Abstract Dialectical Frameworks

Origin. Brewka, Woltran (2010). Generalized argumentation. Acceptance conditions. Beyond attack. Flexible argumentative structure.

Models. Arguments with conditions. Generalized links. Acceptance functions. Boolean conditions.

Formalism.

ADF: D = ⟨S, L, C⟩ where: S: statements (arguments). L ⊆ S × S: links. C = {Cₛ}_{s∈S}: acceptance conditions.

Acceptance condition: Cₛ: 2^{par(s)} → {in, out}. par(s) = {r : (r,s) ∈ L}: parents. Boolean function. When s accepted.

Link types (derived): (r,s) supporting if Cₛ monotone increasing in r. (r,s) attacking if Cₛ monotone decreasing in r. Can be both/neither. Generalized.

Interpretation: v: S → {t, f, u}. Three-valued. t: true/in. f: false/out. u: undecided.

Operator: Γ_D(v)(s) based on Cₛ and v of parents. t if Cₛ true for all completions of v. f if Cₛ false for all completions. u otherwise.

Semantics: Admissible: v ≤_i Γ_D(v). Complete: v = Γ_D(v). Grounded: least fixpoint. Preferred, stable: adapted.

Relation to Dung: Dung AFs: special case. Cₛ = ¬r₁ ∧ ... ∧ ¬rₙ. Only attacks. ADFs generalize.

Bipolar ADFs: Links only support or attack. No mixed links. Common subclass.

Symbols.

Symbol	Unicode	Meaning
Cₛ	—	acceptance condition
par(s)	—	parents of s
Γ_D	—	characteristic operator
ADF	—	abstract dialectical framework

Metatheory. Generalized acceptance. Three-valued semantics. Fixpoint theory. Dung subsumption.

Applies to. Argumentation. Non-monotonic reasoning. Flexible dependencies. Mixed support/attack.

Limitations. Complexity higher than Dung. Condition specification. Less intuitive. Implementation harder.