have so:

Proofs of Chapter 14: Decomposability and Hierarchy

Proofs of Section 14.1: SCC ra f

Proof of Proposition 33 on page 126. Given that x ̸ = y, x ≡ RAF y iff we have the following facts:

• ∃p ∈ Paths ra f (RAF ) such that p = (x, ..., e n-1 , y) and y = t(e n-1 ).

• ∃p ′ ∈ Paths ra f (RAF ) such that p ′ = (y, ..., o m-1 , x) and x = t(o m-1 ).

Let c be the sequence formed by the concatenation of p and p ′ as follows: If |U| = 1 then trivially U is included in some S ∈ SCCS ra f (RAF ). Let consider that |U| > 1. Let e i ∈ U and e j ∈ U such that e i ̸ = e j . Given that c ∈ ClosedWalk ra f (RAF ), then following Proposition 33 on page 126 we have: e i ≡ RAF e j . As a consequence, e i and e j belong to the same S ∈ SCCS ra f (RAF ). We prove so that: U is included in some S ∈ SCCS ra f (RAF ). ■ And:

{{(a, in), (b, out)} , {(b, in), (a, out)} , {(b, und), (a, und)}}

Let consider Ω 2 the partition of RAF 2 corresponding to Ω 1 in which all attacks are in the same part as their sources. We have:

), the RAF-compliant partition of AF 3 corresponding to Ω 2 . We have:

We can observe that:

(a, in), (¬a, out), (γ, in), (¬γ, out), (a.γ, in), (b, out), (¬b, in), (β , in), (¬β , out), (b.β , out)

(a, in), (¬a, out), (γ, in), (¬γ, out), (a.γ, in), (b, out), (¬b, in), (β , in), (¬β , out), (b.β , out)

(a, out), (¬a, in), (γ, in), (¬γ, out), (a.γ, out), (b, in), (¬b, out), (β , in), (¬β , out), (b.β , in)

(a, und), (¬a, und), (γ, in), (¬γ, out), (a.γ, und), (b, und), (¬b, und), (β , in), (¬β , out), (b.β , und)

As a consequence the preferred semantics is not bottom-up decomposable w.r.t. S ra f -c-USCC .

Note:

Other examples than Example 92 on the next page could be constructed for any wanted semantics, any selector, any AF and partition. Example 92 just highlights the idea behind Lemmas 14 and 15 and Propositions 43 and 45 on page 160, on page 161, on the following page and on page 287. Lemma 14. Let σ be a complete-based AF semantics. Let AF 1 = ⟨A 1 , K 1 ⟩ be an AF and Ω 1 = ω

⟩) be the AF corresponding to the flattening of RAF 2 . Let Ω 3 = ω 3 1 , ..., ω 3 n be the partition of AF 3 such that:

The following property holds:

With:

Proof of Lemma 14. Given that all attacks in RAF 2 are valid and that σ is complete-based, we have:

Following the definition of Raf2Af and given that σ is complete-based, we have:

, we have so:

Let σ be a complete-based AF semantics. Let AF 1 = ⟨A 1 , K 1 ⟩ be an AF and Ω 1 = ω 1 1 , ..., ω 1 n be any partition of AF 1 . Let RAF 2 = Af2Raf(AF 1 ) be the non recursive RAF corresponding to AF 1 , with RAF 2 = ⟨A 2 , K 2 , s 2 ,t 2 ⟩ (see Definition 80 on page 121). Let Ω 2 = ω 2 1 , ..., ω 2 n be the partition of RAF 2 such that:

⟩) be the AF corresponding to the flattening of RAF 2 . Let Ω 3 = ω 3 1 , ..., ω 3 n be the partition of AF 3 such that:

Let R ∈ {⊆, ⊇, =} be a binary relation over sets. The following property holds:

j being a labelling of J 3 j Note: Unlike in Lemma 14 on the previous page, there is no constraint on ℓ J 3 j following the definition of the decomposability of an AF semantics. See Definitions 19 and 22 on page 19.

Proof of Lemma 15. Given that σ is complete-based, we have:

Furthermore, given that all attacks in RAF 2 are valid and that σ is complete-based, we have:

From Equations (18.151) and (18.152), from the definition of Raf2Af and from the fact that σ is complete-based, we have:

Now, from Lemma 14 on page 285, we have:

Finally, from Equations (18.153) and (18.154) on the previous page and on this page, we prove so that: Let prove that the reciprocal proposition is also true.

Let σ be an AF complete-based semantics that is not top-down (resp. bottom-up, fully) decomposable. With R a binary relation over sets being respectively "⊆", "⊇", "=", we have so:

Let σ be an AF complete-based semantics that is not top-down (resp. bottom-up, fully) decomposable w.r.t. S USCC . With R a binary relation over sets being respectively "⊆", "⊇", "=", we have so:

n ) be a partition of AF 1 such that they satisfy Equation (18.162). Let RAF 2 = Af2Raf(AF 1 ) be the non recursive RAF corresponding to AF 1 , with RAF 2 = ⟨A 2 , K 2 , s 2 ,t 2 ⟩ (see Definition 80 on page 121). Let Ω 2 = ω 2 1 , ..., ω 2 n be the partition of RAF 2 such that: ∀ω 2 j ∈ Ω 2 , ω 2 j = ω 1 j ∪ α|α ∈ K 2 s.t. s(α) ∈ ω 1 j . Let AF 3 = Raf2Af(RAF 2 ) (with AF 3 = ⟨A 3 , K 3 ⟩) be the AF corresponding to the flattening of RAF 2 . Let Ω 3 = ω 3 1 , ..., ω 3 n be the partition of AF 3 such that: ∀ω 3 j ∈ Ω 3 , ω 3 j = ω 2 j ∪ ¬x|x ∈ (A 2 ∪ K 2 ) ∩ ω 2 j ∪ s(α).α|α ∈ K 2 ∩ ω 2 j . Following Lemma 15 on page 287, we have:

Notice that Ω 3 is a RAF-compliant partition of AF 3 (i.e. Ω 3 ∈ S D-ra f -c (AF 3 )). As a consequence, following the definition of Ω 1 , of Ω 2 and of Ω 3 , we have: Ω 3 ∈ S ra f -c-USCC (AF 3 ).

It follows then that, if σ is not top-down (resp. bottom-up, fully) decomposable w.r.t. S USCC , the following statement holds: ∃AF ∈ {Raf2Af(RAF )|RAF ∈ Φ ra f } and Ω ∈ S ra f -c-USCC (AF ) s.t.

L σ (AF ) R ℓ 1 ∪ ... ∪ ℓ n ℓ i ∈ F a f σ (AF ↓ ω i , J i , ℓ J i , K J i ) (18.164)

Considering the contrapositive of the previous implication, we prove so that: