2.2 Argumentation without Preferences
2.2.2 Assumption-Based Argumentation (ABA)
Definition 2.4. An ABA framework is a tuple (L, R, A,¯¯¯), where:
• (L, R) is a deductive system with L a language (a set of sentences5)
and R a set of rules of the form ϕ0 ← ϕ1, . . . , ϕm, with m > 0 and
ϕi ∈ L for i ∈ {0, . . . , m};
– ϕ0 is referred to as the head of the rule, and
– ϕ1, . . . , ϕm is referred to as the body of the rule;
– if m = 0, then the rule ϕ0 ← ϕ1, . . . , ϕm is said to have an empty
body, and is written as ϕ0 ← >, where > 6∈ L;
• A ⊆ L is a non-empty set whose elements are referred to as assump- tions;
• ¯¯¯ : A → L is a total contrary mapping: for α ∈ A, the sentence α is referred to as the contrary of α.6
5
We make no presuppositions about the logical form of sentences in the language.
6The contrary mapping can also be defined to assign non-empty sets of sentences to each
assumption, as in e.g. [FT14]. The two ways of defining the contrary mapping yield semantically equivalent outcomes (with respect to the ABA semantics defined later in this section). We chose the original formulation, as in [BDKT97], for notational convenience.
Note well that contraries are assigned to assumptions only, because as- sumptions in ABA represent defeasible information, which can be argued against (via contraries).
Observe also that contraries are not in general symmetric, in the sense that if α is an assumption with the contrary α, then α need not (and, in general, will not) be the contrary of α; indeed, α need not (and, in general, will not) itself be an assumption, and hence need not have a contrary. Thus, the contrary mapping is generally different from (classical) negation, which may or may not be present in the language L.
The contrary mapping in ABA can be seen as a generalisation of negation as failure (NAF) [Cla78], in the sense that an assumption α is acceptable as long as its contrary α is not acceptable, in the same vain as a NAF literal not a holds as long as the literal a cannot be shown to hold. For example, in the LP instance of ABA [BDKT97], every NAF literal not a in a logic program is an assumption in the instantiated ABA framework, with the contrary not a = a. Naturally thus, just like NAF in LP, the contrary mapping in ABA is not symmetric.
In the remainder of this section, unless specified differently, we assume as given a fixed but otherwise arbitrary ABA framework (L, R, A,¯¯¯).
Definition 2.5. A deduction for ϕ ∈ L supported by S ⊆ L and R ⊆ R, denoted by S `Rϕ, is a finite tree such that:
• the root is labelled by ϕ;
• the leaves are labelled by > or elements from S;
• for each non-leaf node labelled by ψ ∈ L, its children are labelled by the elements ψ1, . . . , ψmof the body of some rule ψ ← ψ1, . . . ψm from
R with head ψ;
• R is the set comprising exactly those rules ψ ← ψ1, . . . ψm.
The deduction relation ` ⊆ ℘(L)×L associated with (L, R, A,¯¯¯) is defined thus: for S ⊆ L and ϕ ∈ L,
S ` ϕ iff ∃ S `Rϕ for some R ⊆ R.
Definition 2.6. The conclusion operator Cn : ℘(L) → ℘(L) associated with (L, R, A,¯¯¯) is defined thus: for S ⊆ L,
That is, conclusions of S constitute the set of sentences for which deduc- tions supported by subsets of S exist. Although conclusions are defined for any set of sentences, we are usually interested in the conclusions of sets of assumptions, because ABA semantics is defined through set of assumptions. Semantics of ABA frameworks is defined in terms of sets of assumptions meeting desirable requirements. One such requirement is being closed under deduction of assumptions, defined next.
Definition 2.7. Let (L, R, A,¯¯¯) be given and let S ⊆ L.
• The closure Cl (S) of S is the set of assumptions that can be deduced from S, i.e. Cl (S) = Cn(S) ∩ A.
• S is closed just in case S = Cl (S).
• (L, R, A,¯¯¯) is flat just in case every A ⊆ A is closed.
Example 2.2. Consider an ABA framework (L, R, A,¯¯¯) with • the language L = {α, β, x, y, z},
• the set of rules R = {β ← α, z ← >}, • the set of assumptions A = {α, β}, • the contraries given by α = x, β = y. The following are deductions in (L, R, A,¯¯¯):
∅ `{z←>}z, {α} `∅α, {β} `∅ β, {α} `{β←α} β.
