=Paper=
{{Paper
|id=Vol-2777/paper43
|storemode=property
|title=Towards a General Model for Abstract Argumentation Frameworks
|pdfUrl=https://ceur-ws.org/Vol-2777/paper43.pdf
|volume=Vol-2777
|authors=Stefano Ferilli
|dblpUrl=https://dblp.org/rec/conf/aiia/Ferilli20
}}
==Towards a General Model for Abstract Argumentation Frameworks==
https://ceur-ws.org/Vol-2777/paper43.pdf
Towards a General Model for Abstract
Argumentation Frameworks?
Stefano Ferilli
Dipartimento di Informatica – Università di Bari Aldo Moro, Bari, Italy
stefano.ferilli@uniba.it
Abstract. In its original definition, the Abstract Argumentation frame-
work considers atomic claims and a binary attack relationship among
them, based on which different semantics would select subsets of claims
consistently supporting the same position in a dispute or debate. While
attack is obviously the core relationship in this setting, in more complex
(and in many real-world) situations additional information may help,
or might even be crucial, in determining such positions, and especially
those that are going to win the debate. Examples are bipolarity (consid-
ering also the support relationship between pairs of claims) and weights
(assigning different importance to different elements of the framework).
These additional features have often been considered separately, yielding
incompatible or anyhow disjoint models for argumentation frameworks.
In this paper we propose a model that unifies all these perspectives,
and further extends them by allowing to express contextual information
associated to the arguments, in addition to their relationships.
Keywords: Abstract Argumentation · Argumentation Frameworks
1 Introduction
Argumentation is the inferential strategy for practical and uncertain reasoning
aimed at coping with partial and inconsistent knowledge, in order to justify one
of several contrasting positions in a discussion [14]. A typical case is a debate
in which each participant tries to support one position with suitable claims (the
arguments), also attacking the arguments put forward by others to support com-
peting positions, and defending his position from the attacks of the others. Since
different forms of disputes (or anyway situations with contrasting evidence) are
ubiquitous in real life, the availability of automated techniques for carrying out
argumentation would be extremely useful. Hence, the birth of a specific branch
of Artificial Intelligence aimed at developing models, approaches, techniques and
systems for dealing with different aspects of argumentative reasoning.
Abstract argumentation, in particular, focuses on the resolution of the dis-
pute based only on ‘external’ information about the arguments (notably, the
?
Copyright c 2020 for this paper by its authors. Use permitted under Creative Com-
mons License Attribution 4.0 International (CC BY 4.0).
inter-relationships among them), neglecting their internal structure or interpre-
tation. Traditional Abstract Argumentation Frameworks (AFs for short) can ex-
press only attacks among arguments. While already useful to tackle many cases
(because the attack relationship is indeed the very core and driving feature in
a debate), this is obviously a significant limitation in expressiveness. So, several
lines of research tried to overcome such a limitation by introducing additional
features to be considered in the argumentation frameworks. Most famous are the
possibility of expressing supports between arguments (in addition to attacks), or
the ‘strength’ of attacks (in the form of a number). These extensions were mainly
developed independently of each other, so that they cannot be straightforwardly
combined into a more powerful framework encompassing all of them.
This paper proposes a general framework that brings to a cooperation of the
different features of the single frameworks, yielding a much more powerful model
to carry out abstract argumentation. It can simulate any of those frameworks,
and also provides for the additional possibility of assigning a degree of ‘strength’
also to the arguments, not just to their relationships. We call it Generalized Ar-
gumentation Framework, or GAF. With respect to our previous preliminary work
in this direction, here we fix some notational issues, reorganize the model formal-
ization and abstract away from details of specific computational approaches that
can be applied on it. Indeed, we stress the fact that our aim is not proposing any
evaluation strategy or computational procedure, but a model that can be spe-
cialized and tailored to different contexts and domains, and on which theoretical
investigation can be carried out for defining semantics and evaluation strategies.
We believe our proposal can be taken as a reference, both for porting solutions
developed for previous partial extensions, and for developing new solutions that
fully exploit its extended expressive power. Also, we show that our model can
be easily expressed using matrix representations, which might bring significant
improvements in efficiency in computing the argumentation outcomes thanks to
the use of matrix operations.
