seminar abstract argumentation - tu dresden€¦ · parmenides-system for e-democracy: facilitates...
TRANSCRIPT
![Page 1: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/1.jpg)
Artificial Intelligence, Computational Logic
SEMINAR ABSTRACTARGUMENTATION
Introduction to Formal Argumentation∗
slides adapted from Stefan Woltran’s lecture on Abstract Argumentation
Sarah Gaggl
Dresden, 1st November 2013
![Page 2: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/2.jpg)
Roadmap for the Lecture• Introduction• Abstract Argumentation Frameworks• Complexity• Implementation Techniques• Extensions of Abstract Argumentation Frameworks• Students’ Topics
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 2 of 81
![Page 3: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/3.jpg)
Introduction
Argumentation:. . . the study of processes “concerned with how assertions are proposed,discussed, and resolved in the context of issues upon which several divergingopinions may be held”.[Bench-Capon and Dunne, Argumentation in AI, AIJ 171:619-641, 2007]
Formal Models of Argumentation are concerned with• representation of an argument• representation of the relationship between arguments• solving conflicts between the arguments (“acceptability”)
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 3 of 81
![Page 4: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/4.jpg)
Introduction
Argumentation:. . . the study of processes “concerned with how assertions are proposed,discussed, and resolved in the context of issues upon which several divergingopinions may be held”.[Bench-Capon and Dunne, Argumentation in AI, AIJ 171:619-641, 2007]
Formal Models of Argumentation are concerned with• representation of an argument• representation of the relationship between arguments• solving conflicts between the arguments (“acceptability”)
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 4 of 81
![Page 5: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/5.jpg)
Introduction (ctd.)
Increasingly important area• “Argumentation” as keyword at all major AI conferences• dedicated conference: COMMA; several workshops• specialized journal: Argument and Computation (Taylor & Francis)• two text books:
• Besnard, Hunter: Elements of Argumentation. MIT Press, 2008• Rahwan, Simari (eds.): Argumentation in Artificial Intelligence.
Springer, 2009.
Applications• PARMENIDES-system for E-Democracy: facilitates structured arguments
over a proposed course of action [Atkinson et al.; 2006]• IMPACT project: argumentation toolbox for supporting open, inclusive and
transparent deliberations about public policy• Decision support systems, etc.
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 5 of 81
![Page 6: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/6.jpg)
Introduction (ctd.)
Increasingly important area• “Argumentation” as keyword at all major AI conferences• dedicated conference: COMMA; several workshops• specialized journal: Argument and Computation (Taylor & Francis)• two text books:
• Besnard, Hunter: Elements of Argumentation. MIT Press, 2008• Rahwan, Simari (eds.): Argumentation in Artificial Intelligence.
Springer, 2009.
Applications• PARMENIDES-system for E-Democracy: facilitates structured arguments
over a proposed course of action [Atkinson et al.; 2006]• IMPACT project: argumentation toolbox for supporting open, inclusive and
transparent deliberations about public policy• Decision support systems, etc.
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 6 of 81
![Page 7: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/7.jpg)
The Overall Process
Steps• Starting point:
knowledge-base
• Form arguments
• Identify conflicts
• Abstract frominternal structure
• Resolve conflicts
• Draw conclusions
Example
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 7 of 81
![Page 8: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/8.jpg)
The Overall Process
Steps• Starting point:
knowledge-base
• Form arguments
• Identify conflicts
• Abstract frominternal structure
• Resolve conflicts
• Draw conclusions
Example∆ = {s, r, w, s→ ¬r, r→ ¬w, w→ ¬s}
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 8 of 81
![Page 9: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/9.jpg)
The Overall Process
Steps• Starting point:
knowledge-base
• Form arguments
• Identify conflicts
• Abstract frominternal structure
• Resolve conflicts
• Draw conclusions
Example∆ = {s, r, w, s→ ¬r, r→ ¬w, w→ ¬s}
〈{s, s→¬r},¬r〉 〈{r, r→¬w},¬w〉
〈{w, w→¬s},¬s〉
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 9 of 81
![Page 10: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/10.jpg)
The Overall Process
Steps• Starting point:
knowledge-base
• Form arguments
• Identify conflicts
• Abstract frominternal structure
• Resolve conflicts
• Draw conclusions
Example∆ = {s, r, w, s→ ¬r, r→ ¬w, w→ ¬s}
〈{s, s→¬r},¬r〉 〈{r, r→¬w},¬w〉
〈{w, w→¬s},¬s〉
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 10 of 81
![Page 11: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/11.jpg)
The Overall Process
Steps• Starting point:
knowledge-base
• Form arguments
• Identify conflicts
• Abstract frominternal structure
• Resolve conflicts
• Draw conclusions
Example∆ = {s, r, w, s→ ¬r, r→ ¬w, w→ ¬s}
F∆ :
α β
γ
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 11 of 81
![Page 12: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/12.jpg)
The Overall Process
Steps• Starting point:
knowledge-base
• Form arguments
• Identify conflicts
• Abstract frominternal structure
• Resolve conflicts
• Draw conclusions
Example∆ = {s, r, w, s→ ¬r, r→ ¬w, w→ ¬s}
F∆ :
α β
γ
pref (F∆) ={∅}
stage(F∆) ={{α}, {β}, {γ}
}
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 12 of 81
![Page 13: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/13.jpg)
The Overall Process
Steps• Starting point:
knowledge-base
• Form arguments
• Identify conflicts
• Abstract frominternal structure
• Resolve conflicts
• Draw conclusions
Example∆ = {s, r, w, s→ ¬r, r→ ¬w, w→ ¬s}
〈{s, s→¬r},¬r〉 〈{r, r→¬w},¬w〉
〈{w, w→¬s},¬s〉
Cnpref (F∆) = Cn(>)Cnstage(F∆) = Cn(¬r ∨ ¬w ∨ ¬s)
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 13 of 81
![Page 14: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/14.jpg)
The Overall Process (ctd.)
