=Paper=
{{Paper
|id=Vol-2012/paper_12
|storemode=property
|title=Constructing and Evaluating Bipolar Weighted Argumentation Frameworks for Online Debating Systems
|pdfUrl=https://ceur-ws.org/Vol-2012/AI3-2017_paper_12.pdf
|volume=Vol-2012
|authors=Andrea Pazienza,Stefano Ferilli,Floriana Esposito
|dblpUrl=https://dblp.org/rec/conf/aiia/PazienzaFE17a
}}
==Constructing and Evaluating Bipolar Weighted Argumentation Frameworks for Online Debating Systems ==
https://ceur-ws.org/Vol-2012/AI3-2017_paper_12.pdf
Constructing and Evaluating
Bipolar Weighted Argumentation Frameworks
for Online Debating Systems
Andrea Pazienza, Stefano Ferilli, and Floriana Esposito
Dipartimento di Informatica – Università di Bari
name.surname @uniba.it
Abstract. Discussions on social Web platforms carry a lot of informa-
tion which is more and more difficult to analyze. Given a virtual com-
munity of users that discuss a particular topic of interest, an important
task is to extract a model of the whole debate in order to automati-
cally evaluate what are the most reliable claims. This paper proposes
to approach this task using abstract argumentation, and define 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., at-
tacks) relations between arguments. It also provides a way to assess an
acceptability degree for each argument by means of the strength propa-
gation 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.
1 Introduction
People in all societies argue, discuss, and debate not only to convince others of
their own opinions, but because they want to explore the differences between
their own understanding and the conceptualizations of others, and learn from
them. Being one of the primary intellectual activities of the human mind, de-
bating therefore naturally involves a wide range of conceptual capabilities and
activities, ones that have only in part been studied from a computational per-
spective. One of the most influential computational models of argument was pre-
sented by Dung’s Argumentation Frameworks [5] (in short, AF), which is roughly
a directed graph where the vertices are the abstract arguments and the directed
edges correspond to attacks between them. As there are no restrictions on the
attack relation, cycles, self–attackers, and so on, are all allowed. Arguments do
not have any particular structure and the precise conditions for their acceptance
are defined by the semantics. Semantics produce acceptable subsets of the argu-
ments, called extensions, that correspond to various positions one may take based
on the available arguments. However, within the argumentation process, the con-
struction of proper AFs can cause much more concern than expected. Therefore,
basic AFs may not necessarily be the best target systems for the instantiation.
In order to address this problem, a research direction is to extend AFs by equip-
ping relations with more expressive concepts such as support relations, giving
rise to Bipolar Argumentation Frameworks (BAFs) [3], and weighted attacks,
giving rise to Weighted Argumentation Frameworks (WAFs) [6]. Nevertheless, a
single framework that takes both advantages of BAFs and WAFs is still lacking.
Different 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 quantified.
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 ar-
guments, 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/sup-
port 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 briefly 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 Background and Related Work
Let us start by providing the basics of Abstract Argumentation.
Definition 1 An AF is a pair F = hA, Ri, where A is a finite set of arguments
and R ⊆ A × A. Given a, b ∈ A, the relation aRb means that a attacks b.
An argumentation semantics is the formal definition of a method ruling the argu-
ment evaluation process. The most basic concepts shared by all argumentation
semantics in the literature are conflict-freeness and defense.
Definition 2 Let F = hA, Ri be an AF, and S ⊆ A:
– S is conflict-free if @a, b ∈ S s.t. aRb;
– a ∈ A is defended by S if ∀b ∈ A : bRa ⇒ ∃c ∈ S s.t. cRb;
– fF : 2A 7→ 2A s.t. fF (S) = {a | a is defended by S} is called the characteristic
function of F ;
– S is admissible if S is conflict-free and S is defended by itself, i.e. ∀a ∈
S, ∀b ∈ A : bRa ⇒ ∃c ∈ S s.t. cRb.
Then, standard acceptability semantics, introduced by Dung [5], characterize
admissible sets of arguments:
Definition 3 Let F = hA, Ri be an AF and S ⊆ A be an admissible set. Then,
S is a:
– complete extension iff S = fF (S);
– grounded extension iff S is the ⊆-minimal complete extension;
– preferred extension iff S is a ⊆-maximal complete extension;
– stable extension iff ∀a ∈ A, a ∈
/ S, ∃b ∈ S s.t. bRa.
A Bipolar AF (BAF ) [3] is an extension of Dung’s AF in which two kinds of
interactions between arguments are possible: the attack relation and the support
relation. These two relations are independent (i.e., the support relation is not
defined using the attack relation) and lead to a bipolar representation of the
interaction between arguments.
