... produced¹

Here, the arguments are under the form of an ``Explanation-Conclusion Pair''. This is one possible way to compute arguments. See also [17,25,20,22,23,12,15,2].

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...#tex2html_wrap_inline2845#²

Weights being probabilities, the weight of an argument is the probability of the conjunction of the formulae of the argument, and the weight of $\phi_4$ is the probability of the disjunction of $A_1$ and $A_2$.

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... etc.³

Here, we consider only the interactions corresponding to attacks between arguments. There exist also some other types of interactions (for example, arguments which reinforce other arguments instead of attacking them, see [14,24]). For this kind of interaction, graduality has not been considered.

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... arguments⁴

Here, the initial knowledge base is useless.

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... arguments⁵

For example, using [4]'s valuation, we can decide that all the arguments whose value is $> 0.5$ are selected, because $0.5$ is the mean value of the set of values; Another possibility, with different valuations (interaction-based or intrinsic), is to accept an argument when its value is better than the value of each of its attackers.

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...#tex2html_wrap_inline2921#⁶

${\mathcal{R}}^-(A) = \{A_i \in {\mathcal{A}}\vert A_i {\mathcal{R}}A\}$.

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...#tex2html_wrap_inline2927#⁷

${\mathcal{R}}^+(A) = \{A_i \in {\mathcal{A}}\vert A {\mathcal{R}}A_i\}$.

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... root⁸

The word ``root'' is used in an informal sense (it just means that there are in the graph some paths leading to this node). This term and other terms (leaf, branch, path, ...) which are used in this document are standard in graph theory but may have a different definition. They are usual terms in the argumentation domain. Please see Definition 1 in order to know their precise meaning in this document. These definitions simply take into account the fact that the directed edges of our graph link attackers to attacked argument).

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... attackers⁹

$A$ is a leaf iff ${\mathcal{R}}^-(A) = \varnothing$.

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... path¹⁰

We will assume that there exists an infinity of such paths. This assumption greatly simplifies the handling of leaves later in the paper.

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...cycle¹¹

This definition of a cycle corresponds to the definition of an elementary cycle in graph theory (an elementary cycle does not contain 2 edges with the same initial extremity, or the same ending extremity).

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... equivalent¹²

In [9]'s work, direct attackers (resp. defenders) are also indirect attackers (resp. defenders) which is not true in our definitions.

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... arguments¹³

We pursue a work initiated in [7] and propose some improvements.

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... ratio¹⁴

The golden ratio is a famous number since the antiquity which has several interesting properties in several domains (architecture, for example).

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... complete¹⁵

A complete preordering on ${\mathcal{A}}$ means that any two elements of ${\mathcal{A}}$ are comparable.

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... arguments¹⁶

$v(A) = g^2(h(g^2(V_{\mbox{\scriptsize Max}}, g(V_{\mbox{\scriptsize Max}})))$ and $v(B_1) = g(h(g^2(V_{\mbox{\scriptsize Max}}), g(V_{\mbox{\scriptsize Max}})))$.

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...#tex2html_wrap_inline3641#¹⁷

Otherwise it is false : $(0^{\infty} \star (p)) \oplus k = (p + k)$, whereas $(0^{\infty} \oplus k) \star ((p) \oplus k) = (k) \star (p + k) = (k, p + k)$.

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... tuples¹⁸

This definition is different from the definition given in [7]. The ideas are the same but the formalisation is different.

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... both¹⁹

The proof is the following:.

• If $A$ is not a leaf, at least one of the tuples is not empty, because there exists at least one branch whose length is $> 0$ leading to $A$ (see Definitions 8 and 10).
• And, if $A$ is a leaf, there also exists at least one defence branch because the path from $A$ to $A$ is allowed and its length is $0$ (in fact, there are an infinity of such paths - see Definition 1) and no attack branch leading to the leaf (see Definition 10).

So, the value of a leaf is $[0^{\infty},()]$, and it is impossible that $v_p(A) = v_i(A) = ()$.

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... exists²⁰

The operator mod is the modulo function.

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... tuple²¹

The proof is the following:.

• $x'_1$ contains only even integers.
• For each k, $x'_k \neq 0^{\infty}$ since $x'_k$ is the result of the addition of a tuple and an integer.
• If $x'_1$ is not empty, let $e_1$ denote the least even integer present in $x'_1$. As $x'_1 = x'_2 \oplus 2$, $x'_2$ is not empty and $e_2$ will denote the least integer present in $x'_2$. We have $e_1 = e_2 + 2$. So, we are able to build a sequence of positive even integers $e_1, e_2, \ldots$, which is strictly decreasing. That is impossible. So, $x'_1 = ()$.

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...#tex2html_wrap_inline4481#²²

We will also use the notation $B \preceq A$ defined by: $B \preceq A$ iff $A \succeq B$.

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... than²³

With the valuation proposed in [4], we obtain: $v(A) = \frac{3}{4}$ and $v(A') = \frac{1}{2}$.

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... acceptability²⁴

This work has been presented in a workshop [6].

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... cases²⁵

The terminology used in this section is also used in the domain of nonmonotonic reasoning, see [19]: the word uni comes from the word universal which is a ``synonym'' of the word skeptical, and the word exi comes from the word existential which is a ``synonym'' of the word credulous. We have chosen to use the words uni and exi because they recall the logical quantificators $\forall$ (for all) and $\exists$ (exists at least one).

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... coincide²⁶

If there is only one extension then the fact that $A$ belongs to all the extensions is equivalent to the fact that $A$ belongs to at least one extension. Moreover, with only one extension containing $A$, all the attackers of $A$ do not belong to an extension. So, $A$ is cleanly-accepted.

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... stable²⁷

This corresponds to the consistent argumentation system proposed by [9].

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... attackers²⁸

This idea is also used in the notion of ``defeat'' proposed by [3]. So, there is a link between a ``well-defended argument'' and an argument which is not ``attacked'' in the sense of [3] by its direct attackers. Note that, in [3], the valuation is an extra knowledge added in the argumentation framework. In contrast, here, the $v$-preference is extracted from the attack graph.

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... cycles²⁹

So, $({\mathcal{A}}, {\mathcal{R}})$ is well-founded.

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... identical''³⁰

Proof: let $g$ be a non-increasing function, let $\alpha$ and $\beta$ be two fixpoints of $g$. If $\alpha \neq \beta$, we may suppose that $\alpha > \beta$, so $g(\alpha) \leq g(\beta)$ (since $g$ is non-increasing), so $\alpha \leq \beta$ (since $\alpha$ and $\beta$ are fixpoints of $g$), which is in contradiction with the assumption $\alpha > \beta$.

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... case³¹

We work case by case in order to avoid the complex cases in which we have several simultaneous simple modifications. For example, the modification of the length of a branch which changes the status of the branch (an even integer replaced by an odd integer) is a complex case corresponding to two simple cases: the removal of a branch with a given status, then the addition of a new branch with a different status.

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...#tex2html_wrap_inline5979#³²

${\mathcal{R}}'$ is the restriction of ${\mathcal{R}}$ to ${\mathcal{A}}'$ if and only if ${\mathcal{R}}' = \{(a,b) \vert a {\mathcal{R}}b, a \in {\mathcal{A}}', b \in {\mathcal{A}}'\}$.

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