A principle-based robustness analysis of admissibility-based argumentation semantics
Abstract
The principle-based approach is a methodology to classify and analyse argumentation semantics. In this paper we classify seven of the main alternatives for argumentation semantics using a set of new robustness principles. These principles complement Baroni and Giacomin’s original classification and deal with the behaviour of a semantics when the argumentation framework changes due to the addition or removal of an attack between two arguments. We distinguish so-called persistence principles and monotonicity principles, where the former deal with the question of whether a labelling or extension of an argumentation framework under a given semantics persists after a change, and the latter with the question of whether new labellings or extensions are created after a change. We furthermore show in which sense labelling-based and extension-based semantics lead to subtly different principles and results. Our results can be used for choosing a semantics for a particular application, or to guide the search for new argumentation semantics, but they have been used also in the design of algorithms.
1.Introduction
Many semantics have been proposed in the literature on abstract argumentation. One way to compare and evaluate them is to look at examples of argumentation frameworks that bring to light important differences between semantics or highlight a particular strength or weakness of a semantics. While such an example-based approach may produce valuable insights, a deeper understanding of the differences and similarities of semantics requires a more systematic study of their properties. Baroni et al. introduced the so called principle-based approach [4,60], which takes as a starting point principles – that may or may not be satisfied by a semantics – representing various intuitions and requirements, which are used to evaluate a semantics or to guide the search for new semantics. These principles range from basic properties of extensions such as conflict-freeness and admissibility to principles that reflect desirable behaviour in certain classes of examples, such as crash-resistance, non-interference and directionality [4,30]. Semantics for argumentation may also be characterised in terms of strong equivalence conditions, which identify the syntactical differences that are semantically indistinguishable under a given semantics [7–10,18,54], or in terms of their signature, which identifies the set of all possible sets of extensions or labellings that can be expressed under a given semantics [42,44,47]. The general approach of focusing on general properties in the evaluation of a semantics has also been applied to ranking-based semantics [2,19,24] and instantiated argumentation formalisms [27,29].
In this paper we extend the principle-based investigation of semantics for abstract argumentation and study principles that characterise the robustness of extensions or labellings under a semantics with respect to the removal or addition of attacks. These principles are relative to the status assigned to the two arguments between which the attack is added or removed. The set of different statuses we can distinguish depends on whether we look at labelling-based or extension-based semantics. A labelling-based semantics σ maps every argumentation framework to a set of labellings, which are functions that assign to each argument a label
For the extension-based setting we define the same set of principles except that the status of an argument is either in an extension (or, equivalently, labelled
We investigate the principles discussed here with respect to all the admissibility-based semantics that have been defined in the literature. Thus, we focus on the complete, grounded, preferred and stable semantics, which were defined by Dung in terms of extensions [40] and by Caminada in terms of labellings [28], as well as the semi-stable [30], ideal [41] and eager [26] semantics. The investigation of non-admissible semantics is planned for a future publication.
The results we obtain provide insight into the robustness of the evaluation of an argumentation framework under a given semantics when the argumentation framework changes. While we do not argue that any of the principles we investigate must necessarily be satisfied by a semantics in general, some principles can be regarded as reflecting reasonable or desirable requirements. Let us demonstrate this point by considering
The principles we investigate are also useful in the design of algorithms. For example, Niskanen et al. [51] use persistence principles in the design of an algorithm for computing extensions of incomplete argumentation frameworks, where one can specify that attacks between certain arguments may or may not exist. Persistence and monotonicity principles are also useful in addressing enforcement problems in abstract argumentation [11,12,25,34,53]. This is the problem of determining minimal sets of changes to an argumentation framework in order to enforce some result, such as the acceptance of a given set of arguments. Because persistence and monotonicity principles can be used to determine which changes to the attack relation of an argumentation framework do or do not change its evaluation, these principles can be used to guide the search for sets of changes in the enforcement problem. This idea has already been used for extension enforcement under the grounded semantics [53].
