In its original de nition, the Abstract Argumentation framework considers atomic claims and a binary attack relationship among them, based on which di erent 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 (considering also the support relationship between pairs of claims) and weights (assigning di erent importance to di erent 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 uni es all these perspectives, and further extends them by allowing to express contextual information associated to the arguments, in addition to their relationships.
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 [
Abstract argumentation, in particular, focuses on the resolution of the dispute based only on `external' information about the arguments (notably, the ? Copyright c 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). inter-relationships among them), neglecting their internal structure or interpretation. Traditional Abstract Argumentation Frameworks (AFs for short) can express 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 signi cant 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 di erent 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 Argumentation Framework, or GAF. With respect to our previous preliminary work in this direction, here we x some notational issues, reorganize the model formalization and abstract away from details of speci c 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 specialized and tailored to di erent contexts and domains, and on which theoretical investigation can be carried out for de ning 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 signi cant improvements in e ciency 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 dene 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 condence 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
The original (and now classical) Abstract Argumentation setting was proposed
by Dung [
In this setting, no direct agreement between arguments can be expressed. Agreement can only indirectly be derived based on the attack relationship, yielding the notion of defense: De nition 2 Let F = hA; Ri be an AF, and S A: { 2 A is defended by S if 8 2 A : R ) 9 2 S s.t. R ; { fF : 2A 7! 2A s.t. fF (S) = f j is defended by Sg 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 de nition 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 [
Several works tried to overcome the limitations of the classical AFs by
generalizing them in di erent ways. The most investigated limitations were the
possibility of expressing only attacks between pairs of arguments, and the inability
of distinguishing di erent degrees of `strength' for the single attacks. Research
on the former led to the so-called Bipolar AFs (or BAF s) [
This was the reason behind some attempts to de ne extensions encompassing
both possibilities. Speci cally, [
De nition 3 A BWAF is a triplet F = hA; R^ ; wR^ i, where A is a nite set of arguments, R^ A A and wR^ : R^ 7! [ 1; 0[ [ ]0; 1] assigns a weight to each relation instance. Within R^ , the attack sub-relation is de ned as R^ att = ^ fh ; i 2 R j wR^ (h ; i) 2 [ 1; 0[ g, while the support sub-relation is de ned as R^ sup = fh ; i 2 R j wR^ (h ; i) 2 ]0; 1] g.
^ 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
arguments, but also the arguments themselves may have di erent degrees of `strength'
or `reliability'. E.g., according to [
[
In this section we formalize our generalized model that extends traditional AFs
with bipolarity, weights on both attacks and supports, and weights on the
arguments. It comes with no embedded solutions for the use of such components.
Rather, it provides a exible way for representing di erent possible
interpretations and perspectives on them, and a basis to implement di erent evaluation
procedures, including those proposed by previous works. As said, we will present
the new model by referring and comparing it to [
De nition 5 A Generalized Argumentation Framework (GAF) is a tuple F = hA; S(A); wA; wRi, where: { A is a nite 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 de ning, case by case, what S(A) is4, and how it a ects 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, di erently from all previous models, the relationship between arguments 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
R^ = f( ; ) 2 A
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 immediately 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 de nition of attack and support strength might be added, as well. 4 E.g., S(A) = (K; conf) in T-BWAFs.
+ + + + + + 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 argument; 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 de ned for BWAFs5: De nition 6 Given a GAF F = hA; S(A); wA; wRi and a sequence of arguments hx0; x1; : : : ; xni such that 8i = 0; : : : ; n : xi 2 A, we say that: { x0 g-defends xn i { x0 g-attacks xn i
Qn Qni=1 wR(xi 1; xi) > 0
i=1 wR(xi 1; xi) < 0 Note that, while in BWAFs these notions are de ned only for sequences of arguments 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; R^ ; wR^ i, two arguments x0; xn 2 A and a path hx0; x2; : : : ; xn; i from x0 to xn: {{ xx00 bbww--datetfaecnkdss xxnn ii QQinin==11wwRR^^((xxii 11;;xxii))<>00 So, the GAF formalization also provides a computational means to determine whether or not two arguments a ect each other along a given path.
Also, GAFs allow to easily compute statistics on the direct attacks and supports for an argument: De nition 7 Given a GAF F = hA; S(A); wA; wRi and an argument x0 2 A, we can compute: { the number of attacks received by x0 as Px2A;wR(x;x0)<0 1 { the number of supports received by x0 as Px2A;wR(x;x0)>0 1 { the direct justi cation balance of x0 as Px2A 1 sign(wR(x; x0)) { the cumulative weighted attack received by x0 as Px2A;wR(x;x0)<0 wR(x; x0) { the cumulative weighted support received by x0 as P { the weighted direct justi cation balance of x0 as Px2A;wR(x;x0)>0 wR(x; x0) x2A wR(x; x0)
Compared to traditional weighted frameworks (WAFs), where the weight of an attack could be any number, bounding the absolute weights within xed minimum and maximum values intuitively allows one to identify a level of strength at which the attacking argument `fully' defeats the attacked one (or the supporting argument `fully' supports the supported one). The speci c [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 con rms 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; wRi can be de ned, including only the portion of information that they are able to express.
