Evaluation of Arguments from Support Relations: Axioms and Semantics

1 Introduction

Argumentation is a social activity whose aim is to increase (or decrease) the acceptability of a given standpoint for an audience by putting forward arguments. The standpoint may be a claim which can be true or false, an action to be performed, a goal to be reached, etc. Argumentation has gained great interest in Artificial Intelligence. It is used for decision making (e.g., [Amgoud and Prade, 2009]), defeasible reasoning (e.g., [Bondarenko et al., 1997]), and negotiation (e.g., [Reed, 1998]). Interested reader can find more on applications of argumentation in [Rahwan and Simari(eds.), 2009].

Whatever the application, an argumentation-based formalism, called argumentation framework, is generally defined as a set of arguments, attacks amongst the arguments, and a semantics for evaluating the arguments. A semantics assesses to what extent an argument is acceptable. Examples of semantics are those proposed by Dung [1995] and ranking semantics (e.g., [Amgoud and Ben-Naim, 2013; Besnard and Hunter, 2001; da Costa Pereira et al., 2011; Matt and Toni, 2008]).

An attack has a negative effect since its aim is weakening its target. Cayrol and Lagasquie [2005] have pointed out another meaningful relation between arguments, the so-called support. Unlike attack, this relation has positive effect and aims at strengthening its target. Several works have thus been done on the evaluation of arguments in argumentation frameworks where supports and attacks coexist [Oren and Norman, 2008; Boella et al., 2010; Brewka and Woltran, 2010;

Nouioua, 2013; Polberg and Oren, 2014]. They have extended Dung's semantics [1995] for accounting for supports. Surprisingly enough, when the attack relation is empty, these works consider all the arguments as equally acceptable. Thus, supported arguments are as acceptable as non-supported ones. This means that supports are not fully harnessed. Furthermore, these works assumed that arguments have the same basic strength, an assumption hardly ever satisfied in practice. Each argument has a basic strength representing the weight of its source [Parsons et al., 2011], or the importance degree of the value it promotes [Bench-Capon, 2003], or the importance degrees of the goals it supports [Amgoud and Prade, 2009], or the certainty degrees of its premises [Benferhat et al., 1993], ▷ ▷ ▷.

In this paper, we investigate for the first time argumentation frameworks where arguments interact only in a positive way (i.e., by supporting each other). Furthermore, arguments may have different basic strengths. Such frameworks have interesting practical applications like recommendation letters. A recommendation letter contains a general recommendation justified by a list of arguments, some of them support others. The evaluation of each argument, by combining its basic strength and the overall strengths of its supporters, gives insights on the weight of the general recommendation. Another application is the evaluation of newspapers' articles, namely argumentative essays. From a theoretical perspective, defining semantics for evaluating arguments on the basis of supports allows a better understanding of the role and impact of supports, and how they should be considered.

The contribution of the paper is threefold. First, we provide a set of axioms that a semantics should satisfy. Most of the axioms are mandatory, except three which represent pairwise incompatible choices. Second, we investigate some properties of semantics that satisfy the axioms. Third, we propose three semantics, each of which satisfies all the mandatory axioms and one of the three optional axioms.

The paper is structured as follows: Section 2 introduces the argumentation graphs we are interested in. Section 3 presents the axioms. Section 4 is devoted to properties of semantics. Section 5 defines three semantics. The last section concludes.

2 Basic Concepts

A support argumentation framework, called also support argumentation graph throughout the paper, is a set of arguments and a binary relation representing supports amongst arguments. Arguments are abstract entities whose internal structure is not specified. Each argument has an intrinsic strength which is expressed by a numerical value in the interval [0↪ 1]. The value 0 means the argument is worthless while 1 means the argument is very strong. Before introducing the graphs we are dealing with, let us first define the notion of weighting.

Definition 1 (Weighting) A weighting on a set L is a function from L to [0↪ 1].

Let us now introduce support argumentation graphs.

Definition 2 (Support Argumentation Graph) A support argumentation graph (SAF) is an ordered tuple A = hA, w, Si, where A is a non empty finite set of arguments, w is a weighting on A, and S ✓ A ⇥ A is a support relation. For a↪ b 2 A, the notation aSb means a supports b.

