1. Argumentation Logics Lecture 3: Abstract argumentation semantics (3) Henry Prakken Chongqing May 28, 2010

2. Contents Review of grounded, stable and preferred semantics Labelling-based Stable and preferred semantics Extension-based Correspondence between labelling-based and extension-based semantics Concluding remarks on semantics of abstract argumentation.

3. Status of arguments: abstract semantics (Dung 1995) INPUT: an abstract argumentation theory AAT =  Args,Defeat  OUTPUT: An assignment of the status ‘in’ or ‘out’ to all members of Args So: semantics specifies conditions for labeling the ‘argument graph’.

5. Possible labeling conditions Every argument is either ‘in’ or ‘out’. 1. An argument is ‘in’ iff all arguments defeating it are ‘out’. 2. An argument is ‘out’ iff it is defeated by an argument that is ‘in’. Produces unique labelling with: But produces two labellings with: ABC ABAB

6. Two solutions Change conditions so that always a unique status assignment results Use multiple status assignments: and ABC ABAB ABC AB

7. A problem(?) with grounded semantics We have: We want(?): AB C D AB C D

8. Multiple labellings AB C D AB C D

9. Stable status assignments (Below is AAT =  Args,Defeat  implicit) A stable status assignment assigns to all members of Args either the status In or Out (but not both) such that : 1. An argument is In iff all arguments defeating it are Out. 2. An argument is Out iff it is defeated by an argument that is In. A is justified if A is In in all s.a. A is overruled if A is Out in all s.a. A is defensible if A is In in some but not all s.a.

10. Stable status assignments: a problem A stable status assignment assigns to all members of Args either the status In or Out (but not both) such that : 1. An argument is In iff all arguments defeating it are Out. 2. An argument is Out iff it is defeated by an argument that is In. AB C

11. Stable status assignments: a problem A stable status assignment assigns to all members of Args either the status In or Out (but not both) such that : 1. An argument is In iff all arguments defeating it are Out. 2. An argument is Out iff it is defeated by an argument that is In. AB C

12. Stable status assignments: a problem A stable status assignment assigns to all members of Args either the status In or Out (but not both) such that : 1. An argument is In iff all arguments defeating it are Out. 2. An argument is Out iff it is defeated by an argument that is In. AB C

13. Stable status assignments: a problem A stable status assignment assigns to all members of Args either the status In or Out (but not both) such that : 1. An argument is In iff all arguments defeating it are Out. 2. An argument is Out iff it is defeated by an argument that is In. AB C

14. Stable status assignments: a problem A stable status assignment assigns to all members of Args either the status In or Out (but not both) such that : 1. An argument is In iff all arguments defeating it are Out. 2. An argument is Out iff it is defeated by an argument that is In. AB C D

15. Status assignments A status assignment assigns to zero or more members of Args either the status In or Out (but not both) such that: 1. An argument is In iff all arguments defeating it are Out. 2. An argument is Out iff it is defeated by an argument that is In. Let Undecided = Args / (In  Out): A status assignment is stable if Undecided = . In is a stable argument extension A status assignment is preferred if Undecided is  -minimal. In is a preferred argument extension A status assignment is grounded if Undecided is  -maximal. In is the grounded argument extension

16. AB C D 1. An argument is In iff all arguments defeating it are Out. 2. An argument is Out iff it is defeated by an argument that is In. Grounded s.a. minimise node labelling Preferred s.a maximise node labelling

17. AB C D 1. An argument is In iff all arguments defeating it are Out. 2. An argument is Out iff it is defeated by an argument that is In. Grounded s.a. minimise node labelling Preferred s.a maximise node labelling

18. AB C D E 1. An argument is In iff all arguments defeating it are Out. 2. An argument is Out iff it is defeated by an argument that is In. Grounded s.a. minimise node labelling Preferred s.a maximise node labelling

19. Correspondence between labelling- based and extension-based semantics of abstract argumentation Bart Verheij (1996) Hadassah Jakobovits (1999) Martin Caminada (2006)

