# On the Sensitivity of Extension Semantics to Similarity

Leila Amgoud

► To cite this version:

Leila Amgoud. On the Sensitivity of Extension Semantics to Similarity. 19th European Conference on Logics in Artificial Intelligence, JELIA 2025, Sep 2025, Kutaisi, Georgia. pp.63-77, 10.1007/978-3-032-04587-4_5. hal-05346286

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# On the Sensitivity of Extension Semantics to Similarity

Leila Amgoud

CNRS - IRIT, Toulouse, France

amgoud@irit.fr

November 25, 2025

## Abstract

This paper presents the first systematic study of how similarity between arguments affects their evaluation under extension semantics. We define a set of principles that govern how similarity should be taken into account by a semantics, and introduce a broad family of parameterized extension semantics that adhere to these principles. Each member of this family generalizes a classical extension semantics (such as preferred semantics) while disregarding redundant attacks. We identify classes of argumentation graphs in which similarity has no effect on the acceptability status of arguments, and we characterize the conditions under which a family instance either improves an argument's acceptability or coincides with the semantics it generalizes. Finally, we discuss different types of selection functions, which specify the subset of attacks to disregard.

## 1 Introduction

An argumentation framework consists of a graph and a semantics. In the graph, nodes represent arguments, reflecting justifications for claims, while edges represent attacks, modeling conflicts between arguments. The semantics provides a formal method for evaluating the acceptability of each argument within the graph. Numerous semantics have been proposed (see [7] for a survey). The earliest are the extension-based semantics introduced by Dung [13], which identify sets of arguments that can be jointly accepted. More recently, gradual semantics have been introduced [10]; they assign each argument a (numerical or qualitative) value reflecting its degree of strength.

The broad spectrum of theoretical and practical applications of argumentation has brought attention to the concept of *similarity* between arguments, namely *semantical similarity*. The latter concerns the degree to which two arguments are similar in terms of their *content*—their justifications and claims. Consider the following example involving two arguments, $b_1, b_2$, both supporting the claim that *Paul is not a suitable candidate for the lecturer position*.

$b_1$: He has limited teaching experience. $b_2$: He taught only a few hours before.
These arguments are semantically similar as they convey the same information. Existence of similarity introduces new research questions regarding its role in argu-mentation frameworks. We distinguish two main roles for similarity. The first is to enhance the reliability of the information conveyed by similar arguments. In our example, this could lead to merging the two arguments into a single argument with a higher degree of trustworthiness [11]. This approach aligns with practices in belief merging, where claims supported by multiple sources are generally considered more plausible. While this approach is reasonable, its implementation in argumentation frameworks poses challenges. Similar arguments may interact differently with other arguments, so merging them would require a careful redefinition of the graph's structure.

The second role of similarity is to constrain the influence that attacks have on the strength of their targets. Within the context of gradual semantics, previous work [3, 5] has shown that ignoring similarity among attackers can lead to skewed evaluations of arguments. Such inaccuracies can ultimately distort the conclusions provided by argumentation-based systems such as decision systems. These studies have developed methods for integrating similarity into existing gradual semantics. Surprisingly, there is no work in the literature on the integration of similarity into extension semantics. This paper seeks to address this gap through five main contributions:

1. It proposes three principles that extension semantics that incorporate similarity into the evaluation process should satisfy.

2. It defines a broad family of parameterized extension semantics that satisfy the proposed principles. Each member of this family is determined by two parameters: a classical extension semantics $\delta$ (e.g., grounded semantics [13]), and a selection function $\Delta$, which identifies redundant attacks to be removed from the argumentation graph. The evaluation is then performed on the resulting subgraph using the base semantics $\delta$. Each instance of this family is sensitive to similarity, generalizes the underlying semantics $\delta$, and has the potential to improve its evaluations by accounting for redundancy introduced by similar arguments.

3. It shows that in *coherent graphs*—where similar arguments share the same set of attackers—similarity has no impact on argument evaluation. However, in non-coherent graphs, disregarding similarity among attackers can lead to distorted evaluations of their targets.

4. It characterizes the conditions under which a new semantics improves the acceptability status of an argument or coincides with the semantics ($\delta$) it generalizes.

5. It discusses various strategies that can underpin the design of selection functions, offering guidance on how to systematically eliminate redundant attacks.

The paper is organized as follows: Section 2 introduces background, Section 3 defines extended argumentation graphs, and three principles that semantics would satisfy, Section 4 defines the novel family of parameterized semantics and analyses their properties, Section 5 discusses examples of selection functions, Section 6 recalls some related work and the last section is devoted to concluding remarks and perspectives.
<figure><img src="image_2.png" /><figcaption>Table 1: Examples of argumentation graphs $G_i = \langle \mathcal{A}_i, \mathcal{R}_i \rangle$, $i \in \{1, \dots, 4\}$.</figcaption></figure>

## 2 Background

The cornerstone of argumentation theory is the notion of *argument*—a reason supporting a particular standpoint. Arguments are often interconnected through an *attack relation*, which captures the conflicts that arise between pairs of arguments. These interrelations are naturally represented in a graphical form. In this paper, we denote by *Args* the set of all possible arguments, by *a*, *b*, *c*, ... arguments and by *r*, *r*₁, *r*₂, ... attacks.

**Definition 1** An argumentation graph (AG) is a pair $\mathbb{G} = \langle \mathcal{A}, \mathcal{R} \rangle$, where $\mathcal{A} \subseteq \text{Args}$, $\mathcal{A} \neq \emptyset$, and $\mathcal{R} \subseteq \mathcal{A} \times \mathcal{A}$ is an attack relation. Let AG be the set of all possible AGs.

A pair $(b, a) \in \mathcal{R}$ is read “b attacks a”. We say also that a set $\mathcal{E} \subseteq \mathcal{A}$ attacks $a \in \mathcal{A}$ if $\exists b \in \mathcal{E}$ such that $(b, a) \in \mathcal{R}$.

In the seminal paper [13], conflicts between arguments are resolved using an *extension semantics*—a formal method that defines sets of arguments that are jointly *acceptable*, with each such set referred to as an *extension*.

**Definition 2** An extension semantics is a function $\delta$ mapping every $\mathbb{G} = \langle \mathcal{A}, \mathcal{R} \rangle \in \text{AG}$ into $\text{Ext}_\delta(\mathbb{G}) \subseteq \mathbb{P}(\mathcal{A})^1$. Every member of $\text{Ext}_\delta(\mathbb{G})$ is called an extension.

There are several criteria for defining extensions, thus acceptable sets, including those introduced by Dung [13] for defining his main semantics: complete (*co*), grounded (*gr*), preferred (*pr*), and stable (*st*). They are based on *conflict-freeness* and *defence*, where for $\mathcal{E} \subseteq \mathcal{A}$,

* $\mathcal{E}$ is conflict-free iff $\forall a, b \in \mathcal{E}$ such that $(a, b) \in \mathcal{R}$. Else, $\mathcal{E}$ is said to be $\mathcal{R}$-conflicting.

