1 Introduction
In recent years, explainable AI
(XAI) has received much attention, mostly directed at new techniques for explaining decisions of (subsymbolic) machine learning algorithms
[13]. However, explanations traditionally also play an important role in (symbolic) knowledgebased systems [8]. Computational argumentation is one research area in symbolic AI that is frequently mentioned in relation to XAI. For example, arguments can be used to provide reasons for or against decisions [8, 1, 12]. The focus can also be on the argumentation itself, where it is explained whether and why a certain argument or claim can be accepted under certain semantics for computational argumentation [5, 6, 7, 14]. It is the latter type of explanations that is the subject of this paper.Abstract argumentation frameworks, as introduced in [4], consist of sets of arguments (abstract entities) and an attack relation between them. To determine the conclusions of a framework their corresponding extensions – sets of arguments that can collectively be considered as acceptable – are calculated under different semantics [4]. Many of the wellknown and most studied semantics are based on the notion of defense: an argument is defended by a set of arguments if that set attacks all its attackers.
In this paper we investigate explanations for argumentationbased conclusions (i.e., why an argument is (not) part of an or all extensions), by applying a basic framework for explanations on top of abstract argumentation frameworks which can be evaluated by any extensionbased semantics. The explanations are defined in terms of sets of relevant arguments that are part of extensions and explain the (non)acceptance in terms of defense. We consider an argument relevant for another argument if the arguments are connected by means of the attack relation. By requiring relevance of an explanation, arguments that do not (in)directly attack or defend the considered argument (and therefore do not influence the acceptability of that argument) will not be part of the explanation. Since defense is a central notion in many Dungstyle semantics, the explanations thus defined can be applied to all the common semantics (e.g., complete, grounded, preferred).
One of the important characteristics of explanations provided by humans is that they select the explanation from a possible infinite set of explanations, using criteria such as simplicity, necessity and sufficiency [12]. In this paper we look at how to select minimal^{1}^{1}1Interpreting [12] simplicity as minimality, necessary and sufficient explanations for the (non)acceptance of an argument.
After introducing some preliminary notions concerning (defense) in abstract argumentation frameworks, we provide a basic framework with which explanations for both acceptance and nonacceptance of an argument can be provided, given any existing extensionbased semantics that is based on the notion of defense. We show that these explanations are wellbehaved with respect to different Dungstyle semantics (e.g. explanations under grounded semantics are never supersets of explanations under preferred semantics), and discuss notions of minimality introduced in [5] applied to explanations in our framework.
We continue by discussing the notions of necessity and sufficiency, introducing sufficient and necessary explanations for (non)acceptance. For these explanations we show when they exists and how the explanations provided by the basic framework are related to necessary and sufficient explanations. Furthermore, we show how our notions of necessity and sufficiency relate to the notions of minimality from [5]. We conclude with discussing related and future work.
2 Preliminaries
An abstract argumentation framework (AF) [4] is a pair , where is a set of arguments and is an attack relation on these arguments. Such a framework can be viewed as a directed graph, in which the nodes represent arguments and the arrows represent attacks between arguments, see e.g., Figure 1.
Example 1.
Figure 1 represents where and .
Given an AF, Dungstyle semantics [4] can be applied to it, to determine what combinations of arguments (called extensions) can collectively be accepted.
Definition 1.
Let be an AF, a set of arguments and let . Then attacks if there is an such that ; defends if attacks every attacker of ; is conflictfree if there are no such that ; and is admissible () if it is conflictfree and it defends all of its elements.
An admissible set that contains all the arguments that it defends is a complete extension () of . The grounded extension () is the minimal (with respect to ) complete extension. A preferred extension () is a maximal (with respect to ) complete extension. A stable extension () is a complete extension that attacks every argument not in it. denotes the set of all the extensions of under the semantics .
Where is an AF, a semantics and , it is said that is skeptically [resp. credulously] accepted if [resp. ]. These acceptability strategies are denoted by [resp. ]. is said to be credulously [resp. skeptically] nonaccepted in if for all [resp. some] , . We will say that an argument is accepted [resp. nonaccepted] if the strategy is arbitrary or clear from the context.
The notions of attack and defense can also be defined between arguments and can be generalized to indirect versions:
Definition 2.
Let be an AF, and for some . Then defends if: there is some such that and , in this case directly defends ; or defends and defends , in this case indirectly defends . It is said that defends in if defends and .
Similarly, attacks if: , in this case directly attacks ; or attacks some and defends , in this case indirectly attacks .
We will require that an explanation for an argument is relevant, in order to prevent that explanations contain arguments that do not influence the acceptance of .
Definition 3.
Let and . It is said that is relevant for if (in)directly attacks or defends and it does not attack itself. A set is relevant for if all of its arguments are relevant for .
Example 2.
In (from Figure 1) and attack each other and both defend themselves from this attack. The grounded extension is and for . None of the arguments from is skeptically accepted, while all of the arguments in are credulously accepted for . The argument defends directly and indirectly and, similarly, it attacks directly and indirectly. The arguments and are relevant for and but not relevant for each other.
In order to define relevant explanations, we focus on the notion of defense. Note that many of the wellknown Dungstyle semantics result in defended sets of arguments. We define two notions that will be used in the basic definitions of explanations. The first, used for acceptance explanations, denotes the set of arguments that defend the argument , while the last, used for nonacceptance explanations, denotes the set of arguments that attack and for which there is no defense in the given extension.
Definition 4.
Let be an AF, and an extension for some semantics .

