Discussions on social Web platforms carry a lot of information which is more and more di cult to analyze. Given a virtual community of users that discuss a particular topic of interest, an important task is to extract a model of the whole debate in order to automatically evaluate what are the most reliable claims. This paper proposes to approach this task using abstract argumentation, and de ne a new argument system, called Bipolar Weighted Argumentation Framework. It is able to capture all the useful information from a discussion thread, including the strength of positive (i.e., supports) and negative (i.e., attacks) relations between arguments. It also provides a way to assess an acceptability degree for each argument by means of the strength propagation of indirect relations ending to it, and a strategy to build such a framework from an online debate with a hierarchical structure. A model obtained from a real life discussion (a Reddit thread) is discussed and qualitatively evaluated.
People in all societies argue, discuss, and debate not only to convince others of
their own opinions, but because they want to explore the di erences between
their own understanding and the conceptualizations of others, and learn from
them. Being one of the primary intellectual activities of the human mind,
debating therefore naturally involves a wide range of conceptual capabilities and
activities, ones that have only in part been studied from a computational
perspective. One of the most in uential computational models of argument was
presented by Dung's Argumentation Frameworks [
Di erent levels of agreement or disagreement may exist in real discussions with respect to an argument, whereby AFs, BAFs and WAFs may be unable to model certain situations. Moreover, there may exist indirect attack/support relations that in the aforementioned frameworks are not quanti ed.
In the present work we propose a new AF extension, called Bipolar Weighted Argumentation Framework (BWAF), which combines all the properties of BAFs and WAFs. In order to refashion the interactions that may subsist between arguments, BWAFs are able to express weighted directed relations and to compute the propagated strength of indirect relations. We also provide a general strategy to build such a model from an Online Debating System (ODS). We use Text Similarity and Sentiment Analysis techniques to identify weighted attack/support relations. Finally, we show the use of this framework in a real discussion thread taken from Reddit.
This paper is organized as follows. Section 2 brie y recalls the background on abstract argumentation and subsequent generalizations of Dung's Framework. Section 3 introduces our proposal, i.e. BWAF with its useful properties. Section 4 describes a strategy to build a BWAF from an ODS and shows its application to a Reddit discussion. Finally, Section 5 concludes the paper. 2
Let us start by providing the basics of Abstract Argumentation. De nition 1 An AF is a pair F = hA; Ri, where A is a nite set of arguments and R A A. Given a; b 2 A, the relation aRb means that a attacks b. An argumentation semantics is the formal de nition of a method ruling the argument evaluation process. The most basic concepts shared by all argumentation semantics in the literature are con ict-freeness and defense.
De nition 2 Let F = hA; Ri be an AF, and S A: { S is con ict-free if @a; b 2 S s.t. aRb; { a 2 A is defended by S if 8b 2 A : bRa ) 9c 2 S s.t. cRb; { fF : 2A 7! 2A s.t. fF (S) = fa j a is defended by Sg is called the characteristic function of F ; { S is admissible if S is con ict-free and S is defended by itself, i.e. 8a 2
S; 8b 2 A : bRa ) 9c 2 S s.t. cRb.
Then, standard acceptability semantics, introduced by Dung [
{ complete extension i S = fF (S); { grounded extension i S is the -minimal complete extension; { preferred extension i S is a -maximal complete extension; { stable extension i 8a 2 A; a 2= S; 9b 2 S s.t. bRa.
A Bipolar AF (BAF ) [
De nition 4 A BAF is a triplet B = hA; Ratt; Rsupi, where A is a set of arguments, Ratt is a binary relation on A called attack relation and Rsup is another binary relation on A called support relation. Given a; b 2 A, aRattb (resp., aRsupb) means that a attacks b (resp., a supports b).
In BAFs, new kinds of attack emerge from the interaction between the direct attacks and the supports: there is a supported attack i there is a sequence of supports followed by one attack, while, there is an indirect attack i there is an attack followed by a sequence of supports. Taking into account sequences of supports and attacks leads to the following de nitions applying to sets of arguments.
De nition 5 Let B = hA; Ratt; Rsupi be a BAF. A set S A set-attacks
b 2 A, i there exists a supported attack or an indirect attack for b from an
element of S. A set S A set-supports b 2 A, i there exists a sequence of
supports for b from an element of S. A set S A defends a 2 A, i for each
argument b 2 A, if fbg set-attacks a, then b is set-attacked by S.
