Combining Extension-based and Ranking-based Semantics for Abstract Argumentation Elise Bonzon, Jérôme Delobelle, Sébastien Konieczny, Nicolas Maudet

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Elise Bonzon, Jérôme Delobelle, Sébastien Konieczny, Nicolas Maudet. Combining Extension-based Semantics and Ranking-based Semantics for Abstract Argumentation. 16th International Conference on Principles of Knowledge Representation and Reasoning, Oct 2018, Tempe, United States. ￿hal- 01900735￿

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Elise Bonzon Jer´ omeˆ Delobelle Sebastien´ Konieczny Nicolas Maudet LIPADE CRIL, CNRS LIP6, CNRS Universite´ Paris Descartes, France Universite´ d’Artois, France Sorbonne Universite,´ 75005 Paris, France [email protected] {delobelle,konieczny}@cril.fr [email protected]

Abstract and Lagasquie-Schiex 2005; Amgoud and Ben-Naim 2013; Grossi and Modgil 2015; Pu et al. 2015; Amgoud et al. 2016; Two kinds of semantics exist for abstract argumentation. Patkos, Bikakis, and Flouris 2016; Bonzon et al. 2016b)), Extension-based semantics evaluate the acceptability of sets where the aim is to (comparatively) evaluate each argu- of arguments, while ranking-based semantics evaluate the strength of each argument. They on different aspects ment in an argumentation system. Ranking-based semantics of the information conveyed by argumentation systems. Af- are functions that map each argumentation framework to a ter discussing pros and cons of both approaches, we study ranking (usually a total pre-order) on its arguments. This how to combine them, in order to take benefits from both. We ranking represents the comparative strength of each argu- propose six new families of semantics for abstract argumen- ment. Thus, conversely to extension-based (and labelling- tation combining extension-based and ranking-based seman- based) semantics, this approach does not evaluate sets of tics. More precisely we propose to refine the ranking-based arguments but each argument individually, based on its sit- semantics using information coming from extension-based uation in the argumentation graph. A related kind of se- semantics acceptability of arguments, and to modify the ex- mantics are grading-based semantics (see e.g. (Besnard and tensions chosen by extension-based semantics using prefer- Hunter 2001; Matt and Toni 2008; Leite and Martins 2011; ential information coming from ranking-based semantics. da Costa Pereira, Tettamanzi, and Villata 2011)), where a numerical value is assigned to each argument. The evalua- Introduction tion is numerical instead of ordinal, but the aim is still to evaluate each argument individually. Clearly, if one defines Argumentation is the process of confronting conflicting ar- a grading-based semantics, then this straightforwardly in- guments. In the abstract argumentation framework (Dung duces a corresponding ranking-based semantics. 1995), the classical semantics are extension-based seman- tics. These semantics aim at evaluating which sets of argu- Thus we may opt for two kinds of evaluations of argu- ments can be accepted together. These extensions are usually ments: at the level of of arguments (with extension-based based on the conflict-freeness principle (two arguments in an or labelling-based semantics) or at the level of single ar- extension can not attack each other) and on the self-defense guments (with ranking-based or grading-based semantics). principle (an extension has to defend each of its attacked ar- These two ways to evaluate the information encoded in an gument). Thus, these semantics evaluate sets of arguments argumentation framework are interesting, and are useful for in a binary way (sets of arguments are or are not extensions different applications. The second approach is much more for a given semantics). recent and more work is needed to better understand the no- In (Caminada 2006), labelling-based semantics have been tions and look for meaningful new semantics. But even the introduced to associate different labellings to the arguments first kind of evaluation, although studied for a long time, still of any argumentation framework. A labelling is a need some work for understanding their underlying princi- that maps each argument to the set {in, out, undec}, where ples (it worths mentioning that conversely to other reasoning in means that the argument is accepted for the labelling, out tasks like inference (Kraus, Lehmann, and Magidor 1990; means that the argument is rejected, and undec means that Makinson 1994) or revision (Alchourron,´ Gardenfors,¨ and the argument is undecided. So these semantics still perform Makinson 1985; Gardenfors¨ 1988; Katsuno and Mendelzon an evaluation of sets of arguments, just like extension-based 1991), there are no postulates for characterizing rational ar- semantics. And it has been shown that all extension-based gumentation semantics and no representation theorem). semantics correspond to some labelling-based semantics. The starting point of this work is the observation that More recently, it has been argued that this binary or these two kinds of evaluation are in a sense orthogonal. They ternary evaluation can be too rough for