Argumentation framework
In artificial intelligence and related fields, an argumentation framework is a way to deal with contentious information and draw conclusions from it using formalized arguments.
In an abstract argumentation framework,[1] entry-level information is a set of abstract arguments that, for instance, represent data or a proposition.
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.
That explains why the semantics coincide, and the accepted arguments are:
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.
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.
The unique reinstatement labelling that corresponds to the system
In the general case when several extensions are computed for a given semantic
, the agent that reasons from the system can use several mechanisms to infer information:[6] 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
the set of arguments accepted skeptically under 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 reasoning 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[7]).
There exists several criteria of equivalence between argumentation frameworks.
[9] The abstract framework of Dung has been instantiated to several particular cases.
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.
In this case, the attack relation is not given in an explicit way, as a subset of the Cartesian product
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).
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 on the difference of these values.
Formally, a value-based argumentation framework is a tuple
is a preference relation (transitive, irreflexive and asymmetric) on
In assumption-based argumentation (ABA) frameworks, arguments are defined as a set of rules and attacks are defined in terms of assumptions and contraries.
Formally, an assumption-based argumentation framework is a tuple
,[10][11][12] where As a consequence of defining an ABA, an argument can be represented in a tree-form.
, is a tree with nodes labelled by sentences in
