1 Introduction
Explainable AI (XAI) has become an important research direction in AI [24]
. AI systems are being applied in a variety of reallife situations in different domains and with different users. It is therefore essential that such systems are able to give explanations that provide insight into the underlying decision models and techniques, so that users can understand, trust and validate the system, and experts can verify that the system works as intended. Most of the research in XAI is directed at explaining decisions of subsymbolic machine learning algorithms (cf.
[25]), but explanations also play an important role in clarifying the decisions of symbolic algorithms [15], particularly as such algorithms are allpervasive in everyday systems.One area in symbolic AI that has seen a number of realworld applications is formal argumentation [1]. Two central concepts in formal argumentation are abstract argumentation frameworks [10], sets of arguments and the attack relations between them, and structured or logical argumentation frameworks [4], where arguments are constructed from a knowledge base and a set of rules, and the attack relation is based on the individual elements in the arguments. Common for argumentation frameworks, abstract and structured, is that we can determine their extensions, sets of arguments that can collectively be considered as acceptable, under different semantics [10]. The combination of an argumentation framework and its extensions is a global explanation: what can we conclude from the model as a whole? However, often we would prefer simpler, more compact explanations for the acceptability of an individual argument, a local explanation for a particular decision or conclusion [11]). A number of methods for determining local explanations for the (non)acceptability of arguments have been proposed [7, 12, 13, 14, 16, 26]. What is still lacking, however, is an argumentationbased interpretation of contrastive explanations.
Contrastiveness is central to local explanations [17, 18, 19]: when people ask ‘Why P?’, they often mean ‘Why P rather than Q?’ – here is called the fact and is called the foil [17]. The answer to the question is then to explain as many of the differences between fact and foil as possible. Like for XAI in general, much of the research on contrastive explanations is done in the context of machine learning (e.g. [9, 27, 28]). In the literature on formal argumentation, there has been no such work, the existing work focusing on ‘Why is argument A (not) acceptable?’ instead of the contrastive question ‘Why is argument A acceptable and argument B not?’ (or vice versa). While there are other forms of contrastive questions, we choose this one since it is intuitive, allows for a variety of foils and it can be interesting for both expert and lay users of an argumentationbased application.
To study contrastive explanations for argumentationbased conclusions, we extend the basic framework from [7]. With that framework, explanations for accepted and nonaccepted arguments or formulas can be formulated in a variety of ways, e.g., by only returning necessary or sufficient arguments [8]. The main idea of the introduced contrastive explanations is that these return the common elements of the basic acceptance explanation of the fact and the basic nonacceptance explanation of the foil (or vice versa). By taking into account the foil, the explanation is tailored towards a specific explanation direction and other possible explanations that are not relevant for this direction are not included. We show that in almost all situations these explanations are meaningful, i.e., that such common elements exist. Moreover, in the context of a reallife application at the Dutch National Police, we show that contrastive explanations can be smaller than the basic explanations, by requesting an explanation for a specific exception to a rule rather than all possible exceptions. Additionally we show that, due to the explicit notion of conflict within argumentation, we can provide contrastive explanations when the foil is not explicitly known. This is an advantage of formal argumentation, since determining the foil is a challenge for an AI system.
The paper is structured as follows: we briefly discuss some directly related work and then recall abstract argumentation as introduced in [10]. Then, in Section 4, the framework from [7] is recalled and some new results for acceptance and nonacceptance explanations are shown. In Section 5 contrastive explanations are introduced and it is shown how, in formal argumentation, the foil can be determined when it is not explicitly stated. In Section 6 the introduced contrastive explanations are applied to ASPIC [23] and in Section 7 a reallife application is discussed. We conclude in Section 8.
2 Related Work
XAI has been investigated in many directions, for a variety of approaches to AI, including formal argumentation. As mentioned in the introduction, we are interested in contrastive local explanations for conclusions derived from formal argumentation, where the idea is that the proposed method can be applied to any Dungstyle argumentation framework to generate contrastive explanations. While contrastive explanations for learningbased decisions have been investigated extensively (see [27] for a recent overview), there are no results on contrastive explanations for argumentationbased conclusions.
In the direction of local explanations for argumentationbased conclusions some research already exists. For example, [14] introduce explanations for claims as triples of sets of dialectical trees for abstract argumentation and DeLP and Fan and Toni introduced explanations as dispute trees for accepted arguments in abstract argumentation and ABA in [12] and for nonaccepted arguments in abstract argumentation in [13]. Even more recently, explanation semantics, where accepted arguments are labeled with sets of explanation arguments, were introduced in [16] and explanations for nonaccepted arguments as minimal subframeworks are studied in [20, 26].
For this paper, we take the framework from [7], as it is the only one that allows for acceptance and nonacceptance explanations in terms of sets of arguments. While acceptance and nonacceptance is necessary when defining contrastive explanations (see Section 5), explanations in terms of sets of arguments make it easier to process the explanations. Additionally, unlike the other frameworks, the explanations from [7] make it possible to present explanations derived from a structured setting in terms of elements of arguments (e.g., premises or rules), rather than full arguments. Therefore, to the best of our knowledge, this is the first research on contrastive local explanations for conclusions derived from either abstract or structured formal argumentation.
3 Preliminaries
In this paper we focus on explanations for conclusions that can be derived from Dungstyle argumentation frameworks. Due to space restrictions this section is very compact, see, e.g., [10] for a more gentle introduction.
An abstract argumentation framework (AF) [10] is a pair , where is a set of arguments and is an attack relation on these arguments. An argumentation framework can be viewed as a directed graph, in which the nodes represent arguments and the arrows represent attacks between arguments.
Example 1.
Figure 1 represents where and .
Dungstyle semantics [10] can be applied to an AF, to determine what combinations of arguments (called extensions) can collectively be accepted.
Definition 1.
For , 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. We denote by the set of all arguments attacked by .
An admissible set that contains all the arguments that it defends is a complete extension (). The grounded extension () of is the minimal (with respect to ) complete extension; a preferred extension () of is a maximal (with respect to ) complete extension; and a semistable extension () of is a complete extension where is maximal. An extension will be denoted by and denotes the set of all the extensions of under the semantics .
In what follows, given an argumentation framework , we will denote by the set of all extensions of of which is a member and by the set of all extensions of of which is not a member.
Definition 2.
Let be an argumentation framework, and . It is said that, for , is, w.r.t. :