some applica- both can be used to extract some information about the sta- tions, for example for online debate platforms (Leite and tus/strength/situation of (sets of) arguments. Instead of see- Martins 2011), and the need of a more focused evalu- ing these approaches as mutually exclusive, one natural idea ation of each argument has been put forward. This led is to try to take the best of both worlds and combine them. to the idea of ranking-based semantics (see e.g. (Cayrol We believe that studying the potential of such a combination, as we initiate in this work, can be very fruitful for developing between arguments and thus obtain a consistent set of for- argumentation semantics. mulas. In this work we propose six new families of semantics However, in other applications, some of these proper- for abstract argumentation combining extension-based se- ties can be discussed. Recently, online debate platforms are mantics and ranking-based semantics. More precisely, we emerging on the internet. On these debate platforms, agents propose to refine ranking-based semantics using information argue for or against a particular topic (in the form of a coming from extension-based semantics acceptability of ar- question or an affirmation) or other existing arguments. Of- guments, and to modify the extensions chosen by extension- ten, the goal is not to find the arguments which can be ac- based semantics using preferential information coming from cepted together but to evaluate how accepted is the ques- ranking-based semantics. tion/affirmation. But more generally, when one faces many More precisely in the next section we will discuss the arguments, having a more detailed evaluation of arguments differences between the evaluation of arguments obtained than the binary accepted/rejected obtained with extension- by extension-based semantics and ranking-based semantics. based semantics may be useful. Leite and Martins (2011) We will then recall the necessary background on abstract emphasize the limitations of classical acceptability seman- argumentation. We will next show how to modify ranking- tics for this kind of applications. In addition, to accurately based semantics by taking into account information coming represent the opinions of thousands of users, it could be from extension-based semantics. We propose four ways to more appropriate to evaluate arguments using degrees of do that. The first one is by focusing only on the acceptabil- acceptability or gradual acceptability. With ranking-based ity status of each argument (given by the extension-based semantics, we can precisely obtain a very detailed evalua- semantics). The second one is based on a more precise eval- tion of the strength of each argument. This can be useful for uation of the acceptability status of each argument from (Wu these debate platforms, but also to select best arguments in and Caminada 2010). The third and fourth ones are modifi- all kinds of debates (persuasion, deliberation, etc.). cations of a particular ranking-based method, the Propaga- However we can see as a drawback the fact that the eval- tion method (Bonzon et al. 2016b), where we allow a more uation of each argument is not linked at all with its ac- fined-gained distinction of arguments using these accept- ceptance status: being an argument with a good evaluation ability status. Concerning the other way, i.e. how to modify does not mean that this argument should be accepted (un- extension-based semantics using ranking-based semantics, der extension-based semantics), and even if we define “ac- we show first that ranking-based semantics can be used to ceptance” with respect to the ranking, there are no natural evaluate the extensions, and to select only the best of them. threshold to make a distinction between accepted and non- Then we discuss the possibility to take the ranks given by accepted argument. Defining a ranking-based semantics that ranking-based semantics as a preferential information in a is compatible with the acceptance status of an extension- preference-based argumentation framework (Amgoud and based semantics would be a solution. So we propose to build Cayrol 2002) in order to select only the most convincing this kind of semantics by refining ranking-based semantics attacks. using extension-based semantics. Conversely a drawback of extension-based semantics is that they do not allow a very detailed evaluation of argu- Extension-based vs. Ranking-based semantics ments. It is for instance impossible to give a better evalua- Extension-based semantics (Dung 1995) are closely related tion to an unattacked argument than to all the arguments that to models of programs so they exhibit an all-or-nothing this argument defends, whereas the acceptability of the latter evaluation of sets of arguments. Amgoud and Ben-Naim depends on the acceptability of the unattacked argument. So (2013) underlines some characteristics specific to extension- one can use the detailed evaluation of arguments in order to based semantics: Killing: The impact of an attack from an modify extension-based semantics, for instance by selecting argument y to an argument x is drastic, that is, if y belongs only the best extensions with