Intuitively, the GAF can be de ned by setting: { S(A) = f?g, 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 di erent models: BWAF hA; R^ ; wR^ i: wR( ; ) =
^ w ^ ( ; ) if ( ; ) 2 R 0 R otherwise WAF hA; R^; wR^ i: (by normalizing the attack weights into [0; 1] |of course, also the justi cation thresholds used in this model must be normalized into the same range) AF hA; R^i:
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; wRi, corresponding frameworks can be de ned 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; R^; wR^ i with R^ = f( ; ) 2 A
wR^ = wRjR^ WAF hA; R^; wR^ i with R^ = f( ; ) 2 A
wR^ = wRjR^ BAF hA; R^att; R^supi with R^att = f( ; ) 2 A
f( ; ) 2 A A j wR^ ( ; ) > 0g
AF hA; R^i with R^ = f( ; ) 2 A
To fully exploit the extended expressive power of GAFs, the two components
S(A) and wA must be de ned. In particular, S(A) must be preliminarily de ned,
since it is also used in the de nition of wA. While the knowledge engineer is
totally free in de ning such component, we still believe that the features proposed
in [
For this reason, having de ned the overall GAF model, we propose here a rst 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 de ne T-GAFs6, that introduce these rst two components in S(A): U the nite set of members of the community, possibly including the entities who put forward the arguments, and T a nite set of topics that may be involved in an argumentation. For practical purposes, we propose to consider T as always including an additional dummy topic > associated to the general authority and trust of a user, independent of speci c topics. So, formally, T = T [ f>g, where T is the set of speci c 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 de ned, as well. Possible features to combine in wA are: 1. the subjective con dence 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 argument (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 con dence about a speci c claim, and (3) the degree of con dence 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 con dent 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 con dent" (which might be translated into an entity's con dence degree on that argument of 0:7), and di erent degrees of con dence 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 de ned 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 certi cations 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 de nition8: De nition 8 (Community Trust Graph) Given community U , a Community 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 2 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 de nition, we consider a complete graph for the sake of formalization 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) 2 [ 1; 1]
(where the range [ 1; 1] was chosen for compliance with wE ).
8 Compared to [
So, a T-GAF includes a CTG for each topic T 2 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 ) 2 [ 1; 1]
Finally, given speci c de nitions 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 [
2 [0; 1] and the following notational correspondence was applied:
{ wc(u; ) conf(u; ), called the `User Argument Con dence' in [
V +( ) V +( )+V ( ) otherwise if V +( ) = V ( ) = 0 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 2 U is de ned as:
authority(u) = max(mv6=inufwE (v; u); wU (u)g) 11 Let F = hA; R^; wA; w ^ ; K; confi be a T-BWAF, a 2 A an argument, u 2 U a user,
R i 2 T a topic, the intrinsic Argument Strength is de ned as strength(a) = conf(u; a) authorityi(u) + (1 ) wA(a); with which takes values in [ 1; 1] (speci cally, 1 means that all votes are negative, and +1 means that all votes are positive).
where V +( ) and V ( ) denote, respectively, the number of positive and negative votes for argument 2 A. In the T-GAF model, they can be expressed in terms of wc as follows:
V +( ) = jfu 2 U j wc(u; ) > 0gj
V ( ) = jfu 2 U j wc(u; ) < 0gj 5
As for BWAFs in [
De nition 9 Let F = hA; S(A); wA; wRi be a GAF with jAj = n. Then, the General Argumentation Matrix of F is an n n matrix MF = [mij ] such that 8 i; j 2 A : mij = wR( i; j ) 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 representation also for Community Trust Graphs: De nition 10 Let G = hU ; E ; wE i be a CTG with jU j = n. Then, the Community Trust Matrix of G is an n n matrix MG = [mij ] such that Example 2. The GAF G in Figure 2-a has the following matrix representation: 8ui; uj 2 U : mij = wE (ui; uj ) MG = 2 0 0:4 6 0 0 66 0 0 64 0 0 0 0 0 0 0:7 3 0:6 0 0 7 0 0 0 77 0:5 0 0:3 75 0 0 0 13 In BWAFs, given F = hA; R^ ; wR^ i a BWAF with jAj = n, the Signed Weighted Argumentation Matrix of F is de ned as a n n matrix MF = [mij ] such that 8 i; j 2 A : mij = ( wR^ (h i; ji) if h i; ji 2 R^ 0 otherwise 0:4
0:6 0:3 1 2 5
0:5 4 5 6 5 5 7 8 b) 1 2 5 5 a)
0:7 3
5 where weights were normalized with respect to max w ^ ( ; ) = 5. ; 2A R MW =
MB = Note that the graph for G can also be interpreted as a Community Trust Graph, where f ; ; ; ; g are the members in the community, solid edges denote negative 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:
The GAF matrix representation for the WAF W in Figure 2-b is: Finally, the GAF matrix representation of the AF F in Figure 2-c is: MF = 2 0 6 0 0 66 0 64 0 0 0 0
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:
MGA =
w w w w w 6
The classical de nition of Abstract Argumentation Frameworks considers only attacks between arguments, based on which di erent evaluation strategies (`semantics') have been proposed to identify the subsets of arguments that consistently support the same position in a dispute or debate (`extensions'), and possibly determine the winning position. However, in complex situations, additional information may be important to properly describe the debate and take better decisions. This led to the de nition of extended frameworks, among which bipolar (considering also supports among arguments), and weighted ones (allowing to assign di erent 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 information. In particular, we propose that the extended framework includes at least information about authority of users, their subjective con dence in the arguments, and the mutual truth of members in the community. Here we do not propose speci c semantics for GAFs. However, since the previous models can be represented as GAFs, the semantics de ned for the previous models can be also applied to GAFs. Moreover, new ones can be de ned that exploit its extended expressiveness. The de nition of GAFs allows a straightforward matrix representation, that allows the use of matrix operations to improve e ciency in the evaluation of arguments, and perhaps to de ne new semantics.
In the future, we will de ne 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.