## Local approach (generic valuation)

Some existing proposals can already be considered as examples of local valuations.

In [13]'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.

Definition 4 ([13]'s labellings)   Let be an argumentation system. A complete labelling of is a function such that:
1. If then such that
2. If then ,

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:

• If has no attacker .
• If then such that .
• If ( , ) then .
• If then , .

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 [13] 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.

Definition 5 ([13]'s labellings - continuation)   The complete labelling is rooted iff , if then such that .

The rooted semantics enables to clarify the links between all the other semantics introduced in [13] and some semantics introduced in [9].

Example 3   On the following example:

For even, we obtain and .

For odd, we obtain and

Another type of local valuation has been introduced recently in [4] 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. [4] introduces the following function :

• if , then
• if with ,

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 [4]'s function and [13]'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:

• one for aggregating the values of all the direct attackers of the argument; so, this function computes the value of the direct attack'';
• the other for computing the effect of the direct attack'' on the value of the argument: if the value of the direct attack'' increases then the value of this argument decreases, if the value of the direct attack'' decreases then the value of this argument increases.

Let be a totally ordered set with a minimum element ( ) and a subset of , that contains and with a maximum element .

Definition 6 (Generic gradual valuation)   Let be an argumentation system. A valuation is a function such that:

1. ,
2. , if , then
3. , if , then

with such that ( denotes the set of all finite sequences of elements of )

• For any permutation of ,
• if then
and such that
• is non-increasing (if then )

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):

Property 1   The function satisfies for all :

Moreover, if is strictly non-increasing and , the previous inequalities become strict.

A second property shows that the local valuation induces an ordering relation on arguments:

Property 2 (Complete preordering)   Let be a valuation in the sense of Definition 6. induces a complete15 preordering on the set of arguments defined by: iff .

A third property handles the cycles:

Property 3 (Value in a cycle)   Let be an isolated cycle of the attack graph, whose length is . If is odd, all the arguments of the cycle have the same value and this value is a fixpoint of the function . If is even, the value of each argument of the cycle is a fixpoint of the function .

The following property shows the underlying principles satisfied by all the local valuations defined according to our schema:

Property 4 (Underlying principles)   The gradual valuation given by Definition 6 respects the following principles:

P1
The valuation is maximal for an argument without attackers and non maximal for an attacked and undefended argument.
P2
The valuation of an argument is a function of the valuation of its direct attackers (the direct attack'').
P3
The valuation of an argument is a non-increasing function of the valuation of the direct attack''.
P4
Each attacker of an argument contributes to the increase of the valuation of the direct attack'' for this argument.

The last properties explain why [13,4] are instances of the local valuation described in Definition 6:

Every rooted labelling of in the sense of [13] can be defined as an instance of the generic valuation such that:
• with ,
• ,
• ,
• defined by , ,
• and is the function .

Property 6 (Link with [4])   The gradual valuation of [4] can be defined as an instance of the generic valuation such that:
• ,
• ,
• ,
• ,
• defined by
• and defined by .

Note that, in [4], the valued graphs are acyclic. However, it is easy to show that the valuation proposed in [4] can be generalised to graphs with cycles (in this case, we must solve second degree equations - see Example 5).

Example 4   Consider the following graph:

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 [4]:

• ,
• ,
• ,
• ,
• .

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).

Example 5 (Isolated cycle)   Consider the following graph reduced to an isolated cycle:

.

A generic valuation gives fixpoint of .

If we use the instance proposed by [4], 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