respect to this evaluation. to an extension, then x is automatically excluded from that We explore the two paths in the following. extension; Existence: One successful attack against an argu- ment x has the same effect on an argument as any number Background Notions of successful attacks. Indeed, one such attack is sufficient In this section, we briefly recall some key elements of ab- to “kill” x, several attacks cannot kill x to a greater extent; stract argumentation frameworks. Absoluteness: The three possible statuses of the arguments Definition 1 An argumentation framework (AF) is a pair (accepted, rejected or undecided) are absolute, that is, they AF = hA, Ri with A a finite set of arguments and R ⊆ make sense even without comparing them with each other; A × A is the attack relation between arguments. A set of Flatness: All the arguments with the same status have the arguments S ⊆ A attacks an argument y ∈ A, if there exists same level of acceptability. For example, all the accepted x ∈ S, such that (x, y) ∈ R. S defends z ∈ A against its (respectively rejected) argument cannot be distinguished, i.e attacker y if S attacks y. no accepted argument are more acceptable than another ac- cepted argument. This kind of evaluation can be useful to Abstract argumentation frameworks can be represented define arguments from logical formulas. Here, the killing by directed graphs, where the nodes represent the arguments and existence consideration seem essential to capture the and the edges represent the attack relation between two ar- fact that one attack is lethal and prevent any contradiction guments. Let us now introduce some useful notions. Definition 2 (Path) Let AF = hA, Ri be an argumenta- Definition 7 Let AF = hA, Ri be an argumentation frame- tion framework and x, y ∈ A.A path P from y to x, noted work. A reinstatement labelling L is: P (y, x), is a sequence hx , . . . , x i of arguments such that 0 n • a complete labelling; x0 = x, xn = y and ∀i < n, (xi+1, xi) ∈ R. The length of the path P is n (the number of attacks it is composed of) • a grounded labelling if in(L) is minimal (w.r.t. ⊆); and is denoted by lP = n. • a preferred labelling if in(L) is maximal (w.r.t. ⊆); Depending on the length of a path between two argu- • a stable labelling if undec(L) = ∅. ments, the argument at the beginning of this path can be an We denote by Lσ(AF ) the set of reinstatement labellings of attacker and/or a defender (i.e., an argument which attacks AF for the semantics σ ∈ {co, pr, st, gr}. an attacker) of the argument at the end of the path. AF Definition 3 (Defender/Attacker) A defender (resp. at- For an argumentation framework with at least one tacker) of x is an argument situated at the beginning extension (resp. reinstatement labelling), we say that an of an even-length (resp. odd-length) path. Let R (x) = argument is skeptically accepted if it belongs to all of n AF ’s extensions (resp. it is labelled in in all of AF ’s {y | ∃P (y, x) with lP = n} be the multiset of arguments that are bound by a path of length n to the argument x. Thus, reinstatement labellings). An argument is credulously accepted if it belongs to at least one of AF ’s extensions an argument y ∈ Rn(x) is a direct attacker of x if n = 1 or a direct defender of x if n = 2. (resp. it is labelled in in at least one of AF ’s reinstatement labellings). Given a semantics σ, we denote by saσ(AF ) Extension/Labelling-based semantics (resp. caσ(AF )) the set of skeptically (resp. credulously) accepted arguments in AF . In Dung’s framework (Dung 1995), several acceptability se- mantics have been defined to select sets of arguments, called A more fine-grained notion of a justification status has extensions, which can be conjointly accepted (w.r.t some cri- also been introduced in (Wu and Caminada 2010) with a teria depending on the chosen semantic) for a given argu- labelling-based justification status of the arguments in an ar- mentation framework. gumentation framework. Concretely, the justification status Definition 4 Given an argumentation framework AF = of an argument consists of the set of labels that could rea- hA, Ri. A set of arguments S ⊆ A is conflict-free in AF sonably be assigned to the argument w.r.t. the complete se- if ∀x, y ∈ S, (x, y) ∈/ R. A conflict-free set S is admissi- mantics. ble if it defends all its arguments against each of their direct Definition 8 (Justification status) Let AF = hA, Ri be an attackers. An admissible set S is: argumentation framework and x ∈ A. The justification sta- • a complete extension if each argument defended by S be- tus of x is the outcome yielded by the function JS : A → {in,out,undec} longs to S; 2 such that JS(x) = {L(x) | L ∈ Lco(AF )}. • a preferred extension if it is a ⊆-maximal admissible set For example, if an argument is labelled either in or undec of AF ; in all the complete labellings then the justification status of • a stable extension if it attacks each argument in A\S; this argument is {in, undec}. Thus, there are 6 possible sta- • the single grounded extension if it is the ⊆-minimal com- tuses to be considered: {in}, {out}, {undec}, {in, undec}, plete extension of AF . {out, undec} and {in, out, undec}.

