Agent-Based Systems Bargaining Alternating offers Michael Rovatsos - PowerPoint PPT Presentation

Agent-Based Systems Agent-Based Systems Where are we? Last time . . . Agent-Based Systems • Bargaining • Alternating offers Michael Rovatsos • Negotiation decision functions mrovatso@inf.ed.ac.uk • Task-oriented domains • Bargaining for resource allocation Today . . . Lecture 13 – Argumentation in Multiagent Systems • Argumentation in Multiagent Systems 1 / 18 2 / 18 Agent-Based Systems Agent-Based Systems Argumentation Different modes of argument • Agents may have mutually contradicting beliefs • At least four different modes of arguments can be identified - I believe p ; you believe ¬ p between humans: - I believe p , p → q ; you believe ¬ q 1. Logical mode (deductive, proof-like, concerned with making correct • How can agents reach agreements about what to believe ? inferences) • Argumentation provides principled techniques for deciding what 2. Emotional mode (appeals to feelings, attitudes, etc.) to believe in the face of inconsistencies 3. Visceral mode (physical, social aspects) • We achieve this by comparing arguments that can be compiled 4. Kisceral mode (appeals to the intuitive, mystical or religious) from the agents’ beliefs • Different types are used in different situations (e.g. logical mode • Arguments usually present beliefs and describe reasonable (hopefully) in courts of law) justifications 3 / 18 4 / 18

Agent-Based Systems Agent-Based Systems Abstract Argumentation Terminology • Lets consider some meaningful properties for rationally justified • We can decide what to believe while looking at arguments at the abstract level (Dung, 1995): sets of arguments - Disregarding their internal structures, e.g. arguments a , b , c , d • A set of arguments S is conflict-free if if there are no arguments a , - Focus on the attack relation, e.g. a attacks b or a → b b in S such that a attacks b , e.g. - Not concerned with the origin of arguments or the attack relation r • An abstract argumentation system A = � X , →� is defined by q p - a set of arguments X (just a collection of objects), s - →⊆ X × X a binary attack relation on arguments ∅ , { p } , { q } , { r } , { s } , { r , s } , { p , r } , { p , s } , { p , r , s } • Example: �{ p , q , r , s } , { ( r , q ) , ( s , q ) , ( q , p ) }� • An argument a is acceptable with respect to a set S of arguments r Arguments: p , q , r , s iff for each argument a ′ : if a ′ attacks a then a ′ is attacked by some q p Attacks: r → q , s → q , q → p argument in S s • A conflict-free set of arguments S is admissible iff each argument • Which arguments can we consider to be rationally justified? in S is acceptable w.r.t. S There is no universal definition for acceptability e.g. ∅ , { r } , { s } , { r , s } , { p , r } , { p , s } , { p , r , s } 5 / 18 6 / 18 Agent-Based Systems Agent-Based Systems Preferred Extensions Grounded Extensions (I) • Preferred extensions are maximal (w.r.t. set inclusion) admissible sets, e.g. { p , r , s } is a preferred extension, but not ∅ or { p } • An alternative notion of acceptability is provided by the notion of • Preferred extensions help determine which arguments should be grounded extension accepted but are not always useful: • The (unique) grounded extension can be built incrementally: 1 Arguments that are not attacked are “in” Preferred extensions are not necessarily unique a b 2 Delete from the graph every argument that is attacked by an e.g. { a } and { b } here argument that is in the grounded extension and go to Step 1 - Iterate until there are no more changes to the argument graph a The only preferred extension may be the empty set b • The grounded extension c - always exists and - is guaranteed to be unique, but • An argument is sceptically accepted if it is a member of every - may be empty (if no arguments are free of attackers initially) preferred extension • An argument is credulously accepted if it is a member of at least one preferred extension 7 / 18 8 / 18

