A New Labelling Algorithm for Generating Preferred Extensions of

Abstract Argumentation Frameworks

Samer Nofal

, Katie Atkinson

, Paul E. Dunne

and Ismail Hababeh

Department of Computer Science, German Jordanian University, Jordan

Department of Computer Science, University of Liverpool, U.K.

Keywords:

Abstract Argumentation, Argumentation Semantics, Labelling Semantics, Argumentation Algorithm, La-

belling Algorithm, Preferred Semantics.

Abstract:

The ﬁeld of computational models of argument aims to provide support for automated reasoning through algo-

rithms that operate on arguments and attack relations between them. In this paper we present a new labelling

algorithm that lists all preferred extensions of an abstract argumentation framework. The new algorithm is

enhanced by a new pruning strategy. We veriﬁed our new labelling algorithm and showed that it enumerates

preferred extensions faster than the old labelling algorithm.

1 INTRODUCTION

The study of computational argumentation is a major

ﬁeld of artiﬁcial intelligence, see for example (Atkin-

son et al., 2017; Modgil et al., 2013). Abstract ar-

gumentation frameworks of (Dung, 1995) (AFs for

short) are directed graphs with nodes representing ab-

stract arguments while directed edges denote attacks

between arguments. In spite of their simplicity, AFs

are an effective mechanism for decision making in

different domains, see for example (Longo and Don-

dio, 2014; Bench-Capon et al., 2015; Tamani et al.,

2015).

A fundamental issue arises in the context of AFs

concerning identifying which arguments are collec-

tively accepted in a given AF. To this end, an argu-

mentation semantics deﬁnes rules under which one

can compute sets of accepted arguments, what are so-

called extensions. In the literature we ﬁnd several pro-

posals motivating different argumentation semantics.

These varied semantics give a wide range of selection

among which one can choose what best ﬁt the needs

of the target application, see (Baroni et al., 2011) for

a comprehensive review of argumentation semantics.

In this paper we are concerned with the problem of

listing all extensions of a given AF under preferred

argumentation semantics, which is one of the most

studied semantics. We give a precise deﬁnition for

preferred semantics in section 2.

A labelling algorithm for listing preferred exten-

sions of a given AF is basically a search algorithm that

expands an abstract binary tree typically in a depth-

ﬁrst manner. The nodes of the tree represent different

states of the input AF, what are so-called labellings.

Labellings of a given AF are deﬁned by a total func-

tion that maps arguments from the AF to elements

from a predeﬁned set of statuses, what are so-called

argument labels.

The objective of this paper is to present a new,

more efﬁcient labelling algorithm that lists all pre-

ferred extensions of a given AF. The current state-

of-the-art labelling algorithm for preferred extension

enumeration is presented in (Nofal et al., 2016). How-

ever, in this paper we enhance the algorithm of (Nofal

et al., 2016) by a more efﬁcient pruning strategy to

speed up preferred extension generation. We imple-

mented our new algorithm and veriﬁed that the new

pruning strategy resulted in a faster preferred exten-

sion enumeration. Although we focus on the prob-

lem of listing preferred extensions, we believe that

the notions of the new pruning strategy are transfer-

able (with appropriate adjustments) to other compu-

tational problems in the context of abstract argumen-

tation frameworks.

In section 2 we give a necessary background ma-

terial. In section 3 we recall the state-of-the-art la-

belling algorithm for generating all preferred exten-

sions. In section 4 we present our new labelling algo-

rithm for listing preferred extensions. We verify the

efﬁciency of the new algorithm in section 5. We con-

clude the paper in section 6.

340

Nofal, S., Atkinson, K., Dunne, P. and Hababeh, I.

A New Labelling Algorithm for Generating Preferred Extensions of Abstract Argumentation Frameworks.

DOI: 10.5220/0007737503400348

In Proceedings of the 21st International Conference on Enterprise Information Systems (ICEIS 2019), pages 340-348

ISBN: 978-989-758-372-8

2 PRELIMINARIES

In the following deﬁnition we recall the notion of ab-

stract argumentation frameworks as introduced in the

seminal work of (Dung, 1995).

Deﬁnition 1 (Abstract Argumentation Frameworks).

An abstract argumentation framework AF is a pair

(A,R) where A is a set of abstract arguments and

R ⊆ A × A is called the attack relation.

