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A SAT-based Approach forComputing Extensions inAbstract Argumentation
Federico Cerutti, Paul E. Dunne, Massimiliano Giacomin, Mauro Vallati
TAFA-2013Sunday 4th August, 2013
c© 2013 Federico Cerutti <[email protected]>
Summary
Background in abstract argumentationSAT encodings of complete labellings with interesting theoreticalpropertiesAn algorithm exploiting SAT solvers for enumerating preferredextensionsEmpirical evaluation of the algorithm
Background
Definition
Given an AF Γ = 〈A,R〉:a set S ⊆ A is conflict–free if @ a,b ∈ S s.t. a→ b;an argument a ∈ A is acceptable with respect to a set S ⊆ A if∀b ∈ A s.t. b→ a, ∃ c ∈ S s.t. c→ b;a set S ⊆ A is admissible if S is conflict–free and every element ofS is acceptable with respect to S;a set S ⊆ A is a complete extension, i.e. S ∈ ECO(Γ), iff S isadmissible and ∀a ∈ A s.t. a is acceptable w.r.t. S, a ∈ S;a set S ⊆ A is a preferred extension, i.e. S ∈ EPR(Γ), iff S is amaximal (w.r.t. set inclusion) admissible set.
N.B.: EPR(Γ) ⊆ ECO(Γ)
Background
Definition
Let 〈A,R〉 be an argumentation framework. A total functionLab : A 7→ {in, out, undec} is a complete labelling iff it satisfies thefollowing conditions for any a ∈ A:Lab(a) = in⇔ ∀b ∈ a−Lab(b) = out;Lab(a) = out⇔ ∃b ∈ a− : Lab(b) = in;Lab(a) = undec⇔ ∀b ∈ a−Lab(b) 6= in ∧ ∃c ∈ a− : Lab(c) =undec;
From [Caminada, 2006], preferred extensions are in one-to-onecorrespondence with those complete labellings maximizing the set ofarguments labelled in.
An Approach for Expressing the CompleteLabelling as a SAT Problem
Given an AF Γ = 〈A,R〉, ΠΓ is a boolean formula (complete labellingformula) such that each satisfying assignment of the formulacorresponds to a complete labelling:
k = |A|φ : {1, . . . , k} 7→ A is a bijection (the inverse map is φ−1)For each argument φ(i) we define three boolean variables:
Ii, which is true when argument φ(i) is labelled in, false otherwise;Oi, which is true when argument φ(i) is labelled out, falseotherwise;Ui, which is true when argument φ(i) is labelled undec, falseotherwise;
V(Γ) , ∪1≤i≤|A|{Ii, Oi, Ui} (set of variables for the AF Γ)
SAT Encoding of Complete Labelling: C1
Lab is a total function;If a is not attacked, Lab(a) = in;Lab(a) = in⇔ ∀b ∈ a−Lab(b) = out;Lab(a) = out⇔ ∃b ∈ a− : Lab(b) = in;Lab(a) = undec⇔ ∀b ∈ a−Lab(b) 6= in ∧ ∃c ∈ a− : Lab(c) =undec.
SAT Encoding of Complete Labelling: C1
∧i∈{1,...,k}
((Ii ∨ Oi ∨ Ui) ∧ (¬Ii ∨ ¬Oi)∧(¬Ii ∨ ¬Ui) ∧ (¬Oi ∨ ¬Ui)
)∧
∧{i|φ(i)−=∅}
(Ii ∧ ¬Oi ∧ ¬Ui) ∧∧
{i|φ(i)−6=∅}
Ii ∨
∨{j|φ(j)→φ(i)}
(¬Oj)
∧∧
{i|φ(i)−6=∅}
∧{j|φ(j)→φ(i)}
¬Ii ∨ Oj
∧∧
{i|φ(i)−6=∅}
∧{j|φ(j)→φ(i)}
¬Ij ∨ Oi
∧
∧{i|φ(i)−6=∅}
¬Oi ∨ ∨{j|φ(j)→φ(i)}
Ij
∧
∧{i|φ(i)−6=∅}
∧{k|φ(k)→φ(i)}
Ui ∨ ¬Uk ∨
∨{j|φ(j)→φ(i)}
Ij
∧
∧{i|φ(i)−6=∅}
∧{j|φ(j)→φ(i)}
(¬Ui ∨ ¬Ij)
∧¬Ui ∨
∨{j|φ(j)→φ(i)}
Uj
∧
SAT Encoding of Complete Labelling: Ca1
Lab is a total function;If a is not attacked, Lab(a) = in;Lab(a) = in⇔ ∀b ∈ a−Lab(b) = out;Lab(a) = out⇔ ∃b ∈ a− : Lab(b) = in;Lab(a) = undec⇔ ∀b ∈ a−Lab(b) 6= in ∧ ∃c ∈ a− : Lab(c) =undec.