skeptically accepted iff ;

credulously accepted iff ;

skeptically nonaccepted iff ;

credulously nonaccepted iff .
We will denote skeptical [resp. credulous] (non)acceptance by [resp. ] and when or is clear from the context or not relevant simply write accepted and nonaccepted.
The notions of attack and defense can also be defined between arguments and can be generalized to indirect versions: for : 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 does not attack itself. A set is relevant for if all of its arguments are relevant for . A relevant argument for is conflictrelevant for if (in)directly attacks and it is defendingrelevant for if (in)directly defends .
Example 2.
In and attack each other and both defend themselves. Example conflictfree sets are and . There are four preferred and semistable extensions: , , and and is the grounded extension.
No argument is skeptically accepted and all arguments are credulously accepted and skeptically nonaccepted for . Argument defends itself and directly, it attacks directly and indirectly, it is conflictrelevant for and and defendingrelevant for and .
4 The Basic Framework
In this section we recall the basic framework of explanations from [7] and present some new results. The explanations in that paper are defined in terms of two functions: , which determines the arguments that are in the explanation and , which determines what elements of these arguments the explanation presents. To avoid clutter, we instantiate immediately with the following functions, while instantiations of are discussed in Section 6:^{1}^{1}1We refer the interested reader to [7] for suggestions of other variations of these functions.
Definition 4.
Let be an AF, and for some semantics . Then:

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

, denotes the set of all (in)direct attackers of that are not defended by .
All three functions (i.e., , and ) result in relevant sets of arguments for . In particular, all arguments in and are defendingrelevant and all arguments in are conflictrelevant.
4.1 Acceptance Explanations
Let be an AF and let . If is accepted w.r.t. a semantics and an acceptance strategy then an acceptance explanation can be requested. The explanation depends on the acceptance strategy: for a skeptical reasoner the explanation has to account for the acceptance of the argument in each extension, while for a credulous reasoner explaining the acceptance of the argument in one extension is sufficient.
Definition 5 (Argument acceptance explanation).
Let be an AF, let be an argument that is accepted given some semantics and an acceptance strategy ( or ). Then:


.
The explanation returns all the arguments that defend in at least one of the extensions, while the explanation is a set of arguments that defend in one extension.
Example 4 (Example 3 continued).
In we have that: ; and .
Next we show some properties of the acceptance explanations. Proposition 1 shows that the defending arguments of an argument also defend the arguments defended by , while Proposition 2 shows that an explanation for an argument is only empty when it is not attacked.
Proposition 1.
Let be an AF, for and let . Then:

if , then ;

if and , then .
Proof.
Let be an AF, for and let . Suppose that . By definition of it follows that . Let such that . Then there is some such that and defends against this attack. However, since defends , it follows that attacks as well, from which it follows that defends as well. Therefore . The second item follows immediately. ∎
Proposition 2.
Let be an AF and let be such that is accepted w.r.t. and . Then iff there is no such that .
Proof.
Let be an AF and let be such that is accepted w.r.t. and .
Suppose that . Then for each , . Hence there is no attacker of that is defended by some argument from . Since , is defended against its attackers. It follows that is not attacked at all.
Now suppose that is not attacked. Then there is no argument that defends . Therefore, for any , . It follows that . ∎
4.2 Nonacceptance Explanations
In order to explain a contrast between an accepted and nonaccepted argument, we need nonacceptance explanations as well. Therefore, in this section, basic definitions for explanations of nonaccepted arguments are recalled. There are again two types of explanations.
Definition 6 (Argument nonacceptance explanation).
Let be an AF, let be an argument that is not accepted w.r.t. and . Then:
Thus, a nonacceptance explanation contains all the arguments 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 (Example 3 continued).
Consider again . We have that: and .
The next proposition, the counterpart of Proposition 2, shows that a nonacceptance explanation is never empty.
Proposition 3.
Let be an AF and let be such that is nonaccepted w.r.t. and . Then .
Proof.
Let and be such that is nonaccepted w.r.t. and . Assume that , then there is no argument . Hence, for each , . It follows that there is no such that . But then, by the completeness of it follows that . A contradiction. Therefore . ∎
That the above proposition does not hold for follows since not every admissible extension contains all the arguments that it defends. Take for example an AF with arguments and and no attacks between them. Then is an admissible extension, thus is skeptically nonaccepted, yet . In fact: .
4.3 Comparing Acceptance and Nonacceptance
When looking at Examples 4 and 5 for and one can observe that acceptance and nonacceptance explanations are related. In this section we formalize this observation. In particular, we show that nonacceptance explanations contain the acceptance explanations of (1) the direct attackers; (2) the directly attacked arguments; and (3) the indirectly attacked arguments.
Proposition 4.
Let be an argumentation framework, let for some and let such that and indirectly attacks . Then:

where , : ;

when : ;

where and , : for all .
Proof.
Let be an AF, for some and .

Let be such that . If , we are done, hence, let . Then, by the proof of Proposition 2 there is some such that and (in)directly attacks . Since attacks , it follows that defends and that (in)directly attacks . Since , does not defend against the attack from and therefore .