We refer to (x,y) ∈ R as x attacks y (or y is at-

tacked by x). We denote by {x}

−

respectively {x}

the subset of A containing those arguments that attack

(respectively are attacked by) the argument x, and so

we use {x}

to represent the set {x}

∪ {x}

−

. For a

set of arguments S ⊆ A, we deﬁne

−

≡ { y ∈ A | ∃ x ∈ S s.t. y ∈ {x}

−

}

≡ { y ∈ A | ∃ x ∈ S s.t. y ∈ {x}

}

We denote by S

the union S

∪S

−

. We say S ⊆ A

attacks T ⊆ A (or T is attacked by S) if and only if

∩ T 6=

0. S ⊆ A attacks x ∈ A (or x is attacked by

S) if and only if x ∈ S

. Given a subset S ⊆ A, then

• x ∈ A is acceptable w.r.t. S if and only if for every

y ∈ {x}

−

, there is some z ∈ S for which y ∈ {z}

• S is conﬂict free if and only if for each (x,y) ∈

S × S, (x,y) /∈ R.

• S is admissible if and only if it is conﬂict free and

every x ∈ S is acceptable w.r.t. S.

• S is a preferred extension if and only if it is a max-

imal (w.r.t. set inclusion) admissible set.

In this paper we are concerned with the follow-

ing problem: given an AF H = (A,R), enumerate the

preferred extensions of H.

We give now a general account of a labelling algo-

rithm that generates all preferred extensions. As said

earlier, a labelling algorithm expands a conceptual bi-

nary search tree in a depth-ﬁrst way. The algorithm

forks to a left node if it decides to include an argu-

ment in a current under-construction extension. On

the other hand the algorithm forks to a right node if

it decides to exclude an argument from the current

under-construction extension. Once no argument is

left un-included or un-excluded, the algorithm back-

tracks if the current under-construction extension is

not admissible. Equally, the algorithm backtracks to

ﬁnd another extension. Take the AF of ﬁgure 1, then

a labelling algorithm would generate the search tree

visualized in ﬁgure 2.

In the next section we recall a precise deﬁnition

of the state-of-the-art labelling algorithm for listing

preferred extensions.

Figure 1: Argumentation Framework 1 (AF

3 THE STATE-OF-THE-ART

LABELLING ALGORITHM

FOR PREFERRED

EXTENSIONS

We recall the state-of-the-art labellling algorithm,

presented in (Nofal et al., 2016), for listing all pre-

ferred extensions of a given AF. In section 2 we pre-

sented an extension-based deﬁnition for preferred ar-

gumentation semantics. Alternatively, preferred se-

mantics can be described in terms of labellings, which

are mappings that relate every argument in a given

AF to a label in {in,out,undec}. For example, let

(A,R) be an AF and S ⊆ A be an admissible set

then the equivalent labelling for S is described by

a total mapping Lab : A → {in,out,undec} where

S = {x | Lab(x) = in}, S

= {x | Lab(x) = out} and

A \ (S ∪ S

) = {x | Lab(x) = undec}. For a thor-

ough presentation on labelling semantics see (Cami-

nada and Gabbay, 2009). Although a 3-label mapping

is probably sufﬁcient for characterizing extensions,

additional labels have been found useful for enhanc-

ing the efﬁciency of extension enumeration. Thereby

the labelling algorithms of (Nofal et al., 2016; No-

fal et al., 2014b; Nofal et al., 2014a) use instead a

5-label total function that maps arguments to labels

from {in,out,undec,blank,must out}.

As noted earlier, a labelling algorithm for pre-

ferred extension enumeration is basically a depth-ﬁrst

search that explores a conceptual binary tree. At the

root node of the search tree, all arguments of the given

AF are initially labelled according to the following

speciﬁcation.

Deﬁnition 2 (Initial Labelling). Let H = (A, R) be an

AF and S ⊆ A be the set of self-attacking arguments.

Then the initial labelling of H is deﬁned by the union:

{(x,blank) | x ∈ A \ S} ∪ {(y,undec) | y ∈ S}.

At any node of the search tree the algorithm tran-

sitions to a left node by selecting an argument x with

Lab(x) = blank and subsequently modify argument

labels as speciﬁed in the following deﬁnition.

