# bilateral_gradual_semantics_for_weighted_argumentation__bdb81d29.pdf

Bilateral Gradual Semantics for Weighted Argumentation

Zongshun Wang, Yuping Shen

Institute of Logic and Cognition, Department of Philosophy, Sun Yat-sen University, P.R. China wangzsh7@mail2.sysu.edu.cn, shyping@mail.sysu.edu.cn

Abstract argumentation is a reasoning model for evaluating arguments. Recently, gradual semantics has received considerable attention in weighted argumentation, which assigns an acceptability degree to each argument as its strength. In this paper, we aim to enhance gradual semantics by nonreciprocally incorporating the notion of rejectability degree. Such a setting offers a bilateral perspective on argument strength, enabling more comprehensive argument evaluations in practical situations. To this end, we ﬁrst provide a set of principles for our semantics, taking both the acceptability and rejectability degrees into account, and propose three novel semantics conforming to the above principles. These semantics are deﬁned as the limits of iterative sequences that always converge in any given weighted argumentation system, making them preferable for real-world applications.

Introduction Abstract argumentation framework (AF) is a well-studied model for reasoning and decision-making in conﬂict situations (Dung 1995; Amgoud and Prade 2009; Bench-Capon and Dunne 2007). An AF is a directed graph whose nodes represent arguments and arrows represent attacks between arguments. Evaluating arguments is a central topic in abstract argumentation and commonly achieved through various semantics. For instance, the original extension semantics (Dung 1995) looks for sets of arguments that are jointly acceptable, and the justiﬁcation status of arguments is classiﬁed as sceptically/credulously accepted and rejected (Baroni, Caminada, and Giacomin 2011). Similarly, the labelling semantics (Caminada and Gabbay 2009) assigns each argument a label from {accepted, rejected, undecided}. However, such qualitative settings may not always be sufﬁcient in practical applications (Leite and Martins 2011; Polberg and Hunter 2018; Amgoud, Doder, and Vesic 2022), as an argument may have quantitative properties. In recent years, the gradual semantics has emerged as a prominent approach for evaluating arguments (Amgoud et al. 2017; Amgoud and Doder 2018; Amgoud and David 2021; Amgoud, Doder, and Vesic 2022; Baroni, Rago, and Toni 2018; Oren et al. 2022; Prakken 2021). This method provides a ﬁne-grained evaluation scheme which introduces

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the notion of acceptability degree to describe argument strength. It generally assigns each argument a numerical value as its acceptability degree that satisﬁes a set of desirable properties. In this paper, we aim to enhance gradual semantics by non-reciprocally incorporating the notion of rejectability degree. Such a setting offers a bilateral perspective on argument strength, in which positive and negative strength are described separately and non-interchangeably. Bilateral evaluative processes that are non-reciprocal extensively exist in societal phenomena. Evidence in Cognitive Science suggests that in such situations, strength of positivity and negativity should be conceptualized and measured separately (Cacioppo and Berntson 1994; Cacioppo, Gardner, and Berntson 1997). In the literature, gradual semantics assigns each argument an acceptability degree by aggregating a basic weight together with the acceptability degrees of its attackers. Our bilateral gradual semantics non-reciprocally introduces rejectability degree in the sense that on the one hand it additionally inﬂuences the acceptability degree of an argument, on the other hand it is solely determined by the acceptability degree of attackers. This is illustrated in Table 1.

Degree Source of Stength

basic weight acceptability acceptability degree of attackers rejectability degree of attackers

rejectability acceptability degree of attackers

Table 1: Non-reciprocity of Bilateral Gradual Semantics

Simply speaking, we also assume acceptability degree and rejectablility degree are generally antagonistic, but not simply polar opposites. To see why non-reciprocal effects in evaluating arguments do exist, one may imagine scenarios in which safer arguments (i.e., with minor criticisms/lower rejectability degree) are preferred, even if other arguments have higher acceptability degree. For instance, a policy may be abandoned after polling if a threshold for the number of people who are against this policy is reached. Consider the following policy polling example:

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Scenario 1 (S1): (a) Increasing corporate taxes to ﬁnance new infrastructure. (x) Increasing corporate taxes slows economic growth. Scenario 2 (S2): (b) Increasing individual taxes to ﬁnance new infrastructure. (y) Increasing individual taxes imposes a heavy burden. In S1, argument x attacks argument a and in S2, argument y attacks argument b. Suppose in S1 there are 0.2M votes for x, 0.6M votes for a, the rest votes are neutral . In S2, 0.6M votes for y, 0.8M votes for b, the rest are neutral . The basic weights w(x), w(a), w(y) and w(b) are simpliﬁed to 0.2, 0.6, 0.6 and 0.8, respectively.

x : 0.2 a : 0.6

y : 0.6 b : 0.8

Figure 1: Policy Polling Scenario

Now we calculate argument strength under bilateral semantics. As x, y have no attackers, acc(x) = w(x) = 0.2, rej(x) = 0, acc(y) = w(y) = 0.6, rej(y) = 0. By our tailored h-categorizer based equations we have:

acc(a) = w(a) 1+ acc(x) 1+rej(x) = 0.6 1+ 0.2

rej(a) = acc(x) 1+acc(x) = 0.2 1+0.2 = 1

acc(b) = w(b) 1+ acc(y) 1+rej(y) = 0.8 1+ 0.6

rej(b) = acc(y) 1+acc(y) = 0.6 1+0.6 = 3

Note that a and b are indistinguishable in terms of acceptability degree ( 1

2). But b has a rejectability degree ( 3

8) higher than a ( 1

6), for b suffers a stronger attack from y compared to that suffered by a from x. As mentioned above, a policymaker may abandon b and choose a for the latter receives less attacks. It is worth mentioning that the gradual semantics in (Amgoud et al. 2017) also provides the same acceptability degree ( 1

2) for a and b, leaving them indistinguishable as well. Our bilateral semantics also contributes to the notation of defense, which plays a central role in classical semantics (Dung 1995; Baroni, Caminada, and Giacomin 2011) however remains not yet discussed in the context of gradual semantics. In classic semantics, an argument is accepted only if it is defended, i.e., all of its attackers are rejected. That is to say, a rejected attacker has no impact on weakening its target arguments. In this paper, the bilateral semantics naturally leads to a gradual version of defense : an attacker with a higher rejectability degree has less impact on weakening the acceptability of its target arguments. The technical contributions of the paper are as follows. Firstly, we introduce bilateral gradual semantics to measure argument strength through the acceptability and rejectability degrees in weighted argumentation graph (WAG).

Then we follow the well-studied principle-based approach (Amgoud, Doder, and Vesic 2022; van der Torre and Vesic 2017) to: (i) Present a set of desirable properties for semantics concerning acceptability and rejectability degrees, which generally serves as the guidelines for establishing semantics. (ii) Investigate the compatibility and links between the above properties. (iii) Deﬁne iterative sequences according to the desirable properties and prove their convergence for any WAG. (iv) Propose AR-max-based semantics, AR-card-based semantics and AR-hybrid-based semantics, which are deﬁned as the limits of the corresponding iterative sequences. (v) Show that the three semantics satisﬁed most of desirable principles. Finally, the paper ends with discussions and conclusions.

Preliminaries

A weighted argumentation graph (WAG) (Amgoud et al. 2017) consists of a set of arguments and a set of associated attacks. Each argument is assigned a basic weight from the real interval [0, 1].

Deﬁnition 1 (WAG) A weighted argumentation graph (WAG) is a triple G = A, w, R , where A is a non-empty ﬁnite set of arguments, w is a function from A to [0, 1], R A A.

Notations: Let G = A, w, R be a WAG and a, b A. w(a) denotes the basic weight of argument a. (b, a) R means that b attacks a. Att G(a) denotes the set of all attackers of a, i.e., Att G(a) = {b A | (b, a) R}. We say that a is non-attacked if Att G(a) = . We abbreviate Att G(a) as Att(a) when the context is clear. For G = A, w, R and G = A , w , R s.t. A A = , G G = A A , w , R R where for any a A (resp. a A ), w (a) = w(a) (resp. w (a) = w (a)). In the paper, arguments are evaluated through bilateral gradual semantics which assigns both acceptability and rejectability degrees to each argument. We use the real interval [0, 1] as the scale for the acceptability degree and [0, 1) as the scale for the rejectability degree. Dropping the maximal value from [0, 1] will simplify the discussion in establishing principles for the rejectability degree.