The paper is organized as follows. After recalling basic concepts of abstract
AFs and discussing related works in the next section, in Section 3 we will de-
fine the new generalized model and show how it maps onto existing AFs. Then,
in Section 4 we propose a specialization of it that allows to consider user con-
fidence in the arguments and trust in the other users. Section 5 discusses the
advantages of expressing our model using matrix-based representations, while
Section 6 concludes the paper.
2 Basics & Related Work
The original (and now classical) Abstract Argumentation setting was proposed
by Dung [7]. It can express only the attack relationship between pairs of argu-
ments, as the core feature indicating inconsistency in the available information:
Definition 1 An argumentation framework ( AF for short) is a pair F = hA, Ri,
where A is a finite set of arguments and R ⊆ A × A is an attack relationship
(meaning that, given α, β ∈ A, if αRβ then α attacks β).
In this setting, no direct agreement between arguments can be expressed. Agree-
ment can only indirectly be derived based on the attack relationship, yielding
the notion of defense:
Definition 2 Let F = hA, Ri be an AF, and S ⊆ A:
– α ∈ A is defended by S if ∀β ∈ A : βRα ⇒ ∃γ ∈ S s.t. γRβ;
– fF : 2A 7→ 2A s.t. fF (S) = {α | α is defended by S} is the characteristic
function of F .
So, an argument may ‘defend’ other arguments by attacking their attacker (or,
in other words, attacking an attacker amounts to a defense).
An argumentation semantics is the formal definition of a method ruling the
argument evaluation process. In particular, extension-based semantics determine
which subset(s) of arguments in an AF, called extensions, can stand together
and possibly be considered as the ‘winners’ of the dispute expressed by the
AF. On the other hand, ranking-based semantics [1] individually evaluate single
arguments rather than sets of arguments, and, given an AF, determine a ranking
of the available arguments in the form of a pre-order (reflexive and transitive
relation). We will not delve further into semantics in the following, since the aim
of this paper is providing a unified framework in which the existing semantics
can be transposed, and new ones can be developed, leveraging its additional
features.
Several works tried to overcome the limitations of the classical AFs by gen-
eralizing them in different ways. The most investigated limitations were the pos-
sibility of expressing only attacks between pairs of arguments, and the inability
of distinguishing different degrees of ‘strength’ for the single attacks. Research
on the former led to the so-called Bipolar AFs (or BAF s) [6], allowing two kinds
of interactions between arguments, expressed respectively by the attack relation
and the support relation. Research on the latter led to the so-called Weighted
AFs (or WAF s) [8], allowing to specify a numeric weight for each attack between
arguments, indicating its relative strength. BAFs and WAFs cannot be immedi-
ately combined, because the computational procedures for WAFs are specified
only for attacks, and are not simply applicable to supports if no strategy for
combining overall attack and support assessment is provided.
This was the reason behind some attempts to define extensions encompassing
both possibilities. Specifically, [11] proposed a formal extension of the framework
(named Bipolar Weighted Argumentation Framework, or BWAF) and a gradual
evaluation strategy, while [4] extended their previous work on graph-based com-
putational strategies for unipolar AFs. BWAFs embed the notions of attack and
support into the weights, by considering negative weights for attacks and positive
weights for supports.
Definition 3 A BWAF is a triplet F = hA, R̂, wR̂ i, where A is a finite set
of arguments, R̂ ⊆ A × A and wR̂ : R̂ 7→ [−1, 0[ ∪ ]0, 1] assigns a weight to
each relation instance. Within R̂, the attack sub-relation is defined as R̂att =
{hα, βi ∈ R̂ | wR̂ (hα, βi) ∈ [−1, 0[ }, while the support sub-relation is defined as
R̂sup = {hα, βi ∈ R̂ | wR̂ (hα, βi) ∈ ]0, 1] }.
Weight 0 is not considered, since it would mean the absence of an attack or
support relation. Note that this weighting scheme neatly distinguishes attacks
from supports: a support is not considered as just the complement of an attack,
but they are two distinct concepts, and only after determining the concept to
be used (as the sign of the weight) the weight makes sense. This allows BWAFs
to be consistent with previous bipolar approaches to Abstract Argumentation.
