=Paper=
{{Paper
|id=Vol-2710/paper17
|storemode=property
|title=A Labelling Semantics for Weighted Argumentation Frameworks
|pdfUrl=https://ceur-ws.org/Vol-2710/paper17.pdf
|volume=Vol-2710
|authors=Stefano Bistarelli,Carlo Taticchi
|dblpUrl=https://dblp.org/rec/conf/cilc/BistarelliT20
}}
==A Labelling Semantics for Weighted Argumentation Frameworks==
https://ceur-ws.org/Vol-2710/paper17.pdf
A Labelling Semantics for
Weighted Argumentation Frameworks?
Stefano Bistarelli1[0000−0001−7411−9678] and
Carlo Taticchi2[0000−0003−1260−4672]
1
University of Perugia, Italy - stefano.bistarelli@unipg.it
2
Gran Sasso Science Institute, Italy - carlo.taticchi@gssi.it
Abstract. Argumentation Theory provides tools for both modelling and
reasoning with controversial information and is a methodology that is go-
ing to be proposed as a way to give explanations to results provided using
machine learning techniques. In this context, labelling-based semantics
for Abstract Argumentation Frameworks (AFs) allow for establishing the
acceptability of sets of arguments, dividing them into three partitions: ac-
ceptable, rejected and undecidable (instead of classical Dung two sets IN
and OUT partitions). This kind of semantics have been studied only for
classical AFs, whilst the more powerful weighted and preference-based
framework has been not studied yet. In this paper, we define a novel
labelling semantics for Weighted Argumentation Frameworks, extending
and generalising the crisp one.
Keywords: Argumentation theory · labelling-based semantics · weighted
argumentation framework.
1 Introduction
Argumentation and its applications are receiving increasing interest in many
fields of AI. For instance, argumentative processes are used in [21] to interpret
online debates, while in [26] an argumentation system is devised to support ex-
pert opinion. Argumentation is also used to aid machine learning (see [16] for
a survey) for both improving performances (e.g., classification accuracy) and
providing explanations to the results. Argumentation problems are modelled
through Abstract Argumentation Frameworks (AFs in short) [18], that consist
of directed graphs in which the nodes are arguments that contain abstract in-
formation and the edges represent attack relations.
The acceptability of an argument of an AF can then be established following
different criteria, formalised through the extension-based [18] and the labelling-
based semantics [13]. Through the reasoning on the acceptability of the argu-
ments according to a notion of defence, one can divide the set of arguments into
two separated subsets, respectively containing acceptable and non-acceptable
?
Copyright c 2020 for this paper by its authors. Use permitted under Creative Com-
mons License Attribution 4.0 International (CC BY 4.0).
2 S. Bistarelli and C. Taticchi
arguments. However, for certain applications of argumentation (especially those
in which defeating an argument leads to the reinstatement of another one [13]),
it is convenient to consider more degrees of acceptability in order for one to
be able to further differentiate among arguments. The labelling defined in [13]
refines the concept of acceptable argument and builds on the classical semantics
for providing an additional acceptance status through the assignment of labels
to the arguments.
In order to increase the expressiveness of AFs, attack relations between ar-
guments can be endowed with a value (a weight) which indicates the strength
of the attacks themselves. In this kind of frameworks, called weighted AFs, the
acceptability criteria for the arguments also need to consider the weight of in-
coming and outgoing attacks. In two recent works [9, 10] the attacks from an
argument to a set of arguments are grouped together as if they were a unique
attack; in particular, the authors consider a weighted notion of defence that
takes into account the weight associated to each attack, also generalising the
approaches of [17] and [22]. In all these works, extension-based semantics have
been used to identify sets of acceptable arguments. The correspondence between
extension-based semantics and the labelled ones has been proved and showed
important for the crisp framework [25]. The addition of such mapping for the
weighted argumentation is an important result in the area that will be the initial
brick for many additional result in the field. In this work, we extend the notion
of labelling to Weighted Argumentation Frameworks and we provide a defini-
tion that generalises the original labelling [13]. For each weighted semantics, we
give the conditions under which a labelling corresponds to a set of extensions.
