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 [26]) 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:
but also | ||
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