Parametrized Gradual Semantics Dealing with Varied Degrees of Compensation

1 Introduction

An argumentation system is made of an argumentation graph and a semantics. The former is a graph whose nodes are arguments, and the edges represent attacks between arguments. Each argument may have a basic weigh, which represents its intrinsic strength, for example the degree of trustworthiness of the argument's source [da Costa Pereira et al., 2011]. A semantics is a formal method that evaluates the strength of every argument in the graph. This evaluation is crucial since it determines the outcomes of the system. Consequently, a plethora of semantics have been proposed in the literature. The very first ones are the extension semantics introduced in [Dung, 1995]. They calculate sets of jointly acceptable arguments, called extensions, which they use to classify arguments as sceptically accepted, credulously accepted and rejected. In [Cayrol and Lagasquie-Schiex, 2005] another type of semantics, called gradual, has been introduced with the purpose of refining the above cited semantics. Examples of such semantics are Trust-based [da Costa Pereira et al., 2011], h-Categorizer [Besnard and Hunter, 2001; Pu et al., 2014], social semantics [Leite and Martins, 2011].

Due to the large number of semantics in the literature, formal properties, or principles, have been proposed to analyse the underpinnings of each semantics and to compare pairs of semantics in [Amgoud et al., 2017]. Three of those properties, namely Cardinality Precedence (CP), Quality Precedence (QP) and Compensation, describe three incompatible strategies that a semantics may follow whenever it faces a dilemma when comparing an argument that is attacked by a few strong attackers with another argument that is attacked by numerous weaker attackers.

Consider the argumentation graph G1 in Table 1 on a multiple criteria decision problem where A,B stand for buying the houses h1 and h2 respectively, and argument Z for the house h2 being very expensive. Furthermore, let Y1,X1,X2 stand for: the house h1 is far from the school s, there is a metro station just in front of h1, there is another school in the neighbourhood of h1. Assume that the remaining arguments Y2, Y3 refer to other criteria in the same spirit as Y1 and they are attacked by X3, . . . ,X6. The arguments Y1, Y2 and Y3 are attacked while Z is not. Hence, a reasonable semantics would declare Z as stronger than any Yi, i = 1, 3. It follows that A has more attackers than B but its attackers are weaker than the sole attacker of B. The question is which of the two arguments A and B is stronger. A semantics which follows (CP) (resp. (QP)) would declare A weaker than B (resp. A stronger than B). A semantics satisfying Compensation might rather declare the two arguments equally strong.

Compensation is clearly based on a degree, which is a tuple that indicates: i) to what extent an attacker is weak, ii) the number of weak attackers needed to compensate a strong one. However, its corresponding principles in [Amgoud et al., 2017; Amgoud et al., 2022] do not refer to any compensation degree. They state that in case of dilemma, a semantics may compensate without specifying when this may happen, which makes them the less informative properties in the literature. It has been shown in [Amgoud et al., 2022] that there are two gradual semantics in the literature that satisfy the property: Trust-based [da Costa Pereira et al., 2011] and weighted hcategorizer [Amgoud et al., 2017]. However, no insight is given on their degree(s) of compensation.

In [Amgoud et al., 2016], the authors focused on a specific class of degrees of compensation, and proposed a parameterised family of gradual semantics that extend h-Categorizer [Pu et al., 2014]. The aim is that every value of the parameter defines a semantics that compensates at a degree from the class. However, it turns out that the semantics satisfy the principle only when the number of attackers of A (in G1) is fixed to be greater than the number of attackers of each Yi. Hence, compensation is possible in this graph while it is not possible in G2. This seems ad-hoc since there is no intuitive reason for such a restriction. Furthermore, it is not clear what semantics do in such graphs.

