Relations between Global Semantics and Local Semantics
Abstract
Based on the notion of sub-frameworks and their semantics, in this chapter, we introduce the relations between the semantics of an argumentation framework and those of its sub-framework, which lays a foundation for the establishment of some efficient approaches for computing the semantics of argumentation.
Keywords
directionality of argumentation; global semantics; local semantics; mappings; semantics combination
5.1 Introduction
In Chapters 2 and 4, we introduce the semantics of an argumentation framework and that of its sub-frameworks (called the global semantics and the local semantics of the argumentation framework, respectively). Now, an important question arises: what are the relations between these two semantics? More specifically, we have the following two sub-questions:
• Is there a mapping from the global semantics to the local semantics, of an argumentation framework? Is this mapping sound and complete?
• Is there a mapping from the local semantics to the global semantics, of an argumentation framework? Is this mapping sound and complete?
We will define in the subsequent sections these two kinds of mappings. The soundness and completeness of these mappings are affected by the types of argumentation semantics. Under some argumentation semantics such as admissible, complete, preferred and grounded, there exist sound and complete mappings. However, under some other argumentation semantics, sound and complete mappings might not exist. Let us consider the following example.
Example 5.1
Under stable semantics, the set of extensions of 
(as shown in Figure 5.1) is empty. Nevertheless, the sub-framework 
has two stable extensions 
and 
.
.The above example shows that under stable semantics, there might not exist a mapping from global semantics to local semantics. In other words, the mapping from global semantics to local semantics might be incomplete. Meanwhile, there are also cases where the mapping from global semantics to local semantics is not sound. Let us consider the following example.
Example 5.2
Let 
and 
, in which 
is an unconditioned sub-framework, and 
a conditioned sub-framework (see Figure 5.2). Under ideal semantics, 
has an extension 
, while 
has an extension 
. In this case, for 
, there is only one partially assigned sub-framework: 
, in which the status of Argument 
is undecided. It follows that 
has an ideal extension 
. So, if we define a mapping according to which the ideal extension of 
is an empty set, then the mapping is not sound.
and its sub-frameworks.Given that under some argumentation semantics the mappings between global semantics and local semantics might be incomplete and/or unsound, in the subsequent sections and chapters, we only focus on several argumentation semantics (including admissible, complete, grounded and preferred) under which complete and sound mappings exist.
In addition, since there is a bijective correspondence between sets of extensions and a set of labellings under complete, grounded and preferred semantics, in this chapter, each mapping is formulated in terms of the extension-based approach or the labelling-based approach, rather than both approaches.
5.2 Mapping Global Semantics to Local Semantics
In terms of the extension-based approach, given an argumentation framework, a mapping from global semantics to local semantics is to restrict an extension of the argumentation framework to those of its sub-frameworks. More specifically, for every extension of an argumentation framework, from the perspective of a sub-framework, only a part of the extension corresponds to the sub-framework. In other words, we may restrict the extension to the sub-framework (unconditioned or conditioned). Formally, we have the following definition.
Definition 5.1
Restricting an extension to a sub-framework
Let 
be an argumentation framework, and 
be a sub-framework of it. Let 
be the set of extensions of 
under a given semantics 
. When 
, for all 
, the restriction of 
to 
is defined as 
.
Notice that when 
and 
.
Example 5.3
Consider an argumentation framework 
in Figure 5.3. Let 
and 
. It follows that 
and 
, while 
and 
. Under complete semantics, 
has three extensions 
and 
. According to Definition 5.1, we have:
is the restriction of 
to 
;
is the restriction of 
to 
;
is the restriction of 
to 
;
is the restriction of 
to 
;
is the restriction of 
to 
;
is the restriction of 
to 
.
.Based on the notion of restricting an extension to a sub-framework, we may define the mapping from global semantics to local semantics.
First, for an unconditioned sub-framework, the mapping from global semantics to local semantics is closely related to the directionality of argumentation. The basic idea of directionality is that under some argumentation semantics, the status of an argument 
is affected only by the status of its defeaters (which in turn are affected by their defeaters and so on), while the arguments which only receive an attack from 
(and in turn those which are attacked by them and so on) do not have any effect on the status of 
. So, with respect to an unconditioned sub-framework, under admissible, complete, preferred and grounded semantics which satisfy the directionality criterion, the mapping from global semantics to local semantics can be formulated by the following definition [1]:
Definition 5.2
Mapping global semantics to local semantics with respect to an unconditioned sub-framework
Let 
be an argumentation framework, and 
be an unconditioned sub-framework of 
. Under a semantics 
,
(5.1)
Example 5.4
Continue Example 5.3. According to Formula 5.1, 
and 
are complete extensions of 
.
Second, for a conditioned sub-framework 
of an argumentation framework 
, for all 
, if 
is an unattacked set, and 
, then 
could be regarded as a conditioning sub-framework of 
. Given 
, let 
. For all 
, according to Formula 5.1, it holds that 
is an extension of 
under semantics 
, and 
is a partially assigned sub-framework of 
. Based on these notions, for a conditioned sub-framework 
, the mapping from global semantics to local semantics can be formulated by the following proposition:
Proposition 5.1
Let 
be an argumentation framework. Let 
and 
be subsets of 
such that 
and 
. Let 
. Under a semantics 
, if and only if 
is an extension of 
.
Since the soundness and completeness of the mapping from global semantics to local semantics for an unconditioned sub-framework (Formula 5.1) have been verified in [1], in this book, we only present the proofs of the soundness and completeness of the mapping from global semantics to local semantics for a conditioned sub-framework.
When proving Proposition 5.1, we found that the soundness of restricting an extension to a conditioned sub-framework under preferred and grounded semantics as well as the completeness of restricting an extension to a conditioned sub-framework under all four semantics depend on the property of the mappings from local semantics to global semantics. So, we present in this section the soundness of restricting an extension to a conditioned sub-framework under admissible and complete semantics, and then the two remaining parts at the end of Section 5.3.2.
Proof
In this part, we prove the soundness of restricting an extension to a conditioned sub-framework under admissible and complete semantics, i.e., for all 
, if 
, then 
is an extension of 
.
• Under admissible semantics, 
is an admissible set. Meanwhile, according to Formula 5.1, 
is admissible. Since 
is admissible, it holds that 
is conflict-free and each argument in 
is acceptable with respect to 
. In order to prove that 
is admissible with respect to 
, since 
is obviously conflict-free, we only need to verify that each argument in 
is acceptable with respect to 
and 
, in which 
is a status assignment of 
with respect to 
, where 
is accepted with respect to 
is rejected with respect to 
and 
is undecided with respect to 
.
Since each argument in 
is acceptable with respect to 
, for all 
is acceptable with respect to 
. Hence, for all 
, if 
then 
such that 
. Since 
and 
is not attacked by the arguments in 
, it holds that 
. It follows that:
• If 
, then: since 
is attacked by the arguments in 
, it holds that 
, and therefore 
(satisfying the second condition of acceptability of arguments in a partially assigned sub-framework, in Definition 4.9).
• Else, if 
, then: since 
is attacked by the arguments in 
or 
, it holds that 
or 
(satisfying the first condition of acceptability of arguments in a partially assigned sub-framework, in Definition 4.9).
As a result, 
is acceptable with respect to 
and 
.
• Under complete semantics, 
is a complete extension. Based on the proof of the previous item, it holds that 
is admissible with respect to 
. We need only to prove that each argument 
in 
that is acceptable with respect to 
and 
is in 
. Since 
is only possibly attacked by the arguments in 
or 
, when 
is acceptable with respect to 
and 
, we have the following two cases:
• If 
is attacked by 
, then according to the first condition of acceptability of arguments in a partially assigned sub-framework (Definition 4.9), 
, such that 
, or 
, such that 
and 
; and
• If 
is attacked by 
, then according to the second condition of acceptability of arguments in a partially assigned sub-framework (Definition 4.9), 
is rejected with respect to 
, i.e., 
, such that 
.