* $\mathcal{E}$ defends $a \in \mathcal{A}$ iff $\forall b \in \mathcal{A}$ such that $(b, a) \in \mathcal{R}$, $\exists c \in \mathcal{E}$ such that $(c, b) \in \mathcal{R}$.

* $\mathcal{E}$ is admissible iff it is conflict-free and defends all its elements.

Let us now recall the definition of extension under the above-mentioned semantics. Let $\mathcal{E} \subseteq \mathcal{A}$ be conflict-free.

<footnote n="1">¹$\mathbb{P}(\mathcal{A})$ is the powerset, i.e., the set of all partitions of $\mathcal{A}$.</footnote>
• $\mathcal{E}$ is a complete extension iff it defends its elements and contains all the arguments it defends.

• $\mathcal{E}$ is a *grounded* extension iff it is the subset-minimal complete extension.

• $\mathcal{E}$ is a *preferred* extension iff it is a subset-maximal complete extension.

• $\mathcal{E}$ is a *stable* extension iff for any $a \in A \setminus \mathcal{E}$, $\mathcal{E}$ attacks $a$.

Since extension semantics may yield multiple extensions for a given argumentation graph, an *acceptability status* is assigned to each individual argument by aggregating across all extensions. In the literature (e.g., [10]), a four-valued scale {s, c, u, r} is commonly used for this purpose. This scale is equipped with a total ordering $\succeq$, where $s \succ c \succ u \succ r$, and $x \succeq y$ (respectively $x \succ y$) is interpreted as “*x* is at least as acceptable as *y*” (respectively “*x* is strictly more acceptable than *y*”).

**Definition 3** Let $\delta$ be an extension semantics, $G = (\mathcal{A}, \mathcal{R}) \in \mathbf{AG}$ and $a \in \mathcal{A}$. If $\mathrm{Ext}_{\delta}(G) = \emptyset$, then $\mathrm{Acc}_{\delta}(a, G) = \mathbf{u}$. Else:

• $\mathrm{Acc}_{\delta}(a, G) = \mathbf{s}$ iff $a \in \bigcap_{\mathcal{E} \in \mathrm{Ext}_{\delta}(G)} \mathcal{E}$. $a$ is said to be sceptically accepted in $G$.

• $\mathrm{Acc}_{\delta}(a, G) = \mathbf{c}$ iff $\exists \mathcal{E}, \mathcal{E}' \in \mathrm{Ext}_{\delta}(G)$ such that $a \in \mathcal{E}$ and $a \notin \mathcal{E}'$. $a$ is said to be credulously accepted in $G$.

• $\mathrm{Acc}_{\delta}(a, G) = \mathbf{r}$ iff $a \notin \bigcup_{\mathcal{E} \in \mathrm{Ext}_{\delta}(G)} \mathcal{E}$ and $\exists \mathcal{E} \in \mathrm{Ext}_{\delta}(G)$ such that $\mathcal{E}$ attacks $a$. $a$ is said to be rejected in $G$.

• $\mathrm{Acc}_{\delta}(a, G) = \mathbf{u}$ iff $a \notin \bigcup_{\mathcal{E} \in \mathrm{Ext}_{\delta}(G)} \mathcal{E}$ and $\not\exists \mathcal{E} \in \mathrm{Ext}_{\delta}(G)$ such that $\mathcal{E}$ attacks $a$. $a$ is said to be undecided in $G$.

$\mathrm{Acc}_{\delta}(a, G)$ denotes the acceptability status of the argument $a$ in the graph $G$ under the semantics $\delta$.

**Example 1** Let us analyse the four graphs from Table 1 under preferred semantics.

• $\mathrm{Ext}_{\mathrm{pr}}(G_1) = \{\{b_1, b_2\}\}$.

- $\mathrm{Acc}_{\mathrm{pr}}(b_1, G_1) = \mathrm{Acc}_{\mathrm{pr}}(b_2, G_1) = \mathbf{s}$,

- $\mathrm{Acc}_{\mathrm{pr}}(a, G_1) = \mathrm{Acc}_{\mathrm{pr}}(c, G_1) = \mathbf{r}$.

• $\mathrm{Ext}_{\mathrm{pr}}(G_2) = \{\{b_1\}, \{b_2\}\}$.

- $\mathrm{Acc}_{\mathrm{pr}}(b_1, G_2) = \mathrm{Acc}_{\mathrm{pr}}(b_2, G_2) = \mathbf{c}$,

- $\mathrm{Acc}_{\mathrm{pr}}(a, G_2) = \mathbf{r}$.

• $\mathrm{Ext}_{\mathrm{pr}}(G_3) = \{\{a, c_2\}\}$.

- $\mathrm{Acc}_{\mathrm{pr}}(a, G_3) = \mathrm{Acc}_{\mathrm{pr}}(c_2, G_3) = \mathbf{s}$,

- $\mathrm{Acc}_{\mathrm{pr}}(b, G_3) = \mathbf{r}$.

- $\mathrm{Acc}_{\mathrm{pr}}(c_1, G_3) = \mathbf{u}$.

• $\mathrm{Ext}_{\mathrm{pr}}(G_4) = \{\{c_1\}, \{a, c_2\}\}$.

- $\mathrm{Acc}_{\mathrm{pr}}(a, G_4) = \mathrm{Acc}_{\mathrm{pr}}(c_1, G_4) = \mathrm{Acc}_{\mathrm{pr}}(c_2, G_4) = \mathbf{c}$,

- $\mathrm{Acc}_{\mathrm{pr}}(b_1, G_4) = \mathrm{Acc}_{\mathrm{pr}}(b_2, G_4) = \mathbf{r}$.

- $\mathrm{Acc}_{\mathrm{pr}}(c_3, G_4) = \mathbf{u}$.
Below are some useful notations for the rest of the paper.

**Notations:** Let $\mathbb{G} = \langle \mathcal{A}, \mathcal{R} \rangle \in \mathbf{AG}$ and $a, b \in \mathcal{A}$. Let $r \in \mathcal{R}$ such that $r = (b, a)$. The functions $\mathbf{Sc}(r)$ and $\mathbf{Tg}(r)$ return the source $b$ and the target $a$ of the attack $r$ respectively. The function $\mathbf{aAtt}(a, \mathbb{G})$ returns the set of arguments attacking $a$ in $\mathbb{G}$. For instance, in Table 1, $\mathbf{aAtt}(a, \mathbb{G}_1) = \{b_1, b_2\}$. The function $\mathbf{rAtt}(a, \mathbb{G})$ returns the set of all attacks targeting $a$ in $\mathbb{G}$. Hence, $\mathbf{rAtt}(a, \mathbb{G}_1) = \{(b_1, a), (b_2, a)\}$. Let $X \subseteq \mathbf{rAtt}(a, \mathbb{G})$. We define $\mathbb{G} \ominus X = \langle \mathcal{A}, \mathcal{R} \setminus X \rangle$ (i.e., the AG where the attacks in $X$ are removed), and $\mathbb{G} \downarrow X = \langle \mathcal{A}, \mathcal{R}' \rangle$ where $\mathbf{rAtt}(a, \mathbb{G} \downarrow X) = X$ and $\forall b \in \mathcal{A} \setminus \{a\}, \mathbf{rAtt}(b, \mathbb{G} \downarrow X) = \mathbf{rAtt}(b, \mathbb{G})$ (i.e., the AG where the set of attacks on $a$ is $X$). Consider the graph $\mathbb{G}_2$ in Table 1 and the argument $a$. Note that $\mathbf{rAtt}(a, \mathbb{G}_2) = \{(b_1, a), (b_2, a)\}$. For $X = \{(b_1, a)\}, \mathbb{G}_2 \ominus X = \langle \{a, b_1, b_2\}, \{(b_1, b_2), (b_2, b_1), (b_2, a)\} \rangle$.