This paper investigates gradual semantics that satisfy compensation at various degrees. Its contributions are five-fold:

1. It proposes a large parameterised family of gradual semantics. The family is based on a parameter α, which takes values from the interval (0,+∞), each of which leads to a different semantics. The smaller its value, the bigger the influence of the number of attacks. Conversely, the greater its value, the bigger the influence of the quality of attackers. Thus, α determines the strategy that a semantics follows to solve dilemmas.
2. It shows that the family is the first which unifies multiple semantics that share some principles but differ in their behaviour regarding solving dilemmas. Indeed, it proves that when α approaches 0, the corresponding semantics satisfies (CP) while when it approaches +∞, the semantics satisfies (QP). All the other values of α guarantee (Compensation) with different degrees.
3. It proves that the new family encompasses the semantics from [Amgoud et al., 2016] which compensate.
4. It discusses three classes of compensation situations, i.e., three parameterised compensation degrees, and shows that the new semantics satisfies all of the variants of Compensation at some degree.
5. It shows that neither of the existing semantics satisfies all the variants of compensation, meaning that the new family has the greatest compensatory power.

The paper is organised as follows: Section 2 recalls the background, Section 3 defines the novel family of semantics, Section 4 introduces three classes of compensation degrees, Section 5 analyses existing gradual semantics, and the last section is devoted to some related work and concluding remarks. All proofs are provided in supplementary material.

2 Background

In the paper, we are interested in weighted argumentation graphs. Their nodes are arguments, each of which has a basic weight, and edges represent attacks (i.e., conflicts) between arguments. For the sake of simplicity, weights of arguments are elements of the unit interval [0, 1]. The greater the value, the stronger the argument.

Definition 1 (Weighted Graph). A weighted argumentation graph is a tuple G = ⟨A,w,R⟩, where A is a non-empty finite set of arguments, w : A → [0, 1], and R ⊆ A × A. Let AG denote the set of all weighted graphs.

For a, b ∈ A, w(a) is the basic weight of a, (a, b) ∈ R means a attacks b.

Notations: Let G = ⟨A, w, R⟩ ∈ AG and a ∈ A. We denote by w ≡ 1 the case where all arguments have a weight equal to 1 and we call such a graph flat. AttG(a) denotes the set {b ∈ A | (b, a) ∈ R} of direct attackers of a in G. Let G′ = ⟨A′ ,w′ ,R′ ⟩ ∈ AG such that A∩A′ = ∅. G⊕G′ = ⟨A∪A′ , w′′ , R∪R′ ⟩ ∈ AG such that ∀x ∈ A (resp. x ∈ A′ ), w′′ (x) = w(x) (resp. w′′ (x) = w′ (x)).

A gradual semantics is a function that assigns a value from a given ordered scale to each argument. Different scales can be used, but as in most papers on gradual semantics, we use the unit interval of reals [0, 1] with the interpretation: the greater the value, the stronger the argument.

Definition 2 (Gradual Semantics). A gradual semantics is a function S assigning to any G = ⟨A, w, R⟩ ∈ AG a weighting DegSG on A, i.e., DegSG : A → [0, 1]. For any a ∈ A, DegSG(a) is the strength of a in G under semantics S.

Let us now discuss the possible strategies that a semantics may follow when it faces a conflict between the strength and the quantity of attackers. Three strategies have been identified in [Amgoud et al., 2017; Amgoud et al., 2022], they depend on whether quantity or quality is more important. The first, called Cardinality Precedence, states that a great number of attackers has more effect on an argument than just a few. In the graph G1, the argument A would be weaker than B.

Principle 1 (Cardinality Precedence). A semantics S satisfies cardinality precedence (CP) iff, for any G = ⟨A, w, R⟩ ∈ AG, for all a, b ∈ A, if

• w(a) = w(b), DegSG(a) > 0,
• |{x ∈ AttG(a) | DegSG(x) > 0}| < |{y ∈ AttG(b) | DegSG(y) > 0}|,

then DegSG(a) > DegSG(b).

Quality Precedence gives more importance to the strength of attackers. In the graph G1, the argument A would be stronger than B.

