1. A Recursive Approach to Argumentation: Motivation and Perspectives
Pietro Baroni, Massimiliano Giacomin
{baroni,
DEA - Dipartimento di Elettronica per l’Automazione
Università degli Studi di Brescia (Italy)
NMR 2004, Whistler BC, Canada

2. Aim of the work Context: Dung’s abstract argumentation framework
Grounded Semantics
Stable semantics
Preferred Semantics
Argumentation semantics
Problems with odd-length cycles: [Baroni & Giacomin, ECSQARU 03]: Solution based on a recursive definition of extensions
[Baroni & Giacomin, ECAI 04]: General Recursive Schema including Dung’s semantics
In this paper:
Role of the Recursive Schema to support novel proposals: definition of four new semantics overcoming the problems

3. Dung’s Argumentation Framework
AF = <A, >
attack relation
arguments
Semantics
Defeat graph
Extensions
Defeat Status
Justified arguments : belong to all extensions

4. The core of Dung’s theory: Admissibility
Acceptability
 acceptable w.r.t. (“defended by”) S
all defeaters of  are defeated by S  S
Admissibility of a set S
conflict-free
every element acceptable w.r.t. S (defends all of its elements) IF
Complete
extension
contains all acceptable elements w.r.t. itself

5. The Grounded Semantics
Grounded extension:
Least complete extension
(unique status approach)
Most skeptical semantics
Undefeated
Defeated
Provisionally Defeated
Defeat status

6. The Preferred Semantics
Preferred extensions
Maximal complete extensions = max Set:
is conflict-free
defends all of its elements
(multiple status approach)
Defeat status of arguments
UNDEFEATED: belongs to all of the extensions
DEFEATED: out from all of the extensions
PROVISIONALLY DEFEATED: belongs to at least one extension, does not belong to all of the extensions

7. Floating arguments b b g g d d a a b g d Preferred Semantics a VS. b
Grounded Semantics g d a

8. Preferred Semantics: a problematic example involving odd-length cycles
NB: grounded semantics yields all arguments
provisionally defeated VS b d g a f1 f2
[ECSQARU’03]: solution based on a recursive definition along SCCs

9. Generalizing the use of recursion: SCC-based computation
Computation of extensions along Strongly Connected Components S3 S4 S1
SCCs form an acyclic graph
E = S (ES) S2 S5
Possible choices of extension elements in a SCC S only depend on choices made in SCCs that precede S in the graph

10. A General Recursive Schema: “Edge conditions”
SD(E) : nodes attacked by E
SU(E) : nodes defended by E
SP(E) : nodes not attacked nor defended by E S
Choices in preceding components
SD(E)
SP(E)
SU(E)
Possible choices in [SP(E)SU(E)] expressed as a function
FG ( SU(E) )
AF[SP(E)SU(E)]

11. A General Recursive Schema: Recursive Computation
FG (SU(E)) computed recursively
AF[SP(E)SU(E)]
AF[SP(E)SU(E)]
More than one SCC in AF[SP(E)SU(E)] : function computed recursively on each of them
SD(E)
SP(E)
SU(E)
One SCC only in AF[SP(E)SU(E)] : base of the recursion: “base function”
FG* (SU(E))
AF[SP(E)SU(E)]
Semantics specified by a base function defined on single-SCC argumentation frameworks

12. A General Recursive Schema: Definition of SCC-recursive semantics
Given AF = <A, >,
E  A is an extension iff E  FGAF(A)
For all C  A, E  FGAF(C) iff
in case |SCC(AF)|=1 or A= E  FG*AF(C)
otherwise:  S  SCC(AF) (E  S)  FG (SU(E)C)
AF[SP(E)SU(E)]
Definition parametric w.r.t. FG*AF(C): includes complete, grounded and preferred semantics (see ECAI’04)

13. Looking for a new notion of extension: Exploring the space of SCC-Recursiveness
Exploring SCC-recursive semantics  exploring alternative base functions
Preferred extensions = maximal admissible sets
Maintain admissibility  give up maximality
Give up admissibility
NEW NOTION
OF EXTENSION

14. Beyond Preferred Semantics
S2P(E)
S2U(E) b g f1 f2 a S2 S1
{2} 
E admissible  (E  S1) =   1  E
Complete
Extensions
{} if C  A
FG* ({2}) = {}
FG*AF(C) =
AF[S2]
? otherwise

15. Beyond Preferred Semantics: AD1
{} if C  A
FG*AF(C) =
P E AF otherwise
correctly deals with the above example, but… b
A unique (preferred) extension g d
 and  undefeated,  and  defeated a

16. Beyond Preferred Semantics: AD2
g unique defeater of a and f
g is not “fully attacked” (b  E) g f a
{} if C  A
FG*AF(C) =
{E | E is maximal in AS*AF } otherwise
AS*AF = {F  A : F admissible and  g: g F F includes all attackers of g}

17. Beyond Preferred Semantics: adherence to Dung’s framework
AD2 correctly deals with all the examples considered so far (e.g. floating arguments)
AD1 fails in one example
AD1 and AD2 extensions are complete extensions  adhere to Dung’s fundamental notion of admissibility
Other alternative: relax the admissibility constraint…

18. b g a b b b g g g a a a Beyond Admissibility  one node at most
Non empty extensions
NO CONTRADICTIONS
 one node at most
SYMMETRY
 all nodes treated equally b g a b b b g g g a a a
Maximal conflict-free sets

19. Beyond Admissibilty: CF1 and CF2 semantics
FG*AF(C) = Maximal Conflict-Free Sets of AFC S
SD(E)
SP(E)
SU(E)
CF2
FG*AF(C) = Maximal Conflict-Free Sets of AF
SP(E) U SU(E)
Role of “defence” completely ignored S
SD(E)

20. CF1 and CF2 semantics: an example
First alternative for (ES1)
S2D(E)
S2U(E) b g f1 f2 a S2 S1

21. CF1 and CF2 semantics: an example (2)
Second alternative for (ES1) b g f1 f2 a S1
S2=S2U(E) b g f1 f2 a

22. CF1 and CF2 semantics: an example (3)
Third alternative for (ES1)
S2P(E)
S2U(E) b g f1 f2 a S1 S2 b
S2P(E)
S2U(E)
CF2
only! g f1 f2 a

23. Comparing SCC-Recursive Semantics
Any SCC-Recursive semantics agrees with grounded semantics: each extension includes the grounded extension (provided a simple condition on the base function is verified)
AD vs. CF: different treatment of odd-length cycles b b g d f g d f a a b
Empty set is the only admissible set
CF1 and CF2: enforce floating defeat and acceptance g d f a

24. Comparing SCC-Recursive Semantics: Self-defeating arguments
a self-defeating: should not prevent b from being justified
{b} should be the unique extension a b
Any admissibility-based semantics fails (and CF1 too)
Treatment by CF2: a b
Max C.F.Set: 
S2=S2P(E) Max C.F.Set:
{b}

25. Conclusions A general recursive schema:
SCC-recursiveness as a unifying notion
SCC-recursiveness as a “source” of novel proposals
Overcoming problems of preferred semantics
Most satisfactory approach: CF2, which fully departs from the notion of admissibility
Future work:
General properties of the SCC-recursive schema
Further exploration and comparison between semantics
Examples in different application domains
[Incremental] algorithms based on the SCC-recursive schema