We define a semantics as a function assigning for every argument in a SAF a value between 0 and 1. This value, called strength degree, represents the overall strength of the argument, i.e., the strength issued from the aggregation of the basic strength of the argument and the overall strengths of its supporters. The higher the degree, the stronger the argument.

Definition 3 (Semantics) A semantics is a function F transforming any SAF A = hA↪ w↪ Si into a weighting on A. For a 2 A, DegFA(a) denotes the image of argument a by F(A), and is called strength degree of a.

Below is the list of all notations used in the paper.

Notations: Let A = hA↪w↪ Si be a SAF and a 2 A. We denote by SuppA(a) the set of all supporters of a in A (i.e. SuppA(a) = ¶b 2 A ♣ bSa♦), by Supp+A(a) the set of arguments supported by a (i.e. Supp+A(a) = ¶b 2 A ♣ aSb♦). For any two SAFs A = hA↪w↪ Si and A0 = hA0 ↪w0 ↪ S 0 i, A⊕A0 is the SAF hA[A0 ↪ w00 ↪ S [S 0 i where for any x 2 A (resp. x 2 A0 ), w00 (x) = w(x) (resp. w00 (x) = w0 (x)).

3 Axiomatic Foundations of Semantics

We propose a set of 17 axioms that shed light on foundational concepts and principles behind a semantics. Some of the axioms are dual to those proposed in [Amgoud and Ben-Naim, 2016] in case of attack graphs. Before presenting the first axiom, let us recall the definition of isomorphism.

Definition 4 (Isomorphism) Let A = hA, w, Si and A0 = hA0 , w0 , S 0 i be two SAFs. An isomorphism fromA to A0 is a bijective function f fromA to A0 such that:

• ÷ 8a↪ b 2 A, aSb iff f(a)S 0 f(b).
• ÷ 8a 2 A, w(a) = w0 (f(a)), and

The first basic axiom ensures that the strength degree of an argument does not depend on the argument's identity.

Axiom 1 (Anonymity) A semantics F satisfies anonymity iff, for any two SAFs A = hA↪ w↪ Si and A0 = hA0 ↪ w0 ↪ S 0 i, for any isomorphism f fromA to A0 , the following property holds: 8 a 2 A↪ DegFA(a) = DegFA0(f(a))▷

The second basic axiom, called independence, states the following: the strength degree of an argument a should be independent of any argument that is not connected to a (i.e., there is no path from that argument to a).

Axiom 2 (Independence) A semantics F satisfies independence iff, for any two SAFs A = hA↪w↪ Si and A0 = hA0 ↪w0 ↪ S 0 i such that A \ A0 = ;, the following property holds: 8 a 2 A↪ DegFA(a) = DegFA⊕A0(a)▷

Let us consider the following running example.

Example 1 Let A be the support argumentation graph depicted below.

Assume that w(a) = w(b) = w(d) = w(d0 ) = w(e) = w(e0 ) = 13 , w(x) = 0▷2 and w(c) = 34 . Independence ensures that the degree of a is independent of x's since the two arguments are not linked.

The next axiom, called non-dilution, states that supporting arguments has no impact on its own strength degree.

Axiom 3 (Non-Dilution) A semantics F satisfies non dilution iff, for any two SAFs A = hA, w, Si and A0 = hA0 , w0 , S 0 i such that A = A0 , w = w0 and S 0 = S [ ¶(a↪ b)♦ with Supp+A(b) = ;, the following holds: for all x 2 A ∩ ¶b♦, DegFA(x) = DegFA0 (x).

Example 1 (Cont) Non-dilution ensures that the degree of a should not take into account the fact that a supports d.

It is worth mentioning that non dilution is not mandatory in reputation systems. PageRank, the reputation system used by Google search engine for ranking web pages, violates the axiom. It considers that the more an agent (a web page) supports other agents, the less it is credible.

The following axiom, called dummy, states that arguments that get value 0 have no impact on the arguments they support.