Principle 2 (Quality Precedence). A semantics S satisfies quality precedence (QP) iff, for any G = ⟨A,w,R⟩ ∈ AG, for all a, b ∈ A, if

• w(a) = w(b), DegSG(a) > 0,
• ∃y ∈ AttG(b) such that DegSG(y) > 0 and ∀x ∈ AttG(a), DegSG(y) > DegSG(x),

The third strategy is based on compensation. Its basic idea is that several weak attacks have the same impact as few strong attacks on an argument. For instance, in the graph G1, the three weak attackers of A may compensate the strong attacker of B, and thus A is as strong as B. Compensation is very specific compared to other principles. It is clearly based on a degree which is a tuple indicating i) to what extent an attack is weak, and ii) the number of weak attackers needed to compensate a strong one. An exact definition of compensation would fix the degree, however this is not an obvious task since there are a lot of possibilities. Leaving the degree unfixed, as in the definition below from [Amgoud et al., 2022], leads to a principle with a high degree of granularity, but it does not state when a semantics compensates.

Principle 3 (Compensation). A semantics S satisfies compensation iff it violates both CP and QP.

Remark: It is worth noticing that (CP), (QP) and Compensation compare two groups of arguments (attackers). In [Amgoud and Ben-Naim, 2013], the so-called group comparison (GC) has been proposed for ranking pairs of arguments on the basis of their attackers. It states that an argument A is ranked at least as high as B if the attackers of B are at least as numerous and well-ranked as those of A. This relation is not applicable in the context of compensation. Consider for instance the graph G1. (GC) declares the sets {Y1, Y2, Y3} and {Z} as incomparable. Similarly, both in G2 and G3 the groups of attackers would be incomparable with respect to (GC). In fact, (GC) declares two sets as equally strong only when they have the same cardinality and their arguments are equally strong. Hence, it is orthogonal to compensation.

3 Novel Family of Gradual Semantics

This section introduces a novel family of gradual semantics which compensate. The family evaluates the strengths of arguments in weighted argumentation graphs. It is based on a parameter α which takes values from (0,+∞), where each value defines a semantics. We define the strength of an argument in an iterative way, starting from its basic weight in the considered weighted graph, and updating its value in each step, using the values of its attackers from the previous step.

Definition 3 (Parameterised Semantics Sα). Let G = ⟨A, w,R⟩ ∈ AG, and let us assume an ordering on A, i.e., let A = {a1, . . . , an}. Let us define the sequence of n-tuples s0(α,G), s1(α,G), s2(α,G), . . . in the following way:

• s0(α,G) = (w(a1),w(a2), . . . ,w(an))
• si+1(α,G) = F α(si(α,G)), where the update operator

is defined by:

• s∗ (α,G) = limn→+∞sn(α,G).

Then DegSα G is defined by

It is trivial to check that the values of DegSα G do not depend on the ordering of the set A. The exponent min{1, 1/α} in the function that defines F α k is our technical solution of the research problem: define a family of semantics that can compensate at every degree (see Section 4), while the extreme values of α favor quantity and quality (Theorem 5) and the semantics satisfy desirable principles (Theorem 6).

The following result shows that Sα is always well defined. Theorem 1. For every α ∈ (0, +∞) and every G ∈ AG, the sequence {sn(α, G)}+∞ n=0 converges.

Example 1. The tables 2 and 3 summarise the strengths (DegSα G (.)) of arguments in the graphs G2 and G3 respectively, for α = 2 and for α = 0.5. Note that the semantics S0.5 considers the two arguments A and B as equally strong.

In what follows, we show that Sα can be equivalently defined in a more intuitive way, by calculating the strength of an argument based on its basic weight and strengths of its attackers in the way defined by the set of equations (2) (one equation per argument). Our choice to define Sα as above is because Definition 3 directly provides an algorithm for calculation the strength with arbitrary precision.

Theorem 2. For every G = ⟨A, w, R⟩ ∈ AG and every a ∈ A, the following holds:

We strengthen Theorem 2 by showing that DegSα G is the only function satisfying the equation (2). This characterisation result means that (2) is an alternative definition of the semantics, which is useful for checking some of its properties in a convenient way.

