AFs Labelling Definition

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Definition: Let AF = (A, R) be an argumentation framework.
A labelling is a total function L : A → {IN, OUT, UNDEC}.

Definition: Let L be a labelling for AF = (A, R) and x ∈ A, then:

1. x is legally IN iff x is labelled IN and every y that attacks x (yRx) is labelled OUT;
2. x is legally OUT iff x is labelled OUT and there is at least one y that attacks x and y is labelled IN;
3. x is legally UNDEC iff x is labelled UNDEC, there is no y that attacks x such that y is labelled IN, and it is not the case that: for all y, y attacks x implies y is labelled OUT.

The rules above defining legal labellings, aim to capture one’s intuitive understanding of when arguments should be accepted or rejected. Here it is the case that an argument is IN, only if all its attackers are OUT. An argument is OUT if it has at least one attacker that is IN. And an argument is UNDEC if it cannot be IN or OUT.

Definition: A labelling L is a grounded labelling iff {x|L(x)=IN} is subset minimal.

In the following you may find an algorithm to evaluate grounded labelling.

Reference: https://webspace.science.uu.nl/~woerk003/Labellings_and_their_Abstract_Argumentation_Framework_Semantics.pdf

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