Axiom 4 (Dummy) A semantics F satisfies dummy iff, for any SAF A = hA, w, Si, for all a↪ b 2 A, if

• ÷ SuppA(a) = SuppA(b) [ ¶x♦ such that DegFA(x) = 0,
• ÷ w(a) = w(b),

The next axiom, called monotony, ensures that an argument becomes stronger when its set of supporters gets bigger.

Axiom 5 (Monotony) A semantics F satisfies monotony iff, for any SAF A = hA↪ w↪ Si, for all a↪ b 2 A, if

• ÷ SuppA(b) ✓ SuppA(a), then DegFA(a) ≥ DegFA(b)▷
• ÷ w(a) = w(b),

Example 1 (Cont) Monotony ensures DegFA(d0 ) ≥ DegFA(d) ≥ DegFA(b), and DegFA(e0 ) ≥ DegFA(e).

The next axiom, called equivalence, ensures that the strength degree of an argument depends on the strength degrees of its direct supporters and its basic strength.

Axiom 6 (Equivalence) A semantics F satisfies equivalence iff, for any SAF A = hA↪ w↪ Si, for all a↪ b 2 A, if

• ÷ 9f, a bijective function, from SuppA(a) to SuppA(b) s.t 8x 2 SuppA(a), DegFA(x) = DegFA(f(x)),
• ÷ w(a) = w(b),

The axiom called coherence, states that the impact of support is proportional to the basic strength of its target.

Axiom 7 (Coherence) A semantics F satisfies coherence iff, for any SAF A = hA↪ w↪ Si, for all a↪ b 2 A, if

• ÷ DegFA(b) < 1,
• ÷ w(a) > w(b),
• ÷ SuppA(a) = SuppA(b),

The minimality axiom ensures that if an argument is not supported, its overall strength is equal to its basic strength.

Axiom 8 (Minimality) A semantics F satisfies minimality iff, for any SAF A = hA↪w↪ Si, for any argument a 2 A, if SuppA(a) = ;, then DegFA(a) = w(a).

Example 1 (Cont) From minimality, DegFA(a) = 13 and DegFA(x) = 0▷2.

The following axiom, called strengthening, defines formally the role of supports. It states that a support strengthens its target by increasing its overall strength (possibly only by an infinitesimal amount). This is particularly true when the support emanates from (even slightly) acceptable argument (i.e., an argument s.t. DegFA(▷) > 0). If the degree of the argument is already 1, the supports are useless.

Axiom 9 (Strengthening) A semantics F satisfies strengthening iff, for any SAF A = hA↪w↪ Si, for any argument a 2 A, if w(a) < 1 and 9b 2 SuppA(a) s.t. DegFA(b) > 0, then DegFA(a) > w(a).

The next axiom, called strengthening soundness, states that the only way of increasing the overall strength of an argument is by supporting the argument with an acceptable one.

Axiom 10 (Strengthening Soundness) A semantics F satisfies strengthening soundness iff, for any SAF A = hA↪ w↪ Si, for any argument a 2 A, if DegFA(a) > w(a) then 9b 2 SuppA(a) s.t. DegFA(b) > 0.

The three previous axioms are about the role of supports, which is strengthening arguments. The following axioms introduce two key factors that may impact the overall strength of an argument: the number of supporters and their quality. The more numerous the acceptable supporters of an argument, the stronger the argument.

Axiom 11 (Counting) A semantics F satisfies counting iff, for any SAF A = hA↪ w↪ Si, for all a↪ b 2 A such that

• ÷ w(a) = w(b), DegFA(b) < 1,

Example 1 (Cont) Counting ensures DegFA(d0 ) > DegFA(d) > DegFA(b), and DegFA(e0 ) > DegFA(e).

The quality of acceptable supporters is another factor impacting the overall strength of an argument. The next axiom, called reinforcement, states that if the overall strength of a supporter is increased, then its target is strengthened further provided that its strength degree is not already maximal (1).