Some researchers pointed out that not only the relationships among argu-
ments, but also the arguments themselves may have different degrees of ‘strength’
or ‘reliability’. E.g., according to [2], the intrinsic strength of an argument may
come from different sources: the certainty degree of its reason [3], the impor-
tance of the value it promotes if any [5], the reliability of its source [10]. In this
line of thought, albeit there is no agreement in the literature about the possibil-
ity of using contextual information in an AF, [12] further extended the BWAF
framework into the Trust-affected Bipolar Weighted Argumentation Framework
(or T-BWAF), introducing the possibility of weighting also the arguments by de-
termining their intrinsic strength as the result of several factors, internal to the
argument (the authority of the source of the argument and its own confidence
in the validity of the argument) or external to it (the trust of a community in
the source of the argument1 ).
Definition 4 A T-BWAF is a tuple F = hA, R̂, wA , wR̂ , K, confi, where A, R̂
and wR̂ are as in BWAFs, wA : A 7→ [0, 1] assigns a weight to each argument,
K = {Ti }i∈T is a set of Trust Users Graphs2 for a set of topics T and users U,
and conf : U × A 7→ [0, 1] is the User Argument Confidence function.
where the additional components with respect to BWAFs are wA , K, and conf.
While the perspectives according to which T-BWAFs assess the arguments’
strength are sensible (authority, confidence and trust), unfortunately the result-
ing framework is totally integrated with the specific evaluation solutions pro-
posed in the paper. This motivates the work in this paper, that generalizes [12]
so as to abstract away from the specific computational approaches.
[13] proposed a matrix representation for BWAFs, showing how to use it for
computing some traditional semantics and definining a new semantics specifically
associated to such a representation. We propose the use of matrix representations
also for our extended framework.
1
We are aware of other works about trust in argumentation, but since they are later
than [12] and do not refer it, we assume there is no sufficient relationship to this
work, which builds on [12].
2
A Trust Users Graph is a directed weighted graph T = hU, E, wU , wE i where:
– U is a set of users,
– E = U × U (a complete graph),
– wU : U 7→ [0, 1] assigns a weight to each user, expressing his subjective confidence
about the topic;
– wE : E 7→ [0, 1], given a pair of users (u1 , u2 ) ∈ E, expresses the trust that u1 has
for u2 (0 meaning full distrust, 0.5 full ignorance, and 1 full trust).
3 The Generalized Argumentation Framework
In this section we formalize our generalized model that extends traditional AFs
with bipolarity, weights on both attacks and supports, and weights on the ar-
guments. It comes with no embedded solutions for the use of such components.
Rather, it provides a flexible way for representing different possible interpreta-
tions and perspectives on them, and a basis to implement different evaluation
procedures, including those proposed by previous works. As said, we will present
the new model by referring and comparing it to [12], which is the most compre-
hensive model proposed so far. In recalling the elements of [12], we also reorganize
their definitions and presentation in order to make it more comfortable and fix
some formal and notational issues of the original work.
Definition 5 A Generalized Argumentation Framework (GAF) is a tuple F =
hA, S(A), wA , wR i, where:
– A is a finite set of arguments,
– S(A) is a system providing external information on the arguments3 in A,
– wA : A × S(A) 7→ [0, 1] assigns a weight to each argument, to be considered
as its intrinsic strength, also based on S(A), and
– wR : A × A 7→ [−1, 1] assigns a weight to each pair of arguments.
It is up to the knowledge engineer defining, case by case, what S(A) is4 , and
how it affects the assessment of the ‘intrinsic’ reliability of arguments. For those
who are not comfortable with the use of contextual information in an AF, S(A)
can simply be empty. They might still accept the use of wA for expressing some
kind of ‘intrinsic’ strength of the arguments, or ignore wA as well.
