A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks

In Computational Argumentation, understanding how an Abstract Argumentation Framework (AF) is instantiated is crucial for capturing and possibly exploiting structural and dynamic aspects of the argumentative process. To facilitate the reconstruction of this process and, precisely, to determine the order in which arguments are presented, we introduce the notion of semantics-aware evaluation order. This approach relies on two fundamental concepts: syntactic dependence, derived from attack relations within the AF, and semantic dependence, which assesses the impact of an argument on the acceptability of others. Integrating these dependencies provides a more meaningful sequence for presenting arguments, improving its alignment with real-life scenarios and enriching the exploration of debate dynamics.

1. A Semantics-Aware
Evaluation Order for Abstract
Argumentation Frameworks
Stefano Bistarelli and Carlo Taticchi
CILC 2024

2. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Some motivations
•AFs simplify complex reasoning processes and predispose them to automation
•Too simple to capture certain aspects of human reasoning, like dynamics
2

3. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Some motivations
•AFs simplify complex reasoning processes and predispose them to automation
•Too simple to capture certain aspects of human reasoning, like dynamics
•Example:
a.“Everyone should eat more vegetables to reduce the risk of chronic diseases”
b.“Excess consumption of certain vegetables can lead to nutrient overdoses”
c.“Vegetable-induced nutrient overdoses are rare and manageable with a varied diet”
2

4. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Some motivations
•AFs simplify complex reasoning processes and predispose them to automation
•Too simple to capture certain aspects of human reasoning, like dynamics
•Example:
a.“Everyone should eat more vegetables to reduce the risk of chronic diseases”
b.“Excess consumption of certain vegetables can lead to nutrient overdoses”
c.“Vegetable-induced nutrient overdoses are rare and manageable with a varied diet”
•How do we trace the order of the sentences that makes the most sense?
2

5. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Some motivations
•AFs simplify complex reasoning processes and predispose them to automation
•Too simple to capture certain aspects of human reasoning, like dynamics
•Example:
a.“Everyone should eat more vegetables to reduce the risk of chronic diseases”
b.“Excess consumption of certain vegetables can lead to nutrient overdoses”
c.“Vegetable-induced nutrient overdoses are rare and manageable with a varied diet”
•How do we trace the order of the sentences that makes the most sense?
•Proposed solution: capture arguments’ dependency
2

6. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Overview
•Dependency Graphs & Feasible Evaluation Order for AFs
•Semantic dependency
•Semantics-Aware Evaluation Order + some examples
•ASP Implementation for Minimal Invariant Attack Sets
•Conclusion and Future Work
3

7. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Dependency Graphs
•Represent the dependencies of various elements
•We look for a Correct Evaluation Order:
•if x is evaluated before y, then x must not depend on y
4

8. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Dependency Graphs
•Represent the dependencies of various elements
•We look for a Correct Evaluation Order:
•if x is evaluated before y, then x must not depend on y
4
1
2
4
5
3

9. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
AFs as Dependency Graphs
•AFs can be interpreted as Dependency Graphs
•Finding a Correct Evaluation Order = obtain the reasoning process that
generated the graph
5

10. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
AFs as Dependency Graphs
•AFs can be interpreted as Dependency Graphs
•Finding a Correct Evaluation Order = obtain the reasoning process that
generated the graph
•Issue: a Correct Evaluation Order cannot be found when the graph has
circular dependencies
5
?
?
?

11. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
•Treat any cycle as an agglomeration of nodes whose evaluation order is not
influential
•Feasible Evaluation Order: for all arguments and not in circular
dependencies between them, if is evaluated before , then and all
arguments in circular dependences with must not depend on and all
arguments in circular dependences with
xy
y y x
x y
y
Feasible Evaluation Order
1
6
3
2
41
5
6
[1] Stefano Bistarelli, Carlo Taticchi: Deriving Dependency Graphs from Abstract Argumentation Frameworks. AI*IA 2023: 17-29

12. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
•Treat any cycle as an agglomeration of nodes whose evaluation order is not
influential
•Feasible Evaluation Order: for all arguments and not in circular
dependencies between them, if is evaluated before , then and all
arguments in circular dependences with must not depend on and all
arguments in circular dependences with
xy
y y x
x y
y
Feasible Evaluation Order
1
6
3
2
41
5
6
[1] Stefano Bistarelli, Carlo Taticchi: Deriving Dependency Graphs from Abstract Argumentation Frameworks. AI*IA 2023: 17-29
ISSUE: b, c and d
can be evaluated
in any order!

13. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Semantic dependency
•Idea: since the inclusion of does not alter the acceptability state of the other
arguments within the cycle, it can be evaluated without being constrained by
the dependency arising from the attacks it conducts
b
7
An argument is OUT if it is attacked by at least one IN

14. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Semantic dependency
•Idea: since the inclusion of does not alter the acceptability state of the other
arguments within the cycle, it can be evaluated without being constrained by
the dependency arising from the attacks it conducts
b
•Let , , and
semantically depends on if and only if
F=!Arg,R"a,b#Arg G=!Arg,R!{(b,a)}"
a b L
F
(a)$F
G
(a)
• is an invariant attack if is semantically independent of (b,a) a b
7

15. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
•Removing multiple invariant attacks is not guaranteed, in general, to leave the
labelling unaltered.
•Example: removing the both invariant attacks and changes the
label of from OUT to UND
(e,d)(f,d)
d
8

16. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
•Removing multiple invariant attacks is not guaranteed, in general, to leave the
labelling unaltered.
•Example: removing the both invariant attacks and changes the
label of from OUT to UND
(e,d)(f,d)
d
8

17. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
•Removing multiple invariant attacks is not guaranteed, in general, to leave the
labelling unaltered.
•Example: removing the both invariant attacks and changes the
label of from OUT to UND
(e,d)(f,d)
d
8

18. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
•Removing multiple invariant attacks is not guaranteed, in general, to leave the
labelling unaltered.
•Example: removing the both invariant attacks and changes the
label of from OUT to UND
(e,d)(f,d)
d
8

19. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
•Removing multiple invariant attacks is not guaranteed, in general, to leave the
labelling unaltered.
•Example: removing the both invariant attacks and changes the
label of from OUT to UND
(e,d)(f,d)
d
•Consider and let be
the set of attacks between arguments within a cycle. Then is an
invariant attack set within cycles of if, given we have that
.
F=!Arg,R" R={(a,b)%(a,b)#R&b#CiD(a)}
I'R
F G=!Arg,R!I"
L
F
(a)=F
G
(a)
8
(={),{(b,d)},{(d,c)},{(b,d),(d,c)}}

20. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
•Removing multiple invariant attacks is not guaranteed, in general, to leave the
labelling unaltered.
•Example: removing the both invariant attacks and changes the
label of from OUT to UND
(e,d)(f,d)
d
•Consider and let be
the set of attacks between arguments within a cycle. Then is an
invariant attack set within cycles of if, given we have that
.
F=!Arg,R" R={(a,b)%(a,b)#R&b#CiD(a)}
I'R
F G=!Arg,R!I"
L
F
(a)=F
G
(a)
8
(={),{(b,d)},{(d,c)},{(b,d),(d,c)}}

21. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
•Removing multiple invariant attacks is not guaranteed, in general, to leave the
labelling unaltered.
•Example: removing the both invariant attacks and changes the
label of from OUT to UND
(e,d)(f,d)
d
•Consider and let be
the set of attacks between arguments within a cycle. Then is an
invariant attack set within cycles of if, given we have that
.
F=!Arg,R" R={(a,b)%(a,b)#R&b#CiD(a)}
I'R
F G=!Arg,R!I"
L
F
(a)=F
G
(a)
8
(={),{(b,d)},{(d,c)},{(b,d),(d,c)}}

22. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
•Removing multiple invariant attacks is not guaranteed, in general, to leave the
labelling unaltered.
•Example: removing the both invariant attacks and changes the
label of from OUT to UND
(e,d)(f,d)
d
•Consider and let be
the set of attacks between arguments within a cycle. Then is an
invariant attack set within cycles of if, given we have that
.
F=!Arg,R" R={(a,b)%(a,b)#R&b#CiD(a)}
I'R
F G=!Arg,R!I"
L
F
(a)=F
G
(a)
8
(={),{(b,d)},{(d,c)},{(b,d),(d,c)}}

23. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Semantics-Aware Evaluation Order
•Let be an AF, and the set of all invariant attack sets within
cycles of . A semantics-aware evaluation order for is a numbering
such that is a feasible evaluation order for ,
where and
F=!Arg,R" (
F F
n:Arg*+ n G=!Arg,R!I"
I#(
9

24. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Semantics-Aware Evaluation Order
•Let be an AF, and the set of all invariant attack sets within
cycles of . A semantics-aware evaluation order for is a numbering
such that is a feasible evaluation order for ,
where and
F=!Arg,R" (
F F
n:Arg*+ n G=!Arg,R!I"
I#(
•
such that with ,I- #( "ArC(!Arg,R!I- ")"<"ArC(G)" ArC(F)=
!
C#Cycles(F)
C
9
obtaining as
few arguments
as possible in
cycles

25. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Semantics-Aware Evaluation Order
•Let be an AF, and the set of all invariant attack sets within
cycles of . A semantics-aware evaluation order for is a numbering
such that is a feasible evaluation order for ,
where and
F=!Arg,R" (
F F
n:Arg*+ n G=!Arg,R!I"
I#(
•
such that with ,I- #( "ArC(!Arg,R!I- ")"<"ArC(G)" ArC(F)=
!
C#Cycles(F)
C
• such that and ,I- - #( I- - .I"ArC(!Arg,R!I- - ")"="ArC(G)"
9
obtaining as
few arguments
as possible in
cycles
while removing
as few invariant
attacks as
possible

26. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Semantics-Aware Evaluation Order
•Let be an AF, and the set of all invariant attack sets within
cycles of . A semantics-aware evaluation order for is a numbering
such that is a feasible evaluation order for ,
where and
F=!Arg,R" (
F F
n:Arg*+ n G=!Arg,R!I"
I#(
•
such that with ,I- #( "ArC(!Arg,R!I- ")"<"ArC(G)" ArC(F)=
!
C#Cycles(F)
C
• such that and ,I- - #( I- - .I"ArC(!Arg,R!I- - ")"="ArC(G)"
9
1
2
3
4 5
6
obtaining as
few arguments
as possible in
cycles
while removing
as few invariant
attacks as
possible

27. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Example
a.Anna should rent the apartment she found
b.The apartment seems to have humidity problems
c.The owner is committed to solving structural problems in the apartment
d.A nightclub is set to open nearby shortly
e.Laws forbid the opening of nightclubs in the area
f.The owner, planning to sell the property soon, is unlikely to fund long-term repairs
g.Due to legal constraints, the apartment cannot be sold immediately
10
1
2
3
4
5
6
7

28. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Example
a.Anna should rent the apartment she found
b.The apartment seems to have humidity problems
c.The owner is committed to solving structural problems in the apartment
d.A nightclub is set to open nearby shortly
e.Laws forbid the opening of nightclubs in the area
f.The owner, planning to sell the property soon, is unlikely to fund long-term repairs
g.Due to legal constraints, the apartment cannot be sold immediately
10
(a,d,e,c,b,f,g) is not a semantics-aware evaluation order
(={),{(b,f)},{(f,c)},{(b,f),(f,c)}}
1
2
3
4
5
6
7

29. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Example
a.Anna should rent the apartment she found
b.The apartment seems to have humidity problems
c.The owner is committed to solving structural problems in the apartment
d.A nightclub is set to open nearby shortly
e.Laws forbid the opening of nightclubs in the area
f.The owner, planning to sell the property soon, is unlikely to fund long-term repairs
g.Due to legal constraints, the apartment cannot be sold immediately
10
(a,d,e,b,c,f,g) is a semantics-aware evaluation order
(a,d,e,c,b,f,g) is not a semantics-aware evaluation order
(={),{(b,f)},{(f,c)},{(b,f),(f,c)}}
1
2
3
4
5
6
7 4
5

30. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Integration with tcla
•Automatising the instantiation of an AF
•Output example:
14

31. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Conclusion
•The semantics-aware evaluation order introduces arguments in a meaningful
sequence, simulating what might have happened during the AF’s instantiation
•Arguments that receive attacks are evaluated first
•Inside cycles, an argument conducting invariant attacks is selected for
evaluation
15

32. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Conclusion
•The semantics-aware evaluation order introduces arguments in a meaningful
sequence, simulating what might have happened during the AF’s instantiation
•Arguments that receive attacks are evaluated first
•Inside cycles, an argument conducting invariant attacks is selected for
evaluation
•Issues:
•a cycle may contain no invariant attack
•multiple arguments conducting invariant attacks may be candidates within
a cycle for initial evaluation
15

33. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Future Perspectives
•Refine the evaluation process by using three assumptions:
16

34. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Future Perspectives
•Refine the evaluation process by using three assumptions:
1.each new argument inserted must keep the AF connected
16
1
2
3
4
5
6

35. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Future Perspectives
•Refine the evaluation process by using three assumptions:
1.each new argument inserted must keep the AF connected
2.freshly added arguments are considered IN
16
1
2
3
4
5
6

36. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Future Perspectives
•Refine the evaluation process by using three assumptions:
1.each new argument inserted must keep the AF connected
2.freshly added arguments are considered IN
3.each new argument must change the acceptability of some other
argument in the AF
16
1
2
3
4
5
6

37. A Semantics-Aware
Evaluation Order for Abstract
Argumentation Frameworks
Stefano Bistarelli and Carlo Taticchi
Thank you for your attention!

38. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
Minimal Invariant Attack Sets
•We provide an ASP implementation to find minimal invariant attack sets
•First, we define path, arguments in cycles, and the removal choice rule
11

39. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
•Then, we compute the grounded labelling
•(other semantics can be also defined)
12

40. CILC 2024 A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks
•Finally, we establish semantic equivalence and minimise with respect to
•number of arguments in cycles
•removed attacks
13