: Argumentation Framework with Recursive Attacks
Abstract
The issue of representing attacks to attacks in argumentation is receiving an increasing attention as a useful conceptual modelling tool in several contexts. In this paper we present , a formalism encompassing unlimited recursive attacks within argumentation frameworks. satisfies the basic requirements of definition simplicity and rigorous compatibility with Dung’s theory of argumentation. This paper provides a complete development of the formalism complemented by illustrative examples and a detailed comparison with other recursive attack formalizations.
keywords:
Argumentation frameworks, Argumentation semantics, Argument attack relationnumbers,sort&compress
1 Introduction
An argumentation framework ( in the following), as introduced in the seminal paper by Dung dung1995 (), is an abstract structure consisting of a set of elements, called arguments, whose origin, nature and possible internal organization is not specified, and by a binary relation of attack on the set of arguments, whose meaning is not specified either. This abstract formalism has been shown to encompass a large variety of more specific formalisms in areas ranging from nonmonotonic reasoning to logic programming and game theory, and, as such, is widely regarded as a powerful tool for theoretical analysis. Several variations of the original formalism have been proposed in the literature. On one hand, some approaches enrich the original framework with additional concepts, necessary to modelling in a “natural” way specific reasoning situations. This is the case, for instance, of preferencebased argumentation amgoud&cayrol2002 (); amgoud&kaci2007 (), where a preference ordering among arguments is considered, of valuebased argumentation benchcapon2003 (), where a value is associated to arguments in order to account for the concept of preference (an investigation on the relations between preferencebased and valuedbased argumentation is given in kaci&vandertorre2007 ()), of bipolar argumentation amgoudetal2008 (); cayroletal2010 (), where a relation of support between arguments is considered besides that of attack, or of weighted argument systems dunneetal2009 (), where a weight indicates the relative strength of attacks. On the other hand, some proposals investigate generalized versions of the original definition (in particular, of the notion of attack), without introducing any additional concept within the basic scheme, as in modgil2007 (); benchcapon&modgil2008 (); modgil2009 (). This paper lies in the latter line of investigation and pursues the goal of generalizing the notion of attack by allowing an attack, starting from an argument, to be directed not just towards an argument but also towards any other attack. This will be achieved by a recursive definition of the attack, that leads to the proposal of a new framework called (Argumentation Framework with Recursive Attacks).
The paper is organized as follows. Section 2 recalls the basic notions and fundamental properties of Dung’s argumentation framework. Section 3 introduces the definition of accompanied by a discussion of its motivations and objectives. In particular, is required to parallel the semantics notions of Dung’s theory and their fundamental properties while extending them to recursive attacks in an intuitively plausible and formally simple way. Section 4 introduces the generalized version of the basic notions of defeat, conflictfree set, acceptable argument, characteristic function, and admissible set showing that the relevant requirements stated in Section 3 hold. Section 5 extends to the definitions of complete, grounded, preferred, stable, semistable and ideal semantics, showing that their required properties, paralleling the traditional ones, hold. Sections 6 and 7 deal with further relationships between and . The former shows that when an coincides with an (since no attacks to attacks are present) all the generalized notions are fully compatible with the original ones. In the latter a method to express an as an is provided. Section 8 draws a detailed comparison of with the related formalisms Extended Argumentation Framework () and Higher Order Argumentation Framework (HOAF). Finally Section 9 summarizes the main contributions of the paper and discusses directions for future research.
2 Background notions
In Dung’s theory an argumentation framework () is a pair where is a set of arguments and is a binary relation on it.
The terse intuition behind this formalism is that arguments may attack each other
and useful formal definitions and theoretical investigations may be built on this simple basis.
In particular, the notions recalled in Definition 1 lie at the heart
of the definitions of Dung’s argumentation semantics
Definition 1.
Given an :

a set is Dconflictfree if s.t. ;

an argument is Dacceptable with respect to a set (or, equivalently, is defended by ) if s.t. , s.t. ;

the function such that is called the Dcharacteristic function of ;

