An overview of the three main paradigms for deriving semantics through acceptability of arguments in AFS: extension-based, labelling-based and ranking-based.

1. ACCEPTABILITY PARADIGMS
IN ABSTRACT ARGUMENTATION
FRAMEWORKS
Sistemi con Vincoli
e Rappresentazione della Conoscenza

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OVERVIEW
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OVERVIEW
▸ Extension-Based Semantics
▸ Labelling-Based Semantics
▸ Ranking-Based Semantics
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a b c d e

4. EXTENSION
BASED
SEMANTICS
1

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EXTENSION-BASED SEMANTICS
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conflict-free: the arguments do not attack each other

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conflict-free: the arguments do not attack each other
E ∈ Scf(G) if and only if there are no a, b ∈ 𝒜 such that (a, b) ∈ ℛ
EXTENSION-BASED SEMANTICS

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a b c d e
CF = {{}, {a}, {b}, {c}, {d}, {a, c}, {a, d}, {b, d}}
conflict-free: the arguments do not attack each other
EXTENSION-BASED SEMANTICS

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a b c d e
admissible: is conflict-free and the arguments are defended
EXTENSION-BASED SEMANTICS

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a b c d e
admissible: is conflict-free and the arguments are defended
CF = {{}, {a}, {b}, {c}, {d}, {a, c}, {a, d}, {b, d}}
EXTENSION-BASED SEMANTICS
E ∈ Sadm(G) if each a ∈ E is defended by E

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a b c d e
ADM = {{}, {a}, {c}, {d}, {a, c}, {a, d}}
admissible: is conflict-free and the arguments are defended
CF = {{}, {a}, {b}, {c}, {d}, {a, c}, {a, d}, {b, d}}
EXTENSION-BASED SEMANTICS

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a b c d e
complete: is admissible and contains all the defended arguments
EXTENSION-BASED SEMANTICS

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a b c d e
complete: is admissible and contains all the defended arguments
ADM = {{a}, {c}, {d}, {a, c}, {a, d}}
EXTENSION-BASED SEMANTICS
E ∈ Scom(G) if E ∈ Sadm(G) and ∀a ∈ 𝒜 defended by E, a ∈ E

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a b c d e
COM = {{a}, {a, c}, {a, d}}
complete: is admissible and contains all the defended arguments
ADM = {{a}, {c}, {d}, {a, c}, {a, d}}
EXTENSION-BASED SEMANTICS

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preferred: maximal admissible w.r.t. set inclusion
EXTENSION-BASED SEMANTICS

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preferred: maximal admissible w.r.t. set inclusion
ADM = {{a}, {c}, {d}, {a, c}, {a, d}}
EXTENSION-BASED SEMANTICS
E ∈ Sprf(G) if E ∈ Sadm(G) and ∄E′ ∈ Sadm(G) such that E ⊂ E′

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a b c d e
PRE = {{a, c}, {a, d}}
preferred: maximal admissible w.r.t. set inclusion
ADM = {{a}, {c}, {d}, {a, c}, {a, d}}
EXTENSION-BASED SEMANTICS

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a b c d e
grounded: minimal complete w.r.t. set inclusion
EXTENSION-BASED SEMANTICS

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grounded: minimal complete w.r.t. set inclusion
COM = {{a}, {a, c}, {a, d}}
EXTENSION-BASED SEMANTICS
E = Sgde(G) if E ∈ Scom(G) and ∄E′ ∈ Scom(G) such that E′ ⊂ E

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GDE = {{a}}
grounded: minimal complete w.r.t. set inclusion
COM = {{a}, {a, c}, {a, d}}
EXTENSION-BASED SEMANTICS

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stable: is conflict-free and attacks any other argument
EXTENSION-BASED SEMANTICS

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CF = {{}, {a}, {b}, {c}, {d}, {a, c}, {a, d}, {b, d}}
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a b c d e
stable: is conflict-free and attacks any other argument
EXTENSION-BASED SEMANTICS
E ∈ Sstb(G) if ∀a ∈ 𝒜∖E, S attacks a

