3.3 Preference Handling Principles
4.1.3 WCP and Flatness
In this section, we investigate what effect enforcing WCP on an ABA+ framework has with regards to its flatness.
First, observe that, technically, enforcing WCP on a flat ABA+framework may necessarily yield a non-flat ABA+ framework, as illustrated next. Example 4.1. Consider (L, R, A,¯¯¯, 6) with
• L = {α, β, γ, b, c}, • R = {β ← α}, • A = {α, β, γ},
• α = γ, β = b, γ = c, • α < β.
This ABA+ framework is flat. Now, we find {α} ` β and α < β, but for
no S ⊆ {β} we have S ` α. Thus, (L, R, A,¯¯¯, 6) violates WCP. In order to enforce WCP, we need to add some rules so as to obtain S ` α, for some S ⊆ {β}. However, note that whatever new rules we add, the support S of S ` α will be such that S ` γ, because α = γ. Also note that as S ⊆ {β} and γ ∈ Cn(S), S will not not be closed. Therefore, enforcing WCP on (L, R, A,¯¯¯, 6) will necessarily yield a non-flat ABA+ framework.
We will see in Section 4.2 that flat ABA+subject to WCP exhibits certain desirable properties, such as the Fundamental Lemma. Hence, it is desirable
to be able to preserve flatness while enforcing WCP. To this end, we next show how we can, without loss of generality, assume that enforcing WCP preserves flatness. This is done by syntactically slightly modifying ABA+ frameworks without changing their behaviour semantically, and by appro- priately adding new rules for instances of WCP. We begin by describing the syntactic modification.
Instead of defining the contrary mapping ¯¯¯ : A → L to map assumptions into the elements of the language, we can assign new symbols (not in L to begin with) for contraries of assumptions. More concretely, for each assumption α we take a new symbol αc not already in L, and define a contrary mapping C so that C(α) = αc. Then, for the sentence a ∈ L that was intended as the contrary α of α to begin with, we add the rule αc← a. Formally:
Definition 4.4. For (L, R, A,¯¯¯, 6), its contrary-equivalent ABA+frame- work is (L ∪ Lc, R ∪ Rc, A, C, 6) with
• Lc= {αc6∈ L : α ∈ A},
• Rc= {αc← a : α ∈ A, a ∈ L, α = a}, • C : A → Lcsuch that C(α) = αc ∀α ∈ A.
In the remainder of this section, unless specified otherwise, we assume (L ∪ Lc, R ∪ Rc, A, C, 6) to be contrary-equivalent to (L, R, A,¯¯¯, 6). For
notational convenience, `c denotes the deduction relation (Section 2.2.2
Definition 2.5), Cncdenotes the conclusions operator (Section 2.2.2 Defini-
tion 2.6), and cdenotes the <-attack relation (Section 3.1 Definition 3.2)
associated with (L ∪ Lc, R ∪ Rc, A, C, 6).
First observe that deduction relations ` and `c, as well as attack relations < and c, coincide in the following sense.
Lemma 4.3. Let (L, R, A,¯¯¯, 6) be an ABA+ framework and let S ⊆ A and ϕ ∈ L. It holds that:
• S ` ϕ iff S `cϕ; • <= c.
Proof. Suppose S ` ϕ. That is, S `R ϕ for some R ⊆ R. As S ⊆ A and R ⊆ R ∪ Rc, we have S `Rc ϕ too. That is, S `cϕ.
Suppose instead S `c ϕ. That is, S `R
0
c ϕ for some R0 ⊆ R ∪ Rc. As
To show that the <-attack relations are the same, suppose first A < B.
If it is a normal attack, then A0 `R β, A0 ⊆ A, R ⊆ R, β ∈ B and α0 6<
β ∀α0 ∈ A0. By the previous point and construction of Rc, A0`R∪{β
c←β}
c βc.
As α0 6< β ∀α0 ∈ A0, we have A
c B via normal attack. If, on the other
hand, A <B via reverse attack, then B0 ` α, B0 ⊆ B, α ∈ A and β0 < α
for some β0 ∈ B0. Likewise we have B0 `cαc. As β0 < α for some β0 ∈ B0,
we have A cB via reverse attack.
