Commonsense Reasoning and Argumentation 13/14 HC 6 Abstract Argumentation Semantics (2)

Commonsense Reasoning and Argumentation 13/14HC 6Abstract ArgumentationSemantics (2) Henry Prakken February 24, 2014

Overview • Extension-based definitions (Dung 1995) • Self-defeat • (Arguing about defeat relations) • (Default logic as argumentation)

Status assignments (or ‘labellings’) Given an AAT = Args,Defeat: A status assignment of AAT 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 that defeat it are Out. 2. An argument is Out iff some argument that defeats it is In. Let Undecided = Args / (In Out): A status assignment is stable if Undecided =  (stable semantics) A status assignment is preferred if In is -maximal (preferred semantics). A status assignment is grounded if In is -minimal (grounded semantics) S Args is a stable/preferred/grounded argument extension iff S = In for some stable/preferred/grounded labelling

Grounded extension (1) A is acceptable wrtS (or SdefendsA) if all defeaters of A are defeated by S S defeats A if an argument in S defeats 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.

Grounded extension (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 (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 Proposition:S is the grounded extension iff S is the grounded argument extension

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: S is a stable argument extension iff S is a stable extension 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.

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: S is a preferred argument extension iff S is a preferred extension 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.

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 A B E D C Admissible?

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 A B E D C S is preferred if it is maximally admissible Preferred?

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 A B E D C S is groundeded if it is the smallest set s.t. A  S iff S defends A Grounded?

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 B F E D C

Properties • Every admissible set is included in a preferred extension • There exists at least one preferred extension • ...

Self-defeating arguments Grounded semantics: A is justified if A is In in the grounded s.a. A is overruled if is Outin the grounded s.a. A is defensible if A is undecided in the grounded s.a. Preferred semantics: A is justified if A is Inin all preferred s.a. A is overruled if A is Out or undecided in all preferred s.a. A is defensible if A is Inin some but not all preferred s.a. In grounded and preferred semantics self-defeating arguments are not always overruled They can make that there are no stable extensions

Arguing about defeat relations • Standards for determining defeat relations are often: • Domain-specific • Defeasible and conflicting • So determining these standards is argumentation! • Recently Modgil (AIJ 2009) has extended Dung’s abstract approach • Arguments can also attack attack relations

Will it rain in Calcutta? Modgil 2009 BBC says rain CNN says sun B C

Will it rain in Calcutta? Modgil 2009 Trust BBC more than CNN T BBC says rain CNN says sun B C

Will it rain in Calcutta? Modgil 2009 Trust BBC more than CNN T BBC says rain CNN says sun B C S Stats say CNN better than BBC

Will it rain in Calcutta? Modgil 2009 Trust BBC more than CNN T BBC says rain CNN says sun B C R Stats more rational than trust S Stats say CNN better than BBC

Default-logic argumentation theories • Every default theory  has an associated abstract argumentation theory AAT() as follows: • Args = {|  is a finite a-process of } • AdefeatsB iff   In(A) for some   Out(B) • (An a-process is any process with consistency and successfulness checks ignored)

Relation with stable semantics • Notation (for a given default theory ): • For every set S Args: Concs(S) = {|   In(A) for some A  S} • For every set E of wffs: Args(E) = {A  Args| for all k  Out(A): E |- k} • Proposition: For every default theory : • If S is a stable extension of AAT() then Concs(S) is a Reiter-extension of • If E is a Reiter-extension of  then Args(E) is a stable extension of AAT()

Commonsense Reasoning and Argumentation 13/14 HC 6 Abstract Argumentation Semantics (2)