We introduce two generic notions related to consistency and reinstatement in an abstract labelling setting, based on a relation of intolerance between the labelled elements and two specific relations, called incompatibility and reinstatement violation, between the labels assigned to them. This way, the approach allows a spectrum of consistency and reinstatement requirements depending on the actual choice of these relations. As a first application to formal argumentation, we show that traditional Dung's semantics can be expressed as combinations of diferent consistency and reinstatement requirements in this context. Consistency, Reinstatement, Argumentation semantics, Argument justification In formal argumentation, the presence of conflicts between arguments is a key aspect that calls for mechanisms able to produce sensible reasoning outcomes. In particular, these outcomes are typically required to satisfy two somewhat dual properties, i.e. those that have intuitively to do with the notion of consistency and those reflecting a requirement of completeness. For instance, in abstract argumentation semantics [1, 2] either extensions or labellings are typically required to satisfy the property of conflict-freeness, and it is also desired that arguments are accepted when all of their attackers are definitely outright, as indicated by the definition of complete extensions in [1] and the various notions of reinstatement introduced in [3].
(M. Giacomin) an incompatibility relation and a reinstatement violation relation between the labels are at the basis of the proposed notions of consistency and reinstatement. To show the application of the proposed concepts in the context of abstract argumentation, we prove in Section 3 that Dung’s traditional semantics can be expressed as combinations of diferent consistency and reinstatement requirements, particularly with diferent incompatibility and reinstatement violation relations. The relationships of this work with previous literature and various perspectives of future development are finally discussed in Section 4. 2. Generalizing Consistency and Reinstatement for
Labelling-based Systems
In a variety of contexts, the assessments of entities of various kind are expressed by assigning
them a label. In order to provide a common ground to characterize such diferent contexts, in
[
In the following, we introduce the notions of the model from the top level to the bottom level. Definition 1. A set of assessment classes is a set equipped with a total order ≤ (i.e. a reflexive, transitive and antisymmetric relation such that any two elements are comparable) and including a maximum and a minimum element (i.e. an element ∈ such that ∀ ′ ∈ it holds that ′ ≤ or ≤ ′, respectively) which are assumed to be distinct.
In the following we will abbreviate the term ‘set(s) of assessment classes’ as sac(s). Intuitively, the order is meant to capture an abstract distinction between diferent levels of positivity of the assessment, with 1 ≤ 2 meaning that 2 corresponds to an at least as positive assessment as 1. In the following we will mostly use a tripolar sac 3 = {pos, mid, neg} with neg ≤ mid ≤ pos and the intuitive meaning that pos corresponds to a definitely positive assessment, neg to a definitely negative assessment, and mid to an intermediate situation. The basic idea, expressed by the following definition, is that a sac is used to classify the elements of a set of labels according to their level of positivity. Note that the elements of a sac are called classes because in general more than one label can be mapped to the same class.
Definition 2. Given a set of assessment classes , a -classified set of assessment labels is a set Λ equipped with a total function Λ ∶ Λ → . The total preorder induced on Λ by Λ will be denoted by ⪯ where 1 ⪯ 2 if Λ( 1) ≤ Λ( 2). As usual, 1 ≺ 2 will denote 1 ⪯ 2 and 2 ⪯̸ 1
The fact that ⪯ is a total preorder is shown in the following proposition.
Proposition 1. Given a set of assessment classes and a -classified set of assessment labels Λ, the relation ⪯ as introduced in Definition 2 is reflexive and transitive, and for any 1, 2 ∈ Λ, 1 ⪯ 2 or 2 ⪯ 1. □
It is easy to see that ⪯ is not necessarily an order, since diferent labels can be classified with the same assessment class, thus antisymmetry does not hold.
We will abbreviate the term ‘set(s) of assessment labels’ as sal(s) and omit ‘ -classified’, when is not ambiguous. Also, to distinguish preorders referring to diferent sals, given a sal Λ we will denote the relevant preorder as ⪯Λ.
The notion of labelling based on a sal is the usual one.
Diferent sals can be used to express assessments in distinct, but possibly related,
evaluation contexts. For instance, in the context of argument acceptance evaluation based on the
labelling-based version of Dung’s semantics [
We are now ready to introduce the generalized notions of consistency and reinstatement in this formal context. Intuitively, they correspond to dual notions aimed at satisfying somehow conflicting goals: • An inconsistency arises when two elements of a set which cannot stand each other are assigned labels which are ‘too positive’ altogether. Correspondingly, consistency is satisfied whenever this situation does not hold for any couple of elements. • Reinstatement is violated when an element of a set is assigned a label which is ‘too negative’, i.e. a negative label is assigned without a suficient reason. A suficient reason holds if another element which cannot stand together is assigned a suficiently positive label. Correspondingly, reinstatement holds whenever a suficiently positive label is assigned to any element such that all of its conflicting elements are negatively assessed.
It can be seen that consistency and reinstatement are dual properties. In particular, a skeptical assessment which assigns the most negative label to all elements trivially satisfies consistency, but violates reinstatement. Conversely, assigning the most positive label to all elements trivially satisfies reinstatement, but violates consistency whenever two elements cannot stand each other.
According to this informal introduction, both inconsistency and reinstatement violation can be understood as arising from two components: an intolerance relation at the level of the assessed elements, indicating who cannot stand whom, and a relation at the level of the labels indicating which pairs of assessments correspond to a violation if ascribed to a pair of elements connected by the intolerance relation.
Definition 4. Given a set , an intolerance relation on is a binary relation int ⊆ × , where ( 1, 2) ∈ int indicates that 1 is intolerant of 2 and will be denoted as 1 ⊙ 2, while ( 1, 2) ∉ int will be denoted as 1 ⊖ 2.
Note that we do not make any assumption on the intolerance relation, in particular it needs not to be symmetric.
To exemplify, in languages equipped with negation, typically intolerance between language
elements coincides with negation (a symmetric relation where each element has exactly one
opposite), however more general forms of contrariness have been considered in argumentation
contexts, where the corresponding intolerance relation may not be symmetric and allows the
existence of multiple contraries for an element [
Due to the dual nature of consistency w.r.t. reinstatement, violations at the level of the labellings are modelled by distinct relations, namely an incompatibility relation and a reinstatement violation relation on assessment labels, respectively. In the following we will assume that each of these relations on assessment labels is always induced by a corresponding relation on assessment classes.
Definition 5. Given a sac , an incompatibility relation on is a relation inc ⊆ × , where ( 1, 2) ∈ inc indicates that 1 is incompatible with 2 and will be denoted as 1⊡ 2, while ( 1, 2) ∉ inc will be denoted as 1⊟ 2. Given a -classified sal Λ, we define the induced incompatibility relation inc′ ⊆ Λ × Λ as follows: for every 1, 2 ∈ Λ, ( 1, 2) ∈ inc′ if ( Λ( 1), Λ( 2)) ∈ inc. With a little abuse of notation we will also denote ( 1, 2) ∈ inc′ as 1⊡ 2, and analogously for 1⊟ 2. Definition 6. Given a sac , a reinstatement violation relation on is a relation rv ⊆ × , where ( 1, 2) ∈ rv indicates that 1 is not suficiently positive to justify 2 and will be denoted as 1⊡ 2, while ( 1, 2) ∉ rv will be denoted as 1⊟ 2. Given a -classified sal Λ, we define the induced reinstatement violation relation rv′ ⊆ Λ × Λ as follows: for every 1, 2 ∈ Λ, ( 1, 2) ∈ rv′ if ( Λ( 1), Λ( 2)) ∈ rv. With a little abuse of notation we will also denote ( 1, 2) ∈ rv′ as 1⊡ 2, and analogously for 1⊟ 2.
From the intuitions underlying the concepts of consistency and reinstatement, some rather natural properties can be identified for incompatibility and, in a dual manner, for reinstatement violation relations on . The following definition introduces these properties for incompatibility relations.
Definition 7. Given a sac , let inc be an incompatibility relation on . We say that inc is well-founded if it satisfies the following properties:
As to the second property, assume by contradiction that ∃ ∈ such that ⊡ min() . By the monotonicity property and the definition of min() , ∀ ′ ∈ it holds that ⊡ ′, violating the third condition of Definition 7. The other condition can be proved in the same way. □
According to the above proposition, we can identify for any sac the minimal well-founded incompatibility relation as inc = {(max(), max())} .
