Computational Aspects of Abstract ArgumentationPhD Defense, TU Wien (Vienna)
Wolfgang Dvo°áksupervised by Stefan Woltran
Institute of Information Systems,Database and Arti�cial Intelligence Group
Vienna University of Technology
April 11, 2012
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 1
1. Prolog
The Argumentation Process
Steps
Starting point:knowledge-base
Form arguments
Identify con�icts
Abstract frominternal structure
Resolve con�icts
Draw conclusions
Example
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 2
1. Prolog
The Argumentation Process
Steps
Starting point:
knowledge-base
Form arguments
Identify con�icts
Abstract frominternal structure
Resolve con�icts
Draw conclusions
Example
∆ = {⇒ x ,→ ¬x , x → y ,⇒ y ,⇒ ¬y}
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 2
1. Prolog
The Argumentation Process
Steps
Starting point:knowledge-base
Form arguments
Identify con�icts
Abstract frominternal structure
Resolve con�icts
Draw conclusions
Example
∆ = {⇒ x ,→ ¬x , x → y ,⇒ y ,⇒ ¬y}
⇒ x → ¬x
⇒ x → y
⇒ ¬y⇒ y
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 2
1. Prolog
The Argumentation Process
Steps
Starting point:knowledge-base
Form arguments
Identify con�icts
Abstract frominternal structure
Resolve con�icts
Draw conclusions
Example
∆ = {⇒ x ,→ ¬x , x → y ,⇒ y ,⇒ ¬y}
⇒ x → ¬x
⇒ x → y
⇒ ¬y⇒ y
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 2
1. Prolog
The Argumentation Process
Steps
Starting point:knowledge-base
Form arguments
Identify con�icts
Abstract from
internal structure
Resolve con�icts
Draw conclusions
Example
∆ = {⇒ x ,→ ¬x , x → y ,⇒ y ,⇒ ¬y}
F∆ : a b
c
de
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 2
1. Prolog
The Argumentation Process
Steps
Starting point:knowledge-base
Form arguments
Identify con�icts
Abstract frominternal structure
Resolve con�icts
Draw conclusions
Example
∆ = {⇒ x ,→ ¬x , x → y ,⇒ y ,⇒ ¬y}
F∆ : a b
c
de
prf (F∆)={{ b , d}, { b , e}}
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 2
1. Prolog
The Argumentation Process
Steps
Starting point:knowledge-base
Form arguments
Identify con�icts
Abstract frominternal structure
Resolve con�icts
Draw conclusions
Example
∆ = {⇒ x ,→ ¬x , x → y ,⇒ y ,⇒ ¬y}
prf (F∆)={{ b , d},{ b , e}}
CS(F∆)={ ¬x }
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 2
1. Prolog
The Argumentation Process
Remarks
Main idea dates back to [Dung, 1995]; has then been re�ned byseveral authors (Prakken, Gordon, Caminada, etc.)
Abstraction allows to compare several Knowledge Representation(KR) formalisms on a conceptual level
Main Challenge
All Steps in the argumentation process are, in general, intractable.
This calls for:
careful complexity analysis (identi�cation of tractable fragments)re-use of established tools for implementations (reduction method)
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 3
1. Prolog
The Argumentation Process
Remarks
Main idea dates back to [Dung, 1995]; has then been re�ned byseveral authors (Prakken, Gordon, Caminada, etc.)
Abstraction allows to compare several Knowledge Representation(KR) formalisms on a conceptual level
Main Challenge
All Steps in the argumentation process are, in general, intractable.
This calls for:
careful complexity analysis (identi�cation of tractable fragments)re-use of established tools for implementations (reduction method)
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 3
1. Prolog
Dung's Abstract Argumentation Frameworks
a b
c
de
Main Properties
Abstract from the concrete content of arguments and only considerthe relation between them
Semantics select subsets of arguments respecting certain criteria
Simple, yet powerful, formalism
Most active research area in the �eld of argumentation.
�plethora of semantics�
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 4
1. Prolog
Topics of the thesis
Complexity Analysis
Complexity classi�cation of standard reasoning tasks in abstractargumentation
Towards Tractability
Graph classes as tractable fragmentsFixed-parameter tractability
Intertranslatability of argumentation semantics
Translations between semantics as an reduction approach withinargumentation
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 5
2. Abstract Argumentation
Dung's Abstract Argumentation Frameworks
De�nition
An argumentation framework (AF) is a pair (A,R) where
A is a set of arguments
R ⊆ A× A is a relation representing the con�icts (�attacks�)
Example
F=( {a,b,c,d,e} , {(a,b),(c,b),(c,d),(d,c),(d,e),(e,e)} )
b c d ea
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 6
2. Abstract Argumentation
Basic Properties
Con�ict-Free Sets
Given an AF F = (A,R).A set S ⊆ A is con�ict-free in F , if, for each a, b ∈ S , (a, b) /∈ R.
Example
b c d ea
cf (F ) ={{a, c},
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 7
2. Abstract Argumentation
Basic Properties
Con�ict-Free Sets
Given an AF F = (A,R).A set S ⊆ A is con�ict-free in F , if, for each a, b ∈ S , (a, b) /∈ R.
Example
b c d ea
cf (F ) ={{a, c},
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 7
2. Abstract Argumentation
Basic Properties
Con�ict-Free Sets
Given an AF F = (A,R).A set S ⊆ A is con�ict-free in F , if, for each a, b ∈ S , (a, b) /∈ R.
Example
b c d ea
cf (F ) ={{a, c}, {a, d},
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 7
2. Abstract Argumentation
Basic Properties
Con�ict-Free Sets
Given an AF F = (A,R).A set S ⊆ A is con�ict-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},
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 7
2. Abstract Argumentation
Basic Properties
Con�ict-Free Sets
Given an AF F = (A,R).A set S ⊆ A is con�ict-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}, ∅
}
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 7
2. Abstract Argumentation
Basic Properties
Admissible Sets [Dung, 1995]
Given an AF F = (A,R). A set S ⊆ A is admissible in F , if
S is con�ict-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, thereexists a c ∈ S , such that (c, b) ∈ R.
Example
b c d ea
adm(F ) ={{a, c},
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 8
2. Abstract Argumentation
Basic Properties
Admissible Sets [Dung, 1995]
Given an AF F = (A,R). A set S ⊆ A is admissible in F , if
S is con�ict-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, thereexists a c ∈ S , such that (c, b) ∈ R.
Example
b c d ea
adm(F ) ={{a, c},
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 8
2. Abstract Argumentation
Basic Properties
Admissible Sets [Dung, 1995]
Given an AF F = (A,R). A set S ⊆ A is admissible in F , if
S is con�ict-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, thereexists a c ∈ S , such that (c, b) ∈ R.
Example
b c d ea
adm(F ) ={{a, c}, {a, d},
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 8
2. Abstract Argumentation
Basic Properties
Admissible Sets [Dung, 1995]
Given an AF F = (A,R). A set S ⊆ A is admissible in F , if
S is con�ict-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, thereexists a c ∈ S , such that (c, b) ∈ R.
Example
b c d ea
adm(F ) ={{a, c}, {a, d}, {b, d},
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 8
2. Abstract Argumentation
Basic Properties
Admissible Sets [Dung, 1995]
Given an AF F = (A,R). A set S ⊆ A is admissible in F , if
S is con�ict-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, thereexists 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}, ∅
}
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 8
2. Abstract Argumentation 2.1. Argumentation Semantics
Semantics
De�nition
An extension-based semantics is a function σ mapping each AF F to aset of extensions σ(F) ⊆ 2AF .
If for each F , |σ(F)| = 1 then we call σ a unique status semantics,otherwise multiple status semantics.
We consider 9 semantics, namely:
naive groundedstable admissiblecomplete resolution-based groundedpreferred semi-stablestage
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 9
2. Abstract Argumentation 2.1. Argumentation Semantics
Semantics
Grounded Extension [Dung, 1995]
Given an AF F = (A,R). The unique grounded extension of F is de�nedas the outcome S of the following �algorithm�:
1 put each argument a ∈ A which is not attacked in F into S ; if nosuch argument exists, return S ;
2 remove from F all (new) arguments in S and all arguments attackedby them and continue with Step 1.
Example
b c d ea
grd(F ) ={{a}}
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 10
2. Abstract Argumentation 2.1. Argumentation Semantics
Semantics
Grounded Extension [Dung, 1995]
Given an AF F = (A,R). The unique grounded extension of F is de�nedas the outcome S of the following �algorithm�:
1 put each argument a ∈ A which is not attacked in F into S ; if nosuch argument exists, return S ;
2 remove from F all (new) arguments in S and all arguments attackedby them and continue with Step 1.
Example
b c d ea
grd(F ) ={{a}}
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 10
2. Abstract Argumentation 2.1. Argumentation Semantics
Semantics
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
prf (F ) ={{a, c}, {a, d}, {a}, {c}, {d}, ∅
}
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 11
2. Abstract Argumentation 2.1. Argumentation Semantics
Semantics
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
prf (F ) ={{a, c}, {a, d}, {a}, {c}, {d}, ∅
}
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 11
2. Abstract Argumentation 2.1. Argumentation Semantics
Semantics
Stable Extensions [Dung, 1995]
Given an AF F = (A,R). A set S ⊆ A is a stable extension of F , if
S is con�ict-free in F
for each a ∈ A \ S , there exists a b ∈ S , such that (b, a) ∈ R
Example
b c d ea
stb(F ) ={{a, c}
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 12
2. Abstract Argumentation 2.1. Argumentation Semantics
Semantics
Stable Extensions [Dung, 1995]
Given an AF F = (A,R). A set S ⊆ A is a stable extension of F , if
S is con�ict-free in F
for each a ∈ A \ S , there exists a b ∈ S , such that (b, a) ∈ R
Example
b c d ea
stb(F ) ={{a, c}
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 12
2. Abstract Argumentation 2.1. Argumentation Semantics
Semantics
Stable Extensions [Dung, 1995]
Given an AF F = (A,R). A set S ⊆ A is a stable extension of F , if
S is con�ict-free in F
for each a ∈ A \ S , there exists a b ∈ S , such that (b, a) ∈ R
Example
b c d ea
stb(F ) ={{a, c}, {a, d},
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 12
2. Abstract Argumentation 2.1. Argumentation Semantics
Semantics
Stable Extensions [Dung, 1995]
Given an AF F = (A,R). A set S ⊆ A is a stable extension of F , if
S is con�ict-free in F
for each a ∈ A \ S , there exists a b ∈ S , such that (b, a) ∈ R
Example
b c d ea
stb(F ) ={{a, c}, {a, d}, {b, d},
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 12
2. Abstract Argumentation 2.1. Argumentation Semantics
Semantics
Stable Extensions [Dung, 1995]
Given an AF F = (A,R). A set S ⊆ A is a stable extension of F , if
S is con�ict-free in F
for each a ∈ A \ S , there exists a b ∈ S , such that (b, a) ∈ R
Example
b c d ea
stb(F ) ={{a, c}, {a, d}, {b, d}, {a}, {b}, {c}, {d}, ∅,
}
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 12
2. Abstract Argumentation 2.1. Argumentation Semantics
Semantics
Semi-Stable Extensions [Caminada, 2006, Verheij, 1996]
Given an AF F = (A,R). For a set S ⊆ A, de�ne the range S+ = S ∪{a | ∃b ∈ S with (b, 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+
Example
b c d ea
sem(F ) ={{a, c}, {a, d}, {a}, {c}, {d}, ∅
}
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 13
2. Abstract Argumentation 2.1. Argumentation Semantics
Semantics
Semi-Stable Extensions [Caminada, 2006, Verheij, 1996]
Given an AF F = (A,R). For a set S ⊆ A, de�ne the range S+ = S ∪{a | ∃b ∈ S with (b, 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+
Example
b c d ea
sem(F ) ={{a, c}, {a, d}, {a}, {c}, {d}, ∅
}
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 13
2. Abstract Argumentation 2.1. Argumentation Semantics
Semantics
Some Relations
For 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 semi-stable extension of F is also a preferred one.
4 Each stable extension of F is also a semi-stable one.
stb(F ) ⊆ sem(F ) ⊆ prf (F ) ⊆ adm(F ) ⊆ cf (F )
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 14
2. Abstract Argumentation 2.1. Argumentation Semantics
Parametrised Ideal Semantics
Generalising [Dung et al., 2007, Caminada, 2007] we de�ne:
De�nition
Given an AF F = (A,R). A set S ⊆ A is a ideal set w.r.t. base semanticsσ of F , if
I1. S ∈ adm(F)
I2. S ⊆⋂
E∈σ(F)
E
We say that S is an ideal extension of F w.r.t. σ, if S is a ⊆-maximalideal set (of F) w.r.t. σ.
For typical base semantics there is a unique ideal extension.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 15
3. Complexity Analysis
Complexity Analysis
Why doing Complexity Analysis?
Complexity Theoretic View: To understand the Computational Coststhat underlie a certain reasoning problem.
Knowledge-Representation View: Measuring Expressivness of aformalism.
Practitioners View: For applying the Reduction Approach, i.e.encoding a problem in other formalisms, the target formalism mustbe at least of the same complexity.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 16
3. Complexity Analysis
Decision Problems on AFs
Credulous Acceptance
Credσ: Given AF F = (A,R) and a ∈ A; is a contained in at least oneσ-extension of F?
Skeptical Acceptance
Skeptσ: Given AF F = (A,R) and a ∈ A; is a contained in everyσ-extension of F?
If no extension exists then all arguments are skeptically accepted and noargument is credulously accepted.
Ideal Acceptance
Idealσ: Given AF F = (A,R) and a ∈ A; is a contained in the idealextension (w.r.t. base-semantics σ) of F?
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 17
3. Complexity Analysis
Decision Problems on AFs
Credulous Acceptance
Credσ: Given AF F = (A,R) and a ∈ A; is a contained in at least oneσ-extension of F?
Skeptical Acceptance
Skeptσ: Given AF F = (A,R) and a ∈ A; is a contained in everyσ-extension of F?
If no extension exists then all arguments are skeptically accepted and noargument is credulously accepted.
Ideal Acceptance
Idealσ: Given AF F = (A,R) and a ∈ A; is a contained in the idealextension (w.r.t. base-semantics σ) of F?
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 17
3. Complexity Analysis
Further Decision Problems
Verifying an extension
Verσ: Given AF F = (A,R) and S ⊆ A; is S a σ-extension of F?
Does there exist an extension?
Existsσ: Given AF F = (A,R); Does there exist a σ-extension for F?
Does there exist a nonempty extension?
Exists¬∅σ : Does there exist a non-empty σ-extension for F?
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 18
3. Complexity Analysis
Further Decision Problems
Verifying an extension
Verσ: Given AF F = (A,R) and S ⊆ A; is S a σ-extension of F?
Does there exist an extension?
Existsσ: Given AF F = (A,R); Does there exist a σ-extension for F?
Does there exist a nonempty extension?
Exists¬∅σ : Does there exist a non-empty σ-extension for F?
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 18
3. Complexity Analysis
Complexity Landscape (State-of-the-Art)
σ Credσ Skeptσ Idealσ Verσ Existsσ Exists¬∅σ
cf in P trivial ? in P trivial in P
naive in P in P ? in P trivial in P
grd in P in P ? in P trivial in P
stb NP-c coNP-c ? in P NP-c NP-c
adm NP-c trivial ? in P trivial NP-c
com NP-c in P ? in P trivial NP-c
resGr NP-c coNP-c ? in P trivial in P
prf NP-c ΠP2 -c in ΘP
2 coNP-c trivial NP-c
sem in ΣP2 in ΠP
2 ? coNP-c trivial NP-c
stg ? ? ? ? ? ?
Table: State-of-the art complexity landscape for abstract argumentation.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 19
3. Complexity Analysis
Complexity Analysis - Contributions
We contribute in three directions:
Exact complexity classi�cations for semi-stable and stage semantics
Complexity analysis for ideal reasoning
Generic complexity results referring to the complexity of otherreasoning tasks (membership and hardness results)
Exact complexity classi�cations for concrete base semantics
P-completeness classi�cation for tractable problems
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 20
3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics
Complexity of semi-stable and stage semantics
Theorem
Cred sem is ΣP2 -complete and Skeptsem is ΠP
2 -complete.
Hardness is via the following reduction:Given a QBF 2
∀ formula Φ = ∀Y ∃ZC , we de�ne FΦ = (A,R), where
A = {ϕ, ϕ, b} ∪ C ∪ Y ∪ Y ∪ Y ′ ∪ Y ′ ∪ Z ∪ ZR = {(c, ϕ) | c ∈ C} ∪ {(ϕ, ϕ), (ϕ, ϕ), (ϕ, b), (b, b)} ∪
{(x , x), (x , x) | x ∈ Y ∪ Z} ∪{(y , y ′), (y , y ′), (y ′, y ′), (y ′, y ′) | y ∈ Y } ∪{(l , c) | l ∈ C , c ∈ C}.
One can show that Φ is valid i� ϕ is skeptically accepted w.r.t. sem, i� ϕis not credulously accepted w.r.t. sem.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 21
3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics
Complexity of semi-stable and stage semantics
Theorem
Cred sem is ΣP2 -complete and Skeptsem is ΠP
2 -complete.
Hardness is via the following reduction:Given a QBF 2
∀ formula Φ = ∀Y ∃ZC , we de�ne FΦ = (A,R), where
A = {ϕ, ϕ, b} ∪ C ∪ Y ∪ Y ∪ Y ′ ∪ Y ′ ∪ Z ∪ ZR = {(c, ϕ) | c ∈ C} ∪ {(ϕ, ϕ), (ϕ, ϕ), (ϕ, b), (b, b)} ∪
{(x , x), (x , x) | x ∈ Y ∪ Z} ∪{(y , y ′), (y , y ′), (y ′, y ′), (y ′, y ′) | y ∈ Y } ∪{(l , c) | l ∈ C , c ∈ C}.
One can show that Φ is valid i� ϕ is skeptically accepted w.r.t. sem, i� ϕis not credulously accepted w.r.t. sem.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 21
3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics
Complexity of semi-stable and stage semantics
Φ = ∀y1, y2 ∃z3, z4 (y1 ∨ y2 ∨ z3) ∧ (y2 ∨ z3 ∨ z4) ∧ (y1 ∨ y2 ∨ z4).
ϕ
c1 c2 c3
bϕ
y1 y1 y2 y2 z3 z3 z4 z4
y ′1
y ′1
y ′2
y ′2
true assignment τ : τ(y1) = f , τ(y2) = f , τ(z3) = f , τ(z4) = f
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22
3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics
Complexity of semi-stable and stage semantics
Φ = ∀y1, y2 ∃z3, z4 (y1 ∨ y2 ∨ z3) ∧ (y2 ∨ z3 ∨ z4) ∧ (y1 ∨ y2 ∨ z4).
ϕ
c1 c2 c3
bϕ
y1 y1 y2 y2 z3 z3 z4 z4
y ′1
y ′1
y ′2
y ′2
true assignment τ : τ(y1) = f , τ(y2) = f , τ(z3) = f , τ(z4) = f
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22
3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics
Complexity of semi-stable and stage semantics
Φ = ∀y1, y2 ∃z3, z4 (y1 ∨ y2 ∨ z3) ∧ (y2 ∨ z3 ∨ z4) ∧ (y1 ∨ y2 ∨ z4).
ϕ
c1 c2 c3
bϕ
y1 y1 y2 y2 z3 z3 z4 z4
y ′1
y ′1
y ′2
y ′2
true assignment τ : τ(y1) = f , τ(y2) = f , τ(z3) = f , τ(z4) = f
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22
3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics
Complexity of semi-stable and stage semantics
Φ = ∀y1, y2 ∃z3, z4 (y1 ∨ y2 ∨ z3) ∧ (y2 ∨ z3 ∨ z4) ∧ (y1 ∨ y2 ∨ z4).
ϕ
c1 c2 c3
bϕ
y1 y1 y2 y2 z3 z3 z4 z4
y ′1
y ′1
y ′2
y ′2
true assignment τ : τ(y1) = f , τ(y2) = f , τ(z3) = t, τ(z4) = f
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22
3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics
Complexity of semi-stable and stage semantics
Φ = ∀y1, y2 ∃z3, z4 (y1 ∨ y2 ∨ z3) ∧ (y2 ∨ z3 ∨ z4) ∧ (y1 ∨ y2 ∨ z4).
ϕ
c1 c2 c3
bϕ
y1 y1 y2 y2 z3 z3 z4 z4
y ′1
y ′1
y ′2
y ′2
true assignment τ : τ(y1) = t, τ(y2) = f , τ(z3) = t, τ(z4) = f
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22
3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics
Complexity of semi-stable and stage semantics
Φ = ∀y1, y2 ∃z3, z4 (y1 ∨ y2 ∨ z3) ∧ (y2 ∨ z3 ∨ z4) ∧ (y1 ∨ y2 ∨ z4).
ϕ
c1 c2 c3
bϕ
y1 y1 y2 y2 z3 z3 z4 z4
y ′1
y ′1
y ′2
y ′2
true assignment τ : τ(y1) = f , τ(y2) = f , τ(z3) = f , τ(z4) = f
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22
3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics
Complexity of semi-stable and stage semantics
Φ = ∀y1, y2 ∃z3, z4 (y1 ∨ y2 ∨ z3) ∧ (y2 ∨ z3 ∨ z4) ∧ (y1 ∨ y2 ∨ z4).
ϕ
c1 c2 c3
bϕ
y1 y1 y2 y2 z3 z3 z4 z4
y ′1
y ′1
y ′2
y ′2
true assignment τ : τ(y1) = f , τ(y2) = f , τ(z3) = f , τ(z4) = f
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22
3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics
Complexity of semi-stable and stage semantics
Φ = ∀y1, y2 ∃z3, z4 (y1 ∨ y2 ∨ z3) ∧ (y2 ∨ z3 ∨ z4) ∧ (y1 ∨ y2 ∨ z4).
ϕ
c1 c2 c3
bϕ
y1 y1 y2 y2 z3 z3 z4 z4
y ′1
y ′1
y ′2
y ′2
true assignment τ : τ(y1) = f , τ(y2) = f , τ(z3) = t, τ(z4) = f
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22
3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics
Complexity of semi-stable and stage semantics
Φ = ∀y1, y2 ∃z3, z4 (y1 ∨ y2 ∨ z3) ∧ (y2 ∨ z3 ∨ z4) ∧ (y1 ∨ y2 ∨ z4).
ϕ
c1 c2 c3
bϕ
y1 y1 y2 y2 z3 z3 z4 z4
y ′1
y ′1
y ′2
y ′2
true assignment τ : τ(y1) = f , τ(y2) = f , τ(z3) = t, τ(z4) = f
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22
3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics
Complexity of semi-stable and stage semantics
Φ = ∀y1, y2 ∃z3, z4 (y1 ∨ y2 ∨ z3) ∧ (y2 ∨ z3 ∨ z4) ∧ (y1 ∨ y2 ∨ z4).
ϕ
c1 c2 c3
bϕ
y1 y1 y2 y2 z3 z3 z4 z4
y ′1
y ′1
y ′2
y ′2
true assignment τ : τ(y1) = f , τ(y2) = f , τ(z3) = t, τ(z4) = f
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22
3. Complexity Analysis 3.2. Overview
Complexity Landscape
σ Credσ Skeptσ Idealσ Verσ Existsσ Exists¬∅σ
cf in L trivial trivial in L trivial in L
naive in L in L P-c in L trivial in L
grd P-c P-c P-c P-c trivial in L
stb NP-c coNP-c DP-c in L NP-c NP-c
adm NP-c trivial trivial in L trivial NP-c
com NP-c P-c P-c in L trivial NP-c
resGr NP-c coNP-c coNP-c P-c trivial in P
prf NP-c ΠP2 -c in ΘP
2 coNP-c trivial NP-c
sem ΣP
2-c ΠP
2-c ΠP
2-c coNP-c trivial NP-c
stg ΣP
2-c ΠP
2-c ΠP
2-c coNP-c trivial in L
Table: Complexity of abstract argumentation.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 23
4. Towards Tractability
Towards Tractability
Tractability for Abstract Argumentation
Increasing interest for reasoning in argumentation frameworks (AFs).
Many reasoning tasks are computationally intractable.
As AFs can be considered as graphs,
there are several graph classes where some in general hard problemshave been shown to be tractable (Tractable Fragments)there is broad range of graph parameters we can consider to identifytractable fragments (Fixed-Parameter Tractability)
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 24
4. Towards Tractability 4.1. Tractable Fragments
Tractable Fragments
We study four tractable fragments proposed by the literature:
acyclic AFs [Dung, 1995]
AFs without even length cycles (noeven)[Dunne and Bench-Capon, 2001]
symmetric AFs [Coste-Marquis et al., 2005]
bipartite AFs [Dunne, 2007]
We complement existing results by
generalising them to all semantics under our considerations,
classifying them w.r.t. P-completeness,
solving an open problem concerning resolution-based groundedsemantics and bipartite AFs.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 25
4. Towards Tractability 4.1. Tractable Fragments
Tractable FragmentsThe P-hardness for acyclic, noeven, and bipartite is by the following:
Theorem
Credgrd is P-complete even for acyclic bipartite AFs.
Hardness is by a reduction from the Mon. Circuit Value Problem (β, a)
x y
∨
∧
Monotone Boolean Circuit β
a
x
y
∨
∧
x
y
∨
∧1
∧2
AF Fβ,a, with a(x) = 0, a(y) = 1
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 26
4. Towards Tractability 4.1. Tractable Fragments
Tractable FragmentsThe P-hardness for acyclic, noeven, and bipartite is by the following:
Theorem
Credgrd is P-complete even for acyclic bipartite AFs.
Hardness is by a reduction from the Mon. Circuit Value Problem (β, a)
x y
∨
∧
Monotone Boolean Circuit β
a
x
y
∨
∧
x
y
∨
∧1
∧2
AF Fβ,a, with a(x) = 0, a(y) = 1
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 26
4. Towards Tractability 4.2. Fixed-Parameter Tractability
Fixed-Parameter Tractability
Often computational costs primarily depend on some problemparameters rather than on the mere size of the instances.
Many hard problems become tractable if some problem parameter is�xed or bounded by a �xed constant.
In the arena of graphs important parameters are tree-width andclique-width. They have served as the key to many �xed-parametertractability (FPT) results.
We are looking for algorithms with a worst case runtime that mightbe exponential in the parameter but is polynomial in the size of theinstance.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 27
4. Towards Tractability 4.2. Fixed-Parameter Tractability
Fixed-Parameter Tractability
Positive Results:
We show FPT results for the parameters
tree-width and
clique-width
via meta-theorems by Courcelle (1987) and Courcelle, Makowsky &Rotics (2000), and MSO encodings of the argumentation semantics.
Negative Results:
We show that typical reasoning tasks remain intractable if we boundthe parameter cycle-rank.
We extend this result to the parameters directed path-width,DAG-width, Kelly-width, and directed tree-width.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 28
4. Towards Tractability 4.2. Fixed-Parameter Tractability
Negative Results
De�nition
An AF F = (A,R), has cycle rank 0 (cr(F ) = 0) i� F is acyclic, andcr(F ) ≤ 1 i� each strongly connected component of F can be madeacyclic by removing one argument.
Theorem
When restricted to AFs which have a cycle-rank of 1
1 Cred sem remains ΣP2 -hard, and
2 Skeptsem remains ΠP2 -hard.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 29
4. Towards Tractability 4.2. Fixed-Parameter Tractability
Negative Results
De�nition
An AF F = (A,R), has cycle rank 0 (cr(F ) = 0) i� F is acyclic, andcr(F ) ≤ 1 i� each strongly connected component of F can be madeacyclic by removing one argument.
Theorem
When restricted to AFs which have a cycle-rank of 1
1 Cred sem remains ΣP2 -hard, and
2 Skeptsem remains ΠP2 -hard.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 29
4. Towards Tractability 4.2. Fixed-Parameter Tractability
Negative Results
Proof.
Recall the reduction from the hardness proof:
ϕ
c1 c2 c3
y1 y1 y2 y2 z3 z3 z4 z4
y ′1
y ′1
y ′2
y ′2
bϕ
every framework of the form FΦ has cycle-rank 1.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 30
4. Towards Tractability 4.3. Overview
Tractability Results
stb adm com resGr prf sem stg
acyclic X X X X X X Xnoeven X X X X X X 7bipartite X X X 7 X X Xsymmetric 7 X X X X 7 7
bounded tree-width X X X X X X Xbounded clique-width X X X X X X Xbounded cycle-rank 7 7 7 7 7 7 7
bounded directed path-width 7 7 7 7 7 7 7bounded Kelly-width 7 7 7 7 7 7 7bounded DAG-width 7 7 7 7 7 7 7
bounded directed tree-width 7 7 7 7 7 7 7
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 31
5. Intertranslatability of Argumentation Semantics
Intertranslatability of Argumentation Semantics
Why consider translations between Argumentation Semantics ?
�Plethora� of Argumentation Semantics
Reduction approach within argumentation:
Given a translation for semantics σ to semantics σ′ we can reusesophisticated solver for σ′ for semantics σ.
Translationfor σ ⇒ σ′
Solverfor σ′ Filter
AF F Tr (F) σ′(Tr (F)) σ(F)
Figure: Generalising Argumentation Systems via Translations
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 32
5. Intertranslatability of Argumentation Semantics
Translations
De�nition
A Translation Tr is a function mapping (�nite) AFs to (�nite) AFs.
We want translations to satisfy certain properties:
Basic Properties of a Translation Tr
e�cient: for every AF F , Tr (F ) can be computed using logarithmicspace wrt. to |F |
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 33
5. Intertranslatability of Argumentation Semantics
Translations
De�nition
A Translation Tr is a function mapping (�nite) AFs to (�nite) AFs.
We want translations to satisfy certain properties:
Basic Properties of a Translation Tr
e�cient: for every AF F , Tr (F ) can be computed using logarithmicspace wrt. to |F |
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 33
5. Intertranslatability of Argumentation Semantics
Translations
Next we connect translations with semantics.
�Levels of Faithfulness� (for semantics σ, σ′)
exact: for every AF F , σ(F ) = σ′(Tr (F ))
faithful: for every AF F , σ(F ) = {E ∩ AF | E ∈ σ′(Tr (F ))} and|σ(F )| = |σ′(Tr (F ))|.
Translationfor σ ⇒ σ′
Solverfor σ′
AF F Tr (F) σ′(Tr (F)) = σ(F)
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 34
5. Intertranslatability of Argumentation Semantics
Translations
Next we connect translations with semantics.
�Levels of Faithfulness� (for semantics σ, σ′)
exact: for every AF F , σ(F ) = σ′(Tr (F ))
faithful: for every AF F , σ(F ) = {E ∩ AF | E ∈ σ′(Tr (F ))} and|σ(F )| = |σ′(Tr (F ))|.
Translationfor σ ⇒ σ′
Solverfor σ′
AF F Tr (F) σ′(Tr (F)) = σ(F)
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 34
5. Intertranslatability of Argumentation Semantics
Translations
Next we connect translations with semantics.
�Levels of Faithfulness� (for semantics σ, σ′)
exact: for every AF F , σ(F ) = σ′(Tr (F ))
faithful: for every AF F , σ(F ) = {E ∩ AF | E ∈ σ′(Tr (F ))} and|σ(F )| = |σ′(Tr (F ))|.
Translationfor σ ⇒ σ′
Solverfor σ′ E ∩ AF
AF F Tr (F) σ′(Tr (F)) σ(F)
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 34
5. Intertranslatability of Argumentation Semantics 5.1. Translations for Argumentations Semantics
Example Translation 1
De�nition
For AF F , let Tr1(F ) = (A∗,R∗) where A∗ = AF ∪ A′F andR∗ = RF ∪ {(a, a′), (a′, a), (a′, a′) | a∈AF}, with A′F = {a′ | a∈AF}.
Example
a b c d e
a′ b′ c ′ d ′ e′
Result:
Tr1 is an exact translation for prf ⇒ sem.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 35
5. Intertranslatability of Argumentation Semantics 5.1. Translations for Argumentations Semantics
Example Translation 2
De�nition
For AF F , Tr6(F ) = (A∗,R∗) where A∗ = AF ∪ AF ∪ RF andR∗ = RF ∪ {(a, a), (a, a) | a ∈ AF} ∪ {(r , r) | r ∈ RF} ∪{(a, r) | r = (y , a) ∈ RF} ∪ {(a, r) | r = (z , y) ∈ RF , (a, z) ∈ RF}.
Example
a b c d e
a b c d e
(a, b) (c, b) (d, c) (c, d) (d, e) (e, e)
Result:
Tr6 is a faithful translation for adm⇒ stb.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 36
5. Intertranslatability of Argumentation Semantics 5.2. Impossibility Results
Impossibility Results
Proposition
There is no exact translation for
adm⇒ σ with σ ∈ {stb, prf , sem}com⇒ adm
Proposition
There is no e�cient faithful translation for sem⇒ σ, σ∈{adm, stb},unless ΣP
2 = NP.
Follows from complexity results.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 37
5. Intertranslatability of Argumentation Semantics 5.2. Impossibility Results
Impossibility Results
Proposition
There is no exact translation for
adm⇒ σ with σ ∈ {stb, prf , sem}com⇒ adm
Proposition
There is no e�cient faithful translation for sem⇒ σ, σ∈{adm, stb},unless ΣP
2 = NP.
Follows from complexity results.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 37
5. Intertranslatability of Argumentation Semantics 5.3. Overview
Hierarchies of intertranslatability
grounded
stable admissible complete
preferred stage
semi-stable
Exact Intertranslatability
grounded
admissible, complete, stable
preferred stage
semi-stable
Faithful Intertranslatability
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 38
6. Summary
Summary
We complemented existing Complexity Analysis by
exact classi�cations for semi-stable and stage semantics
our studies on ideal reasoning
P-completeness classi�cations
Towards tractable instances we studied Tractable Fragments as wellas Fixed-Parameter Tractability.
We complemented studies of Tractable Fragments
Fixed-Parameter Tractability results for tree-width and clique-width.
By the Intertranslatability of semantics we applied the reductionapproach within abstract argumentation presenting
translations between argumentation semantics
negative results showing that certain translations are impossible
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 39
6. Summary
PublicationsComplexity Analysis:
Dvo°ák, W. and Woltran, S. (2010).Complexity of semi-stable and stage semantics in argumentationframeworks. Inf. Process. Lett., 110(11):425�430.
Dvo°ák, W., Dunne, P. E., and Woltran, S. (2011).Parametric properties of ideal semantics. IJCAI 2011
Towards Tractability:
Dvo°ák, W., Pichler, R., and Woltran, S. (2012).Towards �xed-parameter tractable algorithms for abstractargumentation. Arti�cial Intelligence, 186(0):1 � 37.
Dvo°ák, W., Ordyniak, S., and Szeider, S. (2012).Augmenting tractable fragments of abstract argumentation.Arti�cial Intelligence, in press.
Intertranslatability:
Dvo°ák, W. and Woltran, S. (2011).On the intertranslatability of argumentation semantics. J. Artif.Intell. Res. (JAIR), 41:445�475.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 40
7. Bibliography
Bibliography I
Caminada, M. (2006).Semi-stable semantics.In Dunne, P. E. and Bench-Capon, T. J. M., editors, Proceedings ofthe 1st Conference on Computational Models of Argument(COMMA 2006), volume 144 of Frontiers in Arti�cial Intelligenceand Applications, pages 121�130. IOS Press.
Caminada, M. (2007).Comparing two unique extension semantics for formalargumentation: ideal and eager.In Proceedings of the 19th Belgian-Dutch Conference on Arti�cialIntelligence (BNAIC 2007), pages 81�87.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 41
7. Bibliography
Bibliography II
Coste-Marquis, S., Devred, C., and Marquis, P. (2005).Symmetric argumentation frameworks.In Godo, L., editor, Proceedings of the 8th European Conference onSymbolic and Quantitative Approaches to Reasoning withUncertainty (ECSQARU 2005), volume 3571 of Lecture Notes inComputer Science, pages 317�328. Springer.
Dung, P. M. (1995).On the acceptability of arguments and its fundamental role innonmonotonic reasoning, logic programming and n-person games.Artif. Intell., 77(2):321�358.
Dung, P. M., Mancarella, P., and Toni, F. (2007).Computing ideal sceptical argumentation.Artif. Intell., 171(10-15):642�674.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 42
7. Bibliography
Bibliography III
Dunne, P. E. (2007).Computational properties of argument systems satisfyinggraph-theoretic constraints.Artif. Intell., 171(10-15):701�729.
Dunne, P. E. and Bench-Capon, T. J. M. (2001).Complexity and combinatorial properties of argument systems.Technical report, Dept. of Computer Science, University of Liverpool.
Dvo°ák, W., Dunne, P. E., and Woltran, S. (2011).Parametric properties of ideal semantics.In Walsh, T., editor, IJCAI 2011, Proceedings of the 22ndInternational Joint Conference on Arti�cial Intelligence, Barcelona,Catalonia, Spain, July 16-22, 2011, pages 851�856. IJCAI/AAAI.
Dvo°ák, W., Ordyniak, S., and Szeider, S. (2012a).Augmenting tractable fragments of abstract argumentation.Arti�cial Intelligence, (0):�.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 43
7. Bibliography
Bibliography IV
Dvo°ák, W., Pichler, R., and Woltran, S. (2012b).Towards �xed-parameter tractable algorithms for abstractargumentation.Arti�cial Intelligence, 186(0):1 � 37.
Dvo°ák, W. and Woltran, S. (2010).Complexity of semi-stable and stage semantics in argumentationframeworks.Inf. Process. Lett., 110(11):425�430.
Dvo°ák, W. and Woltran, S. (2011).On the intertranslatability of argumentation semantics.J. Artif. Intell. Res. (JAIR), 41:445�475.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 44
7. Bibliography
Bibliography V
Verheij, B. (1996).Two approaches to dialectical argumentation: admissible sets andargumentation stages.In Meyer, J. and van der Gaag, L., editors, Proceedings of the 8thDutch Conference on Arti�cial Intelligence (NAIC'96), pages357�368.
Computational Aspects of Abstract Argumentation (PhD Defense) Slide 45