This paper extends results presented earlier [56], which focused on addition and removal persistence as well as addition monotonicity, but did not consider removal monotonicity and did not include results for the ideal and eager semantics. In the current paper we additionally examine persistence and monotonicity principles from the extension-based perspective, where we distinguish arguments that are accepted from those that are not, but where no distinction is made between rejected and undecided arguments. In case of addition and removal monotonicity, this different perspective yields a set of principles that are distinct from the labelling-based ones.
The layout of this paper is as follows. In Section 2 we recall the necessary definitions concerning abstract argumentation theory. Then we first focus on labelling-based semantics. The addition and removal persistence principles are discussed in Section 3 and 4, while addition and removal monotonicity principles are discussed in Section 5 and 6. We then discuss, in Section 7, persistence and monotonicity principles for extension-based semantics, and summarise and contrast our results in Section 8. We discuss directions for future work in Section 9, survey related work in Section 10, and conclude in Section 11.
2.Preliminaries
An argumentation framework is a directed graph represented by a set A of arguments and a binary relation ⇝ over A called the attack relation [40]. We assume throughout this paper that a set
Definition 1.
Let
Given an argumentation framework
A labelling-based semantics maps every argumentation framework to a set of labellings, which are functions that map every argument of an argumentation framework to a label. All the labelling-based semantics considered in this paper are defined using three possible labels:
Definition 2.
A labelling of an argumentation framework
Definition 3.
A labelling-based semantics σ defines a function
A complete labelling is a labelling in which an argument is accepted if and only if all attackers are rejected, and an argument is rejected if and only if some attacker is accepted. All the semantics we consider in this paper are based on complete labellings.
Definition 4.
Let
Given two labelling L,
Definition 5.
Let
We furthermore write
Definition 6.
Given an argumentation framework
(1)
(2)
The following definition introduces the labelling-based complete (
Definition 7.
Let
We now state two well-known facts about the semantics defined in Definition 7 that will be used throughout this paper. First of all, under all semantics except for the stable semantics, every argumentation framework has at least one labelling.
Proposition 1.
Let F be an argumentation framework and
The grounded, ideal and eager semantics are unique status semantics, meaning that they map every argumentation framework to a unique labelling.
Proposition 2.
Let F be an argumentation framework and let
3.Addition persistence
We say that a semantics σ satisfies
Definition 8.
Let σ be a semantics and let
Table 1 shows the full overview of addition persistence under the semantics we consider. A gray cell means that the combination is satisfied and a white cell means that it is violated. The white and gray cells refer to the counterexamples and proofs provided in this section.
Table 1
Addition persistence for labelling-based semantics
We can distinguish two types of addition persistence principles. The first are
The remaining combinations form the second type. These are
The grounded semantics violates
The preferred semantics violates
The semi-stable, eager and ideal violate all addition persistence principles. For the semi-stable semantics, the
In the remainder of this section, we present the counterexamples for the white cells in Table 1 and proofs for the gray cells.
Example 1
Example 1(Failure of OO ss ea
In the argumentation framework shown in Fig. 1 the labelling
Fig. 1.
Example 2
Example 2(Failure of OO id
In the argumentation framework shown in Fig. 1 the labelling
Example 3
Example 3(Failure of OU ss ea
In the argumentation framework shown in Fig. 2 the labelling
Fig. 2.
Example 4
Example 4(Failure of OU id
In the argumentation framework shown in Fig. 2 the labelling
Example 5
Example 5(Failure of UO ss ea
In the argumentation framework shown in Fig. 3 the labelling
Fig. 3.
Example 6
Example 6(Failure of UO id
In the argumentation framework shown in Fig. 3 the labelling
Example 7
Example 7(Failure of UU pr id ss ea
In the argumentation framework shown in Fig. 4 the labelling
Fig. 4.
Example 8
Example 8(Failure of OI gr id
In the argumentation framework shown in Fig. 5 the labelling
Fig. 5.
Example 9
Example 9(Failure of OI ss ea
In the argumentation framework shown in Fig. 5 the labelling
Example 10
Example 10(Failure of UI gr co pr ss id ea
In the argumentation framework shown in Fig. 6 the labelling
Fig. 6.
Example 11
Example 11(Failure of IO ss ea
In the argumentation framework shown in Fig. 7 the labelling
Fig. 7.