;

denotes the set of arguments that (in)directly defend in ;

, denotes the set of all (in)direct attackers of for which no defense exists from .
Note that by definition explanations in terms of or result in relevant explanations.
Example 3 (Example 2 continued).
Consider again the AF . Then: , , while and .
We introduce the following notation to keep the notation of the explanations general and short.
Notation 1.
Let be an AF, and . Then, for some :

denotes the set of extensions of of which is a member;

denotes the set of extensions of of which is not a member;
Now the basic framework for explanations in formal argumentation can be defined.
3 Basic Explanations
In this section we present the basic framework with which a variety of explanations can be provided. Explanations are defined in terms of a function , which determines how “far away” we should look when considering attacking and defending arguments as explanations. can be adjusted according to, for example, the application. Examples of adjustments are shown in Sections 5.1 and 5.2.
3.1 Basic Explanations for Acceptance
We define two types of acceptance explanations: explanations provide all the reasons why an argument can be accepted by a skeptical reasoner, while explanations provide one reason why an argument can be accepted by a credulous reasoner. For the purpose of this section let . This means that the presented explanations will be based on the set of arguments that defend the requested argument.
Definition 5 (Argument explanation).
Let be an AF and let be an argument that is accepted, given some and an acceptance strategy ( or ). Then:
Let .
provides for each extension the arguments that defend in , and contains the arguments that defend in one of the extensions.
Remark 1.
A nonempty set for an attacked argument does not guarantee that is defended against all its attackers. In the AF we have that but (since it is not conflictfree with nor with ). In the above definition of explanations this issue does not occur, since there it is required that the argument is part of the extension and thus, by definition of the semantics, it is defended against all its attackers by that extension.
The next proposition shows that the acceptance explanations, under , are wellbehaved when compared to Dungstyle semantics. For example it shows that explanation under grounded semantics results in smaller and more skeptical explanations than under preferred semantics.
Proposition 1.
Let be an argumentation framework, let and let , then:

For all and for : .

.

.

For each there is an such that
Proof.
Let be an argumentation framework, let be accepted and let .

Let , then (in)directly defends and , therefore . It follows that for all . Hence as well.

Let , then there is some such that . Since it follows that for as well. Therefore and .

Let , then for some . Therefore and as well.

