Starting from previous work concerning the notions of consistency and reinstatement in an abstract labelling setting, we propose an enhanced model where the above mentioned notions are integrated with the one of skepticism. Skepticism makes it possible to identify the labellings prescribed by argumentation semantics involving global requirements among those satisfying a combination of local consistency and reinstatement requirements. We then consider the issue of determining argument justification by synthesizing an evaluation labelling from those prescribed by the semantics, and we analyze the evaluation function most commonly used in the literature against a number of possible desiderata. Overall, we obtain a general model, based on the foundational notions of positiveness and skepticism, able to capture a variety of instances of diferent reasoning stages in abstract argumentation.
1. Introduction (M. Giacomin) admissible, complete and stable labellings. However, other labellings cannot be characterized by just the local constraints enforcing consistency and reinstatement, since their definition explicitly or implicitly involves conditions at global level. For instance, the grounded labelling corresponds to the complete labelling minimizing the set of positively assessed arguments, while the preferred labellings maximize them. To fill this gap, we resort in this paper to a formal notion of skepticism, based on the observation that grounded and preferred semantics can be put in correspondence with diferent attitudes in justifying arguments. Our analysis on the role of skepticism is then extended to the assessment of argument justification status, which is derived from the extensions or labellings prescribed by the adopted argumentation semantics.
Altogether, this paper aims to provide a first framework where positiveness and skepticism are integrated, drawing some conceptual considerations and identifying some perspectives for further work.
The paper is organized as follows. Section 2 provides some background on the notion of
consistency and reinstatement, while Section 3 proposes a model where they are integrated
with a notion of skepticism, thus allowing the characterization of grounded, preferred and
semi-stable semantics. Section 4 deals with argument justification in the proposed unifying
setting. A number of intuitive desiderata based on the notions of positiveness and skepticism
is proposed, and the traditional approach to assess argument justification is analyzed against
them, pointing out several limitations. Finally, Section 5 draws some concluding remarks and
discusses future avenues of research.
2. Consistency and reinstatement
To provide a general characterization of labelling-based assessments of entities of various kind,
we have introduced in [
At the top level, the notion of assessment classes provides a reference point to characterize diferent assessment labels. These classes have an underlying order reflecting a level of positiveness of the assessment, with 1 ≤ 2 meaning that 2 corresponds to an at least as positive assessment as 1.
Definition 1. A set of assessment classes (abbreviated as sac(s) in the following) 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.
At an intermediate level, assessment labels are taken from a predefined set and classified on the basis of a sac, thus inheriting the relevant positiveness degree.
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, also called positiveness preorder, 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 [
At the bottom level, a generic set of entities is considered, with the entities related by an intolerance relation, indicating who cannot stand whom (as an example, this relation might coincide with classical negation if the considered language is equipped with it). These entities can be assessed by the usual notion of labelling, i.e. assigning each entity a label. Definition 3. 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.
Definition 4. Given a sal Λ and a set , a Λ-labelling of is a function ∶ → Λ Λ-labelling of and a label ∈ Λ , we define () = { ∈ ∣ () = } . . Given a
Labellings are typically required to satisfy two dual properties. On the one hand, in order to satisfy consistency two elements which cannot stand each other should not be assigned labels which are ‘too positive’ altogether. On the other hand, to avoid unjustified overly negative evaluations, labellings should satisfy reinstatement, i.e. a too negative label should not be assigned to an element unless another intolerant element is assigned a suficiently positive label. The corresponding violations at the level of the labellings are modelled by distinct relations, namely an incompatibility relation and a reinstatement violation relation on assessment labels, induced by two corresponding relations 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.
Some rather natural properties can be identified for incompatibility and, in a dual manner, for reinstatement violation relations on .
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: • inc is 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
′ 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 ∃ 3 ∈ such that 3⊟ 1
The first property of each definition reflects the intuitions underlying the dual concepts of consistency and reinstatement. In particular, inconsistency arises from a sort of ‘excess of simultaneous positiveness’ in the assessment of some elements linked by intolerance, while reinstatement violation is due to an ‘excess of cautiousness’ in assigning positive labels. The second and third properties are the same in both definitions, and require that the intolerance relation between elements of is not void of any efect and that each label is attainable, respectively.
