A SCC Recursive Meta-Algorithm for Computing Preferred Labellings in Abstract Argumentation

A SCC Recursive
Meta-Algorithm for Computing
Preferred Labellings in Abstract
Argumentation
Federico Cerutti, Massimiliano Giacomin, Mauro Vallati, Marina Zanella
KR-2014 — Monday 21st July, 2014

Background on Dung’s AF
SCC-Recursiveness
Exploiting the SCC-Recursiveness
Empirical Results
Conclusions

Background
Definition
Given an AF = hA;Ri, with R A A:
- a set S A is conflict–free if @ a; b 2 S s.t. a ! b;
- an argument a 2 A is acceptable with respect to a set S A if 8b 2 A s.t. b ! a, 9 c 2 S s.t. c ! b;
- a set S A is admissible if S is conflict–free and every element of S is
acceptable with respect to S;
- a set S A is a complete extension, i.e. S 2 ECO(), iff S is admissible
and 8a 2 A s.t. a is acceptable w.r.t. S, a 2 S;
- a set S A is the grounded extensions, i.e. S 2 EGR(), iff S is the
minimal (w.r.t. set inclusion) complete set;
- a set S A is a preferred extension, i.e. S 2 EPR(), iff S is a maximal
(w.r.t. set inclusion) complete set.

Background
Definition
Let hA;Ri be an AF: Lab : A7! fin; out; undecg is a complete labelling iff
8a 2 A:
- Lab(a) = in , 8b 2 aLab(b) = out;
- Lab(a) = out , 9b 2 a : Lab(b) = in.
Let S A a conflict–free set: the corresponding labelling is
Ext2Lab(S) Lab, where
- Lab(a) = in , a 2 S
- Lab(a) = out , 9 b 2 S s.t. b ! a
- Lab(a) = undec , a =2 S ^ @ b 2 S s.t. b ! a
Proposition ([Caminada, 2006])
Given an an AF = hA;Ri, Lab is a complete (grounded, preferred)
labelling of if and only if there is a complete (grounded, preferred)
extension S of such that Lab = Ext2Lab(S).

SCC-Recursiveness: Path-Equivalence

SCC-Recursiveness: Partial Order of the SCCs

SCC-Recursiveness: Recursiveness of the
approach

SCC-Recursiveness
Definition
A given argumentation semantics is SCC-recursive if for any
argumentation framework = hA;Ri, E() = GF(;A) 2A. For
any = hA;Ri and for any set C A, E 2 GF(;C) if and only if
- E 2 BF(;C) if jSCCj = 1
- 8S 2 SCC (E S) 2 GF(#Sn(EnS)+;U(S;E) C) otherwise
where
- BF(;C) is a function, called base function, that, given an
argumentation framework = hA;Ri such that jSCCj = 1 and
a set C A, gives a subset of 2A
- U(S;E) = fa 2 S n (E n S)+ j 8b 2 (a n S); b 2 E+g

SCC-Recursiveness and semantics restricted to
a set of arguments
Definition
Given an AF = hA;Ri and a set C A, a set E A is:
- an admissible set of in C if and only if E is an admissible set of
and E C
- a complete extension of in C if and only if E is an admissible
set of in C, and every argument a 2 C which is acceptable with
respect to E belongs to E
- the grounded extension of in C if and only if it is the least
(with respect to set inclusion) complete extension of in C
- a preferred extension of in C if and only if it is a maximal
(with respect to set inclusion) complete extension of in C.

Research Questions
How can we exploit the SCC-Recursiveness in an algorithm?
Is it worth doing it?

Background on Dung’s AF
SCC-Recursiveness
Exploiting the
SCC-Recursiveness
Empirical Results
Conclusions

The Meta-Algorithm
Proposition
Let hA;Ri be an argumentation
framework and let C A a set of
arguments. Considering the grounded
labelling Lab of in C and the set U
including the undec-labelled arguments
according to Lab, it holds that
LPR(;C) = fLab [ E j E 2
LPR(#U;C U)g.

The Meta-Algorithm
I = ff ; gg
O = ;

The Base Case: Complete Labelling in C
Definition
Let = hA;Ri be an argumentation framework and C A be a set of arguments. A
total function Lab : A7! fin; out; undecg is a complete labelling of in C iff it satisfies
the following conditions for any a 2 C:
C: Lab(a) = in , 8b 2 aLab(b) = out;
L2
L1
C: Lab(a) = out , 9b 2 (a C) : Lab(b) = in;
L3
C: Lab(a) = undec , 8b 2 (a C);Lab(b)6= in ^ 9c 2 a :
Lab(c) = undec;
and the following conditions for any a 2 (A n C):
L1
: Lab(a) = out , 9b 2 (a C) : Lab(b) = in;
AnCL2
AnC: Lab(a) = undec , 8b 2 (a C);Lab(b)6= in.
Proposition
Given an an AF = hA;Ri and a set C A, Lab satisfies the above conditions if and
only if there is a complete extension S of in C such that Lab = Ext2Lab(S).