In the following, we de ne the semantics for acceptability in BAFs [
admissible (resp. s-admissible, c-admissible) subset of A; { a stable extension i S is con ict-free and 8a 2= S, S set-attacks a. -maximal d
A Weighted AF (WAF ) [
De nition 7 A WAF is a triplet W = hA; R; wi, where hA; Ri is the standard
Note that allowing 0-weight attacks is counter-intuitive since it can be interpreted as absence of attack relation. In this framework, some inconsistencies are tolerated in subsets S of arguments, provided that the sum of the weights of attacks between arguments of S does not exceed a given inconsistency budget 2 R+. Hence, given an inconsistency budget , the meaning is that attacks up to a total weight of are neglected. Dung's argument systems assume an inconsistency budget of 0, while, by relaxing this constraint, WAFs can achieve more solutions.
De nition 8 Let W = hA; R; wi be a WAF. Given an inconsistency budget
not exceed , i.e., sub(R; w; ) = fT
R j w(h 1; 2i)
g:
X h 1; 2i2T
Thus, intuitively, any set of arguments is consistent at some cost, and the cost required to make a set of arguments consistent immediately gives us a preference ordering over sets of arguments. Admissibility is de ned in the standard way, and standard semantics are considered leading to various notions of -extensions which echo Dung's ones (i.e., grounded, preferred, stable extensions). De nition 9 Given a WAF W = hA; R; wi, let ES be an extension-based semantics. For 2 R+, the subset ESw(hA; R; wi; ) of 2A is given as w ES (hA; R; wi; ) = fS
A j 9T 2 sub(R; w; ) ^ S 2 ES (hA; R n T ig where ES (hA; Ri) = fS ability semantics S.
A set S 2 ESw(hA; R; wi; ) will be denoted as -S set (extension), so that we refer to -admissible sets, -grounded extensions, -preferred extensions, etc. So, for example, S is -admissible if 9T 2 sub(R; w; ) such that S is admissible in the AF hA; R n T i.
Very recently, a weighted bipolar framework has been proposed in [
While, the work proposed in [
For applications involving a large number of arguments, it can be
problematic to have only two levels of evaluations (arguments are either accepted or
rejected). For instance, such a limitation can be questionable when using
argumentation for debate platforms on the Web for a discussion. In order to x these
problems, a solution consists in using semantics that distinguish arguments not
with the classical accepted/rejected evaluations, but with a larger number of
levels of acceptability. Another way to select a set of acceptable arguments is
to rank arguments from the most to the least acceptable ones. Ranking-based
semantics [
framework F = hA; Ri a ranking SF on A, where SF is a preorder (a re exive and transitive relation) on A. Given two arguments a; b 2 A; a SF b means that a is at least as acceptable as b. 3
Among the various argumentation systems proposed in literature, the instantiation of the proper argumentation system is dependent from the experience of an expert. The need for a unique, general, extended argumentation system is required for two main reasons: (i) theoretical : in order to deal not only with argumentative reasoning without the human expertise for instantiation, but also with other areas of Arti cial Intelligence, such as decision-making, planning, machine learning, dialogue, natural language processing, and multi-agent systems; (ii) practical : in order to facilitate a more direct and natural instantiation from a lot of applications, such as Online Debating Systems (ODS), Social Network analysis and moderation, virtual communities, legal cases, nancial debates.
In these respects, two main generalizations of argumentation systems are detected to cope with all the above requirements: BAF and WAF. Therefore, by taking the best of both worlds we would be able to represent an argumentation system which combines BAFs and WAFs, extending them in a new framework while still resulting compatible with them. Then, the Bipolar Weighted Argumentation Framework (BWAF) is proposed as a further generalization of Dung-style AFs. The idea behind it is to allow not only weighted attack relations between abstract arguments, but also weighted support relations. This is achieved by assigning to each relation a weight which can be positive or negative. More formally: De nition 11 A BWAF is a triplet G = hA; R^ ; wR^ i, where A is a nite set of arguments, R^ A A and wR^ : R^ 7! [ 1; 0[ [ ]0; 1] is a function assigni^ng a weight to each relation. Attack relations are de ned as R^ att = fha; bi 2 R j w ^ (ha; bi) 2 [ 1; 0[ g and support relations as R^ sup = fha; bi 2 R j wR^ (ha; bi) 2 ^
R ]0; 1] g.
