# The framework of [9] and its graphical representation

We consider the abstract framework introduced in [9]. 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.

Example 1
The system defines the following graph with the root8 :

Definition 1 (Graphical representation of an argumentation system)   Let be the attack graph associated with the argumentation system , we define:

Leaf of the attack graph
A leaf of is an argument without attackers9.

Path in the attack graph
A path from to is a sequence of arguments such that:
• ,
• ,
• ...,
• ,
• .
The length of the path is (the number of edges that are used in the path) and will be denoted by .

A special case is the path10 from to whose length is .

The set of paths from to will be denoted by .

Dependence, independence, root-dependence of a path

Consider 2 paths and .

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 .

Cycles in the attack graph
A cycle11 is a path such that , if , then .

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 [9] but are not strictly equivalent12.

Definition 2 (Direct/Indirect Attackers/Defenders of an argument)   Consider :
• The direct attackers of are the elements of .
• The direct defenders of are the direct attackers of the elements of .
• The indirect attackers of are the elements defined by:

such that , with .
• the indirect defenders of are the elements defined by:

such that , with .

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.

Definition 3 (Attack branch and defence branch of an argument)   Consider , an attack branch (resp. defence branch) for is a path in from a leaf to whose length is odd (resp. even). We say that is the root of an attack branch (resp. a defence branch).

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:

Example 2
 On this graph , we can see: a path from to whose length is 2 (), 2 cycles and , of length 3, which are not isolated (note that is not a cycle with our definition), the two previous cycles are interconnected (in and ), the paths and are independent, the paths and are root-dependent and the paths and are dependent, , , are the leaves of , is an attack branch for whose length is 3, is a defence branch for whose length is 2, , and are the direct attackers of , , (which is already a direct attacker of ) and are the direct defenders of , and are the two indirect attackers of , is the only indirect defender of .

Marie-Christine Lagasquie 2005-02-04