A SCC RecursiveMeta-Algorithm for ComputingPreferred Labellings in Abstract

Argumentation

Federico Cerutti, Massimiliano Giacomin, Mauro Vallati, Marina Zanella

KR-2014 — Monday 21st July, 2014

Background on Dung’s AFSCC-Recursiveness

Exploiting the SCC-RecursivenessEmpirical Results

Conclusions

Background

Definition

Given an AF Γ = 〈A,R〉, with R ⊆ A×A:- 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 ∈ As.t. b→ a, ∃ c ∈ S s.t. c→ b;

- a set S ⊆ A is admissible if S is conflict–free and every element of S isacceptable with respect to S;

- a set S ⊆ A is a complete extension, i.e. S ∈ ECO(Γ), iff S is admissibleand ∀a ∈ A s.t. a is acceptable w.r.t. S, a ∈ S;

- a set S ⊆ A is the grounded extensions, i.e. S ∈ EGR(Γ), iff S is theminimal (w.r.t. set inclusion) complete set;

- a set S ⊆ A is a preferred extension, i.e. S ∈ EPR(Γ), iff S is a maximal(w.r.t. set inclusion) complete set.

Background

Definition

Let 〈A,R〉 be an AF : Lab : A 7→ {in, out, undec} is a complete labelling iff∀a ∈ A:

- Lab(a) = in⇔ ∀b ∈ a−Lab(b) = out;- Lab(a) = out⇔ ∃b ∈ a− : Lab(b) = in.

Let S ⊆ A a conflict–free set: the corresponding labelling isExt2Lab(S) ≡ Lab, where

- Lab(a) = in⇔ a ∈ S- Lab(a) = out⇔ ∃ b ∈ S s.t. b→ a- Lab(a) = undec⇔ a /∈ S ∧ @ b ∈ S s.t. b→ a

Proposition ([Caminada, 2006])Given an an AF Γ = 〈A,R〉, Lab is a complete (grounded, preferred)labelling of Γ if and only if there is a complete (grounded, preferred)extension S of Γ such that Lab = Ext2Lab(S).

An Example

An Example

Background on Dung’s AF

SCC-RecursivenessExploiting the SCC-Recursiveness

Empirical ResultsConclusions

SCC-Recursiveness: Path-Equivalence

SCC-Recursiveness: Partial Order of the SCCs

SCC-Recursiveness: Recursiveness of theapproach

SCC-Recursiveness

Definition

A given argumentation semantics σ is SCC-recursive if for anyargumentation framework Γ = 〈A,R〉, Eσ(Γ) = GF(Γ,A) ⊆ 2A. Forany Γ = 〈A,R〉 and for any set C ⊆ A, E ∈ GF(Γ, C) if and only if

- E ∈ BFσ(Γ, C) if |SCCΓ| = 1- ∀S ∈ SCCΓ (E ∩ S) ∈ GF(Γ↓S\(E\S)+ , UΓ(S,E) ∩ C) otherwise

where- BFσ(Γ, C) is a function, called base function, that, given anargumentation framework Γ = 〈A,R〉 such that |SCCΓ| = 1 anda set C ⊆ A, gives a subset of 2A

- UΓ(S,E) = {a ∈ S \ (E \ S)+ | ∀b ∈ (a− \ S), b ∈ E+}

SCC-Recursiveness and semantics restricted toa set of arguments

Definition

Given an AF Γ = 〈A,R〉 and a set C ⊆ A, a set E ⊆ A is:- an admissible set of Γ in C if and only if E is an admissible set of

Γ and E ⊆ C- a complete extension of Γ in C if and only if E is an admissibleset of Γ in C, and every argument a ∈ C which is acceptable withrespect to E belongs to E

- the grounded extension of Γ in C if and only if it is the least(with respect to set inclusion) complete extension of Γ in C

- a preferred extension of Γ in C if and only if it is a maximal(with respect to set inclusion) complete extension of Γ in C.

Research Questions

How can we exploit the SCC-Recursiveness in an algorithm?

Is it worth doing it?

Background on Dung’s AFSCC-Recursiveness

Exploiting theSCC-Recursiveness

Empirical ResultsConclusions

The Starting Point

The Meta-Algorithm

The Meta-Algorithm

The Meta-Algorithm

The Meta-Algorithm

Proposition

Let 〈A,R〉 be an argumentationframework and let C ⊆ A a set ofarguments. Considering the groundedlabelling Lab∗ of Γ in C and the set Uincluding the undec-labelled argumentsaccording to Lab∗, it holds thatLPR(Γ, C) = {Lab∗ ∪ E | E ∈LPR(Γ↓U , C ∩ U)}.

