In this thesis, we study the dynamics of AFs from multiple perspectives with the purpose of better understanding how dynamic (and concurrent) processes can be handled in the context of argumentation. In this sense, we provide theoretical results, algorithms and tools which can be useful in many dynamic aspects of argumentation. Before arriving to define our concurrent language, we set the theoretical results we need to work with dynamics aspect of argumentation.

1. Handling Dynamic
Aspects of Argumentation
Carlo Taticchi
Ph.D. Thesis Defence - Computer Science - XXXIII Cycle
Supervisor: Prof. Stefano Bistarelli
Internal Advisor: Prof. Luca Aceto

2. Carlo Taticchi Handling Dynamic Aspects of Argumentation
Abstract Argumentation
Frameworks (AFs)
• Consist of a pair
• is defended by if
F = ⟨A, R⟩
c ∈ A D ⊆ A ∀(b, c) ∈ R . ∃a ∈ D ∣ (a, b) ∈ R
2

3. Carlo Taticchi Handling Dynamic Aspects of Argumentation
Abstract Argumentation
Frameworks (AFs)
• Consist of a pair
• is defended by if
F = ⟨A, R⟩
c ∈ A D ⊆ A ∀(b, c) ∈ R . ∃a ∈ D ∣ (a, b) ∈ R
2

4. Carlo Taticchi Handling Dynamic Aspects of Argumentation
Abstract Argumentation
Frameworks (AFs)
• Consist of a pair
• is defended by if
F = ⟨A, R⟩
c ∈ A D ⊆ A ∀(b, c) ∈ R . ∃a ∈ D ∣ (a, b) ∈ R
2

5. Carlo Taticchi Handling Dynamic Aspects of Argumentation
Abstract Argumentation
Frameworks (AFs)
• Consist of a pair
• is defended by if
F = ⟨A, R⟩
c ∈ A D ⊆ A ∀(b, c) ∈ R . ∃a ∈ D ∣ (a, b) ∈ R
2

6. Carlo Taticchi Handling Dynamic Aspects of Argumentation
Weighted AFs (WAFs)
• with
•
• (coincides with Martínez&al2008)
• is w-defended by if
F = ⟨A, R, W,
𝕊
⟩
𝕊
= ⟨S, ⊕ , ⊗ , ⊥ , ⊤ ⟩
𝕊
weighted = ⟨ℝ+
∪ {+∞}, min, + , +∞,0⟩
𝕊
fuzzy = ⟨[0,1], max, min,0,1⟩
d ∈ A D ⊆ A ∀(b, d) ∈ R . ∃a ∈ D ∣ W(a, b) ≤
𝕊
W(b, d)
3
Other
notions of
w-defence
exists in the
literature
0.6 ≤
𝕊
0.8

7. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Semantics for (W)AFs:
‣ is (w-)cf if there are no such that
‣ is (w-)adm if is (w-)cf and each is (w-)defended
by
E ⊆ A a, b ∈ E (a, b) ∈ R
E ⊆ A E a ∈ E
E
4
•
•
Scf = {{}, {a}, {b}, {c}, {d}, {a, c},
{a, d}, {c, d}, {a, c, d}}
Sadm = {{}, {a}, {a, d}, {a, c}, {a, c, d}}

8. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Semantics for (W)AFs:
‣ is (w-)cf if there are no such that
‣ is (w-)adm if is (w-)cf and each is (w-)defended
by
E ⊆ A a, b ∈ E (a, b) ∈ R
E ⊆ A E a ∈ E
E
4
•
•
Sw−cf = {{}, {a}, {b}, {c}, {d}, {a, c},
{a, d}, {c, d}, {a, c, d}}
Sw−adm = {{}, {a}, {a, d}}

9. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Other semantics for AFs:
‣ is com if is adm and defended by ,
‣ is stb if such that
‣ is sst if is com and is maximal
‣ is prf if is adm and is maximal
‣ is gde if is com and is minimal
E ⊆ A E ∀a ∈ A E a ∈ E
E ⊆ A ∀b ∈ A∖E . ∃a ∈ E (a, b) ∈ R
E ⊆ A E E ∪ E+
E ⊆ A E E
E ⊆ A E E
5

10. Carlo Taticchi Handling Dynamic Aspects of Argumentation
Overview
• Invariant Operators and Robustness
• Extending Labellings
• Ranking Arguments in AFs
‣ Ranking claims in CAFs
• A Concurrent Language for Argumentation
• Conclusions and Future Work
6

11. Carlo Taticchi Handling Dynamic Aspects of Argumentation
Robustness
• A property of an AF to withstand changes
• Tries to answer the following questions:
‣ Is it possible to change the outcome of a debate according to
a particular semantics or meaning?
‣ If so, how easy could it be to perform such change?
‣ And which consequences does it bring?
7
Carlo Taticchi. A Study of Robustness in Abstract Argumentation Frameworks. AI*IA 2016 Doctoral Consortium.

12. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Local addition operators add attacks to a given AF
• Semantics may change
• We can compare AFs both w.r.t. their structure and w.r.t. their
semantics
8

13. Carlo Taticchi Handling Dynamic Aspects of Argumentation
Invariant Operators
• We want
• Invariant for con
fl
ict-free sets: add iff
‣
‣
‣
F ≡σ m(F)
a → b
(a, a) ∈ R
(b, a) ∈ R
(b, b) ∈ R
9
Stefano Bistarelli, Francesco Santini, and Carlo Taticchi. On Looking for Invariant Operators in Argumentation
Semantics. FLAIRS 2018.

14. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Invariant operators for admissible sets
• Intuition: when adding a new attack
‣ accepted arguments must not be defeated
‣ rejected arguments must not be defended
10

15. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Invariant operators as a metric to measure robustness
• Local-addition robustness degree of : maximum number of
invariant operators that can be applied to
F k
F
11
Stefano Bistarelli, Francesco Santini, and Carlo Taticchi. Local Expansion Invariant Operators in Argumentation
Semantics. CLAR 2018.

16. Carlo Taticchi Handling Dynamic Aspects of Argumentation
k = 1
• Invariant operators as a metric to measure robustness
• Local-addition robustness degree of : maximum number of
invariant operators that can be applied to
F k
F
11
Stefano Bistarelli, Francesco Santini, and Carlo Taticchi. Local Expansion Invariant Operators in Argumentation
Semantics. CLAR 2018.

17. Carlo Taticchi Handling Dynamic Aspects of Argumentation
k = 2
• Invariant operators as a metric to measure robustness
• Local-addition robustness degree of : maximum number of
invariant operators that can be applied to
F k
F
11
Stefano Bistarelli, Francesco Santini, and Carlo Taticchi. Local Expansion Invariant Operators in Argumentation
Semantics. CLAR 2018.

18. Carlo Taticchi Handling Dynamic Aspects of Argumentation
k = 3
• Invariant operators as a metric to measure robustness
• Local-addition robustness degree of : maximum number of
invariant operators that can be applied to
F k
F
11
Stefano Bistarelli, Francesco Santini, and Carlo Taticchi. Local Expansion Invariant Operators in Argumentation
Semantics. CLAR 2018.

19. Carlo Taticchi Handling Dynamic Aspects of Argumentation
k = 4
• Invariant operators as a metric to measure robustness
• Local-addition robustness degree of : maximum number of
invariant operators that can be applied to
F k
F
11
Stefano Bistarelli, Francesco Santini, and Carlo Taticchi. Local Expansion Invariant Operators in Argumentation
Semantics. CLAR 2018.

20. Carlo Taticchi Handling Dynamic Aspects of Argumentation
k = 4
• Invariant operators as a metric to measure robustness
• Local-addition robustness degree of : maximum number of
invariant operators that can be applied to
F k
F
11
Stefano Bistarelli, Francesco Santini, and Carlo Taticchi. Local Expansion Invariant Operators in Argumentation
Semantics. CLAR 2018.

21. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• rob Interface
12
Stefano Bistarelli, Francesco Faloci, Francesco Santini, and Carlo Taticchi. Studying Dynamics in Argumentation
with Rob. COMMA 2018.

22. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Invariant Operators and Robustness
• Extending Labellings
• Ranking Arguments in AFs
‣ Ranking claims in CAFs
• A Concurrent Language for Argumentation
• Conclusions and Future Work
13
Overview

23. Carlo Taticchi Handling Dynamic Aspects of Argumentation
Reinstatement Labelling
(Caminada2006)
• An argument is labelled:
‣ in if it is attacked only by out arguments
‣ out if it is attacked by at least an in argument
‣ undec otherwise
14

24. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• is w-adm iff:
‣
‣ (Jakobovits&Vermeir1999)
L
L(a) = in ⟹ a−
= a−
|out ∧ ∀R(b, a) . ∃c ∣ L(c) = in ∧ W(c, b) ≤
𝕊
W(b, a)
L(a) = out ⟺ W(a−
|in, a) <
𝕊
⊤
Labelling Semantics for WAFs
15
Stefano Bistarelli and Carlo Taticchi. Extending Labelling Semantics to Weighted Argumentation Frameworks.
FLAIRS 2021 (to appear).

25. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• is w-adm iff:
‣
‣ (Caminada2014)
L
L(a) = in ⟹ a−
= a−
|out ∧ ∀R(b, a) . ∃c ∣ L(c) = in ∧ W(c, b) ≤
𝕊
W(b, a)
L(a) = out ⟹ W(a−
|in, a) <
𝕊
⊤
Labelling Semantics for WAFs
16

26. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Other labelling semantics for WAFs
• Correspondence with extension-based weighted semantics
17
Stefano Bistarelli and Carlo Taticchi. A Labelling Semantics for Weighted Argumentation Frameworks. CILC 2020.

27. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• ConArg Web Interface for Weighted Semantics
18
Stefano Bistarelli and Carlo Taticchi. Preliminary Study on Reinstatement Labelling for Weighted Argumentation
Frameworks. AI3@AI*IA 2019.

28. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Invariant Operators and Robustness
• Extending Labellings
• Ranking Arguments in AFs
‣ Ranking claims in CAFs
• A Concurrent Language for Argumentation
• Conclusions and Future Work
19
Overview

29. Carlo Taticchi Handling Dynamic Aspects of Argumentation
Ranking-Based Semantics
(Amgoud&Ben-Naim2013)
• Transforms an AF into a ranking
• Example:
• Criteria: direct attacks, paths length, rewards
• Good properties
a ≻ d ≻ c ≻ e ≻ b
20

30. Carlo Taticchi Handling Dynamic Aspects of Argumentation
Evaluating Arguments
• Assign a value to each argument, then sort them
• We want to consider the importance of an argument in forming
extensions of a certain semantics
• We use the Shapley Value ϕa(v)
21
Considering adm semantics, is arguably better than and
a b c

31. Carlo Taticchi Handling Dynamic Aspects of Argumentation
•
•
• iff
‣ , or
‣ and
vI
σ(S) =
{
1, if S ∈ in(Lσ)
0, otherwise
vO
σ (S) =
{
1, if S ∈ out(Lσ)
0, otherwise
∀a, b ∈ A, a ≻ϕ
σ b
ϕa(vI
σ(S)) > ϕb(vI
σ(S))
ϕa(vI
σ(S)) = ϕb(vI
σ(S)) ϕa(vO
σ (S)) < ϕb(vO
σ (S))
Ranking with Power Indexes
22
a ≻ϕ
adm
c
Stefano Bistarelli, Paolo Giuliodori, Francesco Santini, and Carlo Taticchi. A Cooperative-game Approach to
Share Acceptability and Rank Arguments. AI3@AI*IA 2018.

32. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Shapley Value ranking example
23
Carlo Taticchi. Power Index-Based Semantics for Ranking Arguments in Abstract Argumentation Frameworks:
An Overview. AI*IA 2019 Doctoral Consortium.

33. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Properties studied w.r.t. (Shapley) and (Banzhaf)
‣ Abstraction
‣ Independence
‣ Void Precedence
‣ Self-contradiction
‣ Cardinality Precedence
‣ Quality Precedence
‣ Non-attacked Equivalence
‣ Totality
ϕ β
24
Stefano Bistarelli and Carlo Taticchi. Power index-based semantics for ranking arguments in abstract
argumentation frameworks. Intell. Artif. 2019.

34. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• ConArg Web Interface for Ranking-Based Semantics
25
Stefano Bistarelli, Francesco Faloci, and Carlo Taticchi. Implementing Ranking-Based Semantics in ConArg.
ICTAI 2019.
Stefano Bistarelli, Francesco Faloci, Francesco Santini, and Carlo Taticchi. A Tool For Ranking Arguments
Through Voting-Games Power Indexes. CILC 2019.

35. Carlo Taticchi Handling Dynamic Aspects of Argumentation
Overview
• Invariant Operators and Robustness
• Extending Labellings
• Ranking Arguments in AFs
‣Ranking claims in CAFs
• A Concurrent Language for Argumentation
• Conclusions and Future Work
26

36. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Arguments in CAFs have claims
• We want to rank the claims
• Replacing arguments with their claims in the ranking does not
work…
Claim Ranking
27
a ≃ d ≻ b ≻ c ⟶ x ≃ z ≻ y ≻ x

37. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Arguments in CAFs have claims
• We want to rank the claims
• Replacing arguments with their claims in the ranking does not
work…
Claim Ranking
27
a ≃ d ≻ b ≻ c ⟶ x ≃ z ≻ y ≻ x

38. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Lexicographic order relation: iff
‣ , or
‣
• Lex-lifting:
A ≽L
B
max(A) ≻ max(B)
max(A) ≽ max(B) ∧ A∖ max(A) ≽L
B∖ max(B)
x ≽ y ⟺ Ax ≽L
Ay
28
a ≃ b ≻ f ≻ e ≻ c ≻ d
Ax = {a, f}, Ay = {b, e}
Ax ≻L
Ay ⟹ x ≻ y
Stefano Bistarelli, Wolfgang Dvorák, Carlo Taticchi, and Stefan Woltran. Ranking-Based Semantics from the
Perspective of Claims. COMMA 2020.

39. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Lexicographic order relation: iff
‣ , or
‣
• Lex-lifting:
A ≽L
B
max(A) ≻ max(B)
max(A) ≽ max(B) ∧ A∖ max(A) ≽L
B∖ max(B)
x ≽ y ⟺ Ax ≽L
Ay
28
a ≃ b ≻ f ≻ e ≻ c ≻ d
Ax = {a, f}, Ay = {b, e}
Ax ≻L
Ay ⟹ x ≻ y
Stefano Bistarelli, Wolfgang Dvorák, Carlo Taticchi, and Stefan Woltran. Ranking-Based Semantics from the
Perspective of Claims. COMMA 2020.

40. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Lexicographic order relation: iff
‣ , or
‣
• Lex-lifting:
A ≽L
B
max(A) ≻ max(B)
max(A) ≽ max(B) ∧ A∖ max(A) ≽L
B∖ max(B)
x ≽ y ⟺ Ax ≽L
Ay
28
a ≃ b ≻ f ≻ e ≻ c ≻ d
Ax = {a, f}, Ay = {b, e}
Ax ≻L
Ay ⟹ x ≻ y
Stefano Bistarelli, Wolfgang Dvorák, Carlo Taticchi, and Stefan Woltran. Ranking-Based Semantics from the
Perspective of Claims. COMMA 2020.

41. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Question: do ranking properties hold on CAFs?
‣ E.g., if satis
fi
es VP, will satisfy VP for claims?
• Answer: some property are only lifted for special classes of CAFs
ax ≃ cy ≻ by y ≻ x
29
well-formed att-unitary

42. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Question: do ranking properties hold on CAFs?
‣ E.g., if satis
fi
es VP, will satisfy VP for claims?
• Answer: some property are only lifted for special classes of CAFs
ax ≃ cy ≻ by y ≻ x
29
Abs Ind VP SC CP QP CT SCT DP
C- all all AU
WF
and AU
none none none none none
AC- all
WF or
AU
all all AU AU AU AU AU

43. Carlo Taticchi Handling Dynamic Aspects of Argumentation
Overview
• Invariant Operators and Robustness
• Extending Labellings
• Ranking Arguments in AFs
‣ Ranking claims in CAFs
• A Concurrent Language for Argumentation
• Conclusions and Future Work
30

44. Carlo Taticchi Handling Dynamic Aspects of Argumentation
ConArg_lang
• Negotiation/debating/persuasion between agents
• Models concurrent processes
• Knowledge base represented through AFs
• Syntax:
‣
‣
‣
‣
A ::= success ∣ insert(Arg, R) → A ∣ rmv(Arg, R) → A ∣ A∥A ∣ ∃x A ∣ E
E ::= Ew
∣ Ef
∣ Ef
+P E
Ew
::= testw
c (a, l, σ) → A ∣ testw
s (a, l, σ) → A ∣ checkw
(Arg, R) → A ∣ Ew
+ Ew
Ef
::= testf
c(a, l, σ) → A ∣ testf
s (a, l, σ) → A ∣ checkf
(Arg, R) → A ∣ Ef
∥GEf
∣ failure
31
Stefano Bistarelli and Carlo Taticchi. A Concurrent Language for Argumentation: Preliminary Notes. DIP 2020.

45. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Operational semantics (1)
32
Stefano Bistarelli and Carlo Taticchi. A Concurrent Language for Argumentation. AI3@AI*IA 2020.

46. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Operational semantics (2)
33

47. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• Operational semantics (3)
34

48. Carlo Taticchi Handling Dynamic Aspects of Argumentation
• ConArg_lang Implementation
35
Stefano Bistarelli and Carlo Taticchi. Introducing a Tool for Concurrent Argumentation. JELIA 2021 (to appear)

49. Carlo Taticchi Handling Dynamic Aspects of Argumentation
Connections with AGM
• Implemented operators:
‣ expansion: increases the
number of labels
‣ contraction: decreases the
number of labels
‣ revision: changes a label from
in to out (and vice versa)
36

50. Carlo Taticchi Handling Dynamic Aspects of Argumentation
Overview
• Invariant Operators and Robustness
• Extending Labellings
• Ranking Arguments in AFs
‣ Ranking claims in CAFs
• A Concurrent Language for Argumentation
• Conclusions and Future Work
37

51. Carlo Taticchi Handling Dynamic Aspects of Argumentation
List of Contributions (1)
• Robustness in AFs
‣ Stefano Bistarelli, Francesco Faloci, Francesco Santini, and Carlo Taticchi. Studying Dynamics in Argumentation
with Rob. COMMA 2018.
‣ Stefano Bistarelli, Francesco Santini, and Carlo Taticchi. On Looking for Invariant Operators in Argumentation
Semantics. FLAIRS 2018.
‣ Stefano Bistarelli, Francesco Santini, and Carlo Taticchi. Local Expansion Invariant Operators in Argumentation
Semantics. CLAR 2018.
‣ Carlo Taticchi. A Study of Robustness in Abstract Argumentation Frameworks. AI*IA 2016 Doctoral Consortium.
• Weighted Labelling
‣ Stefano Bistarelli and Carlo Taticchi. Extending Labelling Semantics to Weighted Argumentation Frameworks.
FLAIRS 2021 (to appear).
‣ Stefano Bistarelli and Carlo Taticchi. A Labelling Semantics for Weighted Argumentation Frameworks. CILC 2020.
‣ Stefano Bistarelli and Carlo Taticchi. Preliminary Study on Reinstatement Labelling for Weighted Argumentation
Frameworks. AI3@AI*IA 2019.
38