Let us turn now to well-founded reinstatement violation relations.
Definition 8. Given a sac , let rv be a reinstatement violation relation on . We say that rv is well-founded if it satisfies the following properties: • rv is dually monotonic, i.e. given 1, 2 ∈ such that 1⊡ 2, for every pair 1′, 2′ ∈ such that 1′ ≤ 1 and 2′ ≤ 2 it holds that 1′⊡ 2
′ • rv is non empty, i.e. rv ≠ ∅ • ∀ 1 ∈ , ∃ 2 ∈ such that 1⊟ 2 and ∃ 2 ∈ such that 2⊟ 1
In order to provide an explanation of these requirements, let us refer again to the simple case depicted in Figure 1.
As to the first condition, we remark that reinstatement violation arises from a sort of ‘excess of cautiousness’ in assigning positive labels, i.e. a too much negative label is assigned to an element even in the absence of a positively assessed element linked by intolerance. Let us then consider the case where, in Figure 1, 1⊡ 2. This situation can be interpreted in two equivalent ways: 1. The label 1 assigned to 1 is too much negative to justify the label 2 assigned to 2 2. The label 2 assigned to 2 is too much negative w.r.t. the ‘not so positive’ label 1 assigned to 1 Accordingly, if 1′ ≤ 1 (i.e. positiveness of 1 does not increase) and 2′ ≤ 2 (i.e. positiveness of 2 does not increase), then it must also hold 1′⊡ 2′.
The second condition, i.e. that rv is non empty, is required to avoid an overly skeptical assessment attitude such that assigning the most negative label to all elements is always allowed, independently of the labels of incompatible elements.
The third condition has an analogous rationale w.r.t. the same condition appearing in Definition 7.
Also in this case two additional intuitive properties of well-founded reinstatement violation relations can be derived. First, two minimally positive labels cannot be ascribed together to conflicting elements 1. Second, the maximally positive label is compatible with any other label, in particular max() ⊟ max() , max() ⊟ min() and min() ⊟ max() .
Proposition 3. Given a sac , let rv be a well-founded reinstatement violation relation on . It then holds that: • min() ⊡ min() • ∄ ∈ such that ⊡ max() or max() ⊡ 1This refers to the simple case of Figure 1, while the general case is handled according to Definition 10. Proof: As to the first property, since rv ≠ ∅ there are 1, 2 ∈ such that 1⊡ 2. Taking into account that inc is dually monotonic, it obviously holds that min() ⊡ min() .
As to the second property, assume by contradiction that ∃ ∈ such that ⊡ max() . By the dual monotonicity property and the definition of max() , ∀ ′ ∈ it holds that ⊡ ′, violating the third condition of Definition 8. The other condition can be proved in the same way. □
According to the above proposition, we can identify for any sac the minimal nonempty reinstatement violation relation as rv = {(min(), min())} .
While we have considered above the particular case involving only a couple of elements of , in order to introduce our generalized notions of inconsistency and reinstatement violation we have to consider the general case of labellings of generic sets.
Let us start from our generalized notion of inconsistency of a labelling.
Definition 9. Given a set equipped with an intolerance relation int, a sac equipped with an incompatibility relation inc, and a -classified sal Λ, a Λ-labelling of is int-inc-inconsistent if ∃ 1, 2 ∈ such that 1 ⊙ 2 and ( 1)⊡( 2) Conversely, we say that a labelling is int-inc-consistent if it is not int-inc-inconsistent, i.e.
∀ 1, 2 ∈ such that 1 ⊙ 2, it holds that ( 1)⊟( 2)
The above definition corresponds to the idea that consistency violation arises from an excess of simultaneous positivity between any couple of incompatible elements, i.e. given 1 ∈ a single 2 satisfying the inc relation is suficient to yield inconsistency.
The following proposition is obvious and will not be proved.
