Strong and Weak Forms of Abstract g Argument Defense Diego C. Martínez Alejandro J. García Guillermo R. Simari Artificial Intelligence R&D Laboratory Department of Computer Science and Engineering Universidad Nacional del Sur República Argentina Artificial Intelligence Research and Development Laboratory Department of Computer Science and Engineering – Universidad Nacional del Sur
Attackers with different strength Fly 1 : Fly Oceanic Airlines because it has the cheapest tickets. Fly 1 stronger NoFly : Do not fly Oceanic Airlines because the accident rate is high and the because the accident rate is high and the onboard service is not good. N Fl NoFly Fly 2 : Fly Oceanic Airlines because the accident rate is normal and the onboard service is improving. Fl Fly 3 : Fly Oceanic Airlines because you Fl O b i Ai li Fly 3 can see some islands in the flight route. Fly 2 Artificial Intelligence Research and Development Laboratory Department of Computer Science and Engineering – Universidad Nacional del Sur
Attackers with different strength Fly 1 : Fly Oceanic Airlines because it has the cheapest tickets. Fly 1 NoFly : Do not fly Oceanic Airlines because the accident rate is high and the because the accident rate is high and the onboard service is not good. equivalent N Fl NoFly Fly 2 : Fly Oceanic Airlines because the accident rate is normal and the onboard service is improving. Fl Fly 3 : Fly Oceanic Airlines because you Fl O b i Ai li Fly 3 can see some islands in the flight route. Fly 2 Artificial Intelligence Research and Development Laboratory Department of Computer Science and Engineering – Universidad Nacional del Sur
Attackers with different strength Fly 1 : Fly Oceanic Airlines because it has the cheapest tickets. Fly 1 NoFly : Do not fly Oceanic Airlines because the accident rate is high and the because the accident rate is high and the onboard service is not good. equivalent N Fl NoFly Fly 2 : Fly Oceanic Airlines because the accident rate is normal and the onboard service is improving. Fl Fly 3 : Fly Oceanic Airlines because you Fl O b i Ai li Fly 3 can see some islands in the flight route. Fly 2 Artificial Intelligence Research and Development Laboratory Department of Computer Science and Engineering – Universidad Nacional del Sur
Attackers with different strength Fly 1 : Fly Oceanic Airlines because it has the cheapest tickets. Fly 1 NoFly : Do not fly Oceanic Airlines because the accident rate is high and the because the accident rate is high and the onboard service is not good. N Fl NoFly Fly 2 : Fly Oceanic Airlines because the accident rate is normal and the onboard service is improving. incomparable Fl Fly 3 : Fly Oceanic Airlines because you Fl O b i Ai li Fly 3 can see some islands in the flight route. Fly 2 Artificial Intelligence Research and Development Laboratory Department of Computer Science and Engineering – Universidad Nacional del Sur
Attackers with different strength Fly 1 : Fly Oceanic Airlines because it has the cheapest tickets. Fly 1 NoFly : Do not fly Oceanic Airlines because the accident rate is high and the because the accident rate is high and the onboard service is not good. N Fl NoFly Fly 2 : Fly Oceanic Airlines because the accident rate is normal and the onboard service is improving. incomparable Fl Fly 3 : Fly Oceanic Airlines because you Fl O i Ai li b Fly 3 can see some islands in the flight route. Fly 2 Artificial Intelligence Research and Development Laboratory Department of Computer Science and Engineering – Universidad Nacional del Sur
The strenght of defenses Fly 1 NoFly Fly 3 Fly 2 Argument Fly 1 is defended by Fly 2 and Fly 3 The defense provided by Fly 2 may be considered stronger than the defense provided by Fly 3 This is the main motivation of this work. We explore this idea in the context of extended argumentation frameworks… Artificial Intelligence Research and Development Laboratory Department of Computer Science and Engineering – Universidad Nacional del Sur
Extended Abstract Frameworks - definition A An extended abstract argumentation framework (EAF) is a triplet t d d b t t t ti f k (EAF) i t i l t < AR , � , C , R > < AR � C R > Binary conflict relation Preference Finite set of Subargument relation between arguments relation for arguments C ⊆ AR × AR conflictive ⊆ arguments Arguments are abstract entities: A B C Arguments are abstract entities: A , B , C ,…. • The symbol � denotes subargument relation : A � B means A is a subargument of B . • In this work, the subargument relation is not relevant for the topic addressed and therefore • we will assume � = ∅ we will assume � ∅ . Artificial Intelligence Research and Development Laboratory Department of Computer Science and Engineering – Universidad Nacional del Sur
