## 3 Background  
### Abstract Argumentation Framework and Stable Extensions  
An abstract argumentation framework consists of a set of arguments and binary attack relations between them. In other words, each AF corresponds to a directed graph where each argument in the framework represents a vertex, and each attack relation represents an edge.  
**Definition 1**.: _(Abstract Argumentation Framework): Given an abstract argumentation framework AF, it is a tuple \(\Gamma=\langle A,R\rangle\), where \(A\) is a set of arguments, and \(R\subseteq A\times A\) is a binary relation representing the attack relations on the set \(A\)._  
In an abstract argumentation framework \(AF\), if \((a,b)\in R\), it means that argument \(a\) attacks argument \(b\) (or \(b\) is attacked by \(a\)), and this can also be denoted as \(a\to b\). Let \(\{a\}^{+}\)denote the subset of arguments attacked by \(a\) and \(\{a\}^{-}\) denote the subset of arguments that attack \(a\). Given a set of arguments \(X\subseteq A\):  
\[X^{-}=\{b\in A:\exists a\in X\,s.t.\,b\in\{x\}^{-}\},\]  
\[X^{+}=\{b\in A:\exists a\in X\,s.t.\,b\in\{x\}^{+}\}.\]  
**Definition 2**.: _(Conflict-Free Set): Given an abstract argumentation framework \(\Gamma=\langle A,R\rangle\), a subset \(X\subseteq A\) is a conflict-free set iff \(\forall a,b\in X\), \(\
exists(a\to b\lor b\to a)\), meaning there is no direct attack relationship between \(a\) and \(b\). The set of conflict-free sets is denoted by \(X_{CF}(\Gamma)\)._  
**Definition 3**.: _(Acceptability of Arguments): Given an abstract argumentation framework \(\Gamma=\langle A,R\rangle\), where \(X\in X_{CF}(\Gamma)\), an argument \(a\in A\) is acceptable with respect to \(X\) iff \(\
exists(a\to X)\wedge(X\to a)\) and, \(\forall\ b\in A\) (\(b\
ot\in X\)), if \(b\to a\), then \(X\to b\) must hold._  
**Definition 4**.: _(Admissible Set): Given an abstract argumentation framework \(\Gamma=\langle A,R\rangle\), a subset \(X\subseteq A\) is an admissible set iff \(X\in X_{CF}(\Gamma)\), and each argument in \(X\) is acceptable with respect to \(X\). The set of admissible sets is denoted by \(X_{AS}(\Gamma)\)._  
**Definition 5**.: _(Complete Extension): Given an abstract argumentation framework \(\Gamma=\langle A,R\rangle\), a set of arguments \(X\) is a complete extension iff \(X\in X_{AS}(\Gamma)\), and every argument acceptable with respect to \(X\) belongs to \(X\). The set of complete extensions is denoted by \(X_{CO}(\Gamma)\)._  
A complete extension includes all arguments that are acceptable with respect to itself and does not allow for the addition of any new acceptable arguments through extension.  
**Definition 6**.: _(Stable Extension): Given an argumentation framework \(\Gamma=\langle A,R\rangle\), a set of arguments \(X\subseteq A\) is a stable extension iff \(X\in X_{CO}(\Gamma)\) and, for every argument \(\forall a\in A\) such that \(a\
otin X\), it holds that \(X\to a\). In other words, a stable extension attacks every argument that does not belong to that extension. The set of stable extensions is denoted by \(X_{ST}(\Gamma)\)._  
Stable extensions can be obtained using a characteristic function derived from complete extensions. Each complete extension corresponds to a stable extension. However, different complete extensions may yield the same stable extension through further extension, hence \(X_{ST}(\Gamma)\subseteq X_{CO}(\Gamma)\).