Comparison Criteria for Argumentation Semantics

Abstract

Argumentation reasoning is a way for agents to evaluate a situation. Given a framework made of conflicting arguments, a semantics allows to evaluate the acceptability of the arguments. It may happen that the semantics associated to the framework has to be changed. In order to perform the most suitable change, the current and a potential new semantics have to be compared. Notions of difference measures between semantics have already been proposed, and application cases where they have to be minimized when a change of semantics has to be performed, have been highlighted. This paper develops these notions, it proposes an additional kind of difference measure, and shows application cases where measures may have to be maximized, and combined.

A Proofs

Proof

(Proof of Proposition 2). From our definition of characterizations, the mapping that associates a semantics $\sigma$ to a set of properties $Prop(\sigma )$ guarantees that a semantics cannot be associated with two different sets of properties, and a same set of properties cannot correspond to different semantics.

The weighted sum on sets of properties obviously defines a distance (in particular, when all weights are identical, we obtain the well-known Hamming distance; other weights just define generalization of Hamming distance). Since we can identify the semantics to the sets of properties, $\delta _{prop}^w$ is a distance.

Proof

(Proof of Proposition 3). From the definition of the $\varSigma$-relation graph,

• the difference between $\sigma _1$ and $\sigma _2$ is 0 iff they are the same node of the graph (i.e. $\sigma _1 = \sigma _2$), so coincidence is satisfied;

• the shortest path between two semantics $\sigma _1, \sigma _2$ has the same length whatever the direction of the path (from $\sigma _1$ to $\sigma _2$, or vice-versa), since we do not consider the direction of arrows, so symmetry is satisfied;

• the shortest path between $\sigma _1$ and $\sigma _3$ is at worst the concatenation of the paths $(\sigma _1,\dots ,\sigma _2)$ and $(\sigma _2,\dots ,\sigma _3)$, or (if possible) a shorter one, so triangular inequality is satisfied.

Proof

(Proof of Proposition 4). Example 6 gives the counter-examples for coincidence and symmetry.

Proof

(Proof of Proposition 5). We consider a given AF F and a set of semantics $\varSigma = \{\sigma _1,\dots ,\sigma _n\}$, such that for all $\sigma _i, \sigma _j \in \varSigma$ with $\sigma _i \ne \sigma _j$, $Ext_{\sigma _i}(F) \nsubseteq Ext_{\sigma _n}(F)$.

Obviously, for any semantics $\sigma _i$, $\delta _F^{d_H,\sum }(\sigma _i,\sigma _i) = 0$. Now, let us assume the existence of two semantics $\sigma _i, \sigma _j \in \varSigma$ such that $\delta _F^{d_H,\sum }(\sigma _i,\sigma _j) = 0$. We just rewrite this, following the definition of the measure: $\sum _{\epsilon \in Ext_{\sigma _i}(F)} \min _{\epsilon ' \in Ext_{\sigma _j}(F)} d_H(\epsilon ,\epsilon ') = 0$. Since all distances are non-negative number, if the sum is equal to zero it means that $\forall \epsilon \in Ext_{\sigma _i}(F)$, $\min _{\epsilon ' \in Ext_{\sigma _j}(F)} d_H(\epsilon ,\epsilon ') = 0$. Because of the properties of the Hamming distance, it means that $\epsilon \in Ext_{\sigma _j}$, and so $Ext_{\sigma _i} \subseteq Ext_{\sigma _j}$. From our starting assumption, we deduce that $\sigma _i = \sigma _j$.

Proof

(Proof of Proposition 6). From the definition of the measure, $\delta _{F,sym}^{d_H,\sum }(\sigma _1,\sigma _2) = 0$ iff $Ext_{\sigma _1}(F) = Ext_{\sigma _2}(F)$. Under our assumptions, this is possible only if $\sigma _1 = \sigma _2$. The other direction is trivial, so coincidence is satisfied. Symmetry is obviously satisfied, since $\sigma _1,\sigma _2$ can be inverted in $\max (\delta _{F}^{d,\otimes }(\sigma _1,\sigma _2),\delta _{F}^{d,\otimes }(\sigma _2,\sigma _1))$.

Proof

(Proof of Proposition 7). Weak coincidence and symmetry are trivial from the definition of the measures.

$$\begin{aligned} \begin{array}{rcl} \delta _{F,sk}^{d}(\sigma _1,\sigma _2) + \delta _{F,sk}^{d}(\sigma _2,\sigma _3) &{} = &{} d(sk_{\sigma _1}(F), sk_{\sigma _2}(F)) + d(sk_{\sigma _2}(F), sk_{\sigma _3}(F)) \\ &{} \ge &{} d(sk_{\sigma _1}(F), sk_{\sigma _3}(F)) = \delta _{F,sk}^{d}(\sigma _1,\sigma _3) \end{array} \end{aligned}$$

The same reasoning apply for the credulous acceptance measure. So both satisfy the triangular inequality. Coincidence is not satisfied by the skeptical acceptance measure. For instance, for each AF F, $\emptyset \in Ext_{cf}(F)$ and $\emptyset \in Ext_{adm}(F)$, so $sk_{cf}(F) = sk_{adm}(F) = \emptyset$, and so $\delta _{F,skep}^{d}(cf,adm) = 0$. The same conclusion holds as soon as two semantics yield the same skeptically or credulously accepted arguments.