Solid Semantics for Abstract Argumentation Frameworks and the Preservation of Solid Semantic Properties

Abstract

In this paper, we propose solid admissibility that is a strengthened version of Dung’s admissibility to obtain the most acceptable set of arguments. Besides, other solid extensions based on solid admissibility are defined. Such extensions not only include all defenders of its elements but also exclude all arguments indirectly attacked and indirectly defended by some argument(s). Furthermore, we characterize solid extensions with propositional formulas. Using these formulas, we aggregate solid extensions by using approaches from judgment aggregation. Especially, although no quota rule preserves Dung’s admissibility for any argumentation framework, we show that there exist quota rules that preserve solid admissibility for any argumentation framework.

Supported by the China Postdoctoral Science Foundation Grant, No. 2019M663352.

1 Introduction

In Dung’s work [12], an argumentation framework (AF) is a directed graph, where nodes represent arguments and edges represent elements of a binary relation. It has been studied widely over the last decades. One of the core notions of AFs is admissibility. An admissible extension is a set of arguments that contains no internal conflict and defends its elements against any attacker.

In this paper, we mainly focus on obtaining the most acceptable arguments in AFs by strengthening Dung’s admissibility. Before discussing this idea, we first illustrate two problems (or drawbacks) observed from the literature. The first one is observed from graded acceptability [14] which provides an approach to rank arguments from the most acceptable to the weakest one(s) by parameterizing the numbers of attackers and counter-attackers. Hence, we can require that a set of arguments is graded-acceptable if it contains at least n counter-attackers for each attacker of its elements. Graded acceptability is flexible as we can tune the parameter n. What if we want to find out in an AF the sets of arguments such that they exactly contain all counter-attackers for each attacker of their elements? It is impossible to achieve this goal by tuning the parameter n, as different attackers may have different numbers of counter-attackers. Consider the following example.

Fig. 1.

Three problematic argumentation frameworks

Example 1

In Fig. 1a, $\{A_{1},C_{1}\},\{A_{2},C_{2}\}$ and $\{A_{1},A_{2},C_{2}\}$ are acceptable under graded semantics when we require that a set of arguments is acceptable if it contains at least one counter-attacker for each attacker of its elements. We should notice that although $\{A_{1},A_{2},C_{2}\}$ contains all counter-attackers when its elements are attacked, $\{A_{2},C_{2}\}$ fails to contain all counter-attackers whenever its elements are attacked. When the requirement is more demanding in the sense that a set of arguments is acceptable if it contains at least two counter-attackers for each attacker of its elements, $\{A_{1},A_{2},C_{2}\}$ is still acceptable under this requirement. Although $\{A_{1},C_{1}\}$ contains all counter-attackers when its elements are attacked, it fails to satisfy this requirement.

The other problem is observed from the situation where some argument indirectly attacks and indirectly defends some argument. There are other semantics [1, 7] that also provide approaches to rank arguments. But their approaches rely on conjectures regarding the processing of cycles. However, Dung indicates in [12] that the presence of cycles could be a problem. In this work, an argument A is controversial w.r.t. an argument B if A indirectly attacks and indirectly defends B. Such arguments could lead to problematic situations. Consider the following example.

Example 2

In Fig. 1b, A is controversial w.r.t. C as A indirectly attacks and indirectly defends C. From a skeptical view, A and C should not occur in the same set of acceptable arguments. But $\{A,C,F\}$ is admissible in Dung’s semantics.

There is no consensus on whether to accept or reject such arguments. Note that any argument in an odd-length cycle is controversial w.r.t. any argument in this cycle. There are many articles aiming to address this problem [3, 4, 6, 11]. For example, Baumann et al. [4] argue that in Fig. 1c, A should be rejected while B should be accepted, since the attack from the self-defeat argument A is not valid. However, Jakobovits and Vermeir [15] state that A and B can be labeled as “undecided” and “rejected”, resp. They argue that since A is “strong enough” to attack A, surely it is strong enough to do the same with B. Compared with these approaches directly facing these disputable situations, our approach is more like bypassing such situations.

Considering the emergence of the problems above, we argue that the most acceptable arguments should satisfy two criteria: (i) they should have defenders as many as possible, and (ii) they should avoid the undesirable interference of some arguments. Dung’s admissibility only requires a weak defense in the sense that only one mandatory defender is enough. An interesting fact is that some problematic situations disappear after Dung’s admissibility is strengthened. In this paper, we propose solid admissibility which satisfies the two criteria. Roughly speaking, a solidly admissible extension is a set of arguments that contains no internal conflict, defends its elements against any attackers, and contains all the defenders. We will show that if an argument A is controversial w.r.t. an argument B, then B will never occur in any solid extension based on solid admissibility. To sum up, such extensions not only contain all defenders of their elements, but also avoid the interference of any argument that is indirectly attacked and indirectly defended by some argument. This conforms to the intuition in practical reasoning or real life in the sense that if an argument has more defenders, then surely it has less controversy.

We apply solid semantics in the field of judgment aggregation, a branch of social choice theory. When a group of agents evaluates which arguments are acceptable in an AF, each of them may report a different extension under a specific solid semantics that represents a individual viewpoint about which arguments are acceptable. We study whether their collective outcome is also a solid extension under this semantics when quota rules are applied. Especially, we show that there exist quota rules that preserve solid admissibility for any AF.

Contribution. Firstly, we propose a family of new semantics for abstract argumentation. Such semantics provide an approach to circumvent a controversial situation in AFs and also capture a feature that graded semantics fail to capture. Secondly, the new semantics have more possibility results for extension aggregation than Dung’s semantics do.

Paper Outline. The rest of this paper is organized as follows. Section 2 reviews the background of abstract argumentation and judgment aggregation. Section 3 defines solid admissibility and shows how the problems are addressed. Section 4 develops more solid semantics and shows the connections among them. Besides, we compare solid semantics with Dung’s semantics. Furthermore, we present propositional formulas that characterize the solid semantics and pave the way for solid extensions aggregation. Section 5 shows preservation results for the solid semantics. Section 6 mainly compares solid semantics with other related semantics. Section 7 concludes this paper and points out future work.

2 Preliminary

2.1 Abstract Argumentation

This part reviews some notions of abstract argumentation [12]. Some definitions are adopted from [14].

