4.2 Properties of Flat ABA + with WCP
4.2.2 Preference Handling
We now revisit the principles from Section 3.3 and analyse whether they are fulfilled by flat ABA+ frameworks satisfying WCP. In particular, we show that flat ABA+ frameworks satisfying WCP fulfil the Principle of Maximal Elements for all <-σ semantics and the Principle of Tolerance for all but <-stable semantics, and consider the remaining principles. Recall, however, that principles of Conflict Preservation (Principle 3.3) and Empty Preferences (Principle 3.4) are fulfilled by generic ABA+frameworks, so we
Exten- sion exists is unique is <- admissible is <- complete is <- preferred <- complete X (Ex. 3.6) X X X X <- preferred X X X X (Ex. 3.6) X <-stable X X X X X <-ideal X X (Ex. 4.3) X X (Ex. 4.3) X <- grounded X (Ex. 3.6) X X (Ex. 4.3) X (Ex. 4.3) X
Table 4.2: Relationship among flat ABA+ semantics. Cells with references to Examples 3.6 (Ex. 3.6) and 4.3 (Ex. 4.3), respectively, indicate the differences from Table 4.1 as well as the relevant examples
, Extension exists is unique is <- admissible is <-complete is <-preferred <- complete X X X X X <- preferred X X X X X <-stable X X X X X <-ideal X X X X X <- grounded X X X X X
Table 4.3: Relationship among flat ABA+ semantics, subject to WCP. Un- derlined marks indicate differences from Table 4.2, sanctioned by Theorem 4.7
do not discuss them here.
Maximal Elements
The Principle of Maximal Elements (Principle 3.5) concerns inclusion in extensions of the most preferred assumptions, as long as they are <-conflict- free. Theorem 4.7 implies that flat ABA+ frameworks satisfying WCP fulfil the Principle of Maximal Elements not only for <-complete, <-stable and <-grounded semantics (Proposition 3.12), but also for <-preferred and <- ideal semantics:
Corollary 4.8. Let (L, R, A,¯¯¯, 6) be a flat ABA+ framework satisfying WCP. Then (L, R, A,¯¯¯, 6) fulfils the Principle of Maximal Elements for <-preferred and <-ideal semantics.
Proof. By Theorem 4.7(ii, v), <-preferred and <-ideal extensions are <- complete for flat (L, R, A,¯¯¯, 6) that satisfies WCP. The claim thus follows from Proposition 3.12.
First BE Principle
The First BE Principle (Principle 3.6) insists that among two sets of as- sumptions that are identical but for a pair of distinct assumptions, as long as the one containing the more preferred assumption is <-conflict-free, the other set should not be an extension. Since generic ABA+ frameworks fulfil the First BE Principle for <-stable semantics (Proposition 3.13), we only need to see if flat ABA+ frameworks satisfying WCP fulfil the princi- ple for any other semantics. However, this is not the case. Indeed, recall (L, R, A,¯¯¯, 6) from Example 3.9, which does not fulfil the First BE Princi- ple for any semantics except <-stable. Note that (L, R, A,¯¯¯, 6) is flat and satisfies WCP. Therefore, the First BE Principle does not hold in flat ABA+ subject to WCP under any semantics bar <-stable. We have argued that this is a desirable feature of ABA+, due to the global/non-global nature of different semantics (see Section 3.3.4).
Second BE Principle
The Second BE Principle (Principle 3.7) insists that an extension remains acceptable after addition of a rule which is non-applicable with respect to that extension, as long as preferences are unchanged. Since generic ABA+ frameworks fulfil the Second BE Principle for <-stable semantics (Proposi- tion 3.14), we only need to see if flat ABA+frameworks satisfying WCP fulfil the principle for any other semantics. However, this is not the case. Indeed, recall (L, R, A,¯¯¯, 6) from Example 3.11, which does not fulfil the Second BE Principle for any semantics except <-stable. Note that (L, R, A,¯¯¯, 6) is flat and (trivially) satisfies WCP. Therefore, the Second BE Principle does not hold in flat ABA+ subject to WCP under any semantics bar <- stable. We have argued that this is a desirable feature of ABA+, due to the global/non-global nature of different semantics (see Section 3.3.5).
Principle of Tolerance
The Principle of Tolerance (Principle 3.8) requires an ABA+ framework to admit extensions whenever its underlying ABA framework does. Trivially, by Theorem 4.7, (L, R, A,¯¯¯, 6) admits <-grounded, <-ideal, <-preferred and <-complete extensions, so we have the following result.
Corollary 4.9. Let (L, R, A,¯¯¯, 6) be a flat ABA+ framework satisfying WCP. Then (L, R, A,¯¯¯, 6) fulfils the Principle of Tolerance for <-grounded, <-ideal, <-preferred and <-complete semantics.
Note that the fact that flat ABA+ frameworks satisfying WCP fulfil this
principle for all but <-stable semantics does not mean that the existence of extensions is predetermined by the underlying ABA frameworks. Rather, it is just a consequence of being flat and satisfying WCP that an ABA+
framework admits extensions under these semantics, just like it is a conse- quence of being flat that an ABA framework admits extensions under all but stable semantics.
Under <-stable semantics, however, even flat ABA+ frameworks satisfy- ing WCP do not fulfil this principle, as the following example shows.
