A parametrized ranking-based semantics compatible with persuasion principles

2.1.Dung’s framework

In his seminal work [15], Dung formalizes argumentation through an abstract argumentation framework in which there is no assumption on the nature of the elements it contains. More precisely, neither the structure nor the origin of the arguments are required. Then, an argumentation framework is composed of a set of abstract arguments and of a relation of conflict between them.

From a mathematical point of view, an argumentation framework is a directed graph where a node represents one argument and an arrow between two nodes represents an attack from an argument to another one. Let us introduce different notions related to the graphs that we will use in this article.

According to the length of a path between two arguments, the argument at the beginning of this path can be an attacker or a defender of the argument at the end of the path.

A branch is a path such that the argument at the beginning of the path is not attacked.

The ancestors’ graph of an argument x is a subgraph of an argumentation framework, that contains x and its the attackers and defenders, as well as all the attack relations between these arguments.

Example 2.1.

Fig. 2.

An argumentation framework.

On the argumentation framework represented in Fig. 2, one can find:

• a path of length 2 ⟨c2,b1,a⟩ from c2 to a whereas ⟨b1,a,b2⟩ is not a path,

• a cycle ⟨a1,a2,a3,a1⟩ of length 3,

• ⟨d2,d2⟩ is a loop,

• c1 and d1 are the attack roots of a,

• c1 and c2 are the defense roots of a,

• ⟨c1,a⟩ is an attack branch for a of length 1, ⟨d1,c3,b2,a⟩ is an attack branch for a of length 3,

• ⟨d1,c3,b2⟩, ⟨c1,a,a1⟩ and ⟨c2,b1,a⟩ are three possible defense branches of length 2,

• b1, b2 and c1 are the direct attackers of a,

• c1, c2 and c3 are the direct defenders of a,

• The argumentation framework AF′, represented in the blue rectangle, is the ancestors’ graph of a.

2.2.Semantics and properties

Dung’s framework comes equipped with various types of semantics used to evaluate the arguments. Among these semantics, the ranking-based semantics aiming to rank-order a set of arguments in an argumentation framework from the most to the least acceptable. Thus, unlike classical semantics which assign an absolute status (accepted, rejected, undecided) to each argument, this semantics compares pairs of arguments. We refer the reader to [1,11] for a complete overview of the existing families of semantics in abstract argumentation and for the differences between these approaches (e.g., definitions, outcomes, applications).

Many ranking-based semantics with different behaviors have been introduced in recent years [2,3,8,13,16,27,28]. These semantics have often been defined along with a set of properties, each of which represents a specific criterion (e.g. quantity or quality of the direct attackers of an argument), aiming to highlight the unique behaviour of the proposed semantics [2,13,24]. Bonzon et al. [9] grouped these properties, generalized and defined new ones in order to propose an axiomatic comparison of existing semantics. Note, however, that none of these properties are mandatory. Indeed, some of these properties are not independent of each other because incompatibilities and dependencies can exist between them [5,9]. These properties are therefore an excellent indication to better understand the behavior of these semantics and remain the only way to compare existing ranking-based semantics with new ones. They are also a first step to the ambitious research question of defining fully characterized classes of ranking-based semantics with respect to a subset of properties.

2.2.4.Direct defenders

Defense Precedence states that, when two arguments have the same number of direct attackers, an argument with at least one direct defender should be strictly more acceptable than an argument without any direct defender.

Property 10

Property 10(Defense Precedence (DP) [2]).

A ranking-based semantics σ satisfies Defense Precedence if and only if for any AF=⟨A,R⟩ and ∀x,y∈A such that |R1(x)|=|R1(y)|, if R2(x)≠∅ and R2(y)=∅ then x≻AFσy.

When DP says that it is better for an argument to be defended, it could be interesting to know if different kinds of defense of an argument have the same impact on it. Two kinds of defense of an argument are then defined: the simple defense and the distributed defense.

Definition 2.10

Definition 2.10(Simple defense and distributed defense [2]).

