# 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 |
|--------|---------|------|---------|
| A | — | Assumptions | Assumable formulas |
| ā | — | Contrary | Contrary of a |
| R | — | Rules | Inference rules |
| ⊢ | U+22A2 | Derives | Proof relation |

**Metatheory.** Generalizes logic programming. Maps to Dung. Captures default, autoepistemic. Complexity known.

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

**Limitations.** Contrary function design. Flat vs non-flat choice. Computational cost. Rule encoding effort.

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