It was shown in [1] that direct attacks on an argument are the primary cause of the argument's loss of strength. The author also identified the sets of attacks *necessary* and those *sufficient* to determine the status of their targets according to a given extension semantics. Specifically, a subset of attacks is considered necessary for an argument if its removal automatically leads to a change in the argument's status.

**Definition 4 (Necessity)** Let $\delta$ be an extension semantics, $\mathbb{G} = \langle \mathcal{A}, \mathcal{R} \rangle \in \mathbf{AG}$, $a \in \mathcal{A}$ and $X \subseteq \mathbf{rAtt}(a, \mathbb{G})$. The set $X$ is necessary for a iff:

* $\mathbf{Acc}_\delta(a, \mathbb{G}) \neq \mathbf{Acc}_\delta(a, \mathbb{G} \ominus X)$,

* $\mathbb{X}' \subseteq X$ such that $X'$ satisfies the above condition.

We denote by $\mathbf{Nec}_\delta(a, \mathbb{G})$ the set of all necessary sets for $a$ in $\mathbb{G}$ under $\delta$.

**Example 1 (Cont).** Let us check the necessary attacks for the argument $a$ in each graph under preferred semantics.

$$ \mathbf{Nec}_{pr}(a, \mathbb{G}_1) = \{(b_1, a), (b_2, a)\} \qquad \mathbf{Nec}_{pr}(a, \mathbb{G}_2) = \{(b_1, a), (b_2, a)\} \\ \mathbf{Nec}_{pr}(a, \mathbb{G}_3) = \{(b, a)\} \qquad \mathbf{Nec}_{pr}(a, \mathbb{G}_4) = \{(b_2, a)\} $$

**Remark:** Note that in the graph $\mathbb{G}_4$, the attack $(b_2, a)$ is necessary for the argument $a$ while it emanates from a rejected argument $(\mathbf{Acc}_{pr}(b_2, \mathbb{G}_4) = r)$. Hence, a necessary attack does not necessarily come from a strong argument.

A set of attacks is sufficient for a given targeted argument if, when taken alone (i.e., removing all other attacks toward the argument), the argument gets the same status.

**Definition 5 (Sufficiency)** Let $\delta$ be an extension semantics, $\mathbb{G} = \langle \mathcal{A}, \mathcal{R} \rangle \in \mathbf{AG}$, $a \in \mathcal{A}$, and $X \subseteq \mathbf{rAtt}(a, \mathbb{G})$. The set $X$ is sufficient for a iff:

* $\mathbf{Acc}_\delta(a, \mathbb{G}) = \mathbf{Acc}_\delta(a, \mathbb{G} \downarrow X)$.

* $\mathbb{X}' \subseteq X$ such that $X'$ satisfies the above condition.

$\mathbf{Suff}_\delta(a, \mathbb{G})$ is the set of all sufficient sets for $a$ in $\mathbb{G}$ under $\delta$.
**Example 1 (Cont).** Let us check the sufficient attacks for the argument *a* in each graph under preferred semantics.

$$ \text{Suff}_{pr}(a, \mathbb{G}_1) = \{\{(b_1, a)\}, \{(b_2, a)\}\} \quad \text{Suff}_{pr}(a, \mathbb{G}_2) = \{\{(b_1, a), (b_2, a)\}\} $$

$$ \text{Suff}_{pr}(a, \mathbb{G}_3) = \{\{(b, a)\}\} \quad \text{Suff}_{pr}(a, \mathbb{G}_4) = \{\{(b_2, a)\}\} $$

## 3 Extended Argumentation Graphs

In addition to an attack relation, we assume the existence of a similarity relation between arguments. This relation captures semantical similarity, that is, how closely two arguments align in terms of their content (reasons and claims). Formally, it is a binary, reflexive, and symmetric relation. It can be derived from a similarity **measure** that quantifies the **degree** of similarity between any pair of arguments (see [4, 12] for examples of measures for logical arguments). However, the raw numerical values produced by these measures are often of limited practical use, as it is unclear which value ranges genuinely indicate meaningful relatedness between arguments. A widely adopted approach in the broader similarity literature (e.g., [17]) involves applying a **threshold**, whereby objects with similarity values exceeding this threshold are deemed similar.

**Definition 6** Let $X \subseteq \text{Args.}$ A similarity relation on $X$ is a binary relation $\text{sim} \subseteq X \times X$ such that:

$$ \bullet \forall a \in X, (a, a) \in \text{sim} \qquad (\text{Reflexivity}) $$

$$ \bullet \forall a, b \in X, (a, b) \in \text{sim} \text{ iff } (b, a) \in \text{sim} \qquad (\text{Symmetry}) $$

$$ \text{We define } \text{sim}^* = \{(a, a) \mid a \in X\}. $$

Note that due to the reflexivity property, $\text{sim}^* \neq \emptyset$ and consequently, $\text{sim} \neq \emptyset$. Let us now introduce extended argumentation graphs.

**Definition 7** An extended argumentation graph (eAG) is a tuple $\mathbb{E} = \langle \mathbb{G}, \text{sim} \rangle$, where $\mathbb{G} = \langle \mathcal{A}, \mathcal{R} \rangle \in \text{AG}$ and $\text{sim}$ is a similarity relation on $\mathcal{A}$. Let eAG be the set of all possible eAGs.

Note that an extended graph $\langle \mathbb{G}, \text{sim}^* \rangle$ coincides with the basic graph $\mathbb{G}$ as all arguments are assumed pairwise dissimilar.

Extension semantics that evaluate arguments in extended argumentation graphs are functions that yield extensions, as specified in Def. 2. These extensions are then aggregated, as described in Def. 3, to derive the acceptability statuses of the arguments.