Example 4.4. Recall (L, R, A,¯¯¯, 6) from Example 3.12, and enforce WCP on it by adding the rule α ← γ to obtain F0 = (L, R0, A,¯¯¯, 6) with
• R0 = {β ← α, γ ← α, γ ← β, α ← γ}, • A = {α, β, γ},
• α < γ.
Just as (L, R, A,¯¯¯, 6), F0comprises the 3-cycle {α} <{β} <{γ} <
{α}, and so does not admit <-stable extensions, whereas its underlying ABA framework (L, R0, A,¯¯¯) admits a unique stable extension {α}. (Both frameworks can be illustrated as in Figure 4.2.)
(L, R0, A,¯¯¯) {γ} {β} {α} (L, R0, A,¯¯¯, 6) {γ} {β} {α}
Figure 4.2: Flat ABA+frameworks from Example 4.4 violating the Principle of Tolerance under <-stable semantics
Since in ABA+ preferences arbitrate among conflicting sets of assump- tions through <-attacks, which in turn determine the extensions, existence of extensions after preferences should not be predetermined by extensions before preferences. Particularly with respect to (<-)stable semantics, pref- erences may resolve conflicts in a way that make previously acceptable as- sumptions untenable in the presence of preferences, leaving no acceptable extensions whatsoever. As discussed in Section 3.3.6, this is an expected feature of ABA+.
Principle of Reduction
The Principle of Reduction (Principle 3.9) requires that addition of new preferences does not generate new extensions and instead possibly reduces the space of acceptable extensions. Example 3.14 shows that flat ABA+ frameworks satisfying WCP do not in general fulfil the Principle of Reduc- tion: (L, R, A,¯¯¯, ∅) is flat, satisfies WCP and has a unique <-σ extension {α}, whereas (L, R, A,¯¯¯, 6) with α < β has a unique <-σ extension {β}. We have argued that this is a desirable behaviour, because preference informa- tion should be used to directly and locally resolve conflicts stemming from the defeasible information, and hence to possibly generate new extensions that were not available in the absence of preference information. Unlike using preferences to select among extensions, our approach is compatible with the dialectical nature of argumentation.
Note that in Example 3.14, the framework (L, R, A,¯¯¯, 6) with new pref- erences does not satisfy WCP—the Principle of Reduction imposes no con- ditions on the newly obtained ABA+ framework. We could thus investigate this principle in the setting where the framework with new preferences does satisfy WCP. To this end, we have the following result concerning <-stable semantics.
Proposition 4.10. Let (L, R, A,¯¯¯, 6) be a flat ABA+ framework satisfying WCP. For any preorder 60⊇ 6, if (L, R, A,¯¯¯, 60) satisfies WCP, then E ⊆ A is a <-stable extension of (L, R, A,¯¯¯, 60) only if E is a <-stable extension
of (L, R, A,¯¯¯, 6).
Proof. Let E ⊆ A be a <-stable extension of F0 = (L, R, A,¯¯¯, 60), where 60⊇ 6 is transitive and F0 satisfies WCP. We need to show that E is a
<-stable extension of F = (L, R, A,¯¯¯, 6) too. Let < and 0< be the <-
attack relations of F and F0, respectively. Suppose for a contradiction that E is not <-stable in F . First note that by Theorem 3.5, as E is <-conflict- free in F0, it is conflict-free in (L, R, A,¯¯¯), and hence <-conflict-free in F . Hence, there is β ∈ A \ E such that E 0< {β} but E 6 < {β}. Since
6 ⊆ 60 and as F and F0 have the same rules, it must be that E 0< {β}
via reverse attack. That is, {β} ` α for some α ∈ E with β <0 α. As F0 satisfies WCP, we must have S ` β with S ⊆ {α} (in both F and F0). But then E <{β} after all, which is a contradiction. Therefore, E must be a
stable extension of F to begin with.
Under other semantics, one can exhibit examples which show that the post-condition in the Principle of Reduction on the ABA+ framework with new preferences to satisfy WCP does not yield satisfaction of the prin- ciple: for <-grounded and <-ideal semantics, FC+ from Example 3.5 has a unique <-grounded and <-ideal extension {β, γ}, whereas the underlying ABA framework FC (considered as having the empty preference relation) of
FC+has a unique <-grounded and <-ideal extension {γ}; for <-preferred se- mantics, F0from Example 4.4 has a unique <-preferred extension ∅, whereas (L, R0, A,¯¯¯, ∅) has a unique <-preferred extension {α}; for <-complete se- mantics, an example of a flat ABA+ framework satisfying WCP that has a unique non-empty <-complete extension, but whose underlying ABA frame- work (considered as having the empty preference relation) has a unique <- complete extension ∅, will be given later, in Example 6.2. As discussed in Section 3.3.7, we maintain this exemplifies a generally desirable behaviour of ABA+.
In summary, we revisited and discussed the principles from Section 3.3 in the context of flat ABA+ with WCP. We showed that flat ABA+ frame- works satisfying WCP fulfil the Principle of Maximal Elements for all <-σ semantics and the Principle of Tolerance for all but <-stable semantics, and otherwise retain the same satisfaction results as generic ABA+frameworks. In the next section, we consider another set of properties, namely those pertaining to non-monotonic inference in ABA+.