We denote by Eσ(AF) the set of extensions of AF for the se- mantics σ ∈ {co(mplete), pr(eferred), st(able), gr(ounded)}. Ranking-based semantics An alternative way to represent the of admissi- A ranking-based semantics allows to rank-order the argu- bility, as well as Dung’s semantics, is by using a labelling- ments from the most to the least acceptable ones. based approach (Caminada 2006). Definition 9 A ranking-based semantics σ associates to σ Definition 5 (Labelling) A labelling of an argumentation any argumentation framework AF = hA, Ri a ranking AF σ framework hA, Ri is a function L : A → {in, out, undec}. on A, where AF is a preorder (a reflexive and transitive re- σ Given a label l ∈ {in, out, undec}, we define l(L) = {x ∈ lation) on A. x AF y means that x is at least as acceptable σ σ σ A | L(x) = l}. as y (x 'AF y is a shortcut for x AF y and y AF x, and x σ y is a shortcut for x σ y and y σ x). The notion of reinstatement labelling ensures that the AF AF AF mapping takes the attack relation into account. A lot of these semantics were proposed (see e.g. (Cayrol Definition 6 (Reinstatement Labelling) and Lagasquie-Schiex 2005; Amgoud and Ben-Naim 2013; Let AF = hA, Ri be an argumentation framework. A la- Grossi and Modgil 2015; Pu et al. 2015; Amgoud et al. 2016; belling L is a reinstatement labelling of AF iff Bonzon et al. 2016b)) with, for each of them, different be- haviour and logical properties. In this work, we will fo- •∀ x ∈ A, L(x) = in iff ∀y ∈ R1(x), L(y) = out; cus on the categoriser-based ranking semantics to illustrate •∀ x ∈ A, L(x) = out iff ∃y ∈ R1(x), L(y) = in; our method. This semantics has been initially introduced in •∀ x ∈ A, L(x) = undec iff @y ∈ R1(x), L(y) = in and (Besnard and Hunter 2001) and defined as a ranking-based ∃z ∈ R1(x), L(z) = undec. semantics in (Pu et al. 2014). Definition 10 Let hA, Ri be an argumentation framework. direct attackers than an other argument b, then b should be The categoriser function Cat : A → ]0, 1] is defined as strictly more acceptable than a;(Quality Precedence, QP) ∀x ∈ A, says that if a has a direct attacker strictly more acceptable ( than any direct attacker of b, then a should be strictly more 1 if R1(x) = ∅ acceptable than b;(Defense Precedence, DP) states that for Cat(x) = 1 1+P Cat(y) otherwise two arguments with the same number of direct attackers, y∈R1(x) a defended argument should be strictly more acceptable Definition 11 The categoriser-based ranking semantics than a non-defended argument; (Distributed-Defense Prece- Cat (Cat) associates to any AF = hA, Ri a ranking AF on dence, DDP) considers that a defense where each defender Cat A such that ∀x, y ∈ A, x AF y iff Cat(x) ≥ Cat(y). attacks a distinct attacker is better than any other; (Counter- Example 1 Let us compute the set of extensions and the re- Transitivity, CT) states that if the direct attackers of b are (i) instatement labellings for σ ∈ {co, pr, st, gr} and the rank- at least as numerous and (ii) as acceptable as those of a, then ing returned by the categoriser-based ranking semantics for a should be at least as acceptable as b, while in its strict ver- the AF depicted in Figure 1. sion (SCT) either (i) or (ii) must be strict, implying a strict comparison between a and b. Global properties specify how the ranking should be af- f fected on the basis of the comparison of attack and defense branches. More precisely: adding a defense branch to any ar- a b c d e gument should increase its acceptability (Strict Addition of Defense Branch, ⊕DB); the same properties have been de- g fined but only when a defense branch is added to an attacked argument (Addition of Defense Branch, +DB); increasing the length of an attack branch of an argument should increase Figure 1: An argumentation framework AF its acceptability (Increase of Attack Branch, ↑AB); adding an attack branch to an argument