An abstract argumentation framework $(AF)$ is a pair $(A,D)$, where $A$ is a set of $arguments$ and $D \subseteq A
\times A$ is a binary relation of $defeat$.
We say that $A$ strictly defeats $B$ if $A$ defeats $B$ while $B$ does
not defeat $A$. A semantics for $AFs$ returns sets of arguments called $extensions$, which are internally coherent
and defend themselves against attacks.
Aspic+ Extensions
Let $\left(A,D\right)$ be an AF. $\forall X \in A, X$ is acceptable with respect to some $S \subseteq A$
iff $\forall Y$ s. t. $(Y,X) \in D \Rightarrow \exists Z \in S$ s. t. $(Z,Y) \in D$.
Let $S \subseteq A$ be conflict free , i.e. there are no $A,B \in S$ such that $(A,B) \in D$. Then:
S is an admissible extension iff $X \in S$ implies $X$ is acceptable w.
r. t. $S$;
S is an complete extension iff $X \in S$ whenever $X$ is acceptable w.
r. t. $S$;
S is a preferred extension iff it is a set inclusion maximal complete
extension;
S is the grounded extension iff it is the set inclusion minimal
complete extension;
S is a stable extension iff it is preferred and $\forall Y \notin S,
\exists X \in S$ s. t. $(X,Y) \in D$.