As a result, for any argument 
that attacks 
, there exists an argument in 
that attacks 
. Therefore, 
is acceptable with respect to 
. Since every argument in 
that is acceptable with respect to 
is in 
, it holds that 
is in 
. Since 
, we have 
or 
. If 
, then 
. Else, if 
, then since 
(in that 
), it holds that 
is in 
(otherwise, 
, contradicting 
). 
Example 5.5
Continue Examples 5.3 and 5.4. Since 
is a conditioning sub-framework of 
. According to three complete extensions of 
, there are three partially assigned sub-frameworks of 
(as shown in Figure 5.4):
,
, and
.Then, according to Proposition 5.1 and the above proof, it holds that:
is a complete extension of 
,
is a complete extension of 
, and
is a complete extension of 
.
.5.3 Mapping Local Semantics to Global Semantics
Given an argumentation framework, a mapping from local semantics to global semantics is to combine sets of extensions (labellings) of a set of sub-frameworks to form a set of extensions (respectively, labellings) of the argumentation framework. Since there are two kinds of sub-frameworks (conditioned and unconditioned) and various dependence relations between different sub-frameworks, there are many types of combinations of sub-frameworks, such as a combination of two unconditioned sub-frameworks, a combination of two conditioned sub-frameworks without dependence relation from one to another, a combination of an unconditioned sub-framework and a conditioned sub-framework in which the latter is fully conditioned by the former, and a combination of an unconditioned sub-framework and a conditioned sub-framework in which the latter is partially conditioned by the former, etc. In this book, we study the first three combinations, in terms of the extension-based approach or the labelling-based approach.
5.3.1 Combining Extensions of Two Unconditioned Sub-Frameworks
Let 
and 
be two unconditioned sub-frameworks of an argumentation framework 
. The (syntactic) combination of them, denoted as 
, is also an unconditioned sub-framework.
Example 5.6
Let 
be an argumentation framework (Figure 5.5). Let 
and 
. Since 
and 
and 
are unconditioned sub-frameworks. Let us consider the following two combined sub-frameworks.
, in which 
.
, in which 
.The former is a combination of two unconditioned sub-frameworks that have no arguments in common, while the latter is a combination of two unconditioned sub-frameworks that have some arguments in common. In this book, the former is regarded as a special case of the latter.
.Let 
denote the intersection of 
and 
. Under a semantics 
under which every argumentation framework has at least one extension, the notion of combined extensions is defined as follows.
Definition 5.3
Combined extensions of two unconditioned sub-frameworks
Let 
and 
be two unconditioned sub-frameworks of an argumentation framework 
, and 
. Let 
be a semantics under which every argumentation framework has at least one extension. The set of combined extensions of 
, denoted as 
, is defined as:
Proposition 5.2
Let 
and 
be two unconditioned sub-frameworks of an argumentation framework 
, and 
. For each 
, it holds that: 
.
Proof
(Soundness): For all 
, it holds that 
.
(1) Under admissible semantics, for all 
and 
, if 
then 
. This is because:
– 
is conflict-free. Assume the contrary. Then, 
and 
, such that 
. Since 
and 
, it holds that 
and 
. Since 
, and 
is an unconditioned sub-framework, it holds that 
. Since 
and 
, it holds that 
. Since 
, it follows that 
, and therefore 
. As a result, 
is not conflict-free. Contradiction.
– 
is acceptable with respect to 
. Since 
(
), then since 
and 
is only attacked by the arguments in 
. Since 
is acceptable with respect to 
, if 
, then 
, such that 
. In other words, 
is acceptable with respect to 
.
(2) Under complete semantics, we need to prove that 
, if 
is acceptable with respect to 
, then 
. For 
(
), since 
is acceptable with respect to 
and only affected by the arguments in 
is acceptable with respect to 
; since every argument that is acceptable with respect to 
is in 
, it holds that 
is in 
.
(3) Under preferred semantics, for all 
and 
, since a preferred extension is also a complete extension, based on the proof in the case of complete semantics, we only need to prove that 
is a maximal (with respect to set-inclusion) complete extension of 
. Assume the contrary. Then, 
, such that 
is a preferred extension of 
and 
. Let 
. It holds that 
. Let 
and 
. Since 
, it holds that 
. There are the following two possible cases.