We consider two semantics as compatible if they return the same ranking of arguments on each graph.

Definition 4 (Compatibility). Let S and S′ be two gradual semantics. We say that S and S′ are compatible if for every graph G = ⟨A, w, R⟩, for all a, b ∈ A, the following holds: DegSG(a) ≥ DegSG(b) iff DegS′ G (a) ≥ DegS′ G (b).

We show that each two distinct semantics from the new family are incompatible, which is expected since they are designed to satisfy the compensation at different degrees. This result shows that each (even a small) change in parameter α impacts the way the semantics ranks the arguments.

Theorem 4. Let α, β ∈ (0,+∞). If α ̸= β, then the semantics Sα and Sβ are not compatible.

In what follows, we characterize the two semantics that take the extreme values of α. Let us first consider the weighted max-based semantics from [Amgoud et al., 2017]. This semantics is defined (if we adopt the terminology from this paper) in the same way as Sα, with the difference that the update function F α from Definition 3 is replaced with the function F max : [0, 1]n → [0, 1]n , F max = (F max 1 , . . . , F max n ), defined by

It was shown in [Amgoud et al., 2017] that the weighted max-based semantics satisfies (QP).

Next we introduce a simplification of a weighted cardbased semantics from [Amgoud et al., 2017]. We define that semantics as in Definition 3, by replacing F α with the function F c defined by

where AttFG(a) is the set of attackers of a that have strictly positive basic weight. Observe that the fixed point of this function is reached after one iteration 1 , and from the form of F c it is clear that the resulting semantics satisfies (CP).

Now we formally show that in the limit cases, when α approaches 0 (resp. +∞) we obtain semantics that satisfy (CP) (respectively (QP)).

Theorem 5. The following two results hold.

• limα→+∞F α ≡ F max
• limα→0+ F α ≡ F c .

We prove next that all of our semantics behave well with respect to principles for gradual semantics proposed in [Amgoud et al., 2017]. We recall those principles in Appendix A. Note that there are other principles in the literature (eg., [Amgoud et al., 2022; Baroni et al., 2019; Bonzon et al., 2021]), but due to lack of space, we focus on the ones that highlight the underpinnings of our semantics.

Theorem 6. For every α ∈ (0,+∞), Sα satisfies all the principles from Appendix A, and compensation.

To sum up, this section introduced a novel parameterised family which unifies multiple gradual semantics that share some principles (in Appendix A) but differ in their behaviour regarding solving dilemmas.

4 Classes of Compensation Degrees

We have seen previously that every semantics (not taking one of the two extreme values of α) satisfies compensation. However, the latter does not inform precisely in which way a semantics compensates as it leaves the compensation degree unfixed. The aim of this section is to shed light on various situations where semantics of the family compensate. There are certainly a lot of possibilities but we focus here on three classes of degrees. The first one has been introduced in [Amgoud et al., 2016] for flat graphs. It defines a strong attack as an attack from an unattacked argument, and a weak attack as one that comes from an argument that is attacked by i unattacked arguments. Before recalling the property, let us introduce a useful notation.

Let us now recall the way Compensation was defined by Amgoud et al. (2016) for flat graphs of the form ⟨A,w ≡ 1,R⟩. It says that an argument receiving one strong attack is as good as an argument receiving n weak attacks.

Definition 5 ((n, i)-Compensation). Let n, i ∈ {1, 2, 3, . . .}. A semantics S satisfies (n, i)-compensation iff for any G = ⟨A, w ≡ 1, R⟩ ∈ AG, for all a, b ∈ A, the following holds: if

• |Att(a)| = n, Att(a) ⊆ C i (G), and
• |Att(b)| = 1, Att(b) ⊆ C 0 (G),

1 Namely, after assigning the basic weights, the iterative application of F c k on an argument yields 0 in all iterations if and only if that argument's initial weight is 0. For other arguments (those having strictly positive initial weight), we obtain the final result in the first iteration, since it is depends only on the number of attackers having positive acceptability degree, and that number does not change.