Let AF=⟨A,R⟩ be an argumentation framework and x∈A. The defense of x is simple if and only if every direct defender of x directly attacks exactly one direct attacker of x (i.e. when |R1(x)|=1 because, by definition, there is only one direct attacker which can be attacked, or ∀y,z∈R1(x), R1(y)∩R1(z)=∅). The defense of x is distributed if and only if every direct attacker of x is attacked by at most one argument (i.e. ∀y∈R1(x), R1(y)=∅ or ∃!z∈R2(x) such that (z,y)∈R).

Distributed-Defense Precedence states that if two arguments have the same number of direct attackers and the same number of direct defenders, then it is preferable for an argument that each of its defender attacks a distinct direct attacker in order to weaken all of them instead of focusing on a specific direct attacker so as to greatly weaken it (but at the price of leaving its others attackers unaffected).

Property 11

Property 11(Distributed-Defense Precedence (DDP) [2]).

A ranking-based semantics σ satisfies Distributed-Defense Precedence if and only if for any AF=⟨A,R⟩ and ∀x,y∈A such that |R1(x)|=|R1(y)| and |R2(x)|=|R2(y)|, if the defense of x is simple and distributed and the defense of y is simple but not distributed, then x≻AFσy.

Example 2.2.

Consider the two argumentation frameworks illustrated in Fig. 3.

Fig. 3.

Distributed-defense precedence.

The two arguments a and b have the same number of direct attackers (R1(a)={a1,a3} and R1(b)={b1,b4}) and the same number of direct defenders (R2(a)={a2,a4} and R2(b)={b2,b3}). The defense of a is simple because R1(a1)∩R1(a3)={a2}∩{a4}=∅ and distributed because a1 and a3 are attacked by exactly one argument (a2 and a4 respectively). Concerning b, its defense is also simple (R1(b1)∩R1(b4)={b2,b3}∩∅=∅) but not distributed because b1 is directly attacked by two arguments (b2 and b3). Thus, the property DDP ensures that a is more acceptable than b.

The property Attack vs Full Defense allows us to make a distinction between the semantics which consider that a defense just cancels the effect of an attack (which can thus have the consequence of maintaining or increasing the acceptability of the targeted argument) and those which consider a defense as a weak attack (the impact of the attack is less important but it still has a negative effect on the targeted argument). Thus, an argument without any attack branch should be ranked higher than an argument only attacked by one non-attacked argument.

Property 12

Property 12(Attack vs Full Defense (AvsFD) [9]).

A ranking-based semantics σ satisfies Attack vs Full Defense if and only if for any acyclic AF=⟨A,R⟩ and ∀x,y∈A, if |B−(x)|=0, |R1(y)|=1 and |R2(y)|=0 then x≻AFσy.

2.3.Global properties

“Global” properties specify if some change related to the branches in an argumentation framework can improve or degrade the ranking of one argument. They help answer questions such as: what is the effect on the acceptability of a given argument with an additional attack branch? Is the effect the same if it is a defense branch? Does the length of the branch matter?

Let us first recall how the addition of an attack branch and the addition of a defense branch to an argument are formally defined.

In order to evaluate the impact of an additional branch on a given argument x of an argumentation framework AF, we “clone” this argumentation framework with an isomorphism γ (see Definition 2.7). Then, the argumentation framework γ(AF) can be modified (by adding one attack or defense branch for example) in order to analyse the impact on γ(x) compared to x. This approach allows us to focus exclusively on the impact of this change.

2.3.1.Addition of a branch

The first property concerns the attack branches and states that adding an attack branch to any argument decreases its level of acceptability.

Property 13

Property 13(Addition of an Attack Branch (+AB) [9]).

A ranking-based semantics σ satisfies Addition of an Attack Branch if and only if for any AF, AF′ and x∈Arg(AF), for every isomorphism γ such that AF′=γ(AF), if AF⋆=AF∪AF′∪P−(γ(x)), then x≻AF⋆σγ(x).