– When 
, since 
, and 
, it holds that 
.
So, 
.
According to Formula 5.1, it holds that 
is a complete extension of 
. As a result, 
is not a preferred extension, contradicting 
.
– When 
, similarly, it turns out that 
is not a preferred extension, contradicting 
.
As a result, 
is a preferred extension of 
.
(4) Under grounded semantics, similar to the proof in the case of preferred semantics, we may verify that 
is the minimal (with respect to set-inclusion) complete extension of 
.
(Completeness): For all 
.
According to Formula 5.1, 
and 
. Meanwhile, since 
and 
, it holds that 
. According to Definition 5.3, 
.
Example 5.7
Continue Example 5.6. Let us consider the sub-frameworks 
and 
. Under preferred semantics, the extensions of 
are 
and 
, while the extensions of 
are 
and 
. Let 
. According to Definition 5.3 and Proposition 5.2, we may combine the extensions of 
and those of 
to obtain the extensions of 
.
is a preferred extension of 
, in that 
, and 
.
is a preferred extension of 
, in that 
, and 
.5.3.2 Combining Extensions of a Conditioned Sub-Framework and Those of an Unconditioned Sub-Framework
Let 
be an argumentation framework. Let 
be a conditioned sub-framework, and 
be an unconditioned sub-framework, of 
. When combining 
and 
, we only consider the case when 
. In this case, we have the following proposition.
Proposition 5.3
Let 
be a conditioned sub-framework and 
be an unconditioned sub-framework, of an argumentation framework 
, such that 
. It holds that:
(5.2)
(5.3)
(5.4)
Proof
First, since 
, it holds that: 
, and 
. Second, since 
is an unattacked set, 
, it follows that 
. Third, since 
and 
is an unattacked set, it holds that 
.
According to Proposition 5.3, the (syntactic) combination of 
and 
is equal to 
, called the combined sub-framework of 
and 
. Semantically, we have the following definition.
Definition 5.4
Combined extensions of a conditioned sub-framework and those of an unconditioned sub-framework
Let 
be an argumentation framework, 
and 
be the sub-frameworks of 
, in which 
. The set of combined extensions of 
, denoted as 
, is defined as follows:
Now, let us first verify the soundness of combining extensions of a conditioned sub-framework and those of an unconditioned sub-framework to form the extensions of a combined sub-framework (the completeness of this kind of combination will be presented at the end of this section).
Proposition 5.4
Let 
be an argumentation framework, 
and 
be sub-frameworks of 
, in which 
. Under a semantics 
, for all 
, it holds that 
.
Proof
• Under admissible semantics, 
and 
: in order to prove that 
, we should prove that: (1) 
is conflict-free, and (2) every argument in 
is acceptable with respect to 
.
(1) 
, there are the following four possible cases:
(a) 
and 
: On the one hand, it holds that 
does not attack 
. Otherwise, assume that 
, then 
is attacked by a conditioning argument that is accepted with respect to 
. According to the second condition of acceptability of arguments in a partially assigned sub-framework, 
is not acceptable with respect to 
and 
, i.e., 
, contradicting 
. On the other hand, 
does not attack 
. Otherwise, 
is attacked by an argument outside 
, contradicting the fact that 
is an unattacked set.
(b) 
and 
: On the one hand, it holds that 
does not attack 
. Otherwise, since 
, there exists 
or 
, such that 
attacks 
. Since 
is an unattacked set, 
. It follow that 
. As a result, 
is attacked by an accepted argument in 
, contradicting 
. On the other hand, since 
is an unattacked set and 
does not attack 
.
(c) 
and 
: On the one hand, it holds that 
does not attack 
(the reason is the same as the one in the previous item). On the other hand, 
does not attack 
. Otherwise, since both 
and 
are in 
, and 
, there exists 
such that 
attacks 
, contradicting 
.