Note that, differently from all previous models, the relationship between ar-
guments is implicit in the GAF model. This is because we consider a complete
graph, where any pair of arguments has a weighted relationship. For practical
purposes, weight 0 can be interpreted as the absence of any (attack or support)
relationship, and ignored when drawing the argumentation graph. The bipolar
relationship considered in BWAFs can be easily extracted as
R̂ = {(α, β) ∈ A × A | wR (α, β) 6= 0}
Not only using negative weights for attacks and positive weights for supports
is quite intuitive (attacking an argument subtracts to its credibility, supporting
it adds to its credibility) and comfortable (the kind of relationship can be imme-
diately distinguished by its sign). Using negative weights for attacks also allows
to straighforwardly translate the traditional assumptions for the bipolar case:
1. attacking the attacker of an argument amounts to defending (i.e., somehow
supporting) that argument (known as reinstatement);
3
This allows us to embed existing proposals in the literature in GAFs. In principle, a
system that plugs external information also in the definition of attack and support
strength might be added, as well.
4
E.g., S(A) = (K, conf) in T-BWAFs.
+ −
− − − +
α β γ α β γ
− +
+ − + +
α β γ α β γ
Fig. 1. Sign rule for attacks and supports
2. attacking the supporter of an argument amounts to attacking that argument;
3. supporting the attacker of an argument amounts to attacking that argument;
4. supporting the supporter of an argument amounts to supporting that argu-
ment;
into mathematical computations, since they clearly correspond to the sign rule
used in mathematics:
followed by Support Attack ×+−
Support Support Attack ≡ + + −
Attack Attack Support −−+
(see Figure 1 for a graphical representation).
This rule also allows to immediately turn the notions of indirect attacks and
defenses into mathematical operations. Indeed, just like a path of relationships
including an even number of attacks amounts to a defense, so the product of
an even number of minus signs gets a plus sign; vice versa, just like a path of
relationships including an odd number of attacks still amounts to an attack, so
the product of an odd number of minus signs gets a minus sign. So, we can report
to GAFs the notions of bw-attacks and bw-defense defined for BWAFs5 :
Definition 6 Given a GAF F = hA, S(A), wA , wR i and a sequence of argu-
ments hx0 , x1 , . . . , xn i such that ∀i = 0, . . . , n : xi ∈ A, we say that:
Qn
– x0 g-defends xn iff Q i=1 wR (xi−1 , xi ) > 0
n
– x0 g-attacks xn iff i=1 wR (xi−1 , xi ) < 0
Note that, while in BWAFs these notions are defined only for sequences of ar-
guments which made up a path in the argumentation graph, in GAFs we may
consider any sequence of arguments, since when it is not associated to a path,
the missing links would have weight 0 and thus would bring the product at 0.
5
Given a BWAF G = hA, R̂, wR̂ i, two arguments x0 , xn ∈ A and a path
hx0 , x2 , . . . , xn , i from x0 to xn :
– x0 bw-defends xn iff Q n
Q
i=1 wR̂ (xi−1 , xi ) > 0
– x0 bw-attacks xn iff n i=1 wR̂ (xi−1 , xi ) < 0
So, the GAF formalization also provides a computational means to determine
whether or not two arguments affect each other along a given path.
Also, GAFs allow to easily compute statistics on the direct attacks and sup-
ports for an argument:
Definition 7 Given a GAF F = hA, S(A), wA , wR i and an argument x0 ∈ A,
we can compute:
P
– the number of attacks received by x0 as x∈A,wR (x,x0 )<0 1
P
– the number of supports received by x0 as x∈A,wR (x,x0 )>0 1
P
– the direct justification balance of x0 as x∈A 1P · sign(wR (x, x0 ))
– the cumulative weighted attack received by x0 as x∈A,wR (x,x0 )<0 −wR (x, x0 )
P
– the cumulative weighted support received by x0 as x∈A,wR (x,x0 )>0 wR (x, x0 )
P
– the weighted direct justification balance of x0 as x∈A wR (x, x0 )
Compared to traditional weighted frameworks (WAFs), where the weight of
an attack could be any number, bounding the absolute weights within fixed min-
imum and maximum values intuitively allows one to identify a level of strength
at which the attacking argument ‘fully’ defeats the attacked one (or the sup-
porting argument ‘fully’ supports the supported one). The specific [0, 1] range
also helps intuition due to its wide use in probability theory.