should decrease its accept- Egr(AF ) = {a} ability (Addition of Attack Branch, +AB); and increasing the Epr(AF ) = {{a, c}, {a, d, f}} length of a defense branch of an argument should decrease Est(AF ) = {{a, d, f}} its acceptability (Increase of Defense Branch, ↑DB). Note Eco(AF ) = {{a}, {a, c}, {a, d, f}} that +DB is indeed restricted to attacked arguments, oth- erwise its incompatibility with VP is obvious. In the same AF has three reinstatement labellings L1, L2 and L3 with: spirit, (Attack vs Full Defense, AvsFD) requires that an ar- in(L1) = {a}, out(L1) = {b}, undec(L1) = {c, d, e, f, g} gument with only defense branches and no attack branch in(L2) = {a, c}, out(L2) = {b, d}, undec(L2) = {e, f, g} should be strictly more acceptable than an argument attacked in(L3) = {a, d, f}, out(L3) = {b, c, e, g}, undec(L3) = ∅ once by a non-attacked argument. Note that all these properties can not be satisfied together

Cat Cat Cat Cat Cat Cat (Bonzon et al. 2016a), but checking which ones are satisfied Cat(AF ) = a AF f AF d AF g AF b AF c AF e by a semantics allow to characterize its behaviour. Many properties have been introduced in the literature (see (Bonzon et al. 2016a; Baroni, Rago, and Toni 2018) Improving Ranking-based semantics using for an overview) aiming to better understand the behavior of Extension-based semantics these ranking-based semantics in various situations. Below Refining ranking-based semantics using acceptance we study how some of our methods stand with respect to status these properties. We give their informal definition and point the reader to (Bonzon et al. 2016a) for the complete ver- The idea here is to constrain the rankings to be compati- sions. Basic general properties are the fact that a ranking on ble with the acceptance status of the arguments. We lexico- a set of arguments should only depend on the attack relation graphically combine a ranking denoting the acceptance sta- (Abstraction, Abs); that the ranking between two arguments tus of the arguments given by an extension-based semantics should be independent of arguments that are not connected and the ranking given by a ranking-based semantics. to either of them (Independence, In); that all arguments can Definition 12 Let AF = hA, Ri be an argumentation 1 2 be compared (Total, Tot); and that all non-attacked argu- framework. Let AF and AF be two rankings on A. The (lex- 2 1 ments should be equally acceptable (Non-attacked Equiva- icographical) refinement of AF by AF gives a new ranking lence, NaE). 1,2 AF such that ∀x, y ∈ A, Local properties confine themselves to the level of direct 1,2 2 2 1 attackers or direct defenders: (Void Precedence, VP) states x AF y iff (x AF y) or (x 'AF y and x AF y) that a non-attacked argument should be strictly more accept- The following definition allows to build a ranking from able than any attacked argument; (Self-Contradiction, SC) the acceptance status given by an extension-based seman- states that an argument that attacks itself should be strictly tics1: an argument skeptically accepted is more acceptable less acceptable than an argument that does not; (Cardinality Precedence, CP) says that if an argument a has strictly more 1Please note that, a priori, any extension-based semantics can than an argument credulously accepted which is more ac- Refining ranking-based semantics using ceptable than a rejected argument. justification status Definition 13 Let AF = hA, Ri be an argumentation Instead of focusing on the acceptability status of the argu- σ framework and σ ∈ {co, pr, st, gr}. Let AF be a rank- ments, we can also build a ranking from the labelling-based σ ing on A such that ∀x, y ∈ A, x AF y iff one of justification status of the arguments, that offers a more fined- the following conditions is satisfied: i) x ∈ saσ(AF ), ii) gained distinction of the arguments with respect to the la- x ∈ caσ(AF )\saσ(AF ) and y∈ / saσ(AF ), iii) x, y∈ / bellings/extensions. However, the definition from (Wu and caσ(AF ) Caminada 2010) (see Definition 8) only concerns the com- plete semantics. It is why we propose to extend the definition Definition 14 (Acceptance-based ranking semantics) Let to all Dung’s semantics. σ1 be a ranking-based semantics and σ2 ∈ {co, pr, st, gr}. Definition 15 (Extended justification status) The acceptance-based ranking semantics ARSσ1,σ2 asso- σ1,σ2 Let AF = hA, Ri be an argumentation framework, σ ∈ ciates to any AF = hA, Ri a ranking AF on A which is σ2 σ1 {co, pr, st, gr} and x ∈ A. The extended justification sta- the refinement of AF by AF . tus of x is the outcome yielded by the function JS : A → Example 2 Let us first compute the arguments skeptically {in,out,undec} 2 s.t. JSσ(x) = {Lσ(x) | L ∈ Lσ(AF )}. and credulously accepted w.r.t. the complete semantics on In addition to the 6 statuses {in}, {out}, {undec}, the AF depicted in Figure 1: saco(AF ) = {a} and {in, undec}, {out, undec} and {in, out, undec}, we must caco(AF ) = {a, c, d, f}. add the status {in, out}, that could not appear for the com- co co co co co co a AF c 'AF d 'AF f AF b 'AF e 'AF g plete semantics, but which may be obtained, for instance with the preferred semantics. Let us recall the ranking returned by the categoriser- With the graph depicted in Figure 2, we include the status based ranking semantics: {in, out} in the hierarchy of the justification statuses. Cat Cat Cat Cat Cat Cat a AF f AF d AF g AF b AF c AF e Thus, when we combine the two rankings, the refinement- {in} based ranking semantics returns the following ranking:

Cat,co Cat,co Cat,co Cat,co Cat,co Cat,co {in, undec} a AF f AF d AF c AF g AF b AF e

One can see on this example that g has quite a good eval- {undec} {in, undec, out} {in, out} Cat uation for the ranking-based semantics AF , whereas it is a rejected argument. In particular it has a better evaluation that c that is credulously accepted (for the complete seman- {out, undec} tics). The combined ranking-based semantics Cat,co allows to force c to be better than g. {out} So now the question is to know whether these modifica- tions change the “rationality” of the ranking-based seman- tics, i.e. do these combined semantics satisfy less logical properties than the original ranking-based semantics? Figure 2: The hierarchy of the extended justification statuses

Proposition 1 Let σ1 be a ranking-based semantics and Thus, we can classify the statuses with the follow- σ2 ∈ {co, pr, st, gr}. Let α be any property among Abs, In, ing ranking: {in} js {in, undec} js {undec}' VP, DP, DDP, SC, ⊕DB, +DB, +AB, ↑AB, ↑DB, Tot, NaE. {in, out, undec}'{in, out} {out, undec} {out}. If σ satisfies the property α, then the semantics ARS js js 1 σ1,σ2 According to this classification, we can say that an argument satisfies the property α. The semantics ARS and σ1,gr is more acceptable than another one if it has a better status. ARSσ1,st satisfy QP, CT and SCT. The semantics ARSσ1,σ2 satisfies the property AvsFD and does not satisfy CP. Definition 16 Let AF = hA, Ri be an argumentation JSσ framework and σ ∈ {co, pr, st, gr}. Let AF be a rank- It is interesting to note that, except for AvsFD and CP, the ing on A such that ∀x, y ∈ A, semantics satisfies the property if the original ranking-based semantics satisfies the properties. Thus, the compliance of JSσ x AF y iff JSσ(x) js JSσ(y) the ranking-based semantics with respect to these properties is preserved using the refinement with the extension-based Definition 17 (Justification-based ranking semantics) semantics. Better than that, it allows the enforcement of Let σ1 be a ranking-based semantics and σ ∈ AvsFD that few semantics satisfy (Bonzon et al. 2016a). So {co, pr, st, gr}. The justification-based ranking se- it is an easy way to obtain new semantics satisfying