**Notations:** Let $\mathbb{E} = \langle \mathbb{G}, \text{sim} \rangle \in \text{eAG}$, where $\mathbb{G} = \langle \mathcal{A}, \mathcal{R} \rangle$, $a \in \mathcal{A}$. All notations defined for simple graphs apply to extended graphs. For instance, $\text{rAtt}(a, \mathbb{E})$ is the set of attacks targeting *a* in $\mathbb{E}$, $\text{Ext}_\delta(\mathbb{E})$ is the set of extensions of $\mathbb{E}$ under semantics $\delta$, and $\text{Acc}_\delta(a, \mathbb{E})$ is the acceptability status of *a* in $\mathbb{E}$ under $\delta$. The function $\text{sAtt}(a, \mathbb{E})$ returns
the set of attacks targeting *a* and whose sources are similar, that is, $\mathbf{sAtt}(a, \mathbb{E}) = \{r \in \mathbf{rAtt}(a, \mathbb{E}) \mid \exists r' \in \mathbf{rAtt}(a, \mathbb{E}) \text{ s.t. } (\mathbf{Sc}(r), \mathbf{Sc}(r')) \in \mathbf{sim}\}$.

In the extended setting, extension semantics are expected to satisfy certain key principles. The first is sensitivity to similarity — meaning the semantics should account for similarity whenever such information is available.

**Principle 1 (Sensitivity to similarity)** *An extension semantics $\delta$ is indifferent to similarity iff $\forall \langle G, sim \rangle \in eAG$, $Ext_\delta(\langle G, sim \rangle) = Ext_\delta(\langle G, sim^* \rangle)$; $\delta$ is said to be sensitive to similarity otherwise.*

The second principle addresses the question: *At which level should similarity be considered?* It posits that similarity is relevant at the level of **redundant attacks**—that is, attacks originating from similar arguments and directed at the same target. Since attacks are the primary mechanism by which arguments are weakened, disregarding redundant attacks helps prevent the target argument from being excessively diminished in strength. The principle states that when there are no similar attackers in a graph, similarity does not play a role.

**Principle 2 (Neutrality)** *An extension semantics $\delta$ satisfies neutrality iff $\forall \langle G, sim \rangle \in eAG$ such that $G = \langle A, R \rangle$ and $\forall a \in A$, $\mathbf{sAtt}(a, G) = \emptyset$, $Ext_\delta(\langle G, sim \rangle) = Ext_\delta(\langle G, sim^* \rangle)$.*

**Example 2** *Consider the extended argumentation graph depicted below.*

<figure><img src="image_3.png" /></figure>

$$ \bullet \mathbf{sim} = \{(b, c), (c, b)\} \cup \mathbf{sim}^*. $$

* The principle of Neutrality implies that an extension semantics must disregard the similarity between b and c.

As a consequence of the neutrality principle, if every argument in an extended graph has at most one attacker, similarity has no effect on argument evaluation.

**Property 1** Let $\delta$ be an extension semantics and $\mathbb{E} = \langle G, sim \rangle$, with $G = \langle A, R \rangle$ and $\forall a \in A$, $|\mathbf{rAtt}(a, \mathbb{E})| \le 1$. If $\delta$ satisfies neutrality, then $\forall a \in A$, $Ext_\delta(\langle G, sim \rangle) = Ext_\delta(\langle G, sim^* \rangle)$.

Let us now turn to situations where similarity should be considered — specifically, cases involving redundant attacks. There are two possible ways to address such redundancy: removing similar arguments or removing their attacks. However, the first approach is problematic, as it risks information loss which may result in counter-intuitive evaluations of certain arguments, as the following example illustrates.

**Example 1 (Cont).** Consider the extended argumentation graph $\mathbb{E}_1 = \langle G_1, sim_1 \rangle$, where $sim_1 = \{(b_1, b_2), (b_2, b_1)\} \cup sim_1^*$. Suppose we remove the argument $b_2$ (along with all its outgoing attacks). This yields the subgraph $G_{11} = \{\{a, b_1, c\}, \{(b_1, c)\}\}$, which has a single preferred extension: $\{b_1, c\}$. Notice that in $G_{11}$, the argument c
is skeptically accepted, whereas in the original graph $G_1$, it is rejected. This apparent improvement in $c$'s status is unjustified and arises solely from the loss of critical information — specifically, the fact that $c$ was attacked.

Removing attacks coming from similar arguments does not lead to information loss, as it only alters the set of arguments targeted by those specific attacks. Moreover, the affected arguments remain under attack, since not all attacks from similar sources are eliminated — at least one is always retained to prevent overvaluation. For example, one could remove the attack $(b_2, a)$, ensuring that $c$ remains rejected in the resulting subgraph, while $a$ continues to be attacked because $(b_1, a)$ is preserved.

The third principle concerns the impact of accounting for similarity. It asserts that removing attacks from similar sources cannot lower an argument's acceptability status. In other words, after such modifications, an argument's status in the graph can either remain the same or improve — but it will never worsen.

**Principle 3 (Monotonicity)** An extension semantics $\delta$ is monotonic iff $\forall E = (G, sim) \in eAG$ such that $G = (A, R), \forall a \in A, Acc_\delta(a, (G, sim)) \succeq Acc_\delta(a, (G, sim^*))$.

**Property 2** If a semantics is indifferent to similarity, then it satisfies neutrality.

Complete, grounded, stable, and preferred semantics do not account for similarity. That is, for an extended argumentation graph $E = (G, sim)$, applying any of these semantics produces the same extension(s) as when applied to the base graph $G$, independently of the similarity relation $sim$. As a result, the four semantics satisfy neutrality and monotonicity in a vacuous way.

**Theorem 1** For any $\delta \in \{co, gr, st, pr\}$, the semantics $\delta$ is indifferent to similarity. It also satisfies neutrality and monotonicity.

# 4 Parameterized Extension Semantics

In this section, we introduce a family of parameterized extension semantics that are sensitive to similarity and generalize the four classic semantics defined in [13]. Each instance in this family is defined by two parameters: a *base extension semantics* $\delta$ (e.g., preferred) and a *selection function* $\lambda$. It proceeds in three main steps:

1. Given an extended argumentation graph $E = (G, sim)$, where $G = (A, R)$, it begins by removing all redundant attacks, keeping only one representative for each. This step may produce several subgraphs $G_1, \dots, G_n \in AG$, where each $G_i = (A, R_i)$ with $R_i \subseteq R$, representing possible contractions of the original graph $G$. For example, consider $E_1 = (G_1, sim_1)$, where $G_1$ is the graph depicted below. If $sim_1 = \{(b_1, b_2)\} \cup sim_1^*$, then the argument $a$ receives two redundant attacks, $(b_1, a)$ and $(b_2, a)$. As a result, two subgraphs $(G_{11} and G_{12})$ are generated: each retains one of these two attacks along with $(b_2, c)$.
<figure><img src="image_4.png" /></figure>

2. The selection function $\Delta$ selects **one** from the generated subgraphs $G_1, \dots, G_n$. In the example, it selects either $G_{11}$ or $G_{12}$.

3. Arguments of the selected subgraph are evaluated using the base semantics $\delta$. In the above example, both subgraphs have a single preferred extension $\{b_1, b_2\}$. Hence, the argument $a$ remains rejected even after removal of redundancy.

Before defining the family of semantics, we introduce four notions. The first is the concept of an **equivalence class of attacks**, which groups together redundant attacks.