Example 12
Example 12(Failure of IO id
In the argumentation framework shown in Fig. 7 the labelling
Example 13
Example 13(Failure of IU
In the argumentation framework shown in Fig. 8 the labelling
Fig. 8.
Example 14
Example 14(Failure of II
In the argumentation framework shown in Fig. 9 the labelling
Fig. 9.
The following two propositions will be used in the proofs that follow. The first states that, given any pair of complete labellings L,
Proposition 3.
Let
Proof.
For the only if direction, assume
For the if direction, assume
The second proposition concerns some facts that we can learn about the grounded labelling of an argumentation framework F on the basis of a complete labelling of F.
Proposition 4.
Let F be an argumentation framework, let
(1) If
(2) If
(3) If
Proof.
Let F be an argumentation framework, let
(1) From the definition of grounded labelling it follows that
(2) From the definition of grounded labelling together with Proposition 3 it follows that
(3) If
Proposition 5.
The
Proof.
Follows directly from the definition of a complete labelling (Definition 4). □
Proposition 6.
The
Proof.
Let
Proposition 7.
The
Proof.
Let
(1)
(2)
(3)
(4)
(a)
(b)
If we repeat these steps until no more arguments are relabelled then the resulting labellingi. if x is
ii. if x is
Proposition 8.
The
Proof.
Let
Proposition 9.
The
Proof.
Follows directly from definition of a stable labelling (Definition 7). □
Proposition 10.
The
Proof.
Let
Proposition 11.
The
Proof.
Let
Proposition 12.
The
Proof.
This follow vacuously because, in a stable labelling, no argument is labelled
4.Removal persistence
We say that a semantics σ satisfies
Definition 9.
Let σ be a semantics and let
Table 2 shows the full overview of removal persistence under the semantics we consider. Like before, gray cells refer to proofs and indicate that removal persistence holds, while white cells refer to counterexamples and indicate that removal persistence does not hold.
Table 2
Removal persistence for labelling-based semantics
We can distinguish three types of removal persistence principles. First of all,
The
The
In the remainder of this section, we present the counterexamples for the white cells in Table 2 and proofs for the gray cells.
Example 15
Example 15(Failure of OO
In the argumentation framework shown in Fig. 10 the labelling
Fig. 10.
Example 16
Example 16(Failure of OO
In the argumentation framework shown in Fig. 10 the labelling
Example 17
Example 17(Failure of OU
In the argumentation framework shown in Fig. 11 the labelling
Fig. 11.
Example 18
Example 18(Failure of OU
In the argumentation framework shown in Fig. 11 the labelling
Example 19
Example 19(Failure of UO
In the argumentation framework shown in Fig. 12 the labelling
Fig. 12.
Example 20
Example 20(Failure of UO
In the argumentation framework shown in Fig. 12 the labelling
Example 21
Example 21(Failure of UU
This case is trivial: consider an argumentation framework consisting of one self-attacking argument. This argument is undecided, but is no longer undecided if the self-attack is removed.
Example 22
Example 22(Failure of OI
In the argumentation framework shown in Fig. 13 the labelling
Fig. 13.
Example 23
Example 23(Failure of OI
In the argumentation framework shown in Fig. 13 the labelling
Example 24
Example 24(Failure of IO
This case is trivial: consider an argumentation framework consisting of two arguments a and b where a attacks b. Under all semantics considered here, a is labelled
Proposition 13.
The
Proof.
Let L be the grounded labelling of
Proposition 14.
The
Proof.
Follows directly from Definition 4. □
Proposition 15.
The
Proof.
Let
Proposition 16.
The
Proof.
Let
Proposition 17.
The
Proof.
Follows directly from Definition 7. □
Proposition 18.
The
Proof.
Let
(1)
(2)
If we repeat these steps until no more arguments are relabelled then the resulting labelling(a) if x is
(b) if x is
Proposition 19.
The
Proof.
Let
Proposition 20.
The
Proof.
Follows directly from Definition 4. □
Proposition 21.
Any complete-based semantics satisfies
Proof.
This holds vacuously because, in a complete labelling, these combinations of labels are never assigned to two arguments of which one attacks the other. □
Proposition 22.
