Semantics of Argumentation
Abstract
In this chapter, we introduce two approaches for defining the semantics of an abstract argumentation framework: extension-based approach and labelling-based approach. The idea underlying the extension-based approach is to identify sets of arguments, called extensions, that can be regarded as collectively acceptable according to some criterion. The idea underlying the labelling-based approach is to assign a label to each argument, according to a certain criterion. It has been verified that under most of argumentation semantics, there exists a bijective correspondence between sets of extensions and a set of labellings of an argumentation framework.
Keywords
abstract argumentation frameworks; argumentation semantics; extensions; justification status; labellings; status of arguments
2.1 Introduction
Given a set of conflicting arguments, whether an argument can be accepted depends on the existence of some counter arguments, which may in turn have counter arguments, and so on. In order to capture the attack relation between arguments and facilitate the status evaluation of arguments, Phan Minh Dung proposed abstract argumentation theory in 1995 [1]. He introduced a notion of abstract argumentation framework. An abstract argumentation framework can be regarded as a directed graph where the nodes represent arguments and the arcs represent attacks. Given such a graph, a fundamental problem is to determine which arguments can be regarded as acceptable. According to existing literature, there are mainly two approaches to deal with this problem: extension-based approach and labelling-based approach. Both approaches aim at defining argumentation semantics. An argumentation semantics can be viewed as a pre-defined criterion, according to which the acceptability of arguments in an argumentation framework can be determined.
In this chapter, after the notion of an abstract argumentation framework is introduced, some mainstream semantics defined by the two approaches are formulated.
2.2 Abstract Argumentation Frameworks
The notion of abstract argumentation frameworks (briefly, argumentation frameworks, or AFs) is central to abstract theory of argumentation. As mentioned above, in order to reflect the conflicting relation of arguments and facilitate the status evaluation of arguments, the structures of arguments and the origins of attacks that are irrelevant to the status evaluation of arguments are omitted. As a result, an argumentation framework is simply represented as a directed graph (called a defeat graph) where nodes represent arguments and arcs represent attacks between them. Formally, an argumentation framework is defined as follows.
Definition 2.1
Abstract argumentation framework
An abstract argumentation framework is a tuple 
, where 
is a set of arguments, and 
is a set of attacks.
In Definition 2.1, we use 
to denote that 
attacks 
(with respect to 
). Meanwhile, throughout this book, we assume that 
is generated by a reasoner at a given time point, and therefore is finite.
Let 
be a set of arguments, and 
be an argument. As usual, we say that 
attacks 
if and only if there exists 
such that 
attacks 
. Meanwhile, we say that 
attacks 
if and only if there exists 
such that 
attacks 
.
In a defeat graph, the nodes that attack a given argument 
are called defeaters (or parents) of 
[2]. Given 
and a set 
of arguments, we use 
to denote the set of outside parents of the arguments in 
(when 
is called an unattacked set).
(2.1)
2.3 Argumentation Semantics
Since there exist conflicts between arguments, to determine whether or not an argument is acceptable, the status of all its defeaters should be considered, while the status of each defeater is in turn determined by the status of its defeaters, and so on. When an argument 
is attacked by an argument 
, and 
is in turn attacked by an argument 
, then we say that 
reinstates 
. Whether an argument is able to reinstate another argument (or itself) is related to some specific criterion, which is called argumentation semantics. Given an argumentation framework, under a certain argumentation semantics, zero or more sets of arguments are considered acceptable.
In existing literature, there are two approaches to formulate the status of arguments in an argumentation framework: extension-based approach and labelling-based approach. The idea underlying the extension-based approach is to identify sets of arguments, called extensions, that can be regarded as collectively acceptable according to some criterion. The idea underlying the labelling-based approach is to assign a label to each argument, according to a certain criterion.
In this chapter, we will introduce some basic notions related to argumentation semantics, defined by the above-mentioned two approaches. Readers may refer to [3] for an excellent introduction.
2.3.1 Extension-based Approach
In the extension-based approach, under different semantics which represent different evaluation criteria, an argumentation framework may have different sets of extensions. In this section, the following semantics will be introduced: admissible, complete, grounded, preferred, stable, semi-stable, eager and ideal.
2.3.1.1 Admissible Extension
The notion of admissible is fundamental to the definitions of other semantics. It is defined on the basis of the following two notions: conflict-free and acceptable.
Conflict-freeness is viewed as a minimal requirement of any argumentation semantics.
