The document presents a formalization of Abstract Argumentation Frameworks with recursive attacks (AFRA) as a generalization of Dung's Argumentation Frameworks that allows attacks to attacks recursively. It introduces AFRA through an example, provides the formal definition, discusses the relationship to Dung frameworks, compares to other approaches that also model recursive attacks, and outlines future work in further developing the theoretical foundations of AFRA and argumentation semantics within this framework.

1. University of Brescia
Dipartimento di Elettronica per l'Automazione
Knowledge Engineering and Human-Computer Interaction Research Group
© 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Encompassing Attacks toEncompassing Attacks to
Attacks in AbstractAttacks in Abstract
Argumentation FrameworksArgumentation Frameworks
Pietro Baroni, Federico Cerutti, Massimiliano Giacomin and Giovanni GuidaPietro Baroni, Federico Cerutti, Massimiliano Giacomin and Giovanni Guida

2. Slide 2 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Summary
 Introduction by example
 Formalization
 Relationship with Dung's Argumentation Framework
 Comparison with other approaches
 Conclusion and future works

3. © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Introduction by exampleIntroduction by example
➢ Formalization
➢ Relationship with Dung's Argumentation Framework
➢ Comparison with other approaches
➢ Conclusion and future works

4. Slide 4 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Example (1)
 Simple decision support problem
 Bob's holidays plan
 Two possible choices:
 Gstaad
 Cuba
 Preferences among the possible choices
 Exception to the preference's general rule

5. Slide 5 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Example (2)
There is a last minute offer
for Gstaad →
I should go to Gstaad
GG
If I go to Cuba, I can not
go to Gstaad
There is a last minute offer
for Cuba →
I should go to Cuba
CC

6. Slide 6 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Example (3)
GG
PP
CC
When it is possible, I prefer
to go to a ski resort
According to this preference,
the possible choice for Cuba can
not attack the one for Gstaad

7. Slide 7 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Example (4)
GG
PP
CC
The preference about
skiing does not apply
in the current situation
NN
Since there were no snowfalls in
Gstaad since a month, it is not
possible to go to ski in Gstaad.

8. Slide 8 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Example (5)
GG
PP
CC
AA NN
It is possible to ski in Gstaad
thanks to artificial snow

9. © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
FormalizationFormalization
➢ Introduction by example
➢ Relationship with Dung's Argumentation Framework
➢ Comparison with other approaches
➢ Conclusion and future works

10. Slide 10 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Argumentation Framework with
Recoursive Attacks (AFRA)

11. Slide 11 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Formalization of the example (1)
GG
PP
CC
AA NN
d
a
b
g
e

12. Slide 12 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Defeat relation

13. Slide 13 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Formalization of the example (2)
GG
PP
CC
AA NN
d
a
b
g
e

14. Slide 14 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Admissibility

15. Slide 15 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Fundamental lemma and
preferred extension

16. Slide 16 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
GG
PP
CC
AA NN
d
a
b
g
e
Formalization of the example (3)

17. © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Relationship with Dung's Relationship with Dung's
Argumentation FrameworkArgumentation Framework
➢ Introduction by example
➢ Formalization
➢ Comparison with other approaches
➢ Conclusion and future works

18. Slide 18 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Correspondence between AFRA
and AF (1)
AA
BB
aa
bb
CC
AA
BBbb
aa
CC
gg
gg

19. Slide 19 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Correspondence in the example
GG
ee
PP
CC
AA NN
gg
dd
aa
bb
GG
PP
CC
AA NN
d
a
b
g
e

20. Slide 20 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Correspondence between AFRA
and AF (2)

21. Slide 21 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
GG
ee
PP
CC
AA NN
gg
dd
aa
bb
Correspondence in the example:
The Preferred Extension
GG
PP
CC
AA NN
d
a
b
g
e

22. © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Comparison with other Comparison with other
approachesapproaches
➢ Introduction by example
➢ Formalization
➢ Relationship with Dung's Argumentation Framework
➢ Conclusion and future works

23. Slide 23 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Argumentation attack model
 Modelling the strength of nodes and attacks with
propagation
 Argumentation attack model [Barringer et al, 2005]
 Recursive definition of “torpedoes”, or attacks,
similarly as AFRA attacks
 Focus on attack network and its numerical valuation
 Dung style semantics issues not considered

24. Slide 24 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
The EAF formalism
 Reasoning about preferences/values can be
formalized through attacks to attacks
 Extended Argumentation Framework [Modgil,
2007;2009]
 Specific assumptions:
 A limited level of recursion
 A constraint on some attacks to be symmetric (when the
involved arguments represent conflicting preferences)
 In the paper we have considered a possible extension
of EAF aimed at overcoming these restrictions

25. Slide 25 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Higher-order AF
 Reasoning about coalitions
 Model of attacks to attacks as an instance of Dung's
Argumentation Framework
 Proposal of a “second order argumentation
framework” with some constraints like EAF
 Proposal of a “higher order argumentation
framework” without such constraints [Boella et al,
2009]

26. © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Conclusion and future Conclusion and future
worksworks
➢ Introduction by example
➢ Formalization
➢ Relationship with Dung's Argumentation Framework
➢ Comparison with other approaches

27. Slide 27 © 2009 Federico Cerutti <federico.cerutti@ing.unibs.it>
Conclusion and future works
 Preliminary investigation about AFRA
 Generalization of Dung's Argumentation Framework
 Attacks to attacks recursively encompassed without
restriction
 Focus on decision support context
 Future works:
 Enlarging the theoretical bases of AFRA
 Investigating the definition of argumentation semantics in
this context