Some Remarks• Main idea dates back to Dung [1995]; has then been refined by several
authors (Prakken, Gordon, Caminada, etc.)• Separation between logical (forming arguments) and nonmonotonic
reasoning (“abstract argumentation frameworks”)• Abstraction allows to compare several KR formalisms on a conceptual
level (“calculus of conflict”)
Main Challenge• All Steps in the argumentation process are, in general, intractable.• This calls for:
• careful complexity analysis (identification of tractable fragments)• re-use of established tools for implementations (reduction method)
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 14 of 81
![Page 15: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/15.jpg)
The Overall Process (ctd.)
Some Remarks• Main idea dates back to Dung [1995]; has then been refined by several
authors (Prakken, Gordon, Caminada, etc.)• Separation between logical (forming arguments) and nonmonotonic
reasoning (“abstract argumentation frameworks”)• Abstraction allows to compare several KR formalisms on a conceptual
level (“calculus of conflict”)
Main Challenge• All Steps in the argumentation process are, in general, intractable.• This calls for:
• careful complexity analysis (identification of tractable fragments)• re-use of established tools for implementations (reduction method)
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 15 of 81
![Page 16: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/16.jpg)
Approaches to Form Arguments
Classical Arguments [Besnard & Hunter, 2001]• Given is a KB (a set of propositions) ∆
• argument is a pair (Φ,α), such that Φ ⊆ ∆ is consistent, Φ |= α and forno Ψ ⊂ Φ, Ψ |= α
• conflicts between arguments (Φ,α) and (Φ′,α′) arise if Φ and α′ arecontradicting.
Example〈{s, s→¬r},¬r〉 〈{r, r→¬w},¬w〉
Other Approaches• Arguments are trees of statements• claims are obtained via strict and defeasible rules• different notions of conflict: rebuttal, undercut, etc.
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 16 of 81
![Page 17: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/17.jpg)
Approaches to Form Arguments
Classical Arguments [Besnard & Hunter, 2001]• Given is a KB (a set of propositions) ∆
• argument is a pair (Φ,α), such that Φ ⊆ ∆ is consistent, Φ |= α and forno Ψ ⊂ Φ, Ψ |= α
• conflicts between arguments (Φ,α) and (Φ′,α′) arise if Φ and α′ arecontradicting.
Example〈{s, s→¬r},¬r〉 〈{r, r→¬w},¬w〉
Other Approaches• Arguments are trees of statements• claims are obtained via strict and defeasible rules• different notions of conflict: rebuttal, undercut, etc.
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 17 of 81
![Page 18: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/18.jpg)
Approaches to Form Arguments
Classical Arguments [Besnard & Hunter, 2001]• Given is a KB (a set of propositions) ∆
• argument is a pair (Φ,α), such that Φ ⊆ ∆ is consistent, Φ |= α and forno Ψ ⊂ Φ, Ψ |= α
• conflicts between arguments (Φ,α) and (Φ′,α′) arise if Φ and α′ arecontradicting.
Example〈{s, s→¬r},¬r〉 〈{r, r→¬w},¬w〉
Other Approaches• Arguments are trees of statements• claims are obtained via strict and defeasible rules• different notions of conflict: rebuttal, undercut, etc.
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 18 of 81
![Page 19: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/19.jpg)
Dung’s Abstract Argumentation Frameworks
Example
α β
γ
Main Properties• Abstract from the concrete content of arguments but only consider the
relation between them• Semantics select subsets of arguments respecting certain criteria• Simple, yet powerful, formalism• Most active research area in the field of argumentation.
• “plethora of semantics”
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 19 of 81
![Page 20: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/20.jpg)
Dung’s Abstract Argumentation Frameworks
Example
α β
γ
Main Properties• Abstract from the concrete content of arguments but only consider the
relation between them• Semantics select subsets of arguments respecting certain criteria• Simple, yet powerful, formalism• Most active research area in the field of argumentation.