Definition 2.2
Conflict-free
Let 
be an argumentation framework, and 
a set of arguments. 
is conflict-free if and only if 
, such that 
.
However, conflict-freeness is too weak a condition to justify that a set of arguments is “collectively acceptable”, since such a set could be attacked by arguments not among its members.
Example 2.1
Let 
be an argumentation framework (Figure 2.1). The following extensions are conflict-free.
;
;
.However, if we require that a set of collectively acceptable arguments should be able to defend itself, then not all of these subsets satisfy the requirement. For instance, the set 
can not defend itself, because Argument 
is attacked by Argument 
, and there exists no argument in 
that attacks 
.
.Acceptability is another important requirement for the arguments that constitute an admissible extension.
Definition 2.3
Acceptable
Let 
be an argumentation framework, and 
a set of arguments. An argument 
is acceptable with respect to 
(also called 
is defended by 
), if and only if 
, such that 
.
Based on the notions of conflict-freeness and acceptability, an admissible extension can be defined as follows.
Definition 2.4
Admissible extension
Let 
be an argumentation framework, and 
a set of arguments. 
is admissible if and only if 
is conflict-free, and each argument in B is acceptable with respect to 
.
Example 2.2
Continue Example 2.1. According to Definition 2.4, the following extensions are admissible:
;
;
.2.3.1.2 Complete Extension
As illustrated in Example 2.2, it is not the case that each admissible extension contains all arguments that are acceptable with respect to the extension. For instance, with respect to 
, the argument 
is acceptable, but it is not in 
.
Given an argumentation framework 
and an admissible extension 
, if there exists an argument 
such that 
is acceptable with respect to 
, then we add 
to 
, and get a new set 
. Then, with respect to 
, if there exists an argument 
such that 
is acceptable with respect to 
, then we add 
to 
, and get a new set 
. In this way, if 
is a finite set, then we will finally get a set that is admissible and contains all arguments that are acceptable with respect to this set. This process can be defined by a function, called characteristic function, defined as follows.
Definition 2.5
Characteristic function
Let 
be an argumentation framework. The characteristic function of 
, denoted as 
, is defined as follows:
,
= 
is acceptable with respect to 
.Given an admissible set 
, if 
, then 
is called a complete extension. Formally, we have the following definition.
Definition 2.6
Complete extension
Let 
be an argumentation framework. We say that 
is a complete extension if and only if 
is admissible and each argument in 
that is acceptable with respect to 
is in 
, i.e., 
.
Example 2.3
Continue Example 2.2. According to Definition 2.6, only 
is a complete extension.
.Under complete semantics, some argumentation frameworks may have more than one extension. Let us consider the following example.
Example 2.4
Let 
be an argumentation framework (Figure 2.2). It has three complete extensions as follows:
;
;
.2.3.1.3 Grounded Extension and Preferred Extension
As mentioned above, under complete semantics, there might exist several extensions. The arguments in different extensions might be conflicting, and therefore questionable. If we only accept those arguments that are least questionable, then we get an extension (called grounded extension) that is the most skeptical among all complete extensions. On the contrary, if we want to accept as many arguments as reasonably possible, then we get sets of extensions (called preferred extensions) that are more credulous than some other complete extensions.
According to Example 2.4, the argument 
is acceptable with respect to all extensions, while the arguments 
and 
are not. So, under grounded semantics, one may only regard 
as an acceptable argument, while under preferred semantics, one may regard 
and 
(or 
and 
) as acceptable arguments.
Definition 2.7
Grounded extension and preferred extension
Let 
be an argumentation framework, and 
be a set of arguments.
is a grounded extension if and only if 
is the minimal (with respect to set-inclusion) complete extension.
is a preferred extension if and only if 
is a maximal (with respect to set-inclusion) complete extension.According to [1], for any argumentation framework, there exists a unique grounded extension, while for some argumentation frameworks, there might exist multiple preferred extensions.
Example 2.5
Continue Example 2.4. According to Definition 2.7, it holds that
is the grounded extension;
and 
are preferred extensions.Grounded extension can be obtained by recursively applying characteristic function from an empty set.
Example 2.6
Consider the argumentation framework in Example 2.1. Let 
. According to Definition 2.5, it holds that
;
;
.Hence, the grounded extension of 
is 
.
Alternatively, the grounded extension of an argumentation framework can be defined as follows.