a set is Dadmissible if is Dconflictfree and every element of is Dacceptable with respect to , i.e. .
An argumentation semantics identifies for any argumentation framework a set of extensions,
namely sets of arguments which are “collectively acceptable”,
or, in other words, able to survive together the conflict represented by the attack relation: for instance
arguments that belong to all of extensions can be considered skeptically justified, while arguments belonging to at least an extension can be considered credulously justified.
We recall that while the grounded and ideal semantics always identify a unique extension for a given argumentation framework
(called grounded and ideal extension, respectively),
the preferred, stable, and semistable semantics can identify several extensions
(called preferred, stable, and semistable extensions, respectively)
Definition 2.
Given an :

a set is a Dcomplete extension if is Dadmissible and s.t. is Dacceptable w.r.t. , ;

a set is the Dgrounded extension if is the least (w.r.t. set inclusion) fixed point
^{4} of the Dcharacteristic function ; 
a set is a Dpreferred extension if is a maximal (w.r.t. set inclusion) Dadmissible set;

a set is a Dstable extension if is Dconflictfree and , s.t. ;

a set is a Dsemistable extension if is a Dcomplete extension with maximal (w.r.t. set inclusion) Drange (given a set , the Drange of , denoted as , is where s.t. );

a set is the Dideal extension if is the maximal (w.r.t. set inclusion) Dideal set (a set is Dideal if it is Dadmissible and s.t. is a Dpreferred extension, ).
It is easy to note that any extension of the above semantics is a Dadmissible set, that is, it is able to defend all of its arguments (in the sense that arguments are Dacceptable w.r.t. the extension itself). The notion of acceptability and the related notion of admissibility are supported by intuition and satisfy a set of fundamental properties which in turn entail several desirable consequences, holding even in the infinite case, such as the existence of preferred extensions as well as the existence and uniqueness of the grounded extension. These properties are recalled in the following proposition dung1995 ().
Proposition 1.
Given an :

the Dcharacteristic function preserves Dconflictfreeness, i.e. given a set , if is Dconflictfree then also is Dconflictfree;

the Dcharacteristic function is monotonic, i.e. if , then ;

Dung’s fundamental lemma: given a Dadmissible set and two arguments that are Dacceptable w.r.t. , it holds that is Dadmissible and is Dacceptable w.r.t. ;

the set of all admissible sets form a complete partial order w.r.t. set inclusion.
3 Motivations and requirements
In Dung’s theory, arguments are regarded as the only entities that may be in conflict with each other and may be defeasible. The issue of extending the framework in such a way as also attacks are allowed to feature these properties has recently received significant attention in the literature. In fact, enabling attacks to attacks and considering them defeasible turns out to provide a useful and intuitively plausible formal counterpart to representation and reasoning patterns commonly adopted in various contexts. For instance, an approach to reasoning about preferences based on attacks to attacks has been introduced in modgil2007 () and extensively developed in modgil2009 (). In boellaetal2008 (); boellaetal2008b () attacks to attacks are considered in the context of reasoning about coalitions, while in barringeretal2005 () attacks to attacks are discussed in connection with the notions of strength, support and temporal dynamics.
The present paper contributes to the research line on formalizing attacks to attacks in argumentation by pursuing the following main objectives:

encompassing an unrestricted recursive notion of attack to attack;

keeping the proposed formalism as simple as possible;

encompassing Dung’s as a special case of the proposed formalism;