AvsFD mantics JRSσ1,σ associates to any AF = hA, Ri a ranking σ1,JSσ JSσ σ1 from standard semantics from the literature. AF on A which is the refinement of AF by AF . be used in our method but for our properties, we choose to only Example 3 We recall that the argumentation framework de- focus on the four classical semantics. picted in Figure 1 has three complete labellings L1, L2 and v L3. From these different complete labellings, the labelling- Definition 20 (P ropa ) The ranking-based seman- based justification statuses of each argument in the AF tics P ropav associates to any argumentation frame- is JS (a) = {in}, JS (b) = {out}, JS (c) = Pv co co co work AF = hA, Ri a ranking AF on A, where JSco(d) = {in, out, undec}, JSco(e) = JSco(g) = v is a valuation function, such that ∀x, y ∈ A, {undec, out} and JS (f) = {in, undec}. So we obtain Pv v v co x  y iff P (x) lex P (y). the following ranking: AF

JSco JSco JSco JSco JSco JSco The ranking returned by the semantics clearly depends on a AF f AF c 'AF d AF e 'AF g AF b the chosen valuation function v. In (Bonzon et al. 2016b) Combined with the ranking returned by the categoriser- the valuation function takes only two values (one for non- based ranking semantics, we obtain the following ranking: attacked and one for attacked arguments). We will now pro- pose more complex functions. Let us first define a valuation a Cat,JScof Cat,JScod Cat,JScoc Cat,JScog Cat,JScoe Cat,JScob AF AF AF AF AF AF function which takes into account the level of acceptability 2 Let σ1 be a ranking-based semantics and of arguments. σ ∈ {co, pr, st, gr}. Let α be any property among Abs, 2 Definition 21 (v ) Let AF = hA, Ri be an argumentation In, VP, DP, DDP, ⊕DB, +DB, +AB, ↑AB, ↑DB, Tot, NaE. ~zσ framework and ~zσ = hα, β, γ, δi be a vector of real number If σ1 satisfies the property α, then the semantics ARSσ ,JS 1 σ2 linked to the semantics σ ∈ {co, pr, st, gr}. The valuation satisfies the property α. The semantics ARSσ1,JSgr and ARS satisfy QP, CT and SCT. The semantics ARS function vσ : A → [0, 1] is defined as ∀x ∈ A, σ1,JSst σ1,JSσ2 satisfies the property AvsFD and does not satisfy CP, SC.  α if R (x) = ∅  1 One can remark that the difference of properties satisfied  β if x ∈ saσ(AF ) and R1(x) 6= ∅ v~zσ (x) = between this semantics and the previous one is minor. In- γ if x ∈ caσ(AF )\saσ(AF )  deed, the only difference concerns the Self-Contradiction δ if x∈ / caσ(AF ) (SC) property and can be explained by the fact that an ar- gument which attacks itself is labelled undec if it is not with 1 ≥ α > β > γ > δ ≥ 0. attacked by other arguments. Thus, this argument is more So a preference is given to non-attacked arguments, then acceptable than an argument directly attacked by a non- to skeptically accepted arguments, then to credulously ac- attacked argument while the skeptical and credulous infer- cepted ones, and the worst ones are rejected arguments. ence functions always considers the two arguments as re- jected. Example 4 Applying the complete semantics on the AF de- picted in Figure 1 allows to say that a is the only argument Refining Propagation semantics using acceptance which is not attacked while c, d and f are credulously (and and justification status not skeptically) accepted and b, e and g are considered as We propose to adapt the propagation principle introduced in rejected. (Bonzon et al. 2016b). The idea of propagation is to give With ~zco = h1, 0.7, 0.3, 0i, we obtain the following table a better initial value to non-attacked arguments than to at- which sums up the valuation of each argument at each step: tacked arguments, in order to improve their impact in the P ~zco a b c d e f g evaluation of arguments. Then these values are propagated i into the argumentation framework. 