maximal (w.r.t. set inclusion), with $S+ := \{b | \exists a \in AR: (a, b) \in Attacks\}$.

To introduce the definitions of *stage2* and *CF2* semantics, let us first define the notion of *strongly connected components* as follows.  [@baroni2018abstract]. In the context of an argumentation framework $AF$, the set of strongly connected components of $AF$ is denoted by $SCCS_{AF}$. For example, the graph of the argumentation framework $(\{a, b, c\}, \{(a, b), (b, a), (b, c)\})$ has the SCCS $\{a, b\}$ and $\{ c \}$.\
\

 [@baroni2018abstract]\[cf-2\] Given an argumentation framework $AF = (AR, Attacks)$, a set of arguments $S \subseteq AR$ is an extension of CF2 semantics ($\sigma_{CF2}$) iff:

-   if $|SCCS_{AF}| = 1$, $S$ is a maximal conflict-free extension of $AF$

-   otherwise, $\forall Args \in SCSS_{AF} (S \cap Args) \in \sigma_{CF2}(AF\downarrow_{UP_{AF}(Args,S)}) \;$, with\
    $AF\downarrow_{UP_{AF}(Args,S)} = (UP, S \cap UP \times UP)$, the *restriction of $AF$ to $UP$* [@BARONI2005162] and for any $S, Args \subset AR, UP_{AF}(Args, S) = \{a \in Args | \nexists b \in S \\ Args: (b, a) \in Attacks \}$.

 [@baroni2018abstract]\[stage-2\] Given an argumentation framework $AF = (AR, Attacks)$, a set of arguments $S \subseteq AR$ is an extensions of stage2 semantics ($\sigma_{stage2}$), iff:

