# Argumentation framework

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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:

- A set of abstract elements called
*arguments*, noted - A binary relation on , called
*attack relation*, noted

For instance, the argumentation system with and contains four arguments ( and ) and three attacks ( attacks , attacks and attacks ).

Dung defines some notions :

- an argument is acceptable with respect to if and only if defends , that is such that such that ,
- a set of arguments is conflict-free if there is no attack between its arguments, formally : ,
- a set of arguments is admissible if and only if it is conflict-free and all its arguments are acceptables with respect to .

### 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 ,

- is a complete extensions of only if it is an admissible set and every argument acceptable argument with respect to belongs to ,
- is a preferred extension of only if it is a maximal element (with respect to the set-theoretical inclusion) among the admissible sets with respect to ,
- is an extension stable of only if it is a conflict-free set that attacks every argument that do not belong in (formally, such that ,
- is the (unique) grounded extension of only if it is the smallest element (with respect to the inclusion) among the complete extensions of .

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

- Every stable extension is preferred,
- Every preferred extension if complete,
- The grounded extension is complete,
- If the system is well founded (there exists no infinite sequence such that ), all these semantics coincide—only one extension is grounded, stable, preferred, and complete.

Some other semantics have been defined.^{[4]}

One introduce the notation to note the set of -extensions of the system .

In the case of the system in the figure above, for every Dung's semantic—the system is well-founded. That explains why the semantics coincide, and the accepted arguments are: and .

#### 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 .

Such a mapping does not make sense without additional constraint. The notion of reinstatement labelling guarantees the sense of the mapping. is a reinstatement labelling on the system 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 above is .

### Inference from an argumentation system

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

*Credulous inference*: the agent accepts an argument if it belongs to at least one of the -extensions—in which case, the agent risks accepting some arguments that are not acceptable together ( attacks , and and each belongs to an extension)*Skeptical inference*: the agent accepts an argument only if it belongs to every -extension. In this case, the agent risks deducing a little information (if the intersection of the extensions is empty or with a very small cardinal).

For these two methods to infer information, one can identify the set of accepted arguments, respectively the set of the arguments credulously accepted under the semantic , and the set of arguments accepted skeptically under the semantic (the 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 :

- : two argumentation frameworks are equivalent if they have the same set of -extensions, that is ;
- : two argumentation frameworks are equivalent if they accept skeptically the same arguments, that is ;
- : two argumentation frameworks are equivalent if they accept credulously the same arguments, that is .

The strong equivalence^{[7]} says that two systems and are equivalent if and only if for all other system , the union of with is equivalent (for a given criterion) with the union of and .^{[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 such that

One calls a consequence of , and a support of .

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

- Relation
*defeater*: attacks if and only if for - Relation
*undercut*: attacks if and only if for - Relation
*rebuttal*: attacks if and only if is a tautology

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 with and similar to the standard framework (a set of arguments and a binary relation on this set), is a non empty set of values, is a mapping that associates each element from to an element from , and is a preference relation (transitive, irreflexive and asymmetric) on .

In this framework, an argument defeats another argument if and only if

- attacks in the "standard" meaning: ;
- and , that is the value advanced by is not preferred to the one advanced by .

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.

## See also

## Notes

- ↑ See Dung (1995)
- ↑ See Besnard and Hunter (2001)
- ↑ see Bench-Capon (2002)
- ↑ For instance,
*Ideal*: see Dung, Mancarella and Toni (2006)*Eager*: see Caminada (2007)

- ↑ see Caminada (2006)
- ↑ see Gärdenfors (1988)
- ↑ see Oikarinen and Woltran (2001)
- ↑ 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

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