Cerutti -- TAFA2013

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.


∧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


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


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.


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}


First complete extension found (not deterministic)


Forcing the search process for finding additional in arguments giventhe found complete


Another complete found. . .


. . . which is also preferred: {a,d, f}


Searching for other complete extensions. . .


. . . for instance {b,d} . . .


. . . 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.

http://www.lri.fr/~simon/?page=glucose

Education

Cerutti -- TAFA2013