Definition 2.8
Grounded extension
Let 
be an argumentation framework, and 
be a set of arguments. 
is a grounded extension if and only if 
is the minimal (with respect to set-inclusion) conflict-free fixed point of the characteristic function 
.
2.3.1.4 Stable Extension and Semi-Stable Extension
Under the above-mentioned semantics, with respect to a given extension, the status of some arguments could be undecided, which means that they are neither accepted, nor rejected, with respect to the extension. Here, we say that an argument is accepted with respect to an extension, if it belongs to this extension; an argument is rejected with respect to an extension if it is attacked by the extension.
Example 2.7
Let 
be an argumentation framework (Figure 2.3). Under admissible, complete, grounded, or preferred semantics, it has only one extension 
. The status of arguments 
and 
is undecided.
.Now, given a set of conflict-free arguments, if we require that there are no arguments that are undecided with respect to this set, then it is called a stable extension. Formally, we have the following definition.
Definition 2.9
Stable extension
Let 
be an argumentation framework, and 
be a set of arguments. 
is a stable extension if and only if 
is conflict-free and 
attacks each argument that does not belong to 
.
In [1], it has been proved that every stable extension is a preferred extension, but not vice versa.
Stable semantics is interesting, since it exactly corresponds to the extensions defined in some traditional non-monotonic formalisms, such as Reiter’s default logic, Moore’s autoepistemic logic and logic programming. However, some argumentation frameworks may have no stable extension.
Example 2.8
Continue Example 2.7. Argumentation framework 
has no stable extension.
Since under stable semantics there is a possibility that stable extensions may not exist, Martin Caminada proposed a revised semantics, called semi-stable semantics. This semantics is “backward compatible” to stable semantics in the sense that it is equivalent to stable semantics in situations where stable extensions exist, and still yields a reasonable result in situations where stable extensions do not exist [4]. Compared to stable semantics, semi-stable semantics does not require that the set of undecided arguments is empty, but merely requires that the set of undecided arguments is minimal.
Definition 2.10
Semi-stable extension
Let 
be an argumentation framework, and 
be a set of arguments. 
is a semi-stable extension if and only if 
is a complete extension of which 
is maximal, in which 
.
It has been verified that every stable extension is a semi-stable extension, and every semi-stable extension is a preferred extension.
Example 2.9
Continue Example 2.7. Argumentation framework 
has a semi-stable extension 
.
2.3.1.5 Ideal Extension and Eager Extension
As illustrated in Example 2.5, an argumentation framework may have several preferred extensions, and some arguments in different extensions might conflict. More specifically, 
and 
are respectively in 
and 
, but they are conflicting. This is because preferred semantics is credulous. In many cases, it is better to adopt a semantics that is more skeptical. The above-mentioned grounded semantics is skeptical. However, it is often overly skeptical, in that in many cases, the grounded extension could be an empty set. In order to treat this problem, another skeptical semantics, called ideal semantics, was proposed in [5]. It defines an ideal extension as an admissible extension that is a subset of every preferred extension. Generally, ideal semantics is less skeptical than grounded semantics, but more skeptical than preferred semantics. Formally, we have the following definition.
Definition 2.11
Ideal extension
Let 
be an argumentation framework, and 
be a set of arguments. 
is ideal if and only if 
is admissible and it is contained in every preferred set of arguments. The ideal extension is the maximal (with respect to set-inclusion) ideal set.
Example 2.10
Continue Example 2.5. According to Definition 2.11, 
is the ideal extension. It is also the grounded extension, and the intersection of the two preferred extensions.
Example 2.10 shows that the ideal extension is the same as the grounded extension. Meanwhile, the intersection of the preferred extensions is equal to the ideal extension. However, this coincidence does not happen in all cases. Please consider the following example (Figure 2.4).
Example 2.11
According to the above definition, we have
is the grounded extension;
and 
are four preferred extensions.
is the ideal extension.In this example, the grounded extension is not equal to the ideal extension. Meanwhile, since the intersection of the two preferred extensions 
is not admissible, it is not an ideal set.
.Besides grounded semantics and ideal semantics, the third skeptical semantics that is very close to ideal semantics is eager semantics. While the ideal extension is the biggest admissible subset of each preferred extension, the eager extension is the biggest admissible subset of each semi-stable extension.
and 
are two semi-stable extensions.
is the eager extension.2.3.2 Labelling-based Approach
In labelling-based approaches, there are usually three labels: IN, OUT and UNDEC, where the label IN means the argument is accepted, the label OUT means the argument is rejected and the label UNDEC means one abstains from an opinion on whether the argument is accepted or rejected [3]. Meanwhile, there could be some other choices for the set of labels. For instance, in [6], four-valued labelling is considered. In this book, we choose three-valued labelling. Formally, labelling is defined as follows [7].