The rest of this paper is structured as follows: in Section 2 we summarise the
main concepts of AFs, providing the definitions for extension-based semantics
considering both weighted and non-weighted cases, and in Section 3 we present
our definition of labelling for Weighted Argumentation Frameworks. Section 4
shows an implementation of the weighted labelling within a tool for argumen-
tation problems. Finally, in Section 6 we conclude the paper discussing some of
the possible future directions that we would like to investigate.
2 Preliminaries
In this section we recall the formal definition of AF and the related semantics
introduced by Dung [18], together with the notion of labelling and labelling-
based semantics [13, 2]. We also give the main definitions for Weighted AFs,
relying on the definition of acceptability of arguments given in [9, 10].
2.1 Abstract Argumentation Frameworks
First of all, we recall the formal definition for an AF [18].
Definition 1 (Abstract Argumentation Framework). An Abstract Argu-
mentation Framework is a pair hA, Ri where A is a set of arguments and R is
a binary relation on A.
A Labelling Semantics for Weighted Argumentation Frameworks 3
Consider two arguments a, b belonging to an AF. We denote with (a, b) ∈ R
(or simply a → b) an attack from a to b; we can also say that b is defeated by a.
We define the sets of arguments that attack (and that are attacked by) another
argument as follows.
Definition 2 (Attacks). Let F = hA, Ri be an AF, a ∈ A and A ⊆ A. We
define the sets a+ = {b ∈ A | a → b}, a− = {b ∈ A | b → a}, A+ = ∪{a+ | a ∈
A} and A− = ∪{a− | a ∈ A}.
In order for b to be acceptable, we require that every argument that defeats
b is defeated in turn by some other argument of the AF. More formally, we have
the following definition.
Definition 3 (Acceptable argument). Given an AF F = hA, Ri, an argu-
ment a ∈ A is acceptable with respect to D ⊆ A if and only if ∀b ∈ A such that
b ∈ a− , ∃c ∈ D such that c ∈ b− , and we say that a is defended by D.
Using the notion of defence as a criterion for distinguishing acceptable ar-
guments in the framework, one can further refine the set of selected “good”
arguments through semantics.
Definition 4 (Extension-based semantics). Let F = hA, Ri be an AF. A
set E ⊆ A is conflict-free in F if and only if there are no a, b ∈ A such that
(a, b) ∈ R. A conflict-free subset E is then
– admissible, if each a ∈ E is defended by E;
– complete, if it is admissible and ∀a ∈ A defended by E, a ∈ E;
– stable, if E ∪ E + = A;
– preferred, if it is admissible and it is maximal (with respect to set inclusion);
– grounded, if it is complete and it is minimal (with respect to set inclusion).
Strong admissibility is introduced in [3] as a refinement of the admissible
semantics.
Definition 5. Let F = hA, Ri be an AF. A set E ⊆ A is strongly admissible if
and only if each a ∈ E is defended by some E 0 ⊆ E \ {a} which in its turn is
again strongly admissible.
The work in [13] describes how to assign labels to the arguments of an AF
in such a way that the set of arguments is partitioned into three subsets, each
representing a different degree of acceptance. Below, we report the labelling
function and the characterisation for the various semantics.
Definition 6 (Labelling for AFs). Let F = hA, Ri be an AF. A labelling L
of F is a total function L : A → {IN, OUT, UNDEC}. For any A ⊆ A, we denote
A|IN , A|OUT and A|UNDEC the set of all the arguments labelled IN, OUT and UNDEC
by L, respectively.
Given a labelling L, it is possible to identify a correspondence with the
extension-based semantics [2]. The set of IN arguments coincides with an ex-
tension of acceptable arguments. We rephrase the semantics in [2] as follows.