ensuring compatibility between the semantics notions in the proposed formalism and those in Dung’s .
As to the first point, in some previous proposals (e.g. modgil2007 (); modgil2009 ()) only one level of recursion is allowed, i.e. attacks attacking other attacks can not in turn be attacked. While this choice may be justified in specific contexts (e.g. reasoning about preferences), we aim at proposing a more general formalism which is able to accommodate various kinds of representation and reasoning needs related to recursive attacks. In particular, further levels of recursive attacks can be considered in the area of modelling decision processes as shown by the following example, which will be used throughout the paper to illustrate the main concepts of the proposed approach.
Suppose Bob is deciding about his Christmas holidays and, as a general rule of thumb, he is willing to buy cheap last minute offers. Suppose two such offers are available, one for a week in Gstaad and another for a week in Cuba. Then, using his behavioral rule, Bob can build two arguments, one, let say , whose premise is “There is a last minute offer for Gstaad” and whose conclusion is “I should go to Gstaad”, the other, let say , whose premise is “There is a last minute offer for Cuba” and whose conclusion is “I should go to Cuba”. As the two choices are incompatible, and attack each other, a situation giving rise to an undetermined choice. Suppose however that Bob has a preference for skiing and knows that Gstaad is a renowned ski resort. The point now is: how can we represent this preference? might be represented implicitly by suppressing the attack from to , but this is unsatisfactory, since it would prevent further reasoning on , as described below. So let us consider as an argument whose premise is “Bob likes skiing” and whose conclusion is “If possible, Bob prefers a ski resort”. might then attack , but this would not be sound since is not actually in contrast with the existence of a good last minute offer for Cuba and the fact that, according to Bob’s general behavioral rule, this provides him with a good reason for going to Cuba. Thus, following modgil2009 (), it seems more reasonable to represent as attacking the attack from to , causing to prevail. Note that the attack from to is not suppressed, but only made ineffective, in the specific situation at hand, due to the attack of .
Assume now that Bob learns that there have been no snowfalls in Gstaad since one month and from this fact he derives that it might not be possible to ski there. This argument (), whose premise is “The weather report informs that in Gstaad there were no snowfalls since one month” and whose conclusion is “It is not possible to ski in Gstaad”, does not affect neither the existence of last minute offers for Gstaad nor Bob’s general preference for ski, rather it affects the ability of this preference to affect the choice between Gstaad and Cuba. Thus argument attacks the attack originated from .
Suppose finally that eventually Bob is informed that in Gstaad it is anyway possible to ski, thanks to a good amount of artificial snow. This leads to building an argument, let say , which attacks , thus in turn reinstating the attack originated from and intuitively supporting the choice of Gstaad. A graphical illustration of this example is provided in Figure 1.
As pointed out by one of the reviewers of this paper, alternative formalizations of this example not involving attacks to attacks are possible. For instance, from the general preference for skiing, represented in the example by argument , one might derive a distinct and more specific argument representing the preference for Gstaad over Cuba. In this case argument (instead of ) would attack through and argument would attack (instead of ) through . Of course, the representation adopted for this example – like any formal representation of a real situation – is a matter of modelling choice. In general, we do not claim that there are indisputable, theoretical reasons for asserting that recursive attacks are strictly necessary. Indeed, technically speaking, extended argumentation frameworks encompassing attacks to attacks do not feature an augmented expressive power with respect to Dung’s formalism, as they can be translated into traditional argumentation frameworks, as shown for instance in modgil2009 () and in Section 7 of the present paper. From a modelling point of view, however, it can be observed that attacks to attacks offer a useful tool supporting a natural representation of some reasoning patterns.
As a further example, consider the case presented in modgil2009 () concerning two agents P and Q exchanging arguments about weather forecasts (see Figure 2). Argument , asserted by agent P, can be synthesized as “Today will be dry in London since the BBC forecast sunshine”, while agent Q asserts argument “Today will be wet in London since CNN forecast rain”. Arguments and have contradictory conclusions and therefore attack each other. Preferences may then be expressed by P and Q in order to resolve this undecided situation. For instance P may state an argument “But the BBC are more trustworthy than CNN”, which expresses a preference for BBC, while Q may reply with an argument “However, statistically CNN are more accurate forecasters than the BBC” expressing a preference for CNN. The two conflicting preferences attack each other and, according to the preference modeling adopted in modgil2009 (), attacks the attack from to , while attacks the attack from to . Agent Q may then state an argument asserting that “Basing a comparison on statistics is more rigorous and rational than basing a comparison on your instincts about their relative trustworthiness”. As argument expresses a preference for over , attacks the attack from to . Now, in order to see how recursive attacks may play a role in this context, consider the following additional argument asserted by P: “However, BBC has recently changed its whether forecast model, no information on the new model is available; therefore statistics on CNN loses prevalence over personal opinion about BBC”. does not attack neither that states the preference for CNN’s weather forecast over BBC’s one based upon statistics, nor , which states the general principle that basing a comparison on statistics is more rigorous and rational than basing a comparison on instincts. Obviously, it does not attack neither , nor , nor . In fact, attacks the assumption that affects the attacks between and : while it is generally accepted that basing a comparison on statistics is more rigorous that basing a comparison on personal intuition, in the case at hand, existing statistics are not decisive for a comparison between the accuracy of CNN and BBC forecasts. In other words provides a good reason for believing that does not attack the attack from to and this can be modelled as an attack from to the attack originating from . Therefore the situation remains undecided and both attacks between and are still in force.
Given the kind of representation needs illustrated above, we pursue the second and third objectives stated above by introducing in a rather straightforward way the fundamental definition of our proposal, namely the concept of argumentation framework with recursive attacks.
Definition 3 ().
An Argumentation Framework with Recursive Attacks () is a pair where:

is a set of arguments;

is a set of attacks, namely pairs s.t. and ( or ).
Given an attack , we say that is the source of , denoted as and is the target of , denoted as .
When useful, we will denote an attack to attack explicitly showing all the recursive steps implied by its definition; for instance means where .
The formalization of Bob’s last minute dilemma in terms of gives a simple illustration of the use of the formalism.
Example 1 (Bob’s last minute dilemma).
Let be an where: and , with , , , , .
As to our third objective, it can be noted that an is also an when does not include pairs such that .
The fourth highlevel objective of “compatibility” concerns the semantics notions which will be introduced in Sections 4 and 5. The underlying idea is that the basic concepts of conflictfreeness, acceptability, admissibility and the various proposals of extensionbased semantics are formally introduced in the context of , by explicitly considering both arguments and attacks. We remark in particular that, according to Definition 3, we regard attacks as entities which are rooted in arguments and, as a consequence, we require that their inclusion in an extension is possible only in case of inclusion of their source argument too. This choice ensures preservation of the main lines of Dung’s wellestablished conceptual framework for semantics definition, while anyway reflecting the extended (in a sense, empowered) role ascribed to attacks in , in particular their defeasibility.
From a more formal perspective, the objective of “compatibility” leads to the following requirements:

the fundamental properties listed in Proposition 1 should still hold for the parallel concepts introduced in the context of ;