Definition 4 A BAF is a triplet B = hA, Ratt , Rsup i, 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 ∈ A, aRatt b
(resp., aRsup b) 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 iff there is a sequence of
supports followed by one attack, while, there is an indirect attack iff there is
an attack followed by a sequence of supports. Taking into account sequences
of supports and attacks leads to the following definitions applying to sets of
arguments.
Definition 5 Let B = hA, Ratt , Rsup i be a BAF. A set S ⊆ A set-attacks
b ∈ A, iff there exists a supported attack or an indirect attack for b from an
element of S. A set S ⊆ A set-supports b ∈ A, iff there exists a sequence of
supports for b from an element of S. A set S ⊆ A defends a ∈ A, iff for each
argument b ∈ A, if {b} set-attacks a, then b is set-attacked by S.
In the following, we define the semantics for acceptability in BAFs [3].
Definition 6 Let B = hA, Ratt , Rsup i be a BAF and S ⊆ A. Then, S is:
– conflict-free iff @a, b ∈ S s.t. {a} set-attacks b;
– safe iff @b ∈ A s.t. S set-attacks b and either S set-supports b or a ∈ S;
– d-admissible iff S is conflict-free and ∀a ∈ S, a is defended by S;
– s-admissible iff S is safe and ∀a ∈ S, a is defended by S;
– c-admissible iff S is d-admissible and closed for Rsup .
– a d-preferred (resp. s-preferred, c-preferred) extension is a ⊆-maximal d-
admissible (resp. s-admissible, c-admissible) subset of A;
– a stable extension iff S is conflict-free and ∀a ∈ / S, S set-attacks a.
A Weighted AF (WAF ) [6] is another extension of Dung’s AF in which
attacks between arguments are associated with a weight, indicating the relative
strength of the attack.
Definition 7 A WAF is a triplet W = hA, R, wi, where hA, Ri is the standard
AF and w : R 7→ R+ is a function assigning real valued weights to attacks.
Note that allowing 0-weight attacks is counter-intuitive since it can be inter-
preted 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
β ∈ R+ ∗ . Hence, given an inconsistency budget β, the meaning is that attacks
up to a total weight of β are neglected. Dung’s argument systems assume an
inconsistency budget of 0, while, by relaxing this constraint, WAFs can achieve
more solutions.
Definition 8 Let W = hA, R, wi be a WAF. Given an inconsistency budget
β ∈ R+∗ , function sub returns the set of subsets T of R whose total weight does
not exceed β, i.e.,
X
sub(R, w, β) = {T ⊆ R | w(hα1 , α2 i) ≤ β}.
hα1 ,α2 i∈T
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 defined in the standard way,
and standard semantics are considered leading to various notions of β-extensions
which echo Dung’s ones (i.e., grounded, preferred, stable extensions).
Definition 9 Given a WAF W = hA, R, wi, let ES be an extension-based se-
mantics. For β ∈ R+ , the subset ESw (hA, R, wi, β) of 2A is given as
ESw (hA, R, wi, β) = {S ⊆ A | ∃T ∈ sub(R, w, β) ∧ S ∈ ES (hA, R \ T i}
where ES (hA, Ri) = {S ⊆ A | S(S)} returns the set of subsets of A for accept-
ability semantics S.
A set S ∈ ESw (hA, R, wi, β) will be denoted as β-S set (extension), so that
we refer to β-admissible sets, β-grounded extensions, β-preferred extensions, etc.
So, for example, S is β-admissible if ∃T ∈ sub(R, w, β) such that S is admissible
in the AF hA, R \ T i.
Very recently, a weighted bipolar framework has been proposed in [2], in which
is tackled the issue of arguments evaluation in a strong different perspective. In
this setting, weights are associated to arguments which have a basic (intrinsic)
strength and its evaluation method transforms it accordingly to attacks and
supports received into an overall strength as acceptability degree. A big drawback
in their semantics is the that it deals only with acyclic graphs.
While, the work proposed in [11] clearly distinguish between the intrinsic
strength of an argument and the strength of relations. In fact, a preliminar
version of the BWAF is already presented and a thorough discussion on how
to handle both intrinsic strength of an argument (coming from the realiabilty
of its source) and the strength of relations with other arguments is accurately
described. As well, [10] associates arguments with weights that express their
source’s authority degree and defines a strategy to combine them in order to
determine which arguments withstand in a dispute concerning a given domain.
For applications involving a large number of arguments, it can be problem-
atic to have only two levels of evaluations (arguments are either accepted or
rejected). For instance, such a limitation can be questionable when using argu-
mentation for debate platforms on the Web for a discussion. In order to fix 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 [1] aim at determining such a ranking among arguments.