Definition 1 (Argumentation framework)

An argumentation framework is a pair $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$, where $\textit{Arg}$ is a finite and non-empty set of arguments, and $\rightharpoonup$ is a binary relation on $\textit{Arg}$.

For any $A,B\in \textit{Arg}$, $A\rightharpoonup B$ (or A attacks B) denotes that $(A,B)\in \rightharpoonup$. For any $B\in \textit{Arg}$, $\overline{B}=\{A \in \textit{Arg}\mid A \rightharpoonup B\}$, namely, $\overline{B}$ denotes the set of the attackers of B. A is an initial argument iff $\overline{A}=\emptyset$. For any $\varDelta \subseteq \textit{Arg}$ and any $B\in \textit{Arg}$, $\varDelta \rightharpoonup B$ denotes that there exists an argument $A\in \varDelta$ such that $A\rightharpoonup B$. For any $\varDelta \subseteq \textit{Arg}$ and any $A\in \textit{Arg}$, $A\rightharpoonup \varDelta$ denotes that there exists an argument $B\in \varDelta$ such that $A\rightharpoonup B$. For any $A, C\in \textit{Arg}$, A is a defender of C iff there exists an argument $B\in \textit{Arg}$ such that $A\rightharpoonup B$ and $B\rightharpoonup C$.

An argument A indirectly attacks an argument B iff there exists a finite sequence $A_{0},\ldots ,A_{2n+1}$ such that (i) $B=A_{0}$ and $A=A_{2n+1}$, and (ii) for each i, $0\leqslant i\leqslant 2n$, $A_{i+1}\rightharpoonup A_{i}$. An argument A indirectly defends an argument B iff there exists a finite sequence $A_{0},\ldots ,A_{2n}$ such that (i) $B=A_{0}$ and $A=A_{2n}$, and (ii) for each i, $0 \leqslant i < 2n$, $A_{i+1}\rightharpoonup A_{i}$. An argument A is controversial w.r.t. an argument B iff A indirectly attacks and indirectly defends B. Note that direct attackers (resp. defenders) are also indirect attackers (resp. defenders).

Definition 2 (Dung’s defense)

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. $\varDelta \subseteq \textit{Arg}$ defends $C\in \textit{Arg}$ iff for any $B\in \textit{Arg}$, if $B\rightharpoonup C$ then $\varDelta \rightharpoonup B$.

Definition 3 (Defense function)

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. The defense function $d:2^\textit{Arg}\longrightarrow 2^\textit{Arg}$ of $\textit{AF}$ is defined as $d(\varDelta ) =\{C\in \textit{Arg}\mid \varDelta \text { defends }C \}$.

Definition 4 (Neutrality function)

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. The neutrality function $n:2^{\textit{Arg}}\longrightarrow 2^{\textit{Arg}}$ of $\textit{AF}$ is defined as $n(\varDelta )=\{B\in \textit{Arg}\mid \text {not }\varDelta \rightharpoonup B\}$.

Definition 5 (Dung’s semantics)

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. For any $\varDelta \subseteq \textit{Arg}$, $(i)$ $\varDelta$ is a conflict-free extension iff $\varDelta \subseteq n(\varDelta )$; $(ii)$ $\varDelta$ is a self-defending extension iff $\varDelta \subseteq d(\varDelta )$; $(iii)$ $\varDelta$ is an admissible extension iff $\varDelta \subseteq n(\varDelta )$ and $\varDelta \subseteq d(\varDelta )$; $(iv)$ $\varDelta$ is a complete extension iff $\varDelta \subseteq n(\varDelta )$ and $\varDelta =d(\varDelta )$; $(v)$ $\varDelta$ is a preferred extension iff $\varDelta$ is a maximal admissible extension; $(vi)$ $\varDelta$ is a stable extension iff $\varDelta = n(\varDelta )$; $(vii)$ $\varDelta$ is the grounded extension iff $\varDelta$ is the least fixed point of the defense function d.

Theorem 1 (Dung, 1995)

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. $(i)$ If $\varDelta \subseteq \textit{Arg}$ is a preferred extension, then $\varDelta$ is complete extension, but not vice versa; $(ii)$ If $\varDelta \subseteq \textit{Arg}$ is an admissible extension, then there exists a preferred extension $\varGamma$ such that $\varDelta \subseteq \varGamma$.

2.2 Integrity Constraints and Judgment Aggregation

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$, Dung’s semantics can be captured by propositional language (denoted as $\mathcal {L}_{\textit{AF}}$), whose literals are arguments in $\textit{Arg}$ [5]. A model is represented by the set of the literals which it satisfies. In other words, a model of a formula is a subset of $\textit{Arg}$: ($i$) for any $A\in \textit{Arg}$, $\varDelta \vDash A$ iff $A\in \varDelta$; ($ii$) $\varDelta \vDash \lnot \varphi$ iff $\varDelta \vDash \varphi$ does not hold; ($iii$) $\varDelta \vDash \varphi \wedge \psi$ iff $\varDelta \vDash \varphi$ and $\varDelta \vDash \psi$.

A property $\sigma$ of extensions can be regarded as a subset of $2^{\textit{Arg}}$, namely, $\sigma \subseteq 2^{\textit{Arg}}$. Then the set of the extensions under a semantics is a property, e.g., completeness is the set of the complete extensions of $\textit{AF}$. For any formula $\varphi$ in $\mathcal {L}_{\textit{AF}}$, we let $\mathrm Mod(\varphi )=\{\varDelta \subseteq \textit{Arg}\mid \varDelta \vDash \varphi \}$, namely, $\mathrm Mod(\varphi )$ denotes the set of all models of $\varphi$. Obviously, $\sigma =\mathrm Mod(\varphi )$ is a property. When using a formula $\varphi$ to characterize such a property, $\varphi$ is referred to as an integrity constraint.

We next introduce a model for the aggregation of extensions [10, 13]. Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. Let $N=\{1,\cdots ,n\}$ be a finite set of agents. Imagining that each agent $i\in N$ reports an extension $\varDelta _{i}\subseteq {\textit{Arg}}$. Then $\boldsymbol{\varDelta }=(\varDelta _{1},\cdots ,\varDelta _{n})$ is referred to as a profile. An aggregation rule is a function $F:(2^{\textit{Arg}})^n\longrightarrow 2^{\textit{Arg}}$, mapping any given profile of extensions to a subset of $\textit{Arg}$.