3.1 Mapping From and To Classical Frameworks
Since one stated objective of our proposal is that it should be able to encompass,
combine and extend less expressive models, a basic requirement is that GAFs
can at least simulate the established models in the literature, namely BWAFs,
WAFs, BAFs, and AFs. The following proposition confirms that our generality
hypothesis holds.
Proposition 1 Given an argumentation framework in any of the less expressive
models (BWAF, WAF, BAF, AF), a corresponding GAF F = hA, S(A), wA , wR i
can be defined, including only the portion of information that they are able to
express.
Intuitively, the GAF can be defined by setting:
– S(A) = {⊥}, i.e., a single uninformative item;
– wA = 1, i.e., the constant function returning 1 for any argument, meaning
full reliability
and wR as follows for the different models:
BWAF hA, R̂, wR̂ i:
wR̂ (α, β) if (α, β) ∈ R̂
wR (α, β) =
0 otherwise
WAF hA, R̂, wR̂ i:
( w (α,β)
− maxR̂w (α,β) if (α, β) ∈ R̂
wR (α, β) = α,β∈A R̂
0 otherwise
(by normalizing the attack weights into [0, 1] —of course, also the justifica-
tion thresholds used in this model must be normalized into the same range)
BAF hA, R̂att , R̂sup i:
−1 if (α, β) ∈ R̂att
wR (α, β) = 1 if (α, β) ∈ R̂sup
0 otherwise
AF hA, R̂i:
1 if (α, β) ∈ R̂
wR (α, β) =
0 otherwise
Conversely, when the additional information provided by GAFs is not needed
for the current purposes, one might be interested in working in one of the simpler
models (e.g., for using existing argument evaluation strategies and tools). The
following proposition shows how a GAF can be reduced to each those models,
by stripping the information they cannot convey and keeping only the portion
that they can express.
Proposition 2 Given a GAF F = hA, S(A), wA , wR i, corresponding frame-
works can be defined for each of the less expressive models (BWAF, WAF, BAF,
AF) by extracting from F only the portion of information that they are able to
express.
Indeed, the less expressive frameworks are extracted from GAFs as follows:
BWAF hA, R̂, wR̂ i with R̂ = {(α, β) ∈ A × A | wR (α, β) 6= 0} ⊆ A × A and
wR̂ = wR |R̂
WAF hA, R̂, wR̂ i with R̂ = {(α, β) ∈ A × A | wR (α, β) < 0} ⊆ A × A and
wR̂ = −wR |R̂
BAF hA, R̂att , R̂sup i with R̂att = {(α, β) ∈ A × A | wR̂ (α, β) < 0} and R̂sup =
{(α, β) ∈ A × A | wR̂ (α, β) > 0}
AF hA, R̂i with R̂ = {(α, β) ∈ A × A | wR̂ (α, β) > 0} ⊆ A × A
4 Adding User and Topic Information
To fully exploit the extended expressive power of GAFs, the two components
S(A) and wA must be defined. In particular, S(A) must be preliminarily defined,
since it is also used in the definition of wA . While the knowledge engineer is
totally free in defining such component, we still believe that the features proposed
in [12] are sensible and useful. Indeed, we expect the interrelations existing in
the community in which the argumentation takes place, and the topic about
which the claims are made, to almost always significantly affect the evaluation
of arguments.
For this reason, having defined the overall GAF model, we propose here a first
specialization of it, which is still very general and abstract, but introduces some
fundamental features that would probably be relevant to most practical cases
of argumentation: community and topics. To model these features, we define
T-GAFs6 , that introduce these first two components in S(A):
U the finite set of members of the community, possibly including the entities
who put forward the arguments, and
T a finite set of topics that may be involved in an argumentation.
For practical purposes, we propose to consider T as always including an addi-
tional dummy topic > associated to the general authority and trust of a user,
independent of specific topics. So, formally, T = T ∪ {>}, where T is the set of
specific topics that may be involved in an argumentation.
