Enforcing WCP

In this section, we discuss how an ABA+ framework that does not satisfy WCP (or the Axiom of Contraposition, for that matter) to begin with, can be modified by adding new rules (not necessarily in a unique way) so as to satisfy the axiom. We call this enforcing WCP (in case of contraposition, this is also sometimes called ‘closing under contraposition’).

To illustrate the idea, recall the ABA+ framework F = (L, R, A,¯¯¯, 6) from Example 3.6. As discussed in Section 4.1.1, it does not satisfy WCP, because there is no deduction S ` α with S ⊆ {β, γ}. Suppose we add the rule α ← β, γ to R to obtain the ABA+framework F0= (L, R∪{α ← β, γ}, A,¯¯¯, 6). In F0, we find {β, γ} ` α and note that there are no other deduc- tions that make the antecedent of WCP true. Hence, F0 satisfies WCP.

Enforcing WCP will amount to such addition of rules in order to obtain a new framework that satisfies WCP. We give a formal account of this next. We call the situation where a deduction satisfies the antecedent of WCP while the consequent is false an instance of WCP. More formally:

Definition 4.1. A ` β is an instance of WCP just in case • ∃α0 ∈ A such that α0 < β, and

• for no α ∈ A which is 6-minimal such that α < β there is Aα` α, for

some Aα⊆ (A \ {α}) ∪ {β}.

In F from Example 3.6, {α, γ} ` β is an instance of WCP.

By contrasting the ways to enforce WCP and contraposition, we can see how less restrictive the former is than the latter. Indeed, we next show how, for WCP to be satisfied, it suffices, for every instance of WCP, to ensure one additional deduction for the contrary of some single 6-minimal assumption among those less preferred than the one whose contrary is deduced. We call any such 6-minimal assumption a witness:

Definition 4.2. Let A ` β be an instance of WCP and let α ∈ A. Then α is a witness to A ` β just in case α is 6-minimal such that α < β.

In F from Example 3.6, α is a witness to the instance of WCP {α, γ} ` β. So a witness to an instance of WCP can be seen as a candidate assumption with regards to which an additional deduction is needed in order to satisfy WCP. This can be achieved by adding a single enforcing rule, defined next.

Definition 4.3. Let A ` β be an instance of WCP with a witness α ∈ A. Say A = {α1, . . . , αn}, where α = αi for some i. The enforcing rule,

denoted by α ← A \ α, β, is the rule α ← α1, . . . , αi−1, αi+1, . . . , αn, β.

In Example 3.6, for the only instance of WCP {α, γ} ` β and its sole witness α, the enforcing rule is α ← β, γ.

WCP can be enforced by adding an enforcing rule for every instance of WCP, as shown next.

Proposition 4.2. Let

R_A`β = {α ← A \ α, β is an enforcing rule :

A ` β is an instance of WCP (in (L, R, A,¯¯¯, 6)) with a witness α ∈ A}

be the set of enforcing rules given an instance A ` β of WCP in a given ABA+ framework (L, R, A,¯¯¯, 6), and let f be a function, defined for finite non-empty sets, that selects any one element from a given set. The ABA+ framework (L, R ∪ R0, A,¯¯¯, 6), where

R0= {f (R<sub>A`β</sub>) : A ` β is an instance of WCP (in (L, R, A,¯¯¯, 6))},

satisfies WCP.

Proof. Let A ` β be an instance of WCP (in (L, R, A,¯¯¯, 6)) with a wit- ness α ∈ A. Any rule α ← A \ α, β = f (R<sub>A`β</sub>) guarantees that in (L, R ∪ R0, A,¯¯¯, 6) we find {α1, . . . , αi−1, αi+1, αn, β} ` α. Note that no

such deduction can result in an instance of WCP (in (L, R ∪ R0, A,¯¯¯, 6)), precisely because the witness α is 6-minimal. Therefore, (L, R∪R0, A,¯¯¯, 6) satisfies WCP.

For illustration, to enforce WCP on F from Example 3.6, add the enforc- ing rule α ← β, γ to R to obtain F0 = (L, R ∪ {α ← β, γ}, A,¯¯¯, 6) which satisfies WCP.

Observe that, as discussed in Section 4.1.1, enforcing WCP amounts to generating additional normal attacks. That is, if A ` β is an instance of WCP and α0 is a witness, then enforcing WCP (as in Proposition 4.2) guar- antees the normal attack (A \ {α}) ∪ {β} < {α}. Note well, however,

that this does not have the same effect as attack reversal, because the latter guarantees the reverse attack {β} < A instead. So, on the one hand,

as opposed to attack reversal, enforcing WCP in general requires extra as- sumptions to generate an <-attack. On the other hand, enforcing WCP produces an <-attack targeted at a particular assumption, as opposed to reversing attacks to in general produce <-attacks on sets of assumptions. As a consequence, enforcing WCP and discarding <-attacks due to preferences would not lead to the same outcomes (semantically) as reversing attacks (with or without WCP).