This definition is somehow limited since it ignores arguments that are attacked by arguments which are themselves attacked. In the graph G3, every attacker Yi of A is attacked by two arguments which are themselves attacked. Hence, the above definition does not apply on A and B even if there is a conflict between the number and the strength of their attackers. The notion of weakness is too specific that it ignores reasonable cases of weak arguments. Furthermore, several weak attackers can only compensate a single strong attacker. We propose another class of degrees in which several weaker attackers may compensate a smaller group of stronger attackers, and the two parameters of compensation are fixed.

Definition 6 ((m, c, n, d)-Compensation). Let m, n ∈ N such that m > n and 0 < c < d ≤ 1. A semantics S satisfies (m, c, n, d)-compensation iff for any G = ⟨A, w, R⟩ ∈ AG, for all a, b ∈ A such that w(a) = w(b), if

• |Att(a)| =m
• for every x ∈ Att(a), DegSG(x) = c
• |Att(b)| = n
• for every x ∈ Att(b), DegSG(x) = d

then DegSG(a) = DegSG(b).

This class is also limited since it assumes that all attackers of an argument have the same strength. In what follows, we relax this constraint. The idea is that two different sets of attackers compensate as long as one set contains more attackers (following Principle 1, we exclude worthless attackers, i.e., those attackers whose degree is 0), while the other contains the arguments that are stronger than any of them.

Definition 7 ((m, ¯ c, n, ¯ d)-Compensation). Let m, n ∈ N such that m > n and ¯ c = (c1, . . . , cm) ∈ [0, 1]m and ¯ d = (d1, . . . , dn) ∈ [0, 1]n such that maxci < maxdj and minci > 0. A semantics S satisfies (m, ¯ c, n, ¯ d)-compensation, ¯ c = (c1, . . . , cm) and ¯ d = (d1, . . . , dn), iff for any G = ⟨A,w,R⟩ ∈ AG, for all a, b ∈ A such that w(a) = w(b), the following holds: if

• {z ∈ Att(a) | DegSG(z) > 0} = {x1, . . . , xm}
• (DegSG(x1), . . . , DegSG(xm)) = (c1, . . . , cm)
• {z ∈ Att(b) | DegSG(z) > 0} = {y1, . . . , yn}

· (DegSG(y1), . . . , DegSG(yn)) = (d1, . . . , dn)

then DegSG(a) = DegSG(b).

It is immediate to see that any of the three classes implies Compensation.

Proposition 1. If a semantics S satisfies (n, i)- (resp. (m, c, n, d)-, (m, ¯ c, n, ¯ d)-) Compensation, then S satisfies Compensation.

We show next that semantics of our family satisfy (n, i)-Compensation. We only exclude the case n = 1, since one attacked argument does not compensate one non-attacked argument.

Theorem 7. For every n, i ∈ N such that n > 1 there exists a unique α ∈ (0,+∞) such that Sα satisfies (n, i)-Compensation.

Instances of the novel family of semantics can satisfy all possible degrees of compensation as specified in Definition 6. This result shows that a smaller number of attackers of some fixed degree can always compensate a greater number of attackers of some fixed weaker (but still positive) degree.

Theorem 8. For every quadruple (m, c, n, d) which satisfies the conditions of Definition 6, there exists a unique α ∈ (0, +∞) such that Sα satisfies (m, c, n, d)-compensation.

Finally, we prove that of our family of semantics can cover all possible degrees of compensation as defined in Definition 7. This result is more general and corresponds to the idea that quantity can always be compensated by quality. Intuitively, it says that any given set of attackers can always be compensated by another set containing an attacker which is stronger than each of them individually, regardless of exact strengths of other attackers.

