A Labelling Semantics for Weighted Argumentation Frameworks

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Argumentation Theory provides tools for both modelling and reasoning with controversial information and is a methodology that is going to be proposed as a way to give explanations to results provided using machine learning techniques. In this context, labelling-based semantics for Abstract Argumentation Frameworks (AFs) allow for establishing the acceptability of sets of arguments, dividing them into three partitions: acceptable, rejected and undecidable (instead of classical Dung two sets IN and OUT partition). This kind of semantics have been studied only for classical AFs, whilst the more powerful weighted and preference-based framework has been not studied yet. In this paper, we define a novel labelling semantics for weighted argumentation frameworks, extending and generalising the crisp one.

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A LABELLING SEMANTICS FOR
WEIGHTED ARGUMENTATION
FRAMEWORKS
Stefano Bistarelli and Carlo Taticchi

A Labelling Semantics for Weighted Argumentation Frameworks
Carlo Taticchi — CILC 2020
OVERVIEW
▸ Background:
• Extension- and Labelling-Based Semantics
• Weighted AFs
▸ Weighted Labelling
▸ Demo
▸ Conclusion
2
ConArg

A Labelling Semantics for Weighted Argumentation Frameworks
Carlo Taticchi — CILC 2020
EXTENSION-BASED SEMANTICS FOR AFS1
3
1Phan 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.

A Labelling Semantics for Weighted Argumentation Frameworks
Carlo Taticchi — CILC 2020
REINSTATEMENT LABELLING2,3
4
IN if it is attacked only by OUT arguments
OUT if it is attacked by at least an IN argument
UNDEC otherwise
2Martin Caminada. On the Issue of Reinstatement in Argumentation. JELIA 2006: 111-123.
3Hadassa Jakobovits and Dirk Vermeir. Robust Semantics for Argumentation Frameworks. J. Log. Comput., 9(2), 215–261, (1999).

A Labelling Semantics for Weighted Argumentation Frameworks
Carlo Taticchi — CILC 2020
LABELLING-BASED SEMANTICS
5
a is IN a is not attacked by any IN
a is OUT a is attacked by some IN
CF

A Labelling Semantics for Weighted Argumentation Frameworks
Carlo Taticchi — CILC 2020 6
a is IN a is only attacked by OUT
a is OUT a is attacked by some IN
ADM
LABELLING-BASED SEMANTICS

A Labelling Semantics for Weighted Argumentation Frameworks
Carlo Taticchi — CILC 2020 7
a is IN a is only attacked by OUT
a is OUT a is attacked by some IN
⟺
⟺
COM
LABELLING-BASED SEMANTICS

A Labelling Semantics for Weighted Argumentation Frameworks
Carlo Taticchi — CILC 2020
WEIGHTED ARGUMENTATION FRAMEWORKS (WAFS)
8
▸ The acceptability of the arguments also depends on the weights
▸ Different notion of defence

A Labelling Semantics for Weighted Argumentation Frameworks
Carlo Taticchi — CILC 2020
WEIGHTED DEFENCE4
9
4Stefano Bistarelli, Fabio Rossi, Francesco Santini. A Collective Defence Against Grouped Attacks for Weighted Abstract Argumentation
Frameworks. FLAIRS Conference 2016: 638-643.
▸
▸ {a, b, d} and {a, b, e} are w-admissible extensions
▸ {a, b, d, e} is not w-admissible (2+3 < 4+4)
D w-defends a ⟺ ∀b attacking a . W(b, D ∪ {a}) ≤ W(D, b)

A Labelling Semantics for Weighted Argumentation Frameworks
Carlo Taticchi — CILC 2020
LABELLING-BASED SEMANTICS FOR WAFS
10
▸ The w-conflict-free and the conflict-free labelling coincides
a is IN a is not attacked by any IN
a is OUT a is attacked by some IN
CF
W-CF

A Labelling Semantics for Weighted Argumentation Frameworks
Carlo Taticchi — CILC 2020
LABELLING-BASED SEMANTICS FOR WAFS
11
a is IN
a is OUT
⟹ a−
= a−
|OUT ∧ ∀b ∈ a−
. wb−|IN
≤𝕊 wb+|IN
⟹ wa−|IN
<𝕊 ⊤
W-ADM

A Labelling Semantics for Weighted Argumentation Frameworks
Carlo Taticchi — CILC 2020
LABELLING-BASED SEMANTICS FOR WAFS
12
a is IN
a is OUT
⟹ a−
= a−
|OUT ∧ ∀b ∈ a−
. wb−|IN
≤𝕊 wb+|IN
⟹ wa−|IN
<𝕊 ⊤
W-ADM