SAT Encoding of Complete Labelling: Ca1
∧i∈{1,...,k}
((Ii ∨ Oi ∨ Ui) ∧ (¬Ii ∨ ¬Oi)∧(¬Ii ∨ ¬Ui) ∧ (¬Oi ∨ ¬Ui)
)∧
∧{i|φ(i)−=∅}
(Ii ∧ ¬Oi ∧ ¬Ui) ∧∧
{i|φ(i)−6=∅}
Ii ∨
∨{j|φ(j)→φ(i)}
(¬Oj)
∧∧
{i|φ(i)−6=∅}
∧{j|φ(j)→φ(i)}
¬Ii ∨ Oj
∧∧
{i|φ(i)−6=∅}
∧{j|φ(j)→φ(i)}
¬Ij ∨ Oi
∧
∧{i|φ(i)−6=∅}
¬Oi ∨ ∨{j|φ(j)→φ(i)}
Ij
∧
(((((((((((((((((((((((hhhhhhhhhhhhhhhhhhhhhhh
∧{i|φ(i)−6=∅}
∧{k|φ(k)→φ(i)}
Ui ∨ ¬Uk ∨
∨{j|φ(j)→φ(i)}
Ij
∧
((((((((((((((((((((((((((((hhhhhhhhhhhhhhhhhhhhhhhhhhhh
∧{i|φ(i)−6=∅}
∧{j|φ(j)→φ(i)}
(¬Ui ∨ ¬Ij)
∧¬Ui ∨
∨{j|φ(j)→φ(i)}
Uj
∧
SAT Encoding of Complete Labelling: Cb1
Lab is a total function;If a is not attacked, Lab(a) = in;Lab(a) = in⇔ ∀b ∈ a−Lab(b) = out;Lab(a) = out⇔ ∃b ∈ a− : Lab(b) = in;Lab(a) = undec⇔ ∀b ∈ a−Lab(b) 6= in ∧ ∃c ∈ a− : Lab(c) =undec.
SAT Encoding of Complete Labelling: Cc1
Lab is a total function;If a is not attacked, Lab(a) = in;Lab(a) = in⇔ ∀b ∈ a−Lab(b) = out;Lab(a) = out⇔ ∃b ∈ a− : Lab(b) = in;Lab(a) = undec⇔ ∀b ∈ a−Lab(b) 6= in ∧ ∃c ∈ a− : Lab(c) =undec.
SAT Encoding of Complete Labelling: C2
Lab is a total function;If a is not attacked, Lab(a) = in;Lab(a) = in ⇒ ∀b ∈ a−Lab(b) = out;Lab(a) = out ⇒ ∃b ∈ a− : Lab(b) = in;Lab(a) = undec ⇒∀b ∈ a−Lab(b) 6= in ∧ ∃c ∈ a− : Lab(c) = undec.
SAT Encoding of Complete Labelling: C3
Lab is a total function;If a is not attacked, Lab(a) = in;Lab(a) = in ⇐ ∀b ∈ a−Lab(b) = out;Lab(a) = out ⇐ ∃b ∈ a− : Lab(b) = in;Lab(a) = undec ⇐∀b ∈ a−Lab(b) 6= in ∧ ∃c ∈ a− : Lab(c) = undec.
Equivalence of the Encodings
PropositionThe encodings C1, C
a1 , C
b1, C
c1, C2, C3 are equivalent.
Let us note that Ca1 and C2 correspond to the alternative definitions
of complete labellings in [Caminada and Gabbay, 2009], where a proofof their equivalence is provided.
Exploiting SAT Solvers for EnumeratingPreferred Extensions
Algorithm 1 Enumerating the preferred extensions of an AF
1: Input: Γ = 〈A,R〉2: Output: Ep ⊆ 2A
3: Ep := ∅4: cnf := ΠΓ
5: repeat
6: prefcand := ∅7: cnfdf := cnf
8: repeat
9: lastcompfound := SS(cnfdf)10: if lastcompfound ! = ε then
11: prefcand := lastcompfound
12: for a ∈ INARGS(lastcompfound) do
13: cnfdf := cnfdf ∧ Iφ−1(a)
14: end for
15: remaining := FALSE
16: for a ∈ A \ INARGS(lastcompfound) do
17: remaining := remaining ∨ Iφ−1(a)
18: end for
19: cnfdf := cnfdf ∧ remaining
20: end if
21: until (lastcompfound ! = ε ∧ INARGS(lastcompfound) ! = A)
22: if prefcand ! = ∅ then
23: Ep := Ep ∪ {INARGS(prefcand)}24: oppsolution := FALSE
25: for a ∈ A \ INARGS(prefcand) do
26: oppsolution := oppsolution ∨ Iφ−1(a)
27: end for
28: cnf := cnf ∧ oppsolution
29: end if
30: until (prefcand ! = ∅)
31: if Ep = ∅ then
32: Ep = {∅}33: end if
34: return Ep
Exploiting SAT Solvers for EnumeratingPreferred Extensions: an Example
Complete extensions:{}, {f}, {d}, {a, f}, {b,d}, {d, f}, {b,d, e}, {a, c, f}, {b,d, f},
{a,d, f}
Exploiting SAT Solvers for EnumeratingPreferred Extensions: an Example
First complete extension found (not deterministic)
Exploiting SAT Solvers for EnumeratingPreferred Extensions: an Example
Forcing the search process for finding additional in arguments giventhe found complete
Exploiting SAT Solvers for EnumeratingPreferred Extensions: an Example
Another complete found. . .
