Extension-based approaches

Acceptability of argument is used to define argumentation semantics. Two different methods are proposed to define semantics: extension-based [Dung, 1995] and labeling-based [Caminada, 2006]. We start by the extension-based approach which defines what an acceptable argument means under some spe- cific semantics. The idea behind the extension-based approach is to identify and select a set of arguments called extensions that can survive a conflict together. Thus, an extension is often represented as a reasonable position or viewpoint in a debate. The reader can find examples of the semantics presented in this thesis (the admissible, complete, grounded, preferred and stable) in Dung [1995] and an intuitive introduction to argumentation se- mantics can be found in Baroni et al. [2011].

Extension-based semantics are defined on the principle of conflict-freeness which translates the idea the arguments in an extension should be able to “stand together”, that is, the arguments of the same extension should not attack each other.

Definition 2.36 (Conflict-freeness). Let F = (A , R) be an argumenta- tion framework. A set of arguments S ⊆ A is conflict-free if and only if there are noa,b ∈ S such that (a,b) ∈ R.

Please note that Definition 2.36 excludes all of the sets containing self- attacking arguments.

Example 2.15 (Example 2.13 cont’d). {a, c} is conflict-free.

A maximal conflict-free set of arguments is called a naive extension. Definition 2.37 (Naive semantics). Let F= (A , R) be an argumentation framework. A set of argumentsS ⊆ A is a naive extension if and only if S is conflict-free and for everyS0⊆ A such that S ⊂ S0,S0 is not conflict-free. A set of non-conflicting arguments can be seen as an agent’s position in a debate, for this position to hold it has to defend all its argument. This corresponds to the notion of admissibility [Dung, 1995].

Definition 2.38 (Admissibility of a set). Let F = (A , R) be an argu- mentation framework. A conflict-free set of argumentsS ⊆ A is admissible if and only if every argumenta ∈ S is acceptable with respect to S.

An admissible set of arguments is a set of non-conflicting arguments that defends all its elements, such set is called an admissible extension. Every argumentation framework has at least one admissible set: the empty set. Example 2.16 (Example 2.13 cont’d). The admissible extensions are: ∅, {a}, {c}, and {a, c}. Note that{b} is not an admissible set since it does not defend itself fromc.

A preferred set of arguments is a maximal set of arguments that is ad- missible. The idea of the preferred semantics is that one wants to accept as many arguments as reasonably possible to have the largest viewpoint on a debate.

Definition 2.39 (Preferred semantics). Let F= (A , R) be an argumen- tation framework. A preferred extension is a maximal (for set inclusion) admissible set of argumentsS ⊆ A .

A stable set of arguments S is a conflict-free set that attacks all of the arguments outside ofS. The idea of the stable semantics is that an argument can only be for or against a viewpoint in a debate and that neutrality is not allowed.

Definition 2.40 (Stable semantics). Let F= (A , R) be an argumentation framework. A stable extension is a conflict-free set of argumentsS ⊆ A such that for everyb ∈ (A \S) there exists an argument a ∈ S such that (a,b) ∈ R.

A complete set of arguments is an admissible set that contains all of the arguments that it defends. The complete semantics refines the admissibility in the sense that one should always accept an argument if it can be defended. Definition 2.41 (Complete semantics). Let F= (A , R) be an argumen- tation framework. An admissible set of arguments S ⊆ A is a complete extension if and only if for everya ∈ A , if S defends a then a ∈ S.

Grounded semantics is the most skeptical (or least committed) of argu- mentation semantics, it is defined based on the notion of complete extension. It is the admissible extension that includes all the arguments it can defend from all attacks.

Definition 2.42 (Grounded semantics). The grounded extension of an argumentation framework is the least (with respect to set-inclusion) complete extension.

In some cases, the stable semantics yields no extensions at all (not even the empty set). That is why a more refined approach was defined: the semi- stable semantics [Caminada et al., 2012]. Please note that this semantics is equivalent to the admissible stage semantics defined by Verheij [1999]. Definition 2.43 (Semi-stable semantics [Caminada et al., 2012]). Let F= (A , R) be an argumentation framework. A semi-stable extension is a complete extension S such that S ∪{b ∈ A | there exists a ∈ S such that (a,b) ∈ R} is maximal (with respect to set inclusion) amongst all complete extensions.

Contrary to the stable extensions, the existence of the semi-stable exten- sions is always guaranteed. Furthermore, a stable semantics is a semi-stable extension and semi-stable extensions coincide with stable extensions when the set of stable extensions is not empty.

An ideal extension is a maximal for set inclusion set of argument that is a subset of each preferred extension. It was shown by Caminada and Pigozzi [2011] that the ideal extension is also a complete extension and thus it is a superset of the grounded extension.

Definition 2.44 (Ideal semantics [Caminada and Pigozzi, 2011]). Given an argumentation framework F = (A , R). An admissible set S is called ideal if and only if it is a subset of each preferred extension. The ideal extension of F is a maximal (with respect to set inclusion) ideal set.

For our purposes, we require some further formal notions. An argumen- tation framework is strongly connected if and only if there is a path from any argumenta to any argument a0.

Definition 2.45 (Strongly connected). Let F= (A , R) be an argumen- tation framework. We say that F is strongly connected if and only if for every a, a0_{∈A such that a , a}0_{, there is a path from argumenta to argument a}0_.

The nodes of an arbitrary directed graph can be partitioned such that the subgraphs, induced by each set of nodes, are maximal strongly connected subgraphs. Each set of such a partition is called a strongly connected com- ponents of this graph. In the rest of this thesis, we will denote bySCC(F), this particular partition of the set of arguments of F.

Definition 2.46 (Component-defeated [Gaggl and Woltran, 2013]). Let F = (A , R) be an argumentation framework and S ⊆ A a set of argu- ments. An argumentb ∈ A is component-defeated by S if there exists a ∈ S such that (a,b) ∈ R and a is not in same the strongly connected component thanb. The set of arguments component-defeated by S in F is DF(S).

All of the above mentioned argumentation semantics are admissibility- based, i.e. the extension returned are admissible sets. Moreover, in the multiple-status semantics (such as complete, preferred, stable and semi- stable), we can notice that odd-length unidirectional attack cycles are han- dled badly. However, in some applications, cycles need to be treated equally independently of their length [Pollock, 2001]. The stage semantics conforms with this idea of “equal cycles treatment” but loses its proximity with the grounded semantics as it was shown that even non attacked arguments can be rejected in some cases [Baroni et al., 2011]. Against this background, the cf2 semantics was designed as a multiple-status semantics that is not admissibility-based, treats cycles equally and which accepted arguments are a superset of those accepted by the grounded semantics.

Definition 2.47 (Cf2 semantics [Gaggl and Woltran, 2013]). Let F= (A , R) be an argumentation framework and S ⊆ A be a set of arguments. S is a cf2 extension of F if and only if:

• in case |SCC(F)| = 1, then S is a maximal conflict free set of F, • otherwise, for every C ∈ SCC(F), (S ∩C) is a cf2 extension of (A∩Y , R ∩

(Y × Y )) where Y = C \ DF(S).

Notation 2.7. Let F be an argumentation framework, we will denote by Extx(F) the set of extensions with respect to the argumentation semantics x for F. We use the abbreviationsc f , a,p, s, c,д, ss, i and c f 2 for respectively conflict-free, admissible, preferred, stable, complete, grounded, semi-stable, ideal and cf2.

Definition 2.48 (Sceptically accepted, credulously accepted and rejected arguments). Let F= (A , R) be an argumentation framework and Extx(F) be the set of extensions with respect to the argumentation semantics x for F. We say that:

• a is sceptically accepted with respect to x if and only if for every ε ∈ Extx(F), a ∈ ε.

• a is credulously accepted with respect to x if and only if for there exists ε1, ε2 ∈Extx(F), such that a ∈ ε1 and a < ε2.

• a is rejected with respect to x if and only if for every ε ∈ Extx(F), a < ε. Example 2.17 (Argumentation semantics). Consider the argumenta- tion framework F= (A , R) such that A ={a,b, c,d} and R = {(e, e), (d, e), (d, c), (c,d), (b, c), (a,b)}, represented in Figure 2.4. We make the following observations:

• The admissible extensions are {d}, {a}, {a,d}, {a, c} and ∅. • The complete extensions are {a}, {a, c},{a,d}and ∅. • The preferred and cf2 extensions are {a, c} and {a,d}. • The stable and semi-stable extension is {a,d}.

• The ideal extension is {a}.

• The least complete extension is {a} which is the ground extension.

a b

c d

e

Figure 2.4: Argumentation framework of Example 2.17

Stable Semi-stable Preferred Complete Admissible Conflict-free Cf2 Grounded Ideal Admissiblity-based SCC-based

Figure 2.5: Inclusion relations between the several argumentation semantics used in this thesis.

Semantics CF DF ADM INCDF MAX AGR UNIQ EXIST Admissible X X X X Complete X X X X X Stable X X X X X Semi-stable X X X X X X Preferred X X X X X X Grounded X X X X X X Cf2 X X X Ideal X X X X X X X

Table 2.2: Argumentation semantics with respect to criteria. X means the criterion is satisfied

In Figure 2.5, we show the inclusion relations between the several ar- gumentation semantics used in this thesis. An arrow from the node A to the node B means that an extension for semantics A is also an extension for semanticsB. In Table 2.2, we summarise the semantics and their essen- tial criteria. The criteria are as follows: CF means that the extensions are conflict-free, DF means that they defend all their elements, INCDF means that they include what they defend, MAX means that they are maximal with respect to inclusion, AGR means that they attack all arguments that are outside of the extension, UNIQ means that there is always one extension and EXIST means that there is always at least one extension. The table is only an illustration and the criteria are not completely dependent as some of them are derivable from others.