1 Introduction
An Abstract Argumentation Framework [Dung1995] (AF), is represented by a pair consisting of a set of arguments and a binary relationship of attack defined among them. Given a framework, it is possible to examine the question on which set(s) of arguments can be accepted, hence collectively surviving the conflicts defined by . A very simple example of AF is , where two arguments and attack each other. In this case, each of the two positions represented by either or
can be intuitively valid. AFs can also provide a basis for handling the evolution of situations in which instances of particular problems undergo changes; variations on the underlying information can be interpreted as modifications in the corresponding graph. Such modifications can be performed through operations of addition or subtraction of nodes and edges in the AF. As one can expect, introducing these changes might lead to obtain different semantics for the considered AF. We can classify the operations it is possible to perform on a framework in two types: the ones that change the semantics of the system and the ones that do not. In this paper, we focus on this latter type of operations (which leave the semantics unchanged), and reducing to the case of addition (or subtraction) of an attack.
Our aim is to study a set of local expansion [Baumann2012] operators with respect to which the semantics is not altered. Due to the dynamic nature of certain problems, settling for a solution (in a particular AF) could not be sufficient to guarantee a good outcome in case the problem evolves. In a dynamic setting, it may happen that new arguments can change the meaning (and the outcome) of the conversation itself. Think, for example, to a negotiation or a persuasion dialogue. With invariant operators at dispose, one could test and possibly “enforce” [Baumann and Brewka2010] the strength of its position. Also, invariant operators could be successfully exploited for computing, in an efficient way, the semantics of an evolving AF.
The main content of this paper is a the notion of “robustness” for AFs. The main idea is that every argument (and set of arguments) is more or less suitable to undergo changes in the belief base [Dix et al.2016]. Robustness gives a measure of how many changes an AF can withstand before changing its semantics. The main part the work we carried out, concerns the study of particular modifying operators for which the semantics is invariant: through such operators it is possible to bring changes in the structure of a framework without changing its meaning.
In particular, we study such operators for conflictfree and admissible sets [Dung1995], since they are at the centre of any classical semantics and they have never been studied before in these terms. Differently from other works done in this direction (see the Related Work section), we consider how difficult is to modify the whole set of extensions instead of a single one, for instance as in [Rienstra, Sakama, and van der Torre2015].
This paper is structured as follows. We first summarise the necessary notions of Abstract Argumentation, by presenting extensionbased [Dung1995] and labellingbased [Caminada2006] semantics. We then describe the characteristics of the invariant operators we want to design and we define such operators for conflictfree and admissible sets. We conclude the paper by showing that our approach is novel w.r.t. the related work, and by providing conclusive thoughts and ideas about future work.
2 Abstract Argumentation Frameworks
By neglecting the internal structure of each argument (e.g., premises and a claim), the framework becomes “abstract”, that is we are not interested in the meaning of arguments any more, but we just focus on their relations and we look for general properties. Hence, snapshots of such discussions can be caught by using Abstract Argumentation Frameworks [Dung1995], namely directed graphs that clearly show the exchange of opinions as attacks between arguments/nodes. Then, working with an AF means to identify subsets of nodes, called extensions, which share certain properties, according to a given semantics. Below, we give the fundamental definitions for AFs and extensionbased semantics.
Definition 1 (Abstract Argumentation Framework [Dung1995]).
An Abstract Argumentation Framework is a pair where is a set of arguments and is a binary relation on , i.e., .
We denote the set of all AFs with the set of argument with . For two arguments , represents an attack directed from against . We can interchangeably use . Moreover, we say that a set of arguments attacks an argument if is attacked by an argument .
Definition 2 (Acceptable argument [Dung1995]).
An argument is acceptable with respect to if and only if s.t. , s.t. and we say that defends .
Dung defines several semantics of acceptance for computing subsets of arguments, called extensions, that share characteristic properties. Respectively, , , , , and stand for conflictfree, admissible, complete, stable, preferred, and grounded extensions.
Definition 3 (Extensionbased semantics).
Let be an AF. A set is conflictfree in , denoted if and only if there are no s.t. . For it holds that:

if each is defended by E;

if and defended by , ;

if , attacks ;

if and s.t. ;

if and s.t. .
For the sake of consistency, In the following we will interchangeably use the terms semantics and sets (as conflictfree and admissible sets), annotating them simply with : even if the results developed in this paper concern sets, the same methodology can be applied to semantics as well (see conclusions). Many of these semantics exploit the notion of defence in order to decide whether an argument is part of an extension or not. Such a phenomenon, for which an argument becomes accepted in some extension after being defended by another argument belonging to that extension, is known as “reinstatement”. This mechanism plays a fundamental role when one needs to understand how a semantics changes in case an AF is modified. We exploit the notion of reinstatement labelling introduced in [Caminada2006].
2.1 Reinstatement Labelling
Reinstatement is the process through which a nonaccepted argument of a framework becomes accepted w.r.t. a certain extension when its attackers are defeated by itself or other arguments. Dung provides several approaches for dealing with reinstatement, like stable, preferred, complete, and grounded semantics. In [Caminada2006], Caminada strengthens Dung’s theory by introducing “reinstatement labelling”, namely a function that partitions the set of arguments of an AF into three classes: “in”, “out” and “undec”. An argument is labelled in if all its attackers are labelled out, and it is labelled out if it has at least an attacker which is labelled in. Otherwise, it is labelled undec.
Caminada proved that reinstatement labelling coincides with Dung’s notion of complete extension. In the same way, further restrictions allow for obtaining stable, grounded and preferred semantic as well. Moreover, the same author specifies an additional semantics, called semistable, and provides a partial ordering between the various semantics. In particular, we know that: every stable extension is a semistable extension, every semistable extension is a preferred extension, every preferred extension is a complete extension, and every grounded extension is a complete extension. In Def. 4 we provide a formal definition of reinstatement labelling.
Definition 4 (AFlabelling [Caminada2006]).
Let be a Dungstyle argumentation framework. A labelling is a function such that s.t. , s.t. and s.t. . We say that is a reinstatement labelling if and only if it satisfies the following:

s.t. , s.t. ;

s.t. , s.t. .
In Fig. 1 we show an example of reinstatement labelling obtained with the web interface of ConArg^{1}^{1}1http://www.dmi.unipg.it/conarg [Bistarelli, Rossi, and Santini2014, Bistarelli, Rossi, and Santini2016b, Bistarelli and Santini2012, Bistarelli and Santini2011a, Bistarelli and Santini2011b]: in (), out () and undec () arguments are coloured in three different colours. Note that there exist more than one possible labelling for an AF. Moreover, there exists a connection between reinstatement labellings and the Dungstyle semantics. This connection is summarised in Tab. 1.
Labelling restrictions  Semantics  
no restrictions  complete  
empty  undec  stable 
maximal  in  preferred 
maximal  out  preferred 
maximal  undec  grounded 
minimal  in  grounded 
minimal  out  grounded 
minimal  undec  semistable 
3 Invariant Operators
A change in an AF can consist in addition (or subtraction) of nodes or edges. In this work we focus on modifications concerning attacks between arguments (in particular additions). This kind of transformation coincides with the notion of local expansion ([Baumann2012]). Introducing this type of changes in an AF may produce or not alterations on sets of extensions. This behaviour depends on two factors: the semantics we choose in computing the extensions and the change in the framework itself.
After a modification, either a set of arguments is no more acceptable w.r.t. to a given semantics, or a new extension is generated, so the semantics of the AF will change in turn. On the contrary, if we consider the case in which extensions are preserved, further non trivial observations can be made for what concerns the semantics of the AF. For instance, even if the subsets of arguments remain unchanged, an admissible set can become also complete, if the right modifications are applied. Formally, an operator can be defined as follows.
Definition 5 (Local expansion operator).
Let be an AF. A local expansion operator is a function s.t. , where .
If we consider those operators taking into account also Dung’s semantics, we can study changes in the AFs from the point of view of sets of extensions. An explanatory example is provided in Fig. 2: in the framework , the extension is both admissible and complete while the extension containing argument is only admissible. After the modification consisting in the addition of attack , the extension in the framework is no longer admissible. On the other hand, after the change on the relations set, the extension that in was just admissible, also becomes complete in .
3.1 Semantics Equivalence
Our purpose is to find local expansion operators that leave the whole semantics unchanged, so instead of considering changes on the semantics induced by modifications on the graph, we look for the set of allowed changes that leave the semantics unmodified. In this way, we define semantics homomorphisms, namely operators with respect to which the semantics is invariant. In order to preserve the whole semantics, it is necessary to ensure that all the sets will not be modified, hence every set of extensions has to be, in turn, invariant with respect to these operators. We say that if two AFs have the same set of extensions w.r.t. a certain semantics, then the two frameworks are equivalent for such semantics.
For this reason, we need the following definitions.
Definition 6 (Semantics inclusion).
Let and be two sets of extensions. We say that if and only if s.t. .
Definition 7 (equivalence).
Let and be two AFs and a semantics. We say that