Exploiting SAT Solvers for EnumeratingPreferred Extensions: an Example
. . . which is also preferred: {a,d, f}
Exploiting SAT Solvers for EnumeratingPreferred Extensions: an Example
Searching for other complete extensions. . .
Exploiting SAT Solvers for EnumeratingPreferred Extensions: an Example
. . . for instance {b,d} . . .
Exploiting SAT Solvers for EnumeratingPreferred Extensions: an Example
. . . from which we compute the preferred extensions {b,d, f}.
Empirical Evaluation: the Experiment
Two SAT solvers considered (separately):
PrecoSAT [Biere, 2009], SAT Competition 2009 winner(Application track) → PS-PRE;Glucose[Audemard and Simon, 2009, Audemard and Simon, 2012] SATCompetition 2011 and SAT Challenge 2012 winner (Applicationtrack) → PS-GLU
Random generated 2816 AF s divided in different classes according to twodimensions:
|A|: ranging from 25 to 200 with a step of 25;generation of the attack relations:
fixing the probability patt that there is an attack for each orderedpair of arguments (self-attacks are included), step of 0.25selecting randomly the number natt of attacks in itthe extreme cases of empty attack relation (patt = natt = 0) and offully connected attack relation (patt = 1, natt = |A|2)
Empirical Evaluation: the Analysis Using theInternational Planning Competition (IPC) Score
For each test case (in our case, each test AF ) let T ∗ be the bestexecution time among the compared systems (if no systemproduces the solution within the time limit, the test case is notconsidered valid and ignored).For each valid case, each system gets a score of1/(1 + log10(T/T ∗)), where T is its execution time, or a score of 0if it fails in that case. Runtimes below 1 sec get by default themaximal score of 1.The (non normalized) IPC score for a system is the sum of itsscores over all the valid test cases. The normalised IPC scoreranges from 0 to 100 and is defined as(IPC/# of valid cases) ∗ 100.
Empirical Evaluation: Comparison of DifferentEncodings
50
60
70
80
90
100
50 100 150 200
IPC
no
rmal
ised
to1
00
Number of arguments
IPC normalised to 100 with respect to the number of arguments
C1
Ca1
Cb1
Cc1
C2
C3
Empirical Evaluation: Comparison with Aspartix,Aspartix Meta, [Nofal et al., 2012]
60
65
70
75
80
85
90
95
100
50 100 150 200
%o
fsu
cces
s
Number of arguments
Percentage of success
ASP
ASP-META
NOF
PS-PRE
PS-GLU
Empirical Evaluation: Comparison with Aspartix,Aspartix Meta, [Nofal et al., 2012]
20
30
40
50
60
70
80
90
100
50 100 150 200
IPC
no
rmal
ised
to1
00
Number of arguments
IPC normalised to 100 with respect to the number of arguments
ASP
ASP-META
NOF
PS-PRE
PS-GLU
Conclusions
Novel SAT-based approach for preferred extension enumeration inabstract argumentationAssessed its performances by an empirical comparison withwell-known state-of-the-art systemsEvidence that different encodings, although theoreticallyequivalent, lead to very different empirical resultsThe proposed approach outperforms the state-of-the-artFuture works (currently ongoing):
Implementation of the other Labelling-based semantics(Grounded, Complete, Stable, Semi-stable)Evaluating different SAT-based search schemaIntegrate the proposed approach in the SCC-recursive schema(encouraging preliminary results!)Wider empirical investigation
References I
[Audemard and Simon, 2009] Audemard, G. and Simon, L. (2009).Predicting learnt clauses quality in modern sat solvers.In Proceedings of IJCAI 2009, pages 399–404.
[Audemard and Simon, 2012] Audemard, G. and Simon, L. (2012).Glucose 2.1.http://www.lri.fr/~simon/?page=glucose.
[Biere, 2009] Biere, A. (2009).P{re,ic}osat@sc’09.In SAT Competition 2009.
[Caminada, 2006] Caminada, M. (2006).On the issue of reinstatement in argumentation.In Proceedings of JELIA 2006, pages 111–123.
[Caminada and Gabbay, 2009] Caminada, M. and Gabbay, D. M. (2009).A logical account of formal argumentation.Studia Logica (Special issue: new ideas in argumentation theory), 93(2–3):109–145.
[Nofal et al., 2012] Nofal, S., Dunne, P. E., and Atkinson, K. (2012).On preferred extension enumeration in abstract argumentation.In Proceedings of COMMA 2012, pages 205–216.