A BWAF can be represented as a directed graph whose nodes represent arguments, relations represent attacks and supports, and weights represent the relative strength of relations. We introduce a few new features to deal with bipolar weighted relations. First, we restrict the value of the relation's weight in a speci c bounded interval, in order to de ne its maximum and minimum attack/support degree. Second, we assign negative real values as weights of attack relations: starting from the fact that having negative weight for a negative interaction relies on a more natural intuition, we would be able to better explore the graph with a new useful interaction paradigm dealing with the notion of defense. Last, but not least, we make the same assumptions also for support relations, assigning them positive real values as weights, following the meaning that positive weights would better represent positive interactions between arguments. As for WAFs, there can be several interpretations of these weights for support relations: { Votes on the supports : represents a weight as the number of the votes in endorsement of the support, in the context of collective decision making. { Implicit strength of the supports : equates weights to subjective beliefs, assigning value true to the supported argument when the supporting argument is true. { Explicit strength of the supports : is the simplest interpretation; weights are used to rank the relative strength of supports between arguments, i.e., the higher the weight, the stronger the support. In this interpretation, one might consider subjective or objective criteria for ranking supports.
In order to de ne acceptability in BWAFs, we propose a new interaction paradigm based on weighted relations. The notion of defense, since it lies at the heart of all argumentative evaluation strategies, still remains the central concept when evaluating the justi cation of sets of arguments. So, we rst generalize the key concept of defense between two arguments. Then, we extend this notion to sets of arguments and subsequently de ne acceptability semantics for BWAFs. The argumentation graph evaluation methods we present are all based on the concept of transitivity which stipulates that relations between any two nodes in the graph can be described by paths between the two nodes. In the simplest case we can ask: If there is a path of signed edges between two nodes in the graph, what relation can we induce between the two nodes? We show that the solution to this question is a multiplication rule in which: (i) it is exempli ed the basic Dung's notion in which even-length paths of attacks means a defense (i.e., the attack of an attack is a defense); (ii) BAF's notions of indirect attack and supported attack are generalized into a single de nition.
De nition 12 Let G = hA; R^ ; wR^ i be a BWAF. Given two arguments a; b 2 A and a path ha; x1; x2; : : : ; xn; bi from a towards b, then: { a bw-defends b if the result of weights multiplications wR^ (ha; x1i) wR^ (hx1; x2i) : : : wR^ (hxn; bi) is positive. { a bw-attacks b if the result of weights multiplications wR^ (ha; x1i) wR^ (hx1; x2i) : : : wR^ (hxn; bi) is negative.
a a 0:7 0:6 (a) Defense b b 0:4 0:5 c c
(b) Support Chain a a 0:3 0:1 b b 0:8 0:9 c c (c) Indirect Attack (d) Supported Attack
As we can notice, the notions of \defense" and \support" can now merge in a unique de nition. Indeed, in Figure 1 we show that the notions of defense (Fig. 1 (a)) and support chain (Fig. 1 (b)) convene the notion of bw-defense since the result of weights' multiplication is positive. On the other hand, the notions of indirect and supported attack (Fig. 1 (c) and Fig. 1 (d)) convene the notion of bw-attack since the result of weights' multiplication is negative.
Taking into account sequences of weighted supports and weighted attacks leads to the following de nitions applying to sets of arguments.
De nition 13 Let G = hA; R^ ; wR^ i be a BWAF. A set S A set-bw-attacks an argument b 2 A, i there exists a bw-attack for b from an element of S. A set S A set-bw-defends an argument a 2 A, i for each argument b 2 A, if fbg set-bw-attacks a, then there exists a bw-defense for a from an element of S.
Notice that the notion of collective defense is still required when evaluating the acceptability of subsets of arguments as in the standard extension-based semantics. Hence, admissibility can be still addressed in the classic way. So, BAF's extension-based semantics can be built upon our bipolar weighted version of defense.
De nition 14 Let G = hA; R^ ; wR^ i be a BWAF and S
{ bw-con ict-free set i @a; b 2 S s.t. fag set-bw-attacks b; { bw-admissible set i S is bw-con ict-free and 8a 2 S, a is set-bw-defended by S; { bw-preferred extension i S is a -maximal bw-admissible subset of A; { bw-stable extension i S is bw-con ict-free and 8a 2= S, S set-bw-attacks a.