Definition 2.12
Labelling
Given an argumentation framework 
and three labels IN, OUT and UNDEC, a labelling is defined as a total function
The set of all labellings of 
is denoted as 
.
Let 
, and 
. A labelling 
is often represented as a triple of the form 
.
One of the criteria for labelling-based semantics is whether a label assigned to an argument is legal. According to Definition 2.12, given a labelling 
, the status assigned to each argument might not be legal. We say that assigning the IN label to an argument 
is legal if and only if all its attackers have been assigned the OUT label; assigning the OUT label to an argument 
is legal if and only if one of its attackers has been assigned the IN label; and assigning the UNDEC label to an argument 
is legal if and only if not all its attacks are labelled OUT and it does not have an attacker that is labelled IN. Based on [8], we have the following definition.
Definition 2.13
Legal labelling
be a labelling of an argumentation framework 
and 
.
is legally IN if and only if 
and for all 
, if 
, then 
.
is legally OUT if and only if 
and there exists 
, such that 
, and 
.
is legally UNDEC if and only if 
and
, if 
, then 
, and
, if 
, then 
.According to the notion of legal labelling, the notion of illegal labelling can be defined as follows.
Definition 2.14
Illegal labelling
Let 
be a labelling of an argumentation framework 
and 
. For 
is illegally 
if and only if 
, but 
is not legally 
.
Based on the notions of labelling as well as legal/illegal labelling, various labelling-based semantics corresponding to the above-mentioned extension-based semantics can be defined as follows.
2.3.2.1 Admissible Labelling
Admissible labeling is defined on the basis of conflict-free labelling. According to Definition 2.2, when a set of arguments is conflict-free, there exists no attack between the arguments in the set. So, given a labelling 
, each argument in 
should not have an attacker in 
. Meanwhile, only those arguments that are attacked by at least one argument in 
are labelled OUT. So, all OUT-labelled arguments are legally OUT.
Definition 2.15
Conflict-free labelling
Let 
be an argumentation framework and 
be a total function. 
is a conflict-free labelling, if and only if each argument that is labelled IN does not have an attacker that is labelled IN, and each argument that is labelled OUT is legally OUT.
Example 2.13
Continue Example 2.1. With respect to 
, the following labellings are conflict-free:
From this example, we may observe that a conflict-free set might correspond to several conflict-free labellings.
According to Definition 2.15, not all IN-labelled arguments are legally IN. Given a conflict-free labelling, if each IN-labelled argument is legally IN, then it is called an admissible labelling.
Definition 2.16
Admissible labelling
Let 
be an argumentation framework and 
be a total function. 
is an admissible labelling, if and only if it is a conflict-free labelling, and each argument that is labelled IN is legally IN.
Example 2.14
Continue Example 2.13. According to Definition 2.16, the following labellings are admissible labellings:
2.3.2.2 Complete Labelling
Given an admissible labelling, if each UNDEC-labelled argument is legally UNDEC, then it is called a complete labelling. Formally, we have the following definition.
Definition 2.17
Complete labelling
Let 
be an argumentation framework and 
be a total function. 
is a complete labelling, if and only if it is an admissible labelling, and each argument that is labelled UNDEC is legally UNDEC.
Example 2.15
Continue Example 2.14. According to Definition 2.17, only 
is a complete labelling.
2.3.2.3 Grounded Labelling and Preferred Labelling
Given a complete labelling, if the set of IN-labelled arguments is minimal (respectively, maximal), then it is called a grounded labelling (respectively, preferred labelling). Formally, we have the following definition.
Definition 2.18
Grounded labelling and preferred labelling
be an argumentation framework and 
be a total function.
is a grounded labelling, if and only if it is a complete labelling, and 
is minimal (with respect to set-inclusion).
is a preferred labelling, if and only if it is a complete labelling, and 
is maximal (with respect to set-inclusion).Example 2.16
Consider the argumentation framework in Example 2.11. According to Definition 2.18, it holds that
is the grounded labelling;
and 
are preferred labellings.2.3.2.4 Stable Labelling and Semi-Stable Labelling
Definition 2.19
Stable labelling and semi-stable labelling
be an argumentation framework and 
be a total function.
is a stable labelling, if and only if it is a complete labelling, and 
.
is a semi-stable labelling, if and only if it is a complete labelling, and 
is maximal (with respect to set-inclusion).Example 2.17
Consider again the argumentation framework in Example 2.11. According to Definition 2.19, it holds that
and 
are semi-stable labellings.2.3.2.5 Ideal Labelling and Eager Labelling
The definition of ideal labelling is more complex. According to [9], in order to define the ideal labelling of an argumentation framework, we need to introduce the following notion [10].