Axiom 12 (Reinforcement) A semantics F satisfies reinforcement iff, for any SAF A = hA↪ w↪ Si, 8a↪ b 2 A, if

• ÷ SuppA(a) ∩ SuppA(b) = ¶x♦,
• ÷ w(a) = w(b), DegFA(b) < 1,
• ÷ SuppA(b) ∩ SuppA(a) = ¶y♦,
• ÷ DegFA(x) > DegFA(y) > 0,

Our next axiom, called boundedness, states that an argument which has a maximal degree (1) keeps the same degree if one of its supporters is strengthened.

Axiom 13 (Boundedness) A semantics F satisfies boundedness iff, for any SAF A = hA↪ w↪ Si, 8a↪ b 2 A such that

• ÷ SuppA(a) ∩ SuppA(b) = ¶x♦,
• ÷ w(a) = w(b),
• ÷ SuppA(b) ∩ SuppA(a) = ¶y♦,
• ÷ DegFA(x) > DegFA(y),

The previous axioms are all mandatory and should be satisfied by any semantics. The remaining axioms are optional. The first one, called imperfection, states that an argument can never get a maximal value if its basic strength is not maximal.

Axiom 14 (Imperfection) A semantics F satisfies imperfection iff, for any SAF A = hA↪w↪ Si, for any a 2 A, if w(a) < 1 then DegFA(a) < 1.

The three last axioms give an overwhelming weight to either the number of supporters, or their quality, or allow some compensation. More precisely, cardinality precedence says that an argument a is stronger than an argument b if the acceptable supporters of a are more numerous than those of b.

Axiom 15 (Cardinality Precedence) A semantics F satisfies cardinality precedence iff, for any SAF A = hA↪w↪ Si, for all a↪ b 2 A, if

• ÷ w(a) = w(b), DegFA(b) < 1,
• ÷ 0 < ♣¶x 2 SuppA(b) ♣ DegFA(x) > 0♦♣ < ♣¶y 2 SuppA(a) ♣ DegFA(y) > 0♦♣,
• ÷ 9 x 2 SuppA(b) s.t 8 y 2 SuppA(a), DegFA(x) > DegFA(y),

The next axiom, quality precedence, prefers the quality to the quantity of supporters. It says that an argument a is stronger than an argument b, if some supporter of a is stronger than any supporter of b.

Axiom 16 (Quality Precedence) A semantics F satisfies quality precedence iff, for any SAF A = hA↪w↪ Si, for all a↪ b 2 A, if

• ÷ w(a) = w(b), DegFA(a) < 1,
• ÷ 0 < ♣¶x 2 SuppA(b) ♣ DegFA(x) > 0♦♣ < ♣¶y 2 SuppA(a) ♣ DegFA(y) > 0♦♣,
• ÷ 9x 2 SuppA(b) s.t 8y 2 SuppA(a), DegFA(x) > DegFA(y),

The very last axiom, called compensation, states that a small number of strong supporters compensate a greater number of weak supporters.

Axiom 17 (Compensation) A semantics F satisfies compensation iff, for any SAF A = hA↪ w↪ Si, for all a↪ b 2 A, if

• ÷ ♣SuppA(a)♣ = n, ♣SuppA(b)♣ =mwith n <m,
• ÷ w(a) = w(b),
• ÷ 8x 2 SuppA(a), DegFA(x) = d, 8y 2 SuppA(b), DegFA(x) = d0 with d > d0 > 0

then DegFA(a) = DegFA(b).

4 Properties

Some axioms are incompatible, that is they cannot be satisfied all together by a semantics.

Proposition 1 There is no semantics which satisfies Cardinality Precedence and Quality Precedence (respectively Compensation). There is no semantics which satisfies Quality Precedence and Compensation. There is no semantics which satisfies both Quality Precedence and Counting. There is no semantics which satisfies at the same time: Independence, Non-Dilution, Equivalence, Reinforcement, Boundedness and Quality Precedence.

The other axioms are compatible as shown in Table 1 summarizing the axioms satisfied by the three semantics we will introduce in the next section. Furthermore, we can show that arguments that are supported by only rejected arguments, have an overall strength which is equal to their basic strength.