**Definition 8** Let $\mathbb{E} = \langle \mathbb{G}, \mathbf{sim} \rangle \in \mathbf{eAG}$ with $\mathbb{G} = \langle A, \mathcal{R} \rangle$. A set $X \subseteq \mathcal{R}$ is an equivalence class iff:

* $\forall r, r' \in X, (\mathbf{Sc}(r), \mathbf{Sc}(r')) \in \mathbf{sim} \text{ and } \mathbf{Tg}(r) = \mathbf{Tg}(r')$,

* $\mathcal{X}' \supset X$ which satisfies the above condition.

$\mathbf{Equiv}(\mathbb{E})$ denotes the set of all equivalent classes in $\mathbb{E}$.

**Example 1 (Cont).** Consider $\mathbb{E}_1 = \langle G_1, \mathbf{sim}_1 \rangle$.

* Let $\mathbf{sim}_1 = \mathbf{sim}_1^*$. Then, $\mathbf{Equiv}(\mathbb{E}_1) = \{E_1, E_2, E_3\}$, where $E_1 = \{(b_1, a)\}$, $E_2 = \{(b_2, a)\}$, and $E_3 = \{(b_2, c)\}$.

* Let $\mathbf{sim}_1 = \{(b_1, b_2), (b_2, b_1)\} \cup \mathbf{sim}_1^*$. Then, $\mathbf{Equiv}(\mathbb{E}_1) = \{E_3, E_4\}$, $E_4 = \{(b_1, a), (b_2, a)\}$.

Note that every attack belongs to a *single* equivalence class. Indeed, equivalence classes are pairwise disjoint and form a partition of the whole set of attacks.

**Proposition 1** Let $\mathbb{E} = \langle \mathbb{G}, \mathbf{sim} \rangle \in \mathbf{eAG}$ with $\mathbb{G} = \langle A, \mathcal{R} \rangle$.

* For all $X, Y \in \mathbf{Equiv}(\mathbb{E})$, $X \cap Y = \emptyset$,

* $\mathcal{R} = \bigcup_{X \in \mathbf{Equiv}(\mathbb{E})} X$,

* If $\mathbf{sim} = \mathbf{sim}^*$, then $\mathbf{Equiv}(\mathbb{E}) = \{\{r\} | r \in \mathcal{R}\}$,

* $| \mathbf{Equiv}(\mathbb{E}) | \le | \mathcal{R} |$,

* $\emptyset \in \mathbf{Equiv}(\mathbb{E}) \text{ iff } \mathcal{R} = \emptyset$.

The second key concept is that of a **set of representatives**, which selects exactly one attack from each equivalence class.
**Definition 9** Let $\mathbb{E} = \langle \mathcal{G}, \text{sim} \rangle \in \text{eAG}$, with $\mathcal{G} = \langle \mathcal{A}, \mathcal{R} \rangle$, and $X \subseteq \mathcal{R}$. We say that $X$ is a set of representatives in $\mathbb{E}$ iff $|X| = |\text{Equiv}(\mathbb{E})|$ and $\forall Y \in \text{Equiv}(\mathbb{E}), X \cap Y \neq \emptyset$. Let $\text{Rep}(\mathbb{E})$ contain all sets of representatives of $\mathbb{E}$.

**Example 1 (Cont).** Consider again $\mathbb{E}_1 = \langle \mathcal{G}_1, \text{sim}_1 \rangle$.

* If $\text{sim}_1 = \text{sim}_1^*$, then $\text{Rep}(\mathbb{E}_1) = \{X_0\}$ where $X_0 = \{(b_1, a), (b_2, a), (b_2, c)\}$.

* If $\text{sim}_1 = \{(b_1, b_2), (b_2, b_1)\} \cup \text{sim}_1^*$, then $\text{Rep}(\mathbb{E}_1) = \{X_1, X_2\}$ where:

- $X_1 = \{(b_1, a), (b_2, c)\}$,

- $X_2 = \{(b_2, a), (b_2, c)\}$.

It is clear that a set of representatives contains exactly one attack among redundant ones. Furthermore, when a graph is free of redundancy, then it has a single set of representatives, which is the whole set of attacks.

**Proposition 2** Let $\mathbb{E} = \langle \mathcal{G}, \text{sim} \rangle \in \text{eAG}$ with $\mathcal{G} = \langle \mathcal{A}, \mathcal{R} \rangle$.

* If $\text{sim} = \text{sim}^*$, then $\text{Rep}(\mathbb{E}) = \{\mathcal{R}\}$.

* $\forall X \in \text{Rep}(\mathbb{E}), \mathbb{1}_r, r' \in X$ such that $(\text{Sc}(r), \text{Sc}(r')) \in \text{sim}$ and $\text{Tg}(r) = \text{Tg}(r')$.

The third notion is the **contraction** of an extended argumentation graph. A contraction is a subgraph whose attacks form a set of representatives, and whose admissible sets remain conflict-free with respect to the original attack relation. This latter condition ensures that no argument and its attackers can be jointly acceptable.

**Definition 10** Let $\mathbb{E} = \langle \mathcal{G}, \text{sim} \rangle \in \text{eAG}$, with $\mathcal{G} = \langle \mathcal{A}, \mathcal{R} \rangle$, and $\mathcal{G}' = \langle \mathcal{A}', \mathcal{R}' \rangle \in \text{AG}$. We say that $\mathcal{G}'$ is a contraction of $\mathbb{E}$ iff $\mathcal{A}' = \mathcal{A}, \mathcal{R}' \in \text{Rep}(\mathbb{E})$, and $\mathbb{1}_E \subseteq \mathcal{A}'$ such that $\mathbb{E}$ is admissible in $\mathcal{G}'$ and $\mathcal{R}$-conflicting. Let $\text{Ctr}(\mathbb{E})$ be the set of all possible contractions of $\mathbb{E}$.

**Example 1 (Cont).** Consider $\mathbb{E}_1 = \langle \mathcal{G}_1, \text{sim}_1 \rangle$ and $\mathbb{E}_2 = \langle \mathcal{G}_2, \text{sim}_2 \rangle$, where $\mathcal{G}_1$ and $\mathcal{G}_2$ are depicted in Table 1. Assume that $\text{sim}_i = \{(b_1, b_2), (b_2, b_1)\} \cup \text{sim}_i^*$, with $i \in \{1, 2\}$.

* $\text{Ctr}(\mathbb{E}_1) = \{\mathcal{G}_{11}, \mathcal{G}_{12}\}$, where $\mathcal{G}_{11} = \mathcal{G}_1 \ominus \{(b_1, a)\}$ and $\mathcal{G}_{12} = \mathcal{G}_1 \ominus \{(b_2, a)\}$.

* $\text{Ctr}(\mathbb{E}_2) = \emptyset$ since both subgraphs $\mathcal{G}_2 \ominus \{(b_i, a)\}$ contain an admissible set $\{a, b_i\}$, which is not conflict-free wrt the initial attack relation $\mathcal{R}_2$.