Definition 2.20
Bigger labelling
Let 
and 
be labellings of an argumentation framework 
. It is said that 
is bigger (also called more or equally committed) than 
(written as 
) if and only if 
and 
.
The relation “
” defines a partial order (reflective, anti-symmetric, transitive) on the labellings of an argumentation framework.
Definition 2.21
Ideal labelling and eager labelling
Let 
be an argumentation framework. The ideal (eager) labelling is the biggest admissible labelling that is smaller than or equal to each preferred labelling (respectively, semi-stable labelling).
Example 2.18
Continue Examples 2.11 and 2.17.
that is smaller than each preferred labelling. It is the ideal labelling of 
.
that is smaller than each semi-stable labelling. It is the eager labelling of 
.2.3.3 Relations Between the Two Approaches
As introduced in [3], under most of argumentation semantics, there is a bijective correspondence between the set of labellings and sets of extensions of an argumentation framework. On the one hand, given an argumentation framework, the labels IN can be understood as identifying the members of an extension. According to [3], we have the following definition.
Definition 2.22
A mapping from a labelling to an extension
Let 
be an argumentation framework, and 
be a labelling of 
. The corresponding set of arguments 
.
On the other hand, given an extension 
of an argumentation framework, if it is conflict-free, then we may construct a labelling such that the arguments belonging to 
are labelled IN, those attacked by some arguments of 
are labelled OUT, and those which neither belong to E nor are attacked by 
are labelled UNDEC. Formally, we have the following definition.
Definition 2.23
A mapping from an extension to a labelling
Let 
be an argumentation framework, and 
be a conflict-free set of arguments. The corresponding labelling, denoted as 
, is defined as 
, in which 
.
As proved in [8], there is a correspondence between admissible sets and admissible labellings.
Proposition 2.1
For any argumentation framework 
, if 
is an admissible labelling of 
then 
is an admissible set of 
; if 
is an admissible set of 
then 
is an admissible labelling of 
.
It should be noticed that the correspondence between admissible sets and admissible labellings is not bijective. This is because different admissible labellings may correspond to the same admissible set. For instance, with respect to 
in Example 2.4, 
and 
are two admissible labellings. Both of them give rise to the same admissible set 
.
Except admissible semantics, under other semantics introduced above, there exists a bijective correspondence between sets of extensions and a set of labellings, of an argumentation framework.
In this book, we use 
to denote a semantics, which could be admissible 
, complete 
, preferred 
, grounded 
, stable 
, semi-stable 
, ideal 
or eager 
. Based on these notations, we have the following proposition.
Proposition 2.2
Let 
be an argumentation framework, and 
be a semantics. For all 
, there exists 
such that 
, and for all 
, there exists 
such that 
.
2.3.4 Relations Between Different Semantics
With respect to the extension-based approach, the relations between different argumentation semantics introduced above can be illustrated in Figure 2.5. According to Definitions 2.4 and 2.6, it holds that a complete extension is an admissible set. According to Definition 2.7, a grounded extension (respectively, a preferred extension) is a complete extension. Meanwhile, as proved in [10], an ideal extension (respectively, eager extension) is also a complete extension. Finally, according to [4], every stable extension is also a semi-stable extension, which in turn is a preferred extension. Given an argumentation framework, its eager extension is a superset of its ideal extension, which is in turn a superset of its grounded extension. So, the grounded semantics is the most skeptical semantics mentioned above. On the other hand, as to the labelling-based approach, we have similar results.
2.3.5 Status of Arguments
According to the extensions (labellings) of an argumentation framework, the status of arguments can be determined. Given a labelling, the status of each argument with respect to this labelling is IN, OUT, or UNDEC.