The following definition formalizes the notions of consistent and inconsistent 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 following definition dually introduces the notions of reinstatement compliant and
uncompliant labelling. Note that a special condition is required for initial elements of , i.e. elements
2 of such that there are no elements 1 with 1 ⊙ 2 (the reader is referred to [
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 • inc is non empty, i.e. inc ≠ ∅ • ∀ 1 ∈ , ∃ 2 ∈ such that 1⊟ 2 and ∃ 3 ∈ such that 3⊟ 1 ∃ 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 (1) (2) (3) (4)
The proposed model is able to capture the labelling-based version of Dung’s semantics [
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 { ∈ ∣ (, ) ∈→} .
Given an abstract argumentation framework = ( , →) , we assume = and the intolerance relation coinciding with the attack relation, i.e. ⊙ if ∈ −. We use the sal ΛIOU = {in, out, und} and the tripolar sac 3 = {pos, mid, neg} with neg ≤ mid ≤ pos. Furthermore, we adopt the classification Λ3IOU = {(in, pos), (out, neg), (und, mid)}, i.e. in corresponds to a definitely positive assessment, out to a definitely negative assessment, and und to an intermediate situation.
The main labelling-based semantics corresponding to the extension-based semantics
introduced in [
It is shown in [
The notion of skepticism between semantics in abstract argumentation has been introduced
in [
Given a
, we denote as ≤ a preorder (i.e. a reflexive and transitive relation) on , intended to reflect the relevant skepticism degree (and thus we will sometimes refer to it as skepticism preorder). Given a -classified set of assessment labels Λ, the skepticism preorder
The fact that ⪯ is a preorder can be easily proved.
Intuitively, 1 ⪯ 2 means that assigning the label 1 to an element ∈ represents a “less decided” choice about its justification w.r.t. assigning 2 to . It is worth pointing out that ⪯ must be clearly distinguished from ⪯, which instead reflects the positiveness degree of the labels. In particular, for two labels 1 and 2 corresponding to definite acceptance and definite rejection, i.e. associated to min() and max() respectively, it typically holds 1 ≺ 2 but they reflects antithetical choices about the justification of an element and thus they can be incomparable w.r.t. skepticism.
As the tripolar sac 3 = {pos, mid, neg} and the relevant sal ΛIOU = {in, out, und} adopted in abstract argumentation, we assume ≤ including the relations mid ≤ pos, mid ≤ neg, mid ≤ mid, neg ≤ neg, and pos ≤
pos. Accordingly, it holds und ≺ in and und ≺ out, while in and out are incomparable w.r.t. ⪯ .
The skepticism preorder between labels can naturally be extended to labellings as follows. 1() ⪯ 2() .
Definition 14.
Let Λ be a sal equipped with a skepticism preorder ⪯ and let be a set. The skepticism preorder between the Λ-labellings of is denoted as ⪯ , where 1 ⪯ 2 if ∀ ∈
Intuitively, a labelling 1 is less committed w.r.t. another labelling 2 if 1 makes a less committed choice w.r.t. 2 for each of the elements of , while an incomparable choice for even a single element makes the labellings 1 and 2 incomparable.
Turning to abstract argumentation, it is easy to see that 1 ⪯ preferred labellings among complete labellings. out( 1) ⊆ out( 2). The skepticism preorder ⪯ allows one to identify the grounded and 2 if in( 1) ⊆ in( 2) and Proposition 2. Assuming ≤ as described above, the set of preferred labellings coincides with the set of complete labellings that are maximal w.r.t. ⪯ complete labelling that is minimal w.r.t. ⪯ .
, and the grounded labelling coincides with the
Proof: It has been shown in [
As to semi-stable labellings, there are (at least) two ways to characterize them by maximizing labellings according to a skepticism relation. labellings in a similar way as preferred labellings.