Proposition 4. Given a set equipped with an intolerance relation int, a sac and a -classified sal Λ, consider two incompatibility relations inc and inc’ such that inc ⊆ inc′. Then, an int-incinconsistent Λ-labelling of is also int-inc’-inconsistent, and an int-inc’-consistent Λ-labelling of is also int-inc-consistent.
Turning to reinstatement violation, duality w.r.t. inconsistency is reflected also in the counterpart of Definition 9. In particular, given 2 ∈ , reinstatement is violated if all the elements 1 that are intolerant w.r.t. 2 do not provide a suficient reason (i.e. are not positive enough) to justify the ‘not so positive’ label assigned to 2. Accordingly, a Λ-labelling of should violate reinstatement if
∃ 2 ∶ ∀ 1 ∈ such that 1 ⊙ 2 it holds that ( 1)⊡( 2) while it should satisfy reinstatement if
∀ 2 ∈ , ∃ 1 ∈ such that 1 ⊙ 2 and ( 1)⊟( 2)
However, both conditions (3) and (4) are unsatisfactory for initial2 elements of , i.e. elements 2 of such that there are no elements 1 with 1 ⊙ 2. Such elements 2 trivially satisfy condition 2We borrow the terminology from abstract argumentation, where initial nodes are those without attackers. (1) (2) (3) (4) (3) and never satisfy condition (4), entailing that no labelling is able to satisfy reinstatement whenever there are initial elements in .
A special condition for initial elements is thus needed.
In this regard, a first option is to impose max() as the unique possible label for initial elements, on the grounds that there are no reasons against the acceptance of initial elements. However, this option looks somehow rigid, since a unique label is prescribed for initial nodes, and from a conceptual point of view it looks strange that initial elements receive a special treatment which completely neglects the reinstatement violation relation.
Another option is to introduce a special relation for initial elements which defines the set of their possible labels, so that such a set can be tuned in the same way as for the reinstatement violation relation. While this solution would achieve the maximum flexibility, it is still characterized by the same conceptual problem concerning a special treatment for initial elements, which would be completely independent from the way labels are selected for non initial elements.
We are thus lead to explore solutions where the set of possible labels for initial nodes is derived from the reinstatement violation relation. In this regard, the following two options for initial elements can be considered: 1. { ∈ Λ ∣ min() ⊟ Λ()} 2. { ∈ Λ ∣ ∀ ∈ , ⊟ Λ()}
Intuitively, according to the first option initial elements are equated to non initial elements where intolerant elements w.r.t. them are all labelled with minimally positive labels. Accordingly, the labels that the reinstatement violation relation allows for initial elements are the same that are allowed for an element 2 such that there is a unique element 1 such that 1 ⊙ 2, and the label assigned to 1 is associated to min() . In a sense, the absence of reasons against the acceptance of 2 is equivalent to a contrary reason with minimal acceptance degree.
The second option allows for initial nodes all those labels that would be allowed whatever the labels of intolerant nodes. The underlying idea is that the absence of reasons against the acceptance of a node 2, i.e. in case 2 is an initial element, must prevent only the labels that would be prevented by the reinstatement violation relation whatever the reason against their acceptance, i.e. when there are intolerant elements whatever their labels.
Interestingly enough, the two options turn out to be equivalent if one adopts a well-founded reinstatement violation relation, as the following proposition shows.
Proposition 5. Given a set equipped with an intolerance relation int, a sac equipped with a well-founded reinstatement violation relation rv, and a -classified sal Λ, it turns out that { ∈ Λ ∣ min() ⊟ Λ()} = { ∈ Λ ∣ ∀ ∈ , ⊟ Λ()} Proof: Let us first prove the ⊆ relation. Let ∈ Λ be a label such that min() ⊟ Λ() . By the definition of minimum, ∀ ∈ , min() ≤ . If by contradiction ⊡ Λ() then by dual monotonicity of rv it would be the case that min() ⊡ Λ() , contradicting the hypothesis that min() ⊟ Λ() .
As to the ⊇ relation, obviously any such that ∀ ∈ , min() ⊟ Λ() . ⊟ Λ() satisfies as a particular case
According to this result, we introduce our generalized notion of reinstatement violation of a labelling as follows.