(d) 
and 
: On the one hand, 
does not attack 
. Otherwise, 
is attacked by a conditioning argument that is accepted with respect to 
, and therefore 
. Contradiction. On the other hand, 
does not attack 
(the reason is the same as the one in the previous item).
Since in all possible cases, neither 
attacks 
nor 
attacks 
is conflict-free.
(2) We should prove that 
, if 
, then 
, such that 
.
(a) 
. So, 
is only attacked by the arguments in 
(there exists no argument in 
that attacks 
). Since 
is acceptable with respect to 
, it holds that: 
, if 
, then 
, such that 
.
(b) 
, there are two possible cases: 
is attacked by the arguments in 
with respect to 
, or by the arguments in 
with respect to 
. So, if 
, then it holds that: 
and 
, or 
and 
. Since 
is acceptable with respect to 
and 
, it holds that:
(b1) if 
and 
, then 
is rejected with respect to 
(corresponding to the second condition of acceptability of arguments in a partially assigned sub-framework), i.e., 
, such that 
; and
(b2) if 
and 
, then 
, such that 
, or 
, such that 
is accepted with respect to 
and 
, i.e., 
, such that 
(corresponding to the first condition of acceptability of arguments in a partially assigned sub-framework).
Putting (a) and (b) together, we have: 
, if 
, then 
, such that 
. Therefore, every argument in 
is acceptable with respect to 
.
• Under complete semantics, 
and 
: in order to prove that 
, we should prove that every argument in 
that is acceptable with respect to 
is in 
. Assume the contrary, i.e. 
, such that 
is acceptable with respect to 
, but 
.
(1) If 
, then 
is only attacked by the arguments in 
. Since 
is acceptable with respect to 
, it holds that 
(there exists no argument in 
that attacks 
), if 
, then 
, such that 
, i.e., 
is acceptable with respect to 
. Since 
, it holds that 
. So, 
is not a complete extension, contradicting 
.
(2) If 
, then 
is attacked by the arguments in 
with respect to 
or by the arguments in 
with respect to 
. Since 
is acceptable with respect to 
, it holds that:
(a) 
, if 
, then: since 
is in turn attacked by the arguments in 
with respect to 
or 
with respect to 
, it holds that 
, such that 
, or 
, such that 
(satisfying the first condition of acceptability of arguments in a partially assigned sub-framework); and
(b) 
, if 
, then: since 
is attacked by the arguments in 
, it holds that 
, such that 
, i.e., 
is rejected with respect to 
(satisfying the second condition of acceptability of arguments in a partially assigned sub-framework).
As a result, 
is acceptable with respect to 
and 
. Since 
, it holds that 
. So, 
is not a complete extension, contradicting 
.
According to (1) and (2), we have: 
, if 
is acceptable with respect to 
, then 
. Therefore, every argument in 
that is acceptable with respect to 
is in 
.
• Under preferred semantics, 
and 
: since a preferred extension is also a complete extension, based on the proof in the case of complete semantics, we only need to prove that 
is a maximal (with respect to set-inclusion) complete extension of 
. Assume that 
is not a maximal complete extension. Then, 
, such that 
is a preferred extension of 
and 
. Let 
. If 
, then 
. In this case, 
is a preferred extension. So, we need only to discuss the case when 
. Let 
and 
. It follows that 
.
(1) When 
, since 
and 
, it holds that 
. According to Formula 5.1, it holds that 
. Therefore, 
is not a preferred extension, contradicting 
.
(2) When 
, if 
, then 
. In this case, since 
and 
, it holds that 
. Since a preferred extension is also a complete extension, according to Proposition 5.1 and the proof of this proposition under complete semantics (in Section 5.2), it holds that 
. So, 
is not a preferred extension, contradicting 
.
As a result, we may conclude that 
is a maximal complete extension (i.e., preferred extension).
• Under grounded semantics, 
and 
: since a grounded extension is also a complete extension, based on the proof in the case of complete semantics, we only need to prove that 
is the minimal (with respect to set-inclusion) complete extension of 
. The proof is similar to the one under preferred semantics (omitted).
