in Definition \[arg-extensions\][^5].

 [@dung1995acceptability; @DungMT07; @caminada2012semi; @caminada2007comparing; @verheij1996two; @baroni2018abstract]\[arg-extensions\]\
Given an argumentation framework $AF = (AR, Attacks)$, an *admissible* argument set $S \subseteq AR$ is a:

-   **stable extension** ($\sigma_{stable}$) of $AF$ iff $S$ attacks each argument that does not belong to $S$.

-   **preferred extension** ($\sigma_{preferred}$) of $AF$ iff $S$ is a maximal (w.r.t. set inclusion) admissible set of $AF$.

-   **complete extension** ($\sigma_{complete}$) of $AF$ iff each argument that is acceptable w.r.t. $S$ belongs to $S$.

-   **grounded extension** ($\sigma_{grounded}$) of $AF$ iff $S$ attacks each argument that does not belong to $S$.

-   **ideal extension** ($\sigma_{ideal}$) of $AF$ iff $S$ is contained in every preferred extension of $AF$.

-   **semi-stable extension** ($\sigma_{semi-stable}$) of $AF$ iff $S$ is a complete extension where $S \cup S+$ is maximal (w.r.t. set inclusion), with $S+ :=  \{b | \exists a \in AR: (a, b) \in Attacks\}$.

-   **eager extension** ($\sigma_{eager}$) of $AF$ iff $S$ is the greatest (w.r.t. set inclusion) admissible set that is a subset of each semi-stable extension.

Given an argumentation framework $AF = (AR, Attacks)$, a *conflict-free* argument set $S \subseteq AR$ is a **stage extension** ($\sigma_{stage}$) of $AF$, iff $S$ is conflict-free and $S \cup S+$ is 