• “plethora of semantics”
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 20 of 81
![Page 21: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/21.jpg)
Dung’s Abstract Argumentation Frameworks
DefinitionAn argumentation framework (AF) is a pair (A, R) where• A is a set of arguments• R ⊆ A× A is a relation representing the conflicts (“attacks”)
ExampleF=( {a,b,c,d,e} , {(a,b),(c,b),(c,d),(d,c),(d,e),(e,e)} )
b c d ea
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 21 of 81
![Page 22: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/22.jpg)
Dung’s Abstract Argumentation Frameworks
DefinitionAn argumentation framework (AF) is a pair (A, R) where• A is a set of arguments• R ⊆ A× A is a relation representing the conflicts (“attacks”)
ExampleF=( {a,b,c,d,e} , {(a,b),(c,b),(c,d),(d,c),(d,e),(e,e)} )
b c d ea
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 22 of 81
![Page 23: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/23.jpg)
Basic Properties
Conflict-Free SetsGiven an AF F = (A, R).A set S ⊆ A is conflict-free in F, if, for each a, b ∈ S, (a, b) /∈ R.
Example
b c d ea
cf (F) ={{a, c},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 23 of 81
![Page 24: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/24.jpg)
Basic Properties
Conflict-Free SetsGiven an AF F = (A, R).A set S ⊆ A is conflict-free in F, if, for each a, b ∈ S, (a, b) /∈ R.
Example
b c d ea
cf (F) ={{a, c},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 24 of 81
![Page 25: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/25.jpg)
Basic Properties
Conflict-Free SetsGiven an AF F = (A, R).A set S ⊆ A is conflict-free in F, if, for each a, b ∈ S, (a, b) /∈ R.
Example
b c d ea
cf (F) ={{a, c}, {a, d},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 25 of 81
![Page 26: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/26.jpg)
Basic Properties
Conflict-Free SetsGiven an AF F = (A, R).A set S ⊆ A is conflict-free in F, if, for each a, b ∈ S, (a, b) /∈ R.
Example
b c d ea
cf (F) ={{a, c}, {a, d}, {b, d},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 26 of 81
![Page 27: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/27.jpg)
Basic Properties
Conflict-Free SetsGiven an AF F = (A, R).A set S ⊆ A is conflict-free in F, if, for each a, b ∈ S, (a, b) /∈ R.
Example
b c d ea
cf (F) ={{a, c}, {a, d}, {b, d}, {a}, {b}, {c}, {d}, ∅
}
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 27 of 81
![Page 28: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/28.jpg)
Basic Properties (ctd.)
Admissible Sets [Dung, 1995]Given an AF F = (A, R). A set S ⊆ A is admissible in F, if• S is conflict-free in F
• each a ∈ S is defended by S in F• a ∈ A is defended by S in F, if for each b ∈ A with (b, a) ∈ R, there
exists a c ∈ S, such that (c, b) ∈ R.
Example
b c d ea
adm(F) ={{a, c},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 28 of 81
![Page 29: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/29.jpg)
Basic Properties (ctd.)
Admissible Sets [Dung, 1995]Given an AF F = (A, R). A set S ⊆ A is admissible in F, if• S is conflict-free in F
• each a ∈ S is defended by S in F• a ∈ A is defended by S in F, if for each b ∈ A with (b, a) ∈ R, there
exists a c ∈ S, such that (c, b) ∈ R.
Example
b c d ea
adm(F) ={{a, c},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 29 of 81
![Page 30: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/30.jpg)
Basic Properties (ctd.)
Admissible Sets [Dung, 1995]Given an AF F = (A, R). A set S ⊆ A is admissible in F, if• S is conflict-free in F
• each a ∈ S is defended by S in F• a ∈ A is defended by S in F, if for each b ∈ A with (b, a) ∈ R, there
exists a c ∈ S, such that (c, b) ∈ R.
Example
b c d ea
adm(F) ={{a, c}, {a, d},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 30 of 81
![Page 31: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/31.jpg)
Basic Properties (ctd.)
Admissible Sets [Dung, 1995]Given an AF F = (A, R). A set S ⊆ A is admissible in F, if• S is conflict-free in F
• each a ∈ S is defended by S in F• a ∈ A is defended by S in F, if for each b ∈ A with (b, a) ∈ R, there
exists a c ∈ S, such that (c, b) ∈ R.
Example
b c d ea
adm(F) ={{a, c}, {a, d}, {b, d},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 31 of 81
![Page 32: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/32.jpg)
Basic Properties (ctd.)
Admissible Sets [Dung, 1995]Given an AF F = (A, R). A set S ⊆ A is admissible in F, if• S is conflict-free in F
• each a ∈ S is defended by S in F• a ∈ A is defended by S in F, if for each b ∈ A with (b, a) ∈ R, there
exists a c ∈ S, such that (c, b) ∈ R.
Example
b c d ea
adm(F) ={{a, c}, {a, d}, {b, d}, {a}, {b}, {c}, {d}, ∅
}
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 32 of 81
![Page 33: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/33.jpg)
Basic Properties (ctd.)