Deﬁnition 2 (Bilateral gradual semantics) A bilateral gradual semantics is a function S transforming any WAG G = A, w, R to a function Deg S G deﬁned from A to [0, 1] [0, 1). For any a A, Deg S G(a) = (σ+ G(a), σ G(a)) where σ+ G(a) and σ G(a) represent the acceptability and rejectability degree of a respectively.

When the context is clear, we simply write Deg (resp. σ+, σ ) instead of Deg S G (resp. σ+ G, σ G). For convenience, we will direct our attention to the individual functions σ+ and σ . The notions of Isomorphism and Path will be used to establish principles.

Deﬁnition 3 (Isomorphism) Let G = A, w, R and G = A , w , R be two WAG. An isomorphism from G to G is a bijective function f from A to A s.t. (i) a A, w(a) = w (f(a)), (ii) a, b A, (a, b) R iff (f(a), f(b)) R .

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Deﬁnition 4 (Path) We say that there is a path from a1 to an iff there is a sequence consisting of {a1, ..., an} s.t. (ai, ai+1) R for any i {1, ..., n 1}.

Principles for Semantics The desirable properties, known as principles, represent the conditions that a semantics usually needs to satisfy in practical applications. The principles help to understand the foundations of semantics, choose appropriate semantics for practical applications, and construct new semantics. Numerous studies have investigated the principles of gradual semantics that consider acceptability degree (Amgoud et al. 2017; Baroni, Rago, and Toni 2019). In this section, we propose novel principles that simultaneously take into account the rejectability degree. Simply speaking, the acceptability degree of an argument is determined by its basic weight as well as the acceptability and rejectability degrees of its attackers. On the other hand, the rejectability degree of an argument is solely determined by the acceptability degree of its attackers. It turns out that such a non-reciprocal setting lays down a solid base for our principles and semantics. Furthermore, the setting is also coherent with the labeling semantics (Caminada and Gabbay 2009; Wang and Shen 2023). We consider nearly two dozen principles. Some basic ones are modiﬁed or adopted from (Amgoud et al. 2017) (e.g., Anonymity, Resilience, Proportionality). In particular, we propose principles from a bilateral view to characterize rejectability degree, including: R-Neutrality, R-Minimality, R-Strengthening, R-Strengthening Soundness, R-Counting, R-Reinforcement. Principles Weakened Defense and Strict Weakened Defense are introduced to generalize the notion of defense for gradual semantics in a natural and intuitive way. Anonymity states that the acceptability and rejectability degrees of an argument are independent of its identity.

Principle 1 (Anonymity) A semantics S satisﬁes Anonymity iff, for any two WAG G = A, w, R and G = A , w , R , for any isomorphism f from G to G , it holds that: a A, σ+ G(a) = σ+ G (f(a)), σ G(a) = σ G (f(a)).

Independence states that the acceptability and rejectability degrees of an argument should be independent of any argument that is not connected to it.

Principle 2 (Independence) A semantics S satisﬁes Independence iff, for any two WAG G = A, w, R and G = A , w , R s.t. A A = , it holds that: a A, σ+ G(a) = σ+ G G(a), σ G(a) = σ G G(a).

Directionality states that the acceptability and rejectability degrees of an argument a can depend on argument b only if there is a path from b to a.

Principle 3 (Directionality) A semantics S satisﬁes Directionality iff, for any two WAG G = A, w, R and G = A, w, R s.t. R = R {(a, b)}, x A, if there is no path from b to x, then σ+ G(x) = σ+ G (x), σ G(x) = σ G (x).

Equivalence states that: (i) the acceptability degree of an argument depends on only the acceptability and rejectability

degrees of its attackers, as well as its basic weight; (ii) the rejectability degree depends on only the acceptability degree of its attackers.

Principle 4 (Equivalence) A semantics S satisﬁes Equivalence iff, for any WAG G = A, w, R , a, b A,

if (i) w(a) = w(b), (ii) there exists a bijective function f from Att(a) to Att(b) s.t. x Att(a), σ+(x) = σ+(f(x)) and σ (x) = σ (f(x)), then σ+(a) = σ+(b); if there exists a bijective function f from Att(a) to Att(b) s.t. x Att(a), σ+(x) = σ+(f(x)), then σ (a) = σ (b).

Resilience states that if the basic weight of an argument is greater than 0, then its acceptability degree cannot be reduced to 0.

Principle 5 (Resilience) A semantics S satisﬁes Resilience iff, for any WAG G = A, w, R , a A, if w(a) > 0, then σ+(a) > 0.