The
Proof.
This holds vacuously because, in a stable labelling, no argument is labelled
5.Addition monotonicity
We now consider addition monotonicity for labelling-based semantics. Suppose x and y are labelled X and Y, respectively, in every σ labelling of an argumentation framework F. The
Definition 10.
Let σ be a semantics and let
Remark 1.
Addition monotonicity is concerned only with how the set of σ labellings of an argumentation framework
For any unique status semantics (cf. Proposition 2) it holds that addition monotonicity coincides with addition persistence.
Proposition 23.
If σ is a unique status semantics then σ satisfies XY addition monotonicity if and only if σ satisfies XY addition persistence.
We now show which of the labelling-based semantics that we consider satisfy or violate addition monotonicity. Table 3 shows the full overview: a gray cell means that the principle is satisfied and a white cell means that it is violated. The white and gray cells refer to the counterexamples and proofs provided in this section.
Table 3
Addition monotonicity for labelling-based semantics
| Addition monotonicity | |||||||||
| Grounded | Prp. 24 | Prp. 24 | Prp. 24 | Prp. 24 | Ex. 32 | Ex. 25 | Prp. 24 | Ex. 26 | Ex. 27 |
| Complete | Prp. 25 | Prp. 25 | Prp. 26 | Ex. 31 | Ex. 32 | Ex. 25 | Prp. 26 | Ex. 26 | Ex. 27 |
| Preferred | Ex. 28 | Ex. 29 | Ex. 30 | Ex. 31 | Ex. 32 | Ex. 25 | Ex. 33 | Ex. 26 | Ex. 27 |
| Stable | Ex. 28 | Prp. 27 | Prp. 27 | Prp. 27 | Ex. 32 | Prp. 27 | Ex. 33 | Prp. 27 | Ex. 27 |
| Semi-stable | Ex. 28 | Ex. 29 | Ex. 30 | Ex. 31 | Ex. 32 | Ex. 25 | Ex. 33 | Ex. 26 | Ex. 27 |
| Ideal | Ex. 2 | Ex. 4 | Ex. 6 | Ex. 7 | Ex. 32 | Ex. 25 | Ex. 12 | Ex. 26 | Ex. 27 |
| Eager | Ex. 1 | Ex. 3 | Ex. 5 | Ex. 7 | Ex. 9 | Ex. 25 | Ex. 11 | Ex. 26 | Ex. 27 |
Three types of addition monotonicity principles can be distinguished. The
The remaining principles form the second type:
In the remainder of this section, we present the counterexamples for the white cells in Table 3 and proofs for the gray cells.
Example 25
Example 25(Failure of UI
Consider an argumentation framework with two arguments a, b, where a is self-attacking. The unique labelling under all but the stable semantics assigns
Example 26
Example 26(Failure of IU
Consider an argumentation framework with two arguments a, b, where b is self-attacking. The unique labelling under all but the stable semantics assigns
Example 27
Example 27(Failure of II
Consider an argumentation framework with two arguments a, b and no attacks. The unique labelling under all semantics assigns
Example 28
Example 28(Failure of OO pr stb ss
Let F be the argumentation framework shown in Fig. 14 (not including the attack drawn with a dotted line). The labelling
Fig. 14.
Failure of
Fig. 15.
Failure of
Example 29
Example 29(Failure of OU pr ss
Let F be the argumentation framework shown in Fig. 15 (not including the attack drawn with a dotted line). The labelling
Example 30
Example 30(Failure of UO
Let F be the argumentation framework shown in Fig. 16 (not including the attack drawn with a dotted line). The labelling
Fig. 16.
Failure of
Example 31
Example 31(Failure of UU co pr ss
Consider an argumentation framework F with two arguments a and b where a attacks b and a is self-attacking. Let L be the labelling
Example 32
Example 32(Failure of OI
Consider an argumentation framework F with two arguments a and b where a attacks b. Let
Example 33
Example 33(Failure of IO
Let F be the argumentation framework shown in Fig. 17 (not including the attack drawn with a dotted line). The labelling
Fig. 17.
Failure of
Proposition 24.
The
Proposition 25.
