We study invariant local expansion operators for conflict-free and admissible sets in Abstract Argumentation Frameworks (AFs). Such operators are directly applied on AFs, and are invariant with respect to a chosen “semantics” (that is w.r.t. each of the conflict free/admissible set of arguments). Accordingly, we derive a definition of robustness for AFs in terms of the number of times such operators can be applied without producing any change in the chosen semantics.

1. University of Perugia
Dep. of Mathematics and Computer Science
Master Degree in Computer Science
Looking for Invariant Operators
in Argumentation
– Ricerca di Operatori Invarianti in Argumentation –
Advisor Candidate
Prof. Stefano Bistarelli Carlo Taticchi
Academic Year 2016-2017

6. Index
1. Background
2. Robustness
3. Invariant Operators
4. Tool
5. Conclusion
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Background September 21, 2017

7. Argumentation Frameworks1
A human-like fashion to deal with knowledge
Definition (AF)
An Abstract Argumentation Framework is a pair G = 〈A, R〉 where
A is a set of arguments and R is a binary relation on A.
1PHAN MINH DUNG. On the Acceptability of Arguments and its Fundamental Role in Nonmonotonic Reasoning, Logic
Programming and n-Person Games. Artif. Intell., 77(2):321–358, 1995.
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Background September 21, 2017

8. Sets of Extensions
Definition (Conflict-free extensions)
Let G = 〈A, R〉 be an AF. A set E ⊆ A is conflict-free in G if there
are no a, b ∈ A | (a, b) ∈ R.
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Background September 21, 2017

9. Sets of Extensions
Definition (Conflict-free extensions)
Let G = 〈A, R〉 be an AF. A set E ⊆ A is conflict-free in G if there
are no a, b ∈ A | (a, b) ∈ R.
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Background September 21, 2017

10. Dung’s Semantics
Definition (Admissible Semantics)
Let G = 〈A, R〉 be an AF. A set E ⊆ A is admissible in G if E is
conflict-free and each a ∈ E is defended by E.
Similar definitions for complete, stable, preferred, and grounded.
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Background September 21, 2017

11. Dung’s Semantics
Definition (Admissible Semantics)
Let G = 〈A, R〉 be an AF. A set E ⊆ A is admissible in G if E is
conflict-free and each a ∈ E is defended by E.
Similar definitions for complete, stable, preferred, and grounded.
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Background September 21, 2017

12. Partial Order Between Semantics
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Background September 21, 2017

13. Reinstatement Labelling2
Every argument can be labelled in, out or undec.
Example (Labelling of an AF)
2MARTIN CAMINADA. On the Issue of Reinstatement in Argumentation. Logics in Artificial Intelligence: 10th European
Conference, JELIA 2006 Liverpool, UK, September 13-15, 2006 Proceedings, pages 111–123, 2006.
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Background September 21, 2017

14. Index
1. Background
2. Robustness
3. Invariant Operators
4. Tool
5. Conclusion
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Robustness September 21, 2017

15. Robustness3,4
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?
3S. BISTARELLI, F. FALOCI, F. SANTINI, AND C. TATICCHI. Robustness in abstract argumentation frameworks. In Proceedings
of the 29th International Florida Artificial Intelligence Research Society Conf. FLAIRS, page 703, 2016.
4CARLO TATICCHI. A Study of Robustness in Abstract Argumentation Frameworks. Proceedings of the Doctoral Consortium
of AI*IA 2016, Genova, Italy, November 29, 2016, pages 11–16. CEUR-WS.org, 2016.
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Robustness September 21, 2017

16. Comparing AFs
We have defined a partial order on the set n of all AFs with n
arguments and all possible combinations of attack relations.
Definition (AFs inclusion w.r.t. attacks)
Let G1 = 〈A, R1〉 and G2 = 〈A, R2〉 be two AFs. We say that G1 is
included in G2 w.r.t. attacks if R1 ⊆ R2 and we write G1 ≤A G2.
G1 G2
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Robustness September 21, 2017

17. Lattice of AFs
Obtained as ( n
, ≤A
)
The highlighted AFs are such that G1 ≤A
1
G2.
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Robustness September 21, 2017

18. Unmanageable Numbers
Lattice ( n
, ≤A
)
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Robustness September 21, 2017

19. Unmanageable Numbers
Lattice ( n
, ≤A
)
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Robustness September 21, 2017

20. Reducing AFs
A first improvement
We only consider AFs which:
• do not have self attacks and
• are not disconnected.
Example (Unsuitable AF)
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Robustness September 21, 2017

