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 $\leq_{lex\infty}$ 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 $\in {\mathcal{T}}$.

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

• $\forall j<i$, $x_j = y_j$ and
• $y_i$ exists and:
  • either the tuple $(x_1,\ldots, x_n, \ldots)$ is finite with a number of elements equal to $i-1$ (so, $x_i$ does not exist),
  • or $x_i$ exists and $x_i < y_i$.

$(x_1,\ldots, x_n, \ldots) =_{lex\infty} (y_1,\ldots, y_m, \ldots)$ 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: $(x_1,\ldots, x_n, \ldots) \leq_{lex\infty} (y_1,\ldots, y_m, \ldots)$ iff

$(x_1,\ldots, x_n, \ldots) =_{lex\infty} (y_1,\ldots, y_m, \ldots)$ or $(x_1,\ldots, x_n, \ldots) <_{lex\infty} (y_1,\ldots, y_m, \ldots)$.

The ordering $<_{lex\infty}$ 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: $(0)$ $<_{lex\infty}$ $(0,0)$ $<_{lex\infty}$ ...$<_{lex\infty}$ $(0, \ldots, 0, \ldots)$ $<_{lex\infty}$ ...$<_{lex\infty}$ $(0, 1)$).

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:

• The ``first step'' compares the number of attack branches and the number of defence branches of each argument. So, we have two criteria (one for the defence and the other for the attack). These criteria are aggregated using a cautious method: we conclude if one of the arguments has more defence branches (it is better according to the defence criterion) and less attack branches than the other argument (it is also better according to the attack criterion). Note that we conclude positively only when all the criteria agree: if one of the arguments has more defence branches (it is better according to the defence criterion) and more attack branches than the other argument (it is worse according to the attack criterion), the arguments are considered to be incomparable.
• Else, the arguments have the same number of defence branches and the same number of attack branches, and a ``second step'' compares the quality of the attacks and the quality of the defences using the length of each branch. This comparison is made with a lexicographic principle (see Definition 13) and gives two criteria which are again aggregated using a cautious method. In case of disagreement, the arguments are considered to be incomparable.

Let us consider some examples:

• $[(2),(1)]$ is better than $[(2),(1,1)]$ because there are less attack branches in the first tupled value than in the second tupled value, the numbers of defence branches being the same (first step).
• $[(2),(1)]$ is incomparable with $[(2,2),(1,1)]$ because there are less defence branches and less attack branches in the first tupled value than in the second tupled value (first step).
• $[(2),(3)]$ is better than $[(2),(1)]$ because there are weaker attack branches in the first tupled value than in the second tupled value (the attack branch of the first tupled value is longer than the one of the second tupled value), the defence branches being the same (second step, using the lexicographic comparison applied on even parts then on odd parts of the tupled values).
• $[(2),(3)]$ is better than $[(4),(3)]$ because there are stronger defence branches in the first tupled value than in the second tupled value (the defence branch is shorter in the first tupled value than in the second tupled value), the attack branches being the same (second step).
• $[(2),(1)]$ is incomparable with $[(4),(3)]$ because there are worse attack branches and better defence branches in the first tupled value than in the second tupled value (second step).

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.

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 ${\mbox{\tt Sort}}(x^p_1, \ldots, x^p_n,x^p_{n+1})$ where $x^p_{n+1}$ is the length of the added branch (resp. $\exists j \in [1..n]$ 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:

$\exists j \in [1..n]$ 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:

• $v(E_1) = v(D_2) = v(D_3) = v(C_4) = v(B_4) = [0^\infty,()]$,
• $v(D_1) = v(C_2) = v(C_3) = v(B_3) = [(),(1)]$,
• $v(C_1) = v(B_2) = [(2),()]$,
• $v(B_1) = [(2),(3)]$,
• $v(A) = [(2,4),(1,3,3)]$.

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:

$$E_1, D_2 \succ C_1 \succ B_1 \succ A \succ D_1, C_2$$

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