Assumption-Based Argumentation

Origin. Bondarenko, Dung, Kowalski, Toni (1997). Structured argumentation. Arguments from assumptions. Rules derive conclusions. Attacks via contraries. Foundation for logic-based argumentation.

Models. Deductive system with assumptions. Arguments: proofs from assumptions. Attacks: contrary assumptions. Maps to abstract frameworks. Computation via logic programming.

Formalism.

ABA framework: (L, R, A, ‾) L: language R: inference rules (non-assumption → formula) A ⊆ L: assumptions ‾: contrary function A → L

Argument: Tree with root claim, leaves assumptions. Applies rules from R. Assumptions S ⊢_R c: argument for c.

Attack: Argument with claim c̄ attacks argument using assumption a where c̄ = ā.

Contrary: ā: contrary of a ∈ A. If ā derivable, a cannot be assumed.

Contraries vs contradictories: Contrary: ā attacks a. Contradictory: mutual attack.

Flat ABA: Assumptions not derived. Maps directly to Dung framework.

Non-flat: Assumptions can have derivations. More expressive.

Semantics: Inherited from Dung: grounded, preferred, stable. Apply to sets of assumptions.

Example: Rule: flies(X) ← bird(X), ¬ab(X) Assumption: ¬ab(X) with contrary ab(X) Attacked by: penguin(X) → ab(X)

Symbols.

Symbol	Unicode	Name	Meaning
‾	U+203E	Overline / Contrary function	Maps each assumption to the formula whose derivation defeats it
ā	U+0101	Latin small letter a with macron	The contrary of the assumption a
̄	U+0304	Combining macron	Written over a symbol to form its contrary
⊢	U+22A2	Turnstile / Derivability	S ⊢_R c : the claim c follows from assumptions S by the rules R
⊆	U+2286	Subset	Assumptions are a distinguished subset of the language
∈	U+2208	Membership	Assumption belongs to A
←	U+2190	Leftwards arrow / Rule	Head holds if body holds
→	U+2192	Rightwards arrow	Rule bodies to heads; also the type of the contrary function
¬	U+00AC	Negation	Negative literal treated as an assumption in the example
L	—	Language	Formulas the framework is built over
R	—	Rules	Inference rules of the deductive system
A	—	Assumptions	Formulas that may be assumed
S	—	Assumption set	Candidate set evaluated by a semantics
c	—	Claim	Conclusion of an argument

Metatheory. Bondarenko, Dung, Kowalski and Toni (1997) presented ABA as a unifying frame rather than a new logic: default logic, autoepistemic logic, nonmonotonic modal logics and the standard logic-programming semantics all arise by choosing the deductive system and the contrary function, with negation-as-failure literals taken as assumptions so that stable extensions are the stable models and the grounded extension is the well-founded model. The reduction to Dung's abstract frameworks is exact in the flat case, where no assumption is the conclusion of a rule, so the extension hierarchy and the fixpoint account of the grounded extension transfer unchanged; non-flat frameworks require the semantics to be redefined over closed sets of assumptions. Dimopoulos, Nebel and Toni (2002) showed that for flat frameworks with a tractable derivability relation the complexity mirrors the abstract case — credulous acceptance NP-complete, skeptical acceptance coNP-complete under stable and Π₂ᵖ-complete under preferred. Dung, Kowalski and Toni (2006) gave dispute derivations, procedures computing on assumptions rather than on constructed arguments, sound in general and complete under acyclicity restrictions.

Applies to. Logic programming semantics. Legal reasoning. Medical reasoning. Dialogue systems.

Limitations. Everything rests on the contrary function, which is an unconstrained input: attack is defined by it alone, nothing requires ā to bear any logical relation to a, and a framework is therefore only as sensible as the mapping supplied with it. Contrariness need not be symmetric, so the formalism has attack but no notion of contradiction, and mutual attack must be arranged by hand. Rule heads are single formulas, so disjunctive conclusions are not expressible and reasoning by cases has to be encoded away. The clean correspondence with Dung's frameworks holds only in the flat case; letting assumptions be derived costs it, and the non-flat semantics must be given afresh over closed assumption sets. Arguments are proof trees, so a finite rule set can generate exponentially many of them, and the semantics inherit the intractability of the abstract case.