Introduction

As shown by [9], argumentation frameworks provide a unifying and powerful tool for the study of several formal systems developed for common-sense reasoning, as well as for giving a semantics to logic programs. Argumentation is based on the exchange and valuation of interacting arguments which support opinions and assertions. It can be applied, among others, in the legal domain, for collective decision support systems or for negotiation support.

The fundamental characteristic of an argumentation system is the interaction between arguments. In particular, a relation of attack may exist between arguments. For example, if the argument takes the form of a logical proof, arguments for a proposition and arguments against this proposition can be advanced. In that case, the attack relation relies on logical inconsistency.

The argumentation process is usually divided in two steps: a valuation of the relative strength of the arguments, followed by the selection of the most acceptable arguments.

In the valuation step, it is usual to distinguish two different types of valuations:

• intrinsic valuation: here, the value of an argument is independent of its interactions with the other arguments. This enables to simply express to what extent an argument increases the confidence in the statement it supports. See [20,16,18,22,1,15,21].

  For example, in [16], using the following knowledge base, composed of (formula, probability) pairs $\{(\phi_1, 0.8)$, $(\phi_2, 0.8)$, $(\phi_3, 0.8)$, $((\phi_1 \land \phi_2 \to \phi_4), 1)$, $((\phi_1 \land \phi_3 \to \phi_4), 1)\}$, two arguments can be produced¹:

  • $A_1 = <\{\phi_1, \phi_2, (\phi_1 \land \phi_2 \to \phi_4)\}, \phi_4>$
  • and $A_2 = <\{\phi_1, \phi_3, (\phi_1 \land \phi_3 \to \phi_4)\}, \phi_4>$.
  Both arguments have the same weight $0.8 \times 0.8 \times 1 = 0.64$, and the formula $\phi_4$ has the weight $0.64 + 0.64 - 0.512 = 0.768$².

• interaction-based valuation: here the value of an argument depends on its attackers (the arguments attacking it), the attackers of its attackers (the defenders), etc.³

  Several approaches have been proposed along this line (see [9,1,13,4]) which differ in the sets of values used. Usually, two values are considered. However, there are very few proposals which use more than two values (three values in [13], and an infinity of values in [4]).

  For example, in [4], the set of values is the interval of the real line $[0,1]$. In this case, with the set of arguments⁴ $\{A_1, A_2, A_3\}$ and considering that $A_1$ attacks $A_2$ which attacks $A_3$, the value of the argument $A_1$ (resp. $A_2$, $A_3$) is $1$ (resp. $\frac{1}{2}$, $\frac{2}{3}$).

Intrinsic valuation and interaction-based valuation have often been used separately, according to the considered applications. Some recent works however consider a combination of both approaches (see [1,14,21]).

Considering now the selection of the more acceptable arguments, it is usual to distinguish two approaches:

• individual acceptability: here, the acceptability of an argument depends only on its properties. For example, an argument can be said acceptable if and only if it does not have any attacker (in this case, only the interaction between arguments is considered, see [12]). In the context of an intrinsic valuation, an argument can also be said acceptable if and only if it is ``better'' than each of its attackers (see [1]).

• collective acceptability: in this case, the acceptability of a set of arguments is explicitly defined. For example, to be acceptable, a set of arguments may not contain two arguments such that one attacks the other (interactions between arguments are used). [9]'s framework is well suited for this kind of approach but allows only for a binary classification: the argument belongs or does not belong to an acceptable set.

It is clear that except for intrinsic valuations, most proposals do not allow for any gradual notion of valuation or acceptability (i.e. there is a low number of levels to describe values and the acceptability is usually binary). Our aim is therefore to introduce graduality in these two steps.

However, the processes of valuation and of selection are often linked together. This is the case when the selection is done on the basis of the value of arguments⁵ or when the selection defines a binary valuation on arguments. We will therefore:

• first consider and discuss the general principles concerning the definition of a gradual interaction-based valuation and then define some valuation models in an abstract argumentation system,
• then, introduce the notion of graduality in the definition of the acceptability using the previously defined gradual valuations, but also some more classical mechanisms.

Some graduality has already been introduced in argumentation systems. For instance, in [21], degrees of justification for beliefs are computed. Arguments are sequences of conclusive and/or prima-facie inferences. Arguments are collected in a graph where a node represents the conclusion of an argument, a support link ties a node to nodes from which it is inferred, and an attack link indicates an attack between nodes. The degree of justification of a belief is computed from the strength of the arguments concluding that belief and the strength of the arguments concluding on an attacker of the belief.

Our work takes place in a more abstract framework since we do not consider any argument structure. Our valuation models are based on interactions between arguments and directly apply to arguments.

We use the framework defined by [9]: a set of arguments and a binary attack relation between arguments. We also use a graphical representation of argumentation systems (see Section 2). The gradualisation of interaction-based valuations will be presented in Section 3. Then, in Section 4, we will consider different mechanisms leading to gradual acceptability, sometimes relying on the gradual valuations defined in Section 3. We will conclude in Section 5.

All the proofs of the properties stated in Sections 3 and 4 will be given in Appendix A.