In relation to the Axiom of Contraposition, the 6-minimality of a witness assumption in enforcing WCP plays a crucial role. On the one hand, the 6-minimality of witnesses clearly distinguishes the outcomes of enforcing WCP and closing under contraposition. Specifically, to satisfy the Ax- iom of Contraposition, for every deduction that leads to violation of the axiom, one needs to generate deductions for the contraries of all the as- sumptions in the support (cf. non-existence of S ` γ with S ⊆ {α, β}, as in the proof of Lemma 4.1). This means adding at least as many, but possibly more, rules as needed to enforce WCP. For instance, we would need to add e.g. the rule γ ← α, β, in addition to α ← β, γ, to R of F from Example 3.6, to obtain a new ABA+ _{framework F}00 _{= (L, R ∪}

{α ← β, γ, γ ← α, β}, A,¯¯¯, 6) that satisfies the Axiom of Contraposition. Recall, by way of contrast, that to enforce WCP on F , it suffices to add the enforcing rule α ← β, γ. Consequently, the 6-minimality of witnesses means that generally, enforcing WCP as in Proposition 4.2, does not amount to contraposing all the rules in a given ABA+ framework.

On the other hand, the 6-minimality of witnesses saves from generating redundant deductions when enforcing the axiom. For instance, consider F = (L, R, A,¯¯¯, 6) with R = {γ ← α, β} containing a single rule, assumptions A = {α, β, γ} and preferences α < β < γ. Then {α, β} ` γ is an instance of WCP. If 6-minimality were not required in the conditions of the consequent of WCP, one could choose β and add the rule β ← α, γ to R so as to generate {α, γ} ` β in F0 = (L, R ∪ {β ← α, γ}, A,¯¯¯, 6). This would result in {α, β} ` γ not being an instance of WCP in the new framework F0.

However, {α, γ} ` β would then also make the antecedent of the WCP true (because α < β) while keeping the consequent false, thus yielding an in- stance of WCP in F0. Consequently, to enforce WCP, one would additionally

need to ensure existence of yet another deduction, for example, {β, γ} ` α, by, for instance, adding the rule α ← β, γ. By contrast, choosing a (neces- sarily 6-minimal) witness to begin with, in this case the only one such being α, enables one to add a single rule, say the enforcing rule α ← β, γ, so as to generate the deduction {β, γ} ` α in F00 = (L, R ∪ {α ← β, γ}, A,¯¯¯, 6). Therefore, the instance of WCP in question is eliminated in F00, no further instances of WCP are obtained in F00, and so WCP is enforced on F by modifying it into F00.

Note again that in the above example, enforcing WCP in the first way by adding two rules amounts to closing the framework under contraposition. Enforcing WCP as in Proposition 4.2 is different, because in the latter case, the additional rules are added for the contraries of (some, even if possibly all) 6-minimal witness assumptions, but not for the contraries of the preference- wise ‘intermediate’ assumptions. In the above example, α as a witness is 6-minimal with α < γ, whereas β is ‘intermediate’ preference-wise in the sense that α < β < γ. To enforce WCP, it suffices to account for the witness α, and there is no need to consider the ‘intermediate’ β, as opposed to what is done to enforce the Axiom of Contraposition.

There are other ways to enforce WCP on a given ABA+ framework. For example, given an instance of WCP A ` β with a witness α ∈ A, one could add the rule α ← > to obtain the deduction ∅ ` α with ∅ = Aα ⊆

(A \ {α}) ∪ {β}, as required to eliminate the instance of WCP in question, at the same time avoiding to create additional instances. This particular way seems rather ad hoc and also quite radical with respect to knowledge representation: it seems very unintuitive to have assumptions immediately ‘rejected’ (by deducing their contraries from the empty set) just because they are involved in the argumentative process of deducing contraries of more preferred assumptions. Intuitively, this way of enforcing WCP seems straightforward from a technical point of view, but strikes as very intrusive, compared with Proposition 4.2, from the conceptual point of view.

Apart from the two opposite endmost ways of enforcing WCP discussed above, there are options ‘in between’, in the sense of aiming Aα to be a

proper non-empty subset of (A ∪ {α}) \ {β} (see Axiom 4.2). We do not, however, provide guidelines on how to enforce WCP; this is up to the user of ABA+ to decide, given a specific situation. Nonetheless, we showed how to enforce WCP in one particular way. We contend, however, that enforcing

WCP as in Proposition 4.2 generally involves a choice among rules to be added. This choice need not be innocent in the sense that different choices may in general lead to different frameworks and distinct reasoning outcomes. Again, we believe that such a choice belongs to the user of ABA+, and enforcing WCP as Proposition 4.2 could simply act as an indication that such a choice is needed.

In the next section, we show that, without loss of generality, the way of enforcing WCP as in Proposition 4.2 also preserves flatness. It may be generally desirable to preserve flatness while enforcing WCP, because, in particular, some desirable properties are guaranteed in flat ABA+ subject to WCP, as we will see in Section 4.2. In particular, we will show that the Fundamental Lemma holds for flat ABA+ subject to WCP, and so it is important that it is possible to enforce WCP on a flat ABA+ framework while preserving flatness.