4 S. Bistarelli and C. Taticchi
Definition 7 (Labelling-based semantics). Let L be a labelling of an AF
F = hA, Ri and a ∈ A. Then
– L is a conflict-free labelling if and only if:
• L(a) = IN =⇒ a− |IN = ∅, and
• L(a) = OUT =⇒ a− |IN 6= ∅
– L is a admissible labelling if and only if:
• L(a) = IN =⇒ a− = a− |OUT , and
• L(a) = OUT =⇒ a− |IN 6= ∅
– L is a complete labelling if and only if:
• L(a) = IN ⇐⇒ a− = a− |OUT , and
• L(a) = OUT ⇐⇒ a− |IN 6= ∅
– L is a stable labelling if and only if:
• L is a complete labelling, and
• A|UNDEC = ∅;
– L is a preferred labelling if and only if:
• L is an admissible labelling, and
• A|IN is maximal among all the admissible labellings
– L is a grounded labelling if and only if:
• L is a complete labelling, and
• A|IN is minimal among all the complete labellings
A labelling for the strongly admissible semantics is given in [14], where the
author relies on a numbering on the arguments to assign the correct labels. In
every labelling of the various semantics, arguments for which not every attacker
is labelled OUT, and no attacker is labelled IN are labelled UNDEC. The admis-
sible labelling that we consider in the definition above coincides with the inter-
pretation given in [14], where IN arguments can attack both OUT and UNDEC
arguments. Different definitions of labelling (as for instance the one given in [20]
and surveyed in [15]) force arguments attacked by an IN to be OUT1 . However,
nothing changes in terms of extensions, since the set of IN arguments remains
the same. Also, note that a complete labelling coincides with the reinstatement
labelling given in [13].
2.2 Weighted Argumentation Frameworks
In order to compute the set of extensions of a particular AF, attack relations
are used to determine the acceptability of the arguments. Since it is not possible
to further diversify the relations among arguments, every attack in the AF has
the same “strength”, that is, the existence or not of an attack is the only thing
that matters in determining the semantics. To overcome this limit, Dung’s AFs
have been extended to Weighted AFs (WAFs) by associating the attacks with a
weight that represents the support of the relation [19].
1
Hence ⇐⇒ is used instead of =⇒ in the second condition for admissible labelling:
L(a) = OUT ⇐⇒ a− |IN 6= ∅.
A Labelling Semantics for Weighted Argumentation Frameworks 5
In order to analyse a WAF in terms of sets of extensions, a definition of
defence is required that encompasses the notion of weighted attack relations.
In [10] the framework is equipped with a c-semiring [7, 8] that provides the
operation for composing the weights in order to estimate the effectiveness of a
defence. The acceptability of an argument is then determined by comparing the
compositions of the attacks with the composition of the defences. C-semirings [7,
8] are absorptive, commutative semiring, that is commutative semirings with
idempotent plus operator (also called tropical semirings) and top element. These
structures allow expressing both the values of the weights and the aggregation
operators and thus are parametric to the desired notion of defence.
Definition 8 (c-semirings). A c-semiring is a tuple S = hS, ⊕, ⊗, ⊥, >i such
that S is a set, >, ⊥ ∈ S, and ⊕, ⊗ : S × S → S are binary operators making the
triples hS, ⊕, ⊥i and hS, ⊗, >i commutative monoids (semi-groups with identity),
satisfying i) ∀s, t, u ∈ S. s ⊗ (t ⊕ u) = (s ⊗ t) ⊕ (s ⊗ u) (distributivity), and ii)
∀s ∈ S. s ⊗ ⊥ = ⊥ (annihilator). Moreover, we have that ∀s, t ∈ S. s ⊕ (s ⊗ t) = s
(absorptiveness). The operator ⊕ also defines a preference relation ≤S over the
set S, such that a ≤S b ⇐⇒ a ⊕ b = b, for a, b ∈ S.
We list some of the most common instances of c-semirings.
– Sboolean = h{false, true}, ∨, ∧, false, truei
– Sfuzzy = h[0, 1], max, min, 0, 1i
– Sprobabilistic = h[0, 1], max, ×, 0, 1i
– Sweighted = hR+ ∪ {+∞}, min, +, +∞, 0i
Different c-semirings can represent different notions of defence for WAF, by
using the operators ⊕ and ⊗ for obtaining an ordering among the values in
S. For simplicity, we refer to these values as weights. Note that the element >
of the c-semiring (e.g., 0 for the weighted and true for the boolean) coincides
with having no relation between two arguments. We denote with WAFS a WAF
endowed with a c-semirings S and we call it a semiring-based WAF.
Definition 9 (WAFS ). A semiring-based WAF is a quadruple hA, R, W, Si,
where S is a c-semiring hS, ⊕, ⊗, ⊥, >i, A is a set of arguments, R the attack
binary-relation on A, and W : A × A −→ S is a binary function. Given a, b ∈ A
and R(a, b), then W (a, b) = s means that a attacks b with a weight s ∈ S.