Definition 10 A Ranking-based semantics S associates to any argumentation
framework F = hA, Ri a ranking SF on A, where SF is a preorder (a reflexive
and transitive relation) on A. Given two arguments a, b ∈ A, a SF b means that
a is at least as acceptable as b.
3 Bipolar Weighted Argumentation Framework
Among the various argumentation systems proposed in literature, the instan-
tiation 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 Artificial Intelligence, such as decision-making, planning, ma-
chine 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, financial 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 argumenta-
tion system which combines BAFs and WAFs, extending them in a new frame-
work 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 rela-
tions 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:
Definition 11 A BWAF is a triplet G = hA, R̂, wR̂ i, where A is a finite set
of arguments, R̂ ⊆ A × A and wR̂ : R̂ 7→ [−1, 0[ ∪ ]0, 1] is a function assigning
a weight to each relation. Attack relations are defined as R̂att = {ha, bi ∈ R̂ |
wR̂ (ha, bi) ∈ [−1, 0[ } and support relations as R̂sup = {ha, bi ∈ R̂ | wR̂ (ha, bi) ∈
]0, 1] }.
A BWAF can be represented as a directed graph whose nodes represent ar-
guments, 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 specific bounded interval, in order to define its maximum and minimum
attack/support degree. Second, we assign negative real values as weights of at-
tack 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 argu-
ments. 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, as-
signing 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 define 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 justification of sets of arguments. So, we first generalize the
key concept of defense between two arguments. Then, we extend this notion to
sets of arguments and subsequently define 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 exemplified
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 definition.
Definition 12 Let G = hA, R̂, wR̂ i be a BWAF. Given two arguments a, b ∈ A
and a path ha, x1 , x2 , . . . , xn , bi from a towards b, then:
– a bw-defends b if the result of weights multiplications wR̂ (ha, x1 i)·wR̂ (hx1 , x2 i)·
. . . · wR̂ (hxn , bi) is positive.
– a bw-attacks b if the result of weights multiplications wR̂ (ha, x1 i)·wR̂ (hx1 , x2 i)·
. . . · wR̂ (hxn , bi) is negative.
−0.7 −0.4 0.3 0.8
a b c a b c
(a) Defense (b) Support Chain
−0.6 0.5 0.1 −0.9
a b c a b c
(c) Indirect Attack (d) Supported Attack
Fig. 1. Multiplication rules: in (a): a bw-defends c with a total weight of −0.7·−0.4 =
0.28; in (b): a bw-defends c with a total weight of 0.3·0.8 = 0.24; in (c): a bw-attacks
c with a total weight of −0.6 · 0.5 = −0.3; in (d): a bw-attacks c with a total weight
of 0.1 · −0.9 = −0.009.
As we can notice, the notions of “defense” and “support” can now merge in a
unique definition. 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 definitions applying to sets of arguments.
Definition 13 Let G = hA, R̂, wR̂ i be a BWAF. A set S ⊆ A set-bw-attacks
an argument b ∈ A, iff there exists a bw-attack for b from an element of S. A
set S ⊆ A set-bw-defends an argument a ∈ A, iff for each argument b ∈ A, if
{b} 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.
Definition 14 Let G = hA, R̂, wR̂ i be a BWAF and S ⊆ A. Then, S is a:
– bw-conflict-free set iff @a, b ∈ S s.t. {a} set-bw-attacks b;
– bw-admissible set iff S is bw-conflict-free and ∀a ∈ S, a is set-bw-defended
by S;
– bw-preferred extension iff S is a ⊆-maximal bw-admissible subset of A;
– bw-stable extension iff S is bw-conflict-free and ∀a ∈ / 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 posi-
tive 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 uneffective 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 different 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 uneffective consistencies for
weighted support relations, we extend the notion of inconsistency budget and
define the acceptability semantics for BWAF.
Definition 15 Let G = hA, R̂, wR̂ i be a BWAF, R̂ = R̂att ∪ R̂sup and α ∈
[−1, 0[, β ∈ ]0, 1] two inconsistency budgets. We define:
X
subatt (R̂att , wR̂ , α) = {X | X ⊆ R̂att ∧ wR̂ (ha, bi) ≥ α}
ha,bi∈X
X
subsup (R̂sup , wR̂ , β) = {Y | Y ⊆ R̂sup ∧ wR̂ (ha, bi) ≤ β}
ha,bi∈Y
ESbw (hA, R̂, wR̂ i, α, β) =
{S ⊆ A | ∃R ∈ {subatt (R̂att , wR̂ , α) ∨ subsup (R̂sup , wR̂ , β)} ∧ S ∈ ES (hA, R̂ \ Ri)}
where ES (hA, R̂i) = {S ⊆ A | S(S)} returns the set of subsets of A for accept-
ability semantics S.
