We consider the abstract framework introduced in . An argumentation system is a set of arguments and a binary relation on called an attack relation: consider and , means that attacks or is attacked by (also denoted by ).
An argumentation system is well-founded if and only if there is no infinite sequence , , ..., , ...such that and .
Here, we are not interested in the structure of the arguments and we consider an arbitrary attack relation.
Notation: defines a directed graph called the attack graph. Consider , the set is the set of the arguments attacking 6and the set is the set of the arguments attacked by 7.
A special case is the path10 from to whose length is .
The set of paths from to will be denoted by .
These two paths will be said dependent iff , such that . Otherwise they are independent.
These two paths will be said root-dependent in iff and , such that .
A cycle is isolated iff , such that and .
Two cycles and are interconnected iff such that .
We use the notions of direct and indirect attackers and defenders. The notions introduced here are inspired by related definitions first introduced in  but are not strictly equivalent12.
If the argument is an attacker (direct or indirect) of the argument , we say that attacks (or that is attacked by ). In the same way, if the argument is a defender (direct or indirect) of the argument , then defends (or is defended by ).
Note that an attacker can also be a defender (for example, if attacks which attacks , and also attacks ). In the same way, a direct attacker can be an indirect attacker (for example, if attacks which attacks which attacks , and also attacks ) and the same thing may occur for the defenders.
Note that this notion of defence is the basis of the usual notion of reinstatement ( attacks , attacks and is ``reinstated'' because of ). In this paper, reinstatement is taken into account indirectly, because the value of the argument and the possibility for selecting will be increased thanks to the presence of .
All these notions are illustrated on the following example:
Marie-Christine Lagasquie 2005-02-04