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.

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.

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

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 ratio ^{14}).
*

*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 .

- ,
- , if , then
- , if , then

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

- For any permutation of ,
- if then

- 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):

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:

**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 .

- ,
- ,
- ,
- ,
- 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).

*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
arguments ^{16}.
*

*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).*

*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