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, 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 or the value-based argumentation frameworks. Abstract argumentation frameworks. Formal framework. Abstract argumentation frameworks, also called argumentation frameworks "à la Dung", are defined formally as a pair: For instance, the argumentation system formula_4 with formula_5 and formula_6 contains four arguments (formula_7 and formula_8) and three attacks (formula_9 attacks formula_10, formula_10 attacks formula_12 and formula_8 attacks formula_12). 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 allows, given an argumentation system, sets  of arguments (called "extensions") to be computed. For instance, given formula_26, There exists some inclusions between the sets of extensions built with these semantics : Some other semantics have been defined. One introduce the notation formula_44 to note the set of formula_45-extensions of the system formula_28. In the case of the system formula_28 in the figure above, formula_48 for every Dung's semantic—the system is well-founded. That explains why the semantics coincide, and the accepted arguments are: formula_9 and formula_8. 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 formula_51. Such a mapping does not make sense without additional constraint. The notion of reinstatement labelling guarantees the sense of the mapping. formula_52 is a reinstatement labelling on the system formula_4 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 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 some argument "in", then it is "out". The unique reinstatement labelling that corresponds to the system formula_28 above is formula_65. Inference from an argumentation system. In the general case when several extensions are computed for a given semantic formula_45, the agent that reasons from the system can use several mechanisms to infer information: For these two methods to infer information, one can identify the set of accepted arguments, respectively formula_73 the set of the arguments credulously accepted under the semantic formula_45, and formula_75 the set  of arguments accepted skeptically under the semantic formula_45 (the formula_45 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 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). Equivalence between argumentation frameworks. There exists several criteria of equivalence between argumentation frameworks. Most of those criteria concern the sets of extensions or the set of accepted arguments. Formally, given a semantic formula_45 : The strong equivalence says that two systems formula_86 and formula_87 are equivalent if and only if for all other system formula_88, the union of formula_86 with formula_88 is equivalent (for a given criterion) with the union of formula_87 and formula_88. Other kinds. 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 formula_93 such that One calls formula_98 a consequence of formula_96, and formula_96 a support of formula_98. In this case, the attack relation is not given in an explicit way, as a subset of the Cartesian product formula_104, 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 on the difference of these values. Formally, a value-based argumentation framework is a tuple formula_116 with formula_1 and formula_3 similar to the standard framework (a set of arguments and a binary relation on this set), formula_119 is a non empty set of values, formula_120 is a mapping that associates each element from formula_1 to an element from formula_119, and formula_123 is a preference relation (transitive, irreflexive and asymmetric) on formula_124. In this framework, an argument formula_9 defeats another argument formula_10 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. Assumption-based argumentation frameworks. 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 formula_133, where As a consequence of defining an ABA, an argument can be represented in a tree-form. Formally, given a deductive system formula_134 and set of assumptions formula_141, an argument for claim formula_149 supported by formula_150, is a tree with nodes labelled by sentences in formula_135 or by symbol  formula_152, such that: An argument with claim formula_12 supported by a set of assumption formula_28 can also be denoted as formula_173