Proposition 2 If a semantics F satisfies Independence, NonDilution, Dummy and Minimality, then for any SAF A = hA↪ w↪ Si, for any a 2 A, if 8x 2 SuppA(a), DegFA(x) = 0, then DegFA(a) = w(a).

From the mandatory axioms, it follows that the strength degree of each argument is at least its basic strength, and at most 1. This means that arguments keep their basic strength if they are not supported or their supporters are all rejected. They are strengthened otherwise.

Theorem 1 If a semantics satisfies Independence, Monotony, Minimality and Strengthening, then for any SAF A = hA↪ w↪ Si, for any a 2 A, DegFA(a) 2 [w(a)↪ 1].

5 Semantics

In this section, we propose three semantics for assessing the overall strength of arguments in support argumentation graphs. In addition to the basic axioms, the first semantics satisfies Quality Precedence, the second satisfies Cardinality Precedence, and the last semantics satisfies Compensation.

5.1 Top-based Semantics

The first semantics satisfies quality precedence, thus it favors the quality of supporters over their cardinality. It is based on a scoring function which follows a multiple steps process. At each step, the function assigns a score to each argument. In the initial step, the score of an argument is its basic strength. Then, in each step, the score is recomputed on the basis of the basic strength as well as the score of the strongest supporter of the argument at the previous step.

Definition 5 (Top-based Function) Let A = hA↪ w↪ Si be a support argumentation graph. We define the top-based function ft from A to [0↪+1) as follows: for any argument a 2 A, for i 2 ¶0↪ 1↪ 2↪ ▷ ▷ ▷♦, if i = 0 then f i t (a) = w(a), otherwise f i t (a) = w(a) + (1 - w(a)) maxb2SuppA(a) f i-1 t (b). By convention, maxb2SuppA(a) f i t (b) = 0 if SuppA(a) = ;.

The value f i t (a) is the score of the argument a at step i. This value may change at each step, however, it converges to a unique value as i becomes high.

Theorem 2 The function f i t converges.

The top-based semantics is based on the previous scoring function. The strength degree of each argument is the limit reached using the scoring function ft.

Definition 6 (Top-based Semantics) The top-based semantics is a function Tbs transforming any support argumentation graph A = hA↪ w↪ Si into a weighting on A such that for any a 2 A, Deg Tbs A (a) = limi!1f i t (a).

We show next that the limit scores of arguments satisfy a nice property, namely the equation of Definition 5.

Theorem 3 For any support argumentation graph A = hA↪ w↪ Si, for any a 2 A,

Let us now illustrate the semantics with an example.

Example 1 (Cont) The strength degrees of the arguments of graph A under semantics Tbs are: Deg Tbs A (a) = 13 , Deg Tbs A (b) = 59 , Deg Tbs A (c) = 89 , Deg Tbs A (d) = 19 27 , Deg Tbs A (d0 ) = 77 81 , Deg Tbs A (e) = 25 27 , Deg Tbs A (e0 ) = 77 81 , and Deg Tbs A (x) = 0▷2. Notice that SuppA(e) = ¶c♦ and SuppA(d) = ¶a↪ b♦. The argument e has thus less supporters but its supporter c is stronger than both supporters of d. Since Tbs satisfies quality precedence, then e is stronger than d. Furthermore, d0 ⇡ e0 ≻ e ≻ c ≻ d ≻ b ≻ a ≻ x, where e ≻ c means e is stronger than c and d0 ⇡ e0 means d0 is as strong as e0 .

The Top-based semantics satisfies quality precedence as well as all the mandatory axioms which are compatible with it.

Theorem 4 Table 1 summarizes the axioms satisfied (violated) by top-based semantics.

From Theorems 1 and 4, it follows that the strength degree of each argument a is in the interval [w(a)↪ 1].

Corollary 1 For any support argumentation graph A = hA↪ w↪ Si, for any a 2 A, Deg Tbs A (a) 2 [w(a)↪ 1].

