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

3. There are 4 strongly connected components: $\{b,c\}$, $\{a\}, \{d\}$, and $\{e\}$. We check if the following sets are CF2-extensions:

   • $\{c, d\}$
     • There are 2 argument component defeated by $\{c,d\}$: $\{a\}$ and $\{e\}$.
     • We verify that:
       • $\mathcal{F}\vert_{\{b,c\} \setminus \{a,e\}} = \mathcal{F}\vert_{\{b,c\}}$ . In there, there is only one strongly connected component and $\{b,c\} \cap \{c,d\} = \{c\}$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{d\} \setminus \{a,e\}} = \mathcal{F}\vert_{\{d\}}$ . In there, there is only one strongly connected component and $\{d\}$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{a\} \setminus \{a,e\}}= \mathcal{F}\vert_{\{e\} \setminus \{a,e\}} = \mathcal{F}\vert_{\emptyset}$ . In there, there is no arguments and $\{a\} \cap \{c,d\} = \{e\} \cap \{c,d\} = \emptyset$ is maximal conflict-free.
   • $\{b,d\}$
     • There are 2 argument component defeated by $\{b,d\}$: $\{a\}$ and $\{e\}$.
     • We verify that:
       • $\mathcal{F}\vert_{\{b,c\} \setminus \{a,e\}} = \mathcal{F}\vert_{\{b,c\}}$ . In there, there is only one strongly connected component and $\{b,c\} \cap \{b,d\} = \{b\}$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{d\} \setminus \{a,e\}} = \mathcal{F}\vert_{\{d\}}$ . In there, there is only one strongly connected component and $\{d\} \cap \{b,d\} =\{d\}$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{a\} \setminus \{a,e\}}= \mathcal{F}\vert_{\{e\} \setminus \{a,e\}} = \mathcal{F}\vert_{\emptyset}$ . In there, there is no arguments and $\{a\} \cap \{b,d\} = \{e\} \cap \{b,d\} = \emptyset$ is maximal conflict-free.
   • $\{c,e\}$
     • There is 1 argument component defeated by $\{c,e\}$: $\{a\}$.
     • We verify that:
       • $\mathcal{F}\vert_{\{b,c\} \setminus \{a\}} = \mathcal{F}\vert_{\{b,c\}}$ . In there, there is only one strongly connected component and $\{b,c\} \cap \{c,e\} = \{c\}$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{a\} \setminus \{a\}}= \mathcal{F}\vert_{\emptyset}$ . In there, there is no arguments and $\{a\} \cap \{c,e\} = \emptyset$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{e\} \setminus \{a\}}= \mathcal{F}\vert_{\{e\}}$ . In there, $\{e\} \cap \{c,e\} = \{e\}$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{d\} \setminus \{a\}} = \mathcal{F}\vert_{\{d\}}$ . In there, there is only one strongly connected component and $\{d\} \cap \{c,e\} = \emptyset$ is not maximal conflict-free.
     • We conclude that $\{c,e\}$ is not a CF2 extension
   • $\{b,e\}$
     • There is 1 argument component defeated by $\{b,e\}$: $\{a\}$.
     • We verify that:
       • $\mathcal{F}\vert_{\{b,c\} \setminus \{a\}} = \mathcal{F}\vert_{\{b,c\}}$ . In there, there is only one strongly connected component and $\{b,c\} \cap \{b,e\} = \{b\}$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{a\} \setminus \{a\}}= \mathcal{F}\vert_{\emptyset}$ . In there, there is no arguments and $\{a\} \cap \{b,e\} = \emptyset$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{e\} \setminus \{a\}}= \mathcal{F}\vert_{\{e\}}$ . In there, $\{e\} \cap \{b,e\} = \{e\}$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{d\} \setminus \{a\}} = \mathcal{F}\vert_{\{d\}}$ . In there, there is only one strongly connected component and $\{d\} \cap \{b,e\} = \emptyset$ is not maximal conflict-free.
     • We conclude that $\{b,e\}$ is not a CF2 extension

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 $Lab$.