The example shows that an extended argumentation graph may sometimes have no contractions. This occurs when all subgraphs built from $\text{Rep}(\mathbb{E})$ yield $\mathcal{R}$-conflicting admissible sets. Conversely, a graph may have as many contractions as there are sets of representatives. Notably, when the similarity relation is $\text{sim}^*$, the graph has exactly one contraction — namely, the base graph itself.

**Proposition 3** Let $\mathbb{E} = \langle \mathcal{G}, \text{sim} \rangle \in \text{eAG}$.
• $0 \le |\text{Ctr}(\mathbb{E})| \le |\text{Rep}(\mathbb{E})|$,

• If $\text{sim} = \text{sim}^*$, then $\text{Ctr}(\mathbb{E}) = \{\mathbb{G}\}$.

The last notion is that of **selection function**, which chooses one argumentation graph among a collection of graphs.

**Definition 11** A selection function is a mapping $\Delta : \mathcal{A}^n \to \mathcal{A}$, $n \ge 1$, such that $\Delta(x_1, \dots, x_n) \in \{x_1, \dots, x_n\}$.

We are now ready to define the **parameterized extended extension semantics**, denoted by $\Delta^\delta$, where $\delta$ serves as the *base semantics* and $\Delta$ acts as the *selection function*. An extended semantics $\Delta^\delta$ qualifies as an extension semantics (Def. 2). The term extended refers to the fact that it generalizes and builds upon $\delta$.

**Definition 12** Let $\delta$ be an extension semantics and $\Delta$ a selection function. An extended semantics $\Delta^\delta$ maps every $\mathbb{E} = (\mathbb{G}, \text{sim}) \in \text{eAG}$, $\mathbb{G} = (\mathcal{A}, \mathcal{R})$, to a set of extensions:

$$ \text{Ext}_{\Delta^{\delta}}(\mathbb{E}) = \left\{ \begin{array}{ll} \text{Ext}_{\delta}(\mathbb{G}) & \text{if} \quad \text{Ctr}(\mathbb{E}) = \emptyset \\ \text{Ext}_{\delta}(\Delta(\mathbb{G}_1, \dots, \mathbb{G}_k)) & \text{else, where } \{\mathbb{G}_1, \dots, \mathbb{G}_k\} = \text{Ctr}(\mathbb{E}). \end{array} \right. $$

The acceptability status of any argument $a \in \mathcal{A}$ under $\Delta^\delta$ is:

$$ \text{Acc}_{\Delta^{\delta}}(a, \mathbb{E}) = \left\{ \begin{array}{ll} \text{Acc}_{\delta}(a, \mathbb{G}) & \text{if} \quad \text{Ctr}(\mathbb{E}) = \emptyset \\ \text{Acc}_{\delta}(a, \Delta(\mathbb{G}_1, \dots, \mathbb{G}_k)) & \text{else, where } \{\mathbb{G}_1, \dots, \mathbb{G}_k\} = \text{Ctr}(\mathbb{E}). \end{array} \right. $$

Note that when an extended argumentation graph has no contraction, any parameterized extended semantics defaults to the evaluation provided by its base semantics $\delta$ on the graph $\mathbb{G}$. Recall that the absence of contractions arises when all subgraphs contain at least one $\mathcal{R}$-conflicting admissible set. Selecting one of these subgraphs would lead to counterintuitive outcomes — for example, inferring the claims of the arguments $A = \{\{x\}, x\}$ and $B = \{\neg x, \neg x \to y\}, y\}$, even though $A$ attacks $B$. Indeed, if $x$ is accepted as true, then $y$ cannot validly follow from the premises of $B$.

Extended semantics strike a **balance** between accounting for similarity — to avoid the undervaluation of arguments due to redundant attacks — and ensuring rational evaluations. The idea is to leverage similarity only when doing so does not introduce counterintuitive outcomes due to the structure of the graph.

**Example 1 (Cont).** Let $\delta = \text{pr}$ and consider the four extended graphs $\mathbb{E}_i = (\mathbb{G}_i, \text{sim}_i)$.

• Let $\text{sim}_1 = \{(b_1, b_2), (b_2, b_1)\} \cup \text{sim}_1^*$. Then, $\text{Ctr}(\mathbb{E}_1) = \{\mathbb{G}_{11}, \mathbb{G}_{12}\}$. Any selection function $\Delta$ may choose either $\mathbb{G}_{11}$ or $\mathbb{G}_{12}$. Both contractions have $\{b_1, b_2\}$ as the preferred extension. Thus, $\forall x \in \mathcal{A}_1, \text{Acc}_{\Delta^{\text{pr}}}(x, \mathbb{E}_1) = \text{Acc}_{\text{pr}}(x, \mathbb{G}_1)$.

• Let $\text{sim}_2 = \{(b_1, b_2), (b_2, b_1)\} \cup \text{sim}_2^*$. Then, $\text{Ctr}(\mathbb{E}_1) = \emptyset$. Hence, $\forall x \in \mathcal{A}_1, \text{Acc}_{\Delta^{\text{pr}}}(x, \mathbb{E}_1) = \text{Acc}_{\text{pr}}(x, \mathbb{G}_1)$.

• Let $\text{sim}_3 = \{(c_1, c_2), (c_2, c_1)\} \cup \text{sim}_3^*$. Note that $\text{Ctr}(\mathbb{E}_3) = \{\mathbb{G}_{31}, \mathbb{G}_{32}\}$, where $\mathbb{G}_{31} = \mathbb{G}_3 \ominus \{(c_1, b)\}$ and $\mathbb{G}_{32} = \mathbb{G}_3 \ominus \{(c_2, b)\}$. Note also that $\text{Ext}_{\text{pr}}(\mathbb{G}_{31}) = \{\{a, c_2\}\}$ and $\text{Ext}_{\text{pr}}(\mathbb{G}_{32}) = \{\{c_2\}\}$.

– If $\Delta(\text{Ctr}(\mathbb{E}_3)) = \mathbb{G}_{31}$, then $\forall x \in \mathcal{A}_3, \text{Acc}_{\Delta^{\text{pr}}}(x, \mathbb{E}_3) = \text{Acc}_{\text{pr}}(x, \mathbb{G}_3)$.
- If $\Delta(\mathrm{Ctr}(\mathbb{G}_3)) = \mathbb{G}_{32}$, then $\mathrm{Acc}_{\Delta^{pr}}(a, \mathbb{E}_3) = \mathrm{Acc}_{\Delta^{pr}}(b, \mathbb{E}_3) = \mathbf{u}$. Notice that the status of $b$ improves as it is rejected in $\mathbb{G}_3$. As a consequence, the status of $a$ deteriorates.

