1. Argumentation Logics Lecture 2: Abstract argumentation grounded and stable semantics Henry Prakken Chongqing May 27, 2010

2. Contents Grounded semantics Definitions (extension-based) A problem(?) Stable semantics Labelling-based (Extension-based) A problem with stable semantics

3. We should lower taxes Lower taxes increase productivity Increased productivity is good We should not lower taxes Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity Prof. P says that … Prof. P has political ambitions People with political ambitions are not objective Prof. P is not objective Increased inequality is good Increased inequality stimulates competition Competition is good USA lowered taxes but productivity decreased

4. AB C D E

5. 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’.

6. 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

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

8. Unique status assignments: Grounded semantics, extension- based (informal) Given AAT =  Args,Defeat , A  Args and S  Args: A is acceptable wrt S (or S defends A) if all arguments in Args that defeat A are defeated by S S defeats A if an argument in S defeats A 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 of AAT

9. AB C D E Is B, D or E defended by S1? Is B or E defended by S2?

10. Acceptability status in grounded semantics (extension-based) A is justified if A is in the grounded extension A is overruled if A is not justified and A is defeated by an argument that is justified A is defensible otherwise (so if it is not justified and not overruled)

11. Exercise: determine grounded extension of AAT = A B C B

12. Self-defeating arguments Intuition: should always be overruled (?) Problem: in grounded semantics they are not always overruled Solution: several possibilities (but intuitions must be refined!) AB

13. Exercise: determine grounded extension of AAT = AB C D

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

15. A problem(?) with grounded semantics AB C D A = Frederic Michaud is French since he has a French name B = Frederic Michaud is Dutch since he is a marathon skater C = F.M. likes the EU since he is European (assuming he is not Dutch or French) D = F.M. does not like the EU since he looks like a person who does not like the EU

16. Multiple labellings AB C D AB C D

17. Stable status assignments (Below AAT =  Args,Defeat  is 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.s.a. A is overruled if A is Out in all s.s.a. A is defensible if A is In in some but not all s.s.a.

18. 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 Now: S is a stable argument extension if (In,Out) is a stable status assignment and S = In. Proposition 4.3.4: S is a stable argument extension iff S is a stable extension

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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