Dung’s Fundamental LemmaLet S be admissible in an AF F and a, a′ arguments in F defended by S in F.Then,
1 S′ = S ∪ {a} is admissible in F
2 a′ is defended by S′ in F
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 33 of 81
![Page 34: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/34.jpg)
Semantics
Naive ExtensionsGiven an AF F = (A, R). A set S ⊆ A is a naive extension of F, if• S is conflict-free in F
• for each T ⊆ A conflict-free in F, S 6⊂ T
Example
b c d ea
naive(F) ={{a, c},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 34 of 81
![Page 35: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/35.jpg)
Semantics
Naive ExtensionsGiven an AF F = (A, R). A set S ⊆ A is a naive extension of F, if• S is conflict-free in F
• for each T ⊆ A conflict-free in F, S 6⊂ T
Example
b c d ea
naive(F) ={{a, c},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 35 of 81
![Page 36: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/36.jpg)
Semantics
Naive ExtensionsGiven an AF F = (A, R). A set S ⊆ A is a naive extension of F, if• S is conflict-free in F
• for each T ⊆ A conflict-free in F, S 6⊂ T
Example
b c d ea
naive(F) ={{a, c}, {a, d},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 36 of 81
![Page 37: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/37.jpg)
Semantics
Naive ExtensionsGiven an AF F = (A, R). A set S ⊆ A is a naive extension of F, if• S is conflict-free in F
• for each T ⊆ A conflict-free in F, S 6⊂ T
Example
b c d ea
naive(F) ={{a, c}, {a, d}, {b, d},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 37 of 81
![Page 38: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/38.jpg)
Semantics
Naive ExtensionsGiven an AF F = (A, R). A set S ⊆ A is a naive extension of F, if• S is conflict-free in F
• for each T ⊆ A conflict-free in F, S 6⊂ T
Example
b c d ea
naive(F) ={{a, c}, {a, d}, {b, d}, {a}, {b}, {c}, {d}, ∅
}
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 38 of 81
![Page 39: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/39.jpg)
Semantics (ctd.)
Grounded Extension [Dung, 1995]Given an AF F = (A, R). The unique grounded extension of F is defined as theoutcome S of the following “algorithm”:
1 put each argument a ∈ A which is not attacked in F into S; if no suchargument exists, return S;
2 remove from F all (new) arguments in S and all arguments attacked bythem (together with all adjacent attacks); and continue with Step 1.
Example
b c d ea
ground(F) ={{a}}
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 39 of 81
![Page 40: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/40.jpg)
Semantics (ctd.)
Grounded Extension [Dung, 1995]Given an AF F = (A, R). The unique grounded extension of F is defined as theoutcome S of the following “algorithm”:
1 put each argument a ∈ A which is not attacked in F into S; if no suchargument exists, return S;
2 remove from F all (new) arguments in S and all arguments attacked bythem (together with all adjacent attacks); and continue with Step 1.
Example
b c d ea
ground(F) ={{a}}
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 40 of 81
![Page 41: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/41.jpg)
Semantics (ctd.)
Complete Extension [Dung, 1995]Given an AF (A, R). A set S ⊆ A is complete in F, if• S is admissible in F
• each a ∈ A defended by S in F is contained in S• Recall: a ∈ A is defended by S in F, if for each b ∈ A with (b, a) ∈ R,
there exists a c ∈ S, such that (c, b) ∈ R.
Example
b c d ea
comp(F) ={{a, c},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 41 of 81
![Page 42: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/42.jpg)
Semantics (ctd.)
Complete Extension [Dung, 1995]Given an AF (A, R). A set S ⊆ A is complete in F, if• S is admissible in F
• each a ∈ A defended by S in F is contained in S• Recall: a ∈ A is defended by S in F, if for each b ∈ A with (b, a) ∈ R,
there exists a c ∈ S, such that (c, b) ∈ R.
Example
b c d ea
comp(F) ={{a, c},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 42 of 81
![Page 43: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/43.jpg)
Semantics (ctd.)
Complete Extension [Dung, 1995]Given an AF (A, R). A set S ⊆ A is complete in F, if• S is admissible in F
• each a ∈ A defended by S in F is contained in S• Recall: a ∈ A is defended by S in F, if for each b ∈ A with (b, a) ∈ R,
there exists a c ∈ S, such that (c, b) ∈ R.
Example
b c d ea
comp(F) ={{a, c}, {a, d},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 43 of 81
![Page 44: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/44.jpg)
Semantics (ctd.)
Complete Extension [Dung, 1995]Given an AF (A, R). A set S ⊆ A is complete in F, if• S is admissible in F
• each a ∈ A defended by S in F is contained in S• Recall: a ∈ A is defended by S in F, if for each b ∈ A with (b, a) ∈ R,
there exists a c ∈ S, such that (c, b) ∈ R.
Example
b c d ea
comp(F) ={{a, c}, {a, d}, {a},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 44 of 81
![Page 45: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/45.jpg)
Semantics (ctd.)
