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.
directionality of argumentation; global semantics; local semantics; mappings; semantics combination
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.
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.
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.
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.
Notice that when and .
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]:
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:
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.
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.
Let and be two unconditioned sub-frameworks of an argumentation framework . The (syntactic) combination of them, denoted as , is also an unconditioned sub-framework.
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.
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.
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.
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).
Based on Proposition 5.4, firstly, let us now prove the two remaining parts of Proposition 5.1 mentioned in Section 5.2.
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:
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.
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 .
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.
The soundness and completeness of combining labellings of two conditioned sub-frameworks are formulated by the following proposition.
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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