A Labelling Semantics for Weighted Argumentation Frameworks
Carlo Taticchi — CILC 2020
LABELLING-BASED SEMANTICS FOR WAFS
13
W-COM
a is IN
a is OUT
⟺ a−
= a−
|OUT ∧ ∀b ∈ a−
. wb−|IN∪{a} ≤𝕊 wb+|IN∪{a}
⟺ wa−|IN
<𝕊 ⊤

A Labelling Semantics for Weighted Argumentation Frameworks
Carlo Taticchi — CILC 2020
NEXT
CONCLUSION
15
▸ Use other notions of defence
▸ Give explicit conditions for
labelling UNDEC arguments
▸ Differentiate UNDEC with “don’t
care” and “don’t know” labels
▸ Define w-strongly admissible
extension and labelling
▸ Tool: www.dmi.unipg.it/conarg
▸ Labelling Semantics for WAFs:
• w-conflict-free
• w-admissible
• w-complete
• w-stable
• w-preferred
• w-grounded
• w-quasi-strongly admissible
▸ ConArg implementation

A LABELLING SEMANTICS FOR
WEIGHTED ARGUMENTATION
FRAMEWORKS
Stefano Bistarelli and Carlo Taticchi
Thanks for your attention!

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Argumentation Theory provides tools for both modelling and reasoning with controversial information and is a methodology that is often used as a way to give explanations to results provided using machine learning techniques. In this context, labelling-based semantics for Abstract Argumentation Frameworks (AFs) allow for establishing the acceptability of sets of arguments, dividing them into three partitions: acceptable, rejected and undecidable (instead of classical Dung two sets IN and OUT partitions). This kind of semantics have been studied only for classical AFs, whilst the more powerful weighted and preference-based framework has been not studied yet. In this paper, we define a novel labelling semantics for Weighted Argumentation Frameworks, extending and generalising the crisp one.

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1. A LABELLING SEMANTICS FOR WEIGHTED ARGUMENTATION FRAMEWORKS Stefano Bistarelli and Carlo Taticchi

2. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 OVERVIEW ▸ Background: • Extension- and Labelling-Based Semantics • Weighted AFs ▸ Weighted Labelling ▸ Demo ▸ Conclusion 2 ConArg

3. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 EXTENSION-BASED SEMANTICS FOR AFS1 3 1Phan 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.

4. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 EXTENSION-BASED SEMANTICS FOR AFS1 3 1Phan 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. Conflict-Free {a}, {c}, {a,d}, …

5. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 EXTENSION-BASED SEMANTICS FOR AFS1 3 1Phan 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. Conflict-Free {a}, {c}, {a,d}, … Admissible {a, d}, {a, d, e}, …

6. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 EXTENSION-BASED SEMANTICS FOR AFS1 3 1Phan 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. Conflict-Free {a}, {c}, {a,d}, … Admissible {a, d}, {a, d, e}, …

7. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 EXTENSION-BASED SEMANTICS FOR AFS1 3 1Phan 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. Conflict-Free {a}, {c}, {a,d}, … Admissible {a, d}, {a, d, e}, … Complete {a, b, d, e}

8. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 REINSTATEMENT LABELLING2,3 4 IN if it is attacked only by OUT arguments OUT if it is attacked by at least an IN argument UNDEC otherwise 2Martin Caminada. On the Issue of Reinstatement in Argumentation. JELIA 2006: 111-123. 3Hadassa Jakobovits and Dirk Vermeir. Robust Semantics for Argumentation Frameworks. J. Log. Comput., 9(2), 215–261, (1999).

9. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 LABELLING-BASED SEMANTICS 5 a is IN a is not attacked by any IN a is OUT a is attacked by some IN CF

10. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 6 a is IN a is only attacked by OUT a is OUT a is attacked by some IN ADM LABELLING-BASED SEMANTICS

11. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 7 a is IN a is only attacked by OUT a is OUT a is attacked by some IN ⟺ ⟺ COM LABELLING-BASED SEMANTICS

12. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 WEIGHTED ARGUMENTATION FRAMEWORKS (WAFS) 8 ▸ The acceptability of the arguments also depends on the weights ▸ Different notion of defence

13. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 WEIGHTED ARGUMENTATION FRAMEWORKS (WAFS) 8 ▸ The acceptability of the arguments also depends on the weights ▸ Different notion of defence

14. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 WEIGHTED ARGUMENTATION FRAMEWORKS (WAFS) 8 ▸ The acceptability of the arguments also depends on the weights ▸ Different notion of defence

15. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 WEIGHTED ARGUMENTATION FRAMEWORKS (WAFS) 8 ▸ The acceptability of the arguments also depends on the weights ▸ Different notion of defence

16. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 WEIGHTED DEFENCE4 9 4Stefano Bistarelli, Fabio Rossi, Francesco Santini. A Collective Defence Against Grouped Attacks for Weighted Abstract Argumentation Frameworks. FLAIRS Conference 2016: 638-643. ▸ ▸ {a, b, d} and {a, b, e} are w-admissible extensions ▸ {a, b, d, e} is not w-admissible (2+3 < 4+4) D w-defends a ⟺ ∀b attacking a . W(b, D ∪ {a}) ≤ W(D, b)

17. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 WEIGHTED DEFENCE4 9 4Stefano Bistarelli, Fabio Rossi, Francesco Santini. A Collective Defence Against Grouped Attacks for Weighted Abstract Argumentation Frameworks. FLAIRS Conference 2016: 638-643. ▸ ▸ {a, b, d} and {a, b, e} are w-admissible extensions ▸ {a, b, d, e} is not w-admissible (2+3 < 4+4) D w-defends a ⟺ ∀b attacking a . W(b, D ∪ {a}) ≤ W(D, b)

18. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 LABELLING-BASED SEMANTICS FOR WAFS 10 ▸ The w-conflict-free and the conflict-free labelling coincides a is IN a is not attacked by any IN a is OUT a is attacked by some IN CF W-CF

19. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 LABELLING-BASED SEMANTICS FOR WAFS 11 a is IN a is OUT ⟹ a− = a− |OUT ∧ ∀b ∈ a− . wb−|IN ≤𝕊 wb+|IN ⟹ wa−|IN <𝕊 ⊤ W-ADM

20. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 LABELLING-BASED SEMANTICS FOR WAFS 11 a is IN a is OUT ⟹ a− = a− |OUT ∧ ∀b ∈ a− . wb−|IN ≤𝕊 wb+|IN ⟹ wa−|IN <𝕊 ⊤ W-ADM

21. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 LABELLING-BASED SEMANTICS FOR WAFS 11 a is IN a is OUT ⟹ a− = a− |OUT ∧ ∀b ∈ a− . wb−|IN ≤𝕊 wb+|IN ⟹ wa−|IN <𝕊 ⊤ W-ADM

22. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 LABELLING-BASED SEMANTICS FOR WAFS 11 a is IN a is OUT ⟹ a− = a− |OUT ∧ ∀b ∈ a− . wb−|IN ≤𝕊 wb+|IN ⟹ wa−|IN <𝕊 ⊤ W-ADM

23. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 LABELLING-BASED SEMANTICS FOR WAFS 12 a is IN a is OUT ⟹ a− = a− |OUT ∧ ∀b ∈ a− . wb−|IN ≤𝕊 wb+|IN ⟹ wa−|IN <𝕊 ⊤ W-ADM

24. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 LABELLING-BASED SEMANTICS FOR WAFS 13 W-COM a is IN a is OUT ⟺ a− = a− |OUT ∧ ∀b ∈ a− . wb−|IN∪{a} ≤𝕊 wb+|IN∪{a} ⟺ wa−|IN <𝕊 ⊤

25. ConArg www.dmi.unipg.it/conarg

26. A Labelling Semantics for Weighted Argumentation Frameworks Carlo Taticchi — CILC 2020 NEXT CONCLUSION 15 ▸ Use other notions of defence ▸ Give explicit conditions for labelling UNDEC arguments ▸ Differentiate UNDEC with “don’t care” and “don’t know” labels ▸ Define w-strongly admissible extension and labelling ▸ Tool: www.dmi.unipg.it/conarg ▸ Labelling Semantics for WAFs: • w-conflict-free • w-admissible • w-complete • w-stable • w-preferred • w-grounded • w-quasi-strongly admissible ▸ ConArg implementation

27. A LABELLING SEMANTICS FOR WEIGHTED ARGUMENTATION FRAMEWORKS Stefano Bistarelli and Carlo Taticchi Thanks for your attention!

A Labelling Semantics for Weighted Argumentation Frameworks

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A Labelling Semantics for Weighted Argumentation Frameworks