Example 2.3.

Let us consider the argumentation framework illustrated in Fig. 4. If σ satisfies +AB then a1, which has no attack branch, should be more acceptable than b2, b3, d3 and b4 which have one attack branch. In addition, a2 should be more acceptable than a4 because both have one defense branch with the same length but a4 has also an attack branch while a2 does not.

Fig. 4.

Argumentation framework with different configurations of branches.

The two following properties concern the defense branches. The first one states that adding a defense branch to any argument increases its level of acceptability.

Property 14

Property 14(Strict Addition of a Defense Branch (⊕DB) [9]).

A ranking-based semantics σ satisfies Strict Addition of a Defense Branch if and only if for any AF,AF′∈AF and x∈Arg(AF), for every isomorphism γ such that AF′=γ(AF), if AF⋆=AF∪AF′∪P+(γ(x)), then γ(x)≻AF⋆σx.

Example 2.3

Example 2.3(cont.).

If σ satisfies ⊕DB then a3 should be more acceptable than a2 which should be more acceptable than a1. Indeed, a3 has one more defense branch than a2 which has one more defense branch than a1. In addition, a4, which has one defense branch and one attack branch, should be more acceptable than b2, b3, d3 and b4 which have no defense branch.

Addition of a Defense Branch is defined in a more specific way: adding a defense branch to any attacked argument (therefore excluding non-attacked arguments) increases its level of acceptability.

Property 15

Property 15(Addition of a Defense Branch (+DB) [9]).

A ranking-based semantics σ satisfies Addition of a Defense Branch if and only if for any AF,AF′∈AF and x∈Arg(AF), for every isomorphism γ such that AF′=γ(AF), if AF⋆=AF∪AF′∪P+(γ(x)) and R1(x)≠∅, then γ(x)≻AF⋆σx.

Example 2.3

Example 2.3(cont.).

If σ satisfies +DB, the same conclusion as ⊕DB can be done except for a1. Indeed, as a1 is not attacked, nothing can be said about its ranking in comparison with the other arguments with respect to +DB.

2.3.2.Increasing the length of a branch

Two additional properties based on the increase of the length of a branch have been introduced. Let us recall that, for these properties, the “meaning” of the initial branch and the one of the increased branch have to be identical. In other words, if the initial branch is an attack (resp. defense) branch then the increased branch must be an attack (resp. defense) branch too. And, to keep the meaning of a branch, the only solution is to add a defense branch to the argument at the beginning of the branch.

Fig. 5.

Argumentation framework with different lengths of branch.

The first property states that increasing the length of an attack branch of an argument increases its level of acceptability.

Property 16

Property 16(Increase of an Attack Branch (↑AB) [9]).

A ranking-based semantics σ satisfies Increase of an Attack Branch if and only if for any AF,AF′∈AF and x∈Arg(AF), for every isomorphism γ such that AF′=γ(AF), if ∃y∈B−(x), y∉B+(x) and AF⋆=AF∪AF′∪P+(γ(y)), then γ(x)≻AF⋆σx.

Example 2.4.

Let us consider the argumentation framework illustrated in Fig. 5. If σ satisfies ↑AB then a2 should be more acceptable than a1 because a2 has an attack branch of length 3 while a1 has an attack branch of length 1.

The second property concerns the defense branch and states that increasing the length of a defense branch of an argument decreases its level of acceptability.

Property 17

Property 17(Increase of a Defense Branch (↑DB) [9]).

A ranking-based semantics σ satisfies Increase of a Defense Branch if and only if for any AF,AF′∈AF and x∈Arg(AF), for every isomorphism γ such that AF′=γ(AF), if ∃y∈B+(x), y∉B−(x) and AF⋆=AF∪AF′∪P+(γ(y)), then x≻AF⋆σγ(x).

Example 2.4

Example 2.4(cont.).

If σ satisfies ↑DB then a3 should be more acceptable than a4 because a3 has a defense branch of length 2 while a4 has a defense branch of length 4.