Deﬁnition 3 (in transitions). Let H = (A, R) be an AF,

Lab : A → {in, out, undec,blank,must out} be a total

A New Labelling Algorithm for Generating Preferred Extensions of Abstract Argumentation Frameworks

341

Figure 2: A search tree that would be expanded by a basic labelling algorithm for listing the preferred extensions of AF

which is depicted in ﬁgure 1.

mapping, and x be an argument with Lab(x) = blank

then in trans(x,H,Lab) is deﬁned by the following ac-

tions:

1. Lab

← Lab.

2. Lab

(x) ← in.

3. for each y ∈ {x}

, Lab

(y) ← out.

4. for each y ∈ {x}

−

with Lab

(y) 6= out, Lab

(y) ←

must out.

5. return Lab

After the algorithm ﬁnished exploring the left sub-

tree, that is induced by an in transition, it expands a

right node by an undec transition as described in the

following deﬁnition.

Deﬁnition 4 (undec Transitions). Let H = (A,R) be

an AF, Lab : A → {in,out,undec,blank,must out} be

a total mapping and x be an argument with Lab(x) =

blank. Then und trans(x,H,Lab) is deﬁned by the fol-

lowing actions:

1. Lab

← Lab.

2. Lab

(x) ← undec.

3. return Lab

A labelling algorithm would reach a leaf node if

there are no blank arguments, we call such leaf nodes

terminal labellings.

Deﬁnition 5 (Terminal Labellings). Let H = (A,R) be

an AF and Lab : A → {in, out,undec,blank,must out}

be a total mapping. Then Lab is a terminal labelling

of H if and only if for each x ∈ A, Lab(x) 6= blank.

We call terminal labellings with {x | Lab(x) = in}

being admissible by admissible labellings. It follows

directly from the deﬁnition of admissible sets that if a

terminal labelling, for a given AF, does not map any

argument to must out then the set {x | Lab(x) = in} is

admissible.

Deﬁnition 6 (Admissible Labellings). Let

H = (A, R) be an AF and Lab : A →

{in,out,undec,blank,must out} be a total map-

ping. Then Lab is an admissible labelling of H if and

only if Lab is terminal and there is no x ∈ A with

Lab(x) = must out.

Conversely we denote by rejected labellings (or

occasionally dead-end labellings) the terminal la-

bellings with {x | Lab(x) = in} being not admissible.

It follows directly from the deﬁnition of admissible

sets that if a terminal labelling, for a given AF, maps

an argument to must out then the set {x | Lab(x) = in}

is not admissible.

Deﬁnition 7 (Rejected Labellings). Let H = (A,R) be

an AF and Lab : A → {in, out,undec,blank,must out}

be a total mapping. Then Lab is rejected if and only

if Lab is terminal and there is x ∈ A with Lab(x) =

must out.

We denote by preferred labellings the admissible

labellings with {x|Lab(x) = in} being inclusion-wise

ICEIS 2019 - 21st International Conference on Enterprise Information Systems

342

maximal among all admissible labellings.

Deﬁnition 8 (Preferred Labellings). Let

H = (A, R) be an AF and Lab : A →

{in,out,undec,must out,blank} be a total map-

ping. Then Lab is a preferred labelling of H if and

only if Lab is admissible and {x | Lab(x) = in} is

maximal (w.r.t. ⊆) among all admissible labellings.

Now we recall the pruning strategy used in the la-

belling algorithm of (Nofal et al., 2016). Note that

the pruning strategy of (Nofal et al., 2016) is centered

around two notions: labelling propagation and hope-

less labellings, which both improved preferred exten-

sion enumeration. Labelling propagation is about in-

ferring argument labels by analyzing the current la-

belling while hopeless labelling are those labellings

that never grow to a preferred labelling.

Deﬁnition 9 (Labelling Propagation). Let

H = (A, R) be an AF and Lab : A →

{in,out,undec,must

out,blank} be a total map-

ping. Then propagate(H,Lab) is deﬁned by the

following actions:

1. while ∃x Lab(x) = blank s.t. ∀y ∈ {x}

−

Lab(y) ∈

{out,must out} do

1.1. Lab(x) ← in

1.2. for each y ∈ {x}

do Lab(y) ← out

Deﬁnition 10 (Hopeless Labellings). Let

H = (A, R) be an AF and Lab : A →

{in,out,undec,must out,blank} be a total map-

ping. Then Lab is a hopeless labelling of H if and

only if there is x ∈ A with Lab(x) = must out such

that for all y ∈ {x}

−

Lab(y) ∈ {out, must out,

undec}.