21. Reducing AFs (Cont’d)
Isomorphic graphs
Theorem
Let G1, G2 ∈ n be two AFs. If G1 and G2 are isomorphic, then the
semantics S(G1) and S(G2) are isomorphic in turn.
G1 G2 G3
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Robustness September 21, 2017

22. Isomorphic Lattices
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Robustness September 21, 2017

23. Isomorphic Lattices
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Robustness September 21, 2017

24. Isomorphic Lattices
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Robustness September 21, 2017

25. Isomorphic Lattices
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Robustness September 21, 2017

26. Isomorphic Lattices
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Robustness September 21, 2017

27. Isomorphic Lattices
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Robustness September 21, 2017

28. Isomorphic Lattices
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Robustness September 21, 2017

29. Isomorphic Lattices
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Robustness September 21, 2017

30. Isomorphic Lattices
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Robustness September 21, 2017

31. Isomorphic Lattices
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Robustness September 21, 2017

32. Some Numbers
A 99% reduction
AFs
before after
n
2 16 2
3 512 13
4 65.536 199
5 33.554.432 10.072
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Robustness September 21, 2017

33. Index
1. Background
2. Robustness
3. Invariant Operators
4. Tool
5. Conclusion
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Invariant Operators September 21, 2017

34. Modifying Operators
Allow to construct AFs adding an attack relations at time.
Definition (Modifying operator)
Let G = 〈A, R〉 ∈ n be an AF. A modifying operator is a function
m : n → n such that m(G) = 〈A, m(R)〉, where m(R) ⊇ R.
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Invariant Operators September 21, 2017

35. Modifying Operators
Allow to construct AFs adding an attack relations at time.
Definition (Modifying operator)
Let G = 〈A, R〉 ∈ n be an AF. A modifying operator is a function
m : n → n such that m(G) = 〈A, m(R)〉, where m(R) ⊇ R.
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Invariant Operators September 21, 2017

36. Modifying Operators
Allow to construct AFs adding an attack relations at time.
Definition (Modifying operator)
Let G = 〈A, R〉 ∈ n be an AF. A modifying operator is a function
m : n → n such that m(G) = 〈A, m(R)〉, where m(R) ⊇ R.
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Invariant Operators September 21, 2017

37. Comparing Semantics
Definition (Semantics inclusion)
Let S and S be two sets of extensions. We say that S ⊆ S if and
only if ∀E ∈ S ∃E ∈ S | E ⊆ E .
Sadm(G1) = {{}, {1}, {1, 3}} Sadm(G2) = {{}, {1}}
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Invariant Operators September 21, 2017

38. Invariant Operators for Conflict-Free Extensions
Example (AF with 4 arguments)
Scf = {{}, {1}, {2}, {3}, {4}, {1, 4}, {2, 3}, {3, 4}}
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Invariant Operators September 21, 2017

39. Invariant Operators for Conflict-Free Extensions
Example (AF with 4 arguments)
Scf = {{}, {1}, {2}, {3}, {4}, {1, 4}, {2, 3}, {3, 4}}
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Invariant Operators September 21, 2017

40. Invariant Operators for Conflict-Free Extensions
Example (AF with 4 arguments)
Scf = {{}, {1}, {2}, {3}, {4}, {1, 4}, {2, 3}, {3, 4}}
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Invariant Operators September 21, 2017

41. Invariant Operators for Conflict-Free Extensions
Example (AF with 4 arguments)
Scf = {{}, {1}, {2}, {3}, {4}, {1, 4}, {2, 3}, {3, 4}}
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Invariant Operators September 21, 2017

42. Invariant Operators for Conflict-Free Extensions
Theorem
Every modifying operator m is nonincreasing w.r.t. the conflict-free
semantics.
Theorem (Invariant conflict-free)
Let G = 〈A, R〉 ∈ n be an AF. We have G ≡cf h(G) if and only if
∀(a, b) ∈ h(R):
• (a, b) ∨ (b, a) ∈ R or
• a ∨ b ∈ A INcf (G).
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Invariant Operators September 21, 2017

43. Invariant Operators for Admissible Semantics
Example (AF with 4 arguments)
Sadm = {{}, {1}, {4}, {1, 4}}
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Invariant Operators September 21, 2017

44. Invariant Operators for Admissible Semantics
Example (AF with 4 arguments)
Sadm = {{}, {1}, {4}, {1, 4}}
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Invariant Operators September 21, 2017