in the case where an is also an , a bijective correspondence between the semantics notions according to the two formalisms should hold.
4 Basic semantic notions for
4.1 Defeat and conflictfree sets
As a starting point for the definition of any semanticsrelated notion we consider the concept of defeat. According to the role played by attacks in we introduce a notion of direct defeat which regards attacks, rather than their source arguments, as the subjects able to defeat arguments or other attacks. This is also coherent with the fact that an attack can be made ineffective by attacking the attack itself rather than its source.
Definition 4 (Direct Defeat).
Let be an , , : directly defeats iff .
Moreover, according to the idea that an attack is strictly related to its source, we introduce a notion of indirect defeat for an attack, corresponding to the situation where its source receives a direct defeat.
Definition 5 (Indirect Defeat).
Let be an and : if directly defeats then indirectly defeats .
Example 01 (continued).
In there are the following direct and indirect defeats: directly defeats ; indirectly defeats ; directly defeats ; indirectly defeats ; directly defeats ; directly defeats ; directly defeats ; indirectly defeats .
As a special, but significant, situation note that in case of a selfattacking argument, exemplified by the with , directly defeats and indirectly defeats itself.
Summing up, a defeat is either a direct or indirect defeat.
Definition 6 (Defeat).
Let be an , , : defeats , denoted as , iff directly or indirectly defeats .
The definition of conflictfree set follows directly, requiring the absence of defeats.
Definition 7 (Conflict–free set).
Let be an , is conflict–free iff s.t. .
The definition of conflictfree set for is formally quite similar to the corresponding one in but they feature substantial differences. A first one, which is quite evident and common to other notions, concerns the fact that a set of arguments and attacks, rather than just a set of arguments is considered. A slightly subtler one, related to the underlying notion of defeat, consists in the fact that in every set of arguments is conflictfree, since only the explicit consideration of attacks gives rise to conflict in this approach. While this may sound peculiar according to the “traditional” view, it is again coherent with the central role played by attacks and, as it will be seen later, does not prevent (indeed it enables) the achievement of the compatibility requirement with .
Example 01 (continued).
Consider : as explained above, is conflictfree as it does not explicitly include any attack. On the other hand, the sets , , are not conflictfree. Note also that Definition 7 encompasses sets consisting of attacks only. For instance the set is not conflictfree since ( indirectly defeats ) and, analogously, .
4.2 Acceptability and characteristic function
The definition of acceptability is formally very similar to the traditional one, apart from the fact of encompassing sets of both arguments and attacks.
Definition 8 (Acceptability).
Let be an , and : is acceptable w.r.t. (or, equivalently is defended by ) iff s.t. s.t. .
Note that while acceptability is defined with reference to a set possibly including both arguments and attacks, only attacks are “effective” as far as acceptability is concerned. In fact it is easy to see that an element (either argument or attack) is acceptable w.r.t. a set if and only if it is acceptable w.r.t. to .
Example 01 (continued).
Considering , it can be seen that is acceptable w.r.t. and w.r.t. , while it is not acceptable w.r.t. . As other examples, is acceptable w.r.t. , and is acceptable w.r.t. .
Lemma 1 shows that the acceptability of an attack implies the acceptability of its source, in accordance with the requirements mentioned in Section 3.
Lemma 1.
Let be an and . If an attack is acceptable w.r.t , then is acceptable w.r.t to .
Proof.
Suppose is not acceptable w.r.t. . Then, s.t. and s.t. . But since and , then ; therefore is not acceptable w.r.t. . Contradiction. ∎
The definition of characteristic function parallels the traditional one.
Definition 9.
The characteristic function of an is defined as follows:
Propositions 2 and 3 show that the fundamental properties of preserving conflictfreeness and being monotonic hold for the characteristic function, as required.
Proposition 2.
Let be an . If is conflictfree, then is also conflictfree.
Proof.
Assume that there are and in such that . By the acceptability of , there exists s.t. . Then, by the acceptability of there is s.t. , contradicting the hypothesis that is conflictfree. Therefore is conflictfree. ∎
Proposition 3.
Let be an . The Function is monotonic w.r.t. set inclusion.
Proof.
Letting , we have to show that , i.e. that every which is acceptable w.r.t. is acceptable w.r.t. . Suppose that is acceptable w.r.t. but not w.r.t. . Then, s.t. and s.t. , which, since , implies s.t. , which contradicts the hypothesis that is acceptable w.r.t. . ∎
4.3 Admissibility
The definition of admissible sets in requires conflictfreeness and acceptability of all set elements, exactly as in .
Definition 10 (Admissibility).
Let be an : is admissible iff it is conflict–free and each element of is acceptable w.r.t. (i.e. ).
As required, a parallel of Dung’s fundamental lemma holds in the context of .
Lemma 2 (Fundamental lemma).
Let be an , an admissible set and elements acceptable w.r.t. . Then:

is admissible; and

is acceptable w.r.t. .
Proof.

is acceptable w.r.t. therefore each element of is acceptable w.r.t. . Suppose is not conflict–free; therefore there exists an element such that either or . From the admissibility of and the acceptability of there exists an element such that or . Since is conflict–free it follows that . But then from the acceptability of there must exist an element such that . Contradiction.

