3.2 Intervention-Based Entailment
3.2.3 Basic Properties
Before continuing we prove some basic properties of intervention-based entail- ment under the semantics that we consider. Some of the properties are rather obvious, but it useful to state them here so that we can refer back to them later on.
Reflexivity
In a Φ-modified argumentation framework, the intended effect of the interven- tion Φ is achieved only if the semantics that we use satisfies certain conditions. A sufficient condition is that every σ labelling of an argumentation framework is also a complete labelling. In terms of intervention-based entailment this means that the relation ||=F
σ satisfies Reflexivity whenever this condition is satisfied.
Definition 3.2.9. Let F ∈ F . A relation ||=F⊆ Int(F ) × lang(F ) satisfies
Reflexivity iff for all Φ ∈ Int(F ),
Φ ||=F α for all α ∈ Φ.
Proposition 3.2.2. Let σ be a labelling-based semantics. If for all F ∈ F , Lσ(F ) ⊆ LCo(F ), then for all F ∈ F , ||=Fσ satisfies Reflexivity.
Proof. Let σ be a semantics and suppose that for F ∈ F , Lσ(F ) ⊆ LCo(F ). Let
F = (A, ) be an argumentation framework and Φ ∈ Int(F ) be an intervention. We prove that ||=F
σ satisfies Reflexivity by showing that, for all α ∈ Φ, Φ ||=Fσ α.
Let κ be an F -mapping and let F0= (A0, 0) = F ⊕κΦ. Let L be a σ-labelling of F0. Our assumption implies that L is a complete labelling of F0. We use the fact that L satisfies out-legality, in-legality, reinstatement and rejection (definition 2.1.8 and 2.1.9).
• Let x ∈ A be an argument such that out(x) ∈ Φ. Definition 3.2.5 then implies that κ(x) ∈ A0, κ(x) is unattacked in F0 and κ(x) 0 x. Re- instatement implies that L(κ(x)) = in and hence rejection implies that L(x) = out.
• Let x ∈ A be an argument such that (¬in(x)) ∈ Φ. Definition 3.2.5 then implies that κ(x) ∈ A0, κ(x) 0 κ(x) and κ(x) 0x. Then in-legality and
out-legality imply that L(κ(x)) = und. Finally, in-legality implies that L(x) 6= in.
This implies that for all α ∈ Φ, L |= α. Via definition 3.2.8 it follows that for all α ∈ Φ, Φ ||=F
σ α.
We now obtain the following.
Proposition 3.2.3. For all F ∈ F , ||=F Co, ||= F Gr, ||= F Pr, ||= F St and ||= F SS satisfy Reflexivity.
Proof. Follows from proposition 3.2.2 and definition 2.1.11 and 2.1.12.
Note that some labelling-based semantics considered in the literature do not satisfy the property that every labelling is complete. Examples are the stage semantics and CF2 semantics [4]. We leave the treatment of intervention under these semantics for future work.
Vacuous Interventions
A vacuous intervention Φ does not lead to any change of an argumentation framework. This means that the consequences generated by a relation ||=F
σ
given the vacuous intervention coincide with the consequences generated given F by the relation |=σ.
Proposition 3.2.4. For all F ∈ F , F |=σφ iff ∅ ||=Fσ φ.
Proof. Follows directly from definition 3.2.5 and 3.2.8.
Relative Strength
The following proposition concerns the relative strength of the different interven- tion-based entailment relations. This is due to the inclusion relations between the corresponding semantics.
Proposition 3.2.5. For all F ∈ F , 1. ||=FCo⊆||=F Gr, 2. ||=FCo⊆||=F Pr⊆||= F SS⊆||= F St.
Consistency of Interventions
Because every argumentation framework has at least one complete, grounded, preferred and semi-stable labelling, every intervention yields consistent conse- quences under the complete, grounded, preferred and semi-stable semantics. Proposition 3.2.6. For all F ∈ F , σ ∈ {Co, Gr, Pr, SS} and Φ ∈ Int(F ), Φ 6||=F
σ ⊥.
Proof. Follows from definition 3.2.8 and proposition 2.1.10.