The
Proof.
Let
Proposition 26.
The
Proof.
Let
Proposition 27.
The
Proof.
This holds vacuously because, in a stable labelling, no argument is labelled
6.Removal monotonicity
We finally consider removal monotonicity in the labelling-based setting. Like before, we require that the initial set of labellings is non-empty (see Remark 1).
Definition 11.
Let σ be a semantics and let
Removal monotonicity coincides with removal persistence for any unique status semantics (cf. Proposition 2).
Proposition 28.
If σ is a unique status semantics then σ satisfies XY removal monotonicity if and only if σ satisfies XY removal persistence.
We now show which of the labelling-based semantics that we consider satisfy or violate removal monotonicity. Table 4 shows the full overview: a gray cell means that the principle is satisfied and a white cell means that it is violated. The white and gray cells refer to the counterexamples and proofs provided in this section.
Table 4
Removal monotonicity for labelling-based semantics
| Removal monotonicity | |||||||||
| Grounded | Prp. 29 | Prp. 29 | Prp. 29 | Ex. 34 | Prp. 29 | Prp. 32 | Ex. 35 | Prp. 32 | Prp. 32 |
| Complete | Prp. 30 | Prp. 30 | Prp. 31 | Ex. 34 | Prp. 30 | Prp. 32 | Ex. 35 | Prp. 32 | Prp. 32 |
| Preferred | Ex. 36 | Ex. 37 | Ex. 38 | Ex. 34 | Ex. 39 | Prp. 32 | Ex. 35 | Prp. 32 | Prp. 32 |
| Stable | Ex. 36 | Prp. 33 | Prp. 33 | Prp. 33 | Ex. 39 | Prp. 32 | Ex. 35 | Prp. 32 | Prp. 32 |
| Semi-stable | Ex. 36 | Ex. 37 | Ex. 38 | Ex. 34 | Ex. 39 | Prp. 32 | Ex. 35 | Prp. 32 | Prp. 32 |
| Ideal | Ex. 16 | Ex. 18 | Ex. 20 | Ex. 34 | Ex. 23 | Prp. 32 | Ex. 35 | Prp. 32 | Prp. 32 |
| Eager | Ex. 15 | Ex. 17 | Ex. 19 | Ex. 34 | Ex. 22 | Prp. 32 | Ex. 35 | Prp. 32 | Prp. 32 |
We can distinguish three types of removal monotonicity principles. The
The
The remaining removal monotonicity principles represent the third type. These are
In the remainder of this section, we present the counterexamples for the white cells in Table 4 and proofs for the gray cells.
Example 34
Example 34(Failure of UU
Consider an argumentation framework with two arguments a, b, where a is self-attacking and a attacks b. The unique labelling under all but the stable semantics assigns
Example 35
Example 35(Failure of IO
Consider an argumentation framework with two arguments a, b, where a attacks b. The unique labelling under all semantics assigns
Example 36
Example 36(Failure of OO pr stb ss
Let F be the argumentation framework shown in Fig. 18 (including the attack drawn with a dotted line). The labelling
Fig. 18.
Failure of
Example 37
Example 37(Failure of OU pr ss
Let F be the argumentation framework shown in Fig. 19 (including the attack drawn with a dotted line). The labelling
Fig. 19.
Failure of
Example 38
Example 38(Failure of UO pr ss
Let F be the argumentation framework shown in Fig. 20 (including the attack drawn with a dotted line). The labelling
Fig. 20.
Failure of
Example 39
Example 39(Failure of OI pr stb ss
Let F be the argumentation framework shown in Fig. 21. has a unique preferred, semi-stable and stable labelling
Fig. 21.
Failure of
Proposition 29.
The
Proposition 30.
The
Proof.
Let
Proposition 31.
The
Proof.
Let
Proposition 32.
Any complete-based semantics satisfies
Proof.
This holds vacuously because, in a complete labelling, these combinations of labels are never assigned to two arguments of which one attacks the other. □
Proposition 33.
The
Proof.