2.4.Equal acceptability

The property Argument Equivalence ensures that the acceptability of an argument depends only on (the structure of) its attackers and defenders. Formally, two arguments with the “similar (isomorphic)” ancestors’ graph (which, as a reminder, is an argumentation framework containing the argument and its attackers and defenders) should have the same ranking.

The property Non-attacked Equivalence is a particular case of Argument Equivalence because it focuses on the comparison between the non-attacked arguments.

Another possibility for detecting when two arguments are equally acceptable consists of taking into account their direct attackers. Suppose that two arguments, x and y, have the same number of direct attackers. If, for each direct attacker of x, there exists a direct attacker of y such that the two attackers are equally acceptable, then x and y are equally acceptable too.

3.1.Procatalepsis

Procatalepsis, or prolepsis, is a figure of speech in which the speaker raises an objection to their own argument and then immediately answers it. The goal is to strengthen this argument by dealing with possible counter-arguments before their audience can raise them [33]. An example of the use of the principle of procatalepsis in practice is presented in the introduction and illustrated by an argumentation framework in Fig. 1. In this kind of persuasion context, it is clearly more convincing to state the more plausible counter-argument to a1 in order to provide some convincing defenses against them, than simply stating a1 alone. These anticipations make it possible to persuade the interlocutor that any attack against a1 is vain. In addition, it becomes difficult for the persuadee to find arguments against a1 if the persuader anticipates most of them. Thus, in terms of ranking, a1 with several defense branches could be seen as strictly more acceptable than a1 without any branch.

In order to formally define the procatalepsis principle in the context of abstract argumentation, we first need to formally define the argumentation frameworks that we call persuasion pitches. A persuasion pitch is a tree shaped argumentation framework where an argument x, called the targeted argument, has only defense branches.

Condition (1) means that the targeted argument has no attack branch while condition (2) states that it has at least one defense branch (the length of defense branches does not matter here). Condition (3) guarantees that the argumentation framework is in the shape of a tree centered on the targeted argument x, i.e. it contains only the arguments used in the pitch which are related to x (i.e. for which a path to x exists).

Let us now formally define the procatalepsis principle in the context of abstract argumentation. We define this property by saying that a ranking-based semantics satisfies Procatalepsis if a number of defense branches is sufficient to make a targeted argument from a persuasion pitch at least as acceptable as when it does not have any (i.e. it is not attacked).

We have deliberately made this definition a little more flexible than its original definition because we think it would be a mistake to claim that adding a unique defense branch will always be sufficient to make the targeted argument at least as acceptable as a non-attacked argument. Indeed, one can for example think that in Fig. 1, the defense branch composed of arguments a2 and a3 is not sufficient on its own to make a1 more acceptable than when it was not attacked and that it is the combination of the two defense branches that makes a1 more acceptable than when it was not attacked. However, this flexibility allows us to keep the idea that the more defense branches there are, the harder it will be for the opponent to find arguments to attack a1.

What is striking is that procatalepsis blatantly contradicts the property Void Precedence (VP), considering that a non-attacked argument is strictly more acceptable than an attacked argument. Recall that, as remarked in [9], this property is satisfied by all existing ranking-based semantics. Thus, no ranking-based semantics has yet been proposed where Void Precedence is not satisfied, implying that there exists no ranking-based semantics which can capture the procatalepsis principle.

3.2.Fading

The fading principle states that long lines of argumentation become ineffective in practice, because the audience easily loses track of the relations between the arguments.

Fig. 6.

Argumentation framework with a long line of arguments.

