Comparison of tupled values

In this section, we define the comparison relation between arguments (so, between some particular tupled values), using the following idea: an argument $A$ is better than an argument $B$ iff $A$ 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 % latex2html id marker 4271
$\leq_{lex\infty}$ on ${\mathcal{T}}$ is defined by:

Definition 13 (Lexicographic ordering on tuples)  
Let $(x_1,\ldots, x_n, \ldots)$ and $(y_1,\ldots, y_m, \ldots)$ be 2 finite or infinite tuples $\in {\mathcal{T}}$.

% latex2html id marker 4281
$(x_1,\ldots, x_n, \ldots) <_{lex\infty} (y_1,\ldots, y_m, \ldots)$ iff % latex2html id marker 4283
$\exists i \geq 1$ such that:

% latex2html id marker 4301
$(x_1,\ldots, x_n, \ldots) =_{lex\infty} (y_1,\ldots, y_m, \ldots)$ iff the tuples contain the same number $p \in \mathds{N}\cup \{\infty\}$ of elements and $\forall i$, $1 \leq i \leq p$, $x_i = y_i$.

So, we define: % latex2html id marker 4311
$(x_1,\ldots, x_n, \ldots) \leq_{lex\infty} (y_1,\ldots, y_m, \ldots)$ iff

% latex2html id marker 4313
$(x_1,\ldots, x_n, \ldots)
=_{lex\infty} (y_1,\ldots, y_m, \ldots)$ or % latex2html id marker 4315
$(x_1,\ldots, x_n, \ldots) <_{lex\infty} (y_1,\ldots, y_m,
\ldots)$.

The ordering % latex2html id marker 4317
$<_{lex\infty}$ 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: $(0)$ % latex2html id marker 4321
$<_{lex\infty}$ $(0,0)$ % latex2html id marker 4325
$<_{lex\infty}$ ...% latex2html id marker 4327
$<_{lex\infty}$ $(0,
\ldots, 0, \ldots)$ % latex2html id marker 4331
$<_{lex\infty}$ ...% latex2html id marker 4333
$<_{lex\infty}$ $(0, 1)$).

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.


\begin{algorithm}
% latex2html id marker 814
[H]
\small
\dontprintsemicolon
\Set...
...7 \% \;
}
}
}
}
}
\caption{Comparison of two tupled values
}
\end{algorithm}

Algorithm 1 defines a partial preordering on the set $v({\mathcal{A}})$:

Property 9 (Partial preordering)   Algorithm 1 defines a partial preordering $\succeq$ on the set $v({\mathcal{A}})$.

The tupled value $[0^{\infty},()]$ is the only maximal value of the partial preordering $\succeq$.

The tupled value $[(),1^{\infty}]$ is the only minimal value of the partial preordering $\succeq$.

Notation: the partial preordering $\succeq$ on the set $v({\mathcal{A}})$ induces a partial preordering on the arguments (the partial preordering on ${\mathcal{A}}$ will be denoted like the partial preordering on $v({\mathcal{A}})$): $A
\succeq B$ if and only if $v(A) \succeq
v(B)$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:

Definition 14 (Adding/removing a branch to an argument)  
Let $A$ be an argument whose tupled value is $v(A) = [v_p(A), v_i(A)]$ with $v_p(A) = (x^p_1, \ldots, x^p_n)$ and $v_i(A) =
(x^i_1, \ldots, x^i_m)$ ($v_p(A)$ or $v_i(A)$ may be empty but not simultaneously).

Adding (resp. removing) a defence branch to $A$ is defined by:

$v_p(A)$ becomes % latex2html id marker 4505
${\mbox{\tt Sort}}(x^p_1, \ldots, x^p_n,x^p_{n+1})$ where $x^p_{n+1}$ is the length of the added branch (resp. % latex2html id marker 4509
$\exists j \in [1..n]$ such that $v_p(A)$ becomes $(x^p_1, \ldots,
x^p_{j-1}, x^p_{j+1}, \ldots, x^p_n)$).

And the same thing on $v_i(A)$ for adding (resp. removing) an attack branch to $A$.

Definition 15 (Increasing/decreasing the length of a branch of an argument)  
Let $A$ be an argument whose tupled value is $v(A) = [v_p(A), v_i(A)]$ with $v_p(A) = (x^p_1, \ldots, x^p_n)$ and $v_i(A) =
(x^i_1, \ldots, x^i_m)$ ($v_p(A)$ or $v_i(A)$ may be empty but not simultaneously).

Increasing (resp. decreasing) the length of a defence branch of $A$ is defined by:

% latex2html id marker 4533
$\exists j \in [1..n]$ such that $v_p(A)$ becomes $(x^p_1, \ldots,
x^p_{j-1}, x'^p_j, x^p_{j+1}, \ldots, x^p_n)$ where $x'^p_j >
x^p_j$ (resp. $x'^p_j < x^p_j$) and the parity of $x'^p_j$ is the parity of $x^p_j$.

And the same thing on $v_i(A)$ for increasing (resp. decreasing) an attack branch to $A$.

Definition 16 (Improvement/degradation of the defences/attacks)  
Let $A$ be an argument whose tupled value is $v(A) = [v_p(A), v_i(A)]$ ($v_p(A)$ or $v_i(A)$ may be empty but not simultaneously). We define:
An improvement (resp. degradation) of the defence
consists in
  • adding a defence branch to $A$ if initially $v_p(A) \neq 0^{\infty}$ (resp. removing a defence branch of $A$);
  • or decreasing (resp. increasing) the length of a defence branch of $A$;
  • or removing the only defence branch leading to $A$ (resp. adding a defence branch leading to $A$ if initially $v_p(A)= 0^{\infty}$);

An improvement (resp. degradation) of the attack
consists in
  • adding (resp. removing) an attack branch to $A$;
  • or decreasing (resp. increasing) the length of an attack branch of $A$.

Property 10 (Underlying principles)   Let $v$ be a valuation with tuples (Definition 10) associated with Algorithm 1, $v$ respects the following principles:
P1$'$
The valuation is maximal for an argument without attackers and non maximal for an argument which is attacked (whether it is defended or not).
P2$'$
The valuation of an argument takes into account all the branches which are rooted in this argument.
P3$'$
The improvement of the defence or the degradation of the attack of an argument leads to an increase of the value of this argument.
P4$'$
The improvement of the attack or the degradation of the defence of an argument leads to a decrease of the value of the argument.

Example 4 (continuation) With the valuation with tuples, we obtain:

So, we have:

$E_1, D_2, D_3, C_4, B_4$
$\succ $
$C_1, B_2$ $E_1, D_2, D_3, C_4, B_4$
$\succ $ but also $\succ $
$B_1$ $A$
$\succ $
$D_1, C_2, C_3, B_3$

$A$ 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:


\begin{displaymath}E_1, D_2 \succ C_1 \succ B_1 \succ A \succ D_1, C_2\end{displaymath}

$A$ is now comparable with all the other arguments (in particular, $A$ is ``worse'' than its defender $C_1$ and than its direct attacker $B_1$).

Marie-Christine Lagasquie 2005-02-04