Assumption-Based Argumentation (ABA) is a versatile non-monotonic reasoning formalism that derives arguments from a set of defeasible assumptions via some given inference rules. Like many other formal reasoning tools, ABA also allows the derivation of contradictions, giving rise to direct and indirect self-contradicting arguments. For Abstract Argumentation, a closely related formalism, semantics based on weak admissibility have been shown to solve this problem while otherwise behaving nicely. Instead of the classic defense by counter-attack, they use a recursive notion of admissibility to check the validity of each incoming attack. We explore how this approach can be realized in the more expressive ABA formalism. Our proposal is to use Abstract Bipolar SETAFs as an in-between to take advantage of the abstract intuition behind weak admissibility without a loss in expressiveness on the side of ABA.
In both Structured and Abstract Argumentation, a key concept is the notion of admissibility. Loosely
speaking, admissibility formalizes that, in a given debate, an acceptable set of arguments should
(i) not contain internal conflicts and (ii) be able to refute arguments raised against . However, as
pointed out in several works [
Clearly, in this scenario we would like to disregard the self-attacking argument that represents distrusting reasonable information. However, no commonly agreed semantics for ABAFs can handle this in a satisfactory way.
As we said in our introduction, ABA is a suitable formalism for modeling a variety of argumentation scenarios like this example due to its simple structure but more than suficient expressiveness. Before we go into further detail, we therefore give a short formal introduction to ABA. Based on a deductive system (ℒ, ℛ), where ℒ is a set of sentences and ℛ a set of inference rules, an ABA framework (ABAF) is a tuple = (ℒ, ℛ, , ), where is a set of assumptions from which we infer according to the rules ℛ, and : → ℒ a contrary function we use to derive attacks on assumptions. In order to conduct defeasible reasoning with an ABAF = (ℒ, ℛ, , ) we consider arguments that can be built by applying rules to assumptions. More precisely, we consider the fact, that some ∈ ℒ can be derived from a set of assumptions ⊆ according to the rules ℛ an (ABA) argument, and denote this by ⊢ . Furthermore, if is the contrary () of some assumption ∈ we say the set attacks any set ⊆ with ∈ .A set of assumptions is conflict-free if it does not attack itself; admissible if it is conflict-free and defends itself, i. e. attacks all of its attackers. Consider the introductory example. Example 2.2. We can model our introductory Example 2.1 with the ABAF = {, , }, ℒ = ∪ { | ∈ }, contraries () = for each assumption and rules ℛ = {( ← ), ( ← , )}.
The assumption is attacked by the set {, } and cannot be defended, because the only attacker of that set is the assumption itself.
At first, we tried to use weak admissibility for AFs in ABA out of the box: the idea was to simply evaluate the argumentation graph corresponding to an ABAF . Given an ABAF = (ℒ, ℛ, , ), the corresponding argumentation graph is the tuple = (, ) where is the set of all ABA arguments ⊢ with ∈ ℒ derived from a set of assumptions ⊆ according to the rules ℛ and attacks ( ⊢ , ⊢ ) if is the contrary of some ∈ . Now if an argument attacks itself in an abstract argumentation framework, it can be ignored under weak admissibility and is neither accepted nor relevant for the acceptance of other arguments. This direct approach, however, does not work as we demonstrate next. Example 2.3. Instantiating the ABAF from Example 2.2 as an AF yields the following argumentation graph :
{} ⊢ {, } ⊢ Instead of disregarding the self-attacking assumption , weak admissibility only allows us to disregard the argument “() ← ” which explicates that this assumption is self-attacking. Consequently, and are accepted whereas is not. This is surprisingly far away from what we want to achieve; after all, weak admissibility handles (abstract) self-attackers quite well.
In [
We restrict ourselves to finite BSAFs, those that have finitely many arguments.
Definition 3.2. Given a BSAF F = (, , ) and a set ⊆ of arguments. With
F () := ∪ {ℎ ∈ | ∃ (, ℎ) ∈ : ⊆ } we call cl F () = ⋃︀≥1 F () the closure of ; is closed if F () = .
We denote by + the set of all arguments attacked by and define ⊕ = ∪ + . A set ⊆ is conflict-free ( ∈ cf (F )) if it does not attack itself; defends ∈ if for each closed set ′ ⊆ attacking , attacks ′; defends ′ if defends each ∈ ′.
A conflict-free set is admissible ( ∈ adm(F )) if is closed and defends itself.
Due to space limitations we omit the introduction of the standard semantics. Before we can give the definition of weakly admissible semantics for this framework, we first need a suitable notion of reduct. Definition 3.3. Given a BSAF ( , , ), with
F = (, , ) and ⊆ , the -reduct of F is the BSAF F = = ∖ (cl ()⊕)
Note that some design choices had to be made for handling collective attacks and supports. Intuitively, an attack/support is removed in the reduct, whenever one attacking/supporting assumption is attacked by the set in question, otherwise it is restricted to the assumptions remaining in the reduct. Furthermore, we add self-attacks in the reduct for sets of assumptions that support an argument attacked by . The -reduct for BSAFs gives us the tools to generalize weak admissibility. Note that the definition is recursive, but well-defined as in each recursion step the reduct contains fewer arguments and we deal only with finite BSAFs.
Definition 3.4. Let F = (, , ) be a BSAF, ⊆ a set of arguments, and F = (, , ) its -reduct. Then is called weakly admissible in F ( ∈ admw(F )) if 1. ∈ cf (F ), closed and 2. for each (, ℎ) ∈ with ℎ ∈ , and ∩ + = ∅ it holds ∄′ ∈ admw(F ) s.t. ∩ ⊆ ′.
When applying this definition to our introductory example, we can now ignore the joint attack involving the self-attacker and accept the argument that climate change is happening. Example 3.5. Recall our introductory Example 2.1 about climate change. The instantiated BSAF is given as follows. • there exists ( ′, ) ∈ and ℎ ∈/ ′.
• there exists ( ′, ) ∈ .
We have adm() = {∅, {}, {}, {, }}.
So our motivating example is indeed handled as desired now. To demonstrate that weak admissibility
behaves well in general, let us consider the paradoxical rule principle from [
Definition 3.6. Given a BSAF F = (, , ). An attack = (, ℎ) ∈ is paradoxical if ̸= ∅ and for every ∈ there is a ′ ⊆ , ′ ̸= ∅ s.t.
A support = (, ℎ) ∈ is paradoxical if ̸= ∅ and for every ∈ there is a ′ ⊆ , ′ ̸= ∅ s.t.
We want weak admissibility to be stable under removal of paradoxical rules, thus the principle states the following requirement. (PRS) Paradoxical Attacks/Supports: Removing a paradoxical attack or support does not alter the models of F , i. e. admw(F ) = admw(F ′) where F ′ = (, ∖ {}, ) (resp. F ′ = (, , ∖ {})).
Proposition 3.7. The weakly admissible semantics for ABA satisfies the Principle of Paradoxical Attacks/Supports.
With this result from [
Successfully formulating weak admissibility for general ABA provides valuable insights towards a
native reduct notion for ABA and an ABA-semantics satisfying long-standing rationality postulates
like non-interference [
The research reported here was partially supported by the Deutsche Forschungsgemeinschaft (grant 550735820). 1–6
The author has not employed any Generative AI tools.