First, grounded on the idea that the definition of semi-stable labellings requires the union of in-labelled and out-labelled arguments to be maximized (and thus there is no distinction between them) we could modify the basic skepticism relation between labels by considering in und ≺ and out comparable, as having the same highest level of commitment. Thus, we can assume the preorder ≤′ instead of ≤ , where ≤′=≤ ∪{(pos, neg), (neg, pos)}. Accordingly, it holds ′ in, und ≺ ′ out, out ⪯ ′ in and in ⪯ ′ out. We can then characterize semi-stable labellings coincides with the set of complete labellings that are maximal w.r.t. ⪯ . Proposition 3. Assuming ≤ ′ as the skepticism preorder between sacs, the set of semi-stable Proof: First, notice that given two labellings 1 and 2, 1 ⪯ arguments labelled und, i.e. with the set of semi-stable labellings. holds that 1() = und, i.e. if und( 2) ⊆ und( 1). Accordingly, the set of complete labellings that are maximal w.r.t. ⪯ coincides with the set of complete labellings that minimize the set of 2 if ∀ ∈ ∶ 2() = und it
Alternatively, it might be observed that giving up incomparability between in and out is not necessary: we can adopt a diferent way to extend the skepticism preorder between labels to the labellings of , enforcing a weaker relation w.r.t. ⪯ .
′ of , 1⪯ 2 if ∀ ∈ , 2() ⊀ 1() .
Definition 15. relation ⪯′ between the Λ-labellings of is defined as follows. Given two Λ-labellings 1 and 2
Let Λ be a sal equipped with a skepticism preorder ⪯ and let be a set. The It has to be remarkes that, in general ⪯′ is not a preorder, since it does not satisfy transitivity. For instance, consider a sal Λ = { 1, 2, 3, 4} and an induced skepticism preorder ⪯ such as 1 ≺ 2 ≺ 3, 1 ≺ 4, 2 and 4 incomparable, 3 and 4 incomparable (this can be induced for instance by a
such that each label corresponds exactly to a class, with a preorder between assessment classes corresponding to ⪯ ). Consider a singleton = {} and three labellings 1, 2 ′ and 3 such that 1() = 3, 2() = 4 and 3() = 2. We have 1⪯ 2 since 2() ⊀ 1() , ′ ′ and 2⪯ 3 since 3() ⊀ 2() . However, it is not the case that 1⪯ 3, since 3() ≺ 1() .
Nevertheless, transitivity holds in case the skepticism preorder on Λ is quasi total, as specified in the following definition. □ □ Definition 16.
Let Λ be a sal equipped with a skepticism preorder ⪯ . We say that ⪯ is quasi total if there is a set ⊆ Λ
such that
Note that if 1, 2 ∈ then, by the second point of Definition of maximally committed labels (and thus labels in
are incomparable each other) such that all the other elements are less committed than them, while non maximal elements are comparable to each other.
The following proposition shows that ⪯ is a preorder when ⪯ is quasi total.
′
It is easy to see that the partial order ≤ assumed for ΛIOU is quasi total, with = { in, out}. is quasi total, then the relation ⪯′ between the Λ-labellings of as defined in Definition Proposition 4. Let Λ be a sal equipped with a skepticism preorder ⪯ and let be a set. If ⪯ 15 is a preorder.
′ We have to show that 1⪯ 3, i.e. according to Definition each ∈ we distinguish two cases.
Proof: We have to show that ⪯′ is reflexive and transitive. () ⪯ () , thus () ⊀ () . According to Definition 15 it then holds ⪯ ′ .
As to reflexivity, consider a Λ-labelling of . Since ⪯ is reflexive, ∀ ∈ it holds that ′ ′ As to transitivity, consider three Λ-labellings 1, 2 and 2 of such that 1⪯ 2 and 2⪯ 3.
15 that ∀ ∈ , 3() ⊀ 1() . For
First, if 1() ⪯ 2() and 2() ⪯ 3() , by transitivity of ⪯ it holds that 1() ⪯ 3() , thus 3() ⊀ 1() .