In WAFs, inconsistency budgets are used to solve the empty extension case. In the same way, we address the case when BWAF's extensions may be empty. This means that we would restrict the bipolar weighted graph under a positive inconsistency budget in order to neglect some support relations. The main question to account for is: why should we not consider supports? Consider to extract arguments from text (from a blog post, a social network discussion, a forum thread, etc.). There may exist some forms of action to stand up for an argument, according to the purpose and audience. Nevertheless, in some cases, the supporting arguments may be advantageous if their strength is high. On the contrary, they may result une ective if their strength is low so that one may decide to avoid these supports. Some cases are: (i) \poor" statistical evidence which can translated into a supporting argument with low strength and it could not be useful; (ii) \low" expert opinion which draws di erent conclusions from the same information showing that opinions may not be as reliable as facts or personal experience; (iii) \bad" visuals which translates important information into a visual to aid in readability but, actually, providing unhelpful visual impact.
Therefore, in order to leave out some poor or une ective consistencies for weighted support relations, we extend the notion of inconsistency budget and de ne the acceptability semantics for BWAF.
De nition 15 Let G = hA; R^ ; wR^ i be a BWAF, R^ = R^ att [ R^ sup and
[ 1; 0[
R subsup(R^ sup; w ^ ; ) = fY j Y
R
R^ att ^ R^ sup ^
X ha;bi2X
X ha;bi2Y wR^ (ha; bi) wR^ (ha; bi) g g
ESbw(hA; R^ ; wR^ i; ; ) = fS
A j 9R 2 fsubatt(R^ att; w ^ ; ) _ subsup(R^ sup; w ^ ; )g ^ S 2 ES (hA; R^ n Ri)g
R R where ES (hA; R^ i) = fS ability semantics S.
The function subatt( ) returns the set of subsets R of R^ att whose total weight does not exceed . While, the function subsup( ) returns the set of subsets R of R^ sup whose total weight does not exceed . Therefore, ESbw(hA; R^ ; wR^ i; ; ) yields a subset of the power set of A whose elements contain only those arguments that are in relation with a weight less than and greater than . A bw-admissible set S 2 ESbw(hA; R^ ; wR^ i; ; ) will be denoted as - -S extension so that we refer to - -admissible sets, - -preferred extensions, - -stable extensions, etc. So, for example, S is - -admissible if 9R 2 subatt(R^ att; w ^ ; ) or R R 2 subsup(R^ sup; w ^ ; ) such that S is admissible in the BAF hA; R^ n Ri.
R
BWAF Ranking Semantics based on Strength Progapation One concern about extension-based semantics is that they do not fully exploit the weight of relations. Indeed, bw-admissibility for subsets of arguments is de ned without considering the e ective strength of relations and how these strengths affect the overall evaluation. The existing semantics declare all justi ed arguments of an argumentation framework as equally accepted. Ranking-based semantics take into account the graduality in this particular case.
In particular, one may wonder how much the weight of attack and support relations a ect the overall strength of a path between two arguments. Moreover, more than one di erent path involving both attack and support relations may exist between two arguments. Additionally, an argument can be involved in many cycles, each of which may contain, in turn, arguments involved in other cycles, and so on. For this reason, we de ne an operator to assess the strength propagation of weighted relations that is able to deal with cycles. De nition 16 Let G = hA; R^ ; wR^ i be a BWAF and a; b 2 A be two arguments such that there exists a simple path ha; : : : ; bi. The strength propagation (in short, sp) from a towards b is de ned as: path weight(ha; : : : ; bi) inf luence(c) where sp(a; b) =
X ha;:::;bi
Y c2ha;:::;bi n Y wR^ (hvi 1; vii) i=2 path weight(hv1; : : : ; vni) = inf luence(v) = ( Qhv1;:::;vki path weight(hv1; : : : ; vki) if v1 = vk = v
1 otherwise
Function path weight( ) computes the strength of a simple path by multiplying every weight relation in it, while function inf luence( ) computes the in uence of an node within the simple path on the basis of cycles to which it belongs. If it belongs to a cycle, it is computed as product of the path cycle starting and ending to it, as 1 otherwise. In the case of sub-cycles eventually involved in a path cycle, we only consider its \maximal " circuit. In this way, cycles and possibly involved sub-cycles are traversed exactly once. Since the evaluation of a cycle in an argument graph is always problematic, its presence within a path may in uence signi cantly the evaluation of its overall strength. So, the in uence of a node in a path, which also belongs to a cycle, may reduce drastically the path strength propagation. Hence, function sp(a; b) detects all the simple paths starting from a towards b along possibly cycles in them and returns a positive value if there exists a (collective) bw-defense. Otherwise, it returns a negative value if there exists a (collective) bw-attack between them. Such a function gives a measure of overall strength of the relations between two arguments as part of a whole discussion.