Definition 6 (Quota rules)

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$, let N be a finite set of n agents, and let $q\in \{1,\cdots ,n\}$. The quota rule with quota q is defined as the aggregation rule mapping any given profile $\boldsymbol{\varDelta }=(\varDelta _{1},\cdots ,\varDelta _{n})\in (2^{\textit{Arg}})^{n}$ of extensions to the set including exactly those arguments accepted by at least q agents: $F_{q}(\boldsymbol{\varDelta })=\{A\in \textit{Arg}\mid \#\{i\in N\mid A\in \varDelta _{i}\}\geqslant q \}$.

The quota rule $F_{q}$ for n agents with $q=\lceil \frac{n+1}{2}\rceil$ (resp., $q=1$, $q=n$) for $\textit{AF}$ is called the strict majority (resp., nomination, unanimity) rule.

Definition 7 (Preservation)

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. Let $\sigma \subseteq 2^{\textit{Arg}}$ be a property of extensions. An aggregation rule $F:(2^{\textit{Arg}})^{n}\longrightarrow 2^{\textit{Arg}}$ for n agents is said to preserve $\sigma$ if $F(\boldsymbol{\varDelta })\in \sigma$ for every profile $\boldsymbol{\varDelta }=(\varDelta _{1},\cdots ,\varDelta _{1})\in \sigma ^{n}$.

Lemma 1 (Grandi and Endriss, 2013)

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. Let $\varphi$ be a clause (i.e., disjunctions of literals) in $\mathcal {L}_{\textit{AF}}$ with $k_1$ positive literals and $k_2$ negative literals. Then a quota rule $F_q$ for n agents preserves the property $\mathrm Mod(\varphi )$ for $\textit{AF}$ iff the following inequality holds: $q\cdot (k_2-k_1)>n\cdot (k_2-1)-k_{1}$.

Lemma 2 (Grandi and Endriss, 2013)

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. Let $\varphi _1$ and $\varphi _2$ be integrity constraints in $\mathcal {L}_{\textit{AF}}$. If an aggregation rule F preserves both $\mathrm Mod(\varphi _1)$ and $\mathrm Mod(\varphi _2)$, then F preserves $\mathrm Mod(\varphi _1\wedge \varphi _2)$, but not vice versa.

3 Solid Admissibility

To obtain the most acceptable arguments that satisfy the two criteria proposed in the introduction, we formally introduce solid admissibility in this section. Arguments in admissible extensions satisfy the criteria. Firstly, we strengthen Dung’s defense. Definition 8 states that a set of arguments solidly defends an argument iff this set defends (in Dung’s sense) this argument and contains all the defenders of each element of this set.

Definition 8 (Solid defense)

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. $\varDelta \subseteq \textit{Arg}$ solidly defends (or s-defends) $C\in \textit{Arg}$ iff for any $B\in \textit{Arg}$, if $B\rightharpoonup C$, then $\varDelta \rightharpoonup B$ and $\overline{B}\subseteq \varDelta$.

Definition 9 (Solid defense function)

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. The solid defense function $d_{s}:2^\textit{Arg}\longrightarrow 2^\textit{Arg}$ of $\textit{AF}$ is defined as follows. For any $\varDelta \subseteq \textit{Arg}$,

$$\begin{aligned} d_{s}(\varDelta ) = \big \{ C \in \textit{Arg}\mid \varDelta \text { s-defends } C \big \} \end{aligned}$$

(1)

Next we show an important property of the solid defense function. It is easy to see that, if a set of arguments s-defends an argument, then any superset of this set also s-defends this argument by Definition 8.

Theorem 2

The solid defense function $d_{s}$ is monotonic.

Proposition 1

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. For any $\varDelta \subseteq \textit{Arg}$, $d_{s}(\varDelta )\subseteq d(\varDelta )$, but not vice versa.

Proposition 1 states that solid defense strengthens Dung’s defense since, if a set of arguments s-defends an argument, then this set also defends this argument. To show the converse does not hold, consider $\textit{AF}_1$ in Fig. 2a. $\varDelta$ defends C. But the attackers of B are not fully included in $\varDelta$. So $\varDelta$ does not s-defend C.

Definition 10

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. For any $\varDelta \subseteq \textit{Arg}$, $\varDelta$ is a s-self-defending extension iff $\varDelta \subseteq d_{s}(\varDelta )$.

In graded semantics [14], a set of arguments $\varDelta$ mn-defends an argument C iff there are at most $m-1$ attackers of C that are not counterattacked by at least n arguments in $\varDelta$, where m and n are positive integers. A set of arguments is mn-self-defending iff it mn-defends each element. We can tune the parameters to obtain defenses with different levels of strength. For example, when $n=1$, the larger m is, the stronger the defense is. In Fig. 1a, $\{A_{1},C_{1}\}$, $\{A_{2},C_{2}\}$ and $\{A_{1},A_{2},C_{2}\}$ are 11-self-defending. But $\{A_{2},C_{2}\}$ fails to contain all defenders of $C_{2}$. One might be tempted to tune the parameters to obtain a stronger defense. Then only $\{A_{1},A_{2},C_{2}\}$ is 12-self-defending. Although $\{A_{1},C_{1}\}$ contains all defenders of $C_{1}$, it is not 12-self-defending. Hence, graded defense can not capture sets of arguments that exactly contain all defenders of their elements by tuning the parameters. However, solid defense can accomplish this, since it is identified that $\{A_{2},C_{2}\}$ is not s-self-defending while $\{A_{1},C_{1}\}$ and $\{A_{1},A_{2},C_{2}\}$ are s-self-defending.

Definition 11

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. For any $\varDelta \subseteq \textit{Arg}$, $\varDelta$ is a s-admissible extension iff $\varDelta \subseteq n(\varDelta )$ and $\varDelta \subseteq d_{s}(\varDelta )$.

Definition 11 states that a set of arguments is a s-admissible extension iff the set is a conflict-free and s-self-defending extension. Next we show that Dung’s Fundamental Lemma has a counterpart in our semantics. The following lemma states that whenever we have a s-admissible extension, if we put into this extension an argument that is s-defended by this extension, then the new set is still a s-admissible extension. The proof is similar to Dung’s proof [12].