Provisional defeat leads to the addition of a self-attacking argument. This means that the resulting argumentation framework has no stable labellings. Thus, un- der the stable semantics, only stable interventions generate consistent conclu- sions.
Proposition 3.2.7. For all F ∈ F and Φ ∈ Int(F ), if Φ is not stable then Φ ||=F
St⊥.
Proof. See preceding discussion.
Enforceability Of Formulas
We already mentioned that any constraint on the labels of arguments that is conflict-free can be made true using only the actions of defeat and provisional defeat. We now make this formal. First a definition: a formula is conflict-free with respect to an argumentation framework if it is satisfied by at least one conflict-free labelling of this argumentation framework.
Definition 3.2.10. Let F ∈ F . A formula φ ∈ lang(F ) is conflict-free with respect to F if and only if there is some L ∈ LCf(F ) such that L |= φ.
The following theorem states that, given an argumentation framework F , every formula that is conflict-free with respect to F is a consequence of some interven- tion. Conversely, every consequence of every intervention is conflict-free with respect to F . This holds under the complete, grounded, preferred and semi- stable semantics, but not under the stable semantics. Note that, for the sake of readability, we have moved some of the longer proofs, including the proof for the following two theorems, to section 3.7.
Theorem 3.2.8. Let F be an argumentation framework and σ ∈ {Co, Gr, Pr, SS}. The following are equivalent.
1. For some Φ ∈ Int(F ), Φ ||=Fσ φ.
2. φ is conflict-free with respect to F .
Under the stable semantics we have the following. Given an argumentation framework F , every formula that is stable conflict-free with respect to F is a consequence of some intervention Φ that is consistent (i.e., for which we do not have Φ ||=FSt ⊥). Conversely, every consequence of every intervention that is consistent is stable conflict-free with respect to F .
Definition 3.2.11. A formula φ ∈ lang(F ) is stable conflict-free with respect to F if and only if there is some L ∈ LCf(F ) such that L−1(und) = ∅ and
L |= φ.
Theorem 3.2.9. Let F be an argumentation framework. The following are equivalent.
1. For some Φ ∈ Int(F ) we have Φ ||=F
Stφ and Φ 6||=FSt⊥.
2. φ is a stable conflict-free with respect to F . Proof. See section 3.7
These results show that, given an argumentation framework F and a semantics σ, any constraint φ, as long as it is (stable) conflict-free, can be translated into an intervention Φ that makes φ true. As the following example demonstrates, however, there may be more than one intervention that makes a given constraint true.
Example 3.2.4. Let F be the argumentation framework shown in figure 3.1 • Let σ ∈ {Co, Gr, Pr, SS, St}. The formula in(f ) is (stable) conflict-free
with respect to F . We have {out(a)} ||=F
σ in(f ), {out(c)} ||=Fσ in(f ) and
{out(e)} ||=F
σ in(f ) (see figure 3.2). That is: {out(a)}, {out(c)} and
{out(e)} all make in(f ) true.
• Let σ ∈ {Co, Gr, Pr, SS}. The formula und(f ) is conflict-free with re- spect to F . We have {¬in(a)} ||=F
σ und(f ), {¬in(c)} ||=Fσ und(f ) and
{¬in(e)} ||=F
σ und(f ) (see figure 3.3). That is: {¬in(a)}, {¬in(c)} and
{¬in(e)} all make f undecided.
This result shows that the actions of defeat and provisional defeat are sufficient to make an argumentation framework satisfy any constraint on the status of the arguments, as long as this constraint is (stable) conflict-free. In the next chapter we look at observation-based entailment, and the selection of (minimal) interventions that make a constraint (now taken to be an observation) true will play a central role. In that setting, these interventions play the role of explanations for the observation, and their selection can be seen as a process of abduction.
Note that Baumann and Brewka have proven a result for extension-based se- mantics that is related to what we prove in theorem 3.2.8 and 3.2.9, namely that every conflict-free set of an argumentation framework can be turned into a (unique) complete extension by adding a new argument attacking existing arguments [12].
a
b
c
d
Figure 3.5: Failure of Transitivity and Contraposition.