Now we give algorithm 1 that lists all preferred

extensions. If algorithm 1 is invoked on a given AF

H, the initial labelling of H and an empty set E, then

it will return E containing all preferred extensions.

Throughout the paper we assume that a call by ref-

erence has to be made to invoke an algorithm or a

procedure. Referring to line 4 one can check the max-

imality of a given labelling Lab by ensuring that for

each preferred extension S ∈ E generated so far it is

the case that {x|Lab(x) = in} 6⊆ S. This is true because

algorithm 1 builds admissible sets in a descending or-

der with respect to set inclusion, which means maxi-

mal sets are visited ﬁrst.

In the following section we develop an improved

algorithm for listing preferred extensions.

Algorithm 1: Old list-preferred-extensions.

input : H = (A, R), E ⊆ 2

Lab : A → {in,out,undec, blank,must out}.

output: E ⊆ 2

1 propagate(H,Lab);

2 if Lab is hopeless then return;

3 if Lab is terminal then

4 if Lab is admissible and maximal then

E ← E ∪{{x | Lab(x) = in}};

5 return;

6 select an argument x with Lab(x) = blank;

7 Lab

← in trans(x, H,Lab);

8 list-preferred-extensions(H, Lab

, E);

9 Lab

← und trans(x,H, Lab);

10 list-preferred-extensions(H, Lab

, E);

4 A NEW LABELLING

ALGORITHM FOR

PREFERRED EXTENSION

ENUMERATION

Our new labelling algorithm is enhanced by a new

pruning strategy that we present in section 4.2. In

section 4.3 we give a new strategy for selecting ar-

guments that induce in and undec transitions. We in-

troduce a new labelling scheme in section 4.1.

4.1 A New Labelling Scheme

In section 3 we employed a 5-label total function

that maps arguments from a given AF to labels from

{in,out,must out, undec,blank}. We add to this

scheme two more labels: must in and must undec.

Hence we use a 7-label total function that maps ar-

guments from a given AF to labels from {in, out,

must out, undec, blank, must in, must undec}. From

now on we refer to this 7-label set as L.

Before we give a precise description for those ar-

guments that are eligible to be labelled with must in

or must undec, we deﬁne ﬁrst two helpful total map-

pings that both will streamline the computations

around deciding such eligibility. Our total functions

are called BLANK and UNDEC. BLANK maps an ar-

gument to the number of the attackers that are labeled

with either blank, must in, or must undec. UNDEC

maps an argument to the number of undec attack-

ers. A similar idea to the essence of BLANK and UN-

DEC has been used in (Modgil and Caminada, 2009)

for computing the grounded extension. We specify

BLANK and UNDEC precisely shortly. Observe that

we denote the set of nonnegative integers by N

A New Labelling Algorithm for Generating Preferred Extensions of Abstract Argumentation Frameworks

343

Deﬁnition 11 (BLANK and UNDEC). Let H = (A,R)

be an AF and Lab : A → L be a total mapping, then

BLANK: A → N

and UNDEC: A → N

are total map-

pings such that for every x ∈ A

BLANK(x) = |{ y ∈ {x}

−

: Lab(y) ∈

{blank, must in, must undec}}|, and

UNDEC(x) = |{ y ∈ {x}

−

: Lab(y) = undec}|.

Now we are ready to deﬁne the conditions under

which an argument becomes eligible to be labelled

with must undec or must in.

A blank argument, say x, can be labelled with

must undec if x has to join (but not yet) the current

set of undec arguments because otherwise the under-

construction set of in arguments, say S, together with

x (i.e. {x} ∪ S) will never grow to a preferred exten-

sion for one of two reasons as speciﬁed in the follow-

ing deﬁnition.