52. Carlo Taticchi Handling Dynamic Aspects of Argumentation
List of Contributions (2)
• Ranking-Based Semantics
‣ Stefano Bistarelli and Carlo Taticchi. Power index-based semantics for ranking arguments in abstract
argumentation frameworks. Intell. Artif. 2019.
‣ Stefano Bistarelli, Francesco Faloci, and Carlo Taticchi. Implementing Ranking- Based Semantics in ConArg.
ICTAI 2019.
‣ Stefano Bistarelli, Francesco Faloci, Francesco Santini, and Carlo Taticchi. A Tool For Ranking Arguments
Through Voting-Games Power Indexes. CILC 2019.
‣ Carlo Taticchi. Power Index-Based Semantics for Ranking Arguments in Abstract Argumentation Frameworks:
An Overview. AI*IA 2019 Doctoral Consortium.
‣ Stefano Bistarelli, Paolo Giuliodori, Francesco Santini, and Carlo Taticchi. A Cooperative-game Approach to
Share Acceptability and Rank Arguments. AI3@AI*IA 2018.
• Ranking-Based Semantics for CAFs
‣ Stefano Bistarelli, Wolfgang Dvorák, Carlo Taticchi, and Stefan Woltran. Ranking-Based Semantics from the
Perspective of Claims. COMMA 2020.
39

53. Carlo Taticchi Handling Dynamic Aspects of Argumentation
List of Contributions (3)
• Concurrent Language for Argumentation
‣ Stefano Bistarelli and Carlo Taticchi. Introducing a Tool for Concurrent Argumentation. JELIA 2021 (to appear).
‣ Stefano Bistarelli and Carlo Taticchi. A Concurrent Language for Argumentation. AI3@AI*IA 2020.
‣ Stefano Bistarelli and Carlo Taticchi. A Concurrent Language for Argumentation: Preliminary Notes. DIP 2020.
• Probabilistic Argumentation Tool
‣ Stefano Bistarelli, Theofrastos Mantadelis, Francesco Santini, and Carlo Taticchi. Probabilistic Argumentation
Frameworks with MetaProbLog and ConArg. ICTAI 2018.
‣ Stefano Bistarelli, Theofrastos Mantadelis, Francesco Santini, and Carlo Taticchi. Using MetaProbLog and
ConArg to compute Probabilistic Argumentation Frameworks. AI3@AI*IA 2018.
40

54. Carlo Taticchi Handling Dynamic Aspects of Argumentation
Ongoing work
• Give a de
fi
nition of w-strongly admissible extension/labelling (submitted to JLC)
• De
fi
ne a complete four state labelling where empty labels are allowed
• Use probabilistic argumentation for explainable machine learning
• Handle processes involving time-critical aspects in the concurrent language
41

55. Carlo Taticchi Handling Dynamic Aspects of Argumentation
Next
• On the Concurrent Language
‣ Consider structured AFs
‣ Investigate the link between AGM
postulates and ConArg_lang
operators
• On Robustness
‣ Design invariant operators for
other semantics
‣ Identify core arguments essential
to preserve the semantics
• On the labelling semantics for WAFs
‣ Give explicit conditions for
labelling the undec arguments
• On Ranking-Based Semantics
‣ Consider other Power Indexes
‣ Devise a ranking-based semantics
directly on claims
‣ Use quantitative ranking functions
42

56. Handling Dynamic
Aspects of Argumentation
Carlo Taticchi
Thank you for your attention!