Immediate from Proposition 3. ∎
The following theorem completes the verification that satisfies all the fundamental properties of Dung’s theory listed in Proposition 1.
Theorem 1.
Let be an . The set of all admissible sets of forms a complete partial order with respect to set inclusion.
Proof.
We have to prove that (i) the set of all admissible sets has a least element and (ii) each chain of admissible sets has a least upper bound. Point (i) immediately follows from the fact that the empty set is admissible, therefore it is obvioulsy the least element. As for (ii), let be a chain of admissible sets: we prove that is admissible, thus obviously a least upper bound of . First, is conflictfree, otherwise such that , entailing that such that and contradicting the admissibility of . Second, suppose that and : we have to prove that such that . The conclusion follows from the fact that such that , and since is admissible such that . ∎
Example 01 (continued).
In there are fourty admissible sets, denoted in the following as . First observe that according to Definition 10 the empty set is admissible for any , thus we have . As to sets consisting of arguments only, note that only unattacked arguments can be admissible by themselves since in defense is carried out by attack elements (for instance requires for its defense). Thus we have , , and of course their union (the adopted numbering is in accordance with Figure 3). Also singletons consisting of (directly or indirectly) unattacked attacks and those able to defend themselves on their own are of course admissible, yielding and (note for instance that is indirectly defeated by and it does not defend itself). Of course any set including only these individually admissible elements is admissible too, giving rise to 11 further admissible sets: , , , , , , , , , , .
Considering now defense by individually admissible attacks we note that defends by indirectly defeating and defends by indirectly defeating , leading to , . Of course the union of these two sets, being conflictfree, is admissible too, leading to . Adding other unattacked elements to any of these three sets preserves admissibility, leading to the following 14 admissible sets: , , , , , , , , , , , . , .
Since , being defended by , in turn defends by directly defeating , the set is admissible. Again, adding unattacked elements gives rise to the following 6 further admissible sets: , , , , , .
5 Semantics for
In this section we define and analyse the semantics corresponding to the ones listed in Definition 2.
5.1 Complete Semantics
The notion of complete extension closely parallels the traditional one by requiring admissibility and the inclusion of any acceptable argument.
Definition 11 (Complete extension).
Let be an . A set is a complete extension if and only if is admissible and every element of which is acceptable w.r.t. belongs to , i.e. .
By inspection of Definitions 10 and 11 it is immediate to see that a complete extension can be equivalently characterized as a conflictfree set which is a fixed point of , i.e. such that .
Example 01 (continued).
In there is exactly one complete extension: .
We introduce also a more articulated (shown in Figure 4) which will be useful for illustration and comparison of the semantics to be introduced in the following.
Example 2.
Let be an , where: , and with , , , , , , , , , .
As to the complete extensions of , note first that the unattacked elements are , and and that defends both and by directly defeating . It follows that is a complete extension. Further note that defends itself, and by indirectly defeating and, analogously, defends itself and by indirectly defeating . This gives rise to two further complete extensions, namely and . All other arguments and attacks in have no defense and hence do not belong to any admissible set or complete extension.
5.2 Grounded Semantics
The definition of grounded semantics parallels Dung’s one: as in his approach, the basic properties of the characteristic function (whose validity we have already proved also in the context of ) ensure the uniqueness of the grounded extension and the fact that it can be equivalently characterized as the least complete extension.
Definition 12 (Grounded extension).
Let be an . The grounded extension of is the least fixed point of .
Lemma 3.
The grounded extension is the least complete extension.
Example 01 (continued).
The grounded extension of is .
Example 02 (continued).
The grounded extension of is .
5.3 Preferred semantics
As expected, preferred extensions are defined as maximal admissible sets.
Definition 13 (Preferred extension).
Let be an . A set is a preferred extension of iff it is a maximal (w.r.t. set inclusion) admissible set.
Theorem 2.
Let be an . For each admissible set of , there exists a preferred extension of such that .
Corollary 1.
Every possesses at least one preferred extension.
It also holds that preferred extensions are complete (and hence can be equivalently characterized as maximal complete extensions).
Lemma 4.
Every preferred extension is a complete extension, but not vice versa.
Example 01 (continued).
In the only preferred extension is the grounded extension, i.e. .
Example 02 (continued).
The preferred extensions of are and , while the complete (and grounded) extension is not preferred since it is not maximal w.r.t. set inclusion.
5.4 Stable semantics
Stable semantics is based, as usual, on the idea that each extension attacks all elements not included in it.
Definition 14 (Stable extension).
Let be an . A set is a stable extension of if and only if is conflictfree and , s.t. .
Stable extensions are also preferred, but not vice versa. In particular, as in , there are cases where no extensions complying with Definition 14 exist.
Lemma 5.
Every stable extension is a preferred extension, but not vice versa.
Proof.
It is easy to see that each stable extension is a maximal complete extension, hence a preferred extension. To show that the reverse does not hold, consider an consisting just of a selfdefeating argument: with , . The empty set is a preferred extension of but clearly is not stable. ∎
Example 01 (continued).
The only stable extension of is .
Example 02 (continued).
The two preferred extensions of are not stable. In particular neither of them includes nor defeats the elements , , , and .
5.5 Semistable semantics
Semistable semantics caminada2006 () is based on the idea of prescribing the maximization of both the arguments included in an extension and those attacked by it, i.e. of maximizing the extension range.
Definition 15 (Range).
Let be an and let be a set of arguments and attacks. The range of , denoted as , is defined as where .
Definition 16 (Semistable extension).
Let be an , a set is a semistable extension iff is a complete extension with maximal (w.r.t set inclusion) range.
Proposition 4 summarizes the relations of semistable with stable and preferred semantics in , paralleling those holding in .
Proposition 4.
For any

