Relationship Among Semantics

In this section, we study relationships among ABA+ semantics, akin to the relationships studied in e.g. [Dun95b, BDKT97, BG07], and show that ex- tensions of ABA+frameworks generally relate in the same way as extensions of ABA frameworks, except for one difference between two semantics.

In terms of relationships among semantics, generic ABA+ _frameworks

exhibit several features exhibited also by generic ABA frameworks. We summarise and prove them next.

Theorem 3.6. Let (L, R, A,¯¯¯, 6) be an ABA+ framework and let E ⊆ A. (i) If E is <-admissible, then there is a <-preferred extension E0 such

that E ⊆ E0.

(ii) If E is <-stable, then it is <-preferred. (iii) If E is <-stable, then it is <-complete.

(iv) If E is <-well founded, then for every <-stable extension E0 it holds that E ⊆ E0.

(v) If E is the intersection of all the <-preferred extensions and E is also <-admissible, then E is <-ideal.

(vi) If E is <-ideal, then it is not <-attacked by any <-admissible set of assumptions.

(vii) If the empty set is closed, then there is a <-preferred extension, as well as an <-ideal extension.

Proof.

(i) Let A0 ⊆ A1 ⊆ . . ., where A0 = E, be an ⊆-increasing sequence of <-

admissible supersets of E. Take its upper bound A =S

i>0Aiand note

that it is <-admissible: if it were either not closed, not <-conflict-free, or did not <-defend itself, then some finite subset (since deductions are

finite) A0 ⊆ A would not be either closed or <-conflict-free, or would not <-defend itself. Now, by Zorn’s Lemma, E has a ⊆-maximally <-admissible superset, i.e. a <-preferred extension containing E. (ii) Being <-stable, E is by definition closed and <-conflict-free. Given

that E < {β} for every β ∈ A \ E, it is clear that E < B for

every closed B ⊆ A such that B < E. Hence, E is <-admissible.

Moreover, E is ⊆-maximally <-admissible, as E ∪ {β} < E ∪ {β}

for any β ∈ A \ E. Thus, E is <-preferred, as required.

(iii) Being <-stable, E is <-admissible by (ii) above. Suppose for a con- tradiction that E <-defends A ⊆ A but A * E. Then α 6∈ E for some α ∈ A. Hence, E < {α}, and so E < A, due to stability. As E

<-defends A, we find E < E, which is a contradiction. Thus, by

contradiction, E contains every assumption set it <-defends, and so is <-complete.

(iv) Being <-well founded, E is by definition contained in every <-complete extension. By (iii) above, every <-stable extension is <-complete. Hence, E is contained in every <-stable extension.

(v) As the intersection of all <-preferred extensions, E is ⊆-maximal set of assumptions contained in every <-preferred extension. Given that E is also <-admissible, it is by definition <-ideal.

(vi) Suppose for a contradiction that B ⊆ A is <-admissible and B <E.

By (i) above, there is a <-preferred extension A such that B ⊆ A. Then, as E is <-ideal, we have E ⊆ A, and hence A < A, which is

a contradiction.

(vii) ∅, being closed, is <-admissible. Hence, by (i) above, there is a <-preferred extension. Still further, the intersection of <-preferred extensions exists, and so it has a ⊆-maximally <-admissible subset, i.e. an <-ideal extension.

In addition to the ‘positive’ properties as in Theorem 3.6, ABA+, being a conservative extension of ABA, inherits various ‘negative’ properties from ABA. For instance (see Example 3.6 below), <-complete, and thus <-well founded, extensions need not exist in general; <-preferred extensions need not be <-stable; <-stable extensions need not exist in general. However, not all properties of ABA semantics are preserved in ABA+_{. In particular,}

(i.e. preferred extensions are complete), in ABA+ <-preferred extensions need not be <-complete, as the following example illustrates.

Example 3.6. Consider an ABA+ framework (L, R, A,¯¯¯, 6) with4 • R = {β ← α, γ},

• A = {α, β, γ}, • α < β < γ.

(L, R, A,¯¯¯, 6) can be depicted graphically as in Figure 3.5.

∅ {α} {β} {γ} {α, β} {α, γ} {β, γ} {α, β, γ}

Figure 3.5: ABA+framework from Example 3.6 with <-preferred extensions that are not <-complete

Here, all the sets {α}, {β} and {γ} are <-unattacked5 (in particular, {β} is <-unattacked because α < β, so that {β} < {α, γ} via a reverse

attack.) Both {α, β} and {β, γ} are also <-unattacked. However, {α, β, γ} <-attacks itself. Therefore, (L, R, A,¯¯¯, 6) has {α, β} and {β, γ} as its all and only <-preferred extensions. However, neither of them is <-complete, because they <-defend assumption sets they do not contain, namely {γ} and {α}, respectively. Indeed, (L, R, A,¯¯¯, 6) has no <-complete extension, and thus no <-well founded extension either.

Note also that neither {α, β} nor {β, γ} is <-stable, because the former does not <-attack {γ}, and the latter does not <-attack {α}. In fact, (L, R, A,¯¯¯, 6) has no <-stable extensions.

Example 3.6 also shows that in contrast to ABA, for ABA+ frameworks the union of all the <-unattacked singleton sets of assumptions need not

4_{From now on, unless specified otherwise, we omit L and ¯¯¯, and adopt the following}

conventions: the contrary α of any assumption α is actually a symbol in L, and, unless α appears in either A or R, it is different from the sentences appearing in A or R; thus, L consists of all the sentences appearing in R, A and {α : α ∈ A}.

5

be <-conflict-free: {α, β, γ} <-attacks itself, whereas none of the singleton assumption sets is <-attacked at all. This leads to the main difference among generic ABA and ABA+ frameworks, namely that <-preferred extensions need not be <-complete, as illustrated in Example 3.6.

The significance of this example and the properties of ABA+ it illustrates is that dealing with preferences in generic ABA+frameworks may invalidate some properties that hold in the absence of preferences. In this case, this applies in particular to the relationship between <-preferred and <-complete semantics. We will revisit this relationship later in Chapter 4, in the context of Weak Contraposition.

Tables 3.1 and 3.2 summarise the relationships among the semantics of ABA and ABA+. (Results covering Table 3.1 can be found in [BDKT97, DMT07, Ton14, ˇCFST]. Results covering Table 3.2 follow from Table 3.1 and results obtained in Section 3.2.)

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 well founded X X X X X

Table 3.1: Relationships among non-flat ABA semantics

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 <-well founded X X X X X

Table 3.2: Relationships among non-flat ABA+ semantics. The only differ- ence from Table 3.1 is underlined

In summary, we established relationships that hold among ABA+seman- tics in contrast to relationships among ABA semantics: extensions of ABA+

frameworks generally relate in the same way as extensions of ABA frame- works, except that <-preferred extensions need not be <-complete.