Deﬁnition 12 (must undec Arguments). Let H =

(A,R) be an AF and Lab : A → L be a total mapping,

and x be an argument with Lab(x) = blank then x is

eligible to be labelled with must undec if it holds that

∃y ∈ {x}

−

with Lab(y) ∈ {blank,undec,must undec}

s.t. BLANK(y)=0,

or it holds that

∃y ∈ {x}

with Lab(y) = undec s.t.

∃z ∈ {y}

with Lab(z) ∈ {undec, must undec} and

BLANK(z)=0 and UNDEC(z)=1.

A blank argument, say x, can be labelled with

must in only if x must join (but not yet) the current

set of in arguments, say S, because otherwise S will

never grow to a preferred extension.

Deﬁnition 13 (must in Arguments). Let H = (A,R)

be an AF and Lab : A → L be a total mapping, and x

be an argument with Lab(x) = blank then x is eligible

to be labelled with must in if it holds that

BLANK(x)=0 and UNDEC(x)

∈ {0, |{y : Lab(y) = undec and y ∈ {x}

}|},

or it holds that

∃y ∈ {x}

with Lab(y) = must out such that

BLANK(y)=1.

By using the new labels (i.e. must in and

must undec) we identify four cases by which an ar-

gument’s label can be deduced from the current la-

belling: in two cases an argument must be eventually

labelled with undec while in the other two cases an

argument must end with the in label. The essence of

these four cases are similar to the ones introduced in

(Doutre and Mengin, 2001), but the implementation

is totally new as we show throughout the paper.

The question now is why we do not label an ar-

gument with in and undec immediately instead of

must in and must undec. Note that labelling an ar-

gument with in or undec may trigger new changes in

the current labelling (e.g. labelling propagation) and

in turn these changes may produce further ones and so

on. Thus, to process these cascading changes more ef-

ﬁciently we use must in and must undec. We explain

more precisely the computational beneﬁt of must in

and must undec next.

4.2 A New Pruning Strategy

We develop a precise description for our new prun-

ing strategy by deﬁning a number of constructs in this

section. We start with two important sets MUST IN

and MUST UNDEC that respectively hold must in

and must undec arguments temporarily until they are

eventually labeled with in and undec. The advantage

of using these sets is that we conﬁne computations

to those arguments that truly need further processing,

and in consequence we avoid scanning all arguments

unnecessarily. One might wonder why we utilize two

related notions for apparently the same purpose, for

example the set MUST IN and the label must in seem

to denote the same arguments. We note that although

the two notions refer to the same set of arguments

but they allow for different levels of access: by using

must in it is computationally easy to check the status

of a speciﬁc argument at any point of the extension

enumeration while the set of MUST IN enables an ef-

ﬁcient access to the collection of those arguments that

need to be ﬁnalized with in. Throughout this section

we show exactly the usage of MUST IN and must in as

well as the two related structures MUST UNDEC nad

must undec. We reﬁne now the initial labelling and

hopeless labellings, taking into account the new la-

bels must in and must undec.

Deﬁnition 14 (New Initial Labelling). Let H = (A,R)

be an AF, S be the set of self attacking arguments and

T be the set of {x : |{x}

−

| = 0}. Then the initial la-

belling of H is deﬁned by the union of the following

sets:

{(x,blank) | x ∈ A \ (S ∪ T )} ∪

{(x,must undec) | x ∈ S} ∪

{(x,must in) | x ∈ T }.

Deﬁnition 15 (New Hopeless Labellings). Let H =

(A,R) be an AF and Lab : A → L be a total map-

ping. Then Lab is a hopeless labelling of H if and

only if there is x with Lab(x) = must out such that

BLANK(x)=0.

We introduce the notion of non-maximal la-

bellings to describe those labellings with an undec (or

must undec) argument being attacked by only out (or

ICEIS 2019 - 21st International Conference on Enterprise Information Systems

344

must out) arguments. This is because by labelling an

argument, say x, with undec we mean to ﬁnd a pre-

ferred extension excluding x. But if at some point x

becomes acceptable to the current set of in arguments,

then we better backtrack because the current labelling

will never be maximal. Such labellings are hope-

less in the sense that they will never grow to a pre-

ferred extension although they might be admissible.

Nonetheless, we call such labellings non-maximal to

distinguish them from the hopeless labellings that are

inevitably not admissible.

