To model the process of abstract argumentation Dung introduced an Abstract Argumentation Framework (Argumentation Framework - AF). In this framework it is possible to model the arguments and their relations. These relations are modeled via binary attacks among two argu-ments. So we can represent that an individual argument is attacking another one or is attacked by it.
Definition 2.2.1(Argumentation Framework [Dung, 1995]). An Argumentation Framework is a pair
AF = (AR, AT)
whereARis a set of arguments, andAT is a binary relation onAR, i.e.AT ⊆AR×AR The meaning of(A, B) ∈ AT for two argumentsA, B ∈AR, is that(A, B)represents an attack ofAagainstB. As the definition of the framework is syntactically the same as for a di-rected graph, frameworks can be directly represented as such, where the nodes are the arguments and the attacks are the directed edges.
Example 2.2.2.
TheAF1 = ({A, B, C, D, E},{(A, B),(C, B),(C, D),(D, C),(D, E),(E, E)}) can also be represented by the following graph:
A B C D E
2.2 Dung’s Abstract Argumentation Frameworks In this example we can see that argumentAis not attacked by anyone, while argumentC and Dhave a mutual attack relation between them. ArgumentEis not only attacked byD, but also attacks itself.
This argumentation framework can model different situations. Maybe the 5 arguments stand for different employees in a company and they have to form a team. The attacks can stand for the preferences such that one employee does not want to work with another one for some distinct reasons. Based on the given AF, everyone will be fine withAin the team and maybeEdoes not want to be in the team (as he attacks himself).
Based on theAF we want to get a selection of arguments which is based on the relations between the different arguments. This selection shall model the reasoning process to determine consistent or reasonable sets which lead to a conclusion. Which arguments are selected is deter-mined by thesemantics, which are defined for the framework.
The most intuitive property is theconflict-free set (Definition 2.2.3), where no arguments are inside the set, which are attacking each other. Therefore there cannot be any direct conflicts among the selected arguments.
Definition 2.2.3(Conflict-free set [Dung, 1995]). LetAF = (AR, AT)be an argumentation framework. A set S ⊆ AR of arguments is said to be conflict-free in AF if there are no arguments AandB inS such thatAattacksB. The set cf(AF)is the set of allconflict-free setsfor the argumentation frameworkAF.
Example 2.2.4(Conflict-free set, based on Example 2.2.2).
cf(AF1) ={∅,{A},{B},{C},{D},{A, C},{A, D},{B, D}}
The notion of aconflict-free setis only avoiding attacks between any pair of selected argu-ments. As it does not care about attacks from the unselected arguments one can state that the constructed set can be attacked, and therefore invalidated easily. Based on the example (Exam-ple 2.2.2 and 2.2.4),{∅}and{A}would be the only twoconflict-free sets, where the selected arguments are not attacked from unselected ones. To ensure that the selected arguments are still prepared against attacks from unselected arguments we can “defend”them by an attack from one selected argument against the unselected one. If an argument inside the conflict-free set M ∈cf(AF)is“defended”, we say that it isacceptable w.r.t.M. The setM1 ⊆ARis said to attack the setM2⊆ARif a relation(a, b)∈AT exists, such thata∈M1andb∈M2.
Definition 2.2.5(Acceptable argument [Dung, 1995]). LetAF = (AR, AT)be an argumenta-tion framework. An argumentA ∈ ARisacceptableinAF with respect to a setS ⊆ ARof arguments iff for each argumentB ∈AR: ifBattacksAthenBis attacked byS.
Definition 2.2.6 (Admissible set [Dung, 1995]). Let AF = (AR, AT) be an argumentation framework. A conflict-free set of argumentsS ⊆ARisadmissibleinAF iff each argument in Sis acceptable with respect toS. The set of admissible sets of the frameworkAF is denoted by adm(AF).
Example 2.2.7(Admissible set, based on Example 2.2.2).
adm(AF1) ={∅,{A},{C},{D},{A, C},{A, D}}
We can see that in Example 2.2.7 someadmissible setsof arguments are included in another one. To define the (credulous) semantics for theAFwe need to represent the maximaladmissible set, which is called thepreferred extension.