Complete Labelling in C and CNF
^
i21(C)

(Ii _ Oi _ Ui) ^ (:Ii _ :Oi)^(:Ii _ :Ui) ^ (:Oi _ :Ui)

^
^
i21(C)
0
@Ii _
0
@
_
(:Oj )
fjj(j)!(i)g
1
A
1
A ^
^
i21(C)
0
@
^
fjj(j)!(i)g
:Ii _ Oj
1
A^
^
i21(C)
0
B@
^
fj21(C)j(j)!(i)g
:Ij _ Oi
1
CA
^
^
i21(C)
0
B@
:Oi _
0
B@
_
fj21(C)j(j)!(i)g
Ij
1
CA
1
CA
^
^
^
^
i21(C)
fkj(k)!(i)g

Ui _ :Uk _
_

fj21(C)j(j)!(i)g
Ij

^
^
i21(C)
fj21(C)j(j)!(i)g
(:Ui _ :Ij )
^

:Ui _
_
fkj(k)!(i)g
Uk

^
^

:Ii ^ (Oi _ Ui) ^ (:Oi _ :Ui)
i21(AnC)

^
^
i21(AnC)
0
B@
^
fj21(C)j(j)!(i)g
:Ij _ Oi
1
CA
^
i21(AnC)
0
B@:Oi _
0
B@
_
fj21(C)j(j)!(i)g
Ij
1
CA
1
C A^
^
i21(AnC)
0
B @Ui _
0
B@
_
fj21(C)j(j)!(i)g
Ij
1
CA
1
CA
^
^
i21(AnC)
0
B@
^
fj21(C)j(j)!(i)g
:Ui _ :Ij
1
CA

Complete Labelling in C and CNF
Proposition
Let hA;Ri be an argumentation framework and C A be a set of
arguments. If Lab is a complete labelling of in C, then the
assignment V() f(Ii;) j Lab((i)) = ing [ f(Oi;) j Lab((i)) =
outg [ f(Ui;) j Lab((i)) = undecg satisfies the ENCall encoding
shown before. Conversely, if V() is a satisfying assignment of the
ENCall encoding, then the labelling Lab f(a; in) j I1(a) 2
V()g [ f(b; out) j O1(b) 2 V()g [ f(c; undec) j U1(c) 2 V()g is a complete labelling of in C.

Analysis Using the International Planning
Competition (IPC) Score
- For each test case (in our case, each test AF) let T be the best
execution time among the compared systems (if no system
produces the solution within the time limit, the test case is not
considered valid and ignored).
- For each valid case, each system gets a score of
1=(1 + log10(T=T )), where T is its execution time, or a score of 0
if it fails in that case.
- The (non normalized) IPC score for a system is the sum of its
scores over all the valid test cases. The normalised IPC score
ranges from 0 to 100 and is defined as
(IPC=# of valid cases) 100.

The Experiment
- SAT approach optimized ([Cerutti et al., 2014] SAT-P) vs
SCC-Recursiveness using SAT for the base case (SCC-P);
- I1: on s.t. jSCCj = 1, SCC-P performs worse than SAT-P;
- I2: there exists a value such that on where jSCCj ,
SCC-P performs better that SAT-P;
- I3: on s.t. jSCCj , the greater jEPR()j, the more SAT-P
performs worse than SCC-P.

First Hypothesis
790 AFs (), s.t. jSCCj = 1 and A = 25 : 25 : 250
IPC value (normalised) for SCC-P and SAT-P when |SCC| = 1, varying ||
100
90
80
70
60
50
40
0 50 100 150 200 250
||
SCC-P SAT-P

Second Hypothesis
720 AFs varying jSCCj in 5:5:45. Size of SCCs N( = 20 : 5 : 40, = 5);
attacks among SCCs N( = 20 : 5 : 40, = 5)
100
90
80
70
60
50
40
IPC value (normalised) for SCC-P and SAT-P when 5 |SCC| 45
0 10 20 30 40 50
|SCC|
SCC-P SAT-P

Second Hypothesis
720 AFs varying jSCCj in 5:5:45. Size of SCCs N( = 20 : 5 : 40, = 5);
attacks among SCCs N( = 20 : 5 : 40, = 5)
100
90
80
70
60
50
40
IPC value (normalised) for SCC-P and SAT-P when 5 |SCC| 45
Remark
For jSCCj = 35, Md(SCC-P) = 8:81,
Md(SAT-P) = 8:53, z = 0:35, p = 0:73;
0 10 20 30 40 50
|SCC|
SCC-P SAT-P

Third Hypothesis
2800 AFs (as before) s.t. jSCCj = 50 : 5 : 80
Median of times for SCC-P and SAT-P when 50 |SCC| 80 varying |EPR()|
900
800
700
600
500
400
300
200
100
0
0 50 100 150 200 250 300
s
|EPR()|
SCC-P SAT-P