In focusing on the argumentation framework depicted in Fig. 6, the fading principle concerns the limit until which the length of a path between an argument and another one is too long to have an impact on the targeted argument. For example, if one considers that the arguments situated at the beginning of the paths with a length greater or equal to 5 have no impact on a given argument, then arguments a6, a7 and a8 have no impact on a1. This limit is however not “radical” in the sense that before the limit, the arguments have the same impact and, after the limit, the arguments have no impact. Indeed, it seems natural to think that a closer attacker (respectively, defender) of an argument has more effect than a further one on the argument. For example, argument a2 should have more impact on a1 than any other argument in this argumentation framework because a2 is the direct attacker of a1. So, the impact is gradually reduced when the length of the path between two arguments increases. This idea of attenuation is already partially captured by two existing properties: Increase of an attack branch (↑AB) and Increase of a defense branch (↑DB) which state that increasing the length of an attack (resp. defense) branch of an argument increases (resp. decreases) its level of acceptability. However, we will explain in Section 6.2 why these two properties are not sufficient as they stand to catch the fading principle.

In practice, the fading principle is supported by the work of [30] which shows (in the context of their study, an extensive analysis of persuasive debates which took place on the subreddit “ChangeMyView”22), that arguments located at a distance greater than 10 from another argument (i.e., 5 rounds of back-and-forth), have no impact in the debate.

4.1.Propagation with attenuation

We propose the extension of the propagation principle introduced in [8] with the elements previously put forward. Recall that the idea of propagation is to assign a positive initial value to each argument in the argumentation framework (arguments may start with the same initial value or start with distinct values where non-attacked arguments have greater value than attacked ones). Then each argument propagates its value into the argumentation framework, alternating the polarity according to the considered path (negatively if it is an attack path, positively if it is a defense one).

But, in order to catch the persuasion principles, we formally redefine the propagation mechanism by including a damping factor δ aiming to decrease the impact of attackers situated further away along a path (the longer the path length i, the smaller the δi).

Fig. 7.

An argumentation framework AF1 with 6 arguments and an even cycle.

Table 1

Computation of the valuation P of each argument from AF1 when ϵ=0.5 and δ=0.4

The following proposition answers the question of convergence of the valuation P. The convergence is guaranteed by the use of the damping factor, but also because the set of arguments which attack or defend an argument for a given length of path is finite and limited by the number of arguments in an argumentation framework.

Let us now compute the propagation number of an argument by using a fixed-point iteration (the outcome is guaranteed with the previous proposition).

We want to insist on the fact that the propagation number assigned to each argument is just used to relate and compare arguments of a given AF, and that it should not be used as an (isolated) measure.

4.2.Variable-depth propagation

Let us now define a ranking-based semantics using the propagation number and taking into consideration the persuasion principles. As stated in the introduction of this section, a solution to catch the procatalepsis principle is to only take into account the roots of the arguments. Formally, it is possible when ϵ=0. Indeed, in this case, the non-attacked arguments propagate their weights (vϵ(y)=1) in the argumentation framework, while attacked arguments have an initial weight of 0. Thus, the propagation number of each argument is only based on the value received by their attack or defense roots. Any pairwise strict comparison (based on propagation number) resulting from this process is fixed. Let us take the example of the argumentation framework illustrated in Fig. 8. On the one hand, a1 and b1 have the same propagation number when ϵ=0 (P0,δ(a)=P0,δ(b)=2δ2) because they both have two defense branches and no attack branch. On the other hand, c1 has three defense branches (P0,δ(c)=3δ2). Therefore, since c1 has more branches of defense than a1 and b1, c1 must be more acceptable than a1 and b1 to be in agreement with the procatalepsis principle.

However, we do not want our semantics to be too “disconnected” from the principles that make the strength of ranking-based semantics: distinguishing arguments by taking into account the quality and the quantity of the arguments that attack or defend them. This is why we apply a second phase to break ties among arguments equally valued in the first phase. From a technical point of view, this means we re-run the propagation phase but this time setting an initial weight ϵ≠0 in order to take into account the attacked arguments. For example, a1 and b1 have the same propagation number when ϵ=0 in the argumentation framework represented in Fig. 8 because they both have exactly two defense branches. However, b1 is directly attacked only once while a1 is directly attacked twice, so one can consider that b1 could be more acceptable than a1.

Fig. 8.