0 1 0 0.3 0.3 0 0.3 0 Similarly to the previous sections we propose here to use 1 1 -1 0 0 -0.3 0.3 -0.3 acceptance status and justification status to allow a more 2 1 -1 1.3 0.3 0.3 0.6 -0.3 fine-grained initial evaluation. Let us formally define the propagation principle. When we lexicographically compare the propagation vector of each argument, we obtain the following ranking: Definition 18 (Propagation) Let AF = hA, Ri be an ar- P P P P P P gumentation framework. Let v : A → [0, 1] be a valuation a ~zco f ~zco c ~zco d ~zco e ~zco g ~zco b function giving an initial weight to each argument. The val- AF AF AF AF AF AF uation P of x ∈ A, at step i ∈ N, is given by: Let us check which properties are satisfied: ( v(x) if i = 0 Proposition 3 Let σ ∈ {co, pr, st, gr}. The semantics P v (x) = P v (x) + (−1)i P v(y) otherwise i i−1 P ropa~zσ satisfies Abs, In, VP,DP, ↑AB, ↑DB, +AB, Tot, NaE y∈Ri(x) and AvsFD. The other properties are not satisfied. The propagation vector of x is denoted P v (x) = v v The set of properties satisfied by P ropa~zσ is close to the hP0 (x),P1 (x),... i. ones satisfied by the semantics P ropa introduced in (Bon- Like in (Bonzon et al. 2016b), we use the lexicographical zon et al. 2016b). The differences come from the AvsFD order to compare the propagation vectors of each argument. property that is satisfied by P ropa~zσ thanks to the distinc- Definition 19 (Lexicographical order) A lexicographical tion done between attacked arguments in the initial evalua- order between two vectors of real numbers V = tion, and from the fact that SCT and CT are not satisfied. 0 0 0 hV1,...,Vni and V = hV1 ,...,Vni is defined as follows: We also define another valuation function by considering 0 0 0 V lex V iff ∃i ≤ n s.t. Vi > Vi and ∀j < i, Vj = Vj . the extended justification status (see Definition 15). a Definition 22 (vJSσ ) Let AF = hA, Ri be an argumenta- tion framework and ~JSσ = hα, β, γ, δ, , ωi be a vector of real number linked to the semantics σ ∈ {co, pr, st, gr}. The valuation function v : A → [0, 1] is defined as ∀x ∈ A, JS~ σ b f d  α if R1(x) = ∅   β if JSσ(x) = {in} and R1(x) 6= ∅  e c g  γ if JSσ(x) = {in, undec} v~ (x) = δ if JSσ(x) ∈ {{undec}, {in, out}, JSσ E (AF ) = {}  {in, undec, out}} gr  Epr(AF ) = {{a, c}, {b, d, f}, {b, e, f}, {b, d, g}, {b, e, g}}   if JSσ(x) = {undec, out}  Est(AF ) = {{a, c}, {b, d, f}, {b, e, f}, {b, d, g}, {b, e, g}} ω if JSσ(x) = {out} Eco(AF ) = {∅, {a, c}, {b, d, f}, {b, e, f}, {b, d, g}, with 1 ≥ α>β>γ>δ>>ω ≥ 0. {b, e, g}} Example 5 Let us recall the justification status labelling Cat Cat Cat Cat Cat Cat of each argument: JSco(a) = {in}, JSco(b) = {out}, Cat(AF ) = b AF a AF c AF g AF d 'AF e AF f JSco(c) = JSco(d) = {in, out, undec}, JSco(e) = JSco(g) = {undec, out} and JSco(f) = {in, undec}. So, Figure 3: An AF with many extensions with ~JSco = h1, 0.8, 0.6, 0.4, 0.2, 0i, we have:

JS~ co Pi a b c d e f g in Figure 3 because, when we focus on the preferred seman- 0 1 0 0.4 0.4 0.2 0.6 0.2 tics (the remark also holds for the stable and complete se- 1 1 -1 0 0 -0.4 0.4 -0.4 mantics) the set of arguments skeptically accepted is empty, 2 1 -1 1.4 0.4 0.6 1 -0.2 while the set of arguments credulously accepted contains all the arguments of AF . There exist works where addi- And, consequently, we obtain the following ranking: tional information (e.g. weight on the attacks, preferences) are used to reduce the number of extensions in a given ar- P~ P~ P~ P~ P~ P~ JSco JSco JSco JSco JSco JSco a AF f AF c AF d AF e AF g AF b gumentation framework (e.g. (Coste-Marquis et al. 2012; Amgoud and Vesic 2014)). However, our goal is to reduce As shown in the following proposition, increasing the the set of extensions without any additional information in number of distinctions between the attacked argument does the argumentation framework. While, in (Konieczny, Mar- not change the properties satisfied by the propagation se- quis, and Vesic 2015), the attack relation is t