Moreover, we require that R(a, b) if and only if W (a, b) .
Given a WAFS we can evaluate the overall weight of all the attacks from a
set of arguments towards another set throughNthe composition operator ⊗ of
the c-semiring S [9, 12]. In particular, we use to indicate the ⊗ operator on a
set of values (indeed ⊗ is a binary operator that composes two weights).
Definition 10 (Attacks). Let F = hA, R, W, Si be a WAF S . A set of argu-
ments B attacks a set of arguments D and the weight of such attack is k ∈ S,
if O
W (B, D) = W (b, d) = k.
b∈B,d∈D
6 S. Bistarelli and C. Taticchi
The previous definition also allows composing the attacks from a set of argu-
ments to another single argument, and from a single argument towards a set of
arguments. The notion of weighted defence (or w-defence) can then be expressed
in the following terms.
Definition 11 (w-defence). Let F = hA, R, W, Si be a WAF S . Then B ⊆ A
w-defends b ∈ A if and only if ∀a ∈ A such that R(a, b), we have that W (a, B ∪
{b}) ≥S W (B, a).
According to [10], by using the notion of w-defence for checking the accept-
ability of the arguments in the weighted framework, it is possible to redefine all
the extension-based semantics presented in Definition 4.
Definition 12 (Extension-based semantics for WAFS ). Given a WAFS
F = hA, R, W, Si, a subset of arguments B ⊆ A is w-conflict-free if W (B, B) =
>. A w-conflict-free subset B is then
– w-admissible, if ∀a ∈ B − . W (a, B) ≥S W (B, a) (that is B w-defend itself
from the arguments in A \ B);
– w-complete, if it is w-admissible and each argument b ∈ A such that B ∪ {b}
is w-admissible belongs to B;
– w-stable, if it is w-admissible and ∀a ∈
/ B. ∃b ∈ B such that W (b, a) ;
– w-preferred, if it is a maximal (with respect to set inclusion) w-admissible
subset of A;
– w-grounded, if it is the maximal (with respect to set inclusion) w-admissible
extension included in the intersection of w-complete extensions;
– w-quasi-strongly admissible2 , if ∀a ∈ B− , ∀b ∈ B. ∃C ⊆ B \ {b} with
W (a, B) ≥S W (C, a).
The definition for w-quasi-strongly admissible extensions, first given in [12],
states that a subset of arguments B is w-strongly admissible when for all b ∈ B,
B is defended by a subset of B that does not include b. In other words, each
argument in B is defended by the rest of the arguments in B.
Contrary from classical AFs, for which we can use the procedure in [13] for
assigning labels to the arguments in such a way that there is a correspondence
between the labelling and the set of extension, no work on this direction has
been done for what concerns the weighted case.
3 Labelling for Weighted AFs
We extend the notion of labelling introduced in [13] to weighted AFs. In partic-
ular, we consider a WAFS and we provide a definition for the labelling. Further-
more, we give the conditions for determining whether a labelling corresponds to
2
The definition for the w-quasi-strongly admissible semantics is introduced in [12],
where the authors refer to it by the term w-strongly admissible. However, differently
from the classical case, the defending set B0 ⊆ B\{a} is not required to recursively be
w-strongly admissible, and thus we considered it more appropriate to use a different
name.
A Labelling Semantics for Weighted Argumentation Frameworks 7
a certain extension. In order to incorporate the notion of weighted defence in
the labelling, we need to take into account the strength of the attack relations.
Definition 13 (Labelling for WAFS ). Let F = hA, R, W, Si be a WAFS . A
labelling L of F is a total function L : A → {IN, OUT, UNDEC}. For any A ⊆ A,
we denote A|IN , A|OUT and A|UNDEC the set of all the arguments labelled IN, OUT
and UNDEC by L, respectively. We also define, for each argument, the weight of
attacks, incoming into and outgoing from an argument, as wa− |IN = W (a− |IN , a)
and wa+ |IN = W (a, a+ |IN ).
According to the definition of collective defence [9] we need to know the
strength resulting from the composition of all the attacks towards anN argument.
