Particular cases leading to compatibility

In the context of an argumentation system with a finite relation
without cycles^{29},
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.

- , 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}.
*

- , , (exi, uni, cleanly) accepted for iff well-defended for .
- If is (exi, uni, cleanly) accepted for then is well-defended for (the converse is false).
- 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:

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

This theorem is a direct consequence of the following lemma:

- 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