if ;

if .
The equivalence we consider is referred as standard in [Oikarinen and Woltran2011]. Adding an attack in an AF can have different consequences. The most intuitive one is that the new attacked argument becomes defeated, and so it is forced to be removed from an extension. If we, instead, consider semantics in which the notion of acceptability is taken into account, defeating an argument could lead to accept another argument. In both the cases in which an argument become acceptable or is removed from an extension, the semantics would change. To distinguish the operators that reduce the set of extensions from those that expand it, we give the following definitions.
Definition 8 (Invariant operators).
A local expansion operator is said nondecreasing w.r.t. the semantics if for all the argumentation frameworks , and it is said nonincreasing if . If is both nondecreasing and nonincreasing, it is an invariant: .
The last case may occur when an attack has no effect on the set of extensions. Our purpose is exactly to find local expansion operators that guarantee this last outcome when adding an attack. In the following, an invariant operator will be referred to as .
It is necessary to understand how extensions react to changes in the AF. Since the main issue to deal with is due to the reinstatement, the idea we develop in order to define an invariant operator is to use the notion of reinstatement labelling. Once the arguments of the AF are labelled (with in, out or undec), there are nine () different ways an edge can be added among nodes, according to labels of the source and the target of the attack.
Definition 9.
Let be an AF and a semantics. The sets of arguments labelled in, out or undec in at least one extension are respectively:

in s.t. ;

out s.t. ;

undec s.t. .
Note that in coincides with the set of arguments of credulously accepted w.r.t. the semantics . In the following subsections we separately study the conflictfree and the admissible semantics.
3.2 Operators for ConflictFree Sets
The conflictfree property is very fragile: introducing a relation between two non conflicting nodes is sufficient to change the conflictfree sets. These sets can only be reduced: no new conflictfree set can be generated after the addition of an attack in the AF. Thus, every operator able to perform the addition of an edge in a graph produces another graph in which the semantics is “smaller” (in the sense that in some extensions of the new AF an argument disappears) or at most equal to the set deriving from . is the set of relations belonging to G, while is the same set after the addition of an attack introduced by . We avoid describing the trivial case in which , and we only consider the effective transformation of adding an attack (identical conclusions can be drawn in case of subtraction).
Proposition 1.
Every local expansion operator is nonincreasing w.r.t. the conflictfree sets.
Proof.
We have to show that for every local expansion operator , with . This comes directly from the definition of conflictfree extension, since is such that . ∎
Corollary 1.
Any local expansion operator which is nondecreasing for conflictfree sets is also invariant:
Proof.
We know from Prop. 1 that every local expansion operator is nonincreasing w.r.t. the conflictfree sets, that is . If is also nondecreasing, we have , and thus it is also invariant (). ∎
We conclude that an operator preserves the semantics only if it adds attacks between arguments which already were in conflict. We define an invariant operator for conflictfree sets with the following theorem.
Theorem 1 (Invariant for conflictfree sets).
Let be an AF. We have if and only if