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

3. There are 4 strongly connected components $\{a,b\}, \{c\}, \{d\}$, and $\{e,f,g\}$. We check if the following sets are CF2-extensions:

   • $\{a,e\}$
     • There are two component defeated by $\{a,e\}: \{c\}$ and $\{d\}$
     • We verify that:
       • $\mathcal{F}\vert_{\{a,b\} \setminus \{c,d\}}= \mathcal{F}\vert_{\{a,b\}}$ . In there, $\{a,b\} \cap \{a,e\} = \{a\}$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{c\} \setminus \{c,d\}} = \mathcal{F}\vert_{\{d\} \setminus \{c,d\}}= \mathcal{F}\vert_{\emptyset}$ . In there, there are no arguments and $\{c\} \cap \{a,e\}= \{d\} \cap \{a,e\} = \emptyset$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{e,f,g\} \setminus \{c,d\}}= \mathcal{F}\vert_{\{e,f,g\}}$ . In there, $\{e,f,g\} \cap \{a,e\} = \{e\}$ is maximal conflict-free.
   • $\{b,e\}$
     • There are two component defeated by $\{b,e\}: \{c\}$ and $\{d\}$
     • We verify that:
       • $\mathcal{F}\vert_{\{a,b\} \setminus \{c,d\}}= \mathcal{F}\vert_{\{a,b\}}$ . In there, $\{a,b\} \cap \{b,e\} = \{b\}$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{c\} \setminus \{c,d\}} = \mathcal{F}\vert_{\{d\} \setminus \{c,d\}}= \mathcal{F}\vert_{\emptyset}$ . In there, there are no arguments and $\{c\} \cap \{b,e\}= \{d\} \cap \{b,e\} = \emptyset$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{e,f,g\} \setminus \{c,d\}}= \mathcal{F}\vert_{\{e,f,g\}}$ . In there, $\{e,f,g\} \cap \{b,e\} = \{e\}$ is maximal conflict-free.
   • $\{a,g,d\}$
     • There is one component defeated by $\{a,g,d\}: \{c\}$.
     • We verify that:
       • $\mathcal{F}\vert_{\{a,b\} \setminus \{c\}}= \mathcal{F}\vert_{\{a,b\}}$ . In there, $\{a,b\} \cap \{a,g,d\} = \{a\}$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{c\} \setminus \{c\}} = \mathcal{F}\vert_{\emptyset}$ . In there, there are no arguments and $\{c\} \cap \{a,g,d\} = \emptyset$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{d\} \setminus \{c\}} = \mathcal{F}\vert_{d}$ . In there, $\{d\} \cap \{a,g,d\} = \{d\}$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{e,f,g\} \setminus \{c\}}= \mathcal{F}\vert_{\{e,f,g\}}$ . In there, there is one SCC and $\{e,f,g\} \cap \{a,g,d\} = \{g\}$ is maximal conflict-free.
   • $\{b,g,d\}$
     • There is one component defeated by $\{b,g,d\}: \{c\}$.
     • We verify that:
       • $\mathcal{F}\vert_{\{a,b\} \setminus \{c\}}= \mathcal{F}\vert_{\{a,b\}}$ . In there, $\{a,b\} \cap \{b,g,d\} = \{b\}$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{c\} \setminus \{c\}} = \mathcal{F}\vert_{\emptyset}$ . In there, there are no arguments and $\{c\} \cap \{b,g,d\} = \emptyset$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{d\} \setminus \{c\}} = \mathcal{F}\vert_{d}$ . In there, $\{d\} \cap \{b,g,d\} = \{d\}$ is maximal conflict-free.
       • $\mathcal{F}\vert_{\{e,f,g\} \setminus \{c\}}= \mathcal{F}\vert_{\{e,f,g\}}$ . In there, there is one SCC and $\{e,f,g\} \cap \{b,g,d\} = \{g\}$ is maximal conflict-free.
   • $\{a,f,d\}$
   • $\{b,f,d\}$

Part 2: Proof dialogues

Consider the argumentation graph below.