In the weighted system, OUT arguments are associated with the of the in-
coming attacks. An argument a with label OUT is attacked by the arguments in
a− |IN with a total strength that is expressed by wa− |IN . With this information,
one can easily compute the acceptability of defended arguments. The main issue
one has to take into account when dealing with the study of semantics in WAFS
is the notion of weighted defences among the arguments. According to the clas-
sical notion of defence, an argument a is defended from the attack of another
argument b if there exists a third argument c that attacks b in turn. On the other
hand, when a weight is assigned to the attacks, the previous condition cannot
ensure alone that the argument a will be defended by c: it can be the case that
the attack c → b is not strong enough to defeat b → a and thus to justify a (see
arguments a, c and d in Figure 1).
Fig. 1. Example of two labellings on a WAFS with a weighted semiring where IN
arguments are highlighted in green, UNDEC in yellow, and OUT in red. When d is IN, e
is UNDEC (and vice versa) because arguments a and b can only defend one of the two.
According to the definition of collective weighted defence
N given in [9], a set
of argument is defended from an N attacker b only if the of all the defending
arguments is stronger than the of the attacks coming from b. This means that
the strength of the attacks of the defending arguments is distributed among the
defended arguments, so it is not guaranteed for two arguments that are separately
w-defended to sill be w-defended when considered together (this is what happens
in the example in Figure 1 with arguments d and e).
In the following, we give a characterisation of the weighted semantics through
the notion of labelling of WAFS . The intuition behind this representation is that
8 S. Bistarelli and C. Taticchi
when an argument a attacked by an OUT b cannot be labelled IN because of
another IN argument that is “consuming” the attacks of the defending arguments
towards b, then a is labelled UNDEC.
Fact 1 (w-conflict-free labelling) The w-conflict-free labelling coincides with
the conflict-free labelling.
Indeed, since attacks are not allowed within a conflict-free set of arguments,
one does not need to consider the weights. We now define the w-admissible
labelling.
Definition 14 (w-admissible labelling). Let L be a labelling of a WAFS F =
hA, R, W, Si and a ∈ A. L is a w-admissible labelling for F if and only if:
– L(a) = IN =⇒ a− = a− |OUT ∧ ∀b ∈ a− . wb− |IN ≤S wb+ |IN
– L(a) = OUT =⇒ wa− |IN
The condition wb− |IN ≤S wb+ |IN makes sure that the composition of the attacks
of the arguments defending a is stronger than the attack of b. For an argument
to be OUT, we require wa− |IN , that is to say that that there must exist
at least an attack coming from an IN argument (as for the classical admissible
labelling). Indeed, > means that there is no attack between two arguments. The
WAFS used in Figure 1 admits six w-admissible labellings, corresponding to the
sets of IN arguments {a}, {b}, {a, b}, {a, b, d} and {a, b, e} (depicted in Figure 1),
and the empty set.
Definition 15 (w-complete labelling). Let L be a labelling of a WAFS F =
hA, R, W, Si and a ∈ A. L is a w-complete labelling for F if and only if:
– L(a) = IN ⇐⇒ a− = a− |OUT ∧ ∀b ∈ a− . wb− |IN ≤S wb+ |IN
– L(a) = OUT ⇐⇒ wa− |IN
The definition of the w-complete labelling is similar to the w-admissible one,
with the exception that the conditions given for IN and OUT arguments are both
necessary and sufficient. The two labellings in Figure 1 represent all and only
w-complete labellings for the considered WAFS .
Definition 16 (w-stable labelling). Let L be a labelling of a WAFS F =
hA, R, W, Si. L is a w-stable labelling for F if and only if
– L is a w-complete labelling and
– A|UNDEC = ∅
According to the classical definition, a stable semantics partitions the ar-
guments in two disjoint sets: one contains the arguments that are either not
attacked or defended by other acceptable arguments, while the other contains
the rest of the arguments (i.e., those that are attacked and not defended). In the
weighted case, we obtain the same kind of partition through Definition 16. The
examples in Figure 1 do not represent w-stable labellings since both of them
have an UNDEC argument (respectively e and d). The labelling in Figure 2 is,
instead, w-stable.
We next present the w-preferred labelling for WAFS .