Theorem 9. For every quadruple (m, ¯ c, n, ¯ d) which satisfies the conditions of Definition 7, there exists α ∈ (0, +∞) such that Sα satisfies (m, ¯ c, n, ¯ d)-compensation.

To sum up, this section provided three classes of compensation degrees which are guaranteed by semantics of the novel family. This sheds light on various compensation situations.

5 Analyzing Existing Gradual Semantics

This section analyses existing semantics that satisfy Compensation. There are three such semantics: the α-BBS semantics from [Amgoud et al., 2016], Trust-based from [da Costa Pereira et al., 2011] and weighted h-Categorizer from [Amgoud et al., 2017]. Note that any gradual semantics, like Iterative Schema from [Gabbay and Rodrigues, 2015], which violates Compensation does not satisfy (n, i)-compensation, (m, c, n, d)-compensation nor (m, ¯ c, n, ¯ d)-compensation due to Proposition 1.

α-BBS semantics. α-BBS is a family of gradual semantics introduced in [Amgoud et al., 2016]. It evaluates arguments of flat graphs, i.e., of the form G = ⟨A,w ≡ 1,R⟩. It is based on a parameter α taking values from the interval (0,+∞). Each value gives birth to one semantics, which assigns a numerical value from [1,+∞) to every argument a ∈ A as follows: If AttG(a) = ∅, then sα(a) = 1, else:

Note that this semantics uses the scale [1,+∞) instead of [0, 1] (see Def. 2). Furthermore, the value sα(a) represents the burden of the argument a (i.e., how heavily it is attacked), and therefore follows the intuition that the smaller the value, the stronger the argument.

Example 2. Consider the argumentation graph G1 from Table 1 and let all the arguments have the weight 1. For α = 1, we obtain sα(A) = sα(B) = 2.

α-BBS semantics have been proposed with the aim of ensuring (n, i)-Compensation (Definition 5) at any degree. It has been shown that the parameter α of those semantics is related to the degree (n, i) as follows.

Property 1 ([Amgoud et al., 2016]). For every n, i ∈ N such that n > i there exists a unique α ∈ (0,+∞) such that α-BBS satisfies (n, i)-Compensation.

Furthermore, it is easy to check that a stronger result holds, namely that the α from the previous property is greater or equal to 1.

Property 2. For every n, i ∈ N such that n > i there exists a unique α ∈ [1,+∞) such that α-BBS satisfies (n, i)-Compensation.

Together, the previous two results also imply that for n > i, there exists no α ∈ (0, 1) such that α-BBS satisfies (n, i)-Compensation.

However, the properties remain silent about cases where n ≤ i (as in the graph G2). In what follows, we show that there is no α-BBS semantics that satisfies (n, i)-Compensation when n ≤ i.

Theorem 10. For every n, i ∈ N such that i ≥ n there does not exist α ∈ (0,+∞) such that α-BBS satisfies (n, i)-Compensation.

We can see that the values of α from (0, 1) are never used by α-BBS to satisfy (n, i)-Compensation, and the values from [1,+∞) can only compensate only for (n, i) such that n > i.

Unlike α-BBS, our novel family has more compensation capabilities as its semantics compensate at any degree (n, i) even when n ≤ i, as stated by Theorem 7.

Example 3. Consider the graph G2 from Table 1. According to Theorem 10, α-BBS cannot compensate on G2. However, Theorem 7 shows that there exists a unique α that yields compensation. Indeed, for α = 0.5, arguments A and B in graph G2 have exactly the same degree, namely DegSα G2 (A) = DegSα G2 (B) = 0.5.

The following result states that for every α ≥ 1, the semantics Sα of our family corresponds to the inverse of the corresponding α-BBS semantics. However, this result does not hold for any α < 1.

Theorem 11. Let G = ⟨A,w,R⟩ ∈ AG such that w ≡ 1, and let a ∈ A. For every α ∈ [1, +∞),

This result shows that the novel family improves the family proposed in [Amgoud et al., 2016] by keeping the semantics that satisfy (n, i)-Compensation and replacing the remaining ones with semantics that satisfy the principle (as shown in Theorem 7). Consequently, instances of α-BBS where α ∈ [1,+∞) inherit the properties of our family regarding the two other classes of compensation degrees.

