Conclusion

In this paper, we have introduced graduality in the two main related issues of argumentation systems:

• the valuation of the arguments,
• the acceptability of the arguments.

Regarding the first issue, we have defined two formalisms introducing an interaction-based gradual valuation of arguments.

• First, a generic gradual valuation which covers existing proposals (for example [4] and [13]). This approach is essentially ``local'' since it computes the value of the argument only from the value of its direct attackers.
• Then, an approach based on a labelling which takes the form of a pair of tuples; this labelling memorises the structure of the graph representing the interactions (the ``attack graph''), associating each branch with its length (number of the edges from the leaf to the current node) in the attack graph (if the length of the branch is an even integer, the branch is a defence branch for the current node, otherwise the branch is an attack branch for the current node). This approach is said to be ``global'' since it computes the value of the argument using the whole attack graph influencing the argument.

We have shown that each of these valuations induces a preordering on the set of the arguments, and we have brought to light the main differences between these two approaches.

Regarding the second issue, two distinct approaches have been proposed:

• First, in the context of the collective acceptability of [9]: three levels of acceptability (uni-accepted, exi-accepted, not-accepted) were already defined. More graduality can be introduced in the collective acceptability using the notion of cleanly-accepted arguments (those whose direct attackers are not-accepted).

• Then, in the context of individual acceptability: using the previously defined gradual valuations, the new notion of well-defended arguments has been introduced (those which are preferred to their direct attackers in the sense of a given gradual valuation $v$).

The first concept induces a refinement of the level of exi-accepted in two sublevels (cleanly-accepted arguments and only-exi-accepted arguments). The gradual valuation allows graduality inside each level of this collective acceptability.

The second concept induces two new levels of acceptability (well-defended arguments and not-well-defended arguments). The gradual valuation also allows graduality inside each level of this individual acceptability.

Regarding our initial purpose of introducing graduality in the definition of acceptability, we have adopted a basic principle:

• acceptability is strongly related to the interactions between arguments (represented on the graph of interactions),
• and an argument is all the more acceptable if it is preferred to its direct attackers.

Then, we have followed two different directions. One is based on a refinement of an existing partition and remains in the framework of Dung's work. The other one is based on the original concept of ``being well-defended'', and deserves further investigation, in particular from a computational point of view.