5.2 Reward-based Semantics

The second semantics, called reward-based semantics, favours the number of supporters over their quality. Its basic idea is the following: an argument receives a reward for each of its supporters. The greater the number of supporters, the smaller the amount of the reward. The reward concerning the last supporter takes into account the quality of the supporters. Note that no particular order of arguments is needed. Furthermore, since arguments having degree 0 have no impact on the arguments they support (Dummy axiom), such supporters are not taken into account. We thus consider only founded supporters. An argument is founded if there exists at least one path leading to it such that the basic strength of the source of the path is not 0. As we will see later, a founded argument has necessarily a positive degree.

Definition 7 (Founded Argument) Let A = hA↪ w↪ Si be a support argumentation graph and a 2 A. The argument a is founded iff there exists a finite sequence ha0↪ a1↪ ▷ ▷ ▷ ↪ ani of arguments such that w(a0) > 0, an = a and for all i = 0↪ 1↪ ▷ ▷ ▷ ↪ n - 1, aiSai+1. It is unfounded otherwise. Let SuppFA(a) denote the set of founded supporters of a.

It is easy to show that the basic strength of an unfounded argument is 0. Furthermore, if at least one supporter of an argument is founded, then the argument is itself founded.

Proposition 3 Let A = hA↪ w↪ Si be a support argumentation graph and a 2 A.

• ÷ a is unfounded iff SuppFA(a) = ; and w(a) = 0.
• ÷ If SuppFA(a) 6= ;, then a is founded.

The reward-based semantics is based on a scoring function which assigns a numerical value to each argument. If an argument is not founded, then it receives a score 0. Otherwise, the function proceeds in multiple steps. In the initial step, it assigns to each argument its basic strength. Then in each step, it recomputes all the scores by taking into account the basic strength, the number of founded supporters and their scores in the previous step.

Definition 8 (Reward-based Function) Let A = hA↪ w↪ Si be a support argumentation graph. We define the rewardbased function fr from A to [0↪+1) as follows: for any argument a 2 A, if a is unfounded, then f i r (a) = 0 for any i 2 ¶0↪ 1↪ 2↪ ▷ ▷ ▷♦. If a is founded, then for i 2 ¶0↪ 1↪ 2↪ ▷ ▷ ▷♦, if i = 0 then f i r (a) = w(a), otherwise

where n = ♣SuppFA(a)♣ and m = Pb2SuppFA(a) f i-1 r (b) n ▷ By convention, Pn-1 j=1 1 2j + m 2n = 0 if SuppFA(a) = ;.

The value f i r (a) is the score of the argument a at step i. This value converges to a unique value as i becomes high.

Theorem 5 The function f i r converges.

The reward-based semantics assigns for each argument a score which is equal to the limit reached by the reward-based function fr.

Definition 9 (Reward-based Semantics) The reward-based semantics is a function Rbs transforming any support argumentation graph A = hA↪ w↪ Si into a weighting on A such that for any a 2 A, Deg Rbs A (a) = limi!1f i r (a).

We show next that the limit scores of arguments satisfy the equation of Definition 8.

Theorem 6 For any support argumentation graph A = hA↪w↪ Si, for any a 2 A, Deg Rbs A (a) = 0 if a is unfounded. Otherwise,

where n = ♣SuppFA(a)♣ and

Let us illustrate the semantics with an example.

Example 1 (Cont) It is worth noticing that all the eight arguments are founded. Their strength degrees under semantics Rbs are: Deg Rbs A (a) = 13 , Deg Rbs A (b) = 49 , Deg Rbs A (c) = 29 36 , Deg Rbs A (d) = 79 108 , Deg Rbs A (d0 ) = 31 36 , Deg Rbs A (e) = 13 27 , Deg Rbs A (e0 ) = 109 144 , and Deg Rbs A (x) = 0▷2. According to Rbs, d0 ≻ c ≻ e0 ≻ d ≻ e ≻ b ≻ a ≻ x.

Theorem 7 Table 1 summarizes the axioms satisfied (violated) by reward-based semantics.

From Theorems 1 and 4, it follows that the strength degree of each argument a is in the interval [w(a)↪ 1].

Corollary 2 For any support argumentation graph A = hA↪ w↪ Si, for any a 2 A, Deg Rbs A (a) 2 [w(a)↪ 1].