Deﬁnition 16 (Non-maximal Labellings). Let H =

(A,R) be an AF and Lab : A → L be a total mapping.

Then Lab is a non-maximal labelling of H if and only

if there exists x with Lab(x) ∈ {undec,must undec}

such that BLANK(x)=0 and UNDEC(x)=0.

Now we describe our new labelling propagation.

Basically, labelling propagation might be invoked at

any point of the search to identify those arguments

that are eligible for must in and must undec labels.

Deﬁnition 17 (New Labelling Propagation). Let H =

(A,R) be an AF, Lab : A → L be a total map-

ping, x ∈ A be an argument, BLANK: A → N

and

UNDEC: A → N

be total mappings, MUST IN ⊆ A

and MUST UNDEC ⊆ A be sets of arguments then

propagate(x, H, Lab, BLANK, UNDEC, MUST IN,

MUST UNDEC) is deﬁned by:

1. for each y ∈ {x}

1.1. if y is eligible for must undec then

1.1.1. Lab(y) ← must undec.

1.1.2. MUST UNDEC ← MUST UNDEC ∪ {y}.

1.2. if y is eligible for must in then

1.2.1. Lab(y) ← must in.

1.2.2. MUST IN ← MUST IN ∪ {y}.

1.3. for each z ∈ {y}

1.3.1. if z is eligible for must undec then

1.3.1.1. Lab(z) ← must undec.

1.3.1.2. MUST UNDEC ← MUST UNDEC ∪ {z}.

1.3.2. if z is eligible for must in then

1.3.2.1. Lab(z) ← must in.

1.3.2.2. MUST IN ← MUST IN ∪ {z}.

1.4. if Lab is hopeless or non-maximal then return

false.

2. return true.

Now we expand in transitions to include labelling

propagation.

Deﬁnition 18 (new in Transitions). Let H = (A,R)

be an AF and Lab : A → L be a total mapping, s

be an argument with Lab(s) ∈ {blank, must in},

BLANK: A → N

and UNDEC: A → N

be to-

tal mappings, MUST IN ⊆ A and MUST UNDEC

⊆ A be sets of arguments then in-trans

(s,H,Lab,BLANK,UNDEC,MUST UNDEC,MUST IN)

is deﬁned by:

1. Lab(s) ← in.

2. for each x ∈ {s}

with Lab(x) 6= out do

2.1. for each y ∈ {x}

2.1.1. if Lab(x) = undec then UNDEC(y) ←

UNDEC(y)-1.

2.1.2. if Lab(x) ∈ {blank, must undec} then

BLANK(y) ← BLANK(y)-1.

2.2. BLANK(x) ← BLANK(x) - 1.

2.3. Lab(x) ← out.

2.4. if propagate(x,H, Lab, BLANK, UNDEC,

MUST IN, MUST UNDEC)=false then return

false.

3. for each x ∈ {s}

−

with Lab(x) 6∈ {out,must out}

3.1. Lab(x) ← must out.

3.2. if propagate(x, H, Lab, BLANK, UNDEC,

MUST IN, MUST UNDEC) = false then return

false.

4. return true.

Similarly, we expand undec transitions to include

labelling propagation.

Deﬁnition 19 (New undec Transitions). Let

H = (A,R) be an AF and Lab : A → L be

a total mapping, x be an argument with

Lab(x) ∈ {blank,must undec}, BLANK: A → N

and UNDEC: A → N

be total mappings, MUST IN

⊆ A and MUST UNDEC ⊆ A be sets of arguments then

und-trans(x,H, Lab,BLANK,UNDEC,MUST UNDEC,

MUST IN) is deﬁned by:

1. Lab(x) ← undec.

2. for each y ∈ {x}

2.1. UNDEC(y) ← UNDEC(y)+1.

2.2. BLANK(y) ← BLANK(y)-1.

3. return propagate(x, H, Lab, BLANK, UNDEC,

MUST IN, MUST UNDEC).

Before we make a new branch (by in and un-

dec transitions) we ﬁnalize the label of must in and

must undec arguments with in and undec respec-

tively. Every time we relabel a must in (respectively

must undec) argument with in (respectively undec)

we may ﬁnd that some blank arguments have be-

come eligible for either must in or must undec. Thus,

a change in some argument’s label may cause other

changes in the current labeling and so forth. We de-

ﬁne this recurrent process by labelling broadcasting.