Two arguments a1 and b1 with two defense branches (but with different configurations) involving a similar propagation number when ϵ=0 and c1 which has three defense branches.

Table 2

Computation of the valuation P of each argument from AF1 when ϵ=0 and δ=0.4

Let us give another example where the second phase is needed to distinguish two arguments.

Example 4.2.

Let us compute the ranking returned by vdp0.5,0.4 for the argumentation framework depicted in Fig. 8, beginning with the case ϵ=0 and then the case ϵ=0.5 (see Table 3).

Table 3

Valuation P for each argument in the argumentation framework depicted in Fig. 8 when ϵ=0 (above) and when ϵ=0.5 (below) with δ=0.4

According to the definition of vdp, we first compare the propagation number of each argument when ϵ=0. The result is the following ranking:

a3≃a5≃b3≃b4≃c3≃c5≃c7a3≃a5≃b3≃≻a3≃a5≃b3≃c1a3≃a5≃b3≃≻a3≃a5≃b3a1≃b1a3≃a5≃b3≃≻a3≃a2≃a4≃c2≃c4≃c6a3≃a5≃b3≃≻a3≃a5≃b3≃b2

We can see that some arguments are still equally acceptable, in particular a1 and b1. So, according to the definition of vdp, we restart the process with a non-zero ϵ (here ϵ=0.5):

a3≃vdp0.5,0.4a5≃vdp0.5,0.4b3≃vdp0.5,0.4b4≃vdp0.5,0.4c3≃vdp0.5,0.4c5≃vdp0.5,0.4c7a3≃vdp0.5,0.4a5≃vdp0.5,0.4b3≃vdp0.5,0.4≻vdp0.5,0.4a3≃vdp0.5,0.4a5≃vdp0.5,0.4b3≃vdp0.5,0.4c1a3≃vdp0.5,0.4a5≃vdp0.5,0.4b3≃vdp0.5,0.4≻vdp0.5,0.4a3≃vdp0.5,0.4a5≃vdp0.5,0.4b3≃vdp0.5,0.4a1a3≃vdp0.5,0.4a5≃vdp0.5,0.4b3≃vdp0.5,0.4≻vdp0.5,0.4a3≃vdp0.5,0.4a5≃vdp0.5,0.4b3≃vdp0.5,0.4b1a3≃vdp0.5,0.4a5≃vdp0.5,0.4b3≃vdp0.5,0.4≻vdp0.5,0.4a3≃vdp0.5,0.4a2≃vdp0.5,0.4a4≃vdp0.5,0.4c2≃vdp0.5,0.4c4≃vdp0.5,0.4c6a3≃vdp0.5,0.4a5≃vdp0.5,0.4b3≃vdp0.5,0.4≻vdp0.5,0.4a3≃vdp0.5,0.4a5≃vdp0.5,0.4b3≃vdp0.5,0.4b2

With this second process, a1 and b1 can be distinguished. Indeed, they have two defense branches of length 2, so during the step where ϵ is 0, they receive the same values from their defense roots. However, one can remark that a1 is directly attacked twice while b1 is directly attacked once. So, during the second process where the initial score of the attacked arguments are also propagated, a1 receives one more negative value than b1 (P10.5,0.4(a1)=0.1<0.3=P10.5,0.4(b1)).

5.1.Parameter ϵ

Recall that the parameter ϵ has a key role to distinguish the two phases aiming to compute the ranking between arguments. However, a concern might be that the value of ϵ might change the ranking obtained. We show that this is not the case:

Please note that even though different values of ϵ do not change the ranking between arguments returned by vdp, this parameter remains useful to distinguish the two steps used in the definition of vdp (see Definition 4.3): the first one where non-attacked arguments are the only arguments to propagate their value in the argumentation graph (ϵ=0) and the second one where all arguments propagate their value (ϵ≠0). However, this is a purely internal artefact without any effect on the outcome of the method. To make this clear, we note vdpδ instead of vdpϵ,δ to describe our parametrized ranking semantics in general.

