Different levels of collective acceptability

Under a given semantics, and following Dung, the acceptability of an argument depends on its membership to an extension under this semantics. We consider three possible cases25:

However, these three levels seem insufficient. For example, what should be concluded in the case of two arguments $A$ and $B$ which are exi-accepted and such that $A {\mathcal{R}}B$ or $B {\mathcal{R}}A$?

So, we introduce a new definition which takes into account the situation of the argument w.r.t. its attackers. This refines the class of the exi-accepted arguments under a given semantics $S$.

Definition 19 (Cleanly-accepted argument)   Consider $A \in {\mathcal{A}}$, $A$ is cleanly-accepted if and only if $A$ belongs to at least one extension of $S$ and $\forall B \in
{\mathcal{A}}$ such that $B {\mathcal{R}}A$, $B$ does not belong to any extension of $S$.

Thus, we capture the idea that an argument will be better accepted, if its attackers are not-accepted.

Property 13   Consider $A \in {\mathcal{A}}$ and a semantics $S$ such that each extension for $S$ is conflict-free. If $A$ is uni-accepted then $A$ is cleanly-accepted. The converse is false.

The notion of cleanly-accepted argument refines the class of the exi-accepted arguments. For a semantics $S$ and an argument $A$, we have the following states:

Example 9   Consider the following argumentation system.

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There are two preferred extensions $\{D, C_2, A, G\}$ and $\{D, C_2, E,
G, I\}$. So, for the preferred semantics, the acceptability levels are the following:
  • $D$, $C_2$ and $G$ are uni-accepted,
  • $I$ is cleanly-accepted but not uni-accepted,
  • $A$ and $E$ are only-exi-accepted,
  • $B$, $C_1$, $F$, $H$ and $J$ are not-accepted.

Note that, in all the cases where there is only one extension, the first three levels of acceptability coincide26. This is the case:

Looking more closely, we can prove the following result (proof in Appendix A):

Property 14   Under the stable semantics, the class of the uni-accepted arguments coincides with the class of the cleanly-accepted arguments.

Then, using a result issued from [10,11] and reused in [8] which shows that, when there is no odd cycle, all the preferred extensions are stable27, we apply Property 14 and we obtain the following consequence:

Consequence 1   Under the preferred semantics, when there is no odd cycle, the class of the uni-accepted arguments coincides with the class of the cleanly-accepted arguments.

Finally, the exploitation of the gradual interaction-based valuations (see Section 3) allows us to define new levels of collective acceptability.

Let $v$ be a gradual valuation and let $\succeq$ be the associated preordering (partial or complete) on ${\mathcal{A}}$. This preordering can be used inside each acceptability level (for example, the level of the exi-accepted arguments) in order to identify arguments which are better accepted than others.

Example 9 (continuation) Two different gradual valuations are applied on the same graph:

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With the instance of the generic valuation proposed in [4] (see Section 3.1), we obtain the following comparisons:

\begin{displaymath}D, C_2 \succ I \succ E \succ G \succ J \succ C_1,F \succ A \succ H \succ B\end{displaymath}

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With the global valuation with tuples presented in Section 3.2, we obtain the following comparisons:

\begin{displaymath}D, C_2 \succ G \succ B \succ F, C_1\end{displaymath}

\begin{displaymath}D, C_2 \succ A \succ E\end{displaymath}

\begin{displaymath}D, C_2 \succ H \succ E\end{displaymath}

\begin{displaymath}D, C_2 \succ I\end{displaymath}

\begin{displaymath}D, C_2 \succ J\end{displaymath}

So, all the arguments belonging to a cycle are incomparable with $G$, $B$, $F$, $C_1$ and, even between them, there are few comparison results.

If we apply the preordering induced by a valuation without respecting the acceptability levels defined in this section, counter-intuitive situations may happen. In Example 9, we obtain:

These counter-intuitive situations illustrate the difference between the acceptability definition and the valuation definitions (even if both use the interaction between arguments, they do not use it in the same way).

Marie-Christine Lagasquie 2005-02-04