Definition 10. Given a set equipped with an intolerance relation int, a sac equipped with a reinstatement violation relation rv, and a -classified sal Λ, a Λ-labelling of is int-rvuncompliant if □ (5) (6) (7) ∃ 2 ∈ ∶ { min() ⊡ Λ(( 2)) ∀ 1 ∈ such that 1 ⊙ 2 it holds that ( 1)⊡( 2) if 2 is initial otherwise Conversely, we say that a labelling is int-rv-compliant if it is not int-rv-uncompliant, i.e. ∀ 2 ∈ { min() ⊟ Λ(( 2)) ∃ 1 ∈ such that 1 ⊙ 2 and ( 1)⊟( 2) if 2 is initial otherwise
A corresponding result w.r.t. Proposition 4 holds.
Proposition 6. Given a set equipped with an intolerance relation int, a sac and a -classified sal Λ, consider two reinstatement violation relations rv and rv’ such that rv ⊆ rv′. Then, an int-rvuncompliant Λ-labelling of is also int-rv’-uncompliant, and an int-rv’-compliant Λ-labelling of is also int-rv-compliant.
Proof: If is int-rv-uncompliant, there is an argument which satisfies one of the two cases for 2 of condition (5) w.r.t. rv. Since rv ⊆ rv′, obviously this case would be satisfied also adopting rv’. The result concerning compliant labellings follows from the fact that a labelling is int-rv-compliant if it is not int-rv-uncompliant. □
A final comment can be devoted to the constraints imposed by consistent labellings on the possible labels for initial elements. In particular, according to Definition 9 a labelling is int-inc-consistent if ∀ 2 ∈ the following condition holds:
∀ 1 ∶ 1 ⊙ 2, ( 1)⊟( 2) If 2 is initial, then condition (7) is trivially satisfied, i.e. the possible labels for 2 are unconstrained. However, one may wonder whether a diferent outcome would be obtained by modifying Definition 9 so as to enforce a specific treatment for initial elements, similarly to int-rv-compliant labellings (see Definition 10). Taking into account the intuition behind consistency, the following two options for the possible labels of initial elements can be considered: 1. { ∈ Λ ∣ min()⊟ Λ()} 2. { ∈ Λ ∣ ∃ ∈ ∶ ⊟ Λ()}
Similarly to the counterpart condition in the case of reinstatement, the first option equates initial elements to non initial elements where intolerant elements w.r.t. them are all labelled with minimally positive labels. Again, the absence of intolerant elements (and thus of any simultaneous positivity) is considered equivalent to intolerant elements with minimally positive labels. The second option allows for initial nodes all those labels that would be allowed by at least one label assigned to an intolerant node. The underlying idea is that the absence of simultaneous positivity must leave the maximal freedom in choosing the labels for initial elements, thus any label that is allowed in case there is an intolerant element, whatever its label, must also be allowed for initial elements. It is easy to see that, adopting a well-founded incompatibility relation, both options do not enforce any constraint on the possible labels of initial elements. As to the first option, by Proposition 2 there is no ∈ such that min()⊡ , entailing that ∀ ∈ Λ min()⊟ Λ() . Of course, the second option enforces a weaker constraints w.r.t. the first one, as it is evident by considering = min() , thus it must allow as the first option all of the labels in Λ. Summing up, explicitly considering initial elements would not bring any modification to Definition 9, which thus conceptually corresponds to the dual counterpart of Definition 10. From a theoretical perspective, these considerations support the well-foundedness and generality of both definitions.
The generic definitions of inconsistency and reinstatement violation we have introduced
are ‘tunable’ as their instances can be ‘adjusted’ varying the incompatibility and reinstatement
violation relations, and possibly also the underlying intolerance relation and -classification,
giving rise to a family of alternative (in)consistency and reinstatement (violation) notions. In
particular, diferent combinations of (in)consistency and reinstatement (violation) notions are
able to capture diferent argumentation semantics in Dung’s framework, as discussed next.
