Argumentation Theory allows for reasoning with uncertain and controversial information, and provides tools for deciding which arguments (for instance, of a debate) can be accepted together. The strength of an argument and its attacks can be expressed through weighted argumentation frameworks; in this case, the selection criteria, called semantics, used to identify the sets of acceptable arguments, need to take into account the information given by the weights. In this paper, we conduct an initial study on a novel labelling semantics for weighted argumentation frameworks, extending and generalising the crisp one.

1. PRELIMINARY STUDY ON
REINSTATEMENT LABELLING FOR
WEIGHTED ARGUMENTATION FRAMEWORKS
Stefano Bistarelli and Carlo Taticchi

2. Preliminary Study on Reinstatement Labelling for WAFsCarlo Taticchi — November 19, 2019
OVERVIEW
▸ Extension-Based Semantics
▸ Reinstatement Labelling
▸ Weighted AFs
▸ Weighted Labelling
▸ Conclusion
2
ConArg

3. Preliminary Study on Reinstatement Labelling for WAFsCarlo Taticchi — November 19, 2019
EXTENSION-BASED SEMANTICS1
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.
AdmissibleConflict-Free Complete
{a}, {c}, {d}, … {a, d}, {a, d, e}, … {a, b, c, d}

4. Preliminary Study on Reinstatement Labelling for WAFsCarlo Taticchi — November 19, 2019
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).
Correspondence
between
extension and
labelling

5. Preliminary Study on Reinstatement Labelling for WAFsCarlo Taticchi — November 19, 2019
LABELLING-BASED SEMANTICS
5
a is IN a is not attacked by any IN
a is OUT a is attacked by some IN
CF

6. Preliminary Study on Reinstatement Labelling for WAFsCarlo Taticchi — November 19, 2019 6
a is IN a is only attacked by OUT
a is OUT a is attacked by some IN
ADM
LABELLING-BASED SEMANTICS

7. Preliminary Study on Reinstatement Labelling for WAFsCarlo Taticchi — November 19, 2019 7
a is IN a is only attacked by OUT
a is OUT a is attacked by some IN
⟺
⟺
COM
LABELLING-BASED SEMANTICS

8. Preliminary Study on Reinstatement Labelling for WAFsCarlo Taticchi — November 19, 2019
WEIGHTED ARGUMENTATION FRAMEWORKS (WAFS)
8
▸ The acceptability of the arguments also depends on the weights
▸ Different notion of defence

9. Preliminary Study on Reinstatement Labelling for WAFsCarlo Taticchi — November 19, 2019
SEMIRING-BASED WAFS3
9
▸
▸
▸
▸
WAF 𝕊 : ⟨𝒜, R, W, 𝕊⟩
𝕊 : ⟨S, ⊕ , ⊗ , ⊥ , ⊤ ⟩
𝕊weighted = ⟨ℝ+
∪ {+∞}, min, + , + ∞,0⟩
W(ℬ, 𝒟) =
⨂
b∈ℬ,d∈𝒟
W(b, d)
2Stefano Bistarelli, Francesco Santini. A Common Computational Framework for Semiring-based Argumentation Systems. ECAI 2010: 131-136.

10. Preliminary Study on Reinstatement Labelling for WAFsCarlo Taticchi — November 19, 2019
WEIGHTED DEFENCE4
10
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 the w-complete extensions
▸ {a, b, d, e} is not w-complete (2+3 < 4+4)

11. Preliminary Study on Reinstatement Labelling for WAFsCarlo Taticchi — November 19, 2019
LABELLING-BASED SEMANTICS FOR WAFS
11
▸ 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
CFW-CF

12. Preliminary Study on Reinstatement Labelling for WAFsCarlo Taticchi — November 19, 2019
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

13. Preliminary Study on Reinstatement Labelling for WAFsCarlo Taticchi — November 19, 2019
LABELLING-BASED SEMANTICS FOR WAFS
13
a is IN
a is OUT
⟹ a−
= a−
|OUT ∧ ∀b ∈ a−
. wb−|IN
≤ 𝕊 wb+|IN
⟹ wa−|IN
< 𝕊 ⊤
W-ADM

14. Preliminary Study on Reinstatement Labelling for WAFsCarlo Taticchi — November 19, 2019
LABELLING-BASED SEMANTICS FOR WAFS
14
W-COM
a is IN
a is OUT
⟺ a−
= a−
|OUT ∧ ∀b ∈ a−
. wb−|IN∪{a} ≤ 𝕊 wb+|IN∪{a}
⟺ wa−|IN
< 𝕊 ⊤

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

16. Preliminary Study on Reinstatement Labelling for WAFsCarlo Taticchi — November 19, 2019
CONCLUSION
16
▸ Tool available at www.dmi.unipg.it/conarg
▸ Preliminary work on Labelling
Semantics for WAFs
• Conflict-free
• Admissible
• Complete
• Preferred
• Stable

17. Preliminary Study on Reinstatement Labelling for WAFsCarlo Taticchi — November 19, 2019
NEXTCONCLUSION
17
▸ Weighted Labelling for
• (Quasi-)Strongly-
Admissible
• Grounded
▸ Tool available at www.dmi.unipg.it/conarg
▸ Preliminary work on Labelling
Semantics for WAFs
• Conflict-free
• Admissible
• Complete
• Preferred
• Stable

18. PRELIMINARY STUDY ON
REINSTATEMENT LABELLING FOR
WEIGHTED ARGUMENTATION FRAMEWORKS
Stefano Bistarelli and Carlo Taticchi
Thanks for your attention!