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

Contents Review of grounded semantics Stable semantics Definitions A problem(?) Stable semantics Labelling-based Extension-based A problem with stable semantics

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

A B E C D

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’. Should capture reinstatement: A B C

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’. Works fine with: But not with: A B C A B

Two solutions A B C A B A B C A B A B Change conditions so that always a unique status assignment results Use multiple status assignments: and A B C A B A B C A B A B

Unique status assignments: Grounded semantics (informal) The endpoint (or union) of a sequence s.t.: S0: the empty set Si+1: Si + all arguments acceptable wrt Si ... A is acceptable wrt S (or S defends A) if all defeaters of A are defeated by S S defeats A if an argument in S defeats A

A B C D E Is B, D or E defended by S1? Is B or E defended by S2? This example: first ask which are acceptable wrt the empty set. Other examples of acceptability: A <- B <-C and A <-> B Is B, D or E defended by S1? Is B or E defended by S2?

Grounded semantics (formal 1) Let AAT be an abstract argumentation theory F0AAT =  Fi+1AAT = {A  Args | A is acceptable wrt FiAAT} F∞AAT = ∞i=0 (Fi+1AAT) Problem: does not always contain all intuitively justified arguments. 10

Grounded semantics (formal 2) Let AAT = Args,Defeat and S  Args FAAT(S) = {A  Args | A is acceptable wrt S} Since FAAT is monotonic (and since ...), FAAT has a least fixed point. Now: The grounded extension of AAT is the least fixed point of FAAT An argument is (w.r.t. grounded semantics) justified on the basis of AAT if it is in the grounded extension of AAT. Proposition 4.2.4 (AAT implicit): A  F∞  A is justified If every argument has at most a finite number of defeaters, then A  F∞AT  A is justified 11

Acceptability status with unique status assignments A is justified if A is In A is overruled if A is Out and A is defeated by an argument that is In A is defensible otherwise 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!) 13

A problem(?) with grounded semantics We have: We want(?): A B A B C C D D

A problem(?) with grounded semantics 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 D

A problem(?) with grounded semantics A = Frederic Michaud is French since Alice says so B = Frederic Michaud is Dutch since Bob says so 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 D E = Alice and Bob are unreliable since they contradict each other

Multiple labellings A B A B C C D D

Stable status assignments (Below is AAT = Args,Defeat implicit) A stable status assignment is a partition of Args into sets In and Out such that: 1. An argument is in In iff all arguments defeating it are in Out. 2. An argument is in Out iff it is defeated by an argument that is in 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. 18

Stable extensions Dung (1995): Now: 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 Suppose (In,Out) is a ssa: To be proven: (1) In is conflict-free. Suppose A,B  In and B defeats A. Then A,B  Out: contradiction. So there are no such A and B. (2) All A  Out are defeaed by In. This follows from condition (1) of ssa. Suppose S is a stable extension. To be proven: (S,Args/S) is an ssa. It is immediate that S Args/S = . Next it is to be proven that If A  S, then all defeaters of A are in Args/S This follows from conflict-freeness of S. If A  Args/S, then A is defeated by an argument in S. This follows from the definitions of Args/S and stable extensions. Picture: oval with In and Out, and under that S and Args/S. 19

Stable status assignments: a problem A stable status assignment is a partition of Args into sets In and Out such that: 1. An argument is in In iff all arguments defeating it are in Out. 2. An argument is in Out iff it is defeated by an argument that is in In. A B C 20

Stable status assignments: a problem A stable status assignment is a partition of Args into sets In and Out such that: 1. An argument is in In iff all arguments defeating it are in Out. 2. An argument is in Out iff it is defeated by an argument that is in In. A B C 21

Stable status assignments: a problem A stable status assignment is a partition of Args into sets In and Out such that: 1. An argument is in In iff all arguments defeating it are in Out. 2. An argument is in Out iff it is defeated by an argument that is in In. A B C 22

Stable status assignments: a problem A stable status assignment is a partition of Args into sets In and Out such that: 1. An argument is in In iff all arguments defeating it are in Out. 2. An argument is in Out iff it is defeated by an argument that is in In. A B C 23

Stable status assignments: a problem A stable status assignment is a partition of Args into sets In and Out such that: 1. An argument is in In iff all arguments defeating it are in Out. 2. An argument is in Out iff it is defeated by an argument that is in In. Why bad: add a fourth argument! A B C 24