4.2 The NAKED tool
5.1.2 Characterising ranking changes
In the rest of this section, we consider an argumentation framework and one of its cores constructed either using./1 or./2. We give a necessary and suffi- cient condition for obtaining an equal (with respect to the set of arguments) ./1 and ./2 induced core from its original argumentation framework. Then, for those argumentation frameworks where the induced core is different, we provide sufficient conditions for characterising the difference between the ranking of the core and the one of its original argumentation framework. More precisely:
1. We provide a sufficient condition for an argument’s rank to in- crease in the induced core. The new ranking of these arguments is further characterised by a sufficient condition on their respective po- sitions. This is done via theCP postulate characterisation.
2. We provide a sufficient condition for an argument’s rank to remain unchanged in the induced core. This is done via theNaE postulate characterisation.
3. Last, we provide a sufficient condition for an argument’s rank to decrease in the induced core. This is done via theCP and SCT pos- tulates characterisation.
Identity of induced core. We begin by introducing the notation needed for the rest of this section.
Definition 5.7 (Different core). Let us consider an argumentation frame- work F= (A , R) and one of its cores c = (A0, R0) with respect to an equiv- alence relation, we denote by Xc (or X if the core is obvious) the set of arguments that have been deleted, namely A = A0∪Xc and Xc ∩A0 = ∅. If Xc , ∅ then the core is said to be different from F, otherwise it is not different from F.
The next proposition gives a necessary and sufficient condition for all cores using./1 of an argumentation framework F to be not different from F. Proposition 5.6 (Not different core characterisation ./1). Let KB = (F, R, N) be a knowledge base and FKB be the corresponding argumentation framework. We have thatCore./1(FKB) ={FKB} if and only if for every R-
consistent subsetY of F, there are no y1,y2 such thatSatR(Y ) |= y1, SatR(Y ) |= y2,y1 , y2 andy1 y2.
Please refer to Section 7.2.3 on page xiv for the proof of Proposition 5.6. Similarly, we show a necessary and sufficient condition for all core using ./2 of an argumentation framework F to be no different than F.
Proposition 5.7 (Not different cores characterisation./2). LetKB = (F, R, N) be a knowledge base and FKB be the corresponding argumentation framework. We have Core./2(FKB) = {FKB} if and only if both the two
following items are satisfied:
• there are no f1, f2 ∈F such that f1 f2 and f1 , f2
• for every R-consistent subset Y ⊆ F, there are no y1,y2 such that SatR(Y ) |= y1, SatR(Y ) |= y2,y1, y2 andy1 y2.
Please refer to Section 7.2.3 on page xiv for the proof of Proposition 5.7. Rank increase. From now on, we consider an argumentation framework F = (A , R) and c0 = (A0, R0), one of its cores for equivalence relation ./1 or ./2. An interesting property is that for each attack from an argument removed by the core towards an argument of the core, we can find an attack that comes from an argument of the core towards that same argument. Proposition 5.8 (Attack equivalence). Let us consider F = (A , R), c0= (A0, R0) one of its cores for equivalence relation ./
1 or./2,E ={(a,b) ∈ R | a ∈ Xc0 andb ∈ A0} andE0= {W ⊆ R | W is maximal for set inclusion
such that for every (wi, wj), (wk, wl) ∈ W , wi ./ wk, wj = wl,{wi, wk} ⊆ Xc0
andwj, wl < Xc0}.
The function f : R0 →E0, that associates to each attack (a0,b0) ∈ R0 a set of attacksW ∈ E0 such that for every (wi, wj) ∈ W , wi ./ a0 andwj = b0, is surjective.
Proof. Let us considerW ∈ E0 and an element (wi, wj) ∈ W . Then since c0 is a core of F for ./1 (respectively ./2), we have that there exists a unique z ∈ ¯wi ./2∩A0 (respectively ¯wi ./1∩A0). Then, using Proposition 5.2, we get
that (z, wj) ∈ R0.
This proposition means that the modification of the ranking is induced mainly by a quantitative loss. We now introduce the notion of graph iso- morphism which will be used to clone our argumentation frameworks.