or,

in.
Proof.
We show that all conflictfree extensions are preserved if the above condition holds and vice versa.
“”:
We have an such that . If the condition is not satisfied, then it would exists in a relation between two arguments belonging to the same extension in and so . Contradiction.
“”:
Suppose that the condition hold. If , then and are already in conflict and do not appear together in any conflictfree extension of . Thus, no extension will be lost in when adding an attack between those arguments. On the other hand, having in means that one of the two arguments is never in and so there is no way to change the conflictfree extensions set by adding an attack between and .
∎
If we take into account any of the semantics defined by Dung, we can conclude that adding an attack between two arguments belonging to a certain set always requires those arguments to be removed from such set, changing the semantics in turn. Hence, denying attacks between nodes within same set (which, then, do not attack each other in ) is a necessary condition in order to leave the semantics unchanged in .
3.3 Operators for Admissible Sets
Contrary to the conflictfree sets, for the admissible ones it is not possible to provide a theorem for the inclusion between semantics without taking into account reinstatement. Since arguments can be defended and consequently accepted with respect to a certain extension, we need to consider different types of interactions in order to find an operator capable of maintaining the semantics unchanged. Reinstatement labelling provides a powerful means to overcome the issue of comprehending how arguments defend each other inside an extension. Indeed, labellings are a more expressive way than extensions to suggest the acceptance of arguments. We exploit the notion of in, out and undec arguments to define the invariant operator for the admissible sets. In order to preserve this semantics, we have to guarantee that neither existent extensions will be destroyed, nor new one will be created. To achieve this, an operator has to ensure that extensions in the set remain conflictfree, in arguments are not defeated from outside and out and undec arguments do not become acceptable. We distinguish between modifications that reduce the semantics from modifications that enlarge it.
Theorem 2.
Let be an AF. A local expansion operator is nondecreasing w.r.t. the admissible set if and only if , there does exists a labelling of such that

or

, and s.t. or

.
Proof.
We have to show that if then the condition holds and vice versa.
“”:
An operator is nondecreasing w.r.t. admissible sets. Suppose that there exists an attack relation such that in some labelling of we have or . In both these cases, the admissible extension corresponding to the labelling is lost in and thus , so we have a contradiction. In the case that and and , if and s.t. , then the extension containing the only argument would be lost. Contradiction.
“”:
If the condition holds, it is not possible that an extension in is also in . Consider any labelling of G. If is in or undec and is not in, then the addition of an attack cannot make an admissible extension of to become unacceptable in . If instead is out, it means that it is already defeated, so every argument belonging to some admissible extension of remains acceptable w.r.t. such extension also in .
∎
Theorem 3.
Let be an AF. A local expansion operator is nonincreasing w.r.t. the admissible set if and only if , there does not exists a labelling of such that

and s.t. and or

and s.t. or

and s.t. and or

, there is an odd length sequence of attacks from
to and s.t. there is an odd length sequence of attacks from to but not from to .
Proof.
We show evidence that no new admissible extensions are generated for applying the operator if the conditions of the theorem are satisfied and vice versa.
“”:
Suppose that . If there exists a labelling for which and s.t. and then arguments and would become acceptable together, forming a new admissible extension. The same would happen whenever and s.t. or in the case and s.t. and . If instead the last condition does not hold, then would be defended from all the incoming attacks and so it would be accepted in some admissible extension of . In all these cases we reach a contradiction.
“”:
We will see that if the conditions hold, it is not possible that a new admissible extension can be generated. For every labelling of , a nonincreasing operator is allowed to add an attack between arguments and only in the following cases:

and s.t. and ;

and s.t. ;

and s.t. and ;

and there is no odd length sequence of attacks from to ;

and s.t. there is an odd length sequence of attacks from to but not from to .

.
Case 4 means that is not responsible for being out, so the attack is not sufficient to make acceptable in a new admissible extension. In case 5, even if defeats , it will not become admissible without also defeating . In all remaining cases no arguments can be defended by (neither itself), so no new admissible extensions will form. ∎
Corollary 2.
Let be an AF. A local expansion operator is invariant w.r.t. the admissible set, and we write , if and only if , there does not exists a labelling of such that

, or

and s.t. , or

and s.t. and , or

, and s.t. , or

, there is an odd length sequence of attacks from to and s.t. there is an odd length sequence of attacks from to but not from to , or

.
Proof.
The proof of this corollary is straightforward and comes from the proofs of Th. 2 and Th. 3. In particular, if a labelling of satisfying the properties above does not exists, then the local expansion operator is both nondecreasing (for Th. 2) and nonincreasing (for Th. 3) w.r.t. the admissible semantics. Then is invariant w.r.t. the admissible semantics, because the modification on the set of relations does not allow any change in the semantics. Vice versa, if is invariant w.r.t. the admissible semantics, then and must hold. If a labelling exists such that at least one of the given properties is satisfied, then could be neither nondecreasing, nor nonincreasing (or both), according to Th. 2 and Th. 3. Thus, such a labelling can not exist. ∎
Below, we provide an example of how the conditions given in 2 allow to modify an AF without changing its semantics.
Example 1.
Consider the AF in Fig. 3. The following attacks does not preserve the admissible semantics.

: since both 1 and 4 belong to , the extension would be deleted from ;

: 4 would defend 3 from 2, so the extension would be accepted in ;

: 4 does not attack 4 and it is not attacked by any other out node, so the extension would be deleted from ;