Lemma 3 (s-fundamental lemma)

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$, a s-admissible extension $\varDelta \subseteq \textit{Arg}$, and two arguments $C,\ C'\in \textit{Arg}$ which are s-defended by $\varDelta$. Then ($i$) $\varDelta '=\varDelta \cup \{C\}$ is s-admissible and ($ii$) $\varDelta '$ s-defends $C'$.

As we have strengthened Dung’s admissibility, the second problem mentioned in the introduction can be addressed now. From a skeptical view, it is not cautious to accept an argument that is indirectly attacked and indirectly defended by some argument. The following theorem states that such arguments never occur in s-admissible extensions.

Theorem 3

Given $\textit{AF}=\langle \textit{Arg}, \rightharpoonup \rangle$ and a s-admissible extension $\varDelta \subseteq \textit{Arg}$. If an argument $A\in \textit{Arg}$ is controversial w.r.t. an argument $B\in \textit{Arg}$, then $B \notin \varDelta$.

Proof

Assume that A is controversial w.r.t. B. Suppose for the sake of a contradiction that $B\in \varDelta$. Since A indirectly defends B and $\varDelta$ is s-self-defending, we also have $A\in \varDelta$. Besides, since A indirectly attacks B, there exists a finite sequence $A_{0},\ldots ,A_{2n+1}$ such that (i) $B=A_{0}$ and $A=A_{2n+1}$, and (ii) for each i, $0\leqslant i\leqslant 2n$, $A_{i+1}\rightharpoonup A_{i}$. If $n=0$, then $A_{1}\rightharpoonup A_{0}$, namely, $A\rightharpoonup B$. This contradicts the conflict-freeness of $\varDelta$. If $n\ne 0$, then $A_{2n}$ indirectly defends B. Since $\varDelta$ is s-self-defending, $A_{2n}\in \varDelta$ . Again, the fact $A_{2n+1}\rightharpoonup A_{2n}$ (i.e., $A\rightharpoonup A_{2n}$) contradicts the conflict-freeness of $\varDelta$. So we conclude that $B\notin \varDelta$.

Note that in Theorem 3, A is not excluded from $\varDelta$, since A could be an initial argument. It is not reasonable to reject an unattacked argument. Consider $\textit{AF}_2$ in Fig. 1b. We can see that $\{A,C,F\}$ is admissible. However, since A is controversial w.r.t. C, $\{A,C,F\}$ is not s-admissible. $\{A\}$ is still s-admissible. The problem of odd-length cycles has been widely studied. It is thorny to assign a status to an argument in an odd-length cycle, since any argument in an odd-length cycle is controversial w.r.t. any argument in this cycle. There is no consensus on this problem. Interestingly, the following corollary states that such arguments never occur in s-admissible extensions. Moreover, once there is a path from an odd-length cycle to some argument, this argument will never occur in any s-admissible extension since any argument in the odd-length cycle is controversial w.r.t. it.

Corollary 1

Given $\textit{AF}=\langle \textit{Arg}, \rightharpoonup \rangle$ and a s-admissible extension $\varDelta \subseteq \textit{Arg}$. If an argument $A\in \textit{Arg}$ is in an odd-length cycle, then $A\notin \varDelta$.

4 Solid Semantics

We start by developing some solid semantics based on solid admissibility in this section. These semantics strengthen Dung’s semantics in the sense that for a solid extension $\varDelta$, there exists a Dung’s extension $\varGamma$ such that $\varDelta$ is a subset of $\varGamma$. Moreover, we will show connections among solid extensions and compare solid extensions with Dung’s extensions. These solid semantics can be characterized by propositional formulas.

Definition 12 (Solid semantics)

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. For any $\varDelta \subseteq \textit{Arg}$, $(i)$ $\varDelta$ is a s-complete extension iff $\varDelta \subseteq n(\varDelta )$ and $\varDelta =d_{s}(\varDelta )$; $(ii)$ $\varDelta$ is a s-preferred extension iff $\varDelta$ is a maximal s-admissible extension; $(iii)$ $\varDelta$ is a s-stable extension iff $\varDelta = n(\varDelta )$ and for any argument $A\notin \varDelta$, $\overline{A}\subseteq \varDelta$; $(iv)$ $\varDelta$ is the s-grounded extension iff $\varDelta$ is the least fixed point of $d_{s}$.

Here are some comments for the definition above. We define these solid extensions by using the neutrality function n and the solid defense function $d_{s}$, like Dung’s extension in Definition 5. A s-complete extension is a fixed point of $d_{s}$ which is also a conflict-free extension. In other words, a s-complete extension is a s-admissible extension that contains all arguments s-defended by it. A s-preferred extension has maximality and solid admissibility. A set of arguments $\varDelta$ is a s-stable extension whenever it is a fixed point of n and all attackers of any argument outside of $\varDelta$ are in $\varDelta$. The s-grounded extension is unique.

Fig. 2.

Four argumentation frameworks

Next we present some connections among solid semantics. A s-stable extension is a s-preferred extension. And a s-preferred extension is a s-complete extension. Besides, the s-grounded extension is the least s-complete extension.

Theorem 4

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. For any $\varDelta \subseteq \textit{Arg}$, $(i)$ if $\varDelta$ is a s-preferred extension, then $\varDelta$ is a s-complete extension, but not vice versa; $(ii)$ if $\varDelta$ is a s-stable extension, then $\varDelta$ is a s-preferred extension, but not vice versa; $(iii)$ The s-grounded extension is the least s-complete extension.

Proof

Take a set of arguments $\varDelta \subseteq \textit{Arg}$. ($i$) Suppose that $\varDelta$ is a s-preferred extension. Then according to Definition 11 and the second item of Definition 12, we have $\varDelta \subseteq n(\varDelta )$ and $\varDelta \subseteq d_{s}(\varDelta )$. It suffices to show $d_{s}(\varDelta )\subseteq \varDelta$. Suppose that an argument $C\in d_{s}(\varDelta )$, namely, $\varDelta$ s-defends C. Then $\varDelta \,\cup \,\{C\}$ is s-admissible by Lemma 3. Suppose for the sake of a contradiction that $C\notin \varDelta$. This contradicts the maximality of $\varDelta$. Hence, $C\in \varDelta$. It follows that $\varDelta =d_{s}(\varDelta )$. So $\varDelta$ is s-complete extension by the first item of Definition 12. Next we show the converse does not hold. Consider $\textit{AF}_2$ in Fig. 2b. Take two sets $\varDelta =\{A\}$ and $\varGamma =\{A,C\}$. Then $\varDelta$ is a s-complete extension. However, we can see that $\varGamma$ is s-admissible and $\varDelta \subset \varGamma$. Hence, $\varDelta$ is not a maximal s-admissible extension (i.e., not s-preferred).

