Part 1a: Extension-based and labelling semantics

Consider the argumentation graph $\mathcal{F} = (A,R)$ with $A = \{a,b,c,d,e\}$ and $R = \{ (b,a), (b,c), (c,b), (c,a), (a,d), (d,e)\}$.

1. Calculate stable, preferred, complete and grounded extensions.

   1. The set of preferred/stable semantics are $\{\{b,d\}, \{c,d\}\}$
   2. The complete extensions are $\{\{b,d\}, \{c,d\}, \emptyset\}$
   3. The grounded extension is $\emptyset$

2. For each extension obtained, write the equivalent labelling $Lab$.

   Extension	$\mathtt{in}(Lab)$	$\mathtt{out}(Lab)$	$\mathtt{undec}(Lab)$
   $\{b,d\}$	$\{b,d\}$	$\{c,a,e\}$	$\emptyset$
   $\{c,d\}$	$\{c,d\}$	$\{b,a,e\}$	$\emptyset$
   $\emptyset$	$\emptyset$	$\emptyset$	$\{a,b,c,d,e\}$

Part 1b: Extension-based and labelling semantics

Consider the argumentation graph below.

1. Calculate grounded, stable, preferred, semi-stable, complete and ideal extensions.
   1. The preferred/semi-stable extensions are $\{\{b\}, \{a\}\}$
   2. There are no stable extensions.
   3. The ideal/grounded extensions are $\emptyset$
   4. The complete extensions are $\{\emptyset, \{a\},\{b\} \}$
2. For each extension obtained, write the equivalent labelling $Lab$.

Part 2: Proof dialogues

Consider the argumentation graph below.