3.3 KLM Properties
3.3.7 Even-Cycle-Free Argumentation Frameworks
The next class of argumentation frameworks are those containing no even-length directed cycles. Whereas odd-length directed cycles have the nature of a para- dox, even-length directed cycles have the nature of a dilemma, in the sense that they force one to select one of two possibilities. In this section we look at the behaviour of argumentation frameworks that contain no odd-length directed cycles. We call these argumentation frameworks even-cycle-free.
Definition 3.3.12. Let F = (A, ) be an argumentation framework. A se- quence x0, . . . , xn of arguments is called an even cycle if and only if n is even;
x0 = xn; xi xi+1 for all 0 ≤ i < n; and for all 0 ≤ i, j < n s.t. i 6= j,
xi 6= xj.3 We say that F is even-cycle-free if it contains no even cycles.
It was proven by Dvorak [47] (strengthening a result obtained by Dunne and Bench-Capon [14]) that these argumentation frameworks have a unique com- plete extension, and hence a unique complete labelling.
Proposition 3.3.13. If F is even-cycle-free then |LCo(F )| = 1.
Proof. (Adapted from [47]) We prove it by contraposition. Let F = (A, ) be an argumentation framework. Suppose |LCo(F )| 6= 1. Because |LCo(F )| > 1 it
follows that there are two labellings L, L0 ∈ LCo(F ) such that L 6= L0. Suppose
furthermore that L is the grounded labelling of F . Then there is some x0 ∈ A
such that L0(x0) = in and L(x0) 6= in. Because L(x0) 6= in, there is a x1 ∈ A
such that x1 x0 and L(x1) 6= out. Furthermore, we have L0(x0) = in and
thus L0(x1) = out and hence there is a x2∈ A such that x2 x1and L0(x2) =
in. Because L(x1) 6= out, we furthermore have L(x2) 6= in. Inductively, we
obtain a sequence x0, x1, x2, . . . such that x0 x1 x2. . .; for each i that is
even, L0(i) = in; and for each i that is odd, L0(i) = out. Now let n be the smallest integer such that x0= xn (finiteness of A ensures existence of n). We
then have that n is even, and that, for all 0 ≤ i, j < n s.t. i 6= j, xi6= xj. It then
follows that x0, . . . , xn is an even cycle and hence F is not even-cycle-free.
We now state two immediate consequences of this fact. The first is that even- cycle-freeness ensures that intervention-based entailment under the grounded, complete, preferred and semi-stable semantics coincides. The second is that intervention-based entailment under the grounded and stable semantics coincide as far as interventions are concerned that do not yield inconsistent conclusions under the stable semantics.
Proposition 3.3.14. If F is even-cycle-free then ||=FGr=||=FCo=||=FPr=||=FSS . Proposition 3.3.15. If F is even-cycle-free then for all Φ ∈ Int(F ), if Φ 6||=FSt
⊥ then for all φ ∈ lang(F ), Φ ||=F
Stφ iff Φ ||= F Grφ.
Let us start with the complete semantics. We have seen that Rational Monotony fails under the complete semantics. But because rational monotony is sat- isfied under the grounded semantics, proposition 3.3.14 implies that rational monotony is satisfied in the even-cycle-free case under the complete semantics. Theorem 3.3.16. For all F ∈ F , if F is even-cycle-free then ||=FCo satisfies Rational Monotony.
What about the preferred and semi-stable semantics? Recall that, in the gen- eral case, Cautious Monotony fails under the preferred and semi-stable seman- tics; Cut is satisfied under the preferred semantics but not under the semi- stable semantics; and the preferred and semi-stable semantics both fail Rational Monotony. Proposition 3.3.14 implies that, in the even-cycle-free case, the pre- ferred and semi-stable semantics satisfy Cautious Monotony, Cut and Rational Monotony.
Theorem 3.3.17. For all F ∈ F , if F is even-cycle-free then ||=FPr and ||=FSS satisfy Cautious Monotony, Cut and Rational Monotony.
Proof. This follows immediately from proposition 3.3.14 together with the fact that for all F ∈ F , ||=F
Grsatisfies Cautious Monotony, Cut and Rational Mono-
tony (theorem 3.3.1 and 3.3.3 and proposition 3.3.9).
Under the stable semantics, even-cycle-freeness does not ensure satisfaction of (Stable) Cautious Monotony. Here, (Stable) Cautious Monotony still fails if an initial premise entails inconsistency while a strengthening of this premise does not. This is demonstrated by the following example.
b
a
c
Figure 3.11: Failure of Cautious Monotony due to inconsistency under the stable semantics.
Example 3.3.8. Let F be the argumentation framework shown in figure 3.11. Note that F is even-cycle-free. Because F has no stable labelling we have ∅ ||=F
St
⊥ and hence, trivially, ∅ ||=F
Stout(a). Cautious Monotony implies that we have
{out(a)} ||=F
St ⊥ but this is not the case, because F ⊕κ{out(a)} does have a
stable labelling, and hence {out(a)} 6||=F St⊥.
Finally, proposition 3.3.15 implies that even-cycle-freeness ensures satisfaction of Rational Monotony under the stable semantics. Note that the stable semantics fails Rational Monotony in the general case (see example 3.3.5).
Theorem 3.3.18. For all F ∈ F , if F is even-cycle-free then ||=F
St satisfies
Rational Monotony.
Proof. Let F ∈ F and assume that F is even-cycle-free. Suppose Φ ||=F Stφ and
Φ 6||=F
St ¬α. Then Φ 6||=FSt⊥ and hence proposition 3.3.15 implies Φ ||=FGrφ and
Φ 6||=F
Gr¬α. Because ||=FGrsatisfies Rational Monotony (theorem 3.3.9) it follows
that Φ ∪ {α} ||=F
Grφ. If Φ ∪ {α} ||=FSt⊥, it trivially follows that Φ ∪ {α} ||=FStφ,
and we are done. If we have Φ ∪ {α} 6||=F
St⊥ then the fact that Φ ∪ {α} ||=FGrφ
implies, via proposition 3.3.15, that Φ ∪ {α} ||=FStφ.
Even-cycle-freeness does not ensure satisfaction of Loop under any of the se- mantics that we consider, as the counterexample that we used to demonstrate the failure of Loop relies on an argumentation framework that is even-cycle-free. Table 3.3 summarizes the results obtained here concerning even-cycle-free argu- mentation frameworks (i.e., theorem 3.3.16, 3.3.17 and 3.3.18).
Co Gr Pr SS St
(Stable) Cautious Monotony 3 3 3 3 7
(Stable) Cut 3 3 3 3 3
Rational Monotony 3 3 3 3 3
Loop 7 7 7 7 7
Table 3.3: Properties satisfied by ||=F
σ for every even-cycle-free F ∈ F .