In the previous section we already discussed the differences between most of the semantics, especially the basic semantics defined by Dung are very well known. As the focus of this work is mainly on naive-based semantics and out of them thecf2 semantics, we point out here some special properties and differences between those semantics, where our analysis will be mostly example-driven, and we classify the semantics w.r.t. their subset-relation.
The first example we consider in this context shows one significant difference betweencf2 and stage semantics.
Figure 2.7: The argumentation frameworkF from Example 7.
Example 7. Let F = (A, R) with A = {a, b, c}andR = {(a, b), (b, c), (c, b),(c, c)}as in Figure 2.7. Then, the above defined semantics yield the following extensions.
• stable(F) =∅;
• adm(F) ={{},{a}};
• pref(F) =grd(F) ={{a}}; and
• naive(F) ={{a},{b}}.
Regardingcf2, we check for the two naive setsS={a}andT ={b}if they arecf2 extensions ofF. AsF has two SCCsC1 ={a}andC2 ={b, c},DF(S) ={b}andDF(T) =∅. We first check ifS∈cf2(F)as in Definition 13 .
• (S∩C1)∈cf2(F|C1)holds as{a} ∈naive(F|C1), and
• (S∩C2)∈cf2(F|C2 − {b})holds as∅ ∈naive(F|{c}).
Thus,S∈cf2(F). Next we make the check for the setT.
• (T∩C1)6∈cf2(F|C1)because(T∩C1) =∅andnaive(F|C1) ={{a}},
• (T∩C2)∈cf2(F|C2)holds as{b} ∈naive(F|C2).
As the first check forT failed, we obtain thatT 6∈cf2(F).
Regarding stage semantics, both setsSandT are stage extensions. If we have a closer look at the setT, we see thatTR+={b, c}and there is noU ∈cf(F)s.t.UR+⊃TR+. ✸ The AF of this example shows that stage semantics can accept an extension which does not include the grounded extension. Moreover, the stage extension T = {b} is attacked by the unattacked argumenta. This can be seen as a drawback and, besides naive sets, stage semantics is the only one considered so far showing this behavior. In Chapter 4 we introduce the new semanticsstage2 which repairs this drawback.
For stable semantics we already mentioned that it is the only semantics where it can be the case, that there does not exist any extension. This is due to the fact that the requirements for stable semantics are very strong. Furthermore, stable semantics is the only one falling into both categories, the admissible-based and the naive-based semantics.
Next we consider in more detail thecf2 semantics, as it has some special properties which clearly differ from the admissible-based semantics. Especially the treatment of odd- and even-length cycles is more uniform in the case ofcf2 semantics.
Figure 2.8:The modified AFF′from Example 5.
Figure 2.9: FrameworkF. Figure 2.10:Modified FrameworkG.
Figure 2.11:AFF from Example 9.
For our framework from Example 5 we obtain{g, i}as the only preferred extension. This comes due to the fact that in an odd-length cycle, as we have it in this example none of the argumentsa, bandccan be defended. We modify the framework in the sense that we include a new argumentx which makes the cycle even, as illustrated in Figure 2.8. Then, we obtain totally different preferred extensions, namely{b, x, g, i},{b, x, f, h}and{a, c, d, e, g, i}which are conform with thecf2 extensions of the modified AFF′.
The main motivation behind selecting arguments out of an odd-length cycle is to see the arguments as different choices and to be able to choose between them. Then, there is no need for defense. Consider the following example which illustrates this idea [88].
Example 8. Suppose there are three witnessesA,BandC, whereAstates thatBis unreliable, B states thatCis unreliable andCstates thatAis unreliable. Moreover,Chas a statementS.
The graph of the frameworkF is illustrated in Figure 2.9. Any admissible-based semantics re-turns the empty set as its only extension. But if we have four rather than three witnesses, let’s call the fourth oneX, as in the AFGpictured in Figure 2.10, the situation changes, and the preferred extensions ofGare {a, c, s}and{b, x}. On the other hand, the naive-based semantics return stage(F) =cf2(F) ={{b},{a, s},{c, s}}andstage(G) =cf2(G) ={{a, c, s},{b, x}}. ✸ One special case of an odd-length cycle are self-attacking arguments.
Example 9. Consider the AF F as in Figure 2.11. Then, the empty set is the only preferred extension, whereas{a}is acf2 extension. The motivation behind selecting{a}as a reasonable
Figure 2.12:AF from Example 10.
extension is that it is not necessary to defendaagainst the attack fromb, asbis a self-attacking
argument. ✸
Till now, we only mentioned positive properties of thecf2 semantics compared to admissible-based semantics. The next example will show a more questionable behavior.
Example 10. Consider the AFF in Figure 2.12. We obtain
• stage(F) =pref(F) =stable(F) ={{a, c, e},{b, d, f}}, but
• cf2(F) =naive(F) ={{a, d},{b, e},{c, f},{a, c, e},{b, d, f}}.
In this example we have an even-length cycle and thecf2 semantics produce three more exten-sions. This does not really coincide with the motivation for a symmetric treatment of odd- and even-length cycles, as now the results differ significantly for an even-length cycle. ✸ One suggestion to repair the undesired behavior from Example 10, is to check in Definition 13 for the case|SCCs(F)| = 1whetherS ∈ stage(F) instead of S ∈ naive(F). In Chapter 4 we formalize this modification of cf2 semantics and introduce a new semantics, the stage2 semantics [44].
As pointed out in Example 6, there is no particular relation between thecf2and the preferred semantics, but the stage and thecf2 semantics coincide for this framework. The following ex-amples will show that in general there is no particular relation between stage andcf2 extensions as well.
Example 11. Consider the AFF in Figure 2.14. Here {a, c}is the only stage extension ofF (it is also stable). Concerningcf2 semantics, note thatF is built from a single SCC. Thus, thecf2 extensions are given by the naive sets ofF, which are{a, c}and{a, d}. Thus, we have
stage(F)⊂cf2(F). ✸
As an example for a frameworkF such thatcf2(F) ⊂stage(F), consider the AF from Exam-ple 7, wherecf2(F) ={{a}}butstage(F) ={{a},{b}}.
The relations between the introduced semantics are illustrated in Figure 2.13, an arrow from semanticsσ to semanticsτ encodes that each σextension is also aτ extension [7, 10, 14, 17, 25, 26, 37, 38, 96].
Finally, we consider a class of frameworks where stable and preferred semantics coincide, the so called coherent AFs [37].
Figure 2.13: Relations between semantics.
Figure 2.14:AFF from Example 11.
Definition 14. An AFF iscoherentif each preferred extension ofF is a stable extension ofF.
It follows that coherent AFs are odd-cycle free [37]. Furthermore in coherent AFs also semi-stable and stage semantics coincide with preferred [47]. Whereas this does not hold for cf2 semantics as one can see in Example 10. There, F is coherent but cf2(F) 6= σ(F), where σ ={stable,stage,pref,semis}.