* Let $\mathrm{sim}_4 = \{(b_1, b_2), (b_2, b_1)\} \cup \mathrm{sim}_4^*$. Then, $\mathrm{Ctr}(\mathbb{E}_4) = \{\mathbb{G}_{41}, \mathbb{G}_{42}\}$, where $\mathbb{G}_{41} = \mathbb{G}_4 \ominus \{(b_1, a)\}$ and $\mathbb{G}_{42} = \mathbb{G}_4 \ominus \{(b_2, a)\}$. Note also that $\mathrm{Ext}_{pr}(\mathbb{G}_{41}) = \{{\{c_1\}, \{a, c_2\}}\}$ and $\mathrm{Ext}_{pr}(\mathbb{G}_{42}) = \{{\{a, c_1\}, \{a, c_2\}}\}$.

- If $\Delta(\mathrm{Ctr}(\mathbb{G}_4)) = \mathbb{G}_{41}$, then $\forall x \in \mathcal{A}_4, \mathrm{Acc}_{\Delta^{pr}}(x, \mathbb{E}_4) = \mathrm{Acc}_{pr}(x, \mathbb{G}_4)$.

- If $\Delta(\mathrm{Ctr}(\mathbb{G}_4)) = \mathbb{G}_{42}$, then $\mathrm{Acc}_{\Delta^{pr}}(a, \mathbb{E}_4) = \mathbf{s}$. The status of $a$ is improved due to the removal of redundancy: $\mathrm{Acc}_{\Delta^{pr}}(a, \mathbb{E}_4) > \mathrm{Acc}_{pr}(a, \mathbb{G}_4) = \mathbf{c}$.

We show that the extended extension semantics satisfy the three proposed principles. In particular, monotonicity and sensitivity to similarity are guaranteed when the base semantics is any of Dung's standard semantics [13].

**Theorem 2** Let $\Delta^\delta$ be an extended extension semantics.

* $\Delta^\delta$ satisfies neutrality.

* If $\delta \in \{\mathrm{co}, \mathrm{gr}, \mathrm{pr}, \mathrm{st}\}$, then $\Delta^\delta$ satisfies sensitivity to similarity and monotonicity.

Every instance in the family generalizes its base semantics, as they coincide in cases with no similarity, in graphs without redundant attacks, or when each argument has at most one attacker.

**Theorem 3** Let $\Delta^\delta$ be an extended semantics, $\mathbb{E} = \langle\mathbb{G}, \mathrm{sim}\rangle \in \mathrm{eAG}$, with $\mathbb{G} = \langle\mathcal{A}, \mathcal{R}\rangle$. If at least one of the following conditions holds:

* $\mathrm{sim} = \mathrm{sim}^*$,

* $\forall a \in \mathcal{A}, \mathrm{sAtt}(a, \mathbb{G}) = \emptyset$,

* $\forall a \in \mathcal{A}, |\mathrm{rAtt}(a, \mathbb{G})| \le 1$,

then $\mathrm{Ext}_{\Delta^\delta}(\mathbb{E}) = \mathrm{Ext}_\delta(\mathbb{G})$.

Extended extension semantics coincide also with their base semantics in the case of *coherent* extended argumentation graphs—that is, graphs in which similar arguments have exactly the same set of attackers. In $\mathbb{E}_1 = \langle\mathbb{G}_1, \mathrm{sim}_1\rangle$, if $(b_1, b_2) \in \mathrm{sim}_1$, then $\mathbb{E}_1$ is not coherent as $b_1$ and $b_2$ do not have identical sets of attackers.

**Definition 13** Let $\mathbb{E} = \langle\mathbb{G}, \mathrm{sim}\rangle \in \mathrm{eAG}$ with $\mathbb{G} = \langle\mathcal{A}, \mathcal{R}\rangle$. $\mathbb{E}$ is coherent iff $\forall a, b \in \mathcal{A}$, if $(a, b) \in \mathrm{sim}$, then $\{\mathrm{Sc}(r) \mid r \in \mathrm{rAtt}(a, \mathbb{G})\} = \{\mathrm{Sc}(r') \mid r' \in \mathrm{rAtt}(b, \mathbb{G})\}$.

We next show that extended semantics based on complete, stable, preferred, or grounded semantics are indifferent to similarity in coherent graphs. Such graphs always admit contractions, and moreover, all contractions yield the same extensions, which coincide with those of the base graph.
**Theorem 4** Let $\Delta^\delta$ be an extended semantics, with $\delta \in \{co, gr, pr, st\}$, and $\mathbb{E} = \langle \mathbb{G}, \mathit{sim} \rangle \in \mathit{eAG}$, with $\mathbb{G} = \langle \mathcal{A}, \mathcal{R} \rangle$. If $\mathbb{E}$ is coherent, then the following hold.

* $\mathrm{Ctr}(\mathbb{E}) = \{\langle \mathcal{A}, \mathcal{R}' \rangle \in \mathit{AG} \mid \mathcal{R}' \in \mathrm{Rep}(\mathbb{E})\}$,

* For any $\mathbb{G}' \in \mathrm{Ctr}(\mathbb{E})$, $\mathrm{Ext}_\delta(\mathbb{G}') = \mathrm{Ext}_\delta(\mathbb{G})$.

We now investigate how the acceptability status of an argument changes when moving from a base extension semantics $\delta \in \{co, gr, pr, st\}$ to its corresponding extended semantics $\Delta^\delta$. We focus on cases where similarity influences the evaluation—specifically, when contractions do exist. The first result establishes that an argument retains the same status under both semantics if and only if the set of attacks preserved by the contraction selected by $\Delta$ contains a sufficient set under $\delta$.

**Theorem 5** Let $\Delta^\delta$ be an extended semantics, with $\delta \in \{co, gr, pr, st\}$, $\mathbb{E} = \langle \mathbb{G}, \mathit{sim} \rangle \in \mathit{eAG}$, $\mathbb{G} = \langle \mathcal{A}, \mathcal{R} \rangle$, $\Delta(\mathrm{Ctr}(\mathbb{E})) = \langle \mathcal{A}, \mathcal{R}' \rangle$, $a \in \mathcal{A}$ s.t $\mathit{sim} \subseteq \mathit{aAtt}(a, \mathbb{G}) \times \mathit{aAtt}(a, \mathbb{G})$.

$$ \mathrm{Acc}_{\Delta^{\delta}}(a, \mathbb{E}) = \mathrm{Acc}_{\delta}(a, \mathbb{G}) \quad \text{iff} \quad \exists S \subseteq \mathcal{R}' \text{ such that } S \in \mathrm{Suff}_{\delta}(a, \mathbb{G}). $$

**Example 1 (Cont).** Recall that $\mathrm{Suff}_{pr}(a, \mathbb{G}_1) = \{(b_1, a)\}, \{(b_2, a)\}\}$. Hence, each of the two contractions $\mathbb{G}_{11}, \mathbb{G}_{12}$ of $\mathbb{E}_1 = \langle \mathbb{G}_1, \mathit{sim}_1 \rangle$ keeps one sufficient set, which explains why the two corresponding extended semantics reject $a$ as in $\mathbb{G}_1$. For $\mathbb{E}_3 = \langle \mathbb{G}_3, \mathit{sim}_3 \rangle$, $\mathrm{Suff}_{pr}(b, \mathbb{G}_3) = \{(c_2, b)\}$. Hence, the contraction $\mathbb{G}_{31}$ keeps the attack $(c_2, b)$ leading a stable status for $b$.

The second result characterizes the conditions under which the acceptability status of an argument changes. It shows that such a change occurs if and only if a **necessary set** under $\delta$ is among the attacks removed by the contraction selected by $\Delta$.

**Theorem 6** Let $\Delta^\delta$ be an extended semantics, with $\delta \in \{co, gr, pr, st\}$, $\mathbb{E} = \langle \mathbb{G}, \mathit{sim} \rangle \in \mathit{eAG}$, $\mathbb{G} = \langle \mathcal{A}, \mathcal{R} \rangle$, $\Delta(\mathrm{Ctr}(\mathbb{E})) = \langle \mathcal{A}, \mathcal{R}' \rangle$, $a \in \mathcal{A}$ s.t $\mathit{sim} \subseteq \mathit{aAtt}(a, \mathbb{G}) \times \mathit{aAtt}(a, \mathbb{G})$.

$$ \mathrm{Acc}_{\Delta^{\delta}}(a, \mathbb{E}) \succ \mathrm{Acc}_{\delta}(a, \mathbb{G}) \quad \text{iff} \quad \exists S \in \mathrm{Nec}_{\delta}(a, \mathbb{G}) \text{ such that } S \cap \mathcal{R}' = \emptyset. $$

**Example 1 (Cont).** Recall that $\mathrm{Nec}_{pr}(b, \mathbb{G}_3) = \{(c_2, b)\}$. The contraction $\mathbb{G}_{32}$ of $\mathbb{E}_3 = \langle \mathbb{G}_3, \mathit{sim}_3 \rangle$ removes the necessary attack $(c_2, b)$, leading a status change for $b$.

# 5 Selection Functions

The key idea behind the novel semantics $\Delta^\delta$ is to retain only one attack among a set of redundant ones. The function $\Delta$ selects which attacks to keep. In what follows, we present three families of **strategies** that $\Delta$ may follow. All of them begin by evaluating arguments in the initial graph under the base semantics $\delta$.

**Strategies based on the status of arguments:** These strategies remove redundant attacks depending on the acceptability status of their sources. The idea is to eliminate attacks originating from either: i) the weakest attackers, or ii) the strongest ones. Consider the graph $\mathbb{G}_3$, where $(c_1, c_2) \in \mathit{sim}_3$. Note that $\mathrm{Acc}_{pr}(c_2, \mathbb{G}_3) = s \succ$
Acc<sub>pr</sub>(c<sub>1</sub>, G<sub>3</sub>) = u. Removing the weakest (c<sub>1</sub>) results in the same status of b and thus for a. However, discarding the strongest (c<sub>2</sub>) leads to an improved status for b (passing from rejected to undecided). In the graph G<sub>4</sub>, both b<sub>1</sub>, b<sub>2</sub> are rejected under preferred semantics. Thus, this strategy allows to choose any of the two attacks.

**Strategies based on types of attacks:** These strategies consider the type of attack, rather than the status of its source. They assume the following hierarchy of importance: necessary > sufficient > dummy, where a dummy attack is an attack that has no impact on its target [1]. A reasonable strategy would remove the weakest available set of redundant attacks. In G<sub>4</sub>, the attack (b<sub>1</sub>, a) is dummy, while (b<sub>2</sub>, a) is necessary. Therefore, Δ would preserve the necessary attack and remove the dummy one.

**Mixed strategies:** Mixed strategies combine the previous two approaches. They first identify the weakest sets of redundant attacks based on their type, then refine the choice by comparing the status of their sources.

# 6 Related Work

Despite its importance and notable influence on the evaluation of arguments—and consequently, on the outcomes of argumentation-based systems—the notion of similarity has received relatively limited attention within the argumentation community. Existing research addressing this concept can be broadly categorized into three main areas.

The first category focuses on *measuring similarity* between arguments. This includes studies on logical arguments [2, 4, 6, 8, 12] and textual arguments [16, 18, 15].

The second category investigates how similarity can be used to *modify the structure of an argumentation graph* before applying any semantics. A representative example is [9], where similarity is employed as a pre-processing step to alter the graph’s relational structure. Specifically, this work prevents attack relations between similar arguments. Consider for instance the graph G<sub>2</sub> in Table 1 and assume that sim<sub>2</sub> = sim<sub>2</sub><sup>*</sup> ∪ {(b<sub>1</sub>, b<sub>2</sub>), (b<sub>2</sub>, b<sub>1</sub>)}. The approach in [8, 9] starts by removing the two attacks between the arguments b<sub>1</sub> and b<sub>2</sub>, then evaluates the resulting sub-graph using any extension-based semantics. Thus, it does not remove the **redundant attacks**, namely the two targeting the argument a. Consequently, it violates the Neutrality principle but satisfies the Sensitivity to Similarity principle, due to the removal of attacks between similar arguments.

The third category explores how to incorporate similarity directly into the *semantics*. Two works—[3, 5]—have addressed this challenge. These studies focus on gradual semantics in abstract argumentation. They propose a set of desirable properties and offer mechanisms for extending existing gradual semantics to respect similarity while satisfying those properties. However, these approaches do not consider extension-based semantics, which are the focus of our work. It is also important to highlight that the aforementioned studies are grounded in a specific class of gradual semantics that satisfy the equivalence property—i.e., the acceptability status of an argument is determined solely by the statuses of its direct attackers. In contrast, extension semantics do not satisfy this property, which necessitates a fundamentally different approach. Accordingly, our work introduces a novel method for integrating similarity into extension
semantics.

Some works in the literature have studied *incomplete* argumentation graphs, where there is uncertainty regarding the presence of certain arguments and/or attacks (e.g., [14]). To evaluate arguments under such uncertainty, a widely adopted approach involves completing the graph by adding the uncertain elements, resulting in multiple possible completions. These completions are then aggregated to determine the status of each argument. While our parameterized semantics compute contractions, the objectives are fundamentally different. In the case of similarity, the issue is not uncertainty but redundancy—specifically, the presence of similar attacks—which should be minimized as much as possible. Each contraction in our framework corresponds to an alternative, similarity-aware solution.

# 7 Conclusion

We presented the first study on the impact of similarity between arguments on their evaluation under extension semantics. We introduced a set of principles that extension semantics should satisfy in the presence of similarity, and proposed a family of semantics that adhere to them. Furthermore, we identified structural conditions—specifically, coherent versus non-coherent graphs—that determine whether similarity can be safely ignored or must be taken into account. We identified also the conditions under which similarity affects the acceptability status of arguments, and when it does not.

This work lends itself to a number of developments in order to improve its generality and the compromise rational evaluation/removal of redundancy of the semantics. The idea is to develop a principled framework for selection functions. Another line of research consists of investigating alternative methods for addressing similarity, including the merging of similar arguments into a single argument.

# Acknowledgments

This work was funded by the French program AI Cluster – ANR-23-IACL-0002.

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