Now, also based on these components in S(A), some components to be used
in wA can be defined, as well. Possible features to combine in wA are:
1. the subjective confidence that the members of the community (including the
entity which posits the argument) have in an argument;
2. the recognized authority degree of the entity putting forward an argument
on the topic of the argument7 ;
3. the trust that the community of entities involved in the argumentation have
in the entity putting forward an argument, relative to the topic of the argu-
ment (indeed, not just the quality of evidence, but also the credibility of the
entity positing it is important).
While (2) expresses the degree of expertise of an entity about a topic (e.g.,
medicine), (1) expresses the degree of confidence about a specific claim, and (3)
the degree of confidence by which a user’s opinions about a topic are taken into
consideration by other users. E.g., if Joe, a MD, posits the argument “I am quite
confident that a vaccine for COVID-19 disease will be available before the end
of year 2020”, we may consider: via (1), a degree of uncertainty expressed by Joe
himself about the validity of the argument, in the phrase “quite confident” (which
might be translated into an entity’s confidence degree on that argument of 0.7),
and different degrees of confidence of the various members of the community
with respect to that claim (some will more or less agree with Joe, some will
more or less disagree); via (2), a degree of authority of Joe about medicine (let’s
6
We call it T-GAF for analogy to T-BWAF, since it is aimed at introducing the same
components as T-BWAF, but again at generalizing them so that several frameworks,
including T-BWAFs, can be expressed in the model.
7
E.g., the education or skill level of the user on that topic —opinions of experts in a
topic are typically more convincing than those of novices or outsiders of the topic.
say it’s 0.8, since Joe is a MD); and via (3), a degree of trust of the community
for Joe as a doctor (many people might consider him not a very good doctor).
The 3 features above are formalized by the following functions:
1. wc : U × A 7→ [0, 1] where 1 means certainty, according to the classical
probabilistic interpretation.
2. wa : U × T 7→ [0, 1] where 1 means maximum authority of the user in the
topic, and 0 absolutely no authority.
3. wt : U × T 7→ [−1, 1] where −1 means total distrust, 0 means no opinion,
and 1 means full trust
Functions 1 and 2 might be defined extensionally, by directly associating a value
to each input pair based on the available information. E.g., feature 1 is quite
subjective, and the values might be obtained by asking the single members of
the community; feature 2 might be assessed based on the formal certifications
owned by the arguer about the given topic (e.g., BSc, MS, PhD, etc.) Feature
3 is more complex, because it must be based on a formal model of trust that
might involve many direct and indirect trust evaluations between the members
of the community. We propose a graph-based formal model of trust based on the
following definition8 :
Definition 8 (Community Trust Graph) Given community U, a Commu-
nity Trust Graph (or CTG) for U is a complete directed weighted graph G =
hU, E, wE i where:
– U is the set of members in the community,
– E = U × U is the complete set of edges,
– wE : E 7→ [−1, 1] is a function that, given two members u1 , u2 ∈ U, expresses
the trust wE (u1 , u2 ) that member u1 has for member u2 (where −1 means
total distrust, 0 means no opinion, and 1 means full trust).
Like for the GAF definition, we consider a complete graph for the sake of for-
malization simplicity and for allowing a more straightforward translation of the
graph into matrix representation. Again, for practical purposes, edges having
0 weight can be ignored and removed from the graphical representation. Using
a [−1, 1] range for trust provides the same computational advantages as in the
case of GAF. Indeed, the sign rule can again leveraged to handle the fact that,
if u distrusts v and in turn v distrusts s, then this might be taken as a hint that
u might somehow trust s.
Given a community U, and a CTG G for U, the overall trust for each member
of U according to G, possibly based on the direct and indirect trust information
expressed by G, can be determined by evaluating a function, say t(u, G) ∈ [−1, 1]
(where the range [−1, 1] was chosen for compliance with wE ).
8
Compared to [12], here we adopt a [−1, 1] range for trust, which is more intuitive
and provides computational advantages, and dismiss the weights on the nodes, that
in our model are recovered by function wα . Indeed, the authority of a user is not
necessarily related to the subjective trust the community has in the user.
