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Semi-Stable Semantics for Abstract Dialectical Frameworks

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dialectical frameworks (ADFs) have been introduced as a formalism for modeling and evaluating argumentation allowing general logical satisfaction conditions. Different criteria that have been used to settle the acceptance of arguments are called semantics. However, the notion of semi-stable semantics as studied for abstract argumentation frameworks has received little attention for ADFs. In the current work, we present the concepts of semi-two-valued models and semi-stable models for ADFs. We show that these two notions satisfy a set of plausible properties required for semi-stable semantics of ADFs. Moreover, we show that semi-two-valued and semi-stable semantics of ADFs form a proper generalization of the semi-stable semantics of AFs, just like two-valued model and stable semantics for ADFs are generalizations of stable semantics for AFs.
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Semi-Stable Semantics for
Abstract Dialectical Frameworks
Atefeh Keshavarzi Zafarghandi ,Rineke Verbrugge ,Bart Verheij
Department of Artificial Intelligence, Bernoulli Institute,
University of Groningen, The Netherlands
{A.Keshavarzi.Zafarghandi, L.C.Verbrugge, Bart.Verheij}@rug.nl
Abstract
Abstract dialectical frameworks (ADFs) have been intro-
duced as a formalism for modeling and evaluating argumen-
tation allowing general logical satisfaction conditions. Dif-
ferent criteria that have been used to settle the acceptance of
arguments are called semantics. However, the notion of semi-
stable semantics as studied for abstract argumentation frame-
works has received little attention for ADFs. In the current
work, we present the concepts of semi-two-valued models
and semi-stable models for ADFs. We show that these two
notions satisfy a set of plausible properties required for semi-
stable semantics of ADFs. Moreover, we show that semi-
two-valued and semi-stable semantics of ADFs form a proper
generalization of the semi-stable semantics of AFs, just like
two-valued model and stable semantics for ADFs are gener-
alizations of stable semantics for AFs.
1 Introduction
Formalisms of argumentation have been introduced to model
and evaluate argumentation. Abstract argumentation frame-
works (AFs) as introduced by Dung (1995) are a core for-
malism in formal argumentation. A popular line of research
investigates extensions of Dung’s AFs that allow for a richer
syntax (see, e.g. Brewka, Polberg, and Woltran 2014).
In this work, we investigate a generalisation of
Dung’s AFs, namely, abstract dialectical frameworks
(ADFs) (Brewka et al. 2018), which are known as an ad-
vanced abstract formalism for argumentation covering sev-
eral generalizations of AFs (Brewka, Polberg, and Woltran
2014; Polberg 2017; Dvoˇ
r´
ak, Keshavarzi Zafarghandi, and
Woltran 2020). This is accomplished by acceptance con-
ditions which specify, for each argument, its relation to its
neighbour arguments via propositional formulas. These con-
ditions determine the links between the arguments which
can be, in particular, attacking or supporting.
In formal argumentation one is interested in investigat-
ing ‘How is it possible to evaluate arguments in a given
formalism? Answering this question leads to the introduc-
tion of several types of semantics. In AFs, one starts with
selecting a set of arguments without any conflicts. Conflict-
freeness is a main characteristic of all types of semantics of
AFs. Very often a new semantics is an improvement of an
already existing one by introducing further restrictions on
the set of accepted arguments or possible attackers. A list
of semantics of AFs is presented in (Dung 1995), namely
conflict-free, admissible, complete, preferred, and stable se-
mantics. Further semantics for AFs have been introduced
later on, for instance, stage semantics (Verheij 1996), semi-
stable semantics first in (Verheij 1996) (under a different
name) then further investigated in (Caminada 2006), ideal
semantics (Dung, Mancarella, and Toni 2007), and eager se-
mantics (Caminada 2007b). Each semantics presents a point
of view on accepting arguments.
Most of the semantics of AFs have been defined for
ADFs and it has been shown that semantics of ADFs are
generalizations of semantics of AFs (Brewka et al. 2018;
Gaggl, Rudolph, and Straß 2021). In this work, we focus on
semi-stable semantics for ADFs, in a way that follows the
same idea of semi-stable semantics of AFs. To this end, we
first present a weaker version of the two-valued models of
ADFs, which we call semi-two-valued models. Then we de-
fine semi-stable models for ADFs as a special case of semi-
two-valued models of a given ADF. The relation between
semi-two-valued and semi-stable models is similar to the re-
lation between two-valued and stable models for ADFs. The
difference is that a stable model is chosen among two-valued
models, however, a semi-stable model will be chosen among
semi-two-valued models of a given ADF.