Definition 2.2.8(Preferred extension [Dung, 1995]). Apreferred extensionof an argumentation frameworkAF is a maximal (with respect to set inclusion) admissible set ofAF.
Example 2.2.9(Preferred extension, based on Example 2.2.2). Our example has two preferred extensionEAF1
1 ={A, C}andEAF2
1 ={A, D}.
Another, even more restricted semantics is thestable extension.
Definition 2.2.10(Stable extension [Dung, 1995]). LetAF = (AR, AT)be an argumentation framework. A conflict-free set of argumentsS ⊆ ARis called astable extensioninAF iff S attacks each argument which does not belong toS.
Example 2.2.11(Stable extension, based on Example 2.2.2).
Here the stable extension isSEAF1 ={A, D}.
It is obvious to see that thestable extensionis also apreferred extension in the examples.
Indeed one of the results by Dung states thatevery stable extension is a preferred extension, but not vice versa.[Dung, 1995].
In contrast to the credulous semantics another approach is used to introduce the skeptical seman-tics. This one is thegrounded extension(Definition 2.2.13), which is based on a characteristic fix-point function.
Definition 2.2.12 (Characteristic function [Dung, 1995]). The characteristic function FAF : 2AR→2ARof an argumentation frameworkAF = (AR, AT)is defined as follows:
FAF(S) ={A|Ais acceptable with respect toS}.
Definition 2.2.13(Grounded extension [Dung, 1995]). The grounded extensionGEAF of an argumentation frameworkAF is the unique least fixed point ofFAF.
Additionally there is another semantics defined, which relates the preferred extensions and the grounded extensions (i.e. relating credulous and skeptical semantics). This is achieved with thecomplete extension.
Definition 2.2.14(Complete extension). An admissible setS ⊆ ARof arguments is called a complete extensionof an argumentation frameworkAF = (AR, AT)iff each argument, which is acceptable with respect toS, belongs toS.
Example 2.2.15(Grounded and complete extensions, based on Example 2.2.2). In our example the grounded extension would beGEAF1 = {A} and the complete extensions are CEAF1 = {{A},{A, C},{A, D}}.
2.2 Dung’s Abstract Argumentation Frameworks The mentioned relation between the extensions is summarized in Theorem 2.2.16.
Theorem 2.2.16([Dung, 1995]).
(I) Each preferred extension is a complete extension, but not vice versa.
(II) The grounded extension is the least (with respect to set inclusion) complete extension.
(III) The complete extensions form a complete semilattice1with respect to set inclusion.
It is obvious that theAFis only capable of modeling binary attacks directly among two ar-guments. More sophisticated relations are represented via distinct structures which are specially tailored for a prior chosen semantics. In the following we will describe a few approaches for suchsemantic structureswhich are primary based on the stable extension.
Example 2.2.17(Semantic structure for a support relation).
A supp B
The aboveAFmodels a support between the nodeAand the nodeBunder the stable extension.
We might not be interested in thesupport-argument “supp”and remove it from the set of argu-ments in the final conclusion.
The above structure has the following stable extension:SE ={A, B}
Example 2.2.18(Semantic structure for a distributed attack). Here we want to give an example for a distributed attack (i.e. two argumentsAandBattack together one argumentC, but only if both are accepted) under the stable extension.
A B
C
An Bn
Ah Bh
H
1A partial order(S,≤)is a complete semilattice iff each nonempty subset ofShas a greatest lower bound and each increasing sequence ofShas a lower upper bound.
Here we have the following stable extensions (the interesting nodes are underlined):
SE1 ={A, B, H}
SE2 ={C, An, Bn, Ah, Bh} SE3 ={B, C, An, Ah} SE4 ={A, C, Bn, Bh}
Example 2.2.17 shows that we can define a support with the help of anotherhelperargument, which forces the stable extension to selectB if we have selectedA. Example 2.2.18 tries to define a structure which pictures a distributed attack. Here both attacking arguments,AandB, need to be selected by the semantics to have an applicable attack against the third argumentC.
These examples shall only motivate that it is possible to represent more relations between the arguments than only an attack. We still have to keep in mind that we need special structures which are bound to a specific semantics. For further observations and discussions about relation of the semantics of Dung’s AF, see [Baroni et al., 2011a].