5.2.Damping factor δ

5.2.1.Controlling the scope of influence of the arguments

The parameter δ is defined as the damping factor allowing us to decrease the impact of the argument when the length of the path increases. Following this, there intuitively exists a length such that the impact of arguments situating at the beginning of this path is negligible compared to the nearest arguments. Thus, the role of this parameter is to choose the scope of influence of the arguments in the argumentation framework, in addition to guarantee the convergence of the valuation P. For instance, with a value of δ close to 0, only the nearest arguments (so a small part of the argumentation framework) are taken into consideration to compute the different propagation numbers, whereas with a value of δ close to 1, (almost) all the argumentation framework will be inspected. Consequently, two different values of δ can produce different rankings for the same argumentation framework. Following the principle of the fading effect, it is natural to assume that arguments located at a long distance from another argument become ineffective. In terms of design, it seems very interesting to have the ability to control this parameter so as to specify a maximal depth after which arguments see their influence on the value of others vanish.

To better understand how to take the fading principle into account in using δ, let us detail the algorithm used to compute the propagation numbers.

• 1) A positive number is assigned to each argument: ∀a∈A, P0ϵ,δ(a)=1 if a is non-attacked or P0ϵ,δ(a)=ϵ otherwise,

• 2) We increase the step i by 1 and we add (or subtract) the score computed during the previous step (Pi−1ϵ,δ(a)) and the attenuated weights (vϵ and δi) received from defenders (or attackers) at the beginning of a path with a length of i (Ri(a)):

  Piϵ,δ(a)=Pi−1ϵ,δ(a)+(−1)iδi∑b∈Ri(a)vϵ(b)

• 3) If, between two steps, the difference, for all valuations P, is smaller than a fixed precision threshold μ (i.e. ∀a∈A, |Piϵ,δ(a)−Pi−1ϵ,δ(a)|<μ) then the process is stopped33 and the last values correspond to the propagation number of each argument. If it is not the case, we go back to 2).

Thus, given a precision threshold, one can choose δ according to the maximal expected depth.

Proposition 4.

Let AF=⟨A,R⟩ be an argumentation framework, i∈N∖{0} be the maximal depth and μ be the precision threshold. If δ<μmaxa∈A(|Ri(a)|)i then, for all a∈A, the sequence {Piϵ,δ(a)}i=0+∞ converges before step i+1.

Example 5.1

Example 5.1(cont.).

Consider the argumentation framework AF1 depicted in Fig. 7. Suppose that one considers that the maximal depth should be 5. In using the previous formula with a precision μ=0.0001, then δ should be smaller than 0.000125≃0.1379. Thus, a value close to this limit, for instance δ=0.137, ensures that only the arguments until a depth of 5 (included) are considered.

Through the formula given in Proposition 4, we can determine, for each maximal depth, which value of δ should be used. For example, Fig. 9 represents the arguments which propagate their initial value to f (from AF1 illustrated in Fig. 7) before the convergence and the associated interval of values of δ. One can remark that AF1 contains a cycle ⟨a,d,f,e,a⟩, it is why f can receive (according to the value of δ) a value from itself.

If the function defined in Proposition 4 allows us to select an appropriate δ in order to capture a given maximal expected depth (representing the fading effect), a legitimate question could concern the interval of values of δ to ensure the convergence at a specific depth i. However, it is not possible to answer this question in the general case, because of the diversity of argumentation frameworks. For example, suppose that one wants to consider a length up to 5 (no more no less). With an argumentation framework without cycles and with a maximal path smaller than 5, the process will be obviously stopped before. Thus the condition cannot be respected. It is for this reason that we only focus on the maximal depth.

Fig. 9.

Arguments which propagates their value to f according to the value of δ.