Complete Extension [Dung, 1995]Given an AF (A, R). A set S ⊆ A is complete in F, if• S is admissible in F
• each a ∈ A defended by S in F is contained in S• Recall: a ∈ A is defended by S in F, if for each b ∈ A with (b, a) ∈ R,
there exists a c ∈ S, such that (c, b) ∈ R.
Example
b c d ea
comp(F) ={{a, c}, {a, d}, {a}, {c}, {d}, ∅
}
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 45 of 81
![Page 46: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/46.jpg)
Semantics (ctd.)
Properties of the Grounded ExtensionFor any AF F, the grounded extension of F is the subset-minimal completeextension of F.
RemarkSince there exists exactly one grounded extension for each AF F, we often writeground(F) = S instead of ground(F) = {S}.
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 46 of 81
![Page 47: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/47.jpg)
Semantics (ctd.)
Properties of the Grounded ExtensionFor any AF F, the grounded extension of F is the subset-minimal completeextension of F.
RemarkSince there exists exactly one grounded extension for each AF F, we often writeground(F) = S instead of ground(F) = {S}.
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 47 of 81
![Page 48: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/48.jpg)
Semantics (ctd.)
Preferred Extensions [Dung, 1995]Given an AF F = (A, R). A set S ⊆ A is a preferred extension of F, if• S is admissible in F
• for each T ⊆ A admissible in F, S 6⊂ T
Example
b c d ea
pref (F) ={{a, c}, {a, d}, {a}, {c}, {d}, ∅
}
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 48 of 81
![Page 49: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/49.jpg)
Semantics (ctd.)
Preferred Extensions [Dung, 1995]Given an AF F = (A, R). A set S ⊆ A is a preferred extension of F, if• S is admissible in F
• for each T ⊆ A admissible in F, S 6⊂ T
Example
b c d ea
pref (F) ={{a, c}, {a, d}, {a}, {c}, {d}, ∅
}
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 49 of 81
![Page 50: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/50.jpg)
Semantics (ctd.)
Stable Extensions [Dung, 1995]Given an AF F = (A, R). A set S ⊆ A is a stable extension of F, if• S is conflict-free in F
• for each a ∈ A \ S, there exists a b ∈ S, such that (b, a) ∈ R
Example
b c d ea
stable(F) ={{a, c}
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 50 of 81
![Page 51: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/51.jpg)
Semantics (ctd.)
Stable Extensions [Dung, 1995]Given an AF F = (A, R). A set S ⊆ A is a stable extension of F, if• S is conflict-free in F
• for each a ∈ A \ S, there exists a b ∈ S, such that (b, a) ∈ R
Example
b c d ea
stable(F) ={{a, c}
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 51 of 81
![Page 52: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/52.jpg)
Semantics (ctd.)
Stable Extensions [Dung, 1995]Given an AF F = (A, R). A set S ⊆ A is a stable extension of F, if• S is conflict-free in F
• for each a ∈ A \ S, there exists a b ∈ S, such that (b, a) ∈ R
Example
b c d ea
stable(F) ={{a, c}, {a, d},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 52 of 81
![Page 53: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/53.jpg)
Semantics (ctd.)
Stable Extensions [Dung, 1995]Given an AF F = (A, R). A set S ⊆ A is a stable extension of F, if• S is conflict-free in F
• for each a ∈ A \ S, there exists a b ∈ S, such that (b, a) ∈ R
Example
b c d ea
stable(F) ={{a, c}, {a, d}, {b, d},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 53 of 81
![Page 54: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/54.jpg)
Semantics (ctd.)
Stable Extensions [Dung, 1995]Given an AF F = (A, R). A set S ⊆ A is a stable extension of F, if• S is conflict-free in F
• for each a ∈ A \ S, there exists a b ∈ S, such that (b, a) ∈ R
Example
b c d ea
stable(F) ={{a, c}, {a, d}, {b, d}, {a}, {b}, {c}, {d}, ∅,
}
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 54 of 81
![Page 55: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/55.jpg)
Semantics (ctd.)
Some RelationsFor any AF F the following relations hold:
1 Each stable extension of F is admissible in F
2 Each stable extension of F is also a preferred one
3 Each preferred extension of F is also a complete one
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 55 of 81
![Page 56: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/56.jpg)
Semantics (ctd.)
Semi-Stable Extensions [Caminada, 2006]Given an AF F = (A, R). A set S ⊆ A is a semi-stable extension of F, if• S is admissible in F
• for each T ⊆ A admissible in F, S+ 6⊂ T+
• for S ⊆ A, define S+ = S ∪ {a | ∃b ∈ S with (b, a) ∈ R}
Example
b c d ea
semi(F) ={{a, c}, {a, d}, {a}, {c}, {d}, ∅
}
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 56 of 81
![Page 57: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/57.jpg)
Semantics (ctd.)
