Dung's Argumentation Framework

Question

Given an extension of Dung's argumentation framework, and all definitions therein, let $$AF_1 = \langle A,Def \rangle$$ be a framework where $$A = \{ A, B, C, D, E, F, G, H \}$$ and $$Def = \{A def G, D def C, D def E, C def F, F def C, E def F, F def E\}$$ where $def$ is the binary defeats relation. The set $\{A, B, D, F, H\}$ is given as the unique preferred and stable extension of $AF_1$. It is also given as a grounded extension, and I don't understand how/why. Why can't $\{A\}$, $\{D\}$, $\{B\}$, or $\{H\}$ be grounded extensions?

Answer 1

The grounded extension (there is always exactly one) is sometimes defined as least fixed point of the characteristic function. The characteristic function $charF_F(X)$ delivers all arguments defended by $X$. Since the empty set defends itself, the grounded extension is well-defined.

In your case arguments $A$, $B$, $D$ and $H$ do not have any attackers. Which means that $charF_F(\emptyset)=\{A,B,D,H\}$. The set $\{A,B,D,H\}$ now attacks arguments $C,E,G$. Argument $F$ is attacked only by $C$ and $E$ and hence defended by $\{A,B,D,H\}$. Then $charF_F(\{A,B,D,H\})=\{A,B,D,F,H\}$. Finally $\{A,B,D,F,H\}$ attacks all other arguments (i.e. $C,E,G$) and thus is a fixed point of the characteristic function, by construction even the least fixed point = the grounded extension, and in this case the only preferred and the only stable extension.

If grounded and preferred semantics coincide, then also complete semantics yields the same unique extension. As to your question, maybe interesting, the sets $\{A\}$, $\{B\}$, $\{D\}$, $\{H\}$ (and by the way also $\{F\}$ and all combinations of these sets) are admissible, i.e. defend themselves.