Since the only conclusion that can be drawn from ∅ is z, Cn(∅) = {z}. Since no assumptions can be deduced from the empty set of assumptions, the closure of ∅ is Cl (∅) = ∅. The conclusions of {α} are Cn({α}) = {α, β, z}, because both assumptions α and β can de deduced from {α} and ∅ ⊆ {α}. The closure of {α} is Cl ({α}) = Cn({α}) ∩ A = {α, β}. Since {α} is not equal to its closure, it is not closed. As not every set of assumptions is closed, (L, R, A,¯¯¯) is not flat.
Note that Cl ({β}) = {β}, so {β} is closed. Similarly, {α, β} is closed too.
We often use the term ‘flat ABA’ to refer to the class of flat ABA frame- works. We will see later that flat ABA exhibits additional properties to those exhibited by ABA in general.
The remaining desirable requirements met by sets of assumptions, as semantics for ABA frameworks, are given in terms of a notion of attack between sets of assumptions, defined as follows.
Definition 2.8. Let (L, R, A,¯¯¯) be given and let A, B ⊆ A. A attacks B, denoted A B, just in case A0` β for some β ∈ B and A0⊆ A.
To define ABA semantics, we use the following auxiliary notions.
Definition 2.9. Let (L, R, A,¯¯¯) be given and let E, A ⊆ A. • E is conflict-free just in case E 6 E;
• E defends A just in case for every closed B ⊆ A such that B A it holds that E B;
• E is admissible just in case E is closed, conflict-free and defends itself.
Defence as above can be said to be defined ‘setwise’, in the sense that a set E ⊆ A defends a set A ⊆ A (that contains possibly more than one as- sumption) against attacks on the whole of A. Originally [BDKT97], defence in ABA was defined with respect to a single assumption, in that E ⊆ A defended {α} for some α ∈ A whenever for every closed B ⊆ A such that B {α} it held that E B. Defence with respect to a set of assumptions was then extended so that E ⊆ A defended A ⊆ A whenever E defended every α ∈ A. In other words, defence in ABA was originally defined ‘point- wise’, as formulated next:
E ⊆ A defends A ⊆ A pointwise just in case for all α ∈ A, for every closed B ⊆ A such that B {α} it holds that E B.
The notions of defence (‘setwise’) and defence pointwise are equivalent in the following sense:7
Lemma 2.1. Let (L, R, A,¯¯¯) be given and let E, A ⊆ A. E defends A iff E defends A pointwise.
Proof. Suppose first E defends A. Let α ∈ A and consider B {α} with B ⊆ A closed. By definition of attack, we find B A. As E defends A, it holds that E B. Since α ∈ A and B {α} with B ⊆ A closed were arbitrary, we conclude that E defends A pointwise.
Suppose now that E defends A pointwise. Consider B A with B ⊆ A closed. By definition of attack, there is B0 ` α for some α ∈ A and B0⊆ B. Hence, B {α}. As E defends A pointwise, it holds that E B. Since B A with B ⊆ A closed was arbitrary, we conclude that E defends A.
7
Due to equivalence, in what follows we will use the standard (‘setwise’) notion of defence as in Definition 2.9, because it is more appropriate for ABA+, as we will see in Section 3.1.
ABA semantics is defined as follows.
Definition 2.10. Let (L, R, A,¯¯¯) be given and let E ⊆ A. E is:
• complete just in case E is admissible and contains every set of as- sumptions it defends;
• preferred just in case E is ⊆-maximally admissible;
• stable just in case E is closed, conflict-free and for every β ∈ A \ E it holds that E {β};
• ideal just in case E is ⊆-maximal among sets of assumptions that are admissible and contained in all preferred sets of assumptions;
• well founded just in case E is the intersection of all complete sets of assumptions.
A complete/preferred/stable/ideal/well founded set E of assumptions is also called a complete/preferred/stable/ideal/well founded extension of (L, R, A,¯¯¯).
Note that ideal sets of assumptions were originally defined in [DMT07] in the context of flat ABA frameworks only. The original definition naturally generalises to generic, possibly non-flat, ABA frameworks as given above. Note also that, in the case of flat ABA frameworks, the term grounded is conventionally used instead of well founded (e.g. in [DMT07]): we will adopt this convention too.
In this work we focus on the five semantics given above. To the best of our knowledge, other semantics have not been defined for, or investigated in, ABA, except for the semi-stable semantics [Cam06, Ver96], which was recently defined [CSAD13, ST15] and investigated for (flat) ABA [ST17]. We leave defining (for ABA+) and investigating other semantics, among them semi-stable, for future work.
We will also make use of the so-called defence operator, defined next.