Trust-based semantics. Trust-based semantics (TB) has been proposed in [da Costa Pereira et al., 2011] to evaluate the strength of arguments in weighted argumentation graphs, where the basic weight assigned to an argument represents the degree of reliability of its source. For any graph G = ⟨A,w,R⟩, any a ∈ A, the strength of a is the limit reached by the scoring function f defined as follows:

While it has been shown in [Amgoud et al., 2022] that (TB) satisfies Compensation, the following result shows that it is not able to compensate at any of the three classes of compensation degrees.

Theorem 12. The following properties hold:

• There exist no n, i ∈ {1, 2, 3, . . .} such that Trust-based semantics satisfies (n, i)-Compensation.
• There exist no m, c, n, d such that Trust-based semantics satisfies (m, c, n, d)-Compensation.
• There exist no m, c, n, d such that Trust-based semantics satisfies (m, ¯ c, n, ¯ d)-Compensation.

Weighted h-Categoriser Introduced in [Amgoud et al., 2017], weighted h-Categoriser evaluates arguments in weighted graphs as follows: For any graph G = ⟨A,w,R⟩, any a ∈ A, Deg Hbs G (a) = w(a) if AttG(a) = ∅, else:

It has been shown in [Amgoud et al., 2022] that Hbs satisfies Compensation. It has also been shown in [Amgoud et al., 2016] that in case of flat graphs, it satisfies (n, i)-Compensation when n = i + 1.

Property 3 ([Amgoud et al., 2016; Amgoud et al., 2022]).

• Hbs satisfies Compensation.
• Hbs satisfies (n, i)-Compensation if and only if n = i + 1.

The following result shows that under some conditions, Hbs can compensate at the two other classes of degrees.

Theorem 13. The following properties hold:

• Hbs satisfies (m, c, n, d)-compensation if and only if m · c = n · d
• Hbs satisfies (m, ¯ c, n, ¯ d)-compensation if and only if ∑mi=1 ci = ∑nj=1 dj

To sum up, this section showed that the novel family offers semantics with greater compensation capabilities. They allow compensation at more degrees than the existing gradual semantics. Furthermore, the family encompasses some semantics of α-BBS as well as Hbs.

6 RelatedWork and Conclusion

This paper investigated gradual semantics, which have been initiated by Cayrol and Lagasquie-Schiex (2005). Several semantics and approaches have been proposed since then (eg., [Besnard and Hunter, 2001; da Costa Pereira et al., 2011;

Leite and Martins, 2011; Rago et al., 2016; Potyka, 2018; Amgoud et al., 2022]) and several efforts have also been made to study principles underlying gradual semantics (eg., [Amgoud and Ben-Naim, 2013; Bonzon et al., 2016; Baroni et al., 2019; Yun et al., 2020; Amgoud et al., 2022]. This paper focused on one of them, namely compensation. The latter is one of the possible strategies to follow when facing a dilemma between the quantity and the quality of attackers.

In this paper we pointed out some limits of existing principles that formalise the notion of compensation, and proposed ways for getting rid of them. The idea is to have insight on the ways semantics compensate. For that purpose, we defined three classes of compensation degrees. We also proposed a large family of gradual semantics which encompasses semantics that are able to compensate at any of those degrees. We have shown that there is no other gradual semantics in the literature that is able to do the same job.

Let us draw the reader's attention to the fact that the notion of compensation is different from of accrual [Prakken, 2005; Lucero et al., 2009]. Compensation is related to the fact that the effect of few strong attackers can be similar to that of more weaker attackers. Accrual focuses on claims, and shows how different reasons supporting the same claim may be aggregated into a single argument.