Let , then there is some such that . By Definition 1 there is some , let . Note that and that . ∎
3.2 Basic Explanations for NonAcceptance
Understanding why something is not accepted might sometimes be just as important as understanding why something is accepted. We therefore discuss basic definitions for explanations of nonaccepted arguments. Since we focus on the notion of defense and admissibilitybased semantics, an argument is not accepted if it is attacked and there is no defense for this attack by an accepted argument. In this section, let .
Definition 6 (Nonacceptance explanation).
Let be an AF and let be an argument that is not accepted, given some and some .
Intuitively, a nonacceptance explanation contains all the arguments in that attack and for which no defense exists in: some extensions (for ) of which is not a member; all extensions (for ). That for only some extensions have to be considered follows since is skeptically nonaccepted as soon as , while is credulously nonaccepted when .
Example 5.
The next proposition is the nonacceptance counterpart of Proposition 1 and shows how nonacceptance explanations are related to each other under different semantics.
Proposition 2.
Let be an AF, let be nonaccepted, and let . Then:

.

.
Proof.
Let be an argumentation framework, let be nonaccepted and let .

Suppose that is not accepted w.r.t. and , then for some , . Note that . Therefore, for all , and . Hence .
The case for is similar and left to the reader. Note that, by assumption, is not accepted w.r.t. and , otherwise is not defined. ∎
Proposition 1 together with the above result shows that the choice of the semantics influences the size of the explanation in a similar way as the choice of semantics influences the size and number of extensions themselves. This is useful to know, since it shows that the explanations behave in a predictable way and that semantics can be chosen as usual. In the next section we will look at minimal explanations.
4 Minimality
As mentioned in the introduction, humans select the explanation from all the possible explanations, using criteria such as simplicity, necessity and sufficiency [12]. One way to look at simplicity is minimality. In [5] two notions of minimality were introduced (as well as two notions of maximality, but we are in this paper only interested in minimality): minimality (i.e., minimality w.r.t. )^{2}^{2}2Where minimality w.r.t. is applied to the size of sets: denotes and compactness (i.e., minimality w.r.t. ). In our setting we can formulate different minimal explanations for (non)acceptance as follows, where .
Example 6 (Examples 4 and 5 continued).
Recall that, for , . Of these possible explanations both and are minimal, but only is minimal. Similarly, we had that , there are two minimal explanations: and but only is also a minimal explanation.
These notions of minimality are already useful in restricting the size of an explanation. As we have seen in the example above, if . It is therefore no longer the case that could be when considering minimal explanations. However, these notions do not say anything about necessity, sufficiency or even relevance. In the next section we therefore look further into restricting the size of explanations, this time based on relevance, sufficiency and necessity.
5 Necessity and Sufficiency
Necessity and sufficiency in the context of philosophy and cognitive science are discussed in, for example, [10, 11, 15]. Intuitively, an event is sufficient for if no other causes are required for to happen, while is necessary for , if in order for to happen, has to happen as well. In the context of logical implication (denoted by ), one could model sufficiency by and necessity by [9]. In the next sections we formulate these logical notions in our argumentation setting.
5.1 Necessity and Sufficiency for Acceptance
In the context of argumentation, where explanations are sets of arguments, a set of accepted arguments is sufficient if it guarantees, independent of the status of other arguments, that the considered argument is accepted, while an accepted argument is necessary if it is impossible to accept the considered argument without it.
Definition 7.
Let be an AF and let be accepted (w.r.t. and or ). Then:

is sufficient for the acceptance of if is relevant for , is conflictfree and defends against all its attackers;

is necessary for the acceptance of if is relevant for and if for some , then .
Example 7 (Example 6 continued).
In both and are sufficient for the acceptance of but neither is necessary, while for , is both sufficient and necessary.
In order to use the above notions as variations of , we introduce:^{3}^{3}3 and are defined for an argument and the empty set. We do so because requires an argument and an extension.

denotes the set of all sufficient sets of arguments for the acceptance of ;

denotes the set of all arguments that are necessary for the acceptance of .
Remark 2.
When , is the same for any semantics . This is the case since the definition of sufficiency and necessity is not defined w.r.t. .
Example 8 (Example 7 continued).
For we have that for and for . This means that the credulous acceptance of can be explained by the existence of the arguments and , which are both sufficient and necessary for to be accepted.
Next we show some useful properties of sufficient and necessary (sets of) arguments for acceptance. In particular, we show that the sets in are admissible and contain all the needed arguments. We further look at conditions under which and are empty, as well as the relation between and .
Proposition 3.
Let be an AF and let be accepted w.r.t. some and . Then:

For all , ;

iff there is no such that ;

iff there is no such that or .

