In this section, we define the comparison relation between arguments (so, between some particular tupled values), using the following idea: an argument is better than an argument iff has a better defence (for it) and a lower attack (against it).
The first idea is to use a lexicographic ordering on the tuples. This lexicographic ordering denoted by on is defined by:
iff such that:
iff the tuples contain the same number of elements and , , .
So, we define: iff
The ordering is a generalisation of the classical lexicographic ordering (see ) to the case of infinite tuples. This ordering is complete but not well-founded (there exist infinite sequences which are strictly non-increasing: ... ... ).
Since the even values and the odd values in the tupled value of an argument do not play the same role, we cannot use a classical lexicographic comparison. So, we compare tupled values in two steps:
Let us consider some examples:
The comparison of arguments is done using Algorithm 1 which implements the principle of a double comparison (first quantitative, then qualitative) with two criteria (one defence criterion and one attack criterion) using a cautious method.
Algorithm 1 defines a partial preordering on the set :
The tupled value is the only maximal value of the partial preordering .
The tupled value is the only minimal value of the partial preordering .
Notation: the partial preordering on the set induces a partial preordering on the arguments (the partial preordering on will be denoted like the partial preordering on ): if and only if 22.
In order to present the underlying principles satisfied by the global valuation, we first consider the different ways for modifying the defence part or the attack part of an argument:
Adding (resp. removing) a defence branch to is defined by:
becomes where is the length of the added branch (resp. such that becomes ).
And the same thing on for adding (resp. removing) an attack branch to .
Increasing (resp. decreasing) the length of a defence branch of is defined by:
such that becomes where (resp. ) and the parity of is the parity of .
And the same thing on for increasing (resp. decreasing) an attack branch to .
Example 4 (continuation) With the valuation with tuples, we obtain:
So, we have:
is incomparable with almost all the other arguments (except with the leaves of the graph).
Similarly, on the hatched part of the graph, we obtain the following results:
is now comparable with all the other arguments (in particular, is ``worse'' than its defender and than its direct attacker ).
Marie-Christine Lagasquie 2005-02-04