3. Consistency and reinstatement properties in argumentation
semantics
As well-known, in abstract argumentation an argumentation semantics is a formal specification
of a criterion to determine the possible outcomes of a situation of conflict, represented by a
binary relation of attack (denoted as → in the following), between a set of arguments. A
set of arguments and the relevant attack relation are modelled by the traditional notion of
argumentation framework [
Definition 11. An argumentation framework is a pair = ( , →) where is a set of arguments and →⊆ × is a binary relation of attack between them. Given an argument ∈ , we denote as − the set { ∈ ∣ (, ) ∈→} . An argument such that − = ∅ is called initial.
In the extension-based approach to argumentation semantics the conflict outcomes are
expressed as sets of arguments called extensions and, in this context, two somewhat dual notions
corresponding to those introduced in the paper have been exploited in the relevant definitions.
On the one hand, a basic consistency notion called conflict-freeness has been traditionally
considered: a set of arguments is conflict-free if it does not include any pair of arguments , such
that ∈ −. On the other hand, the reinstatement criterion, as well as some of its variants, has
been made explicit in [
In this paper we consider the labelling-based approach to argumentation semantics. In particular, the outcomes are expressed as arguments labellings instead of extensions, i.e. as assignments of labels, taken from a given set, to the set of arguments . Using the set of three labels ΛIOU a correspondence can be drawn between extensions and labellings, while in general the labelling-based approach is more expressive than the extension-based approach.
Combining the generalized notions of consistency and reinstatement with three-valued
labellings enables to identify correspondences between diferent instances of our generalized
notions and diferent semantics. In particular, given an abstract argumentation framework, we
assume that the intolerance relation coincides with the attack relation, i.e. ⊙ if ∈ −, and
use the classification Λ3IOU introduced above. Then, an analysis of labelling-based semantics in
this perspective can be developed, as we do in the following, where we review the definitions
of some fundamental labelling-based semantics [
The simplest semantics notion is the one of conflict-freeness, recalled in Definition 12. Definition 12. Let be a labelling of an argumentation framework = ( , →) free if for each ∈ it holds that: . is conflict→-rv 3-compliant if for each ∈ Proposition 7. Let be a labelling of an argumentation framework = ( , →) it holds that if () = out then ∃ ∈ − ∶ () = . Then, is in.
Proof: Let be →-rv 3-compliant and assume by contradiction that there is an argument ∈ such that () = out and ∀ ∈ − () ≠ in. If is initial, according to Definition 10 it must be the case that min() ⊟ Λ(()) , i.e. taking into account the definition of rv 3 it holds that Λ(()) ∈ { mid, pos}, which contradicts () = out. If is not initial, according to Definition 10 it holds that ∃ ∈ ∶ ⊙ (i.e. ∈ −) and () ⊟() , i.e. taking again into account the definition of rv 3 and the fact that () = out we have that ∃ ∈ ∶ ∈ − and () ∈ { in}, contradicting the initial assumption that ∀ ∈ − () ≠ in.
As to the reverse direction, assume that for each ∈ if () = out then ∃ ∈ − ∶ () = in, and assume by contradiction that is →-rv 3-uncompliant. According to Definition 10, at least one of the following two cases holds.
1. There is ∈ such that is initial and min() ⊡ Λ(()) . Taking into account the definition of rv 3, this entails () = out. By the initial assumption ∃ ∈ − ∶ () = in, contradicting the fact that is initial. 2. There is a non initial argument ∈ such that ∀ ∈ such that 1 ∈ − it holds that () ⊡() . Taking into account the definition of rv 3, it must be the case that () = out and ∀ ∈ such that ∈ −, () ∈ { out, und}. This contradicts the assumption that ∃ ∈ − ∶ () = in. □
It is then immediate to characterize conflict-free labellings in terms of our generalized notions. Proposition 8. The set of conflict-free labellings coincides with the set of labellings which are →-inc 3-consistent and →-rv 3-compliant.
Proof: The proof is immediate taking into account the correspondence between Item 1 in Definition 12 and consistency under inc 3, as well as correspondence between Item 2 and reinstatement compliance under rv 3 ensured by Proposition 7. □
Admissibility of a set of arguments was introduced in [
if for each ∈
Item 1 in Definition 13 is a strengthening of item 1 of Definition 12, while item 2 is the same in both Definition 12 and 13. Interestingly, this strengthening corresponds to the choice of a stronger form of consistency: having an attacker labelled und is forbidden for an argument labelled in, while having an attacker labelled in is allowed for an argument labelled und. This coincides with adopting the following asymmetric incompatibility relation inc 3 = {(pos, pos), (mid, pos)}.
