*Notes taken from ‘On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games’, by Phan Minh Dung (1995)*

**1.1, Argumentation Frameworks**

1, An

*argumentation framework*is a pair AF = (AR, attacks) where AR is a set of arguments, and attacks is a binary relation on AR.

2, A set of arguments is said to be

*conflict-free*if there are no arguments A, B in S such that A attacks B.

3, (1) An argument A is

*acceptable*wrt a set S of arguments iff for each argument B: if B attacks A then B is attacked by S. (2) A conflict-free set of arguments S is

*admissible*iff each argument in S is acceptable wrt S.

The

*(credulous) semantics*of an argumentation framework is defined by the notion of preferred extension.

4, A

*preferred extension*of an argumentation framework AF is a maximal (wrt set inclusion) admissible set of AF.

(Corollary 2)

*Every*argumentation framework possesses

*at least one*preferred extension. Hence, preferred extension semantics is

*always*defined for any argumentation framework.

5, A conflict-free set of arguments S is called a

*stable extension*iff S attacks each argument which does not belong to S.

(Lemma 3) S is a

*stable extension*iff S = { A | A is not attacked by S}

(Lemma 4)

*Every*stable extension is a preferred extension, but not vice versa.

**1.2, Fixpoint Semantics and Grounded (Sceptical) Semantics**

6, The

*characteristic function*, denoted by F, of an argumentation framework AF is defined as follows: F(S) = {A | A is acceptable wrt S}

(Lemma 5) A conflict-free set S of arguments is

*admissible*iff S is a subset of F(S).

It is easy to see that if an argument A is acceptable wrt S then A is also acceptable wrt any superset of S. Thus, it follows immediately that (Lemma 6) F is

*monotonic*(wrt set inclusion).

The

*sceptical semantic*of argumentation frameworks is defined by the notion of grounded extension:

7, The

*grounded extension*of an argumentation framework AF, denoted by GE, is the least fixed-point of F.

8, An admissible set S of arguments is called a

*complete extension*iff each argument which is acceptable wrt S, belongs to S.

Intuitively, the notion of complete extensions captures the kind of

*confident rational agent*who believes in everything he can defend.

(Lemma 7) A conflict-free set of arguments E is a

*complete extension*iff E = F(E).

The

*relations*between preferred extensions, grounded extensions and complete extensions is as follows (Theorem 2):

(1) Each preferred extensions is a complete extension, but not vice-versa.

(2) The grounded extension is the least (wrt set inclusion) complete extension.

(3) …

In general, the intersection of all preferred extensions does not coincide with the grounded extension.

9, An argumentation framework AF = (AR, attacks) is

*finitary*iff for each argument A, there are only finitely many arguments in AR which attack A.

**1.3, Sufficient Conditions for Coincidence between Different Semantics**

10, An argumentation framework is

*well-founded*iff there exists no infinite sequence A0, A1, …, An, … such that for each i, Ai+1 attacks Ai. (Note: This eliminates cycles.)

(Theorem 3)

*Every*well-founded argumentation framework has

*exactly one*complete extension which is grounded, preferred and stable.

11, (1) An argumentation framework AF is said to be

*coherent*if each preferred extension of AF is stable. (2) We say that an argumentation framework AF is

*relatively grounded*if its grounded extension coincides with the intersection of all preferred extensions.

It follows directly from the definition that there exists

*at least one*stable extension in a coherent argumentation framework.

An argument B

*indirectly attacks*A if there exists a finite sequence A0, …, A2n+1 such that (1) A = A0 and B = A2n+1, and (2) for each i, 0<=i<=2n, Ai+1 attacks Ai.

An argument B

*indirectly defends*A if there exists a finite sequence A0, …, A2n such that (1) A = A0 and B = A2n, and (2) for each i, 0<=i<=2n, Ai+1 attacks Ai.

An argument B is said to be

*controversial*wrt A if B indirectly attacks A and indirectly defends A.

An argument is

*controversial*if it is controversial wrt some argument A.

1.12, (1) An argumentation framework is

*uncontroversial*if none of its arguments is controversial. (2) An argumentation framework is

*limited controversial*if there exists no infinite sequence of arguments A0, …, An, … such that Ai+1 is controversial wrt Ai.

(Theorem 4) (1)

*Every*limited controversial argumentation framework is coherent. (2)

*Every*uncontroversial argumentation framework is coherent and relatively grounded.

An argument A is said to be a

*threat*to a set of arguments S if A attacks S and A is not attacked by S. A set of arguments D is called a

*defence*of a set of arguments S if D attacks each threat to S.

(Lemma 9) Let AF be a limited controversial argumentation framework. Then there exists

*at least one*nonempty complete extension E of AF.

(Lemma 10) Let AF be an uncontroversial argumentation framework, and A be an argument such that A is not attacked by the grounded extension GE of AF and A is not in GE. Then (1) There

*exists*a complete extension E1 such that A is in E1, and (2) There

*exists*a complete extension E2 such that E2 attacks A.

(Corollary 11)

*Every*limited controversial argumentation framework possesses

*at least one*stable extension.

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