3.4 Directionality and Noninterference
3.4.1 Directionality
The Directionality Principle
The directionality principle is due to Baroni and Giacomin [6] and was later adapted for labelling-based semantics by Baroni et al. [4]. Intuitively, it ex- presses the idea that the notion of attack is directional: an argument x has an effect on an argument y only if x attacks y. This is formalized by the condition that the unattacked sets of an argumentation framework (sets of arguments not attacked by an argument outside this set) can be evaluated independently of the rest of the argumentation framework. First some definitions.
Definition 3.4.1. [4] Given an argumentation framework F = (A, ) a la- belling L ∈ L(F ) and a set B ⊆ A we denote by L ↓ B the restriction of L by B, which is defined to be the labelling (L ↓ B) : B → {in, out, und} such that (L ↓ B)(x) = L(x), for all x ∈ B. Given a set M ⊆ L(F ), we denote by M ↓ B the set {L ↓ B | L ∈ M }.
Definition 3.4.2. [6] Given an argumentation framework F = (A, ) and a set B ⊆ A we denote by F ↓ B the restriction of F by B, which is defined to be the argumentation framework (B, ∩(B × B)).
Definition 3.4.3. [6] Let F = (A, ) be an argumentation framework and let B ⊆ A. We say that B is unattacked iff there is no x ∈ B and y ∈ A \ B such that y x. We let U (F ) denote the set of unattacked sets of F .
Formally, the condition that unattacked sets can be evaluated independently of the rest of the argumentation framework means that, given an argumentation framework F and unattacked set B ∈ U (F ), the labellings of the restriction of F by B coincide with the labellings of F , restricted by B.
Definition 3.4.4. [4] A labelling-based semantics σ satisfies the directionality principle iff for all F ∈ F and B ∈ U (F ),
Lσ(F ) ↓ B = Lσ(F ↓ B).
We can characterize the directionality principle in terms of a labelling-based entailment relation as follows.
Proposition 3.4.1. A labelling-based semantics σ satisfies the directionality principle if and only if
∀B ∈ U (F ), φ ∈ lang(B), (F ↓ B) |=σφ iff F |=σφ.
Proof. This follows from definition 2.1.17.
Not all semantics satisfy directionality. While the complete, grounded and pre- ferred semantics do, the semi-stable and stable semantics do not. This was shown by Baroni et al. [6].
Proposition 3.4.2. [6] The Co, Gr and Pr semantics satisfy the directionality principle but the SS and St semantics do not.
a
b
c
Figure 3.12: Failure of directionality.
The failure of the directionality principle under the semi-stable and stable se- mantics is demonstrated by the following example.
Example 3.4.1. Let F be the argumentation framework shown in figure 3.12. Let σ ∈ {St, SS}. We have
(Lσ(F ) ↓ {a, b}) = {{(a, out), (b, in)}}.
But we also have {a, b} ∈ U (F ) and
Lσ(F ↓ {a, b}) = {{(a, in), (b, out)}, {(a, out), (b, in)}}.
In terms of labelling-based entailment, this means that we have, for example, F |=σin(b) but not F ↓ {a, b} |=σin(b). This is a violation of directionality.
The Conditional Directionality Property
We now introduce the Conditional Directionality property for intervention-based entailment. Informally speaking, Conditional Directionality states that an in- tervention that applies to an argument x only changes the label of an argument y if there is a directed path from x to y. We make this formal using the relation of structural relevance4.
Definition 3.4.5. Let F = (A, ) be an argumentation framework. We say that x is structurally relevant to y (written x ∗ y) if x = y or if there is a directed path in F from x to y. We similarly say that B ⊆ A is structurally relevant to B0 ⊆ A (written B ∗ B0) iff for some x ∈ B and y ∈ B0 it holds
that x ∗y.
Note that this definition implies that every argument is structurally relevant to itself. We also apply the relation of structural relevance to interventions and formulas. For example, an intervention Φ is structurally relevant to a formula φ whenever the set of arguments occurring in Φ is structurally relevant to the set of arguments occurring in φ.
Definition 3.4.6. Let F = (A, ) be an argumentation framework. We de- note by Args(Φ) (resp. Args(φ)) the set of all arguments occurring in Φ (resp. φ). We slightly abuse notation and write φ ∗ ψ, φ ∗ Ψ and Φ ∗ ψ to mean Args(φ) ∗ Args(ψ), Args(φ) ∗ Args(Ψ) and Args(Φ) ∗ Args(ψ), respectively.
4Not to be confused with the relation of relevance used by Caminada in [27], which is what
The Conditional Directionality property states that an intervention only changes the consequences to which the intervention is structurally relevant. In other words, whether or not a formula is a consequence is independent of any inter- vention not structurally relevant to this formula. Formally, we express this by saying that the consequences of two interventions Φ ∪ Ψ and Φ are the same as far as consequences to which Ψ is not structurally relevant are concerned. Definition 3.4.7. Let F ∈ F . A relation ||=F⊆ Int(F ) × lang(F ) satisfies Conditional Directionality iff for all Φ, Ψ ∈ Int(F ) and φ ∈ lang(F ),
if Ψ 6 ∗φ then Φ ∪ Ψ ||=F φ iff Φ ||=F φ.
The principle of directionality, if satisfied by a labelling-based semantics σ, ensures that ||=F
σ satisfies Conditional Directionality.
Theorem 3.4.3. If σ satisfies the directionality principle then for all F ∈ F , ||=F
σ satisfies Conditional Directionality.
Proof. See section 3.7.
The complete, grounded and preferred semantics all satisfy the directionality principle and hence for all F ∈ F , the relations ||=F
Co, ||= F
Gr and ||= F
Pr satisfy
Conditional Directionality. This does not hold for the semi-stable and stable se- mantics, which do not satisfy the directionality principle. The following example demonstrates unintuitive behaviour as a result of this.
Example 3.4.2. Let F be the argumentation framework shown in figure 3.12. Note that acceptance of b is necessary to minimize undecidedness. Hence we have ∅ ||=F
SS in(b). Furthermore we have {out(c)} 6 ∗in(b), and hence the
intervention {out(c)} should not affect whether or not in(b) is a consequence. However, if we defeat c then acceptance of b is no longer necessary to mini- mize undecidedness. Thus we have {out(c)} 6||=F
SS in(b), which is a violation
of Conditional Directionality. This example shows that under the semi-stable semantics an intervention might affect the labels of arguments to which the in- tervention is not structurally relevant. This example also applies to the stable semantics.