it is maximal w.r.t set inclusion admissible.

-   **complete** iff it is admissible and all arguments defended by $E$ are in $E$.

-   **stable** iff it is conflict–free and for each $a\in A \setminus E$ there exists an argument $b \in E$ s.t. $(b,a) \in R$.

The **characteristic function** $F_{AF} : 2^A \rightarrow 2^A$ is defined as: $F_{AF}(E) = \{a \mid a \text{ is defended by } E \text{ in } AF\}$. The **grounded extension** is the least fixed point of $F_{AF}$.

In the context of this paper, we would also like to recall the notion of range:

Let $E^+$ be the set of arguments attacked by $E$ and $E^-$ the set of arguments that attack $E$ . $E^+ \cup E$ is the **range** of $E$.

Please note the concepts $E^+$ and the $E^-$ sets can be used to redefine defense. This idea will be partially used in creating the semantics of ADFs. Moreover, there is also an alternative way of computing the grounded extension:

The unique **grounded extension** of $AF$ is defined as the outcome $E$ of the following “algorithm”. Let us start with $E=\emptyset$:

1.  put each argument $a \in A$ which is not attacked in $AF$ into 