Let , then, by Proposition 2, . Suppose that directly defends , then there is a such that . Since it follows that . Now suppose that indirectly defends . Then there are , where
is odd, such that
. Since defends and attacks it follows that attacks as well. Hence . Note that, for any a exists. It therefore follows that . 
Let and suppose that for some . By assumption indirectly attacks for all and since , is not defended against this attack by . Therefore . Note that any defends and therefore indirectly attacks as well. It therefore follows that and hence for all . ∎
To see that , take a look at the following example. Intuitively this is the case since, in terms of labeling semantics [2], an argument can be in the extension, attacked by the extension (i.e., out) or attacked by an argument that is not in or out (i.e., undecided).
Example 6.
Let , as shown in Figure 2. There are two preferred extensions: . Here we have that but only such that , for which: .
Knowing how acceptance and nonacceptance explanations are related is useful in the context of contrastive explanations, where explanations are not only about the requested argument, but about arguments that are conflicting with the requested argument as well.
5 Contrastive Explanations
A contrastive explanation explains by explaining why rather than . Important in contrastive explanations is that the difference between fact (i.e., ) and foil (i.e., ) is highlighted. In this paper we assume that fact and foil are not always compatible: it cannot be the case that both and are skeptically accepted. Intuitively, we make this assumption since otherwise there is no contrastive question for fact and foil (i.e., why both and is not contrastive).
In the context of formal argumentation contrastive explanations are modeled by comparing the elements of the basic explanations that explain the acceptance [resp. nonacceptance] of the fact and, at the same time, explain the nonacceptance [resp. acceptance] of the foil.
Definition 7 (Contrastive explanations).
Let be an AF, let (the fact) and let (a set of foils) such that there is no in which for all . Moreover, and if and if and . Contrastive explanations are then defined as in Figure 3.
In words, when there are arguments that cause the fact to be accepted [resp. nonaccepted] and the foil to be nonaccepted [resp. accepted], the contrastive explanation is the set of such arguments, this is the first case. If there are no common causes for the acceptance [resp. nonacceptance] of the fact and the nonacceptance [resp. acceptance] of the foil, the contrastive explanation is a pair of the respective explanations, the second case.
Example 7 (Examples 4 and 5 continued).
For the framework we have that: , and while and .
Thus, while the acceptance of can generally be explained by and (recall Example 4) when compared to the nonacceptance of [resp. ] the acceptance of is explained by [resp. ] alone.
One could consider contrastive explanations more meaningful when these return a set, rather than a pair. This is the case since then there are arguments that influence both the acceptance [resp. nonacceptance] of the fact and the nonacceptance [resp. acceptance] of the foil. The next proposition shows that in most cases the explanation is a set. Only when the accepted argument is not attacked at all or fact and foil are not conflictrelevant is the intersection empty.
Proposition 5.
Let be an AF and . Then implies that ; or is not conflictrelevant for .
Proof.
Let be an AF and let . That when follows immediately. Recall from Proposition 3 that . Suppose that is conflictrelevant for and, without loss of generality, that . Since is requested and, by assumption , there is some such that and . If , by Proposition 4.1 . If indirectly attacks , then by Proposition 4.3 it follows that as well. Since we have that . ∎
In view of the above result, the following conditions are introduced on the fact and foil. By requiring these conditions to hold, meaningful contrastive explanations can be obtained. For this let be an AF and let . Then [resp. ] can be requested when, for each :

is at least credulously accepted [resp. skeptically nonaccepted] and is at least skeptically nonaccepted [resp. credulously accepted];

for each it never holds that ;

either is conflictrelevant for or is conflictrelevant for .
These conditions ensure that fact and foil are incompatible, but still relevant for each other: it is explained what makes the fact accepted [resp. nonaccepted] and, simultaneously causes the foil to be nonaccepted [resp. accepted]. This prevents contrastive explanations for arguments that are not related or conflicting. These conditions are not exhaustive, depending on, e.g., the application, a user might wish to enforce further conditions on fact or foil.
5.1 Nonexplicit Foil
When humans request a (contrastive) explanation the foil is sometimes left implicit, yet the expected explanation does not provide all reasons for the fact happening, but should rather explain the difference between fact and foil. While humans are able to detect the foil based on, e.g., context, this is a challenge for AI systems, including argumentation. In particular, it is impossible to provide one strategy, since different applications entail different foils. For example, if argumentation is applied to determine a yes or no answer (e.g., whether one qualifies for a loan), the foil would be not A. However, if argumentation is applied to choose one element from a larger set (e.g., a medical diagnosis), the foil might be any member of that set.
Since in the definition of contrastive explanations it is necessary to provide a foil, a way to determine the foil is required. This is where one of the advantages of formal argumentation comes in: the explicit nature of conflicts between arguments makes that the foil or a set of foils can be constructed from an AF. Since the relation between arguments is only determined by the attack relation in our setting, it is impossible to distinguish between attackers. To illustrate the possibilities, in the remainder of the paper the foil will consist of all directly attacking arguments.
Definition 8.
Let be an AF and let . Then: .
Example 8.
For we have that: ; and .
Note that, for our running example, the explanations with implicit foil do not change:
In what follows it will be assumed that , for fact , i.e., that a foil exists. Note that, by Definition 8, for any AF and , iff there is no such that . Hence, any argument without a foil is not attacked at all. In such a case a nonacceptance explanation is not applicable and, by Proposition 2, the acceptance explanation is empty. Therefore, this requirement does not restrict our results.
The next proposition shows that the obtained contrastive explanations are meaningful when the first condition of the applicability of contrastive explanations is fulfilled and the foil is defined as in Definition 8.
Proposition 6.
Let be an AF, let be such that and . Then a contrastive acceptance [resp. nonacceptance] explanation can be requested for , when is at least credulously accepted [resp. skeptically nonaccepted] and for all , is at least skeptically nonaccepted [resp. credulously accepted].
Proof.
Let be an AF, let be such that and . To show that:

for each it is never the case that : by definition, and hence, since each is conflictfree, it is never the case that .

either is conflictrelevant for or is conflictrelevant for : by definition, and hence is conflictrelevant for . ∎
Corollary 1.
Let be an AF, let be such that and . Then:

is never of the form ;

iff for all .
Thus, when the foil is determined as in Definition 8, nonacceptance contrastive explanations are pairs if and only if the fact is only attacked by nonattacked arguments.
6 Contrastive Explanations in Structured Argumentation
Since many approaches to structured argumentation result in an abstract argumentation framework (see e.g., [4]), the basic and contrastive explanations as well as the results in this paper are applicable to such approaches as well. However, like in [7], the structure of the arguments within any approach to structured argumentation, makes it possible to refine the explanations. For this we take ASPIC [23].^{2}^{2}2For the sake of simplicity and conciseness we take classical negation (denoted by ) as the contrariness function and we do not considered preferences in this paper.
Aspic
In ASPIC, an argumentation system consisting of a propositional language , a set of rules (of the form for strict rules () and for defeasible rules ()) and the naming convention for defeasible rules and the knowledge base (containing the disjoint sets of axioms () and ordinary premises ()) form an argumentation theory , within which arguments can be constructed:
Definition 9.
An argument on the basis of a knowledge base in an argumentation system is:

if , with , and ;

if are arguments such that there exists a strict/defeasible rule in .
; ; ; .
For a set of argument : .
Attacks on an argument are based on the rules and premises applied in the construction of that argument.
Definition 10.
An argument attacks an argument iff undercuts, rebuts or undermines , where:^{3}^{3}3We let if or .

undercuts (on ) iff for some such that ;

rebuts (on ) iff for some of the form ;

undermines (on ) iff for some .
Abstract argumentation frameworks can be derived from argumentation theories: , where is the set of all arguments constructed from the argumentation theory and iff attacks .
Dungstyle semantics can be applied to such AFs. We denote: , , , .
Basic Explanations for ASPIC
In abstract argumentation the arguments are abstract entities, however, in ASPIC the structure of the arguments is known and can be used in the explanations. To this end we use the function, , which determines the content of an explanation (e.g., explanations can consist of arguments () or of the premises of those arguments ()).^{4}^{4}4See [7] for additional variations for . Formula explanations differ in two ways from the explanations in Section 4: the function is applied; and the arguments for have to be considered (e.g., all accepted arguments for for acceptance and an accepted argument for for acceptance). The basic explanations from Section 4 for formulas are defined by:
Definition 11 (Basic formula explanations).
Let be a formula and be based on . Suppose that is accepted w.r.t. and or . Then:
Suppose now that is nonaccepted w.r.t. and or :
That for the nonacceptance explanation all arguments for have to be accounted for follows since it might be the case that an explanation does not contain one particular argument for but it does contain another.
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