Proportionality states that the higher the basic weight of an argument, the higher its acceptability degree.

Principle 6 (Proportionality) A semantics S satisﬁes Proportionality iff, for any WAG G = A, w, R , a, b A, if (i) w(a) > w(b), (ii) σ+(a) > 0, and (iii) Att(a) = Att(b), then σ+(a) > σ+(b).

The following principles consider the impact of attackers on the acceptability and rejectability degrees of the arguments under attack. We say an argument a is worthless if σ+(a) = 0 and alive if σ+(a) > 0. A-Neutrality (resp. R-Neutrality) states that a worthless attacker has no impact on the acceptability (resp. rejectability) degree of the arguments it attacks.

Principle 7 (A-Neutrality) A semantics S satisﬁes ANeutrality iff, for any WAG G = A, w, R , a, b A, if (i) w(a) = w(b), (ii) Att(a) = Att(b) \ {x} with x Att(b), and (iii) σ+(x) = 0, then σ+(a) = σ+(b).

Principle 8 (R-Neutrality) A semantics S satisﬁes RNeutrality iff, for any WAG G = A, w, R , a, b A, if (i) Att(a) = Att(b)\{x} with x Att(b), and (ii) σ+(x) = 0, then σ (a) = σ (b).

A-Maximality states that the acceptability degree of a non-attacked argument is equal to its basic weight, and R-Minimality states that the rejectability degree of a nonattacked argument is equal to 0.

Principle 9 (A-Maximality) A semantics S satisﬁes AMaximality iff, for any WAG G = A, w, R , a A, if Att(a) = , then σ+(a) = w(a).

Principle 10 (R-Minimality) A semantics S satisﬁes RMinimality iff, for any WAG G = A, w, R , a A, if Att(a) = , then σ (a) = 0.

A-Weakening states that an alive attacker weakens the acceptability degree of an argument to be less than its basic weight, and R-Strengthening states that an alive attacker strengthens the rejectability degree to be greater than 0.

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Principle 11 (A-Weakening) A semantics S satisﬁes AWeakening iff, for any WAG G = A, w, R , a A, if w(a) > 0 and b Att(a) s.t. σ+(b) > 0, then σ+(a) < w(a). Principle 12 (R-Strengthening) A semantics S satisﬁes RStrengthening iff, for any WAG G = A, w, R , a A, if b Att(a) s.t. σ+(b) > 0, then σ (a) > 0. A-Weakening Soundness states that alive attacks are the only source of losing acceptability degree, and RStrengthening Soundness states that alive attacks are the only source of obtaining rejectability degree. Principle 13 (A-Weakening Soundness) A semantics S satisﬁes A-Weakening Soundness iff, for any WAG G = A, w, R , a A s.t. w(a) > 0, if σ+(a) < w(a), then b Att(a) s.t. σ+(b) > 0. Principle 14 (R-Strengthening Soundness) A semantics S satisﬁes R-Weakening Soundness iff, for any WAG G = A, w, R , a A, if σ (a) > 0, then b Att(a) s.t. σ+(b) > 0. A-Counting states that adding an alive attacker leads to a decrease in the acceptability degree of the arguments it attacks, and R-Counting states that adding an alive attacker leads to an increase in the rejectability degree of the arguments it attacks. Principle 15 (A-Counting) A semantics S satisﬁes ACounting iff, for any WAG G = A, w, R , a, b A, if (i) w(a) = w(b), (ii) σ+(a) > 0, (iii) Att(a) = Att(b) \ {x} with x Att(b), and (iv) σ+(x) > 0, then σ+(a) > σ+(b). Principle 16 (R-Counting) A semantics S satisﬁes RCounting iff, for any WAG G = A, w, R , a, b A, if (i) Att(a) = Att(b)\{x} with x Att(b), and (ii) σ+(x) > 0, then σ (a) < σ (b). A-Reinforcement states that increasing an attacker s acceptability degree leads to a decrease in the acceptability degree of the arguments it attacks, and R-Reinforcement states that increasing an attacker s acceptability degree leads to an increase in the rejectability degree of the arguments it attacks. Principle 17 (A-Reinforcement) A semantics S satisﬁes A-Reinforcement iff, for any WAG G = A, w, R , a, b A, if (i) w(a) = w(b), (ii) σ+(a) > 0, (iii) Att(a) \ {x} = Att(b) \ {y} with x Att(a) and y Att(b), (iv) σ+(x) < σ+(y) and σ (x) = σ (y), then σ+(a) > σ+(b). Principle 18 (R-Reinforcement) A semantics S satisﬁes R-Reinforcement iff, for any WAG G = A, w, R , a, b A, if (i) Att(a) \ {x} = Att(b) \ {y} with x Att(a) and y Att(b), (ii) σ+(x) < σ+(y), then σ (a) < σ (b). Principles Weakened Defense and Strict Weakened Defense are provided to reﬁne the classical concept of defense for bilateral gradual semantics. The Weakened Defense principle speciﬁcally applies to situations where arguments have only one attacker, stating that increasing the rejectability of the attacker leads to an increase in the acceptability of its target arguments. Its strict version extends to a general case, stating that increasing the rejectability of any attacker leads to an increase in the acceptability of its target arguments.