The Meta-Algorithm

The Meta-Algorithm

The Meta-Algorithm

The Meta-Algorithm

The Meta-Algorithm

The Meta-Algorithm

The Meta-Algorithm

I = {f ,g}O = ∅

The Meta-Algorithm

The Base Case: Complete Labelling in C

DefinitionLet Γ = 〈A,R〉 be an argumentation framework and C ⊆ A be a set of arguments. Atotal function Lab : A 7→ {in, out, undec} is a complete labelling of Γ in C iff it satisfiesthe following conditions for any a ∈ C:

L1C : Lab(a) = in⇔ ∀b ∈ a−Lab(b) = out;L2C : Lab(a) = out⇔ ∃b ∈ (a− ∩ C) : Lab(b) = in;L3C : Lab(a) = undec⇔ ∀b ∈ (a− ∩ C),Lab(b) 6= in ∧ ∃c ∈ a− :

Lab(c) = undec;and the following conditions for any a ∈ (A \ C):

L1A\C : Lab(a) = out⇔ ∃b ∈ (a− ∩ C) : Lab(b) = in;L2A\C : Lab(a) = undec⇔ ∀b ∈ (a− ∩ C),Lab(b) 6= in.

PropositionGiven an an AF Γ = 〈A,R〉 and a set C ⊆ A, Lab satisfies the above conditions if andonly if there is a complete extension S of Γ in C such that Lab = Ext2Lab(S).

Complete Labelling in C and CNF∧

i∈φ−1(C)

((Ii ∨ Oi ∨ Ui) ∧ (¬Ii ∨ ¬Oi)∧(¬Ii ∨ ¬Ui) ∧ (¬Oi ∨ ¬Ui)

)∧

∧i∈φ−1(C)

Ii ∨

∨{j|φ(j)→φ(i)}

(¬Oj)

∧ ∧i∈φ−1(C)

∧{j|φ(j)→φ(i)}

¬Ii ∨ Oj

∧∧

i∈φ−1(C)

∧{j∈φ−1(C)|φ(j)→φ(i)}

¬Ij ∨ Oi

∧ ∧i∈φ−1(C)

¬Oi ∨ ∨{j∈φ−1(C)|φ(j)→φ(i)}

Ij

∧∧

i∈φ−1(C)

( ∧{k|φ(k)→φ(i)}

(Ui ∨ ¬Uk ∨

( ∨{j∈φ−1(C)|φ(j)→φ(i)}

Ij

)))∧∧

i∈φ−1(C)

(( ∧{j∈φ−1(C)|φ(j)→φ(i)}

(¬Ui ∨ ¬Ij))∧(¬Ui ∨

( ∨{k|φ(k)→φ(i)}

Uk

)))∧

∧i∈φ−1(A\C)

(¬Ii ∧ (Oi ∨ Ui) ∧ (¬Oi ∨ ¬Ui)

)∧

∧i∈φ−1(A\C)

∧{j∈φ−1(C)|φ(j)→φ(i)}

¬Ij ∨ Oi

∧

i∈φ−1(A\C)

¬Oi ∨ ∨{j∈φ−1(C)|φ(j)→φ(i)}

Ij

∧ ∧

i∈φ−1(A\C)

Ui ∨

∨{j∈φ−1(C)|φ(j)→φ(i)}

Ij

∧

∧i∈φ−1(A\C)

∧{j∈φ−1(C)|φ(j)→φ(i)}

¬Ui ∨ ¬Ij

Complete Labelling in C and CNF

Proposition

Let 〈A,R〉 be an argumentation framework and C ⊆ A be a set ofarguments. If Lab is a complete labelling of Γ in C, then theassignment ΦV(Γ) ≡ {(Ii,>) | Lab(φ(i)) = in} ∪ {(Oi,>) | Lab(φ(i)) =out} ∪ {(Ui,>) | Lab(φ(i)) = undec} satisfies the ENCall encodingshown before. Conversely, if ΦV(Γ) is a satisfying assignment of theENCall encoding, then the labelling Lab ≡ {(a, in) | Iφ−1(a) ∈ΦV(Γ)} ∪ {(b, out) | Oφ−1(b) ∈ ΦV(Γ)} ∪ {(c, undec) | Uφ−1(c) ∈ ΦV(Γ)}is a complete labelling of Γ in C.

The Base-Case

The Base-Case

The Meta-Algorithm

Background on Dung’s AFSCC-Recursiveness

Exploiting the SCC-Recursiveness

Empirical ResultsConclusions

Analysis Using the International PlanningCompetition (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.

- 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.

