A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks

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

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

Some motivations
• 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

Some motivations
• Example:
• How do we trace the order of the sentences that makes the most sense?
2

Some motivations
• Example:
• How do we trace the order of the sentences that makes the most sense?
• Proposed solution: capture arguments’ dependency
2

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

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
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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
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AFs as Dependency Graphs
• AFs can be interpreted as Dependency Graphs
• Finding a Correct Evaluation Order = obtain the reasoning process that
generated the graph
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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
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?
?
?

• 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
x y
y y x
x y
y
Feasible Evaluation Order1
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3
2
4
1
5
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[1] Stefano Bistarelli, Carlo Taticchi: Deriving Dependency Graphs from Abstract Argumentation Frameworks. AI*IA 2023: 17-29

• 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
x y
y y x
x y
y
Feasible Evaluation Order1
6
3
2
4
1
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!

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

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 LF
(a) ≠ FG
(a)
• is an invariant attack if is semantically independent of
(b, a) a b
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• 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
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• 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⟩
LF
(a) = FG
(a)
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ℐ = {∅, {(b, d)}, {(d, c)}, {(b, d), (d, c)}}

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⟩ ℐ
F F
n : Arg → ℕ n G = ⟨Arg, R∖I⟩
I ∈ ℐ
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where and
F F
I ∈ ℐ
• such that with
∄I′ ∈ ℐ |ArC(⟨Arg, R∖I′⟩)| < |ArC(G)| ArC(F) =
⋃
C∈Cycles(F)
C
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obtaining as
few arguments
as possible in
cycles

where and
F F
I ∈ ℐ
• such that with
⋃
C∈Cycles(F)
C
• such that and
∄I′′ ∈ ℐ I′′ ⊂ I |ArC(⟨Arg, R∖I′′⟩)| = |ArC(G)|
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obtaining as
few arguments
as possible in
cycles
while removing
as few invariant
attacks as
possible

where and
F F
I ∈ ℐ
• such that with
⋃
C∈Cycles(F)
C
• such that and
∄I′′ ∈ ℐ I′′ ⊂ I |ArC(⟨Arg, R∖I′′⟩)| = |ArC(G)|
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obtaining as
few arguments
as possible in
cycles
while removing
as few invariant
attacks as
possible

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
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Example
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(a,d,e,c,b,f,g) is not a semantics-aware evaluation order
ℐ = {∅, {(b, f )}, {( f, c)}, {(b, f ), ( f, c)}}
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Example
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
ℐ = {∅, {(b, f )}, {( f, c)}, {(b, f ), ( f, c)}}
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Integration with tcla
• Automatising the instantiation of an AF
• Output example:
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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
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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
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Future Perspectives
• Refine the evaluation process by using three assumptions:
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Future Perspectives
1. each new argument inserted must keep the AF connected
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Future Perspectives
2. freshly added arguments are considered IN
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Future Perspectives
2. freshly added arguments are considered IN
3. each new argument must change the acceptability of some other
argument in the AF
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A Semantics-Aware
Evaluation Order for Abstract
Argumentation Frameworks
Stefano Bistarelli and Carlo Taticchi
Thank you for your attention!

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
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• Then, we compute the grounded labelling
• (other semantics can be also defined)
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• Finally, we establish semantic equivalence and minimise with respect to
• number of arguments in cycles
• removed attacks
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A Semantics-Aware Evaluation Order for Abstract Argumentation Frameworks

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