We show its behavior in the following example.
a
Example 1. Consider the BWAF depicted in Figure 2. Suppose that we want to compute the strength propagation for the path from a to d. First, we have to compute the existing path weigths from a to d. There exist two path weights: path weight(ha; b; c; di) = 0:7 0:9 ( 0:4) = 0:252 path weight(ha; e; di) = ( 0:7) 0:3 = 0:21 Since e belongs to a cycle, we need to compute its in uence: inf luence(e) = path weight(he; f; h; f; g; ei) = = ( 0:5) ( 0:5) ( 0:7) ( 0:3) ( 0:1) = 0:00525 Hence, we can compute the strength propagation from a to d: sp(a; d) = path weight(ha; b; c; di) 1 + + path weight(ha; e; di) inf luence(e) = 0:2508975:
With a big number of arguments, and a lot of users participating, it can be problematic to have a boolean evaluation (accepted/rejected). Then, a more informative evaluation is required. In the following, we propose a new semantics, i.e. sp-ranking semantics, which ranks arguments by comparing them with a numerical acceptability degree assigned to each argument. Such a ranking relies on bw-attacks and bw-defenses of each argument in order to evaluate its acceptability rank. We recall that an attack amounts to undermining one of the components of an argument, and has thus a negative impact on its target. Vice versa, a support amounts to stand up for an argument, and has thus a positive impact on its target. An evaluation of the overall acceptability of an argument becomes mandatory, namely for judging how much its conclusion is reliable. Therefore, sp-ranking semantics exploits the strength propagation of couples of arguments linked by path branches. A path branch from a to b is a sequence of nodes ha0; : : : ; aki, with a0 = a; ak = b and 8i < k; hai; ai+1i 2 R^ and a is not attacked nor supported. So, we rst de ne a ranking opeator to assign an acceptability degree to each argument in the BWAF which deals with the strength propagation of path branches ending to them. Then, we rank arguments according to their acceptability degree.
De nition 17 Let G = hA; R^ ; wR^ i be a BWAF, a 2 A an argument, sp( ; a) the strength propagation of path branch ending to a, SP = fsp(x1; a); : : : ; sp(xm; a)g the set of all the strength propagations ending to a and P = fp1; : : : ; png the set of all directed paths towards a in G, with pi = hx; : : : ; ai 2 P; 8i n. The sp-rank function spr : A 7! [0; 2] is de ned as: spr(a) = ( 1 if 8x 2 A : hx; ai 2= R^ 1 Pm n sp(xi;a)2SP 1 + sp(xi; a) otherwise
Our approach is based two principles: the impact of unrelated argument, which play a key role in the (extension-based) acceptability of an argument, and the strength propagation of its attackers and of its supporters. It is worth to clarify why we di erently consider the set P of directed paths and the set SP of strength propagation paths. The strength propagation function sp( ) computes the strength of all the existing multipaths between two nodes in the argumentation graph at once. If there exists at most one path for each couple of nodes, then n = m, otherwise we have that m < n. In this respect, we ensure that the ranking of arguments is always in the interval [0; 2]. Generally, arguments that not receive any attack or support play a key role in the (classical) acceptability. So, we set for them a ranking of 1. In this way, bw-defended arguments will achieve a ranking > 1, otherwise bw-attacked ones will achieve a ranking < 1. In this sense, the ranking of 1 will tip the scale, meaning that we would be able to consider not only an acceptability ranking for arguments, but also a mapping to the classical accepted/rejected evaluation. Finally, we can rank arguments according to their spr( ) function.
De nition 18 The sp-ranking semantics spr associates to any BWAF G = hA; R^ ; wR^ i a ranking sGpr on A such that 8a; b 2 A; a sGpr b i spr(a) spr(b). 4
The task of building an argument system from a debate platform has been
already addressed in [
The set of arguments A is made up of tree nodes representing the targeted discussion. The set of relations R^ is built starting from each comment, which can be in favor or against the argument to which one is replying. In particular, the comments in the rst level of the tree, related to the major claim, can be analyzed to extract the sentiment polarity with respect to that argument. If the polarity is positive, such comments are arguments supporting the major claim, otherwise, if the polarity is negative, they are attacking arguments. Speci cally, we assume that the arguments in the rst level of the tree address the major claim without going \o -topic". Regarding the answer comments to a comment, there can be an attack or a support with respect to it, depending on the extent they address the same argument. In fact, if a comment addresses the same topic of the comment it answers then there is a support, otherwise there is an attack. Formally, the task of weighing the relation between two arguments a; b 2 A can be quanti ed as follows: wR^ (ha; bi) = similarity(ha; bi) sentiment(b) (1) where similarity : A A 7! [0; 1] is a function evaluating the similarity between two arguments, and sentiment : A 7! [ 1; 1] is a function that evaluates the polarity of sentiment associated with an argument. Given a; b 2 A, if w ^ (ha; bi) < 0 then there exists an attack from a towards b, otherwise if
R wR^ (ha; bi) > 0 then there exists a support from a towards b.