A Labelling Semantics for Weighted Argumentation Frameworks 9
Fig. 2. Examples of a WAFS with a w-stable labelling.
Definition 17 (w-preferred labelling). Let L be a labelling of a WAFS F =
hA, R, W, Si. L is a w-preferred labelling for F if and only if
– L is a w-admissible labelling and
– A|IN is maximal among all the w-admissible labellings
As for the classical definition, also in the weighted case the w-preferred ex-
tensions is the largest admissible sets. The WAFS in Figure 1 has only two
w-preferred labellings, both represented in the picture.
Definition 18 (w-grounded labelling). Let L be a labelling of a WAFS F =
hA, R, W, Si and a ∈ A. L is a w-grounded labelling for F if and only if:
– L(a) = IN ⇐⇒ for all w-complete labellings L0 , L0 (a) = IN and
– L(a) = OUT ⇐⇒ wa− |IN
We know from [12] that the w-grounded extension always exists, is unique and
corresponds to any maximal w-admissible extension included in the intersection
of w-complete extensions. None of the labelling in Figure 1 is w-grounded. Indeed
the intersection of IN arguments in the example WAFS is {a, b}, that is neither d
nor e should be IN. Figure 2, instead, shows an example of w-grounded labelling.
Definition 19 (w-quasi-strongly admissible labelling). Let L be a labelling
of a WAFS F = hA, R, W, Si and a ∈ A. L is a w-quasi-strongly admissible
labelling for F if and only if:
– L(a) = IN =⇒ a− = a− |OUT ∧ wb− |IN \{a} ≤S wb+ |IN
– L(a) = OUT =⇒ wa− |IN
We obtain a w-quasi-strongly admissible labelling by imposing that every IN
argument is always defended by other IN arguments. The labelling in Figure 2 is
not a w-quasi-strongly admissible labelling: in fact, the attack of the IN argument
a towards the OUT argument b is not sufficient alone to defend c. On the other
hand, both the labellings in Figure 1 are w-quasi-strongly admissible.
The sets of arguments labelled IN by the above-defined labellings for WAFS
are equivalent to the extensions of the corresponding semantics.
Theorem 1. A labelling L of a WAFS F = hA, R, W, Si is a w-admissible (re-
spectively w-complete, w-stable, w-preferred, w-grounded, w-quasi-strongly ad-
missible) labelling if and only if A|IN is a w-admissible (respectively w-complete,
w-stable, w-preferred, w-grounded, w-quasi-strongly admissible) extension of F .
10 S. Bistarelli and C. Taticchi
Proof. We show for each semantics the correspondence between the IN arguments
and the set of extensions. We refer to Definition 12 for the WAFS semantics.
– (L is w-admissible ⇒ A|IN is w-admissible.) The OUT arguments attacking
A|IN are defeated by A|IN . Thus, A|IN is w-defend from the attacks coming
from A \ A|IN and so it is a w-admissible extension.
– (A|IN is w-admissible ⇒ L is w-admissible.) A|IN w-defends itself from the
attacks of every b ∈ A \ A|IN , so W (A|IN , b) ≤S W (b, A|IN ). Moreover, every
a ∈ A|IN , is IN and thus L is a w-admissible labelling.
– (L is w-complete ⇒ A|IN is w-complete.) When L is w-complete, then it is
also w-admissible and it labels all the arguments w-defended by A|IN as IN.
Hence A|IN is a w-complete extension.
– (A|IN is w-complete ⇒ L is w-complete.) In this case A|IN is a w-admissible
extension where all the w-defended arguments belong to A|IN . Then L is
w-complete labelling.
– (L is w-stable ⇒ A|IN is w-stable.) L is a w-complete labelling in which no
argument is labelled UNDEC. Thus, the set A|IN attacks all the other argu-
ments in A \ A|IN , and so A|IN is a w-stable extension.
– (A|IN is w-stable ⇒ L is w-stable.) We have that the set A|IN is attacking all
the arguments in A\A|IN , so A|UNDEC = ∅. Then, since A|IN is a w-admissible
extension containing all the w-defended arguments, A|IN is a w-complete
extension and L a w-stable labelling.