if stable extensions exist, then they coincide with semistable extensions;

every semistable extension is preferred but not viceversa.
Proof.
As to the first point, note that, by definition, the range of any stable extension coincides with , which is of course the largest possible one. Moreover stable extensions are admissible sets by Lemma 5, hence the conclusion. As to the second point, suppose a semistable extension is not preferred, i.e. there is an admissible set strictly including it: the range of strictly includes the range of , contradicting the hypothesis that is a semistable extension. On the other hand there are preferred extensions which are not semistable as in Example 2 (see below). ∎
Example 01 (continued).
The only semistable extension of is .
Example 02 (continued).
Consider the preferred extensions of . Letting it holds . On the other hand letting it holds . is the only semistable extension of since .
5.6 Ideal semantics
Ideal semantics dungetal2006 () considers the largest admissible set included in all preferred extensions.
Definition 17 (Ideal extension).
Let be an . A set is ideal iff is admissible and s.t. is a preferred extension of , . The ideal extension is the maximal (w.r.t. set inclusion) ideal set.
Definition 17 anticipates the uniqueness of ideal extension shown in the following proposition.
Proposition 5.
The ideal extension is unique.
Proof.
Suppose that there are two distinct maximal ideal sets and complying with Definition 17. Now is included in all the preferred extensions, hence it is conflictfree, and defends all its elements, hence it is also admissible. Therefore is a larger ideal set than and , contradicting the hypothesis. ∎
It can also be seen that the ideal extension includes all acceptable elements, i.e. it is a complete extension.
Lemma 6.
The ideal extension is a complete extension.
Proof.
The ideal extension is admissible by definition, thus it is sufficient to show that it includes any element which is acceptable w.r.t. it. Since the ideal extension is contained in any preferred extension, it is easy to see that is acceptable w.r.t any preferred extension too. By Lemma 4 the preferred extensions are also complete, therefore they must all include . As a consequence, is included in the ideal extension, otherwise including it would give rise by the fundamental lemma (Lemma 2) to a strictly greater admissible set contained in all preferred extensions, contradicting the maximality of the ideal extension. ∎
Since the grounded extension is included in all complete extensions (and hence in all preferred extensions) and is admissible, the ideal extension is a (possibly strict) superset of the grounded extension.
Example 01 (continued).
The ideal extension of is .
Example 02 (continued).
The ideal extension of is .
6 Compatibility with
In this section we prove the satisfaction of the compatibility requirement formally stated at the end of Section 3 for the case where a given is an . To be precise, throughout this section when stating “let be an ”, we will consider that is an such that attacks involve just arguments rather than being directed against other attacks (formally, ).
First of all, it is easy to see that, in this case, a dual property holds w.r.t. Lemma 1.
Lemma 7.
Let be an and . If an argument is acceptable w.r.t. , then any such that is acceptable w.r.t. .
Proof.
We prove that is defended by from any attack. For any such that , does not directly attack since is an . As a consequence, it must be the case that , i.e. : since is acceptable w.r.t. , then s.t. . ∎
As anticipated above, we show that in the case of an the semantics defined in Section 5 reduce to those adopted in the context of the traditional Dung’s framework. Of course, this correspondence can only be established through a mapping, since extensions in , differently from those in the traditional Dung’s framework, include both arguments and attacks. Accordingly, Definition 18 provides a natural way to extend sets of arguments (corresponding to traditional extensions) into sets of arguments and attacks (corresponding to extensions).
Definition 18 ( operator).
Let be an . Given a set of arguments , .
In words, given a set of arguments , the operator completes with all of the attacks arising from it. This operator will play a key role in proving the satisfaction of compatibility requirements for all the considered semantics (Propositions 611). In fact, given a semantics , the compatibility requirement (in the case where a given is an ) will be expressed as a bijective correspondence, through the operator, between (i) extensions prescribed by in the traditional formulation, and (ii) extensions prescribed by in the formulation. More specifically, we will show for each semantics that if a set of arguments is an extension according to then is an extension according to and, vice versa, if a set of arguments and attacks is an extension according to then there is a set of arguments such that and is an extension according to .
The use of the operator to prove these correspondences is supported by Lemmata 1 and 7. The relevant properties shown in Lemma 8 will be exploited in the following.
Lemma 8.
Let be an , and let two sets of arguments. It holds that:

iff

iff

Proof.

As for the direction, let . If is an argument then by definition thus also belongs to . In the other case and by definition , thus and, again by definition, . As for the other direction, if an argument then it belongs to , and since it is an argument then it must be the case that .

Taking into account the previous point, for the direction we have just to show that, considering an argument such that , it holds by definition that but , entailing that . As for the other direction, by the hypothesis . If is an argument then by definition and , if then these conditions hold for : in any case, .

or .
∎
A key role in proving the satisfaction of the compatibility requirement is played by showing in Proposition 6 that the desired bijective correspondence between and extensions holds for the case of complete semantics.
Proposition 6.
Let be an . Then, is a complete extension of iff where is a Dcomplete extension of .
Proof.
. We first show that .
In fact, for any
if then it obviously belongs to .
In the other case, namely , is acceptable w.r.t.
since it belongs to which is a complete extension,
thus by Lemma 1 ,
which by definition of the operator entails .
On the other hand, for any
if then it obviously belongs to ;
in the other case and follows from
Lemma 7 and the fact that is a complete extension.
According to this result, we have to show that is a Dcomplete extension.
First, is Dconflictfree,
otherwise there would exist with ,
i.e. letting we would have
with :
since both and belong to ,
would not be conflictfree, contradicting the hypothesis.
Then, we show that is Dadmissible, i.e.
given , for any such that
such that .
Since , letting
yields with .
Since and is admissible by the hypothesis,
,
which taking into account that is an yields .
Since , ,
and the thesis follows from .
Finally, we prove that is Dcomplete by showing that,
for any which is Dacceptable w.r.t. ,
is acceptable w.r.t. :
by the hypothesis that is a complete extension it then follows that
, i.e. .
Let us then consider an attack such that .
Obviously this is equivalent to ,
and since is Dacceptable w.r.t. ,
.
Letting ,
we have ,
and since
then also .
Summing up, for any such that
such that .
. We have to show that is a complete extension, namely conflictfree,
admissible and including all acceptable elements.
As to the first point, assume by contradiction that
such that .
While by definition, either belongs to or to .
In the first case we have ,
and by definition of both and belong to ,
contradicting the fact that is Dconflictfree.
In the other case, i.e. is an attack, since is an
we have ,
i.e. .
But entails ,
again contradicting the fact that is Dconflictfree.
To show that is admissible, consider a generic
and suppose that such that .
Taking into account that is an and that by definition includes the sources of
all the attacks it includes, it is easy to see that
such that (where is either or ).
Therefore , and since is Dadmissible
.
Letting ,
we have and by definition of
it is the case that