On the other hand, from the perspective of the extension-based approach, the status of arguments is determined by an extension, rather than a labeling. As presented in Definition 2.23, when an extension is conflict-free, it can be mapped to a labelling. Hence, given an argumentation framework and a conflict-free extension of it, for each argument within the argumentation framework, we may differentiate three status: accepted, rejected and undecided, corresponding to IN, OUT and UNDEC in the labelling-based approach.
Informally, an argument is accepted with respect to an extension, if and only if it belongs to this extension; an argument is rejected with respect to an extension, if and only if it is attacked by another argument that is accepted with respect to this extension; an argument is undecided with respect to an extension, if and only if it is neither accepted nor rejected with respect to this extension. Here, the extension should be conflict-free. Otherwise, the above three classes of status could not be differentiated. For instance, consider an argumentation framework 
. Suppose that 
is an extension. It follows that 
is both accepted (because it belongs to 
) and rejected (because it is attacked by 
that is accepted with respect to 
).
Definition 2.24
Status of arguments with respect to an extension
Let 
be an argumentation framework, and 
be an extension that is conflict-free. For all 
:
is accepted with respect to 
, if and only if 
;
is rejected with respect to 
, if and only if there exists 
, such that 
;
is undecided with respect to 
, if and only if 
is neither accepted nor rejected with respect to 
.Beside the status of arguments with respect to an extension or a labelling, we may evaluate the status of arguments with respect to sets of extensions or a set of labellings, call the justification status of arguments. Since under some argumentation semantics, an argumentation framework might have multiple extensions (labellings), the justification status of an argument could be sceptically justified, credulously justified, and indefensible. An argument is sceptically justified, if it belongs to each extension (labelling); an argument is credulously justified, if it belongs to some (at least one) extensions (labellings) and does not belong to some other (at least one) extensions (labellings); an argument is indefensible, if it does not belong to any extension (labelling). Formally, we have the following definition.
Definition 2.25
Justification status of arguments
Given 
and a conflict-free semantics 
, an argument 
is credulously justified if and only if 
, such that 
and 
; it is skeptically justified if and only if 
; and it is indefensible if and only if 
such that 
.
2.4 Conclusions
In this chapter, we have introduced the notion of abstract argumentation frameworks and the semantics of an argumentation framework from the perspective of an extension-based approach and that of a labelling-based approach, respectively. Under most semantics (complete, preferred, grounded, stable, semi-stable, ideal and eager), there is a bijective correspondence between sets of extensions and a set of labellings.
Under a given semantics, an argumentation framework may have a unique extension or multiple extensions. If under a semantics 
, any argumentation framework has only one extension, then it is called a unique-status semantics; otherwise, it is called a multiple-status semantics. Grounded, ideal and eager are three unique-status semantics introduced in this chapter. Among them, the grounded semantics is the most skeptical one, the ideal semantics is the second, and the eager semantics is the third. On the other hand, admissible, complete, preferred, stable and semi-stable are multiple-status semantics. Under stable semantics, an argumentation framework might have an empty set of extension. Semi-stable semantics was proposed to handle this problem.
Based on the definition of argumentation semantics, given an argumentation framework, an important problem is to efficiently compute the status of arguments. In the subsequent chapters, we will focus on this problem.
References
1. Dung PM. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence. 1995;77(2):321–357.
2. Baroni P, Giacomin M. On principle-based evaluation of extension-based argumentation semantics. Artificial Intelligence. 2007;171(10–15):675–700.
3. Baroni P, Caminada M, Giacomin M. An introduction to argumentation semantics. The Knowledge Engineering Review. 2011;26(4):365–410.
4. Caminada M. Semi-stable semantics. In: Proceedings of First International Conference on Computational Models of Argument. 2006;121–130.
5. Dung PM, Mancarella P, Toni F. Computing ideal sceptical argumentation. Artificial Intelligence. 2007;171(10–15):642–674.
6. Jakobovits H, Vermeir D. Robust semantics for argumentation frameworks. Journal of Logic and Computation. 1999;9(2):215–261.
7. Caminada M. On the issue of reinstatement in argumentation. In: Proceedings of the 10th European Conference on Logics in Artificial Intelligence. 2006;111–123.
8. Caminada M, Gabbay Dov. A logical account of formal argumentation. Studia Logica. 2009;93(2–3):109–145.
9. Caminada M. A labelling approach for ideal and stage semantics. Argument & Computation. 2011;2(1):1–21.
10. Caminada M, Pigozzi G. On judgment aggregation in abstract argumentation. Journal of Autonomous Agents and Multi-Agent Systems. 2011;22(1):64–102.