Definition 5.8 (Isomorphism). Let G1, G2 be two directed graphs. Let V (Gi) denote the set of vertices of Gi and E(Gi) the set of its arcs. We say thatγ : V (G1) → V (G2) is an isomorphism from G1 toG2 if and only if for every (x,y) ∈ E(G1), (γ (x),γ (y)) ∈ E(G2). For simplicity purposes, we will also writeG2 = γ (G1).
Using the previous Proposition 5.8, we can have a better understanding as to why some arguments have better ranking in a core than in F with some ranking-based semantics. The reason is because some arguments of the corec0that have equivalent arguments inXc0 (with respect to./2 or./1)
have their attacks amplified by those arguments. Of course, depending on the ranking-based semantics used, having more attackers does not always mean that the ranking of the argument is worst. This concept corresponds to the Cardinality Postulate (CP) postulate [Bonzon et al., 2016; Amgoud and Ben-Naim, 2013].
Definition 5.9 (CP [Amgoud and Ben-Naim, 2013]). Let σ be a ranking-based semantics and Fσ be the ranking obtained after applying σ on F. We say that σ satisfies CP if and only if for every F = (A , R) and for every a,b ∈ A such that |Att−(a)| < |Att−(b)|, it holds that b σ
F a and a σ
F b.
Note that the burden-based and the discussion-based ranking-based se- mantics both satisfy theCP postulate [Amgoud and Ben-Naim, 2013].
We are now interested in the impact of arguments removed by a core on arguments that belong to the core.
Definition 5.10 (Set of arguments attacked by filtrated arguments). Let F be an argumentation framework andc0 be one of its cores. We denote by Jc0 (or J if the core is obvious) the set of arguments of c0 that have at
least one attacker that is equivalent to an argument inXc0. More precisely,
J = {a ∈ A0| there exists (e, a) ∈ R0 and f ((e, a)) , ∅}, where f is the function defined in Proposition 5.8.
Example 5.5. Let F = (A , R) be an argumentation framework and c0 = (A0, R0) be a core of F for an equivalence relation. In this example depicted in Figure 5.3, we haveA ={a,b, c,d, e, i,д, h}andR ={(i, a), (д, a), (c,b), (d,b), (e,b), (h,b)}. Suppose that ¯i = {i,д} and ¯c = {c,d, e}. The core c0 is such thatA0 ={a, i, c,b, h} and J
c0 ={a,b}.
The next proposition states that every argument of the core that is attacked by an argument equivalent to a deleted argument is ranked better in the core.
Proposition 5.9 (Argument rank increase). Let F, F0 be two argumen- tation frameworks,c0 be a core of F with respect to an equivalence relation, γ be an isomorphism such that F0 = F ⊕ γ (c0) and σ be a ranking-based
b b b b b b b A b X g i h e d c a b
Figure 5.3: Representation of an argumentation framework F and one of its coresc0
semantics that satisfies CP. It holds that for every b ∈ Jc0,b σ
F0 γ (b) and
γ (b) σ F0 b.
Proof. Let (a,b) be an attack of c0 such that f ((a,b)) , ∅. It means that there exists an argument a0 ∈ X
c0 such that (a0,b) ∈ R. We thus have
|Att−(γ (b))| < |Att−(b)| and since σ satisfies CP, b σ
F0γ (b) and γ (b) Fσ0b.
In Proposition 5.9, we showed that some arguments of the core may be ranked higher. We now proceed further in this direction by introducing a sufficient condition for characterising the ranking of such arguments. Proposition 5.10 (Rank characterisation). Let a,b ∈ J. If σ satisfies CP and |Att−(a)| − P e ∈Att−(a)∩A0 |f ((e, a))| < |Att−(b)| − P e ∈Att−(b)∩A0 |f ((e,b))| thenb cσ0a and a cσ0b.
Proof. We have for all argumentsa in A0, |Att−(a)|−Pe ∈Att−(a)∩A0|f ((e, a))| =
|Att−(a) ∩ A0|. Thus, we can say that |Att−(a) ∩ A0| < |Att−(b) ∩ A0|. Since σ is a semantics that satisfy CP, b σ
c0 a and a σc0b.