($ii$) Suppose that $\varDelta$ is a s-stable extension. Then applying the third item of Definition 12 yields $\varDelta =n(\varDelta )$. We next verify $\varDelta \subseteq d_{s}(\varDelta )$. Assume that an argument $C\in \varDelta$. To demonstrate $C\in d_{s}(\varDelta )$, suppose that an argument $B\in \textit{Arg}$ attacks C. Then $B\notin \varDelta$ as $\varDelta$ is conflict-free. It follows that $B\notin n(\varDelta )$. Thus, $\varDelta \rightharpoonup B$ by Definition 4. Again, using the third item of Definition 12 yields $\overline{B}\subseteq \varDelta$. Hence, $\varDelta$ s-defends C by Definition 8. Then $C\in d_{s}(\varDelta )$ by Definition 9. It follows that $\varDelta$ is a s-admissible extension. At last, we prove the maximality of $\varDelta$. Suppose for the sake of a contradiction that there exists a s-admissible set $\varGamma$ such that $\varDelta \subset \varGamma$. Then there exists an argument $C'$ such that $C'\in \varGamma$ but $C'\notin \varDelta$. Immediately, we have $\varDelta \rightharpoonup C'$ as $C'\notin \varDelta$. This contradicts conflict-freeness of $\varGamma$. So $\varDelta$ is a maximal s-admissible extension. To show that the converse does not hold, let us consider $\textit{AF}_3$ in Fig. 2c. Let $\varDelta =\{A,C\}$. Then $\varDelta$ is a s-preferred extension. We can see that $D\notin \varDelta$. But the attackers of D are not fully included in $\varDelta$. Hence, $\varDelta$ is not a s-stable extension.

($iii$) As $\textit{Arg}$ is finite, the least fixed point of $d_{s}$ (i.e., the grounded extension) can be computed as $d^{i_{min}}_{s}(\emptyset )$ where $i_{min}$ is the least integer i such that $d^{i+1}_{s}(\emptyset )=d^{i}_{s}(\emptyset )$. Moreover, $d^{i}_{s}(\emptyset )$ is s-admissible by induction on natural number i and Lemma 3. Hence, the least fixed point of $d_{s}$ is a s-complete extension according to the first item of Definition 12. Thus, the least fixed point of $d_{s}$ is a subset of any s-complete extension as as any s-complete extension is a fixed point of $d_{s}$.

Recall that any s-admissible extension contains no argument that is indirectly attacked and indirectly defended by some argument. Since we have showed that the solid extensions in Definition 12 are also s-admissible, they contain no such argument either, according to Theorem 3. Next we present an interesting property that the set of arguments outside of a s-stable extension is conflict-free.

Proposition 2

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. For any $\varDelta \subseteq \textit{Arg}$, if $\varDelta$ is a s-stable extension, then $\textit{Arg}\setminus \varDelta$ is a conflict-free extension.

In the following, we show that solid semantics can be interpreted as a class of strengthenings of Dung’s semantics. In other words, for any solid extension, there exists a Dung’s counterpart such that it is a superset of the solid extension.

Proposition 3

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. For any $\varDelta \subseteq \textit{Arg}$, ($i$) if $\varDelta$ is a s-self-defending extensions, then $\varDelta$ is self-defending extension; ($ii$) if $\varDelta$ is a s-admissible extension, then $\varDelta$ is an admissible extension; ($iii$) if $\varDelta$ is a s-complete extension, then there exists a complete extension $\varGamma$ such that $\varDelta \subseteq \varGamma$; ($iv$) if $\varDelta$ is a s-preferred extension, then there exists a preferred extension $\varGamma$ such that $\varDelta \subseteq \varGamma$; ($v$) if $\varDelta$ is a s-stable extension, then $\varDelta$ is a stable extension; ($vi$) if $\varDelta$ is the s-grounded extension, then $\varDelta$ is a subset of the grounded extension.

Proof

Take a set of arguments $\varDelta \subseteq \textit{Arg}$. ($i$) Suppose $\varDelta$ is s-self-defending. Then from Definition 10 we have $\varDelta \subseteq d_{s}(\varDelta )$. We also have $d_{s}(\varDelta )\subseteq d(\varDelta )$ by Proposition 1. It follows that $\varDelta \subseteq d(\varDelta )$. ($ii$) This item is easily obtained from the first item. ($iii$) Suppose that $\varDelta$ is s-complete. Then $\varDelta$ is s-admissible. Hence, $\varDelta$ is admissible. Therefore, there exists a preferred extension $\varGamma$ such that $\varDelta \subseteq \varGamma$ by Theorem 1. Besides, $\varGamma$ is also a complete extension by Theorem 1. ($iv$) Suppose that $\varDelta$ is s-preferred. Then $\varDelta$ is s-admissible. Hence, $\varDelta$ is admissible. Therefore, there exists a preferred extension $\varGamma$ such that $\varDelta \subseteq \varGamma$ by Theorem 1. ($v$) This item follows from the third item of Definition 12 and the fifth item of Definition 5. ($vi$) Recall that the least fixed point of $d_{s}$ (resp., d) is found by iterating the application of $d_{s}$ (resp., d) from the empty set. Besides, we have $d^{i}_{s}(\emptyset )\subseteq d^{i}(\emptyset )$ by induction on natural number i. And we also have $d^{i}(\emptyset )\subseteq d^{i+1}(\emptyset )$. Therefore, during the process of iteration, the s-grounded extension will be found no later than the grounded extension. So the former is a subset of the latter.

Fig. 3.

An overview of solid semantics and Dung’s semantics. We can see that for any solid extension, there exists a superset that is also a Dung’s extension.

To gain a better understanding of differences between Dung’s extensions and solid extensions, we provide two examples. Example 3 shows that if a set of arguments is a self-defending (resp., a complete, a preferred, a stable, or the grounded) extension, then it may fail to be a s-self-defending (resp., s-complete, a s-preferred, a s-stable or the s-grounded) extension. Example 4 shows that if a set of arguments is a s-complete (resp., a s-preferred or the s-grounded) extension, then it may fail to be a complete (resp., a preferred or the grounded) extension. However, it is easy to see that a s-stable extension must be a stable extension by Definition 5 and Definition 12. We illustrate in Fig. 3 an overview of how the solid extensions are related to each other and Dung’s extensions.