– (L is w-preferred ⇒ A|IN is w-preferred.) The set of arguments labelled IN
by L coincides with a w-admissible extension which is maximal with respect
to the set inclusion. Follows that A|IN is a w-preferred extension.
– (A|IN is w-preferred ⇒ L is w-preferred.) We have that A|IN is a maximal
w-admissible extension, so L is a w-preferred labelling.
– (L is w-grounded ⇒ A|IN is w-grounded.) If L is w-grounded, all the ar-
guments in A|IN are also INin any w-complete labelling, thus A|IN repre-
sents the maximal w-admissible extension included in the intersection of
w-complete extensions.
– (A|IN is w-grounded ⇒ L is w-grounded.) A|IN contains all and only argu-
ments that are included in the intersection of w-complete extensions, so L
is a w-grounded labelling.
– (L is w-strongly admissible ⇒ A|IN is w-strongly admissible.) The OUT ar-
guments attacking any argument a ∈ A|IN are defeated by (A \ {a})|IN .
Thus, any argument in A|IN is w-defend by the other arguments in A|IN
from the attacks coming from A \ A|IN and so A|IN is a w-strongly admissi-
ble extension.
– (A|IN is w-strongly admissible ⇒ L is w-strongly admissible.) Each argument
a ∈ A|IN is w-defends by (A \ {a})|IN from the attacks of every b ∈ a− ∩ (A \
A|IN ), so W ((A \ {a}), b) ≤S W (b, A|IN ). Hence L is a w-strongly admissible
labelling.
t
u
We summarize in Table 3 the conditions specified in Definitions from 14 to 18
for obtaining weighted labellings corresponding to the Dung semantics.
A Labelling Semantics for Weighted Argumentation Frameworks 11
Table 1. Summarisation of the introduced labellings for WAFS .
conditions on IN arguments conditions on OUT arguments other conditions
w-cf L(a) = IN =⇒ a− |IN = ∅ L(a) = OUT =⇒ a− |IN 6= ∅
L(a) = IN =⇒ a− = a− |OUT
w-adm L(a) = OUT =⇒ wa− |IN
∧∀b ∈ a− . wb− |IN ≤S wb+ |IN
− −
L(a) = IN ⇐⇒ a = a |OUT
w-com L(a) = OUT ⇐⇒ wa− |IN
∧∀b ∈ a− . wb− |IN ≤S wb+ |IN
L(a) = IN ⇐⇒ a− = a− |OUT
w-stb L(a) = OUT ⇐⇒ wa− |IN A|UNDEC = ∅
∧∀b ∈ a− . wb− |IN ≤S wb+ |IN
− −
L(a) = IN =⇒ a = a |OUT
w-pre L(a) = OUT =⇒ wa− |IN A|IN max w-adm
∧∀b ∈ a− . wb− |IN ≤S wb+ |IN
0
L(a) = IN ⇐⇒ ∀L w-com,
w-gde L(a) = OUT ⇐⇒ wa− |IN
L0 (a) = IN
− −
L(a) = IN =⇒ a = a |OUT
w-qsa L(a) = OUT =⇒ wa− |IN
∧wb− |IN \{a} ≤S wb+ |IN
The conditions we give for the weighted semantics are a generalization of
the classical case, and all the labellings for WAFS corresponds to the respective
classical semantics when the framework is instantiated with a boolean semiring.
When the WAFS is instantiated with a boolean semiring, all the attacks from an
argument to another are associated with the value f alse and also wa− |IN always
corresponds to f alse.
Theorem 2. The labelling of a WAFS instantiated with a boolean semiring cor-
responds to the classical labelling.
Proof. By Definition 8, the weight of an attack between two arguments in a
WAFS F where S is boolean always correspond to the value f alse. Since the
composition operator is ∧, also the ⊗ of every pair of attacks in F is f alse,
and thus assigning a labelling boils down to checking the existence of attacks
between arguments, as for the crisp case. t
u
It follows that if L is a w-admissible (respectively w-complete, w-stable,
w-preferred, w-grounded) labelling of a WAFS F , then L is an admissible (re-
spectively complete, stable, preferred, grounded) labelling of F .