So, a T-GAF includes a CTG for each topic T ∈ T (let us call it GT ), and
assessing the trust of a user u for T corresponds to computing
wt (u, T ) = t(u, GT ) ∈ [−1, 1]
Finally, given specific definitions for functions wc , wa and wt (for the various
topics), an overall assessment wA of the ‘intrinsic’ reliability of an argument
in the GAF can be obtained by applying a function that combines all these
perspectives together.
Example 1. To show a possible practical application of the GAF, let us express
in the GAF model the aggregation function proposed in [12] for T-BWAFs. For
reference and for the sake of comparison, we will report in footnotes the defini-
tions of the various functions in T-BWAFs. Consider an argument α, posited by
user u and concerning topic T . Then,
wA (α) = β · wc (u, α) · max(min wET (v, u), wa (u)) + (1 − β) · ca(α)
v6=u
where β ∈ [0, 1] and the following notational correspondence was applied:
– wc (u, α) ≡ conf(u, α), called the ‘User Argument Confidence’ in [12]9
– max(minv6=u {wET (v, u)}, wa (uα )) ≡ authority(u), called the ‘Authority De-
gree’ in [12]10 , and also based on the Trust Users Graph as in GAFs, where
t(u, GT ) = minv6=u {wET (v, u)}
– wA (α) ≡ strength(α), called the ‘Argument Strength’ in [12]11
– ca(α), called the ‘Crowd’s Agreement’ of the community in [12], is imple-
mented, following [9], as the Simple Vote Aggregation function12 :
0 if V + (α) = V − (α) = 0
ca(α) = V + (α)
V + (α)+V − (α) otherwise
9
Note that conf : U × A 7→ [0, 1], whereas wc ranges in [−1, 1].
10
Given a Trust Users Graph T = hU, E, wU , wE i, the Authority Degree of a user u ∈ U
is defined as:
authority(u) = max(min{wE (v, u), wU (u)})
v6=u
11
Let F = hA, R̂, wA , wR̂ , K, confi be a T-BWAF, a ∈ A an argument, u ∈ U a user,
i ∈ T a topic, the intrinsic Argument Strength is defined as
strength(a) = α · conf(u, a) · authorityi (u) + (1 − α) · wA (a), with α ∈ [0, 1].
12
As defined in [12], ca(α) takes values in [0, 1], and only considers positive votes.
In the T-GAF framework, it would be more consistent (and a more appropriate
approach in general) to consider also negative votes, using the following formula:
V + (α) − V − (α)
V + (α) + V − (α)
which takes values in [−1, 1] (specifically, −1 means that all votes are negative, and
+1 means that all votes are positive).
where V + (α) and V − (α) denote, respectively, the number of positive and
negative votes for argument α ∈ A. In the T-GAF model, they can be
expressed in terms of wc as follows:
• V + (α) = |{u ∈ U | wc (u, α) > 0}|
• V − (α) = |{u ∈ U | wc (u, α) < 0}|
5 Matrix Representation
As for BWAFs in [13]13 , we propose a matrix representation for GAFs. Indeed, in
addition to providing a comfortable representation that is also consistent with
intuition, matrices also provide an efficient computational tool for supporting
many argument evaluation-related tasks, and may even suggest new semantics,
especially in the extended framework where computations on argument and re-
lationship weights are needed.
Definition 9 Let F = hA, S(A), wA , wR i be a GAF with |A| = n. Then, the
General Argumentation Matrix of F is an n × n matrix MF = [mij ] such that
∀αi , αj ∈ A : mij = wR (αi , αj )
Note that this representation for GAFs is even more straightforward than for
BWAFs, since the 0 value for pairs of arguments having no relationship is explicit
in the formalization of GAFs, while in BWAFs it must be handled as a default
case.