This holds vacuously because, in a stable labelling, no argument is labelled
7.Persistence and monotonicity in the extension-based setting
Most of the semantics we consider in this paper were originally defined in terms of extensions. These extension-based definitions can also be derived from the labelling-based Definitions [31]. Given a semantics σ, a σ-extension of an argumentation framework
Proposition 34.
For every semantics σ we have:
a) σ satisfies
b) σ satisfies
c) σ satisfies
Proof.
We prove this for item a and for addition persistence. Removal persistence, as well as items b and c, follow similarly.
For the only if direction, assume that σ satisfies
For the if direction, assume that σ satisfies
For addition and removal monotonicity things look different. Again, the following fact follows directly from the definition of addition and removal monotonicity.
Proposition 35.
For every semantics σ we have:
a) σ satisfies
b) σ satisfies
c) σ satisfies
Proof.
Similar to the proof of the only if direction in Proposition 34. □
The if direction of the three statements in Proposition 35 does not hold in general, and this is where the extension-based perspective is different from the labelling-based one. For instance, the combination of
Using Proposition 34 we can derive results for
Proposition 36.
The grounded semantics satisfies
Proof.
This follows from the fact that the grounded semantics is a unique status semantics and satisfies
Proposition 37.
The complete semantics satisfies
Proof.
This follows from the fact that the complete semantics satisfies
8.Discussion
In this section, we summarise and discuss the results obtained so far. Table 5 provides an overview of all the labelling-based robustness principles. Table 6 provides a similar overview for the extension-based robustness principles. Note that Table 5 is a duplicate of the tables included in earlier sections, while the results in Table 6 follow from the results on labelling-based principles via Propositions 34 and 35, and otherwise from Propositions 36 and 37.
Table 5
Addition/removal persistence/monotonicity for labelling-based semantics
Table 6
Addition/removal persistence/monotonicity for extension-based semantics
A number of general insights can be extracted from these tables. If we look at the extension-based robustness principles we see that the semi-stable, ideal, and eager do not satisfy any robustness principle, that the preferred semantics satisfies one, the complete semantics three, the grounded semantics four, and the stable semantics five. Moreover, of the few principles satisfied, most are persistence principles. In the labelling-based setting things look very different. Here, almost half of the principles are satisfied. This supports the intuition that the additional discriminative power of labellings is very useful in describing the robustness of semantics.
Dung’s traditional admissibility-based semantics (grounded, complete, preferred and stable) all behave differently with respect to the desirable robustness principles, so these principles are useful to distinguish these semantics in this respect. However, our principles do not distinguish the semi-stable, ideal and eager semantics, as none of these semantics satisfy any of the desirable principles (extension-based or labelling-based). The stable semantics appears to be the best behaved one with respect to the extension-based persistence principles. The same holds for the labelling-based persistence principles. Note however, that all the labelling-based principles involving
Throughout this paper, we made a distinction between persistence and monotonicity principles. We can also combine the two principles. Let us say that a semantics satisfies
Table 7
Addition/removal robustness for labelling-based semantics
Table 8
Addition/removal robustness for extension-based semantics
Note that for the grounded, ideal and eager semantics, which are unique status semantics, robustness coincides with persistence and monotonicity. The preferred, semi-stable, ideal and eager semantics violate all robustness principles, except for the
9.Future work
The results of the robustness principles introduced and analyzed in this paper give rise to three main lines of further research. First, the analysis can be extended to other semantics, such as non-admissible-based semantics. Second, the set of principles discussed in this chapter can be further extended. Third, relations between principles can be studied, such as relations between the robustness principles introduced in this paper and principles discussed in the literature like directionality and SCC decomposability. We consider these three lines of research below.
9.1.Semantics
The first topic for further research is to verify whether other semantics proposed in the literature satisfy our new principles. In some cases this will be easy. For example, given the construction of prudent semantics [33], it is easy to show that most of our principles are not satisfied. However, for naive-based semantics such as the Stage, Stage2 and CF2 semantics [5,43,61], it is often more difficult to see whether our robustness principles are satisfied or not. This is analogous to the case for many other principles, where for naive-based semantics often larger counter-examples are given [60]. Recently, a range of new semantics based on the notion of weak admissibility have been introduced [15]. Dauphin et al. [35] provide a principle-based analysis of these semantics, and also define the closely related qualified and semi-qualified semantics. Whether these new semantics satisfy our robustness principles is also an open question.