Suppose now A c B. If it is a normal attack, then A0 `c βc, A0 ⊆ A,
β ∈ B and α0 6< β ∀α0 ∈ A0. As βc6∈ L, by construction of Rcwe have that
A0 `R∪{βc c←β}βcfor some R ⊆ R, and hence A0 `Rc β. Then A0` β, and as
α0 6< β ∀α0 ∈ A0, we conclude with A < B via normal attack. The case
with A cB via reverse attack is proven analogously.
Therefore, (L, R, A,¯¯¯, 6) and (L ∪ Lc, R ∪ Rc, A, C, 6) are semantically
equivalent:
Proposition 4.4. Let (L, R, A,¯¯¯, 6) be an ABA+ framework and let E ⊆
A. It holds that E is a <-σ extension of (L, R, A,¯¯¯, 6) iff E is a <-σ extension of (L ∪ Lc, R ∪ Rc, A, C, 6). Moreover, Cn(E) = Cnc(E) ∩ L.
Proof. That (L, R, A,¯¯¯, 6) and (L ∪ Lc, R ∪ Rc, A, C, 6) have the same ex-
tensions follows from (the second bullet point of) Lemma 4.3, and from the fact that both ABA+ frameworks share the same set of assumptions. That Cn(E) = Cnc(E) ∩ L follows from the definitions of Cn and Cnc, and from
(the first bullet point of) Lemma 4.3.
We can now specify at least one way of enforcing WCP that preserves flatness, namely to take the ABA+ framework (L ∪ Lc, R ∪ Rc, A, C, 6)
contrary-equivalent to (L, R, A,¯¯¯, 6) and to add the enforcing rules thus. Proposition 4.5. Let (L, R, A,¯¯¯, 6) be a flat ABA+ framework. Let
RA`cβc = {α
c← A \ α, β is an enforcing rule :
A `cβc is an instance of WCP (in (L ∪ Lc, R ∪ Rc, A, C, 6))
with a witness α ∈ A}
be the set of enforcing rules given an instance A `cβcof WCP in the ABA+
Let f be a function, defined for finite non-empty sets, that selects any one el- ement from a given set. The ABA+ framework (L∪Lc, R∪Rc∪R0, A, C, 6),
where
R0= {f (RA`cβc) :
A `cβc is an instance of WCP (in (L ∪ Lc, R ∪ Rc, A, C, 6))},
is flat and satisfies WCP.
Proof. Let F0 = (L ∪ Lc, R ∪ Rc∪ R0, A, C, 6). As (L, R, A,¯¯¯, 6) is flat,
(L ∪ Lc, R ∪ Rc, A, C, 6) is also flat, by (the first bullet point of) Lemma
4.3 and construction of Lc and Rc. By definition of R0, it is plain to see
that F0 is flat too.
Now note that every instance A `Rβ of WCP (in (L, R, A,¯¯¯, 6)) corre- sponds to an instance A `R∪{βc c←β}βcof WCP (in (L ∪ Lc, R ∪ Rc, A, C, 6)),
and vice versa, with the same (possibly multiple) witnesses. So, like in the proof of Proposition 4.2, it is easy to see that F0 satisfies WCP.
So, given a flat ABA+ framework, in order to avoid obtaining a non-flat framework after enforcing WCP, take its contrary-equivalent ABA+frame- work (which is semantically equivalent (Proposition 4.4) and has the same instances of WCP), and add to it the enforcing rules to guarantee satisfac- tion of WCP. This is illustrated in the context of our previous example.
Example 4.2. For F = (L, R, A,¯¯¯, 6) from Example 4.1, its contrary- equivalent ABA+ framework is Fc= (L ∪ Lc, R ∪ Rc, A, C, 6) with
• L = {α, β, γ, b, c}, Lc= {αc, βc, γc},
• R = {b ← α}, Rc= {αc← γ, βc← b, γc← c}, • A = {α, β, γ},
• C(α) = αc, C(β) = βc, C(γ) = γc,
• α < β.
Note that Fc is flat. Just like F , Fc has only one instance of WCP,
namely {α} ` βc, with a unique witness α. So we need S `cαc, for some
S ⊆ {β}, to enforce WCP. Consider the enforcing rule αc← β and add it to (L ∪ Lc, R ∪ Rc, A, C, 6) to obtain F0 = (L∪Lc, R∪Rc∪{αc← β}, A, C, 6).
As intended, F0 is flat and satisfies WCP.
generality, we can assume that enforcing WCP on an arbitrary flat ABA+ framework always yields a flat ABA+ framework that satisfies WCP.