La desesperanza y la derrota

The Noninterference Principle

Noninterference is a weakening of directionality: it states that an argument x has an effect on an argument y only if x and y are connected (in either direction) by an attack. Noninterference was introduced by Caminada [27] and adapted by Baroni et al. [4] for labelling-based semantics. Formally, noninterference is defined by the condition that the isolated sets (sets of arguments not attacked by arguments outside this set and not receiving attacks from outside this set) can be evaluated independently of the rest of the argumentation framework. Definition 3.4.8. [4] Let F = (A, ) be an argumentation framework and let B ⊆ A. We say that B is isolated iff there is no x ∈ B and y ∈ A \ B such that x y or y x. We let I(F ) denote the set of isolated sets of F .

a

b

c

Figure 3.13: A counterexample for noninterference under the stable semantics.

Formally, the condition that isolated sets can be evaluated independently of the rest of the argumentation framework means that, given an argumentation framework F and isolated set B ∈ I(F ), the labellings of the restriction of F by B coincide with the labellings of F , restricted by B. Note that, since an isolated set is also an unattacked set, the directionality principle implies the noninterference principle.

Definition 3.4.9. [4] A labelling-based semantics σ satisfies the noninterfer- ence principle if and only if for all F ∈ F and B ∈ I(F ), Lσ(F ) ↓ B = Lσ(F ↓

B).

The noninterference principle is characterized in terms of a labelling-based en- tailment relation as follows.

Proposition 3.4.4. A labelling-based semantics σ satisfies noninterference if and only if

∀B ∈ I(F ), φ ∈ lang(B), (F ↓ B) |=σφ iff F |=σφ.

Proof. This follows from definition 2.1.17.

The complete, grounded and preferred semantics satisfy the directionality prin- ciple and therefore they also satisfy the noninterference principle. Caminada has shown that the semi-stable semantics also satisfies noninterference, but that the stable semantics does not [27].

Proposition 3.4.5. The Co, Gr, Pr and SS semantics satisfy the noninterfer- ence principle but the St semantics does not.

The failure of the noninterference principle under the stable semantics is demon- strated by the following example.

Example 3.4.3. Let F be the argumentation framework shown in figure 3.13. Note that {a, b} ∈ I(F ). We have LSt(F ) = ∅ and hence

(LSt(F ) ↓ {a, b}) = ∅.

But we also have

LSt(F ↓ {a, b}) = {{(a, in), (b, out)}}.

In terms of labelling-based entailment, this means that we have, for example, F |=St⊥ but not F ↓ {a, b} |=St⊥. This is a violation of noninterference.

The Conditional Noninterference Property

The Conditional Noninterference property for intervention-based entailment is related to the noninterference property of a semantics. Informally speaking, Conditional Noninterference states that an intervention that applies to an ar- gument x only changes the label of an argument y if there is an undirected path from x to y. We make this formal using the relation of structural connectedness. We say that an argument x is structurally connected to an argument y if there is an undirected path between x and y.

Definition 3.4.10. Let F = (A, ) be an argumentation framework. We say that x is structurally connected to y (written x ∗y) if x = y or if there is an undirected path in F from x to y. We similarly say that B ⊆ A is structurally connected to B0 ⊆ A (written B !∗_B0_{) iff for some x ∈ B and y ∈ B}0 _{it holds}

that x !∗y.

It is easy to check that the structural connectedness relation is an equivalence relation, whose classes are exactly the isolated sets of the argumentation frame- work. Like we did for the relation of structural relevance, we extend the relation of structural connectedness to apply to interventions and formulas. If, e.g., it holds that Args(Φ) !∗Args(φ) then we also write Φ !∗φ.

Conditional Noninterference states that an intervention only changes conse- quences to which the intervention is structurally connected. In other words, whether or not a formula is a consequence is independent of any intervention not structurally connected to this formula.

Definition 3.4.11. Let F ∈ F . A relation ||=F_{⊆ Int(F ) × lang(F ) satisfies}

Conditional Noninterference iff for all Φ, Ψ ∈ Int(F ) and φ ∈ lang(F ), if Ψ 6!∗φ then Φ ∪ Ψ ||=F φ iff Φ ||=F φ.

It is easy to see that structural relevance implies structural connectedness, and hence that we have the following.

Proposition 3.4.6. If ||=F _{satisfies Conditional Directionality then ||=}F _satis-

fies Conditional Noninterference. Proof. Suppose ||=F

satisfies Conditional Directionality and assume Ψ 6!∗φ. It then follows that Ψ 6 ∗φ and hence, by Conditional Directionality, Φ ∪ Ψ ||=F _φ

iff Φ ||=F _φ.

Furthermore, if a labelling-based semantics σ satisfies the noninterference prin- ciple then for every argumentation framework F , the relation ||=F

σ satisfies Con-

ditional Noninterference.

Theorem 3.4.7. For any labelling-based semantics σ that satisfies the non- interference principle it holds that for all F ∈ F , ||=F

σ satisfies Conditional

Noninterference. Proof. See section 3.7.

The complete, grounded, preferred and semi-stable semantics all satisfy the noninterference principle and hence for all F ∈ F , the relations ||=F_Co, ||=F_Gr, ||=F

Pr and ||= F

SS satisfy Conditional Noninterference. This does not hold for

the stable semantics, which does not satisfy the noninterference principle. The following example demonstrates unintuitive behaviour as a result of this. Example 3.4.4. Let F be the argumentation framework shown in figure 3.13. We have {out(c)} 6||=F_{St in(b) and {out(c)} 6!}∗in(b). Conditional Noninter- ference would imply that we have ∅ 6||=F_Stin(b) but this is not the case. We have instead ∅ ||=F_St⊥ and hence ∅ ||=F

Stin(b).