The function subatt (·) returns the set of subsets R of R̂att whose total weight
does not exceed α. While, the function subsup (·) returns the set of subsets R of
R̂sup whose total weight does not exceed β. Therefore, ESbw (hA, R̂, wR̂ i, α, β)
yields a subset of the power set of A whose elements contain only those ar-
guments that are in relation with a weight less than α and greater than β. A
bw-admissible set S ∈ ESbw (hA, R̂, wR̂ i, α, β) will be denoted as α-β-S extension
so that we refer to α-β-admissible sets, α-β-preferred extensions, α-β-stable ex-
tensions, etc. So, for example, S is α-β-admissible if ∃R ∈ subatt (R̂att , wR̂ , α) or
R ∈ subsup (R̂sup , wR̂ , β) such that S is admissible in the BAF hA, R̂ \ Ri.
3.1 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 defined
without considering the effective strength of relations and how these strengths af-
fect the overall evaluation. The existing semantics declare all justified 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 affect the overall strength of a path between two arguments. Moreover,
more than one different 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 define an operator to assess the strength
propagation of weighted relations that is able to deal with cycles.
Definition 16 Let G = hA, R̂, wR̂ i be a BWAF and a, b ∈ A be two arguments
such that there exists a simple path ha, . . . , bi. The strength propagation (in
short, sp) from a towards b is defined as:
X Y
sp(a, b) = path weight(ha, . . . , bi) · inf luence(c)
ha,...,bi c∈ha,...,bi
where
n
Y
path weight(hv1 , . . . , vn i) = wR̂ (hvi−1 , vi i)
i=2
(Q
hv1 ,...,vk i path weight(hv1 , . . . , vk i) if v1 = vk = v
inf luence(v) =
1 otherwise
Function path weight(·) computes the strength of a simple path by mul-
tiplying every weight relation in it, while function inf luence(·) computes the
influence of an node within the simple path on the basis of cycles to which it be-
longs. 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 influence significantly the evaluation of its overall strength. So, the influence
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.
0.9 c -0.4
b
0.7 d
0.3
a
-0.7
e -0.1
-0.5 g
-0.3
f -0.5
-0.7 h
Fig. 2. Representation of BWAF for Example 1.
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 influence:
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 , . . . , ak i, with a0 = a, ak = b and ∀i < k, hai , ai+1 i ∈ R̂ and a
is not attacked nor supported. So, we first define 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.
Definition 17 Let G = hA, R̂, wR̂ i be a BWAF, a ∈ A an argument, sp(·, a) the
strength propagation of path branch ending to a, SP = {sp(x1 , a), . . . , sp(xm , a)}
the set of all the strength propagations ending to a and P = {p1 , . . . , pn } the set
of all directed paths towards a in G, with pi = hx, . . . , ai ∈ P, ∀i ≤ n. The
sp-rank function spr : A 7→ [0, 2] is defined as:
(
1 if ∀x ∈ A : hx, ai ∈
/ R̂
spr(a) = 1 Pm
n sp(xi ,a)∈SP 1 + sp(x i , 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 differently 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 argumen-
tation 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.
Definition 18 The sp-ranking semantics spr associates to any BWAF G =
hA, R̂, wR̂ i a ranking spr spr
G on A such that ∀a, b ∈ A, a G b iff spr(a) ≥ spr(b).
4 Construction of a BWAF from an ODS
The task of building an argument system from a debate platform has been
already addressed in [4, 7, 8]. Here, we consider classical thread discussions,
that involve a tree (i.e., hierarchical) structure. A typical online debate thread
would consist of a discussion topic, i.e. a content shared by a user, followed by
comments from other users. In response to a comment there may be an answer,
so comments are organized in a tree discussion where the root node, i.e. the
major claim, is the shared content and the other nodes represent comments.
Each of these comments has as children the comments in response to it. Hence,
considering the similarity between the comments, the sentiment associated with
them and their hierarchical structure, it is possible to extract a BWAF that
models the discussion by identifying weighted attacks and supports depending
on their strength.