Agent-Based Systems Agent-Based Systems Grounded Extensions (II) Example c m d • The characteristic function of an argumentation system g A = � X , →� , is the function F : 2 X → 2 X , which is defined as a k l j follows: b i F ( S ) = { a | a is acceptable w.r.t. S } e • The grounded extension of an argumentation system is the least n p fixed point of the characteristic function F f q h • Consider the sequence: - F 0 = ∅ , • Argument h has no attackers “in” - F i + 1 = { a ∈ X | a is acceptable w.r.t. F i } • Because of this, a is not acceptable “out” - · · · (until no arguments are added to the set) • For same reason p is out • p only attacker of q , thus q is “in” • · · · 9 / 18 10 / 18 Agent-Based Systems Agent-Based Systems Deductive Argumentation Systems Example: Arguments human ( X ) ⇒ mortal ( X ) • “Purest”, most rational kind of argument: in classical logic, human ( Hercules ) argument = sequence of inferences leading to a conclusion father ( Heracles , Zeus ) • Write Γ ⊢ ϕ to denote that sequence of inference steps from premises Γ will allow us to establish proposition ϕ , where Γ is part father ( Apollo , Zeus ) of our overall knowledge base ∆ divine ( X ) ⇒ ¬ mortal ( X ) Example: Γ ⊢ mortal ( Socrates ) where father ( X , Zeus ) ⇒ divine ( X ) Γ = { human ( Socrates ) , human ( X ) ⇒ mortal ( X ) } ¬ ( father ( X , Zeus ) ⇒ divine ( X )) • A deductive argument is a pair � Γ , ϕ � with support Γ and conclusion ϕ where: Examples of arguments: i. Γ ⊂ ∆ , Γ ⊢ ϕ Arg 1 = �{ human ( Heracles ) , human ( X ) ⇒ mortal ( X ) } , mortal ( Heracles ) � ii. Γ is logically consistent Arg 2 = �{ father ( Heracles , Zeus ) , father ( X , Zeus ) ⇒ divine ( X ) , iii. Γ is minimal (i.e. none of its subsets satisfies the above) • Two important classes of arguments: divine ( X ) ⇒ ¬ mortal ( X ) } , ¬ mortal ( Heracles ) � - Tautological arguments : � Γ , ϕ � where Γ = ∅ Arg 3 = �{¬ ( father ( X , Zeus ) ⇒ divine ( X )) } , ¬ ( father ( X , Zeus ) ⇒ divine ( X )) � - Non-trivial arguments : � Γ , ϕ � where Γ is consistent 11 / 18 12 / 18

Agent-Based Systems Agent-Based Systems The Attack Relation Argument Classes The attack relation is defined as follows • For any propositions ϕ and ψ , ϕ attacks ψ iff ϕ ≡ ¬ ψ We can identify five classes of argument type in order of increasing • � Γ 1 , ϕ 1 � rebuts � Γ 2 , ϕ 2 � if ϕ 1 attacks ϕ 2 acceptability • � Γ 1 , ϕ 1 � undercuts � Γ 2 , ϕ 2 � if ϕ 1 attacks some ψ ∈ Γ 2 A1: The class of all arguments that can be constructed • � Γ 1 , ϕ 1 � attacks � Γ 2 , ϕ 2 � if it undercuts or rebuts it A2: The class of all non-trivial arguments that can be constructed Example: A3: The class of all arguments that can be constructed with no Arg 1 = �{ human ( Heracles ) , human ( X ) ⇒ mortal ( X ) } , mortal ( Heracles ) � rebutting arguments Arg 2 = �{ father ( Heracles , Zeus ) , father ( X , Zeus ) ⇒ divine ( X ) , A4: The class of all arguments that can be constructed with no divine ( X ) ⇒ ¬ mortal ( X ) } , ¬ mortal ( Heracles ) � undercutting arguments Arg 3 = �{¬ ( father ( X , Zeus ) ⇒ divine ( X )) } , ¬ ( father ( X , Zeus ) ⇒ divine ( X )) � A5: The class of all tautological arguments that can be constructed - Arguments Arg 1 and Arg 2 are mutually rebutting - Argument Arg 3 undercuts argument Arg 2 13 / 18 14 / 18 Agent-Based Systems Agent-Based Systems Example: Argument Classes Argumentation dialogue systems • Agents engage in dialogue to convince other agents of some state of affairs • Consider two agents 0 and 1 engaging in the following dialogue: Arg 1 = �{ human ( Heracles ) , human ( X ) ⇒ mortal ( X ) } , mortal ( Heracles ) � - Agent 0 attempts to convince 1 of some argument Arg 2 = �{ father ( Heracles , Zeus ) , father ( X , Zeus ) ⇒ divine ( X ) , - Agent 1 attempts to rebut or undercut it divine ( X ) ⇒ ¬ mortal ( X ) } , ¬ mortal ( Heracles ) � - Agent 0 in turn attempts to defeat 1’s argument - And so on . . . Arg 3 = �{¬ ( father ( X , Zeus ) ⇒ divine ( X )) } , ¬ ( father ( X , Zeus ) ⇒ divine ( X )) � • Moves � Player , Arg � are steps in such a dialogue, Player ∈ { 0 , 1 } , Arg ∈ A (∆) (the set of all arguments constructed from ∆ ) - Arg 1 and Arg 2 are mutually rebutting and thus in A2 • A sequence � m 0 , . . . m k � is a dialogue history if - �∅ , divine ( Heracles ) ∨ ¬ divine ( Heracles ) � is in A5 - Player 2 i = 0, Player 2 i + 1 = 1 for all i ≥ 0 - �{ father ( apollo , Zeus ) , father ( X , Zeus ) ⇒ divine ( X ) , divine ( X ) ⇒ - If Player i = Player j and i � = j , then Arg i � = Arg j , ¬ mortal ( X ) } , ¬ mortal ( apollo ) � is in A4 - Arg i + 1 defeats Arg i for all i ≥ 0 • A dialogue ends if no further moves are possible, the winner is Player k 15 / 18 16 / 18