First, in order to record the lengths of the branches leading to the arguments, we use the notion of tuples and we define some operations on these tuples:
Note that we allow infinite tuples, among other reasons, because they are needed later in order to compute the ordering relations described in Section 3.2.4 (in particular when the graph is cyclic).
The operations on the tuples have the following properties:
For any tuple and any integers and , .
For any integer and any tuples and different from 17, .
In order to valuate the arguments, we split the set of the lengths of the branches leading to the argument in two subsets, one for the lengths of defence branches (even integers) and the other one for the lengths of attack branches (odd integers). This is captured by the notion of tupled values:
Using this notion of tupled-values, we can define the computation process of the gradual valuation with tuples18 in the case of acyclic graphs.
If is a leaf then
If has direct attackers denoted by , ..., , ...then
Notes: The choice of the value for the leaves is justified by the fact that the value of an argument memorises all the lengths of the branches leading to the argument. Using the same constraint, either or may be empty but not both19.
Note also that the set of the direct attackers of an argument can be infinite (this property will be used when we take into account an argumentation graph with cycles).
Marie-Christine Lagasquie 2005-02-04