Some of the semantics have become popular in the do-
main of argumentation, such as grounded semantics, pre-
ferred semantics and stable semantics. Each AF has a unique
grounded extension, and one or more preferred extensions.
However, it is possible that an AF does not have any sta-
ble extension. Because of this shortcoming of stable seman-
tics, in order to pick at least one set of arguments, preferred
and grounded semantics become more popular in argumen-
tation. In contrast, stable semantics still enjoys a strong
support in logic programming (1988) and answer set pro-
gramming (1991), since it is preferred to have no outcome
as opposed to an imperfect one. On the one hand, in argu-
mentation a grounded extension presents the least amount
of information about the acceptance of arguments. That is, a
grounded extension collects a set of arguments about which
there is no doubt. In other words, the grounded extension of
a given AF is very skeptical. On the other hand, it is possi-
ble that an AF has a stable extension but the set of preferred
extensions and stable extensions are not equal.
To overcome this deficiency, semi-stable semantics have
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been introduced for AFs. Semi-stable semantics is a way
of approximating stable semantics when a given AF does
not have any stable extension. Key characteristics of semi-
stable semantics in AFs are: 1. It is placed between stable
semantics and preferred semantics; 2. If an AF has at least
one stable extension, then the set of stable extensions and
the set of semi-stable extensions coincide; 3. Each finite AF
has at least one semi-stable extension.1Computational com-
plexity of semi-stable semantics is studied in (Dunne and
Caminada 2008). Furthermore, (Caminada 2007a) presents
an algorithm to compute semi-stable semantics of AFs.
In this paper we propose a notion of semi-stable semantics
for ADFs. First we discuss required properties for such a se-
mantics in order to ensure that our notion is a proper general-
ization of the notion of semi-stable semantics for AFs. Then
we define our notion of semi-stable semantics for ADFs and
study its properties. It turns out that our notion fulfills the
required properties presented in Section 1.1.
1.1 Requirements of Semi-Stable Semantics
For AFs, the property holds that a semi-stable extension is
stable in the AF restricted to the arguments that have a truth
value (accepted/rejected, in/out). This holds in general, and
in particular also for AFs that have no stable extension. In
the current work we follow this same idea to extend the no-
tion of semi-stable semantics of AFs for ADFs.
In ADFs, the notion of stable model is defined based on
the notion of two-valued model. An ADF may have no
stable model. On the one hand, if a given ADF does not
have any two-valued model, then it does not have any stable
model. On the other hand, an ADF may have two-valued
models, while none of them is a stable model. We focus on
the first issue here. To define the notion of semi-stable se-
mantics for ADFs, we follow the same method as for stable
semantics of ADFs. That is, first we introduce the notion of
semi-two-valued semantics. Subsequently, we pick semi-
stable models among semi-two-valued models of a given
ADF. A semi-two-valued model is a complete interpreta-
tion, that is, the number of arguments that are assigned to
unknown is -minimal among all complete interpretations.
Further, a semi-stable model is a semi-two-valued model v
that has a constructive proof for arguments that are assigned
to tin v. We show that the semi-stable semantics/semi-two-
valued model presented in this work will satisfy the follow-
ing conditions, which are akin to the properties of the notion
of semi-stable semantics of AFs.
1. A semi-stable/semi-two-valued model of a given ADF
should maximize the union of the sets of the accepted and
of the rejected/denied arguments among all complete in-
terpretations, with respect to subset inclusion;
2. Each semi-stable/semi-two-valued model is a preferred
interpretation;
3. Each stable model is a semi-stable/semi-two-valued
model;
4. Each finite ADF has at least one semi-two-valued model;
1(Verheij 2003, Example 5.8) shows that existence is not guar-
anteed for infinite AFs. See also (Caminada and Verheij 2010).
5. If an ADF has a stable model, then the set of stable mod-
els coincides with the set of semi-stable models;
6. The notion of semi-stable/semi-two-valued semantics for
ADFs is a proper generalization of semi-stable semantics
for AFs.
This paper is structured as follows. In Section 2, we
present the relevant background of AFs and ADFs. Then, in
Section 3, we present definitions of semi-two-valued/semi-
stable semantics for ADFs. In this section, we show that
the notion of semi-stable semantics and semi-two-valued se-
mantics satisfy the required properties, items 15, presented
above in this section. Further, in Section 4 we show that the
notion of semi-stable/semi-two-valued semantics of ADFs is
a proper generalization of the concept of semi-stable seman-
tics of AFs, cf. the 6th property. In Section 5, we present
the conclusion of our work. Furthermore, we briefly discuss
a related research, in particular, (Alcˆ
antara and S´
a 2018) has
also proposed a notion of semi-stable semantics for ADFs.
2 Background
2.1 Abstract Argumentation Frameworks
We recall the basic notions of Dung’s abstract argumentation
frameworks (AFs) (Dung 1995).
Definition 1. (Dung 1995) An abstract argumentation
framework (AF) is a pair (A, R)in which Ais a set of ar-
guments and RA×Ais a binary relation representing
attacks among arguments.
Let F= (A, R)be an AF. For each a, b A, the relation
(a, b)Ris used to represent that ais an argument attack-
ing the argument b. An argument aAis, on the other
hand, defended by a set SAof arguments (alternatively,
the argument is acceptable with respect to S) (in F) if for
each argument cA, it holds that if (c, a)R, then there
is a sSsuch that (s, c)R(sis called a defender of a).
Different semantics of AFs present which sets of argu-
ments in a given AF can be accepted jointly. In (Dung 1995),
extension-based semantics of AFs are presented; we recall
them in Definition 2. An extension is a set of arguments of
a given AFs. Set SAis called a conflict-free set (exten-
sion) (in F) if there is no pair a, b Ssuch that (a, b)R.
The characteristic function ΓF: 2A7→ 2Ais defined as
ΓF(S) = {a|ais defended by S}.
Definition 2. Let F= (A, R)be an AF. A set Scf(F)is
admissible in Fif SΓF(S);
preferred in Fif Sis -maximal admissible;
complete in Fif S= ΓF(S);
grounded in Fif Sis the -least fixed point of ΓF(S);
stable in Fif aA\S:bSs.t. (b, a)R.
Let σ {cf,adm,grd,prf,com,stb}be different semantics
in the obvious manner. Now we define that the set of all σ
extensions for an AF Fis denoted by σ(F).
Definition 3. (Caminada 2006) Let F= (A, R)be an AF
and let Sbe an extension of F. For aA, we write a+=
{b|(a, b)R}and S+=∪{a+|aS}. Set Sis called
a semi-stable extension iff Sis a complete extension where
SS+is maximal.
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The set of semi-stable extensions of Fis denoted by
semi-stb(F). We recall some of the main properties of
semi-stable semantics of AFs in Theorem 1.
Theorem 1. (Caminada 2006) Let F= (A, R)be a finite
AF, i.e., |A|is finite, and let Sbe an extension of F.
semi-stb(F)6=;
if Ssemi-stb(F), then Sprf(F);
if Sstb(F), then Ssemi-stb(F);
if stb(F)6=, then stb(F) = semi-stb(F).
2.2 Abstract Dialectical Frameworks
In the current section, we briefly restate some of the key
concepts of abstract dialectical frameworks that are derived
from those given in (Brewka et al. 2018; Brewka et al.
2017b; Brewka et al. 2013; Brewka and Woltran 2010).
Definition 4. An abstract dialectical framework (ADF) is a
tuple D= (A, L, C)where:
Ais a finite set of arguments (statements, positions), de-
noted by letters;
LA×Ais a set of links among arguments;
C={ϕa}aAis a collection of propositional formulas
over arguments, called acceptance conditions.
An ADF can be represented by a graph in which nodes in-
dicate arguments and links show the relation among argu-
ments. Each argument ain an ADF is labelled by a proposi-
tional formula, called acceptance condition, ϕaover par (a)
where par(a) = {b|(b, a)L}. The acceptance condi-
tion of each argument clarifies under which condition it can
be accepted. Acceptance conditions indicate the set of links
implicitly, thus, there is no need to explicitly present Lin
the example ADFs. We restrict to the finite setting, thereby
excluding the complication mentioned in footnote 1.
An argument ais called an initial argument if par (a) =
{}. A three-valued interpretation v(for D) is a function v:
A7→ {t,f,u}, that maps arguments to one of the three truth
values true (t), false (f), or undecided (u). Interpretation v
is called trivial, and vis denoted by vu, if v(a) = ufor each
aA. Furthermore, vis called a two-valued interpretation
if for each aAeither v(a) = tor v(a) = f. For x
{t,f,u}we write vx={aA|v(a) = x}.
Truth values can be ordered via the information ordering
relation <igiven by u<itand u<ifand no other pair of
truth values are related by <i. Relation iis the reflexive
closure of <i. Let Vbe the set of all interpretations for an
ADF D. Interpretations can be ordered via iwith respect
to their information content, i.e., for v, w V, if w(a)i
v(a)for each aA, it is said that vis an extension of w,
denoted by wiv.
The characteristic operator ΓDmaps interpretations to
interpretations. Given an interpretation v(for D), the partial
valuation of ϕaby v, is v(ϕa) = ϕv
a=ϕa[b/>:v(b) =
t][b/:v(b) = f], for bpar(a).
Definition 5. Let Dbe an ADF and let vbe an interpretation
of D. Applying ΓDon vleads to v0such that for each aA,
abcd
>a ¬c¬bd
Figure 1: ADF of Example 1
v0is as follows:
v0(a) =
tif ϕv
ais irrefutable (i.e., ϕv
ais a tautology) ,
fif ϕv
ais unsatisfiable,
uotherwise.
The semantics of ADFs are defined via the characteristic op-
erator as in Definition 6.
Definition 6. Given an ADF D, an interpretation vis:
conflict-free in Diff v(s) = timplies ϕv
sis satisfiable
and v(s) = fimplies ϕv
sis unsatisfiable;
admissible in Diff viΓD(v);
preferred in Diff vis i-maximal admissible;
complete in Diff v= ΓD(v);
a (two-valued) model in Diff vis two-valued and
ΓD(v) = v;
the grounded interpretation in Diff vis the least fixed
point of ΓD.
Let σ(D)where σ {cf,adm,grd,prf,com,mod}be the
different semantics in the obvious manner. The set of all σ
interpretations for an ADF Dis denoted by σ(D).
Example 1. An example of an ADF D= (S, L, C )is
shown in Figure 1. To each argument a propositional for-
mula is associated, the acceptance condition of the argu-
ment. For instance, the acceptance condition of c, namely,
ϕc:¬bd, states that ccan be accepted in an interpretation
where bis denied and dis accepted. In Dthe interpretation
v={a7→ u, b 7→ t, c 7→ u, d 7→ u}is conflict-free, since
ϕv
b=a ¬cis satisfiable. However, vis not an admissi-
ble interpretation, because ΓD(v) = {a7→ t, b 7→ u, c 7→
f, d 7→ f}, that is, v6≤iΓD(v).
The interpretation v1={a7→ t, b 7→ u, c 7→ f, d 7→ f},
on the other hand, is an admissible interpretation, since
ΓD(v1) = {a7→ t, b 7→ t, c 7→ f, d 7→ f}and v1i
ΓD(v1). Moreover, in Dthe unique grounded interpretation
v2={a7→ t, b 7→ t, c 7→ f, d 7→ f}is a preferred interpre-
tation of D. In addition, com(D) = {v2}.
The notion of stable semantics for ADFs is defined follow-
ing similar ideas from logic programming. Stable models
extend the concept of minimal model in logic programming
by excluding self-justifying cycles of atoms. The concept
of stable semantics of ADFs has been presented in (Brewka
et al. 2013, Definition 6) and in (Brewka et al. 2017a, Defi-
nition 18); we recall it in Definition 7.
Definition 7. Let Dbe an ADF and let vbe a two-valued
model of D. Then vis a stable model of Dif vt=wt,
where wis the grounded interpretation of the stb-reduct
Dv= (Av, Lv, Cv), where Av=vt,Lv=L(Av×Av),
and ϕa[p/:v(p) = f]for each aAv.
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b
a
c
¬b
b ¬c¬a ¬b
Figure 2: The ADF of Example 2.
a
cbc
>
b ¬c¬a ¬⊥ >
stb-reduct Dv1
¬⊥ ¬b >
stb-reduct Dv2
Figure 3: The reduct of ADF Dof Example 2.
Intuitively, the grounded interpretation collects all the infor-
mation that is beyond any doubt, thus, it is said that there
is a constructive proof for all arguments presented in the
grounded interpretation. Hence, a two-valued model vof D
is a stable interpretation (model), if there exists a construc-
tive proof for all arguments assigned to true in v, in case all
arguments that are assigned to false in vare actually false.
Example 2 clarifies the notion of stable semantics of ADFs.
Example 2. Let D= ({a, b, c},{ϕa:¬b, ϕb:b ¬c, ϕc:
¬a ¬b})be an ADF, depicted in Figure 2. Dhas two
two-valued models, namely v1={a7→ t, b 7→ f, c 7→ t}
and v2={a7→ f, b 7→ t, c 7→ t}. We check whether
they are stable models. To investigate whether v1is a sta-
ble model, first we evaluate the stb-reduct of Dunder v1,
namely Dv1= (Av1, Lv1, Cv1). Here Av1={a, c},
Lv1={(a, c)}, and ϕa:¬⊥ > and ϕc:¬a ¬⊥ >.
The reduct Dv1is depicted in Figure 3 (on the left). Since
the unique grounded interpretation of Dv1is w={a7→
t, c 7→ t}, i.e., wt=vt
1, we have v1stb(D).
We show that v2is not a stable model of D. To this end,
we first evaluate Dv2= (Av2, Lv2, Cv2), where Av2=
{b, c},Lv2={(b, b),(b, c),(c, b)}, and ϕb:b ¬cand
ϕc:¬⊥¬b >, depicted in Figure 3 (on the right). Since
the unique grounded interpretation of Dv2is w={b7→
u, c 7→ t}, i.e., wt6=vt
2, we have v26∈ stb(D). Intuitively,
it holds that v26∈ stb(D), since in v2the acceptance of bde-
pends on bitself, that is, there is a cyclic justification. Thus,
v2violates the main condition of stable semantics that a sta-
ble model should have no self-justifying cycles of atoms.
An ADF may have no stable model. Example 3 presents an
ADF that has a two-valued model, but no stable model.
Example 3. Let D= ({a, b, c},{ϕa:cb, ϕb:c, ϕc:a
b}), depicted in Figure 4. The only two-valued model of D
is v={a7→ t, b 7→ t, c 7→ t}. However, wt={} where w
is the grounded interpretation of Dv, where Dv=D. Thus,
wt6=vt. Hence, vis not a stable model of D.
Definition 8 presents the associated ADF for a given AF.
b
a
c
cb
cab
Figure 4: The ADF of Example 3
Definition 8. For an AF F= (A, R), define the ADF as-
sociated to Fas DF= (A, R, C)with C={ϕa}aAsuch
that for each aAthe acceptance condition is as follows:
ϕa=^
(b,a)R
¬b
The semantics of ADFs generalize those of AFs. Since
in AFs there is no direct support link, stable models (i.e.,
to avoid cyclic support among arguments) and models are
equal in the associated ADF DFof a given AF F.
3 Semi-Stable Semantics for ADFs
Before providing the formal definition of semi-stable seman-
tics for ADFs, we present the intuition why an ADF may
have no stable models. An ADF Dmay not have any stable
model due to either of the following two reasons:
1. mod(D) = , i.e., Ddoes not have any two-valued mod-
els from which to pick a stable model; or,
2. mod(D)6=, but for any vmod(D)it holds that v6∈
stb(D); that is, when there is no constructive proof for
arguments that are assigned to tin vwhere vmod(D).
Nonetheless, there are many cases about which one might
want to draw a conclusion even when a given ADF does not
have any two-valued model or stable model. One option is
focusing on other semantics like preferred and grounded se-
mantics that exist for any ADF. However, a unique grounded
interpretation presents a piece of information about those ar-
guments about which there is no doubt. That is, it is possible
that in a given ADF the grounded interpretation has less in-
formation than each of its stable models. In other words, the
information of the grounded interpretation is too skeptical.
Furthermore, there exists an ADF Dsuch that stb(D)6=
but the set of stable semantics of Dand the set of preferred
semantics of Dare not equivalent, i.e., stb(D)(prf(D).
That is, by preferred semantics some non-stable models may
be introduced, even in the case that a stable model exists.
Example 4 is an instance of an ADF such that stb(D)6=
but stb(D)(prf(D).
Example 4. Let D= ({a, b, c},{ϕa:c ¬b, ϕb:c
¬a, ϕc:b ¬a})be an ADF, depicted in Figure 5. The
set of preferred interpretations of Dis prf(D) = {{a7→
t, b 7→ t, c 7→ t},{a7→ t, b 7→ f, c 7→ f}}. Both of the
preferred interpretations of Dare two-valued models of D.
However, stb(D) = {{a7→ t, b 7→ f, c 7→ f}}. That is,
stb(D)(prf(D).
Furthermore, the unique grounded interpretation of Dis
the trivial interpretation that has strictly less information
than the stable model of D, i.e., {a7→ t, b 7→ f, c 7→ f}.
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b
a
c
c ¬b
c ¬ab ¬a
Figure 5: The ADF of Example 4
Still, it is interesting to present a semantics for ADFs that
is equal to stable semantics if there exists a stable model.