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CF = {{}, {a}, {b}, {c}, {d}, {a, c}, {a, d}, {b, d}}
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a b c d e
STA = {{a,d}}
stable: is conflict-free and attacks any other argument
EXTENSION-BASED SEMANTICS

23. LABELLING
BASED
SEMANTICS
2

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REINSTATEMENT LABELLING
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REINSTATEMENT LABELLING
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IN if it is attacked only by OUT arguments
OUT if it is attacked by at least an IN argument
UNDEC otherwise

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REINSTATEMENT LABELLING
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IN if it is attacked only by OUT arguments
OUT if it is attacked by at least an IN argument
UNDEC otherwise
argteach.herokuapp.com

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conflict-free: for each a ∈ A it holds that
• if a is labelled IN then it does not have an attacker that is IN, and
• if a is labelled OUT then it has at least one attacker that is IN
LABELLING-BASED SEMANTICS
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conflict-free: for each a ∈ A it holds that
• if a is labelled IN then it does not have an attacker that is IN, and
• if a is labelled OUT then it has at least one attacker that is IN
LABELLING-BASED SEMANTICS
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a b c d e

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conflict-free: for each a ∈ A it holds that
• if a is labelled IN then it does not have an attacker that is IN, and
• if a is labelled OUT then it has at least one attacker that is IN
LABELLING-BASED SEMANTICS
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a b c d e

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conflict-free: for each a ∈ A it holds that
• if a is labelled IN then it does not have an attacker that is IN, and
• if a is labelled OUT then it has at least one attacker that is IN
LABELLING-BASED SEMANTICS
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a b c d e

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conflict-free: for each a ∈ A it holds that
• if a is labelled IN then it does not have an attacker that is IN, and
• if a is labelled OUT then it has at least one attacker that is IN
LABELLING-BASED SEMANTICS
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a b c d e

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conflict-free: for each a ∈ A it holds that
• if a is labelled IN then it does not have an attacker that is IN, and
• if a is labelled OUT then it has at least one attacker that is IN
LABELLING-BASED SEMANTICS
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a b c d e

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conflict-free: for each a ∈ A it holds that
• if a is labelled IN then it does not have an attacker that is IN, and
• if a is labelled OUT then it has at least one attacker that is IN
LABELLING-BASED SEMANTICS
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a b c d e

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admissible: for each a ∈ A
• if a is labelled IN then all its attackers are labelled OUT, and
• if a is labelled OUT then it has at least one attacker that is IN
LABELLING-BASED SEMANTICS
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LABELLING-BASED SEMANTICS
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admissible: for each a ∈ A
• if a is labelled IN then all its attackers are labelled OUT, and
• if a is labelled OUT then it has at least one attacker that is IN

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LABELLING-BASED SEMANTICS
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a b c d e
admissible: for each a ∈ A
• if a is labelled IN then all its attackers are labelled OUT, and
• if a is labelled OUT then it has at least one attacker that is IN

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LABELLING-BASED SEMANTICS
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a b c d e
admissible: for each a ∈ A
• if a is labelled IN then all its attackers are labelled OUT, and
• if a is labelled OUT then it has at least one attacker that is IN

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LABELLING-BASED SEMANTICS
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a b c d e
admissible: for each a ∈ A
• if a is labelled IN then all its attackers are labelled OUT, and
• if a is labelled OUT then it has at least one attacker that is IN

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LABELLING-BASED SEMANTICS
39
complete: for each a ∈ A
• a is labelled IN iff all its attackers are labelled OUT, and
• a is OUT iff it has at least one attacker that is IN

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complete: for each a ∈ A
• a is labelled IN iff all its attackers are labelled OUT, and
• a is OUT iff it has at least one attacker that is IN
LABELLING-BASED SEMANTICS
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a b c d e

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LABELLING-BASED SEMANTICS
41
a b c d e
complete: for each a ∈ A
• a is labelled IN iff all its attackers are labelled OUT, and
• a is OUT iff it has at least one attacker that is IN

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LABELLING-BASED SEMANTICS
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a b c d e
complete: for each a ∈ A
• a is labelled IN iff all its attackers are labelled OUT, and
• a is OUT iff it has at least one attacker that is IN

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LABELLING-BASED SEMANTICS
43
a b c d e
complete: for each a ∈ A
• a is labelled IN iff all its attackers are labelled OUT, and
• a is OUT iff it has at least one attacker that is IN

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LABELLING-BASED SEMANTICS
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preferred: the labelling
• is a complete, and
• the set of IN arguments is maximal among all complete labellings

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LABELLING-BASED SEMANTICS
45
preferred: the labelling
• is a complete, and
• the set of IN arguments is maximal among all complete labellings
grounded: the labelling
• is a complete, and
• the set of IN arguments is minimal among all complete labellings

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a b c d e
LABELLING-BASED SEMANTICS
a b c d e
a b c d e

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a b c d e
LABELLING-BASED SEMANTICS
a b c d e
a b c d e
GROUNDED

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a b c d e
LABELLING-BASED SEMANTICS
a b c d e
a b c d e
GROUNDED
PREFERRED

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LABELLING-BASED SEMANTICS
a b c d e
a b c d e
GROUNDED
PREFERRED
PREFERRED

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WHICH IS THE BEST?
LABELLING ACCEPTANCE DEGREES

51. RANKING
BASED
SEMANTICS
3

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RANKING-BASED SEMANTICS
▸ Transforms an Argumentation Framework into a ranking
▸ Criteria: direct attacks, lengths of the incoming paths, rewards
52
≻a d ≻c ≻e ≻ b
ADM = {{}, {a}, {c}, {d}, {a, c}, {a, d}}

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CATEGORIZER
53
▸ For each argument looks at the value of its direct attackers
Cat(x) =
1 if R−
1 (x) = 0
1
1 + ∑y∈R−
1 (x)
Cat(y)
otherwise

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CATEGORIZER
54
▸ For each argument looks at the value of its direct attackers
a
b c
d e
Cat(x) =
1 if R−
1 (x) = 0
1
1 + ∑y∈R−
1 (x)
Cat(y)
otherwise

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CATEGORIZER
55
Cat(x) =
1 if R−
1 (x) = 0
1
1 + ∑y∈R−
1 (x)
Cat(y)
otherwise
▸ For each argument looks at the value of its direct attackers
a
b c
d e
Cat(a) =
1
1 + 1 + 1
1 + 1
1 + 0.5
≈ 0.38
Cat(b) = 1 Cat(c) = 0.5
Cat(d) ≈ 0.65 Cat(e) ≈ 0.53

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CATEGORIZER
56
▸ For each argument looks at the value of its direct attackers
a
b c
d e
1 0.5
0.38
0.65 0.53
Cat(x) =
1 if R−
1 (x) = 0
1
1 + ∑y∈R−
1 (x)
Cat(y)
otherwise
Cat(a) =
1
1 + 1 + 1
1 + 1
1 + 0.5
≈ 0.38
Cat(b) = 1 Cat(c) = 0.5
Cat(d) ≈ 0.65 Cat(e) ≈ 0.53

57. Sistemi con Vincoli e Rappresentazione della ConoscenzaCarlo Taticchi — November 26, 2018
CATEGORIZER
57
▸ For each argument looks at the value of its direct attackers
a
b c
d e
1 0.5
0.38
0.65 0.53
▸ The ranking-based Categoriser associates
to any AF a ranking A ≽Cat
AF on A such that
∀a, b ∈ A, a ≻Cat
AF b iff Cat(a) ≥ Cat(b)
b ≻Cat
d ≻Cat
e ≻Cat
c ≻Cat
a

58. Sistemi con Vincoli e Rappresentazione della ConoscenzaCarlo Taticchi — November 26, 2018
GRADED SEMANTICS
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‣ Generalisation of Dung’s theory
‣ Principles:
• the greater the number of attacks on an argument b,
the weaker is b’s level of justification
• the larger the number of arguments defending a, the
stronger is a’s level of justification

59. Sistemi con Vincoli e Rappresentazione della ConoscenzaCarlo Taticchi — November 26, 2018
GRADED DEFENSE
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‣ is the set of arguments which do not have at least m
attackers that are not counter-attacked by at least n
arguments in X.
dm
n (X)

