$E$; if no such argument exists, return $E$.

2.  remove from $AF$ all (new) arguments in $E$ and all arguments attacked by them (together with all adjacent attacks) and continue with Step 1.

\[prop:dung-grd\]

What we have described above forms a family of the extension–based semantics. However, there exist also labeling–based ones [@CaminadaG09; @article:semintro]. Instead of computing sets of accepted arguments, they generate labelings, i.e. total functions $Lab: A \rightarrow \{in, out, undec\}$. Although we will not recall them here, we would like to draw the attention to the fact that for every extension we can obtain an appropriate labeling and vice versa. This property is particularly important as it does not fully carry over to the ADF setting.

Finally, we would like to recall several important lemmas and theorems from the original paper on AFs [@article:dung].

[**Dung’s Fundamental Lemma**]{} Let $E$ be an admissible extension, $a$ and $b$ two arguments defended by $E$. Then $E' = E \cup \{a\}$ is admissible and $b$ is defended by $E'$.

Every stable extension is a preferred extension, but not vice versa. Every preferred extension is a complete extension, but not vice versa. The grounded extension is the least w.r.t. set inclusion complete 