Example 3

Let us consider $\textit{AF}_1$ in Fig. 2a. Take a set of arguments $\varDelta =\{A,C\}$. Then it is easy to see that $\varDelta$ is a self-defending extension but not a s-self-defending extension. Besides, $\varDelta$ is a complete, a preferred, a stable and the grounded extension. However, according to Theorem 3, C can not be included in any s-admissible extension, since B is controversial w.r.t. C. Hence, $\varDelta$ is neither a s-complete, a s-preferred, a s-stable, nor the s-grounded extension.

Example 4

Let us consider $\textit{AF}_4$ in Fig. 2d. Let $\varDelta =\{A,D,F\}$. Then $\varDelta$ is a s-complete, a s-preferred and the s-grounded extension. However, we can see that argument C which is defended by $\varDelta$ is not included in $\varDelta$. Hence, $\varDelta$ is neither a complete, a preferred, a stable nor the grounded extension.

Before moving to the next section, we characterize the solid semantics in terms of integrity constraints (i.e., propositional formulas) expressed in $\mathcal {L}_{\textit{AF}}$. We say that a set of arguments is a s-reinstating extension iff each argument s-defended by this set belongs to this set. It is worth mentioning that the integrity constrain of conflict-freeness [5] is $\mathrm IC_{\textit{CF}}\equiv \bigwedge _{{A,B\in \textit{Arg}}\atop {A\rightharpoonup B}}\big (\lnot A\vee \lnot B\big )$.

Proposition 4

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. $\varDelta \subseteq \textit{Arg}$ is a s-self-defending, s-reinstating, s-stable, s-admissible, s-complete, s-preferred and s-grounded extension, resp. iff

• $\varDelta \vDash \mathrm IC_{\textit{SS}}$ where $\mathrm IC_{\textit{SS}}\equiv \bigwedge \limits _{C\in \textit{Arg}}\big [ C\rightarrow \bigwedge \limits _{{B\in \textit{Arg}}\atop {B\rightharpoonup C}}\big ( (\bigvee \limits _{{A\in \textit{Arg}}\atop {A\rightharpoonup B}}A)\wedge (\bigwedge \limits _{{A\in \textit{Arg}}\atop {A\rightharpoonup B}}A)\big ) \big ]$;

• $\varDelta \vDash \mathrm IC_{\textit{SR}}$ where $\mathrm IC_{\textit{SR}}\equiv \bigwedge \limits _{C\in \textit{Arg}}\big [\bigwedge \limits _{{B\in \textit{Arg}}\atop {B\rightharpoonup C}}\big ( ( \bigvee \limits _{{A\in \textit{Arg}}\atop {A\rightharpoonup B}}A) \wedge (\bigwedge \limits _{{A\in \textit{Arg}}\atop {A\rightharpoonup B}}A) \big )\rightarrow C\big ]$;

• $\varDelta \vDash \mathrm IC_{\textit{SST}}$ where $\mathrm IC_{\textit{SST}}\equiv \bigwedge \limits _{B\in \textit{Arg}}\big [\big (B\leftrightarrow \bigwedge \limits _{{A\in \textit{Arg}}\atop {A\rightharpoonup B}}\lnot A\big )\wedge \big (\lnot B\rightarrow \bigwedge \limits _{{A\in \textit{Arg}}\atop {A\rightharpoonup B}}A\big )\big ]$;

• $\varDelta \vDash \mathrm IC_{\textit{SA}}$ where $\mathrm IC_{\textit{SA}}\equiv \mathrm IC_{\textit{CF}}\wedge \mathrm IC_{\textit{SS}}$;

• $\varDelta \vDash \mathrm IC_{\textit{SC}}$ where $\mathrm IC_{\textit{SC}}\equiv \mathrm IC_{\textit{SA}}\wedge \mathrm IC_{\textit{SR}}$;

• $\varDelta$ is a maximal model of $\mathrm IC_{\textit{SA}}$;

• $\varDelta$ is the least model of $\mathrm IC_{\textit{SC}}$.

5 Preservation of Solid Semantic Properties

Quota rules are natural rules to be considered when contemplating mechanisms to perform aggregation. They have low computational complexity and satisfy some appealing properties. For instance, they are monotonic and strategy-proof as studied in judgment aggregation. The problem of aggregating extensions submitted by several agents on a given AF is an important and interesting topic. Especially, no quota rule preserves Dung’s admissibility for all AFs [10]. So we wonder what admissible extensions can be preserved. In light of the integrity constraints for solid semantics in Proposition 4, we can investigate the preservation results for solid semantic properties by using the model defined in Sect. 2.2 and quota rules. We analyze that in the scenarios where a set of agents each provides us with a set of arguments that satisfies a specific solid semantics, under what circumstance such solid semantic property will be preserved under aggregation.

5.1 Preserving Solid Self-defence and Solid Admissibility

We start by exploring the circumstances where the s-self-defending property can be preserved. Theorem 5 presents a positive result that every quota rule preserves the s-self-defending property for all AFs.

Theorem 5

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. Every quota rule $F_q$ for n agents preserves the property of being s-self-defending for $\textit{AF}$.

