Argumentation Logics Lecture 2 Abstract argumentation grounded and

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

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

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

A C B D E

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

Possible labeling conditions n 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’. n Works fine with: n But not with: A B C

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

Unique status assignments: Grounded semantics (informal) n The endpoint (or union) of a sequence s. t. : n n S 0: 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 n S defeats A if an argument in S defeats A

A C B D E Is B, D or E defended by S 1? Is B or E defended by S 2?

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

Grounded semantics (formal 2) n Let AAT = Args, Defeat and S Args n n n FAAT(S) = {A Args | A is acceptable wrt S} Since FAAT is monotonic (and since. . . ), FAAT has a least fixed point. Now: n The grounded extension of AAT is the least fixed point of n An argument is (w. r. t. grounded semantics) justified on the basis of AAT if it is in the grounded extension of AAT. FAAT Proposition 4. 2. 4 (AAT implicit): n n A F∞ A is justified If every argument has at most a finite number of defeaters, then A F∞AT A is justified

Acceptability status with unique status assignments n n n 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

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

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

A problem(? ) with grounded semantics A 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 B C D

A problem(? ) with grounded semantics E A 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 E = Alice and Bob are unreliable since they contradict each other B C D

Multiple labellings A B C C D D

Stable status assignments n n (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. n n n 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.

Stable extensions n Dung (1995): n n n arguments outside it Now: n n 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 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

Stable status assignments: a problem n 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