Abstract Argumentation

Origin. Phan Minh Dung introduced abstract argumentation frameworks (1995). Arguments are abstract entities; the focus is on attack relations and acceptable sets. Unified view subsuming logic programming, default logic, and other non-monotonic systems. Foundation for computational argumentation in AI, law, and multi-agent systems.

Models. Arguments attacking other arguments. An argumentation framework is a directed graph: nodes are arguments, edges are attacks. Which arguments can be jointly accepted? Various semantics define acceptable sets (extensions). Abstract: no internal structure of arguments, only attacks matter.

Formalism.

Argumentation framework: AF = (A, R) where:

A: set of arguments
R ⊆ A × A: attack relation
(a,b) ∈ R means "a attacks b"

Key concepts:

S ⊆ A is conflict-free iff no a,b ∈ S with (a,b) ∈ R
a is defended by S iff for all b attacking a, some c ∈ S attacks b
S is admissible iff conflict-free and defends all its members

Extension semantics:

Complete: admissible and contains all arguments it defends
Preferred: maximal complete extension
Grounded: minimal complete extension (unique, skeptical)
Stable: conflict-free and attacks all non-members
Hierarchy: Stable ⊆ Preferred ⊆ Complete ⊇ Grounded

Labeling semantics: Assign each argument: in (accepted), out (rejected), undec (undecided).

in: all attackers are out
out: some attacker is in
undec: neither applies

Reasoning modes:

Credulous: argument is in some extension
Skeptical: argument is in all extensions

Example: A = {a, b, c}, R = {(a,b), (b,a), (b,c)}

a and b attack each other
b attacks c
Preferred extensions: {a,c} and {b}
Grounded: {} (no unattacked argument)
Stable: {a,c} and {b}

Symbols.

Symbol	Unicode	Name	Meaning
→	U+2192	Attacks	Directed attack
↔	U+2194	Mutual attack	Symmetric attack
+	—	Defends	Counterattacks attacker
∈ E	—	In extension	Accepted in E
GE	—	Grounded	Unique grounded extension
PR	—	Preferred	Maximal extensions
ST	—	Stable	Complete attacking
in/out/undec	—	Labels	Acceptance status

Metatheory. Deciding credulous acceptance: NP-complete for preferred/stable, P for grounded. Skeptical preferred: Π₂ᵖ-complete. Stable may have no extension. Preferred always exists (possibly empty). Dung showed: logic programming, default logic can be translated to AF. AFs are abstract; to argue about specific domains, instantiate with structured argumentation.

Applies to. AI reasoning systems. Legal argumentation. Multi-agent dialogue. Decision support. Explainable AI. Deliberation and debate modeling. Defeasible reasoning.

Limitations. Abstract: doesn't say what arguments are or why attacks hold. Structured argumentation (ASPIC+, ABA, DeLP) adds this but increases complexity. No support relation (only attack) in basic AFs; bipolar extensions add support. Static: no dynamics of argument introduction or persuasion. Multiple extensions can be hard to interpret. Scalability: large AFs are computationally challenging.