Third Hypothesis
2800 AFs (as before) s.t. jSCCj = 50 : 5 : 80
Remark
Regression to the function f(x) = a x+b:
SCC-P, a = 0:43, b = 31:33;
SAT-P, a = 2:40, b = 87:53
Median of times for SCC-P and SAT-P when 50 |SCC| 80 varying |EPR()|
900
800
700
600
500
400
300
200
100
0
0 50 100 150 200 250 300
s
|EPR()|
SCC-P SAT-P

Conclusions
1. Efficient algorithmic implementation of the SCC-recursiveness
schema
- [Liao et al., 2013] decomposition but not the recursiveness
- Other [Baumann et al., 2012, Dvorák et al., 2012b,
Baroni et al., 2012]
2. Generalisation of [Cerutti et al., 2014] for the computation of
labellings in sub-frameworks
- [Besnard and Doutre, 2004, Dvorák et al., 2012a,
Arieli and Caminada, 2013]. . .
3. Empirical evaluation:
- with jSCCj = 35 statistical evidence that the
SCC-recursive schema reduces the computational effort of
enumerating the preferred labellings;
- Execution time of the SCC-recursive implementation is less
sensible to the number of labellings

Future Works
- Exploiting and comparing different equivalent encodings of
complete labellings in C including the redundant ones
- Different B-PR algorithm for computing labellings in
sub-frameworks
[Egly et al., 2010, Nofal et al., 2014, Dvorák et al., 2014]
- Meta-algorithm to stable and CF2 semantics (directly fit the
SCC-recursive schema [Baroni et al., 2005]) and to semi-stable
and ideal semantics (relationship with preferred semantics)

References I
[Arieli and Caminada, 2013] Arieli, O. and Caminada, M. W. (2013).
A QBF-based formalization of abstract argumentation semantics.
Journal of Applied Logic, 11(2):229–252.
[Baroni et al., 2012] Baroni, P., Boella, G., Cerutti, F., Giacomin, M., van der Torre, L., and Villata,
S. (2012).
On Input/Output Argumentation Frameworks.
In Proceedings of the 4th International Conference on Computational Models of Arguments
(COMMA 2012), pages 358–365.
[Baroni et al., 2005] Baroni, P., Giacomin, M., and Guida, G. (2005).
SCC-recursiveness: a general schema for argumentation semantics.
Artificial Intelligence, 168(1-2):165–210.
[Baumann et al., 2012] Baumann, R., Brewka, G., Dvorák, W., and Woltran, S. (2012).
Parameterized Splitting: A Simple Modification-Based Approach.
In Erdem, E., Lee, J., Lierler, Y., and Pearce, D., editors, Correct Reasoning, volume 7265 of
Lecture Notes in Computer Science, pages 57–71. Springer Berlin Heidelberg.

References II
[Besnard and Doutre, 2004] Besnard, P. and Doutre, S. (2004).
Checking the acceptability of a set of arguments.
In Proceedings of the 10th International Workshop on Non-Monotonic Reasoning (NMR 2004),
pages 59–64.
[Caminada, 2006] Caminada, M. (2006).
On the Issue of Reinstatement in Argumentation.
In Proceedings of the 10th European Conference on Logics in Artificial Intelligence (JELIA
2006), pages 111–123.
[Cerutti et al., 2014] Cerutti, F., Dunne, P. E., Giacomin, M., and Vallati, M. (2014).
Computing Preferred Extensions in Abstract Argumentation: A SAT-Based Approach.
In Black, E., Modgil, S., and Oren, N., editors, TAFA 2013, volume 8306 of Lecture Notes in
Computer Science, pages 176–193. Springer-Verlag Berlin Heidelberg.
[Dvorák et al., 2012a] Dvorák, W., Järvisalo, M., Wallner, J. P., and Woltran, S. (2012a).
Complexity-Sensitive Decision Procedures for Abstract Argumentation.
In Proceedings of 13th International Conference on Principles of Knowledge Representation
and Reasoning (KR 2012), pages 54–64.

References III
[Dvorák et al., 2014] Dvorák, W., Järvisalo, M., Wallner, J. P., and Woltran, S. (2014).
Complexity-sensitive decision procedures for abstract argumentation.
Artificial Intelligence, 206:53–78.
[Dvorák et al., 2012b] Dvorák, W., Pichler, R., and Woltran, S. (2012b).
Towards fixed-parameter tractable algorithms for abstract argumentation.
[Egly et al., 2010] Egly, U., Alice Gaggl, S., and Woltran, S. (2010).
Answer-set programming encodings for argumentation frameworks.
Argument Computation, 1(2):147–177.
[Liao et al., 2013] Liao, B., Lei, L., and Dai, J. (2013).
Computing Preferred Labellings by Exploiting SCCs and Most Sceptically Rejected Arguments.
In Second International Workshop on Theory and Applications of Formal Argumentation
(TAFA-13).
[Nofal et al., 2014] Nofal, S., Atkinson, K., and Dunne, P. E. (2014).
Algorithms for decision problems in argument systems under preferred semantics.

A SCC Recursive Meta-Algorithm for Computing Preferred Labellings in Abstract Argumentation

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A SCC Recursive Meta-Algorithm for Computing Preferred Labellings in Abstract Argumentation