In order to determine if two arguments are treating the same theme, we
adopt a measure of similarity based on word embeddings. These techniques are
particularly e ective in a large variety of applications related to the similarity
of the contents. Generally speaking, in the context of semantic spaces models,
a word embedding is a dense vector representation of a word. These vectors,
learned from a corpus of text, capture the similarity between words by using them
in the context in which they appear more frequently. In this way, it is satis ed the
distributional hypothesis for two words in which the more semantically similar,
the higher they tend to recur in similar contexts [
Concerning the task to determine the sentiment polarity of a sentence, we
exploited the model presented in [
The procedure of (weighted) argument graph construction just described may embed some noise. Nevertheless, the procedure is simple and computationally fast, so that the graph instantiation is still quite reliable. 4.1
Application to a Reddit Thread In order to identify arguments and their interaction in real cases, we considered the content on the platform Reddit 1, a big community that allows users to share links, opinions, real-time content and information and to discuss and comment on what is published on the platform. In this case, we consider a Reddit discussion of an episode of popular TV series2 divided into several arguments with attacks and supports.
a62
The produced BWAF is made up of 70 arguments and 69 relations, of which 52 attacks and 17 supports. The corresponding graph is reported in Figure 3. By analyzing the thread, it is possible make some considerations useful to judge the quality of the generated BWAF and to assess it with both extension- and ranking-based semantics. Node a0 in Figure 3 acts as the major claim for the whole discussion. While, other nodes in the BWAF represent replies to the major claim and subsequently comments on them. We note that the Equation (1) is 1 www.reddit.com 2 http://bit.ly/2fQxq4I able to capture the meaning of nodes relating to other ones and that the weight of relation helps us in identifying some useful semantics. The bw-stable semantics has a unique extension, which is the following: fa4, a5, a8, a13, a14, a15, a16, a17, a19, a21, a22, a23, a24, a26, a29, a33, a35, a36, a38, a40, a41, a42, a43, a44, a45, a47, a48, a49, a50, a51, a53, a55, a57, a59, a62, a63, a64, a65, a66, a69, a71, a72, a73, a76, a78, a80, a84, a85g.
While, in Table 1 is reported the sp-ranking semantics. It becomes apparent to note that all the arguments belonging to the bw-stable extension hold an acceptability degree 1. A slight di erence in the two semantics is that spranking one yields as defended also arguments a9, a10, and a27. This is because the computation of the (collective) strength propagation of all path branches ending to them takes advantages of weight of relations and, in particular, of weighted support relations. In fact, one limit of extension-based semantics is that they do not fully pro t by support relations. 5
This work proposed a generalization of Dung's AF, the BWAF, which is able to express weighted attack and support relations. We presented a new characterization of the notion of defense, along with an extended version of inconsistency budget, two classes of extension-based semantics and a new ranking-based semantics. Since weighted relations can a ect indirect attacks/supports, we handle this with the de nition of strength propagation, in order to quantify the positive or negative strength of indirect relations between two arguments. Another contribution regards the strategy to construct BWAFs starting from real data. The detection of weighted relations is based on the similarity measure of textual contents and on their associated sentiment polarity. To prove its e ectiveness, we built a BWAF from a Reddit discussion and we examined the resulting model by comparing extension- and ranking-based semantics. It is shown that our framework is able to clearly represent the whole discussion and the sp-ranking semantics better exploits the e ectiveness of (weighted) support relations.
Our work opens some issues for further research. In the phase of evaluation of accepted arguments, one may nd that not all the arguments of a discussion are essential when drawing conclusions. Especially when the cardinality of the set of arguments is high, identifying the most relevant arguments is a tricky task. In huge argumentation graphs, the analysis of its synthesis may favor better interpretability and may allow you to extract semantics that include the strongest arguments.
Acknowledgments This work was partially funded by the Italian PON 2007-2013 project PON02 00563 3489339 `Puglia@Service'.