Principle 19 (Weakened Defense) A semantics S satisﬁes Weakened Defense iff, for any WAG G = A, w, R , a, b A, if (i) w(a) = w(b), (ii) σ+(a) > 0, (iii) Att(a) = {x} and Att(b) = {y}, (iv) σ (x) > σ (y) and σ+(x) = σ+(y), then σ+(a) > σ+(b). Principle 20 (Strict Weakened Defense) A semantics S satisﬁes Strict Weakened Defense iff, for any WAG G = A, w, R , a, b A, if (i) w(a) = w(b), (ii) σ+(a) > 0, (iii) Att(a) \ {x} = Att(b) \ {y} with x Att(a) and y Att(b), (iv) σ (x) > σ (y) and σ+(x) = σ+(y), then σ+(a) > σ+(b). The last three principles offer three strategies when confronted with a question (Fig. 2): which aspect is more signiﬁcant the quality of attackers or the quantity of attackers?

Figure 2: Let all the arguments be assigned the basic weight of 1. The argument x has a strong attacker, whereas the argument a has two weak attackers (each one is attacked). Which one is more acceptable and which one is more rejectable?

Quality Precedence (QP) prioritizes the quality of attackers, which states that (i) the greater acceptability and the lower rejectability of the strongest attacker of an argument, the lower its acceptability, and (ii) the greater acceptability of the strongest attacker of an argument, the greater its rejectability. Principle 21 (Quality Precedence) A semantics S satisﬁes Quality Precedence iff, for any WAG G = A, w, R , a, b A, if (i) w(a) = w(b), (ii) σ+(a) > 0, and (iii) y Att(b) s.t. x Att(a), σ+(x) < σ+(y) and σ (x) > σ (y), then σ+(a) > σ+(b); if y Att(b) s.t. x Att(a), σ+(x) < σ+(y), then σ (a) < σ (b). Cardinality Precedence (CP) gives more importance to the quantity of attackers. It states that the greater the number of alive attackers of an argument, the lower its acceptability and the greater its rejectability. Principle 22 (Cardinality Precedence) A semantics S satisﬁes Cardinality Precedence iff, for any WAG G = A, w, R , a, b A, if (i) w(a) = w(b), (ii) σ+(a) > 0, and (iii) |{x Att(a)|σ+(x) > 0}| < |{y Att(b)|σ+(y) > 0}|, then σ+(a) > σ+(b); if |{x Att(a)|σ+(x) > 0}| < |{y Att(b)|σ+(y) > 0}|, then σ (a) < σ (b). Compensation concerns both the quality and quantity of attackers. Principle 23 (Compensation) A semantics S satisﬁes Compensation iff it violates both Quality Precedence and Cardinality Precedence.