′ ′ In the other case 1() ⪯̸ 2() or 2() ⪯̸ 3() , and since 1⪯ 2 and 2⪯ 3 we have that 1() and 2() are incomparable w.r.t. ⪯ , or 2() and 3() are incomparable w.r.t. ⪯ . By the first point of Definition thus by the second point of Definition 16, { 1(), 2()} ⊆ or { 2(), 3()} ⊆ . In any case 2() ∈ , ′ 16 2() ⪯̸ 3() , and since 2⪯ 3 we have that 2() and 3() are incomparable. By the first point of Definition 16 this can hold only if 3() ∈ .
By the second point of Definition 16 it cannot then be the case that 3() ≺ 1() , i.e. it holds
We can then show that, in the case of abstract argumentation, the relation ⪯ (which is a
′
preorder by Proposition 4) allows one to identify the semi-stable labellings among complete
3() ⊀ 1() .
Proposition 5. The set of semi-stable labellings coincides with the set of complete labellings that
′
Proof: Given two labellings 1 and 2, 1⪯ 2 if ∀ ∈ , 2() ⊀ 1() . If 2() ∈ { in, out}
then it cannot be the case that 2() ≺ 1() , while if 2() = und then 2() ⊀ 1() holds if
′
1() = und. Summing up, 1⪯ 2 if und( 2) ⊆ und( 1). As a consequence, as in the proof
of Proposition 3 the set of complete labellings that are maximal w.r.t. ⪯
coincides with the
set of complete labellings that minimize the set of arguments labelled und, i.e. with the set of
semi-stable labellings.
4. Argument evaluation
As a subsequent step of argumentative reasoning (see e.g. [
Definition 17. Λ -labelling of .
Given two disjoint2 sals Λ and Λ , an evaluation function from Λ to Λ is a mapping evfun which for every set associates to each non-empty set of Λ-labellings of a
A particular case of evaluation function corresponds to deriving the evaluation label of each argument only from the labels assigned to the same argument by the Λ-labellings. The corresponding mapping is modelled by a simple synthesis function.
Definition 18. a mapping syn ∶ 2Λ ⧵ {∅} → Λ . The evaluation function derived from syn, denoted as evfunsyn,
Given two disjoint sals Λ and Λ , a simple synthesis function (ssf) from Λ to Λ is is defined, for every non-empty set of Λ-labellings ℒ and for every ∈ as
evfunsyn(ℒ )() = syn(ℒ ↓()) where ℒ ↓() ≜ {() ∣ ∈ ℒ }
.
In abstract argumentation, the most typical argument evaluation is based on three justification states. In particular, if the semantics prescribes a set ℒ of ΛIOU-labellings of a set of arguments , an argument ∈
is: • skeptically justified if ∀ ∈ ℒ ( ) =
in; • credulously justified if it is not skeptically justified • not justified if ∄ ∈ ℒ ∶ ( ) = in.
3 and ∃ ∈ ℒ ∶ ( ) = in; for every Λ ⊆ ΛIOU as follows: Accordingly, we consider a sal ΛAJ = {SkJ, CrJ, NoJ} and a ssf synAJ from ΛIOU to ΛAJ defined, • synAJ(Λ) = SkJ if Λ = {in}; • synAJ(Λ) = CrJ if Λ ⊋ {in}; • synAJ(Λ) = NoJ otherwise.
As to the classification of ΛAJ, it is intuitive to {(SkJ, pos), (NoJ, neg), (CrJ, mid)}.
This way, it turns out that NoJ ≺
CrJ ≺
SkJ and CrJ ≺ NoJ, CrJ ≺ SkJ, NoJ and SkJ incomparable w.r.t. ⪯ . 2We assume without loss of generality that Λ and Λ are disjoint. Since they are used in diferent stages of the reasoning process, it is always possible to adopt diferent ‘names’ for the labels in Λ and Λ . 3Traditionally credulous justification is regarded as including skeptical justification. We enforce disjoint notions so that argument justification can be properly modelled as a labelling. assume 3 ΛAJ =
It is interesting to define desiderata for evaluation functions based on the notions of positiveness and skepticism and also to consider the issue of preserving consistency and reinstatement properties across the evaluation step.
In order to express desiderata concerning positiveness, we first define a positiveness ordering between labels belonging to diferent , provided that they are classified with the same assessment class. In particular, we will assume that both Λ and Λ are -classified, where is a sac that provides a reference structure to compare labels in Λ ∪ Λ .