Proof

Recall from Proposition 4 that $\mathrm IC_{\textit{SS}}$ is a conjunction of formulas of the form: $\varphi \equiv C\rightarrow \bigwedge _{{B\in \textit{Arg}}\atop {B\rightharpoonup C}}\big ( (\bigvee _{{A\in \textit{Arg}}\atop {A\rightharpoonup B}}A)\wedge (\bigwedge _{{A\in \textit{Arg}}\atop {A\rightharpoonup B}}A)\big )$. Note that this formula is indexed by argument C, let us study the preservation of such formula. If C is an initial argument (an argument not receiving attacks in $\textit{AF}$), then $\varphi \equiv C \rightarrow \top$, which can be simplified to $\varphi \equiv \top$. It follows that in this case, every quota rule $F_{q}$ for n agents preserves $\mathrm Mod(\varphi )$ for $\textit{AF}$. If C has at least one attacker, then we need to take into account the following two cases. The first case is that there exists an attacker B of C such that B is an initial argument. In this case, we have $\varphi \equiv \lnot C$ (this can be easily obtained as $\bigwedge _{{B\in \textit{Arg}} \atop {B\rightharpoonup C}}\big ((\bigvee _{{A\in \textit{Arg}} \atop {A\rightharpoonup B}}A)\wedge (\bigwedge _{{A\in \textit{Arg}} \atop {A\rightharpoonup B}}A)\big )$ is always false if there is an attacker B of C that has no attacker). Then, according to Lemma 1, every quota rule $F_{q}$ for n agents preserves $\mathrm Mod(\varphi )$ for $\textit{AF}$, as the inequality $q\cdot (1-0)>n\cdot (1-1)-0$, which can be simplified to $q>0$, always holds. The other case is that all attackers of C are not initial. Then, we let $\{B_1,\ldots ,B_p\}$ be the set of attackers of C, and let $\{A^{i}_{1},\ldots ,A^{i}_{\ell _{i}}\}$ be the set of attackers of $B_i$ where $1\leqslant i\leqslant p$ and $\ell _{i}=|\overline{B_{i}}|$. Then $\varphi$ can be rewritten as follows: $\varphi \equiv \big ( (\lnot C\vee A_{1}^{1})\wedge \cdots \wedge (\lnot C\vee A_{\ell _{1}}^{1}) \big )\wedge \cdots \wedge \big ( (\lnot C\vee A_{1}^{p})\wedge \cdots \wedge (\lnot C\vee A_{\ell _{p}}^{p}) \big )$. Thus, in this case, $\varphi$ is a conjunction of clauses with one negative literal and one positive literal. We take one such clause $\psi = \lnot C \vee A^i_{l_j}$. According to Lemma 1, every quota rule $F_q$ preserves $\psi$ in this case. Thus, every quota rule for n agents preserves $\varphi$. It follows that all clauses of $\mathrm IC_{\textit{SS}}$ can be preserved by all quota rules. We conclude that every quota rule $F_q$ for n agents preserves the property $\mathrm Mod(\mathrm IC_{\textit{SS}})$ for $\textit{AF}$ by Lemma 2.

Obviously, the strict majority rule, the nomination rule and the unanimity rule as specific quota rules, will preserve s-self-defense. Note that different from the property of being s-self-defending, Dung’s self-defense cannot be preserved by some quota rule. One example is the strict majority rule [10].

We now turn to consider the preservation of solid admissibility. Recall that a set of arguments is s-admissible if it satisfies conflict-freeness and s-self-defense. With Lemma 2, we know that if an aggregation rule preserves both conflict-freeness and s-self-defense, then solid admissibility will be preserved by such rule. The following proposition restates a result for conflict-freeness in [10].

Proposition 5

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. A quota rule $F_q$ for n agents preserves conflict-freeness for $\textit{AF}$ if $q>\frac{n}{2}$.

Theorem 6

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. Any quota rule $F_{q}$ for n agents with $q>\frac{n}{2}$ preserves solid admissibility for $\textit{AF}$.

Applying Theorem 5 and Proposition 5 immediately yields Theorem 6, which shows that any quota rule higher than or equal to the strict majority rule can preserve solid admissibility for arbitrary AF. Recall that no quota rule preserves Dung’s admissibility for all AFs [10]. Thus, we have obtained a positive result, i.e., we know that there exist quota rules that preserve solid admissibility for all AFs. Notably, Chen [9] uses a different model to show that the majority rule guarantees Dung’s admissibility on profiles of solid admissible sets during aggregation of extensions. Theorem 6 entails this result, but not vice versa.

5.2 Preserving Solid Reinstatement and Solid Completeness

We turn to explore solid reinstatement. For convenience, we provide three notations. Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. Firstly, for any $C\in \textit{Arg}$, $\mathscr {D}_{\textit{AF}}(C)$ denotes the set of C’s defenders, i.e., $\mathscr {D}_{\textit{AF}}(C)=\{A\in \textit{Arg}\mid A \text { is a defender of } C\}$. Secondly, we let $\mathcal {E}(\textit{AF})$ denote the set of arguments which are not initial arguments and whose attackers are not initial arguments either, i.e., $\mathcal {E}({\textit{AF}})=\big \{C\in \textit{Arg}\mid \overline{C}\ne \emptyset \text { and for any } B\in \overline{C},\ \overline{B}\ne \emptyset \}$. Thirdly, we let $\mathcal {M}(\textit{AF})$ denote the maximal number of the defenders of an argument in $\mathcal {E}({\textit{AF}})$, i.e., $\mathcal {M}(\textit{AF})=\max \limits _{C\in \mathcal {E}(\textit{AF})}|\mathscr {D}_{\textit{AF}}(C)|$.

Theorem 7

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. A quota rule $F_q$ for n agents preserves the property of being s-reinstating for $\textit{AF}$ if $q\cdot (\mathcal {M}(\textit{AF})-1)>n\cdot (\mathcal {M}(\textit{AF})-1)-1$.

Proof

Assume that $q\cdot (\mathcal {M}(\textit{AF})-1)>n\cdot (\mathcal {M}(\textit{AF})-1)-1$. $\mathrm IC_{\textit{SR}}$ is a conjunction of formulas of the form of $\varphi \equiv \bigwedge _{{B\in \textit{Arg}}\atop {B\rightharpoonup C}}\big ( ( \bigvee _{{A\in \textit{Arg}}\atop {A\rightharpoonup B}}A) \wedge (\bigwedge _{{A\in \textit{Arg}}\atop {A\rightharpoonup B}}A) \big )\rightarrow C$ from Proposition 4. Take an argument $C\in \textit{Arg}$. If C is an initial argument, then $\varphi \equiv C$, which can be regarded as a 1-clause (with 1 positive literal and no negative literal). Applying Lemma 1, any quota rule $F_{q}$ for n agents preserves $\mathrm Mod(\varphi )$ for $\textit{AF}$ in this case, as the inequality $q\cdot (0-1)>n\cdot (0-1)-1$ (which can be simplified to $q<n+1$) always holds.