: the only odd length sequence of attacks toward 2 comes from 1, so 2 would defend itself from 1, generating the admissible extension .
Remark 1.
In order to determine if an operator is invariant w.r.t. the admissible semantics, it is sufficient to consider only labelling in which is maximal, that is the preferred extensions.
In fact, in extensions that are not maximal, some arguments remain labelled undec even if they have different labels in more inclusive extensions (w.r.t. set inclusion). Thus, looking directly at the most inclusive extension allows for establishing rules able to preserve all the sets.
Invariant operators can be used as a metric to measure the robustness of AFs. The idea is that, starting from , different invariant operators can be applied in sequence, until no more exists for the last obtained AF: for example and no exists (as in Fig. 4). Thus, the more operations are allowed for a framework, the more difficult it will be to change the extensions set for such semantics. We define the expansionbased robustness of a graph for a generic as follows.
Definition 10 (Localexpansion robustness).
The localexpansion robustness degree of an AF w.r.t. a semantics is measured as the maximum number of invariant operators that can be applied on s.t. .
4 Related Work
In the following we review the most meaningful works related to what presented in this paper.
Rienstra et al. [Rienstra, Sakama, and van der Torre2015] focus on finding conditions under which the evaluation of an AF remains unchanged when an attack is added or removed. The authors consider grounded, complete, preferred, stable and semistable semantics and, for each of them, a set of properties for which extensions are preserved is given. Those properties are in the form: “given a certain labelling, attacks between two arguments with labels X and Y respectively are allowed (or not) for the semantics ”. Invariance is intended w.r.t. a single extension and not w.r.t. the whole semantics (as we do).
The problem of finding principles stating whether an extension does not change after adding/removing an attack between two arguments is also addressed by Boella et al. in [Boella, Kaci, and van der Torre2009a] and [Boella, Kaci, and van der Torre2009b]. Differently from us, the authors consider only the case in which the semantics of an AF contains exactly one extension, using the grounded semantics as example.
Cayrol et al [Cayrol, de SaintCyr, and LagasquieSchiex2010] studied the impact on the evaluation of an AF when new arguments and attacks are added. They define a number of properties for the change operations according to how the extensions are modified. For instance, a change operation can be “conservative” if the set of extensions is the same after a change.
In [Cayrol, de SaintCyr, and LagasquieSchiex2008] is addressed the problem of revising AFs when a new argument is added. In particular, they focus on the impact of new arguments on the set of initial extensions, introducing various kinds of revision operators that have different effects on the semantics. For instance, Decisive revision, as its name suggests, allows for making a decision by providing a revised extensions set with a unique nonempty extension.
In [Baumann and Brewka2010] the problem of revising argumentation frameworks according to acquisition of new knowledge is taken into account. While attacks among the old arguments remain unchanged, new arguments and attacks among them can be added. In particular, the authors introduce the notion of enforcing, namely the process of modifying an AF (and possibly changing its semantics) in order to obtain a desired set of extensions. This notion departs from our work, in which we instead look for operations that leave the semantics unchanged.
Also Baumann introduces the concepts of update and deletion [Baumann2014], focusing on modifications that retract arguments and attacks form an AF. New notions of equivalence are characterized through the so called kernels, namely functions that delete redundant attacks from a given framework. We instead concentrate on devising operators that permit both to modify AFs without changing their semantics, and to give a measure of how robust is a given AF, w.r.t. changes on the attack relations set.
The concept of desire set is also studied by Boella et al. in [Boella et al.2008] with a work on persuasion in multiagent systems, addressing the problem of choosing arguments to add into a system in order to maximise their acceptability w.r.t. the receiving agent. To this purpose, the notion of “more appealing” argument is introduced: in making the choice of a belief to add, an argument is more appealing than another if it does not interact with previous goals and beliefs of the agent. This is in contrast with our goal that consists in keeping unaltered the set of extension.
The authors of [Croitoru and Kötzing2013] show that every AF can be augmented in a normal form preserving the semantic properties. In such normal form no argument attacks a conflicting pair of arguments. A augmentation is an alteration of an AF that leaves unchanged the semantics . The changes in the AF can involve arguments (the only allowed operation is the addition) and attacks. Due to the process through which is obtained, the normal form there proposed is not a representative for the isomorphism classes we use to construct our lattice.
A different, more restrictive, kind of equivalence is introduced in [Oikarinen and Woltran2011]: two AFs and are considered strongly equivalent to each other when they are equivalent after the conjunction with a third AF . Since our intent is to provide a method for building equivalent AFs through the addition/deletion of attacks on a same framework, the notion of standard equivalence results to be more fitting than the strong equivalence.
5 Conclusions and Future Work
We defined invariant operators for AFs w.r.t. the semantics: these operators allow for performing changes on AFs while preserving the semantics. In particular, we have defined two operators, one for the conflictfree and one for the admissible sets, which can be applied to AFs for adding attack relations without resulting in changes to the set of extensions. The operators we have introduced exploit the notion of reinstatement labelling, and thus can be applied without even being aware of the extensions admitted for a given semantics. Moreover we gave the definition for the semantic equivalence between AFs, and we presented a method for computing the expansionbased robustness degree of a framework.
Our study has a very wide set of future perspectives. First, we plan to design invariant operators w.r.t. the complete, stable, semistable, preferred and grounded semantics (until now studied only w.r.t. single extensions [Rienstra, Sakama, and van der Torre2015]). We would like to find the sets of arguments which are essential to preserve the whole semantics. Every change inside those sets modifies the semantics, while changes outside do not cause any alteration. By removing the noncore part of AFs, it is possible to obtain equivalent frameworks for which the computation of extensions is faster, especially for checking credulous/sceptical acceptance of arguments.
As further work, different notions of equivalence, e.g. local equivalence [Oikarinen and Woltran2011], could be taken into account, and additional modifications of AFs could be considered, as the deletion of attack or the addition/removal of arguments. We also plan to devise a more general notion of robustness, involving the new modifications proposed above. By relaxing the conditions underlying invariant operators, and thus allowing the semantics to change, other operators could be obtained, that allow to reach “compromises”: if two parts of a debate desire two different outcomes in terms of semantics, a compromise can be reached as a third semantics, that is the closest one w.r.t. those desired by both the counterparts. Definitions of closeness could be devised as well. Finally, we want to study local expansion operators also for semiringbased weighted AFs [Bistarelli, Rossi, and Santini2018, Bistarelli, Rossi, and Santini2016a, Bistarelli, Pirolandi, and Santini2009, Bistarelli and Santini2010].
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