Extended Abstract Frameworks - definition A An extended abstract argumentation framework (EAF) is a triplet t d d b t t t ti f k (EAF) i t i l t < AR , � , C , R > < AR � C R > Binary conflict relation Preference Finite set of Subargument relation between arguments relation for arguments C ⊆ AR × AR conflictive ⊆ arguments • The conflict relation states the incompatibility of acceptance between arguments. • It is a symmetric relation. y • It is devoided of any form of argument evaluation. Artificial Intelligence Research and Development Laboratory Department of Computer Science and Engineering – Universidad Nacional del Sur
Extended Abstract Frameworks - definition An extended abstract argumentation framework (EAF) is a triplet A t d d b t t t ti f k (EAF) i t i l t < AR , � , C , R > < AR � C R > Binary conflict relation Preference Finite set of Subargument relation between arguments relation for arguments C ⊆ AR × AR conflictive ⊆ arguments • The preference relation is used to compare conflicting arguments. p p g g • It captures any form of evaluation. For example, an argument may be preferred to other if, • it exposes more specific information., or • it was constructed recently, or it t t d tl • it is proposed by a more reliable agent, or • it is undercutting the other argument, or • it simply satisfies a particular bias. it simply satisfies a particular bias Artificial Intelligence Research and Development Laboratory Department of Computer Science and Engineering – Universidad Nacional del Sur
Extended Abstract Frameworks - defeaters Relation R represents an order on the set of arguments Relation R represents an order on the set of arguments. If A R B but not B R A then A is preferred to B , denoted A � B • If A R B and B R A then A and B are arguments with equal relative • preference, denoted A ≡ B If neither A R B and B R A then A and B are incomparable arguments, • denoted A�B Let A and B be two arguments in AR such that { A B } ∈ C Let A and B be two arguments in AR such that { A , B } ∈ C. � If A is preferred to B , then it is said that A is a proper defeater of B . � If A and B have the equal relative strength , or are incomparable then no proper defeat relation can be established, and it is said that A and B are blocking , g defeaters. Artificial Intelligence Research and Development Laboratory Department of Computer Science and Engineering – Universidad Nacional del Sur
EAF - example AR = { Fly 1 , NoFly, Fly 2 , Fly 3 } AR = { Fly NoFly Fly Fly } � = ∅ < AR , � , C , R > C = {{ Fly 1 ,NoFly }, { Fly 2 ,NoFly }, { Fly 3 ,NoFly } } {{ } { } { } } 1 2 3 NoFly � Fly 1 , Fly 2 ≡ NoFly , Fly 3 � NoFly Fly 1 proper defeat NoFly blocking defeat by blocking defeat by incomparability incomparability equivalence in strength equivalence in strength Fly 3 Fly 2 Artificial Intelligence Research and Development Laboratory Department of Computer Science and Engineering – Universidad Nacional del Sur
Comparing individual defenses Let AF <Args, � ,C,R> be an EAF. Let AF=<Args � C R> be an EAF Let A and B be two arguments in Args. The function pref: Args × Args → {0,1,2} is defined as follows 0 if A � B 1 if A ≡ B pref( A , B ) = 2 if A � B 2 if A � B Let AF=<Args, � ,C,R> be an EAF. A Let A ∈ Args be an argument with defeater B , which is defeated, in turn, by arguments C and D . Then t C d D Th d f t d i t b � C and D are equivalent in force defenders of A if B B pref( C , B )=pref( D , B ) � C is a stronger defender than D if pref( C , B )>pref( D , B ) It is also said that D is a weaker defender than C C D Artificial Intelligence Research and Development Laboratory Department of Computer Science and Engineering – Universidad Nacional del Sur
Stronger Defense Let Φ =<Args � C R> be an EAF <Args, � ,C,R> be an EAF. Let Φ Let A ∈ Args be an argument acceptable with respect to S 1 ⊆ Args. A set of arguments S 2 ⊆ Args is said to be a stronger collective defense of A if A i A is acceptable with respect to S 2 , and t bl ith t t S d • There are no two arguments X ∈ S 1 and Y ∈ S 2 such that X constitutes a stronger defense than Y . • For at least one defender X ∈ S 1 of A , there exists an argument Y ∈ S 2 − S 1 which constitutes a stronger defense of A . S 1 A S 2 No argument in S 1 is a stronger defender than an argument in S 2 . S 2 provides at least one stronger defender than an argument in S 1 . 2 p g g 1 Artificial Intelligence Research and Development Laboratory Department of Computer Science and Engineering – Universidad Nacional del Sur
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