Example 5.8
Continue Example 5.6. Let 
and 
. It follow that 
. Under complete semantics,
, in which 
and 
;
, in which 
;
, in which 
;
, in which 
.According to Definition 5.4, we have 
, in which
,
, and
.Based on Proposition 5.4, firstly, let us now prove the two remaining parts of Proposition 5.1 mentioned in Section 5.2.
Proof
• In the first part, we prove the soundness of restricting an extension to a conditioned sub-framework under preferred and grounded semantics, i.e., for all 
, if 
, then 
is an extension of 
.
According to the proof in Section 5.2, 
, in which 
. Based on this result, we have the following proof.
(1) Under preferred semantics, we only need to verify that 
is maximal. Assume the contrary. Then, 
, such that 
. It follows that 
. According to Formula 5.1, 
is a preferred extension of 
, and 
is a preferred extension of 
. Since 
and 
, according to Proposition 5.4, 
is a preferred extension of 
. As a result, 
is not a preferred extension of 
. Contradiction.
(2) Under grounded semantics, we only need to verify that 
is minimal. The proof is similar to the previous item. Omitted.
• In the second part, we prove the completeness of restricting an extension to a conditioned sub-framework under all four semantics, i.e., for all 
, if 
is an extension of 
, then 
such that 
. According to Proposition 5.4, 
is an extension of 
. Let 
. It holds that 
. For all 
. Let 
. It holds that 
.
According to the soundness and completeness of restricting an extension to a conditioned sub-framework, with respect to a conditioned sub-framework, the mapping from global semantics to local semantics can also be formulated by the following definition:
Definition 5.5
Mapping global semantics to local semantics with respect to a conditioned sub-framework
Let 
be an argumentation framework, and 
and 
be sub-frameworks of 
. Under a semantics 
, for all 
, it holds that
(5.5)
Example 5.9
Let us consider an argumentation framework 
as shown in Figure 5.6. Let 
and 
. It follows that 
. Under preferred semantics, 
has the following four extensions:
,
,
, and
.It holds that 
, and 
. According to Formula 5.5, we have
, in which
,
, and
.
, in which 
.
.Finally, according to the soundness of restricting an extension to a sub-framework (conditioned or unconditioned), we may verify the completeness of combining extensions of a conditioned sub-framework and those of an unconditioned sub-framework to form the extensions of a combined sub-framework.
Proposition 5.5
Let 
be an argumentation framework, 
and 
be sub-frameworks of 
, in which 
. Under a semantics 
, for all 
, it holds that 
.
Proof
Under a given semantics 
, for all 
, let 
and 
. It holds that 
. According to Formulas 5.1 and 5.5, it holds that 
and 
. According to Definition 5.4, it holds that 
.
5.3.3 Combining Labellings of Two Conditioned Sub-Frameworks
In the previous two subsections, by using the extension-based approach, we have presented the semantics combination of two unconditioned sub-frameworks, and of a conditioned sub-framework and an unconditioned sub-framework. In this subsection, we will formulate the semantics combination of two conditioned sub-frameworks, in terms of labelling-based approach.
Let 
be an argumentation framework, and 
and 
be two conditioned sub-frameworks. The (syntactic) combination of the two sub-frameworks is 
.
There are some possible relations between 
and 
and 
, and 
and 
. In this book, we only consider the case where 
and 
.
Let 
. According to the following proposition, 
is equal to 
.
and 
be conditioned sub-frameworks of 
, such that 
and 
. Let 
. It holds that:
,
, and
.Proof
(1) On the one hand, for all 
, there exists 
such that 
. Since 
, we have the following two possible cases. When 
, since 
. It follows that 
. Similarly, when 
, since 
. Hence, 
. On the other hand, for all 
or 
. When 
, there exists 
such that 
. Since 
. It follows that 
and 
. So, 
. Similarly, when 
, it holds that 
.
(2) Since 
and 
, it holds that there are no interactions between 
and 
, i.e., 
and 
. So, 
.
(3) Since 
and 
.
Now, let us define the combination of the labellings of two conditioned sub-frameworks. Let 
and 
be two labellings. The combination of 
and 
is denoted as:
(5.6)
Based on this notion, we have the following definition.
