# Argumentation framework

An argumentation framework, or argumentation system, is a way to deal with contentious information and draw conclusions from it.

In an abstract argumentation framework,[1] entry-level information is a set of abstract arguments that, for instance, represent data or a proposition. Conflicts between arguments are represented by a binary relation on the set of arguments. In concrete terms, you represent an argumentation framework with a directed graph such that the nodes are the arguments, and the arrows represent the attack relation. There exist some extensions of the Dung's framework, like the logic-based argumentation frameworks[2] or the value-based argumentation frameworks.[3]

## Abstract argumentation frameworks

### Formal framework

Abstract argumentation frameworks, also called argumentation frameworks à la Dung, are defined formally as a pair:

Dung defines some notions :

### Different semantics of acceptance

#### Extensions

To decide if an argument can be accepted or not, or if several arguments can be accepted together, Dung defines several semantics of acceptance that allow, given an argumentation system, to compute sets of arguments, called extensions. For instance, given ${\displaystyle S=\langle A,R\rangle }$,

There exists some inclusions between the sets of extensions built with these semantics :

Some other semantics have been defined.[4]

One introduce the notation ${\displaystyle Ext_{\sigma }(S)}$ to note the set of ${\displaystyle \sigma }$-extensions of the system ${\displaystyle S}$.

In the case of the system ${\displaystyle S}$ in the figure above, ${\displaystyle Ext_{\sigma }(S)=\{\{a,d\}\}}$ for every Dung's semantic—the system is well-founded. That explains why the semantics coincide, and the accepted arguments are: ${\displaystyle a}$ and ${\displaystyle d}$.

#### Labellings

Labellings are a more expressive way than extensions to express the acceptance of the arguments. Concretely, a labelling is a mapping that associates every argument with a label in (the argument is accepted), out (the argument is rejected), or undec (the argument is undefined—not accepted or refused). One can also note a labelling as a set of pairs ${\displaystyle ({\mathit {argument}},{\mathit {label}})}$.

Such a mapping does not make sense without additional constraint. The notion of reinstatement labelling guarantees the sense of the mapping. ${\displaystyle L}$ is a reinstatement labelling on the system ${\displaystyle S=\langle A,R\rangle }$ if and only if :

One can convert every extension into a reinstatement labelling: the arguments of the extension are in, those attacked by an argument of the extension are out, and the others are undec. Conversely, one can build an extension from a reinstatement labelling just by keeping the arguments in. Indeed, Caminada[5] proved that the reinstatement labellings and the complete extensions can be mapped in a bijective way. Moreover, the other Datung's semantics can be associated to some particular sets of reinstatement labellings.

Reinstatement labellings distinguish arguments not accepted because they are attacked by accepted arguments from undefined arguments—that is, those that are not defended cannot defend themselves. An argument is undec if it is attacked by at least another undec. If it is attacked only by arguments out, it must be in, and if it is attacked only by arguments in, then it is out.

The unique reinstatement labelling that corresponds to the system ${\displaystyle S}$ above is ${\displaystyle L=\{(a,{\mathit {in}}),(b,{\mathit {out}}),(c,{\mathit {out}}),(d,{\mathit {in}})\}}$.

### Inference from an argumentation system

In the general case when several extensions are computed for a given semantic ${\displaystyle \sigma }$, the agent that reasons from the system can use several mechanism to infer information:

For these two methods to infer information, one can identify the set of accepted arguments, respectively ${\displaystyle Cr_{\sigma }(S)}$ the set of the arguments credulously accepted under the semantic ${\displaystyle \sigma }$, and ${\displaystyle Sc_{\sigma }(S)}$ the set of arguments accepted skeptically under the semantic ${\displaystyle \sigma }$ (the ${\displaystyle \sigma }$ can be missed if there is no possible ambiguity about the semantic).

Of course, when there is only one extension (for instance, when the system is well-founded), this problem is very simple: the agent accepts arguments of the unique extension and rejects others.

The same reasonning can be done with labellings that correspond to the chosen semantic : an argument can be accepted if it is in for each labelling and refused if it is out for each labelling, the others being in an undecided state (the status of the arguments can remind the epistemic states of a belief in the AGM framework for dynamic of beliefs[6]).

### Equivalence between argumentation frameworks

There exists several criterions of equivalence between argumentation frameworks. Most of those criterions concern the sets of extensions or the set of accepted arguments. Formally, given a semantic ${\displaystyle \sigma }$ :

The strong equivalence[7] says that two systems ${\displaystyle S_{1}}$ and ${\displaystyle S_{2}}$ are equivalent if and only if for all other system ${\displaystyle S_{3}}$, the union of ${\displaystyle S_{1}}$ with ${\displaystyle S_{3}}$ is equivalent (for a given criterion) with the union of ${\displaystyle S_{2}}$ and ${\displaystyle S_{3}}$.[8]

## Other kind of argumentation frameworks

The abstract framework of Dung has been instantiated to several particular cases.

### Logic-based argumentation frameworks

In the case of logic-based argumentation frameworks, an argument is not an abstract entity, but a pair, where the first part is a minimal consistent set of formulae enough to prove the formula for the second part of the argument. Formally, an argument is a pair ${\displaystyle (\Phi ,\alpha )}$ such that

In this case, the attack relation is not given in an explicit way, as a subset of the Cartesian product ${\displaystyle A\times A}$, but as a property that indicates if an argument attacks another. For instance,

Given a particular attack relation, one can build a graph and reason in a similar way to the abstract argumentation frameworks (use of semantics to build extension, skeptical or credulous inference), the difference is that the information inferred from a logic based argumentation framework is a set of formulae (the consequences of the accepted arguments).

### Value-based argumentation frameworks

The value-based argumentation frameworks come from the idea that during an exchange of arguments, some can be stronger than others with respect to a certain value they advance, and so the success of an attack between arguments depends of the difference of these values.

Formally, a value-based argumentation framework is a tuple ${\displaystyle VAF=\langle A,R,V,val,valprefs\rangle }$ with ${\displaystyle A}$ and ${\displaystyle R}$ similar to the standard framework (a set of arguments and a binary relation on this set), ${\displaystyle V}$ is a non empty set of values, ${\displaystyle val}$ is a mapping that associates each element from ${\displaystyle A}$ to an element from ${\displaystyle V}$, and ${\displaystyle valprefs}$ is a preference relation (transitive, irreflexive and asymmetric) on ${\displaystyle V\times V}$.

In this framework, an argument ${\displaystyle a}$ defeats another argument ${\displaystyle b}$ if and only if

One remarks that an attack succeeds if both arguments are associated to the same value, or if there is no preference between their respective values.

## Notes

1. See Dung (1995)
2. See Besnard and Hunter (2001)
3. see Bench-Capon (2002)
4. For instance,
• Ideal : see Dung, Mancarella and Toni (2006)
• Eager : see Caminada (2007)
6. see Gärdenfors (1988)
7. see Oikarinen and Woltran (2001)
8. the union of two systems represents here the system built from the union of the sets of arguments and the union of the attack relations

## References

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