The second topic for further research is the consideration of other semantic frameworks such as other labelling conventions. As we have shown, there is a clear distinction between the extension-based results in Table 5 and the labelling-based results in Table 6. Whereas for extension-based semantics very few principles are satisfied, many more principles are satisfied for labelling-based semantics. This can be readily explained: the distinction between out and undecided arguments makes the principles more precise. However, it also raises a question: if we adopt another way to describe the semantics, for example using four valued labelling functions, would that further increase the precision of the analysis? We can invert the question: which set of truth values can be defined such that we have maximal persistence and monotonicity principles?
The third topic of further research is to use the robustness principles to guide the search for new semantics, as in general suggested by van der Torre and Vesic [60]. However, whereas robustness may be desirable in some applications, it may be seen as a drawback in others. For example, the most robust semantics is one in which the set of extensions is always the same. In other words, the search for new semantics may require to make additional requirements explicit, for example in terms of additional principles. Moreover, we need to decide which of our robustness principles can play the role of desiderata in such a search, maybe in the form of meta-principles.
Finally, a fourth topic for further research is to consider robustness for ranking semantics, or other semantics functions. Likewise, one can extend the definition of robustness and the analysis to more expressive argumentation frameworks, for example expressing collective or set attack, higher-order attack, support relation in bipolar argumentation frameworks, or abstract dialectical frameworks.
9.2.Principles
The fifth topic for further research is to define principles by weakening or adapting the existing principles. This can be done to find principles which are more discriminative, or that correspond to particular intuitions. For example, since we are typically interested only in the accepted arguments, we may require that a labelling does not have to be exactly the same, only the in-part needs to remain unchanged. As another example, since self-attacks are known to cause problems in many examples, we may consider only frameworks that do not contain self-attacks, or that more generally do not contain odd loops.
9.3.Relations between principles
The sixth topic for further research is to study logical relations among the principles, independently of the semantics. For example, we may verify whether one property implies another one. In particular, the relation between our principles and existing ones needs to be further studied. The most obvious ones are directionality, and SCC decomposability.
The seventh topic for further research is to consider sets of principles. For example, as common in social choice, there may be sets of principles such that there are no semantics satisfying all of them, or there may be sets of principles such that there is exactly one semantics satisfying all of them. We are not aware of such impossibility or characterisation results in the literature on formal argumentation.
10.Related work
The robustness principles investigated in this paper build on earlier work by Boella et al. [22,23], who studied refinement and abstraction principles, which are conditions under which the evaluation of an argumentation framework remains unchanged when an attack is added or removed. However, they consider only semantics that yield a single extension or labelling. The results we obtained for addition and removal persistence under the grounded semantics coincide with theirs.
Our principles are related to the so called explicit conflict conjecture which was introduced by Baumann et al. [17] in the investigation of the notion of compact argumentation frameworks. These are argumentation frameworks that are minimal in the sense that no argument can be removed from the argumentation framework without affecting the set of extensions. For the definition, let
Cayrol et al. [32] studied the impact on the evaluation of an argumentation framework when new arguments and attacks are added. The main difference with our approach is that they focus on different ways in which the evaluation changes, rather than conditions under which the evaluation does not change. Some types of change that they define are changes leading to a larger, unique, or smaller set of extensions, changes that are monotonic (every extension of the old argumentation framework is included in an extension of the changed argumentation framework) and changes that are monotonous with respect to an argument (every argument included in an extension of the old argumentation framework is also included in an extension of the changed argumentation framework). They then define conditions under which the addition of an argument to an argumentation framework leads to these types of change.
The notion of strong equivalence in abstract argumentation deals with an alternative perspective on robustness in abstract argumentation. Given two argumentation frameworks
Bistarelli et al. [21] study so called local-expansion invariant operators for abstract argumentation frameworks. These operators are functions that take an argumentation framework as input and return the same argumentation framework except that zero or more attacks are added. They then determine conditions under which such operators are invariant with respect to a semantics σ. An operator is invariant with respect to a semantics σ precisely when the set of σ extensions remains unchanged after applying the operator. They focus only on conflict-freeness and admissibility but plan to study further semantics in future work.