Proposition 9. The set of admissible labellings coincides with the set of labellings which are →-inc 3-consistent and →-rv 3-compliant.
Proof: It has been shown in [
Completeness of a set of arguments was introduced in [
In words a complete labelling is an admissible labelling with the additional requirement that an argument which is labelled und must have an und-labelled attacker and no in-labelled attackers. In particular, the last condition amounts to further strengthening the notion of consistency by adopting the incompatibility relation inc 3 = {(pos, pos), (pos, mid), (mid, pos)}, while the first condition is verified if the following reinstatement property is enforced.
() = out then () = in. it holds that if ∀ ∈ −
The following proposition shows that the reinstatement property can be captured by the reinstatement violation relation rv 3 = {(neg, neg), (neg, mid), (mid, neg)}.
Proposition 10. Let be a labelling of an argumentation framework = ( , →) →-rv 3-compliant if it satisfies the reinstatement property and for each ∈ () = out then ∃ ∈ − ∶ () = in.
. Then, is it holds that if Proof: Assume that is →-rv 3-compliant. By Proposition 6, is →-rv 3-compliant, and by Proposition 7 the second condition of the thesis holds. To prove the reinstatement property, let us consider an argument ∈ such that ∀ ∈ −, () = out, and let us show that () = in. If is initial, by Definition 10 it must be the case that min() ⊟ Λ(()) , i.e. taking into account the definition of rv 3 we have that Λ(()) = pos, which holds if () = in. If is non initial, by Definition 10 there is an argument ∈ − such that () ⊟() . Since by the hypothesis () = out, according to the definition of rv 3 it must be the case that () = in.
As to the reverse direction of the proof, assume that satisfies the reinstatement property and the second condition of the hypothesis, and let us prove that is →-rv 3-compliant. According to Definition 10, we can consider an argument and distinguish two cases for it. If is initial, by the reinstatement property () = in, thus Λ(()) = pos which satisfies the required condition min() ⊟ Λ(()) . If is not initial, referring to Definition 10 assume by contradiction that there is no ∈ − such that () ⊟() . This means that ∀ ∈ −, () ⊡() , i.e. ((), ()) ∈ {( out, out), (out, und), (und, out)}. According to the second condition of the hypothesis if () = out then ∃ ∈ − ∶ () = in, which entails that () ≠ out. Then () = und and ∀ ∈ − () = out, contradicting the reinstatement property. □
We can now characterize complete labellings in terms of generalized consistency and reinstatement.
Proposition 11. The set of complete labellings coincides with the set of labellings which are →-inc 3-consistent and →-rv 3-compliant.
Proof: It is shown in [
Stability of a set of arguments can be characterized in several ways, its key feature being that no room is left for undecidedness (an argument is either accepted or attacked by an accepted argument) as indicated by Definition 16.
Definition 16. Let be a labelling of an argumentation framework = ( , →) it is complete and ∄ ∈ ∶ () = und.
This constraint can be put in correspondence with the adoption of the strongest notion of consistency, namely with the choice of the incompatibility relation inc 3 = {(pos, pos), (pos, mid), (mid, pos), (mid, mid)}.
Proposition 12. The set of stable labellings coincides with the set of labellings which are →-inc 3consistent and →-rv 3-compliant.
Proof: It is shown in [
To summarize, conflict-free labellings can be characterized in terms of generalized consistency and reinstatement, admissible labellings can be characterized in terms of strengthening consistency with respect to conflict-freeness without resorting to the traditional notion of defense, while further strengthenings of generalized consistency and reinstatement characterize complete and stable labellings.
In this paper, we have introduced the generalized notions of consistency and reinstatement
encompassing Dung’s traditional semantics. To our knowledge, providing a generalized form
of the notion of consistency has only been considered in [
Several other research work can be envisaged. First of all, in [
Extending the analysis beyond tripolar classifications is another important future
development. For example, more articulated notions of argument justification have been considered in
the literature [