20. Status assignments A status assignment assigns to zero or more members of Args either the status In or Out (but not both) such that: 1. An argument is In iff all arguments defeating it are Out. 2. An argument is Out iff it is defeated by an argument that is In. Let Undecided = Args / (In  Out): A status assignment is stable if Undecided = . In is a stable argument extension A status assignment is preferred if Undecided is  -minimal. In is a preferred argument extension A status assignment is grounded if Undecided is  -maximal. In is the grounded argument extension

21. Grounded extensions again Dung (1995): Construct a sequence such that: S0: the empty set Si+1: Si + all arguments in Args that are defended by Si The endpoint of the sequence is the grounded extension Recall: S is a grounded argument extension if (In,Out) is a grounded status assignment and S = In. Proposition 2.3.16: S is a grounded argument extension iff S is a grounded extension

22. Stable extensions Dung (1995): S is conflict-free if no member of S defeats a member of S S is a stable extension if it is conflict-free and defeats all arguments outside it Recall: S is a stable argument extension if (In,Out) is a stable status assignment and S = In. Proposition 2.3.4: S is a stable argument extension iff S is a stable extension

23. Preferred extensions Dung (1995): S is conflict-free if no member of S defeats a member of S S is admissible if it is conflict-free and all its members are acceptable wrt S S is a preferred extension if it is  -maximally admissible Recall: S is a preferred argument extension if (In,Out) is a preferred status assignment and S = In. Proposition 2.3.13: S is a preferred argument extension iff S is a preferred extension

24. AB C D E S defends A if all defeaters of A are defeated by a member of S S is admissible if it is conflict-free and defends all its members Admissible?

25. AB C D E S defends A if all defeaters of A are defeated by a member of S S is admissible if it is conflict-free and defends all its members Admissible?

26. AB C D E S defends A if all defeaters of A are defeated by a member of S S is admissible if it is conflict-free and defends all its members Admissible?

27. AB C D E S defends A if all defeaters of A are defeated by a member of S S is admissible if it is conflict-free and defends all its members Admissible?

28. AB C D E S defends A if all defeaters of A are defeated by a member of S S is admissible if it is conflict-free and defends all its members Preferred? S is preferred if it is maximally admissible

29. AB C D E S defends A if all defeaters of A are defeated by a member of S S is admissible if it is conflict-free and defends all its members Preferred? S is preferred if it is maximally admissible

30. AB C D E S defends A if all defeaters of A are defeated by a member of S S is admissible if it is conflict-free and defends all its members Preferred? S is preferred if it is maximally admissible

31. AB C D E S defends A if all defeaters of A are defeated by a member of S S is admissible if it is conflict-free and defends all its members Grounded?

32. AB C D E S defends A if all defeaters of A are defeated by a member of S S is admissible if it is conflict-free and defends all its members Grounded?

33. AB C D E 1. An argument is In if all arguments defeating it are Out. 2. An argument is Out if it is defeated by an argument that is In. F

34. Properties Every admissible set is included in a preferred extension The grounded extension is unique Every stable extension is preferred (but not v.v.) There exists at least one preferred extension The grounded extension is a subset of all preferred and stable extensions Every AAT without infinite defeat paths has a unique extension (which is the same in all semantics) Every AAT without defeat cycles of odd length has a stable extension...

35. Self-defeating arguments again Recall (for preferred and stable semantics): A is justified if A is In in all s/p.s.a. A is overruled if A is Out in all s/p.s.a. A is defensible if A is In in some but not in all s/p.s.a. In (grounded and) preferred semantics self-defeating arguments are not always overruled They can make that there are no stable extensions AB

36. Self-defeating arguments again Recall (for preferred and stable semantics): A is justified if A is In in all s/p.s.a. A is overruled if A is Out in all s/p.s.a. A is defensible if A is In in some but not in all s/p.s.a. In (grounded and) preferred semantics self-defeating arguments are not always overruled They can make that there are no stable extensions AB

37. Which semantics is the “right” one? Alternative semantics may each have their use in different contexts E.g. in criminal procedure the burden of proof is on the prosecution, so grounded semantics with justified arguments is suitable. Or in decision making a choice must be made between alternative ways to achieve one’s goals, so preferred semantics with defensible arguments is suitable.