Semi-Stable Extensions [Caminada, 2006]Given an AF F = (A, R). A set S ⊆ A is a semi-stable extension of F, if• S is admissible in F
• for each T ⊆ A admissible in F, S+ 6⊂ T+
• for S ⊆ A, define S+ = S ∪ {a | ∃b ∈ S with (b, a) ∈ R}
Example
b c d ea
semi(F) ={{a, c}, {a, d}, {a}, {c}, {d}, ∅
}
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 57 of 81
![Page 58: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/58.jpg)
Semantics (ctd.)
Stage Extensions [Verheij, 1996]Given an AF F = (A, R). A set S ⊆ A is a stage extension of F, if• S is conflict-free in F
• for each T ⊆ A conflict-free in F, S+ 6⊂ T+
• recall S+ = S ∪ {a | ∃b ∈ S with (b, a) ∈ R}
Ideal Extension [Dung, Mancarella & Toni 2007]Given an AF F = (A, R). A set S ⊆ A is an ideal extension of F, if• S is admissible in F and contained in each preferred extension of F
• there is no T ⊃ S admissible in F and contained in each of pref (F)
Eager Extension [Caminada, 2007]Given an AF F = (A, R). A set S ⊆ A is an eager extension of F, if• S is admissible in F and contained in each semi-stable extension of F
• there is no T ⊃ S admissible in F and contained in each of semi(F)
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 58 of 81
![Page 59: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/59.jpg)
Semantics (ctd.)
Stage Extensions [Verheij, 1996]Given an AF F = (A, R). A set S ⊆ A is a stage extension of F, if• S is conflict-free in F
• for each T ⊆ A conflict-free in F, S+ 6⊂ T+
• recall S+ = S ∪ {a | ∃b ∈ S with (b, a) ∈ R}
Ideal Extension [Dung, Mancarella & Toni 2007]Given an AF F = (A, R). A set S ⊆ A is an ideal extension of F, if• S is admissible in F and contained in each preferred extension of F
• there is no T ⊃ S admissible in F and contained in each of pref (F)
Eager Extension [Caminada, 2007]Given an AF F = (A, R). A set S ⊆ A is an eager extension of F, if• S is admissible in F and contained in each semi-stable extension of F
• there is no T ⊃ S admissible in F and contained in each of semi(F)
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 59 of 81
![Page 60: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/60.jpg)
Semantics (ctd.)
Stage Extensions [Verheij, 1996]Given an AF F = (A, R). A set S ⊆ A is a stage extension of F, if• S is conflict-free in F
• for each T ⊆ A conflict-free in F, S+ 6⊂ T+
• recall S+ = S ∪ {a | ∃b ∈ S with (b, a) ∈ R}
Ideal Extension [Dung, Mancarella & Toni 2007]Given an AF F = (A, R). A set S ⊆ A is an ideal extension of F, if• S is admissible in F and contained in each preferred extension of F
• there is no T ⊃ S admissible in F and contained in each of pref (F)
Eager Extension [Caminada, 2007]Given an AF F = (A, R). A set S ⊆ A is an eager extension of F, if• S is admissible in F and contained in each semi-stable extension of F
• there is no T ⊃ S admissible in F and contained in each of semi(F)
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 60 of 81
![Page 61: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/61.jpg)
Semantics (ctd.)
Properties of Ideal ExtensionsFor any AF F the following observations hold:
1 there exists exactly one ideal extension of F
2 the ideal extension of F is also a complete one
The same results hold for the eager extension and similar variants [Dvorák etal., 2011].
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 61 of 81
![Page 62: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/62.jpg)
Semantics (ctd.)
Resolution-based grounded Extensions[Baroni,Giacomin 2008]A resolution β of an AF F = (A, R) contains exactly one of the attacks (a, b),(b, a) for each pair a, b ∈ A with {(a, b), (b, a)} ⊆ R.
A set S ⊆ A is a resolution-based grounded extension of F, if• there exists a resolution β such that ground((A, R \ β)) = S
• and there is no resolution β′ such that ground((A, R \ β′)) ⊂ S
Example
b c d ea
ground∗(F) ={{a, c},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 62 of 81
![Page 63: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/63.jpg)
Semantics (ctd.)
Resolution-based grounded Extensions[Baroni,Giacomin 2008]A resolution β of an AF F = (A, R) contains exactly one of the attacks (a, b),(b, a) for each pair a, b ∈ A with {(a, b), (b, a)} ⊆ R.
A set S ⊆ A is a resolution-based grounded extension of F, if• there exists a resolution β such that ground((A, R \ β)) = S
• and there is no resolution β′ such that ground((A, R \ β′)) ⊂ S
Example
b c d ea
ground∗(F) ={{a, c},
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 63 of 81
![Page 64: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/64.jpg)
Semantics (ctd.)
Resolution-based grounded Extensions[Baroni,Giacomin 2008]A resolution β of an AF F = (A, R) contains exactly one of the attacks (a, b),(b, a) for each pair a, b ∈ A with {(a, b), (b, a)} ⊆ R.
A set S ⊆ A is a resolution-based grounded extension of F, if• there exists a resolution β such that ground((A, R \ β)) = S
• and there is no resolution β′ such that ground((A, R \ β′)) ⊂ S
Example
b c d ea
ground∗(F) ={{a, c}, {a, d}}
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 64 of 81
![Page 65: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/65.jpg)
cf2 Semantics [Baroni, Giacomin & Guida2005]
Definition (Separation)An AF F = (A, R) is called separated if for each (a, b) ∈ R, there exists a pathfrom b to a. We define [[F]] =
⋃C∈SCCs(F) F|C and call [[F]] the separation of F.
Example
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 65 of 81
![Page 66: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/66.jpg)
cf2 Semantics [Baroni, Giacomin & Guida2005]
Definition (Separation)An AF F = (A, R) is called separated if for each (a, b) ∈ R, there exists a pathfrom b to a. We define [[F]] =
⋃C∈SCCs(F) F|C and call [[F]] the separation of F.
Example
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 66 of 81
![Page 67: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/67.jpg)
cf2 Semantics (ctd.)
Definition (Reachability)Let F = (A, R) be an AF, B a set of arguments, and a, b ∈ A. We say that b isreachable in F from a modulo B, in symbols a⇒B
F b, if there exists a path from ato b in F|B.
Definition (∆F,S)For an AF F = (A, R), D ⊆ A, and a set S of arguments,
∆F,S(D) = {a ∈ A | ∃b ∈ S : b 6= a, (b, a) ∈ R, a 6⇒A\DF b}.
By ∆F,S, we denote the lfp of ∆F,S(∅).
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 67 of 81
![Page 68: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/68.jpg)
cf2 Semantics (ctd.)
Definition (Reachability)Let F = (A, R) be an AF, B a set of arguments, and a, b ∈ A. We say that b isreachable in F from a modulo B, in symbols a⇒B
F b, if there exists a path from ato b in F|B.
Definition (∆F,S)For an AF F = (A, R), D ⊆ A, and a set S of arguments,
∆F,S(D) = {a ∈ A | ∃b ∈ S : b 6= a, (b, a) ∈ R, a 6⇒A\DF b}.
By ∆F,S, we denote the lfp of ∆F,S(∅).
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 68 of 81
![Page 69: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/69.jpg)
cf2 Semantics (ctd.)
cf2 Extensions [G & Woltran 2010]Given an AF F = (A, R). A set S ⊆ A is a cf2-extension of F, if• S is conflict-free in F
• and S ∈ naive([[F −∆F,S]]).
ExampleS = {c, f , h}, S ∈ cf (F).
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 69 of 81
![Page 70: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/70.jpg)
cf2 Semantics (ctd.)
cf2 Extensions [G & Woltran 2010]Given an AF F = (A, R). A set S ⊆ A is a cf2-extension of F, if• S is conflict-free in F
• and S ∈ naive([[F −∆F,S]]).
ExampleS = {c, f , h}, S ∈ cf (F).
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 70 of 81
![Page 71: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/71.jpg)
cf2 Semantics (ctd.)
cf2 Extensions [G & Woltran 2010]Given an AF F = (A, R). A set S ⊆ A is a cf2-extension of F, if• S is conflict-free in F
• and S ∈ naive([[F −∆F,S]]).
ExampleS = {c, f , h}, S ∈ cf (F), ∆F,S(∅) = {d, e}.
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 71 of 81
![Page 72: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/72.jpg)
cf2 Semantics (ctd.)
cf2 Extensions [G & Woltran 2010]Given an AF F = (A, R). A set S ⊆ A is a cf2-extension of F, if• S is conflict-free in F
• and S ∈ naive([[F −∆F,S]]).
ExampleS = {c, f , h}, S ∈ cf (F), ∆F,S({d, e}) = {d, e}.
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 72 of 81
![Page 73: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/73.jpg)
cf2 Semantics (ctd.)
cf2 Extensions [G & Woltran 2010]Given an AF F = (A, R). A set S ⊆ A is a cf2-extension of F, if• S is conflict-free in F
• and S ∈ naive([[F −∆F,S]]).
ExampleS = {c, f , h}, S ∈ cf (F), ∆F,S = {d, e}, S ∈ naive([[F −∆F,S]]).
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 73 of 81
![Page 74: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/74.jpg)
Relations between Semantics
conflict-free
naive
stage
stable
admissible
complete
preferred
semi-stable
ideal eager
grounded
res.b. grounded
cf2
Figure: An arrow from semantics σ to semantics τ encodes thateach σ-extension is also a τ -extension.
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 74 of 81
![Page 75: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/75.jpg)
Characteristics of Argumentation Semantics
Examplea
b d
c
f e
pref (F) ={{a, d, e}, {b, c, e}, {a, b}
}naive(F) =
{{a, d, e}, {b, c, e}, {a, b, e}
}Natural Questions• How to change the AF if we want {a, b, e} instead of {a, b} in pref (F)?• How to change the AF if we want {a, b, d} instead of {a, b} in pref (F)?• Can we have equivalent AFs without argument f ?