The following proposition shows additional basic properties of reward-based semantics.

Proposition 4 Let A = hA↪ w↪ Si be a support argumentation graph and a 2 A.

• ÷ If SuppFA(a) = ;, then Deg Rbs A (a) = w(a).
• ÷ If a is founded, then Deg Rbs A (a) > 0.

5.3 Aggregation-based Semantic

Our last semantics satisfies compensation axiom, i.e., allows a small number of strong supporters to compensate a large number of weaker supporters. This semantics makes use of a scoring function which assigns a numerical value to each argument. Like the two previous functions, the new function proceeds in several steps. It starts by assigning to each argument its basic strength. Then at each step, it recomputes the score of the argument by adding to the basic strength a certain percentage of the scores of its supporters at the previous step.

◦

◦

÷

Table 1: Axioms satisfaction by the three semantics. The symbol ÷ (resp. ◦) means the axiom is satisfied (resp. violated).

Definition 10 Let A = hA↪w↪ Si be a support argumentation graph. We define the aggregation function fg from A to [0↪+1) as follows: for any argument a 2 A, for i 2 ¶0↪ 1↪ 2↪ ▷ ▷ ▷♦, if i = 0, then f i g (a) = w(a) otherwise

The value f i g (a) is the score of the argument a at step i. This value may change at each step, however, it converges to a unique value as i becomes high.

The aggregation-based semantics is based on the previous scoring function. The strength degree of each argument is the limit reached using the scoring function.

Definition 11 (Aggregation-based Semantics) The

aggregation-based semantics is a function Gbs transforming any support argumentation graph A = hA↪w↪ Si into a weighting on A such that for any a 2 A, Deg Gbs A (a) = limi!1f i g (a).

We show next that the limit scores of arguments satisfy a nice property, namely the equation of Definition 10.

Theorem 9 For any support argumentation graph A = hA↪ w↪ Si, for any a 2 A,

Consider again our running example.

Example 1 (Cont) The strength degrees of the arguments of graph A under semantics Gbs are: Deg Gbs A (a) = 13 , Deg Gbs A (b) = 12 , Deg Gbs A (c) = 56 , Deg Gbs A (d) = 21 33 , Deg Gbs A (d0 ) = 0▷74, Deg Gbs A (e) = 21 33 , Deg Gbs A (e0 ) = 0▷72, and Deg Gbs A (x) = 0▷2. Recall that SuppA(e) = ¶c♦ and SuppA(d) = ¶a↪ b♦. The argument e has less supporters but its supporter c is stronger than both supporters of d. Since Gbs satisfies compensation, then d is as strong as e. Tbs promotes the quality, thus it declares e stronger than d. Finally, Rbs promotes the cardinality, thus declares d stronger than e.

Aggregation-based semantics satisfies compensation as well as all the mandatory axioms.

Theorem 10 Table 1 summarizes the axioms satisfied (violated) by aggregation-based semantics.

From Theorems 1 and 4, it follows that the strength degree of each argument a is in interval [w(a)↪ 1].

Corollary 3 For any support argumentation graph A = hA↪ w↪ Si, for any a 2 A, Deg Gbs A (a) 2 [w(a)↪ 1].

It is worth noticing from Table 1, that Tbs is the only semantics that violates Imperfection. Indeed, according to this semantics, an argument whose intrinsic strength is 0 may become very strong (i.e., gets degree 1) due to its supporters. This scenario is not possible with the two other semantics.

6 Conclusion

This paper presented the first study on support argumentation graphs. It tackled the problem of assessing the overall strengths of arguments in such graphs. It proposed a set of axioms guiding the well definition of semantics. Then, it proposed three semantics that satisfy most of the axioms.

Our future work consists of extending these semantics for evaluating arguments in bipolar graphs (graphs containing attacks and supports). Recall that the existing semantics in the argumentation literature do not treat properly supports. For instance, if the attack relation is empty, then they declare all the arguments as acceptable, neglecting thus the supports.

Acknowledgements

This work benefited from the support of CIMI and AMANDE ANR-13-BS02-0004 project funded by the French National Research Agency.