For the same reasons as for GAFs, we propose to use the same matrix rep-
resentation also for Community Trust Graphs:
Definition 10 Let G = hU, E, wE i be a CTG with |U| = n. Then, the Commu-
nity Trust Matrix of G is an n × n matrix MG = [mij ] such that
∀ui , uj ∈ U : mij = wE (ui , uj )
Example 2. The GAF G in Figure 2-a has the following matrix representation:
α β γ δ
α 0 0.4 0 0 −0.7
β 0 0 0.6 0 0
MG = γ 0 0 0 0 0
δ 0 0 −0.5 0 0.3
0 0 0 0 0
13
In BWAFs, given F = hA, R̂, wR̂ i a BWAF with |A| = n, the Signed Weighted
Argumentation Matrix of F is defined as a n × n matrix MF = [mij ] such that
(
wR̂ (hαi , αj i) if hαi , αj i ∈ R̂
∀αi , αj ∈ A : mij =
0 otherwise
0.4 0.6
α β γ
−0.7 −0.5
0.3
δ
a)
α1 α7
5 5
1
5 5
α3 α4 α5 α6
5 5
2
α2 α8
b)
α β γ δ
c)
Fig. 2. Sample argumentation frameworks: (a) a GAF G, (b) a WAF W , (c) an AF F
Note that the graph for G can also be interpreted as a Community Trust Graph,
where {α, β, γ, δ, } are the members in the community, solid edges denote nega-
tive trust between members, and dashed edges represent positive trust between
members (and the weight represent the magnitude of the trust). Under this
interpretation, MG is the matrix representing the community trust.
The GAF G is clearly also a BWAF. Ignoring the weights in G, we have a
BAF B with the following GAF matrix representation:
α β γ δ
0 1 0 0 −1
α
β 0 0 1 0 0
0 0 0 0 0
MB = γ
δ 0 0 −1 0 1
0 0 0 0 0
The GAF matrix representation for the WAF W in Figure 2-b is:
α1 α2 α3 α4 α5 α6 α7 α8
α1 0 0 0 0 0 0 0 0
α2 0 0 0 0 0 0 0 0
α3 −1 −1 0
0 0 0 0 0
α4 0 0 −1 0 −0.2 0 0 0
MW =
α5
0 0 0 −0.4 0 −1 0 0
α6 0
0 0 0 0 0 −1 −1
α7 0 0 0 0 0 0 0 0
α8 0 0 0 0 0 0 0 0
where weights were normalized with respect to max wR̂ (α, β) = 5.
α,β∈A
Finally, the GAF matrix representation of the AF F in Figure 2-c is:
α β γ δ
α0 −1 0 0 0
β 0 0 0 0 0
MF = γ 0 −1 0 −1 0
δ 0 0 −1 0 −1
0 0 0 0 −1
As regards the argument weights assigned by wA , they can be collected in a
vector, indexed exactly like the argumentation matrix rows and columns, which
allows their easy combination through standard matrix operators. E.g., for the
GAF G in Example 2:
A
α β γ δ
MG = wα wβ wγ wδ w
6 Conclusion
The classical definition of Abstract Argumentation Frameworks considers only
attacks between arguments, based on which different evaluation strategies (‘se-
mantics’) have been proposed to identify the subsets of arguments that con-
sistently support the same position in a dispute or debate (‘extensions’), and
possibly determine the winning position. However, in complex situations, addi-
tional information may be important to properly describe the debate and take
better decisions. This led to the definition of extended frameworks, among which
bipolar (considering also supports among arguments), and weighted ones (allow-
ing to assign different importance to the attacks). Since some of these extended
frameworks are partly incompatible, or anyhow disjoint, this paper proposed
GAFs, a general model that encompasses all of them, and further extends them
by allowing to express weights also on arguments, based on contextual informa-
tion. In particular, we propose that the extended framework includes at least
information about authority of users, their subjective confidence in the argu-
ments, and the mutual truth of members in the community. Here we do not
propose specific semantics for GAFs. However, since the previous models can be
represented as GAFs, the semantics defined for the previous models can be also
applied to GAFs. Moreover, new ones can be defined that exploit its extended
expressiveness. The definition of GAFs allows a straightforward matrix repre-
sentation, that allows the use of matrix operations to improve efficiency in the
evaluation of arguments, and perhaps to define new semantics.
In the future, we will define new semantics that can exploit the full expressive
power of GAFs. We also would like to investigate its relationships to other AFs
proposed in the literature, and to identify other relevant specializations of it.
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