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Links Between Principles In this section we brieﬂy present some links between the principles. Some principles are incompatible, i.e., they cannot be satisﬁed at the same time under a given semantics. Proposition 1 The following properties hold: 1. A-Maximality, QP, CP are incompatible. 2. Compensation is not compatible with QP or CP. 3. Independence, Directionality, Equivalence, Resilience, A-Maximality, R-Minimality, A-Reinforcement, R-Reinforcement, A-Weakening, R-Strengthening, ACounting, R-Counting, Weakened Defense and QP are incompatible. 4. CP (resp. Compensation) is compatible with principles 1-20. Implications between principles are presented as below. Proposition 2 Let S be a semantics which satisﬁes Directionality, and Independence. Then: 1. If S satisﬁes A-Maximality, A-Neutrality, then it also satisﬁes A-Weakening Soundness. 2. If S satisﬁes R-Minimality, R-Neutrality, then it also satisﬁes R-Strengthening Soundness. 3. If S satisﬁes Equivalence, A-Maximality, A-Neutrality, AReinforcement, then it also satisﬁes A-Counting and AWeakening. 4. If S satisﬁes Equivalence, A-Maximality, R-Neutrality, RReinforcement, then it also satisﬁes R-Counting and RStrengthening. A number of properties concerning the acceptability degree can be found in (Amgoud et al. 2017). Here we present properties regarding the rejectability degree derived from our principles. To begin with, a set of arguments that are not attacked by any other arguments keeps their acceptability and rejectability degrees unchanged in any graph whenever the semantics satisﬁes Independence and Directionality. Proposition 3 If a semantics S satisﬁes Independence and Directionality, then for any two WAG G = A, w, R and G = A , w , R , if A A , for any a A, w(a) = w (a), and R (A A) = R, then for any a A, σ+ G(a) = σ+ G (a) and σ G(a) = σ G (a). Example 1 Consider two WAG depicted as follows and let all the arguments be assigned the basic weight of 1.

Let S be a semantics which satisﬁes Independence and Directionality. Since x {a, b, c} is not attacked by {d, e}, we have σ+ G1(x) = σ+ G2(x) and σ G1(x) = σ G2(x). If an argument is only attacked by worthless arguments, then its rejectability degree is 0, whenever the semantics satisﬁes Independence, Directionality, Equivalence, R-Minimality, and R-Neutrality.

Proposition 4 If a semantics S satisﬁes Independence, Directionality, Equivalence, R-Minimality, and R-Neutrality, then for any WAG G = A, w, R , a A, if for any x Att(a), σ+(x) = 0, then σ (a) = 0.

Worthless attackers have no effect on the rejectability degree of their target arguments whenever the semantics satisﬁes Independence, Directionality, Equivalence, and RNeutrality.

Proposition 5 If a semantics S satisﬁes Independence, Directionality, Equivalence, and R-Neutrality, then for any WAG G = A, w, R , a, b A, if Att(a) Att(b) and for any x Att(b) \ Att(a), σ+(x) = 0, then σ (a) = σ (b).

Semantics and their Properties In this section, we propose bilateral gradual semantics which correspond to three strategies: QP, CP, and Compensation. As pointed out in (Gabbay and Rodrigues 2015), the best way of deﬁning gradual semantics is to introduce iterative functions that take WAG as inputs and produce sequences of values that eventually converge. We construct our semantics based on the well-studied h-categorizer function family (Besnard and Hunter 2001; Pu et al. 2014; Amgoud et al. 2017; Amgoud, Doder, and Vesic 2022). The resulting functions produce iterative sequences that always converge for any WAG, not just restricted to acyclic ones.

AR-Max-Based Semantics The AR-max-based semantics (ARM) satisﬁes Quality Precedence, prioritizing the quality of attackers over their quantity.

Deﬁnition 5 Let G = A, w, R be a WAG and an argument a A. An iterative sequence {F i(a)}i N is deﬁned as F i(a) = (f i(a), gi(a)) where f i : A [0, 1] and gi : A [0, 1) such that

w(a), i = 0, w(a)

1+ max b Att(a)

fi 1(b) 1+gi 1(b) , i 1.

0, i = 0, max b Att(a) f i 1(b)

1+ max b Att(a) f i 1(b), i 1.

By convention, max b Att(a)

f i 1(b) 1+gi 1(b) = 0 and

max b Att(a) f i 1(b) = 0 if Att(a) = .

Theorem 1 The sequence {F i(a)}i N converges for any a A as i approaches inﬁnity.

Now we deﬁne the ARM semantics according to the limit of the above iterative sequence.

Deﬁnition 6 (ARM semantics) The AR-max-based semantics is a function ARM transforming any WAG G = A, w, R into a function Deg ARM G deﬁned from A to [0, 1] [0, 1) s.t. a A, Deg ARM G (a) = (σ+(a), σ (a)) with σ+(a) = lim i f i(a), σ (a) = lim i gi(a).