Viana and Alcˆantara (2021) defined two new principles: Quality Compensation and Cardinality Compensation, which are weakened versions of Quality Precedence and Cardinality Precedence, respectively. They allow, for example, to use the quality of attacks to decide which argument is stronger in case the overall sum of attacks on two arguments is equal. We do not provide a more fine-grained discussion regarding these principles because our principles are neither inspired by nor related to them.

As a future work, we plan to investigate other reasonable degrees of compensation, and apply our formal approach in multiple criteria decision making.

Appendix A: Principles for Semantics

In what follows, we recall some properties from [Amgoud et al., 2017]. Let S be a semantics.

Anonymity: S satisfies anonymity iff ∀G = ⟨A, w, R⟩, G′ = ⟨A′ , w′ ,R′ ⟩ ∈ AG, for any isomorphism f fromG to G′ , the following holds: ∀ a ∈ A, DegSG(a) = DegSG′ (f (a)).

Independence: S satisfies independence iff ∀G = ⟨A,w,R⟩,G′ = ⟨A′ ,w′ ,R′ ⟩ ∈ AG s.t A ∩ A′ = ∅, the following holds: ∀ a ∈ A, DegSG(a) = DegSG⊕G′(a).

Directionality: S satisfies directionality iff ∀G = ⟨A, w, R⟩ ∈ AG, ∀a, b ∈ A, ∀G′ = ⟨A′ , w′ , R′ ⟩ ∈ AG, s.t.

• A′ = A,
• w′ = w,
• R′ = R ∪ {(a, b)},

it holds that: ∀x ∈ A, if there is no path from b to x, then DegSG(x) = DegSG′ (x).

Equivalence: S satisfies equivalence iff ∀G = ⟨A,w,R⟩ ∈ AG, ∀a, b ∈ A, if

• w(a) = w(b), and
• there exists a bijective function f fromAtt(a) to Att(b) s.t. ∀x ∈ Att(a), DegSG(x) = DegSG(f(x)),

Maximality: S satisfies maximality iff ∀G = ⟨A,w,R⟩ ∈ AG, ∀a ∈ A, if Att(a) = ∅, then DegSG(a) = w(a).

Neutrality: S satisfies neutrality iff ∀G = ⟨A,w,R⟩ ∈ AG, ∀a, b ∈ A, if

• w(a) = w(b), and
• Att(b) = Att(a) ∪ {x} s.t. x ∈ A \ Att(a) and DegSG(x) = 0,

Weakening: S satisfies weakening iff ∀G = ⟨A, w, R⟩ ∈ AG, ∀a ∈ A, if

• w(a) > 0, and
• ∃b ∈ Att(a) s.t. w(b) > 0,

then DegSG(a) < w(a).

Proportionality: S satisfies proportionality iff ∀G = ⟨A, w, R⟩ ∈ AG, ∀a, b ∈ A, if

• Att(a) = Att(b),
• w(a) > w(b),
• DegSG(a) > 0,

Resilience: S satisfies resilience iff ∀G = ⟨A,w,R⟩ ∈ AG, ∀a ∈ A, DegSG(a) = 0 iff w(a) = 0.

Reinforcement: S satisfies reinforcement iff ∀G = ⟨A, w, R⟩ ∈ AG, ∀a, b ∈ A, if

• w(a) = w(b),
• DegSG(a) > 0,
• Att(a) \ Att(b) = {x}, Att(b) \ Att(a) = {y},
• DegSG(y) > DegSG(x),

Counting: S satisfies counting iff ∀G = ⟨A,w,R⟩ ∈ AG, ∀a, b ∈ A, if

• w(a) = w(b),
• Att(b) = Att(a)∪{x}with x /∈ Att(a), DegSG(x) > 0, and DegSG(a) > 0,

Acknowledgements

Srdjan Vesic was supported by the project AGGREEY ANR22-CE23-0005 from the French National Research Agency (ANR). Leila Amgoud was supported by the ANR (ANR-19PI3A-0004) through the AI Interdisciplinary Institute, ANITI, as a part of"Investing for the Future - PIA3" program.