### Particular cases leading to compatibility

In the context of an argumentation system with a finite relation without cycles29, the stable and the preferred semantics provide only one extension and the levels of uni-accepted, exi-accepted, cleanly-accepted coincide.

In this context, there are at least two particular cases leading to compatibility.

First case: It deals with the global valuation with tuples.

Theorem 1   Let be the graph associated with , being an argumentation system with a finite relation without cycles and satisfying the following condition: such that
• , leaf of , only one path from to , with and the length of this path (if is even, this path is a defence branch for , else it is an attack branch),
• all the paths from to are root-dependent in ,
• , a leaf of such that belongs to a path from to .

Let be a valuation with tuples. Let be a semantics {preferred, stable}.

1. , , (exi, uni, cleanly) accepted for iff well-defended for .
2. If is (exi, uni, cleanly) accepted for then is well-defended for (the converse is false).
3. If is well-defended for and if all the branches leading to are defence branches for then is (exi, uni, cleanly) accepted for .

Note that Theorem 1 is, in general, not satisfied by a local valuation. See the following counterexample for the valuation of [4]:

The graph satisfies the condition stated in Theorem 1. The set of well-defended arguments is (so, is not well-defended). Nevertheless, is the preferred extension.

Second case: This second case concerns the generic local valuation:

Theorem 2   Let be an argumentation system with a finite relation without cycles. Let be a semantics {preferred, stable}. Let be a generic local valuation satisfying the following condition :

, (exi, uni, cleanly) accepted for iff well-defended for .

This theorem is a direct consequence of the following lemma:

Lemma 1   Let be an argumentation system with a finite relation without cycles. Let be a semantics {preferred, stable}. Let be a generic local valuation satisfying the condition .
If is exi-accepted and has only one direct attacker then .

If is not-accepted and has only one direct attacker then .

Remark: The condition stated in Theorem 2 is:

• false for the local valuation proposed by [4], as shown in the following graph:

We know that and (see Property 6). We get:

• , ,
• , , so ,
• and nevertheless .

• false for the local valuations defined with such that with (for all the functions strictly non-increasing): see the previous graph where and .

• true for the local valuations defined with (for all the functions ): if then , being the maximum of the ; and, by assumption, , , so in particular for ; so, we get:

.

Marie-Christine Lagasquie 2005-02-04