The Experiment

- SAT approach optimized ([Cerutti et al., 2014] SAT-P) vsSCC-Recursiveness using SAT for the base case (SCC-P);

- I1: on Γ s.t. |SCCΓ| = 1, SCC-P performs worse than SAT-P;- I2: there exists a value χ such that on Γ where |SCCΓ| > χ,SCC-P performs better that SAT-P;

- I3: on Γ s.t. |SCCΓ| > χ, the greater |EPR(Γ)|, the more SAT-Pperforms worse than SCC-P.

First Hypothesis

790 AF s (Γ), s.t. |SCCΓ| = 1 and A = 25 : 25 : 250

40

50

60

70

80

90

100

0 50 100 150 200 250

|Γ|

IPC value (normalised) for SCC-P and SAT-P when |SCCΓ| = 1, varying |Γ|

SCC-P SAT-P

Second Hypothesis720 AF s varying |SCCΓ| in 5:5:45. Size of SCCs N(µ = 20 : 5 : 40, σ = 5);

attacks among SCCs N(µ = 20 : 5 : 40, σ = 5)

40

50

60

70

80

90

100

0 10 20 30 40 50

|SCCΓ|

IPC value (normalised) for SCC-P and SAT-P when 5 ≤ |SCCΓ| ≤ 45

SCC-P SAT-P

Second Hypothesis720 AF s varying |SCCΓ| in 5:5:45. Size of SCCs N(µ = 20 : 5 : 40, σ = 5);

attacks among SCCs N(µ = 20 : 5 : 40, σ = 5)

40

50

60

70

80

90

100

0 10 20 30 40 50

|SCCΓ|

IPC value (normalised) for SCC-P and SAT-P when 5 ≤ |SCCΓ| ≤ 45

SCC-P SAT-P

RemarkFor |SCCΓ| = 35, Md(SCC-P) = 8.81,Md(SAT-P) = 8.53, z = −0.35, p = 0.73;

Third Hypothesis2800 AF s (as before) s.t. |SCCΓ| = 50 : 5 : 80

0

100

200

300

400

500

600

700

800

900

0 50 100 150 200 250 300

s

|EPR(Γ)|

Median of times for SCC-P and SAT-P when 50 ≤ |SCCΓ| ≤ 80 varying |EPR(Γ)|

SCC-P SAT-P

Third Hypothesis2800 AF s (as before) s.t. |SCCΓ| = 50 : 5 : 80

0

100

200

300

400

500

600

700

800

900

0 50 100 150 200 250 300

s

|EPR(Γ)|

Median of times for SCC-P and SAT-P when 50 ≤ |SCCΓ| ≤ 80 varying |EPR(Γ)|

SCC-P SAT-P

RemarkRegression to the function f(x) = a x+ b:SCC-P, a = 0.43, b = 31.33;SAT-P, a = 2.40, b = 87.53

Background on Dung’s AFSCC-Recursiveness

Exploiting the SCC-RecursivenessEmpirical Results

Conclusions

Conclusions

1. Efficient algorithmic implementation of the SCC-recursivenessschema

- [Liao et al., 2013] decomposition but not the recursiveness- Other [Baumann et al., 2012, Dvořák et al., 2012b,Baroni et al., 2012]

2. Generalisation of [Cerutti et al., 2014] for the computation oflabellings in sub-frameworks

- [Besnard and Doutre, 2004, Dvořák et al., 2012a,Arieli and Caminada, 2013]. . .

3. Empirical evaluation:- Γ with |SCCΓ| > χ = 35 statistical evidence that theSCC-recursive schema reduces the computational effort ofenumerating the preferred labellings;

- Execution time of the SCC-recursive implementation is lesssensible to the number of labellings

Future Works

- Exploiting and comparing different equivalent encodings ofcomplete labellings in C including the redundant ones

- Different B-PR algorithm for computing labellings insub-frameworks[Egly et al., 2010, Nofal et al., 2014, Dvořák et al., 2014]

- Meta-algorithm to stable and CF2 semantics (directly fit theSCC-recursive schema [Baroni et al., 2005]) and to semi-stableand 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., Dvořák, 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 ofLecture 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 (JELIA2006), 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 inComputer Science, pages 176–193. Springer-Verlag Berlin Heidelberg.

[Dvořák et al., 2012a] Dvořák, 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 Representationand Reasoning (KR 2012), pages 54–64.

References III

[Dvořák et al., 2014] Dvořák, W., Järvisalo, M., Wallner, J. P., and Woltran, S. (2014).Complexity-sensitive decision procedures for abstract argumentation.Artificial Intelligence, 206:53–78.

[Dvořák et al., 2012b] Dvořák, W., Pichler, R., and Woltran, S. (2012b).Towards fixed-parameter tractable algorithms for abstract argumentation.Artificial Intelligence, 186:1–37.

[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.Artificial Intelligence, 207:23–51.