Definition 2.11. Let (L, R, A,¯¯¯) be given. Define the defence operator Def : ℘(A) → ℘(A) thus: for A ⊆ A, Def (A) = {α ∈ A : A defends {α}}.
We recall the following established facts regarding ABA semantics and the defence operator, in the context of flat ABA: by [BDKT97, Theorem
5.7], for E ⊆ A admissible and S ⊆ Def (E), we have that E ∪S is admissible (this is called the Fundamental Lemma of ABA); by [BDKT97, Theorem 6.2], the grounded extension is the least fixed point of Def obtained by iteratively applying Def to ∅, equal to S
i>0Defi(∅), and it is therefore
complete.
We illustrate various ABA concepts with a formalisation of the Referen- dum example from the Introduction.
Example 2.3 (Example 1.1 as a flat ABA framework). The information given in Example 1.1 can be represented as an ABA framework FZ =
(L, R, A,¯¯¯) (which models Zed’s knowledge) with • L = {α, β, leave, stay},
• R = {leave ← α, stay ← β}, • A = {α, β},
• α = stay, β = leave.
Here, assumptions α and β stand for believing in Ann and Bob, respec- tively. Rules leave ← α and stay ← β represent the statements of Zed’s interlocutors: for instance, leave ← α represents that if Zed were to believe in Ann, the outcome of the referendum would be the Dutch leaving the EU. The contraries indicate which information is conflicting: for instance, the contrary of β being leave models that the Dutch leaving the EU—leave— conflicts with believing in Bob—β.8
Note that FZ is flat: no assumption can be deduced from the empty set
of assumptions, so Cl (∅) = ∅; the only assumptions deducible from {α} and {β} are α and β, respectively, so both {α} and {β} are closed; clearly, A is closed; hence, all sets of assumptions are closed.
In FZ, we find that {α} and {β} attack each other, and both of them at-
tack and are attacked by {α, β}, which also attacks itself. FZcan be graphi-
cally represented as in Figure 2.2 (in illustrations of ABA frameworks, nodes hold sets of assumptions while directed edges (arrows) indicate attacks).
FZ has two preferred extensions {α} and {β}, which are also stable, with
conclusions Cn({α}) = {α, leave} and Cn({β}) = {β, stay}, respectively. FZ has a unique grounded (well founded) and ideal extension ∅, with con-
8
We contend that other ways of formalising such examples in ABA are possible; we chose what seemed a natural and simple representation. Generally, representing knowledge in argumentation (and KR in general) is an open problem, discussion of which is beyond the scope of this thesis.
∅ {α} {β} {α, β}
Figure 2.2: Flat ABA framework FZ from Example 2.3
clusions Cn(∅) = ∅. Furthermore, all of {α}, {β} and ∅ are complete exten- sions.
Similarly, the extended Referendum example can be represented in ABA, but via a non-flat ABA framework, as follows.
Example 2.4 (Example 1.4 as a non-flat ABA framework). The situation where Dan joins the conversation can be represented by a non-flat ABA framework FD, which is FZ from Example 2.3 extended with an additional
rule β ← δ and an additional assumption δ (standing for trust in Dan). Overall, FD has
• L = {α, β, δ, leave, stay, δ},9
• R = {leave ← α, stay ← β, β ← δ}, • A = {α, β, δ},
• α = stay, β = leave.
FD can be depicted as in Figure 2.3.
∅ {α} {β} {α, β}
{δ} {α, δ} {β, δ} {α, β, δ}
Figure 2.3: Non-flat ABA framework FD from Example 2.4
In FD, if a set of assumptions contains δ, then it is closed only if it also
contains β. In particular, {δ} is not closed, and hence (L, R, A,¯¯¯) is not flat. Further, the only admissible sets of FD are ∅, {α}, {β} and {β, δ}.
Also, {β, δ} is a unique stable extension of FD, whereas both {α} and {β, δ}
9
Throughout the thesis, we assume, unless specified otherwise, that the contrary δ of any assumption δ is actually a symbol in the language L (i.e. δ ∈ L), wherefore we do not need to specify (some part of) the contrary mapping separately.
are preferred. Hence, ∅ is a unique ideal extension of FD. Also, {β, δ} is
the only complete extension, so it is a unique well founded extension of FD.
2.3 Argumentation with Preferences
In this section, we provide background on formalisms of argumentation with preferences that are most closely related to ABA+and that we will formally compare ABA+ to. These details will be required in Chapter 6; in addition, details on ASPIC+ will be needed in Chapter 4.