If C is not an initial argument, then we need to consider two cases. The first case is that there exists a C’s attacker B such that B is an initial argument, then $\varphi \equiv \top$. It follows that in this case, any quota rule $F_{q}$ for n agents preserves $\mathrm Mod(\varphi )$ for $\textit{AF}$. The second case is that any C’s attacker B is not an initial argument. Then we can let $\{B_1,\cdots ,B_p\}$ be the set of C’s attackers, and let $\{A^{i}_{1},\cdots ,A^{i}_{\ell _{i}}\}$ be the set of $B_i$’s attackers where $1\leqslant i\leqslant p$ and $\ell _{i}=|\overline{B_{i}}|$. we can reformulate $\varphi$ as follows: $\varphi \equiv \big ((\lnot A^1_1\vee \cdots \vee \lnot A^1_{\ell _1} )\vee \cdots \vee (\lnot A^p_1\vee \cdots \vee \lnot A^p_{\ell _p} )\big )\vee C.$

Hence, $\varphi$ is a $(\sum \limits _{i=1}^{p}\ell _i+1)$-clause (with $\sum \limits _{i=1}^{p}\ell _i$ negative literals and 1 positive literal). By Lemma 1, a quota rule $F_{q}$ for n agents preserves $\mathrm Mod(\varphi )$ if the following inequality holds: $q\cdot (\sum _{i=1}^{p}\ell _i-1)>n\cdot (\sum _{i=1}^{p}\ell _i-1)-1$. Doing so becomes harder as $\sum _{i=1}^{p}\ell _i$ increases. Note that $C\in \mathcal {E}({\textit{AF}})$ in this case and $\sum _{i=1}^{p}\ell _i$ is the number of the defenders of C. Recall that the maximal number of defenders of an argument in $\mathcal {E}({\textit{AF}})$ is $\mathcal {M}(\textit{AF})$. Hence, the maximal value of $\sum _{i=1}^{p}\ell _i$ is $\mathcal {M}(\textit{AF})$. Thus, by Lemma 1, a quota rule $F_{q}$ for n agents preserve $\mathrm Mod(\varphi )$ for $\textit{AF}$ in this case, if the inequality holds for the maximal value of $\sum _{i=1}^{p}\ell _i$ (i.e., $\mathcal {M}(\textit{AF}))$. As we have assumed that $q\cdot (\mathcal {M}(\textit{AF})-1)>n\cdot (\mathcal {M}(\textit{AF})-1)-1$, it follows that a quota rule $F_{q}$ for n agents preserves $\mathrm Mod(\varphi )$ for $\textit{AF}$. Finally, using Lemma 2, we can conclude that a quota rule $F_q$ for n agents preserves $\mathrm Mod(\mathrm IC_{\textit{SR}})$ for $\textit{AF}$ if $q\cdot (\mathcal {M}(\textit{AF})-1)>n\cdot (\mathcal {M}(\textit{AF})-1)-1$.

Recall that the unanimity rule is a quota rule $F_{q}$ for n agents with $q=n$. It is easy to see that the inequality $n\cdot (\mathcal {M}(\textit{AF})-1)>n\cdot (\mathcal {M}(\textit{AF})-1)-1$ always holds. Then according to Theorem 7, the unanimity rule preserves the property of being s-reinstating for any AF. The theorem below for s-completeness is a direct consequence of Theorem 6 and Theorem 7.

Theorem 8

Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$. A quota rule $F_q$ for n agents preserves s-completeness for $\textit{AF}$ if $q>\frac{n}{2}$ and $q\cdot (\mathcal {M}(\textit{AF})-1)>n\cdot (\mathcal {M}(\textit{AF})-1)-1$.

5.3 Preserving Solid Groundedness, Solid Preferredness and Solid Stability

As the s-grounded extension is unique in any AF, any quota rule preserves s-groundedness for any AF. We say that a property $\sigma$ is inclusion maximal if for any $\varDelta _{1},\ \varDelta _{2}\in \sigma$, if $\varDelta _{1}\subseteq \varDelta _{2}$ then $\varDelta _{1}=\varDelta _{2}$. It is easy to see that both the solid preferredness and solid stability are inclusion maximal. Hence, we can investigate these two properties together. Given $\textit{AF}=\langle \textit{Arg},\rightharpoonup \rangle$, let $\sigma$ be an inclusion maximal property of extensions such that $|\sigma |\geqslant 2$, and let n be the number of agents. If n is even, then no quota rule preserves $\sigma$ for $\textit{AF}$. If n is odd, then no quota rule different from the strict majority rule preserves $\sigma$ for $\textit{AF}$. Such results are highly analogous to Theorem 15 in [10]. We omit the proof for this reason.

6 Related Work

Various notions of admissibility are proposed since Dung’s admissibility was introduced in [12]. Baroni and Giacomin introduce the notion of strong admissibility [2] which is stronger than Dung’s admissibility. It captures the idea that any argument in a strongly admissible set neither defend itself nor involve in its own defense. Grossi and Modgil propose Graded admissibility [14], whereby Dung’s admissibility can be strengthened or weakened by parameterizing the numbers of attackers and defenders. Chen [8] proposes concrete admissibility. Differently from solid admissibility, concrete admissibility does not require the existence of defenders, although both of them require containing all defenders. Prudent semantics [11] is another semantics that aims at dealing with controversial arguments. Whenever an argument A is controversial w.r.t. an argument B, both prudent semantics and solid semantics can prevent A and B from occurring in the same extension. But there is a difference between these two semantics: both A and B can occur in a prudent extension independently, however, B is excluded from any s-admissible extension, while A might occur in some s-admissible extension (e.g., A is an initial argument).

7 Conclusion and Future Work

This paper mainly makes contributions to the field of abstract argumentation theory. To address the problems observed in ranking-based argumentation and controversial arguments, we develop solid semantics by strengthening Dung’s semantics. By applying the technique in [5], we capture solid semantics by using propositional formulas. Finally, in virtue of these formulas, we aggregate solid extensions by using quota rules and obtain positive preservation results.

Recall that solid defense requires that all the defenders of an argument C are included in a set of arguments $\varDelta$. It would be interesting to characterize the idea that any percent of the defenders of C are included in a set of arguments $\varDelta$. Moreover, we can also allow a part of attackers to be not attacked. For example, we can try to capture the idea that $\varDelta$ defends C if more than fifty percent of C’s attackers have more than fifty percent of their attackers in $\varDelta$ (i.e., if the majority of C’s attackers have the majority of their attackers in $\varDelta$). Future work could focus on introducing proportionality to the defense of arguments of abstract argumentation. Furthermore, the complexity of reasoning tasks involving solid semantics should be studied.