Finally, we can also find a computational advantage to represent the fading effect. Indeed, as the number of steps needed to find the propagation number of each argument is smaller than (or, in the “worst” case, equal to) when all of the argumentation framework have to be considered, the ranking is computed faster. An example of a practical use of our semantics would be to apply it to the persuasive debates from the subreddit “ChangeMyView”. We would then have to parameterize delta in order to take this aspect into account. In addition to satisfying the fading principle, this will reduce the computation time because it will not be necessary to go through the whole graph to evaluate these arguments.

5.2.2.On the diversity of rankings

As shown in Fig. 10, for a given argumentation framework, different values of δ can produce different rankings. Indeed, when δ∈{0.0001,0.2,0.4,0.6}, vdpδ provides the same ranking, whereas, when δ⩾0.8, c becomes more acceptable than f and d becomes more acceptable than b.

In light of these differences, one may be worried that the diversity of rankings could be so high that the semantics becomes too sensitive to small modifications of the parameter δ. To check this, we applied our variable-depth propagation on 1000 randomly generated argumentation frameworks44 for different values of δ∈{0.001,0.2,0.4,0.6,0.8,0.9}. Then, we measure the dissimilarity degree between two rankings from two different values of δ using the Kendall’s tau coefficient [21]. This coefficient corresponds to the total number of rank disagreements over all unordered pairs of arguments between two rankings. It therefore allows us to obtain a dissimilarity degree between two rankings.

Fig. 10.

An argumentation framework and the rankings returned by vdpδ for different values of δ.

Definition 5.1

Definition 5.1(Kendall’s tau coefficient).

Let AF=⟨A,R⟩ be an argumentation framework and τσ1, τσ2 the rankings returned by the ranking semantics σ1 and σ2 respectively. The Kendall’s tau coefficient between τσ1 and τσ2 is calculated as follows:

K(τσ1,τσ2)=∑{i,j}∈AKi,j‾(τσ1,τσ2)0.5×|A|×(|A|−1)

with:

• Ki,j‾(τσ1,τσ2)=0 if i≻AFσ1j and i≻AFσ2j, or i≺AFσ1j and i≺AFσ2j, or i≃AFσ1j and i≃AFσ2j;

• Ki,j‾(τσ1,τσ2)=1 if i≻AFσ1j and i≺AFσ2j, or j≻AFσ1i and j≺AFσ2i;

• Ki,j‾(τσ1,τσ2)=0.5 if (i≻AFσ1j or i≺AFσ1j and i≃AFσ2j), or (j≻AFσ1i or j≺AFσ1i and j≃AFσ2i).

A Kendall’s tau coefficient of 1 (K(τσ1,τσ2)=1) between two rankings means that both rankings are opposite (i.e. for all arguments x,y∈A, if x≻AFσ1y then y≻AFσ2x) while a Kendall’s tau coefficient of 0 (K(τσ1,τσ2)=0) means that both rankings are identical. So, the smaller the Kendall’s tau coefficient between two rankings, the higher their similarity.

Table 4 contains, for each pair of δ, the average Kendall’s tau coefficient, from the results previously computed, that we multiply by 100 to obtain a percentage of dissimilarity.

Table 4

Percentage of dissimilarity between the rankings from vdpδ with δ∈{0.001,0.2,0.4,0.6,0.8,0.9}

δ	0.001	0.2	0.4	0.6	0.8	0.9
0.001	0	0.06	0.55	4.09	10.62	13.74
0.2	0.06	0	0.52	4.13	10.63	13.64
0.4	0.55	0.52	0	3.71	10.13	13.3
0.6	4.09	4.13	3.71	0	6.82	9.86
0.8	10.62	10.63	10.13	6.82	0	3.16
0.9	13.74	13.64	13.3	9.86	3.16	0

The results show that the obtained rankings stay pretty close since the biggest dissimilarity between the smallest and largest value of δ is 13.74%. This dissimilarity remains overall very small, showing that the semantics remain quite stable as the parameter varies.

The question now is whether these differences are only caused by the fading effect or if δ has an impact on other domains too. In the following section, we will check which properties recalled in Section 2.2 are satisfied by vdpδ to answer this question.