Definition 20. Let be a sac, and let Λ and Λ be two disjoint -classified sals. Let be a set. The positiveness preorder between the Λ-labellings and Λ -labellings of is denoted as ⪯ , where 1 ⪯ 2 if ∀ ∈ , 1() ⪯ 2() . The positiveness preorder between the sets of Λ-labellings and Λ -labellings of is denoted as ⪯ , where ℒ1 ⪯ ℒ2 if ∀ ∈ ℒ 1 ∃ ′ ∈ ℒ2 such that ⪯ ′ and ∀ ′ ∈ ℒ2 ∃ ∈ ℒ 1 such that ⪯ ′.
Intuitively, a labelling 1 is no more positive w.r.t. another labelling 2 if 1 makes a no more positive choice w.r.t. 2 for each of the elements of . The idea of the ⪯ relation between sets of labellings is that every labelling of ℒ1 can be mapped into one at least as positive labelling of ℒ2 and at the same time every labelling of ℒ2 can be mapped into a no more positive labelling of ℒ1. It is easy to see that both ⪯ and ⪯ are preorders, taking into account that ⪯ is a preorder.
We can now express two possible desiderata for an evaluation function based on the positiveness preorder between sets of labellings. First, it is reasonable for an evaluation function to be monotonic with respect to this preorder.
Definition 21. Let be a sac, and let Λ and Λ be two disjoint -classified sals. An evaluation function evfun from Λ to Λ is well-behaved if for any set and for any two non-empty sets of Λ-labellings ℒ1, ℒ2 of such that ℒ1 ⪯ ℒ2, it holds that evfun(ℒ1) ⪯ evfun(ℒ2).
Besides monotonicity, one might require the evaluation function to be reasonably bounded on the basis of the aggregated labellings.
Definition 22. Let be a sac, and let Λ and Λ be two disjoint -classified sals. An evaluation function evfun from Λ to Λ is faithful if for any set and for any non-empty set ℒ of Λ-labellings of , ∃ ∈ ℒ such that evfun(ℒ ) ⪯ and ∃ ∈ ℒ such that ⪯ evfun(ℒ ).
In words, the result produced by evfun is neither strictly greater nor strictly lower than all labellings which are aggregated.
A counterpart of well-behaved and faithful evaluation functions can be introduced by considering the skepticism relation too. As in the case of positiveness, this first requires to consider labels belonging to diferent sals. 2 ∈ Λ2.
Definition 23.
Let be a sac equipped with a skepticism preorder ≤ , and let Λ and Λ be two disjoint -classified sals. The skepticism preorder induced on Λ ∪ Λ by Λ and Λ , is denoted
14 and Definition 15 to the case of diferent sals, introducing two skepticism relations between labellings.
′ and 1⪯ 2 if ∀ ∈ , 2() ⊀ 1() .
Definition 24. disjoint -classified sals. Let be a set. The skepticism preorder ⪯ and the relation ⪯′ between
Let be a sac equipped with a skepticism preorder ≤ , and let Λ and Λ be two diferent ways, e.g. with the relation ⪯ or ⪯′ as in Definition 24.
The following definition introduces the skepticism relation between sets of labellings, assuming a skepticism preorder ⪯ between labellings as a parameter that can be instantiated in Definition 25. disjoint -classified sals. Let be a set and ⪯ a binary relation between the Λ-labellings and Λ -labellings of . The skepticism preorder between the sets of Λ-labellings and Λ -labellings of
Let be a sac equipped with a skepticism preorder ≤ , and let Λ and Λ be two induced by ⪯ is denoted as ⪯ , where ℒ1 ⪯
ℒ2 if ∀ 2 ∈ ℒ2 ∃ 1 ∈ ℒ1 such that 1 ⪯ 2.
It should be noted that Definition 25 does not correspond to the way the positiveness preorder between labellings is extended to sets of labellings (Definition to hold each labelling of ℒ2 must be not less committed than at least a labelling of ℒ1, while ℒ1 can include labellings that are unrelated to those in ℒ2. Intuitively, including additional labellings in ℒ1 leaves open more possibilities as far as the argument justification is concerned, thus corresponding to a less decided assessment.