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Example 2 Applying the ARM semantics to the WAG depicted in Figure 2, we have σ+(a) = 3

4, σ (a) = 1

3, σ+(x) = 1 2, σ (z) = 1

2. So σ+(a) > σ+(x) and σ (a) < σ (x). It illustrates that the ARM semantics prioritizes the quality of attackers over their quantity. The acceptability and rejectability degrees assigned to the arguments by the ARM semantics satisfy the equations deﬁned in Deﬁnition 5, stated as below. Theorem 2 Let G = A, w, R be a WAG. Then for any a A, Deg ARM G (a) = (σ+(a), σ (a)) where

σ+(a) = w(a)

1 + max b Att(a)

σ+(b) 1+σ (b) ,

σ (a) = max b Att(a) σ+(b)

1 + max b Att(a) σ+(b).

Since the ARM semantics focuses on the strongest attacker, it violates Strict Weakened Defense, A-Counting, RCounting, A-Reinforcement, and R-Reinforcement. The theorem below says that the ARM semantics satisﬁes all the rest principles that are compatible with QP. Theorem 3 The ARM semantics violates Strict Weakened Defense, A-Counting, R-Counting, A-Reinforcement, RReinforcement, CP and Compensation. It satisﬁes all the remaining principles.

AR-Card-Based Semantics The AR-card-based semantics (ARC) satisﬁes Cardinality Precedence, giving more importance to the quantity of attackers. Moreover, the semantics considers only founded attackers, i.e., attackers with basic weight greater than 0. This is because unfounded arguments are worthless and their attacks are ineffective. Deﬁnition 7 Let G = A, w, R be a WAG and a A. The argument a is founded iff w(a) > 0. It is unfounded otherwise. Let Att G(a) (simpliﬁed as Att (a)) denote the set of the founded attackers of a. Now we introduce the following iterative sequences for AR-card-based semantics. Deﬁnition 8 Let G = A, w, R be a WAG and an argument a A. An iterative sequence {F i(a)}i N is deﬁned as F i(a) = (f i(a), gi(a)) where f i : A [0, 1] and gi : A [0, 1) such that

w(a), i = 0, w(a)

1+|Att (a)|+ 1

fi 1(b) 1+gi 1(b) , i 1.

0, i = 0, |Att (a)|+ 1

b Att (a) f i 1(b)

1+|Att (a)|+ 1

b Att (a) f i 1(b), i 1.

with n = |A|. By convention, P

f i 1(b) 1+gi 1(b) = 0 and P

b Att (a) f i 1(b) = 0 if Att (a) = .

Theorem 4 The sequence {F i(a)}i N converges for any a A as i approaches inﬁnity.

The ARC semantics is deﬁned according to the limit of the above iterative sequence.

Deﬁnition 9 (ARC semantics) The AR-card-based semantics is a function ARC transforming any WAG G = A, w, R into a function Deg ARC G deﬁned from A to [0, 1] [0, 1) s.t. a A, Deg ARC G (a) = (σ+(a), σ (a)) with σ+(a) = lim i f i(a), σ (a) = lim i gi(a).

Example 3 Applying the ARC semantics to the WAG depicted in Figure 2, we have σ+(a) = 5 16, σ (a) = 7 10, σ+(x) = 1

3, σ (x) = 2

3. So σ+(a) < σ+(x) and σ (a) > σ (x). It illustrates that the ARC semantics prioritizes the quantity of attackers over their quality.

The acceptability and rejectability degrees assigned to the arguments by the ARC semantics satisfy the equations deﬁned in Deﬁnition 8, stated as below.

Theorem 5 Let G = A, w, R be a WAG and n = |A|. Then for any a A, Deg ARC G (a) = (σ+(a), σ (a)) where

σ+(a) = w(a)

1 + |Att (a)| + 1

σ+(b) 1+σ (b)

|Att (a)| + 1

b Att (a) σ+(b)

1 + |Att (a)| + 1

b Att (a) σ+(b).

The following theorem says that the ARC semantics satisﬁes all the principles that are compatible with CP.

Theorem 6 The ARC semantics satisﬁes all the principles except QP and Compensation.

AR-Hybrid-Based Semantics

The AR-hybrid-based semantics (ARH) satisﬁes Compensation, taking both the quantity and quality of attackers into account.