Deﬁnition 20 (Labelling Broadcasting). Let H =

(A,R) be an AF and Lab : A → L be a total mapping,

A New Labelling Algorithm for Generating Preferred Extensions of Abstract Argumentation Frameworks

345

BLANK: A → N

and UNDEC: A → N

be total map-

pings, MUST IN ⊆ A and MUST UNDEC ⊆ A be sets

of arguments then broadcast(Lab,H, BLANK, UNDEC,

MUST IN, MUST UNDEC) is deﬁned by:

1. while MUST IN 6=

0 or MUST UNDEC 6=

0 do

1.1. while MUST IN 6=

0 do

1.1.1. remove an argument x from MUST IN.

1.1.2. if in-trans(x, H, Lab, BLANK, UNDEC,

MUST IN, MUST UNDEC)=false then return

false.

1.2. while MUST UNDEC 6=

0 do

1.2.1. remove an argument x from MUST UNDEC.

1.2.2. if und-trans(x, H, Lab, BLANK, UNDEC,

MUST IN, MUST UNDEC)=false then return

false.

2. return true.

Using the new in transition, undec transition, and

labelling broadcasting, we present algorithm 2. Let

H = (A,R) be an AF, Lab be the initial labelling of H,

and for each x ∈ A let BLANK(x) be equal to |{x}

−

while UNDEC(x) be equal to 0, and let MUST IN be

the set {x : |{x}

−

| = 0}, MUST UNDEC be the set

{x | (x,x) ∈ R}, and E be an empty set then if we call

algorithm 2 on H, Lab, BLANK, UNDEC, MUST IN,

MUST UNDEC, and E, then the algorithm returns E

containing all preferred extensions of H. Figure 3

shows a running of algorithm 2. In the next section

we add a further enhancement, which is a new argu-

ment selection strategy.

4.3 A New Argument Selection Strategy

Referring to line 6 of algorithm 2, there are many pos-

sible selection strategies. One possibility is to pick ar-

guments randomly. Another strategy may depend on

some heuristic measures, such as the number of ad-

jacent arguments, which is the strategy used in (No-

fal et al., 2016). See (Geilen and Thimm, 2017) for

more discussions on heuristic-based selection strate-

gies that depend on the current AF labelling and/or its

underlying graph structure properties. Here we intro-

duce a different selection strategy that relies on argu-

ment history proﬁle as we explain next.

Our selection strategy is simple. Every time we

reach a hopeless labelling because of a must out ar-

gument, say x, we mark such x as a failure point and

later we give priority to the blank attackers of x to

be selected for inducing a transition. In other words,

our selection strategy prioritizes those arguments that

might soon produce a hopeless labelling, instead of

delaying the awareness of hopeless labelling possibly

until a very late point of the search.

Now we come to the speciﬁcations of our selec-

tion strategy. We note that a minor modiﬁcation has

to be introduced to algorithm 2 such that every time

we detect a hopeless labelling because of a must out

argument, say x, the algorithm has to push x on top

of a stack structure denoted by S. Algorithm 3 imple-

ments our selection strategy. Note that the algorithm

either returns a selected argument or it returns -1 to

indicate that the current labelling is either hopeless or

terminal.

5 VERIFYING THE EFFICIENCY

OF THE NEW ALGORITHM

We implemented our new algorithm using

the C++ programming language. The source

code of the implementation can be found at

https://sourceforge.net/projects/argtools. We evalu-

ated the new algorithm using benchmark A of the

second international competition of computational

models of argumentation 2017 (ICCMA17) (ICC,

). Benchmark A includes 350 AFs, for more details

see (ICC, ). We carried out the evaluation on a system

with an intel-core-i7 processor and four gigabytes

of system memory. For each problem instance we

limit the memory space to one gigabyte. In contrast,

ICCMA17 uses a more powerful environment with

an intel-xeon processor and with four gigabytes of

memory being allocated for each problem instance.

However, with respect to the running time we follow

ICCMA17, and hence, set a timeout of 10 minutes

for each problem instance.

The objective of this evaluation is

to verify that the new algorithm enu-

merates preferred extensions faster than the old

algorithm. We found that the new algorithm is able

to solve successfully 233 AFs out of benchmark A.

