Thursday, 22 March 2007

Well-founded frameworks and complete extensions

Dung defines the well-founded frameworks in his Acceptability of Arguments paper as: "Definition 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."

So, does this framework satisfy the definition (where 'C -> B' means argument C attacks argument B):
C -> B
B -> A
D -> A
A -> E


Yes, the framework is well-founded.

If so, what would be its complete extension? I think it would be {C, D}. However, according to Theorem 3 "Every well-founded argumentation framework has exactly one complete extension which is grounded, preferred and stable". This extension {C, D} is grounded and preferred but not stable. Can you see where I've gone wrong in my interpretation?

The (grounded, preferred, stable) extension is {C,D,E}. Indeed, E is defended by D, so it has to be in the grounded extension.

If an argument F were added to the framework such that F attacks C (F -> C), then the (grounded, preferred, stable) extension would be {F, B, D, E}.

If, on the other hand, argument E was removed (and hence the attack 'A -> E') from the original framework above, then the (grounded, preferred, stable) extension would be {C, D}.

Lastly, if F and G were added to the original framework above such that 'F -> C' and 'G -> D', then the (grounded, preferred, stable) extension would be {F, B, G}.

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