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 75 of 81
![Page 76: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/76.jpg)
Some Properties . . .
TheoremFor any AFs F and G, we have• adm(F) = adm(G) =⇒ σ(F) = σ(G), for σ ∈ {pref , ideal};• comp(F) = comp(G) =⇒ ϑ(F) = ϑ(G), for ϑ ∈ {pref , ideal, ground};• no other such relation between the different semantics (adm, pref , ideal,
semi, eager, ground, comp, stable) in terms of standard equivalence holds.
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 76 of 81
![Page 77: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/77.jpg)
Strong Equivalence [Oikarinen & Woltran 2011,
G & Woltran 2011]
DefinitionTwo AFs F and G are strongly equivalent wrt. a semanticsσ ∈ {stable, adm, pref , ideal, semi, comp, ground, stage}, in symbols F ≡σ
s G, iffσ(F ∪ H)=σ(G ∪ H), for each AF H.
• Idea: Find “σ-kernels” of AFs, such that the σ-kernels of F and G coincideiff F ≡σ
s G.• Verification of strong equivalence then reduces to checking
syntactical equivalence
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 77 of 81
![Page 78: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/78.jpg)
Strong Equivalence for Stable Semantics
Kernel for stable semanticsFor AF F = (A, R), we define stable-kernel of F as Fκ = (A, Rκ) with
Rκ = R \ {(a, b) | a 6= b, (a, a) ∈ R}.
TheoremFor any AFs F and G: Fκ = Gκ iff F ≡stable
s G iff F ≡stages G.
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 78 of 81
![Page 79: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/79.jpg)
Exercises1 Give an AF F such that stable(F) = ∅ and semi(F) 6= {∅}.
2 Show that the following statement holds for any AF F.If stable(F) 6= ∅ then stable(F) = semi(F) = stage(F).
3 Select three different semantics σ,σ′,σ′′ out of {pref , ideal, semi, eager,ground, stable} of your choice and provide three pairs of AFs such that
• σ(F1) = σ(G1) but σ′(F1) 6= σ′(G1)• σ′(F2) = σ′(G2) but σ′′(F2) 6= σ′′(G2)• σ′′(F3) = σ′′(G3) but σ(F3) 6= σ(G3)
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 79 of 81
![Page 80: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/80.jpg)
P. Baroni, P. E. Dunne, and M. Giacomin.On the resolution-based family of abstract argumentation semantics and its grounded instance.Artif. Intell., 175(3-4):791–813, 2011.
P. Baroni and M. Giacomin.Semantics of abstract argument systems.In Argumentation in Artificial Intelligence, pages 25–44. Springer, 2009.
P. Baroni, M. Giacomin, and G. Guida.SCC-Recursiveness: A General Schema for Argumentation Semantics.Artif. Intell., 168(1-2): 162–210. Springer, 2005.
T.J.M. Bench-Capon and P.E.Dunne.Argumentation in AI,AIJ 171:619-641, 2007
M. Caminada.Semi-stable semantics.In Proc. COMMA 2006, pages 121–130. IOS Press, 2006.
M. Caminada.Comparing two unique extension semantics for formal argumentation: ideal and eagerIn Proc. BNAIC 2007, pages 81–87, 2007.
P. M. Dung.On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logicprogramming and n-person games.Artif. Intell., 77(2):321–358, 1995.
P. M. Dung, P. Mancarella, and F. Toni.Computing ideal sceptical argumentation.Artif. Intell. 171(10-15):642–674, 2007.
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 80 of 81
![Page 81: SEMINAR ABSTRACT ARGUMENTATION - TU Dresden€¦ · PARMENIDES-system for E-Democracy: facilitates structured arguments over a proposed course of action [Atkinson et al.; 2006] IMPACT](https://reader033.vdocument.in/reader033/viewer/2022042312/5eda7e607cafa20f700c8dc3/html5/thumbnails/81.jpg)
W. Dvorák, P. Dunne, and S. Woltran.Parametric properties of ideal semantics.In Proc. IJCAI 2011, pages 851–856, 2011.
W. Dvorák and S. WoltranOn the intertranslatability of argumentation semanticsJ. Artif. Intell. Res. 41:445–475, 2011
S. Gaggl and S. Woltran.cf2 semantics revisited.In Proc. COMMA 2010, pages 243–2540. IOS Press, 2010.
S. Gaggl and S. Woltran.Strong equivalence for argumentation semantics based on conflict-free sets.In Proc. ECSQARU 2011, pages 38–49. Springer, 2011.
E. Oikarinen and S. Woltran.Characterizing strong equivalence for argumentation frameworks.Artif. Intell. 175(14-15): 1985–2009, 2011.
B. Verheij.Two approaches to dialectical argumentation: admissible sets and argumentation stages.In Proc. NAIC’96, pages 357–368, 1996.
TU Dresden, 1st November 2013 Seminar Abstract Argumentation slide 81 of 81