Deﬁnition 10 Let G = A, w, R be a WAG and an argument a A. An iterative sequence {F i(a)}i N is deﬁned as F i(a) = (f i(a), gi(a)) where f i : A [0, 1] and gi : A [0, 1) such that

w(a), i = 0, w(a)

1+|Att (a)|+ P

fi 1(b) 1+gi 1(b) , i 1.

0, i = 0, |Att (a)|+ P

b Att (a) f i 1(b)

1+|Att (a)|+ P

b Att (a) f i 1(b), i 1.

Theorem 7 The sequence {F i(a)}i N converges for any a A as i approaches inﬁnity.

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Deﬁnition 11 (ARH semantics) The AR-hybrid-based semantics is a function ARH transforming any WAG G = A, w, R into a function Deg ARH G deﬁned from A to [0, 1] [0, 1) s.t. a A, Deg ARH G (a) = (σ+(a), σ (a)) with σ+(a) = lim i f i(a), σ (a) = lim i gi(a).

Figure 3: ARH Semantics Illustration

Example 4 Applying the ARH semantics to the WAG depicted in Figure 3, we have (i) x Att(z) s.t. for any b Att(a), σ+(x) > σ+(a), (ii) |{x Att(a)|σ+(x) > 0}| < |{y Att(b)|σ+(y) > 0}|. But σ (a) = σ (z) = 4 5. Therefore, the ARH semantics satisﬁes Compensation.

The acceptability and rejectability degrees assigned to the arguments by the ARH semantics satisfy the equations deﬁned in Deﬁnition 10, stated as below.

Theorem 8 Let G = A, w, R be a WAG. Then for any a A, Deg ARH G (a) = (σ+(a), σ (a)) where

σ+(a) = w(a)

1 + |Att (a)| + P

σ+(b) 1+σ (b)

|Att (a)| + P

b Att (a) σ+(b)

1 + |Att (a)| + P

b Att (a) σ+(b).

Theorem 9 says that the ARH semantics satisﬁes all the principles that are compatible with Compensation. Theorem 9 The ARH semantics satisﬁes all the principles except QP and CP. Comparisons for principles under ARM, ARC and ARH are summaried in Table 2.

Discussion and Conclusion The gradual semantics has received considerable attention in the literature and a plethora of semantics have been proposed, e.g., trust-based semantics (da Costa Pereira, Tettamanzi, and Villata 2011), iterative-based semantics (Gabbay and Rodrigues 2015), weighted max-based, cardbased and h-categorizer semantics (Amgoud et al. 2017; Amgoud, Doder, and Vesic 2022), Qu AD semantics (Baroni et al. 2015), DF-Qu AD semantics (Rago et al. 2016). However, to the best of our knowledge, none of the existing gradual semantics considers the rejectability degree for argument evaluation. In the probabilistic approach (Hunter, Polberg, and Thimm 2020; Polberg and Hunter 2018), the

ARM ARC ARH Anonymity Independence Directionality Equivalence Resilience Proportionality A-Neutrality R-Neutrality A-Maximality R-Minimality A-Weakening R-Strengthening A-Weakening soundness R-Strengthening soundness A-Counting R-Counting A-Reinforcement R-Reinforcement Weakened Defense Strict Weakened Defense Quality Precedence Cardinality Precedence Compensation

Table 2: Principles under ARM, ARC and ARH semanatics

authors proposed that the acceptability of an argument can be expressed by the degree of believed and disbelieved. Nonetheless, the foundational approach to establishing semantics diverges considerably from our framework. In this paper, we formalized a comprehensive framework to evaluate arguments through the acceptability and rejectability degrees. This novel approach provides new insight into argument strength and gives us a deeper understanding of the status of arguments. We then proposed a set of desirable properties that take into account both acceptability and rejectability degrees. Furthermore, we provided three novel semantics that satisfy most of the desirable properties: ARmax-based semantics, AR-card-based semantics and ARhybrid-based semantics. We believe bilateral gradual semantics can be applied to many areas including argumentative decision-making (Amgoud and Prade 2009), argumentative explainable AI ( ˇCyras et al. 2021), etc., as considering both the positive and negative strength is a common approach in practice. In future work, we shall introduce bilateral gradual semantics to other argumentation systems, such as weighted bipolar argumentation graphs (Amgoud and Ben-Naim 2018) and SETAF (Nielsen and Parsons 2006).

Acknowledgements

This work was partially supported by the National Key Research Institutes Project (19JJD720002) and the National Social Science Fund of China (20BZX106).

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