We can then introduce the notion of well-behaved evaluation function w.r.t. skepticism (which is parameterized w.r.t. the adopted skepticism relation between labellings ⪯ too). 20). In particular, for ℒ1 ⪯ ℒ2 evfun(ℒ2).
of such that ℒ1 ⪯ Definition 26.
Let be a sac, and let Λ and Λ be two disjoint -classified sals. Let ⪯ adopted skepticism relation between labellings. An evaluation function evfun from Λ to Λ is wellbehaved w.r.t. skepticism if for any set and for any two non-empty sets of Λ-labellings ℒ1, ℒ2 ℒ2 (where ⪯ is induced by the relation ⪯ ), it holds that evfun(ℒ1) ⪯ We then consider the counterpart of Definition 22 w.r.t. skepticism (again parameterized w.r.t. the adopted skepticism relation between labellings ⪯ ).
Definition 27. adopted skepticism relation between labellings. An evaluation function evfun from Λ to Λ is faithful w.r.t. skepticism if for any set and for any non-empty set ℒ of Λ-labellings of , ∃ ∈ ℒ such that evfun(ℒ ) ⪯ and ∃ ∈ ℒ
such that ⪯ evfun(ℒ ).
Turning to consistency and reinstatement preservation, it appears desirable that the consistency and reinstatement properties of the original labellings are not lost in the derived justification labelling. Since consistency and reinstatement refers to a specific incompatibility and reinstatement violation relation, respectively, also preservation properties refer to them. Definition 28.
Let be a sac equipped with an incompatibility relation inc, and Λ and Λ be two -classified sets of labels. An evaluation function preserving according to4 inc if for any set equipped with an intolerance relation int and any non-empty int-inc-consistent set ℒ1 of Λ-labellings of it holds that the labelling evfun(ℒ1) is evfun from Λ to Λ is consistency int-inc-consistent.
Definition 29.
Let be a sac equipped with a reinstatement violation relation rv, and Λ and Λ be two -classifed sets of labels. An evaluation function evfun from Λ to Λ is reinstatement preserving according to5 rv if for any set equipped with an intolerance relation int and any non-empty int-rv-compliant set ℒ1 of Λ-labellings of it holds that the labelling evfun(ℒ1) is int-rv-compliant.
It is then interesting to analyze the evaluation function evfunsynAJ typically adopted in abstract argumentation against the requirements introduced so far.
As to the satisfied requirements, it has been shown in [
3 and inc 3, while it is not according to inc 3. A evfunsynAJ fails to satisfy faithfulness.
Example 1. Consider the argumentation framework = ⟨{ , }, {( , ), (, )}⟩ pair of mutually attacking arguments. The preferred labellings of → −inc 3 − preserving according to inc 3. (this can also be inferred by Proposition 1 taking into account that 1 and 2 are also the stable CrJ. However, is not , since (mid, mid) ∈ inc 3. As a consequence, evfunsynAJ is not consistency i.e. including a are 1 and 2, where 1( ) =
Unfortunately, all of the other requirements are unsatisfied. The next example shows that Example 2. Consider the same argumentation framework = ⟨{ , }, {( , ), (, )}⟩ ple 1. The grounded semantics prescribes the unique labelling such that ( ) = () = of Examund. We 4More precisely, consistency preservation should be defined w.r.t. a tuple (, how the labels of Λ and Λ are mapped into assessment classes. However, for ease of notation we focus on the incompatibility relation inc, since the mappings Λ and Λ are usually clear from the context. 5Similarly to the case of consistency preservation, reinstatement preservation should be defined w.r.t. (, Λ, Λ , rv). Λ, Λ , inc), since it also depends on corresponds to the assessment class mid and according to 3 not the case that mid ⪯ neg, it is also not the case that ⪯ , violating the second condition of Definition 22.
Moreover, evfunsynAJ is neither well-behaved nor faithful w.r.t. skepticism, for both of the choices concerning the skepticism relation between labellings.