The set of arguments A is made up of tree nodes representing the targeted
discussion. The set of relations R̂ 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 first 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. Specifically,
we assume that the arguments in the first level of the tree address the major
claim without going “off-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 ∈ A can
be quantified as follows:
wR̂ (ha, bi) = similarity(ha, bi) · sentiment(b) (1)
where similarity : A × A 7→ [0, 1] is a function evaluating the similarity be-
tween two arguments, and sentiment : A 7→ [−1, 1] is a function that evalu-
ates the polarity of sentiment associated with an argument. Given a, b ∈ A,
if wR̂ (ha, bi) < 0 then there exists an attack from a towards b, otherwise if
wR̂ (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 effective 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 satisfied the
distributional hypothesis for two words in which the more semantically similar,
the higher they tend to recur in similar contexts [9]. For this task, we exploited
the GloVe model [12], which is a well-known model for learning word embeddings.
In particular, the similarity between two sentences is computed as the cosine
similarity between the average of the word embedding associated with the token
present in sentences.
Concerning the task to determine the sentiment polarity of a sentence, we
exploited the model presented in [13], which is a well-known model in the field of
sentiment analysis. This model learns complex relationships between the terms
of a sentence to determine the sentiment polarity. In particular, polarity classes
provided by the model are “very positive”, “positive”, “neutral”, “negative” and
“very negative”, for which it has been established the correspondence, respec-
tively, with the numerical values −1, −0.5, 0, 0.5 and 1.
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.
a20 -0.19 a19
-0.45 -0.48
a18
a21
a76
-0.48
a23
-0.42 a17
0.32
a62 a14
a63
a24 a16
a75 -0.49
-0.16 a22
-0.16 a80 -0.48
a8 -0.44
a10
a61 0.16
-0.1 0.16 -0.47
-0.1
a78 a11
0.32 a45 0.16
-0.08 -0.48
a60 a53 a12
a74 a9
-0.47
-0.17 a85
-0.46
-0.12
a59 a84 -0.17
-0.12 a15 a44 a13
a58 -0.1 -0.5 0.17
-0.25 a72
-0.5 0.43
-0.13 -0.25
a66 a43
-0.5 a33
a0 -0.4 a71
-0.5 0.24
0.25
a69 -0.44 a68 a42
a57 -0.5 0.5 a27 -0.48
-0.5 -0.5 -0.13
-0.33 -0.5 0.5 a26 -0.16
a65 0.06 0.22
0.08
a31 -0.45 a32
a56 -0.14
-0.48 a28
a54 a73 a64
a4 -0.15 -0.45 -0.48
a47 -0.16
a48 a29
-0.46 a50 -0.16 -0.16
a36
a37 a34
0.45
a55 0.05 a41 -0.34
0.08 a39
-0.24
a5
-0.32 a38
a51 a49
a40 a35
Fig. 3. BWAF representation for the considered Reddit Thread.
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:
{a4, 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, a85}.
Table 1. sp-ranking semantics
arg spr arg spr arg spr arg spr arg spr arg spr
a4 1.45 a80 1.0 a59 1.0 a38 1.0 a13 1.0 a32 0.8176
a43 1.43 a78 1.0 a57 1.0 a36 1.0 a8 1.0 a34 0.76
a22 1.32 a76 1.0 a55 1.0 a35 1.0 a5 1.0 a11 0.7077
a10 1.108 a73 1.0 a53 1.0 a33 1.0 a58 0.9641 a39 0.68
a19 1.0855 a72 1.0 a51 1.0 a29 1.0 a28 0.9595 a56 0.67
a50 1.08 a71 1.0 a49 1.0 a26 1.0 a18 0.9589 a37 0.66
a48 1.05 a69 1.0 a47 1.0 a24 1.0 a74 0.954 a75 0.58
a9 1.0425 a66 1.0 a45 1.0 a23 1.0 a60 0.9488 a68 0.56
a17 1.0197 a65 1.0 a44 1.0 a21 1.0 a0 0.9225 a20 0.55
a27 1.0035 a64 1.0 a42 1.0 a16 1.0 a31 0.9047 a12 0.54
a85 1.0 a63 1.0 a41 1.0 a15 1.0 a54 0.8492
a84 1.0 a62 1.0 a40 1.0 a14 1.0 a61 0.84
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 difference in the two semantics is that sp-
ranking 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 profit by support relations.
5 Conclusions and Future Work
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 characteri-
zation of the notion of defense, along with an extended version of inconsistency
budget, two classes of extension-based semantics and a new ranking-based se-
mantics. Since weighted relations can affect indirect attacks/supports, we handle
this with the definition of strength propagation, in order to quantify the posi-
tive 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 effectiveness, 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 frame-
work is able to clearly represent the whole discussion and the sp-ranking seman-
tics better exploits the effectiveness of (weighted) support relations.
Our work opens some issues for further research. In the phase of evaluation
of accepted arguments, one may find 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 in-
terpretability 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’.
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