Definition 5.6
Combined labellings of two conditioned sub-frameworks
Let 
and 
be conditioned sub-frameworks of 
, such that 
and 
. Let 
. So, 
is a combined framework of 
and 
. Let 
and 
be two unconditioned sub-frameworks of 
, such that 
and 
. Let 
be a semantics, under which every argumentation framework has at least one labelling. For all 
, let 
.
The soundness and completeness of combining labellings of two conditioned sub-frameworks are formulated by the following proposition.
Proposition 5.7
Based on Definition 5.6, it holds that 
.
Proof
(Soundness): For all 
and 
, it holds that 
.
(1) Under admissible semantics, 
and 
are admissible labellings. According to Propositions 5.2 and 2.1, 
is an admissible labelling. For all 
and 
, we need to verify that 
is an admissible labelling of 
. According to Definition 4.6, if for 
, each argument 
in 
that is labelled 
in 
is legally 
in 
with regard to 
, then 
is an admissible labelling with respect to 
.
Since 
, and 
, it holds that 
(
is either 1 or 2). Since 
is labelled 
in 
and 
is labelled 
in 
. Since 
is an admissible labelling, it holds that 
is legally 
in 
with regard to 
. Since 
is only possibly attacked by the arguments in 
and 
, it holds that 
is legally 
in 
with regard to 
.
(2) Under complete semantics, 
and 
are complete labellings. According to Propositions 5.2 and 2.2, 
is a complete labelling. For all 
and 
, we need to verify that 
is a complete labelling of 
. Since a complete labelling is an admissible labelling, according to Definition 4.6, we only need to verify that each argument 
in 
that is labelled UNDEC in 
is legally UNDEC in 
with regard to 
. Since 
(
is either 1 or 2) and 
is labelled UNDEC in 
is legally UNDEC in 
with regard to 
(
or 2). Since 
is only possibly attacked by the arguments in 
and 
, it holds that 
is legally UNDEC in 
with regard to 
.
(3) Under preferred semantics, 
and 
are preferred labellings. According to Propositions 5.2 and 2.2, 
is a preferred labelling. For all 
and 
, we need to verify that 
is a preferred labelling of 
. Since a preferred labelling is also a complete labelling, based on the proof in the case of complete semantics, we only need to prove that 
is maximal.
Assume the contrary. Then, there exists another complete labelling 
of 
, such that 
. Let 
. It holds that 
.
Let 
and 
. Since 
, it holds that 
or 
. Without loss of generality, let 
. Since 
, and 
, it holds that 
. So, 
.
According to Formula 5.5 and Proposition 2.2, we may infer that 
is a complete labelling of 
. Since 
is not a preferred labelling, contradicting 
.
(4) Under grounded semantics, 
and 
are grounded labellings. For all 
and 
, we need to verify that 
is the grounded labelling of 
. Since a grounded labelling is also a complete labelling, based on the proof in the case of complete semantics, we only need to prove that 
is minimal. The proof is similar to the one in the case of preferred semantics (omitted).
(Completeness): For every labelling 
, it holds that 
is a combined labelling of 
, i.e., 
.
Let 
, for all 
. According to Formula 5.5 and Proposition 2.2, we may infer that 
. Since 
, according to Definition 5.6, it holds that 
.
5.4 Conclusions
This chapter has presented the relations between global semantics and local semantics of an argumentation framework, which lays a foundation for efficient computation of argumentation semantics. The notions related to the global semantics and local semantics also appeared in [2,3] and [4]. In addition, this notion may be extended to the context of multiple argumentation systems. In [5], the authors proposed a notion of merging (different) argumentation systems. Given a set of distinct argumentation frameworks from different agents, they are expanded respectively into partial systems over the set of all arguments considered by the group of agents. Then, a merging operator is used to produce a set of argumentation systems that are as close as possible to the partial systems (to realize a kind of consensus). And then, the acceptability of a set of arguments at the group level is obtained by selecting the extensions of a set of produced (merged) argumentation frameworks at the local level.
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