45. Invariant Operators for Admissible Semantics
Example (AF with 4 arguments)
Sadm = {{}, {1}, {4}, {1, 4}}
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Invariant Operators September 21, 2017

46. Invariant Operators for Admissible Semantics
Example (AF with 4 arguments)
Sadm = {{}, {1}, {4}, {1, 4}}
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Invariant Operators September 21, 2017

47. Invariant Operators for Admissible Semantics
Example (AF with 4 arguments)
Sadm = {{}, {1}, {4}, {1, 4}}
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Invariant Operators September 21, 2017

48. Invariant Operators for Admissible Semantics
a → b Condition
in → in never possible
in → out c ∈ in( ) | (b, c) ∈ R
in → undec c ∈ undec( ) | (c, c) /∈ R and (b, c) ∈ R
out →
in there is no odd length sequence of attacks from b to a
out or ∃c = b | there is an odd length sequence of attacks
undec from c to a, but not from a to c
undec → in never possible
undec → out no condition required
undec → undec no condition required
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Invariant Operators September 21, 2017

49. Index
1. Background
2. Robustness
3. Invariant Operators
4. Tool
5. Conclusion
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Tool September 21, 2017

50. Rob5
A tool for studying robustness
5S. BISTARELLI, F. FALOCI, F. SANTINI, AND C. TATICCHI. A visual tool for studying robustness in abstract argumentation
framework (Demo). Proceedings of the 31st Italian Conference on Computational Logic, Genova, Italy, June 20-22, 2016.
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Tool September 21, 2017

51. Index
1. Background
2. Robustness
3. Invariant Operators
4. Tool
5. Conclusion
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Conclusion September 21, 2017

52. Summing Up
This work is a prosecution of studies in dynamic argumentation6,7.
Our main results have been:
• obtaining a partial order among non isomorphic AFs,
• defining invariant operators for the conflict-free and the
admissible semantics.
6G. BOELLA, S. KACI, AND L. W. N. VAN DER TORRE. Dynamics in Argumentation with Single Extensions: Abstraction Princi-
ples and the Grounded Extension. In Proceedings of 5590 of Lecture Notes in Computer Science, 107–118. Springer, 2009.
7T. RIENSTRA, C. SAKAMA, AND L. W. N. VAN DER TORRE. Persistence and Monotony Properties of Argumentation Seman-
tics. In Theory and Applications of Formal Argumentation - Third International Workshop, TAFA 2015, pages 211–225.
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Conclusion September 21, 2017

53. Further Research
Next possible steps:
• defining invariant operators for the remaining semantics,
• implementing semantics inclusion in Rob,
• focusing on particular classes of graphs8,9 (e.g. bipartite),
• using concurrency10 to manage the dynamic evolution of AFs,
• ...
8http://www.graphclasses.org/classes.cgi
9http://mathworld.wolfram.com/topics/SimpleGraphs.html
10V. A. SARASWAT AND M. C. RINARD. Concurrent Constraint Programming. In Conference Record of the 17th Annual ACM
Symposium on Principles of Programming Languages, pages 232–245. ACM Press, 1990.
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Conclusion September 21, 2017

54. Further Research (Cont’d)
• ...
• bringing the notion of robustness in Belief Revision11 and
Defeasible Reasoning12,
• extending the concept of robustness and semantics inclusion
by considering coalitions of arguments,
• exploiting AFs to address typical cybersecurity13 problems of
systems surveillance.
11M. O. MOGUILLANSKY, N. D. ROTSTEIN, M. A. FALAPPA, A. J. GARCIA, AND G. R. SIMARI. Argument Theory Change Applied
to Defeasible Logic Programming. In Proceedings of the 23rd AAAI Conference on Artificial Intelligence, 2008, pages 132–137.
12J. L. POLLOCK. Defeasible Reasoning. In Cognitive Science, Volume 11, Issue 4, 1987, Pages 481-518, ISSN 0364-0213.
13S. BISTARELLI, F. ROSSI, F. SANTINI, AND C. TATICCHI. Towards visualising security with arguments. In Proceedings of the
30th Italian Conference on Computational Logic, 1459 of CEUR Workshop Proceedings, pages 197–201.
Carlo Taticchi (UniPG) Looking for Invariant Operators in Argumentation - Conclusion September 21, 2017

55. University of Perugia
Dep. of Mathematics and Computer Science
Master Degree in Computer Science
Thanks for your attention!
Questions?
go to Rob
Advisor Candidate
Prof. Stefano Bistarelli Carlo Taticchi
Academic Year 2016-2017