4 Implementation
To complete our study and facilitate the use of weighted labelling semantics
for argumentation-based application, we provide a tool able to represent WAFS
and visualize the computed labellings for various semantics. For this purpose, we
extend ConArg3 [11], a suite of tools for argumentation, with a series of function-
alities for handling weighted argumentation problems. The web interface, which
3
ConArg website: http://dmi.unipg.it/conarg.
12 S. Bistarelli and C. Taticchi
is shown in Figure 3, is implemented in JavaScript and relies on a server-side
solver written in C. In the following, we describe an example of use of the tool
for weighted argumentation.
Fig. 3. ConArg web interface displaying a weighted labelling for a WAFS . The high-
lighted areas corresponds to: 1) semantics selection, 2) representation of weights by
stroke/label, 3) solution selection, 4) semiring selection, 5) input area, 6) output area.
First of all, we use panel 4 of Figure 3 to select a semiring: this determines
both the representation of the AF (for instance classical, weighted, probabilistic)
and the kind of solution provided by the solver. If weighted is chosen, it is possible
to specify a WAFS by either using the input area (panel 5) or directly clicking
on the canvas to draw arguments and attacks. The next step is to select the
semantics (panel 1) for which we want obtain a labelling. Since we selected the
weighted semiring, we will obtain a weighted labelling. The solver computes the
sets of IN arguments, that are then displayed in panel 6. The labellings are
directly visible on the WAFS through the usual colour scheme: IN arguments
are green; any arguments attacked by an IN is red (that stands for OUT); all the
remaining arguments (i.e., the UNDEC ones) are yellow. In case the solver returns
more than one solution for the selected semantics (as happens in Figure 3), we
can choose which labelling to visualise by using panel 3.
5 Related Work
The problem of extending classical AFs with values expressing the strength of
arguments and attacks is widely studied, and many different approaches have
been presented in the literature. In [1], the authors take into account prefer-
ence orderings for comparing arguments, while in [6] the success of an attack
A Labelling Semantics for Weighted Argumentation Frameworks 13
conducted by an argument toward another one depends on an ordering among
the “values” promoted by each argument. A study on bipolar WAFs is con-
ducted in [24], where the authors present an extension for weighted frameworks
that takes into account two different types of relations (one for attack and one
for support). Another formalism based on a notion of strength is given in [5],
were arguments in Quantitative Argumentation Debate Frameworks are eval-
uated through a score system. The main difference with our work lies in the
fact that we take into account the basic definition of WAFs [19], without fur-
ther refinements on the framework level. Moreover, our study is focused on the
interpretation of the labelling in the weighted case.
For what concern the notion of weighted defence, many possible definitions
can be considered: for instance, Martı̀nez, Garcı́a and Simari [22] use the relative
strength of the attacks in order to determine if some defence constraints are
satisfied, while in [17] the authors aggregate the weights of the defence and
check if this value is greater than the weight of the corresponding attack. We, on
the other hand, use the notion introduced in [9], that also generalises the other
two approaches mentioned above.
6 Conclusion and Future Work
With this work, we introduce a labelling for semiring-based WAFs (never done
before), together with a set of labelling conditions corresponding to extensions for
some semantics. We also show that our labelling function generalises the classical
approach for the non-weighted case. We have also developed and made available
online an implementation of the labelling for WAFs. We have considered the
definition of collective defence provided in [9], for which an argument a of a WAFS
is defended by a set of arguments a− |IN when W (a− |IN , a) ≥S W (a, a− |IN ).
As future work, we plan to extend this work in different directions. For in-
stance, since all the definitions we give for weighted semantics are parametric to a
chosen notion of defence, it is possible to obtain labellings for semantics in which
the weighted defence is differently declined. The definitions of the labelling-based
semantics for WAFs, that we give in Section 3, do not include conditions for the
UNDEC since they are obtained from IN and OUT arguments. In this sense, we
would like to investigate the possible advantages of giving explicit conditions for
labelling the UNDEC arguments, similarly to what is done in [23] for classical
AFs. An interesting study could then be carried out on the dont care and dont
know labels, that are used in [4] as further differentiation of UNDEC arguments.
In our context, the difference between the two labels could be made more con-
tinuous by considering the weight on the attack relations. We also plan to give
a definition of w-strongly admissible extension (generalising the one provided
in [3] for the crisp case) and introduce the respective labelling.
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