Example 5.6 (Example 5.5 cont’d). We have that f ((i, a)) = {(д, a)},
f ((c,b)) = {(d,b), (e,b)} and f ((h,b)) = ∅. Thus, we can compute that |Att−(a)|−P
e ∈Att−(a)∩A0|f ((e, a))| = 1 and |Att−(b)|−Pe ∈Att−(b)∩A0|f ((e,b))| =
4 − 2 = 2. We conclude that under a ranking-based semantics σ satisfying CP, b σ
c0a and a cσ0b.
Unchanged rank. We now give a sufficient condition for an argument to keep the same rank. The basic notion behind this is that arguments that are not attacked by others do not undergo a change in their rank. This is true if the Non Attacked Equivalence (NaE) postulate is satisfied, namely if all the non-attacked argument have the same rank.
Definition 5.11 (NaE [Amgoud and Ben-Naim, 2013]). We say that a ranking-based semantics σ satisfies the NaE if and only if for every ar- gumentation framework F = (A , R) and for every a,b ∈ A such that Att−(a) = Att−(b) = ∅, it holds that a σ
F b and b σ F a.
Note that the burden-based, discussion-based, the h-categoriser [Besnard and Hunter, 2001] and the Tuples [Cayrol and Lagasquie-Schiex, 2005] ranking- based semantics satisfy theNaE postulate.
Proposition 5.11 (Unchanged non attacked arguments). Let F and F0 be two argumentation frameworks,c0 = (A0, R0) be a core of F, a ∈ A0, Att−(a) = ∅ and γ be an isomorphism such that F0 = F ⊕ γ (c0). If σ is a ranking-based semantics that satisfies NaE then a Fσ0γ (a) and γ (a) Fσ0 a.
Proof. We know that the corec0 has fewer arguments and attacks than F. Thus, the argument a is not attacked in either c0 or γ (c0). Furthermore, sinceσ satisfies NaE, γ (a) and a are equivalent. Rank decrease. In the next proposition, we introduce a sufficient condition for an argument of the core to have its rank decreased. This condition holds if the semantics used for the ranking satisfies the CP and Strict Counter Transitivity (SCT ) postulates. The SCT postulate specifies that if the at- tackers of an argumentb are at least as numerous and acceptable as those of an argument a and either the attackers of b are strictly more numerous or acceptable than those ofa, then a is strictly more acceptable than b. Definition 5.12 (SCT [Amgoud and Ben-Naim, 2013]). We say that a ranking-based semanticsσ satisfies SCT if and only if for every argumen- tation framework F= (A , R) and for every a,b ∈ A such that there is an in- jective mappingд : Att−(a) → Att−(b) with for every a0∈Att−(a), a0 Fσ д(a0) and (|Att−(a)| < |Att−(b)| or there exists a0 ∈Att−(a), a0 σ
F д(a
0),д(a0) σ F a
0) thenb Fσ a and a Fσ b.
Note that the burden-based, discussion-based and the h-categoriser ranking- based semantics satisfy theSCT postulate.
The idea behind the next proposition is that if an argument’s attackers have their ranks increased, then its rank is reduced.
Proposition 5.12 (Argument rank decrease). Let F and F0 be two argumentation frameworks, c0 = (A0, R0) be a core of F for an arbitrary equivalence relation, a be an argument of A0 witha < Jc0 and γ be an iso-
morphism such that F0= F ⊕ γ (c0). If σ is a semantics that satisfies CP and SCT and Att−(a) ⊆ J
c0 thenγ (a) σ
F0a and a Fσ0γ (a).
Proof. Sincea < Jc0, we have thatAtt−(γ (a)) ={γ (a0)|a0 ∈Att−(a)}and thus
|Att−(a)| = |Att−(γ (a))|. Now, since Att−(a) ⊆ J
c0, we have that for every
b ∈ Att−(b),b σ
F0γ (b) and γ (b) Fσ0b (using Proposition 5.9). Finally, using