In ADFs, to define the notion of stable semantics as it is
in (Brewka et al. 2018), first the notion of two-valued se-
mantics is introduced. Then a two-valued model is called
a stable model if it satisfies the conditions of Definition 7,
i.e., if it does not contain any support cycle. Since in AFs
there is no support cycle, these two notions are equal. That
is, for the associated ADF DFof a given AF Fit holds
that mod(DF) = stb(DF). Due to this distinction between
two-valued models and stable models in ADFs, different lev-
els of semi-stable semantics can be considered in ADFs for
the notion of semi-stable semantics of AFs. Here we fol-
low a similar method as presented in (Brewka et al. 2018;
Brewka et al. 2017b) for stable semantics to present the con-
cept of semi-stable semantics.
The first reason that an ADF Ddoes not have any stable
semantics is that Ddoes not have any two-valued model.
We focus on this issue to present an alternative semantics
for stable semantics of ADFs. In this alternative option, i.e.,
semi-stable semantics for ADFs, we are looking for a semi-
two-valued model, which is a partially two-valued model,
presented in Definition 9, that satisfies the condition of Defi-
nition 7, that is, it does not contain any support cycles among
arguments. These new points of view of acceptance of ar-
guments, which are called semi-two-valued semantics and
semi-stable semantics of ADFs, have to satisfy the require-
ments presented in Section 1.1. The properties in Section 1.1
are akin to the properties of the notion of semi-stable seman-
tics of AFs, presented in Theorem 1.
Definition 9. Let Dbe an ADF and let vbe an interpreta-
tion of D. An interpretation vis a semi-two-valued model
(interpretation) of Dif the following conditions hold:
1. vis a complete interpretation of D;
2. vuis -minimal among all wusuch that wis a complete
interpretation of D.
The set of semi-two-valued models of Dis denoted by
semi-mod(D). Note that when an ADF has a two-valued
model, then the set of semi-two-valued models and the set
of two-valued models coincide, which is shown in Lemma 1.
We introduce the concept of semi-stable models over the no-
tion of semi-two-valued models in Definition 10.
Definition 10. Let Dbe an ADF and let vbe a semi-two-
valued model of D. An interpretation vis a semi-stable
model (interpretation) of Dif the following condition holds:
vt=wts.t wis the grounded interpretation of sub-reduct
Dv= (Av, Lv, Cv), where Av=vtvu,Lv=L
(Av×Av), and ϕa[p/:v(p) = f]for each aAv.
The set of semi-stable models of Dis denoted by
semi-stb(D). Note that in Definition 10, in sub-reduct Dv
we assume that vis a semi-two-valued model (complete in-
terpretation) of D, however, in Definition 7, in sub-reduct
Dvit is assumed that a given interpretation vis a two-valued
model of D. Since in Definition 10, interpretation vis a
semi-two-valued model, it may contain an argument that is
assigned to u. Therefore, in sub-reduct Dvin Definition 10,
we keep those arguments that are assigned to uin vas well,
i.e., Av=vtvu. Arguments that are assigned to uin vwill
remain in ϕa[p/:v(p) = f]for each aAv. Intuitively,
a complete interpretation vis a semi-stable model of Dif
vuis -minimal among complete interpretations of Dand
there exists a constructive proof for arguments which are as-
signed to tin v, in case all arguments which are assigned to
false in vare actually false. Corollary 1 is a direct result of
Definition 3, which defines the notion of semi-stable model
over the set of semi-two-valued models of a given ADF.
Corollary 1. Let Dbe an ADF. Each semi-stable model of
Dis a semi-two-valued model of D.
Example 5 clarifies the notion of semi-stable semantics of
ADFs.
Example 5. Let D= ({a, b, c},{ϕa:¬a, ϕb:c(¬a
c), ϕc:b(ab)})be an ADF, depicted in Figure 6. The
set of preferred interpretations of Dis prf(D) = {{a7→
u, b 7→ t, c 7→ t},{a7→ u, b 7→ f, c 7→ f}}. None of
the preferred interpretations is a two-valued model. Thus, D
does not have any stable model. Both v1={a7→ u, b 7→
t, c 7→ t}and v2={a7→ u, b 7→ f, c 7→ f}are complete
interpretations of D. Furthermore, both v1and v2are semi-
two-valued models of D, since vu
1=vu
2={a}. However,
we show that only v2is a semi-stable model of D. To this
end, we first evaluate sub-reduct Dv2. Since no argument is
assigned to tin v2and only ais assigned to uin v,Av2=
{a}. Thus, Dv2= ({a},{ϕa:