Example 3. Consider a set including a single element , and two sets of labellings ℒ1 = { ′1, 1″} and ℒ2 = { ′2, 2″}, with ′1() = out, 1″() = und, ′2() = in, 2″() = checked that 1″ ⪯ ′2, 1″ ⪯
2″, 1″⪯′ ′2, and 1″⪯′ 2″, thus ℒ1 ⪯ ℒ2 both with ⪯
und. It can be induced by ⪯ evfunsynAJ(ℒ2)() = and with ⪯
CrJ, thus it neither holds evfunsynAJ(ℒ1) ⪯
evfunsynAJ(ℒ2)() nor induced by ⪯′ . On the other hand, evfunsynAJ(ℒ1)() =
NoJ, evfunsynAJ(ℒ1)⪯′ evfunsynAJ(ℒ2)() . This shows that evfunsynAJ is not well-behaved w.r.t. skepticism. easy to see that, letting again be the labelling obtained by evfunsynAJ({}) , neither ⪯ Example 4. Consider again the argumentation framework and the labelling of Example 2. It is nor ′ ⪯ holds. Thus, the first condition of Definition faithful w.r.t. skepticism. 27 is violated, showing that evfunsynAJ is not preserving according to rv 3. Finally, evfunsynAJ is neither reinstatement preserving according to rv 3 nor reinstatement and → −rv
3 − out, 1( ) = → −rv 3 −
Example 5. Consider the argumentation framework = ⟨{, , }, {(, ), (, ), (, ), (, )}⟩ Here preferred and stable semantics prescribe two labellings, namely 1 with 1() = . out, and 2 with 1() = out, 1() = in, 1( ) =
out. Both of them are to the pair (mid, neg) which is forbidden by both rv 3 and rv 3. { 1, 2}. The synthesis produced by evfunsynAJ(ℒ ) for and is CrJ and for is NoJ corresponding (this also derives from Proposition 1). Let ℒ = 5. Conclusions and perspectives In this paper, we have introduced a first model for argument evaluation in Dung’s abstract argumentation, based on the complementary notions of positiveness and skepticism characterizing the assessment labels. The model is graphically represented in Figure 5, where boxes denote reasoning steps and ovals represent (possibly singleton) sets of labellings, and with the parameters instantiating the reasoning steps reported at the top of the boxes. In particular, on the basis of the set of assessment labels Λ all of the possible labellings (i.e. the unconstrained labellings) of the input argumentation framework can be constructed. Then, unconstrained labellings are ifltered by selecting those that satisfy the local constraints expressed by the incompatibility and reinstatement violation relations (see Section 2) and further possibly6 filtered by maximizing or 6In the figure, ’no’ indicates that no filter is applied. This can be the case e.g. of stable semantics, since stable labellings can be directly identified by local constraints filtering.
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E v a l u a t i o n p r e s c r i b e d
SyJ52+Ro/u8K0zBQHPO XmGfY>ACEncDLS9VpkMdW3vT7 l<atexish1_b64="jUNZrqIFwg y n t h e s i s f u n c t i o n l a b e l l i n g l a b e l l i n g s F i g u r e 1 :
A g r a p h i c a l r e p r e s e n t a t i o n o f t h e g e n e r a l m o d e l f o r a r g u m e n t e v a l u a t i o n . m i n i m i z i n g w . r . t .
a s k e p t i c i s m r e l a t i o n ( s e e
S e c t i o n 3 ) .
F i n a l l y , a n e v a l u a t i o n f u n c t i o n i s a p p l i e d t o d e t e r m i n e a n e v a l u a t i o n l a b e l l i n g ( s e e
S e c t i o n 4 ) .
W e r e m a r k t h a t t h e p r o p o s e d m o d e l i s c o n c e p t u a l r a t h e r t h a n b e i n g a n i m p l e m e n t a t i o n a r c h i t e c t u r e , a n d i t r e p r e s e n t s a r if s t s t e p o p e n t o f u r t h e r i n v e s t i g a t i o n s , s o m e o f w h i c h a r e d i s c u s s e d b e l o w . s y n A
J S t a r t i n g f r o m t h e n if a l s t e p , w e h a v e s h o w n t h a t f a i l s t o s a t i s f y m o s t o f t h
e e v f u n d e s i d e r a t a i n t r o d u c e d i n
S e c t i o n 4 .