Sakama studied counterfactuals in the context of abstract argumentation [58]. Such counterfactuals have the form 
between so called interventions and consequences. An intervention is a set of literals of the form 
) if the change represented by Φ leads to the truth of ϕ in all σ labellings of F. Like in Sakama’s approach, the principles defined in this paper for a semantics σ translate to properties that characterise the behaviour of the inference relation 
. Rienstra considers a number of common inference rules such as Cautious Monotonicity and Cut. Cautious Monotonicity states that, if 
and 
, then 
. Cut states that, if 
and 
, then 
. While these are properties that are generally considered desirable, Rienstra showed that they are not satisfied under all semantics. For example, Cautious Monotonicity does not hold under the preferred semantics. This follows from the violation of
Much work has been done in the past decade on what is often referred to as dynamics of argumentation. What unifies this work is its motivation, namely that argumentation is a dynamic process, which means that argumentation frameworks are not static entities but evolve over time. The current paper extends this line of work, as it contributes to the understanding of how admissibility-based semantics behave when the argumentation framework changes. There is a large amount of further work on dynamics of argumentation. For example, the question of whether and how an argumentation framework can be changed in order to enforce some result, such as the acceptance of a given set of arguments, is referred to as the enforcement problem [11,12,20,39,52,53,62]. Change in argumentation has furthermore been related to (or modelled using tools of) theories of belief change, often taking an AGM-inspired approach [13,14,25,34,36–38,45,48–50,57,59]. Computational issues in a dynamic setting have also been studied, such as efficiently (re)computing extensions after an argumentation framework changes [1,6,46]. In general, we believe that the robustness principles studied in this paper are relevant to much of the research on dynamics of argumentation. In the introduction we already discussed ways in which earlier work on robustness principles has been applied in the design of algorithms for the enforcement problem [53] and incomplete argumentation frameworks [51].
11.Summary and conclusion
We have defined a set of general principles for argumentation semantics expressing robustness of labellings and extensions against the addition and removal of attacks, and investigated all the admissibility-based semantics with respect to these principles. The results we obtain provide insight into the behaviour of these semantics in a dynamic context. For example, while the grounded, complete, preferred and stable semantics satisfy some of the desirable robustness principles, the semi-stable, eager and ideal semantics satisfy none. Furthermore, Dung’s traditional semantics (i.e., grounded, complete, preferred and stable) all satisfy different sets of robustness principles, so these principles are useful to distinguish these semantics in this respect.
The robustness principles we investigate are also useful in the design of algorithms. As we have discussed, addition and removal persistence properties have already been applied in the design of algorithms. For instance, knowledge about whether the addition or removal of an attack has an effect on the evaluation of an argumentation framework may be used to optimise search problems connected with the enforcement problem in argumentation [53]. In this context an interesting observation is that the semi-stable, ideal and eager semantics do not satisfy any useful robustness principle, which implies that these semantics cannot benefit from such optimisations, because there are no conditions under robustness against addition or removal of attacks is guaranteed.
We discussed a number of directions for future research, such as extending our analysis by weakening or adapting the robustness principles, and studying relationships between our robustness principles and other principles studied in the literature. We may furthermore consider robustness in the context of more expressive argumentation formalisms, such as frameworks supporting collective or set attacks, higher-order attacks, support relations (bipolar argumentation frameworks) or abstract dialectical frameworks. Finally, we focused in this paper only on admissibility-based semantics. We plan to extend our investigation of robustness to non-admissibility-based semantics in a follow-up publication.
Notes
1 As an example, consider a semantics σ that, given that
Acknowledgement
Tjitze Rienstra is supported by the Deutsche Forschungsgemeinschaft (grant KE 1686/3-1). Leendert van der Torre is supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 690974 for the project MIREL (MIning and REasoning with Legal texts). Beishui Liao is supported by the Convergence Research Project for Brain Research and Artificial Intelligence, Zhejiang University, and the Natural Science Foundation of Zhejiang Province of China under Grant No. LY20F030014.
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