.
Proof.
Let be an AF and let be accepted w.r.t. some and .

Let . Note that, by definition, is conflictfree. If there is some such that then there is some that defends against this attack (i.e., ), a contradiction. If, there is some such that , then indirectly attacks itself. Since there is no such that it follows that is not defended against the attack from . A contradiction with the definition of that it defends against all attackers. Hence is conflictfree.
Now suppose that there is some and some such that . Since (in)directly defends , indirectly attacks . By definition of a sufficient set of arguments defends against . It follows that there is some such that . Hence defends and all its own elements against any attacker. Therefore .

Suppose that , then there is no such that is relevant for and defends against all its arguments. Since is accepted by assumption, it follows that is not attacked at all. Now suppose that there is no such that . Then there is no that is relevant for and hence .

First suppose that . Then there is no argument relevant for (from which it follows that there is no such that ) or there is no such that . Note that for each there is some such that . Since it follows that as well.
For the other direction suppose first that is not attacked at all, then there is no argument relevant for from which it follows that . Now suppose that . By assumption is accepted and is attacked, hence . It follows that for each and for each there is an such that and therefore also an with but . Therefore none of the arguments is necessary: .

In view of the above two items, suppose that is attacked by some argument. Let and suppose that . Then there is some such that . Note that . However, , a contradiction with .∎
The next proposition relates the introduced notions of necessity and sufficiency with . This shows that, although is only one of many options for , it is closely related with these selection criteria and therefore a useful notion from which to start the investigation into explanations within the basic framework.
Proposition 4.
Let be an AF and let be accepted w.r.t. and . Then:

for all , ;

.
Proof.
Let be an AF and let be an argument that is accepted w.r.t. and . Consider both items.

Since is accepted, there is some such that . Let . By definition, is relevant for (i.e., all (in)directly defend and since , ). Now suppose that there is some such that attacks and is not defended by . By assumption . Hence there is a such that . But then (in)directly defends and therefore . Thus defends against all its attackers and therefore .

Let , since is accepted, . Suppose there is some which is not necessary for the acceptance of . Then there is an such that . However, by definition of , . Hence is necessary for . To see that contains all the necessary arguments, assume it does not. Then there is some such that but is necessary for the acceptance of . However, since , there is some such that , but . A contradiction. ∎
5.2 Necessity and Sufficiency for NonAcceptance
When looking at the nonacceptance of an argument , the acceptance of any of its direct attackers is a sufficient explanation. However, other arguments (e.g., some of the indirect attackers) might be sufficient as well. An argument is necessary for the nonacceptance of , when it is relevant and is accepted in the argumentation framework without it. In what follows we will assume that , since otherwise itself is the reason for its nonacceptance.
In order to define sufficiency for nonacceptance we need the following definition.
Definition 8.
Let be an AF and let such that indirectly attacks , via , i.e., . It is said that the attack from on is uncontested if there is no such that for . It is contested otherwise, in which case it is said that the attack from is contested in and that is the contested argument.
This definition is needed since the acceptance of an indirect attacker might already be sufficient for the nonacceptance of an argument, but not every indirect attacker is sufficient for nonacceptance. See also the next example.
Example 9 (Example 8 continued).
For from Example 1 we have that the indirect attack from on is uncontested. This follows since is not attacked and hence, when is accepted, so is . Therefore, both and can be seen as sufficient for the nonacceptance of . However, the attacks from and on are contested in . For this follows since it defends , but is attacked by and, similarly, defends , but is attacked by . Hence, although and indirectly attack , by just accepting one, is not necessarily nonaccepted, therefore neither would be sufficient on its own to make nonaccepted.
For the definition of necessary for nonacceptance we define subframeworks, which are needed since an argument might be nonaccepted since it is attacked by an accepted or by another nonaccepted argument.^{4}^{4}4In terms of labeling semantics (see e.g., [2]) an argument is nonaccepted if it is out (i.e., attacked by an in argument) or undecided.
Definition 9.
Let be an AF and let . Then denotes the AF based on but without .
Since indirect attacks might be sufficient for not accepting an argument, but they also might be contested, the definition of sufficiency for nonacceptance is defined inductively.
Definition 10.
Let be an AF and let be nonaccepted (w.r.t. and or ). Then:

is sufficient for the nonacceptance of if is relevant for and there is a such that:

; or

indirectly attacks and that attack is uncontested; or

indirectly attacks and for every argument in which the attack from on is contested and every such that , there is an that is sufficient for the nonacceptance of .


is necessary for the nonacceptance of if is relevant for and is accepted w.r.t. and resp. in .
Example 10 (Example 9 continued).
For from Example 1 we have that is both necessary and sufficient for the nonacceptance of . Moreover, while and are neither sufficient for the nonacceptance of , is. For the nonacceptance of we have that , and are sufficient, but none of these is necessary.
We define the following notation, to use the above notions as variations of :

, denotes the set of sets of arguments that, when accepted, cause to be nonaccepted;

, denotes the set of all arguments that are necessary for not to be accepted.
Example 11 (Example 10 continued).
For we have that ^{5}^{5}5The explanation could contain other arguments as well (e.g., the explanation could be ). This is the case since is not assumed to be minimal. for while for . This means that the skeptical nonacceptance of can be explained by the existence of the arguments , and , which are all sufficient for the nonacceptance of but none of them is necessary.
The next propositions are the nonacceptance counterparts of Propositions 3 and 4. First some basic properties of sufficiency and necessity for nonacceptance.
Proposition 5.
Let be an AF and let be nonaccepted w.r.t. and . Then:

;

implies that there are at least two direct attackers of .
Proof.
Let be an AF and let be an argument that is not accepted w.r.t. and .

Suppose that . Then there is no that is relevant for and in which (in)directly attacks . It follows that there is no such that . A contradiction with the assumption that is nonaccepted and that .

It follows that there are such that . Assume that but that . Since by assumption in this section , it follows that is not attacked in and should therefore be accepted in any complete extension. Hence . ∎
Now we show how is related to the here introduced notions of sufficiency and necessity for nonacceptance. For this we first need the following lemma:^{6}^{6}6For this lemma was shown in [3].
Lemma 1.
Let , for some and . If there is a such that , then .
Proof.
Let , for some and such that and . Note that is still admissible in since no new attacks are added.
. Now suppose there is some such that but is defended by in . If is not attacked at all in , since , , but then defends in , a contradiction. Hence there is some such that and defends against this attack in , but then would defend in as well. Again a contradiction. Hence is complete in and if was maximally complete in it is still maximally complete in .
. Any argument, other than , attacked by is still attacked by in . Since is still complete, it follows that is also still stable. ∎
Proposition 6.
Let be an AF and let be an argument that is not accepted w.r.t. and . Then:

for all such that , ;

.
Proof.
Let be an AF and let be nonaccepted w.r.t. and . Consider both items:

By definition of , is relevant for . We show that there is a such that . Suppose there is no such , then is not attacked at all or defends against all its direct attackers and therefore against all its attackers, both are a contradiction with the completeness of . Hence there is such a . From which it follows that .

Let and suppose that . Then there is some such that . By assumption, is relevant for and thus (in)directly attacks . From which it follows that there is some such that . By Lemma 1, , a contradiction with the assumption that . ∎
5.3 Necessity, Sufficiency and Minimality
In order to compare the introduced notions of necessity and sufficiency with the notions of minimality known from [5] and recalled in Section 4, we define minimal sufficient sets, where :

, denotes the set of all minimally sufficient sets for the acceptance of .

, denotes the set of all minimally sufficient sets for the nonacceptance of .
Although the notions of minimality are aimed at reducing the size of an explanation, by applying instead the notions of sufficiency and necessity as introduced in this paper, the size of the explanation can be further reduced. To see this, consider the following example:
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