T h e c o u n t e r e x a m p l e s p r o v i d e d s e e m t o s u g g e s t t h a t a n if e r A
J g r a i n e d s e t o f e v a l u a t i o n l a b e l s w o u l d b e n e e d e d w . r . t .
t o .
F o r i n s t a n c e Λ s y n A
J 2 t h e r e a s o n w h y f a i t h f u l n e s s i s n o t s a t i s if e d i s t h a t d o e s n o t d i s t i n g u i s h b e t w e e n e v f u n t h e c a s e w h e r e a n a r g u m e n t i s l a b e l l e d b y a l l l a b e l l i n g s a n d t h e c a s e w h e r e t h e a r g u m e n t i s o u
t l a b e l l e d b y a t l e a s t o n e l a b e l l i n g .
A d i s t i n c t i o n b e t w e e n t h e s e t w o c a s e s w o u l d r e q u i r e a n u n d
A
J a d d i t i o n a l l a b e l i n t o r e p r e s e n t t h e l a t t e r c a s e , t h a t w o u l d b e m a p p e d t o a n a s s e s s m e n t c l a s s Λ c h a r a c t e r i z e d b y a n i n t e r m e d i a t e p o s i t i v e n e s s l e v e l .
S i m i l a r c o n s i d e r a t i o n s a p p l y t o
E x a m p l e 4 c o n c e r n i n g s k e p t i c i s m .
A s t o
E x a m p l e 5 c o n c e r n i n g r e i n s t a t e m e n t p r e s e r v a t i o n i t s e e m s t h a t s y n A
J f a i l s t o d i s t i n g u i s h b e t w e e n t h e c a s e w h e r e a n a r g u m e n t i s l a b e l l e d b e c a u s e o f e v f u n o u t a n a t t a c k e r w h i c h i s l a b e l l e d b y a l l l a b e l l i n g s , a n d t h e c a s e o f w h e r e d i fe r e n
t o lf a t i n g d e f e a t i
n a t t a c k e r s a r e l a b e l l e d b y d if e r e n t l a b e l l i n g s .
D i s t i n g u i s h i n g b e t w e e n t h e s e t w o c a s e s m i g h t i
n r e q u i r e b o t h a if n e r g r a i n e d s e t o f e v a l u a t i o n l a b e l s a n d a n e v a l u a t i o n f u n c t i o n w h i c h i s n o t b a s e d o n a s i m p l e s y n t h e s i s f u n c t i o n .
I n g e n e r a l , d e v i s i n g a s e t o f a s s e s s m e n t l a b e l s o f a n
d Λ a n e v a l u a t i o n f u n c t i o n a b l e t o s a t i s f y m o r e d e s i d e r a t a i s a n i n t e r e s t i n g r e s e a r c h i s s u e . e v f u
n T u r n i n g t o s k e p t i c i s m b a s e d a n d l o c a l c o n s t r a i n t s if l t e r i n g i t w o u l d b e i n t e r e s t i n g t o c o n s i d e r a r g u m e n t a t i o n s e m a n t i c s t h a t a r e n o t d i r e c t l y c a p t u r e d i n t h e c u r r e n t p r o p o s a l .
F o r i n s t a n c e t h e i d e a l s e m a n t i c s p r e s c r i b e s a s t h e u n i q u e l a b e l l i n g t h e m o s t c o m m i t t e d l a b e l l i n g t h a t i s l e s s committed than all preferred labellings. This can be captured by introducing a skepticism-based iflter that receives in input an additional set of labellings w.r.t. the one to be filtered. Furthermore, some variations of the introduced notions can be considered, and diferent combinations of consistency, reinstatement and skepticism-based relations might be explored.
Another interesting issue concerns the set of assessment labels Λ adopted for the semantics
prescribed labellings. While we focused on tripolar labellings, quadripolar labellings have also
been introduced in the argumentation literature [
Acknowledgments This work has been partially supported by the research project GNCS-INdAM CUP E55F22000270001, “Verifica Formale di Dibattiti nella Teoria dell’Argomentazione’’.