Some existing proposals can already be considered as examples of local valuations.
In 's approach, a labelling of a set of arguments assigns a status (accepted, rejected, undecided) to each argument using labels from the set . (resp. , ) represents the ``accepted'' (resp. ``rejected'', ``undecided'') status. Intuitively, an argument labelled with is both supported and weakened.
The underlying intuition is that an argument can only be weakened (label or ) if one of its direct attackers is supported (condition 1); an argument can get a support only if all its direct attackers are weakened and an argument which is supported (label or ) weakens the arguments it attacks (condition 2). So:
Every argumentation system can be completely labelled. The associated semantics is that is an acceptable set of arguments iff there exists a complete labelling of such that .
Other types of labellings are introduced in  among which the so-called ``rooted labelling'' which induces a corresponding ``rooted'' semantics. The idea is to reject only the arguments attacked by accepted arguments: an attack by an ``undecided'' argument is not rooted since an ``undecided'' attacker may become rejected.
The rooted semantics enables to clarify the links between all the other semantics introduced in  and some semantics introduced in .
For even, we obtain and .
For odd, we obtain and
Another type of local valuation has been introduced recently in  for ``deductive'' arguments. The approach can be characterised as follows. An argument is structured as a pair , where support is a consistent set of formulae that enables to prove the formula conclusion. The attack relation considered here is strict and cycles are not allowed. The notion of a ``tree of arguments'' allows a concise and exhaustive representation of attackers and defenders of a given argument, root of the tree. A function, called a ``categoriser'', assigns a value to a tree of arguments. This value represents the relative strength of an argument (root of the tree) given all its attackers and defenders. Another function, called an ``accumulator'', synthesises the values assigned to all the argument trees whose root is an argument for (resp. against) a given conclusion. The phase of categorisation therefore corresponds to an interaction-based valuation.  introduces the following function :
Intuitively, the larger the number of direct attackers of an argument, the lower its value. The larger the number of defenders of an argument, the larger its value.
Example 3 (continuation) We obtain:
, , , , ..., and when (this value is the inverse of the golden ratio14).
So, we have:
If is even
If is odd
Our approach for local valuations is a generalisation of these two previous proposals in the sense that 's function and 's labellings are instances of our approach.
The main idea is that the value of an argument is obtained with the composition of two functions:
Let be a totally ordered set with a minimum element ( ) and a subset of , that contains and with a maximum element .
with such that ( denotes the set of all finite sequences of elements of )
Note that is a logical consequence of the properties of the function .
A first property on the function explains the behaviour of the local valuation in the case of an argument which is the root of only one branch (like in Example 3):
A second property shows that the local valuation induces an ordering relation on arguments:
A third property handles the cycles:
The following property shows the underlying principles satisfied by all the local valuations defined according to our schema:
The last properties explain why [13,4] are instances of the local valuation described in Definition 6:
Note that, in , the valued graphs are acyclic. However, it is easy to show that the valuation proposed in  can be generalised to graphs with cycles (in this case, we must solve second degree equations - see Example 5).
In this example, with the generic valuation, we obtain:
So, we have:
However, the constraints on and are insufficient to compare and with the other arguments.
The same problem exists if we reduce the example to the hatched part of the graph in the previous figure; we obtain , but and cannot be compared with the other arguments16.
Now, we use the instance of the generic valuation proposed in :
So, we have:
However, if we reduce the example to the hatched part of the graph, then the value of is . So, is better than and , but also than ( becomes better than its defender).
A generic valuation gives fixpoint of .
If we use the instance proposed by , and are solutions of the following second degree equation: .
So, we obtain: (the inverse of the golden ratio again).
Marie-Christine Lagasquie 2005-02-04