Note that the implementation of the old algorithm

is included in ArgTools (ﬁrst version), which is a

labelling-based solver that was submitted to the ﬁrst

version of the competition ICCMA15 (Thimm and

Villata, 2017). In fact, the old algorithm did not

solve any AF of benchmark A. In its second version,

submitted to ICCMA17, ArgTools includes some (not

all) aspects of the new algorithm. ArgTools (version

2) enumerated all preferred extensions successfully

for 157 AFs (ICC, ). Another labelling-based solver

is Heureka (Geilen and Thimm, 2017), which also

participated in ICCMA17. Heureka implements the

old algorithm but with a profound heuristic-based

argument selection strategy. Heureka enumerated all

preferred extensions for 178 AFs of benchmark A.

ICEIS 2019 - 21st International Conference on Enterprise Information Systems

346

Algorithm 2: New list-preferred-extensions.

input : H = (A,R), Lab : A → L, BLANK : A → N

, UNDEC : A → N

, MUST IN ⊆ A, MUST UNDEC ⊆ A, E ⊆ 2

output : E ⊆ 2

1 if broadcast(Lab,H,BLANK,UNDEC,MUST IN,MUST UNDEC)=false then return;

2 if Lab is terminal then

3 if Lab is admissible and maximal then

4 E ← E ∪{{x | Lab(x) = in}} ;

5 return;

6 select an argument x with Lab(x) = blank;

7 Lab

← Lab, BLANK

← BLANK, UNDEC

←UNDEC;

8 MUST

← MUST IN, MUST UNDEC

← MUST UNDEC;

9 if in trans(x,H, Lab

,BLANK

,UNDEC

,MUST IN

,MUST UNDEC

) = true then

10 list-preferred-extensions(H, Lab

,BLANK

,UNDEC

,MUST IN

,MUST UNDEC

, E);

11 if und trans(x,H, Lab,BLANK,UNDEC,MUST IN,MUST UNDEC) = true then

12 list-preferred-extensions(H,Lab,BLANK,UNDEC,MUST IN,MUST UNDEC, E);

Figure 3: The search tree that is expanded by algorithm 2 in listing the preferred extensions of AF

depicted in ﬁgure 1.

6 CONCLUSION

We presented a new labelling algorithm that lists all

preferred extensions of a given AF. We evaluated

the new algorithm and our ﬁndings veriﬁed that the

new algorithm enumerates preferred extensions sig-

niﬁcantly faster than the old algorithm. The obtained

speedup is due to the new pruning strategy of the new

algorithm. We plan to study the impact of our new

pruning strategy in the context of other computational

problems in the ﬁeld of abstract argumentation.

Lastly we note that the labelling algorithm of

(Nofal et al., 2014b) for preferred extension enu-

meration enhanced the previous algorithm of (Doutre

and Mengin, 2001; Caminada, 2007). Nevertheless,

the algorithm of (Nofal et al., 2014b) has been im-

proved further in (Nofal et al., 2016) by a look-ahead

strategy. In this work we build on the algorithm

of (Nofal et al., 2016) as we explained throughout

the paper. Another mainstream research concerns

building reduction-based solvers, see some examples

in (Thimm and Villata, 2017). For computational

complexity of abstract argumentation see for exam-

ple (Dunne and Wooldridge, 2009). For a survey on

methods for solving different computational problems

of AFs see the article of (Charwat et al., 2015).

A New Labelling Algorithm for Generating Preferred Extensions of Abstract Argumentation Frameworks

347

Algorithm 3: New selecting an argument for tran-

sitions.

input : H = (A, R), Lab : A → L, S is a stack of

arguments.

output: x ∈ A, S.

1 while S 6=

0 do

2 y ← pop an argument from top of S;

3 if Lab(y) = must out then

4 if BLANK(y) > 0 then return x ∈ {y}

−

with

Lab(x) = blank else return -1;

5 foreach y ∈ A with Lab(y) = must out do

6 if BLANK(y) > 0 then return some x ∈ {y}

−

with Lab(x) = blank else return -1;

7 if ∃x with Lab(x) = blank such that UNDEC(x) > 0

then return x;

8 if ∃x with Lab(x) = blank then return x else return

-1;

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