60. Sistemi con Vincoli e Rappresentazione della ConoscenzaCarlo Taticchi — November 26, 2018
GRADED DEFENSE
60
‣ is the set of arguments which do not have at least m
attackers that are not counter-attacked by at least n
arguments in X.
dm
n (X)
a1 ∈ d1
1(X1)

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GRADED DEFENSE
61
‣ is the set of arguments which do not have at least m
attackers that are not counter-attacked by at least n
arguments in X.
dm
n (X)
a1 ∈ d1
1(X1) a2 ∈ d1
1(X2)

62. Sistemi con Vincoli e Rappresentazione della ConoscenzaCarlo Taticchi — November 26, 2018
GRADED DEFENSE
62
‣ is the set of arguments which do not have at least m
attackers that are not counter-attacked by at least n
arguments in X.
dm
n (X)
a1 ∈ d1
1(X1) a2 ∈ d1
1(X2) a1 ∈ d1
2(X1) ?

63. Sistemi con Vincoli e Rappresentazione della ConoscenzaCarlo Taticchi — November 26, 2018
GRADED DEFENSE
63
‣ is the set of arguments which do not have at least m
attackers that are not counter-attacked by at least n
arguments in X.
dm
n (X)
a1 ∈ d1
1(X1) a2 ∈ d1
1(X2) a1 ∉ d1
2(X1)

64. Sistemi con Vincoli e Rappresentazione della ConoscenzaCarlo Taticchi — November 26, 2018
GRADED DEFENSE
64
‣ is the set of arguments which do not have at least m
attackers that are not counter-attacked by at least n
arguments in X.
dm
n (X)
a1 ∈ d1
1(X1) a2 ∈ d1
1(X2) a1 ∉ d1
2(X1) a2 ∈ d1
2(X2) ?

65. Sistemi con Vincoli e Rappresentazione della ConoscenzaCarlo Taticchi — November 26, 2018
GRADED DEFENSE
65
‣ is the set of arguments which do not have at least m
attackers that are not counter-attacked by at least n
arguments in X.
dm
n (X)
a1 ∈ d1
1(X1) a2 ∈ d1
1(X2) a1 ∉ d1
2(X1) a2 ∈ d1
2(X2)

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GRADED SEMANTICS - RANKING FUNCTIONS
66
‣ Rules:
• the less attackers, the better
• the more counter-attacks, the better
dm
n ⊳ ds
t ⟺ m ≤ s AND t ≤ n

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GRADED SEMANTICS - RANKING FUNCTIONS
67
Exercise

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GRADED SEMANTICS - PARTIAL ORDER
68
‣ Some functions may be incomparable
a3 ∈ d3
3(X3)a3 ∉ d2
2(X3) a4 ∉ d3
3(X4)a4 ∈ d2
2(X4)

69. Sistemi con Vincoli e Rappresentazione della ConoscenzaCarlo Taticchi — November 26, 2018
APPLYING GRADED SEMANTICS
69

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APPLYING GRADED SEMANTICS
70

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APPLYING GRADED SEMANTICS
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b ∈ d1
2(X2)a ∉ d1
2(X1) b ≻pref
a

72. Sistemi con Vincoli e Rappresentazione della ConoscenzaCarlo Taticchi — November 26, 2018
▸ Abstraction
▸ Independence
▸ Self-Contradiction
▸ Void Precedence
▸ Cardinality Precedence
PROPERTIES
72
▸ Quality Precedence
▸ Counter-Transitivity
▸ Strict Counter-Transitivity
▸ Defense Precedence
▸ Total order

73. SUGGESTED READINGS
▸ Phan Minh Dung. On the acceptability of arguments and its fundamental
role in nonmonotonic reasoning, logic programming and n-person games.
Artificial Intelligence, 77(2):321–357.
▸ Martin Caminada. On the Issue of Reinstatement in Argumentation.
JELIA 2006: 111-123.
▸ Leila Amgoud, Jonathan Ben-Naim. Ranking-Based Semantics for
Argumentation Frameworks. SUM 2013: 134-147.
▸ Elise Bonzon, Jérôme Delobelle, Sébastien Konieczny, Nicolas Maudet.
A Comparative Study of Ranking-Based Semantics for Abstract
Argumentation. AAAI 2016: 914-920.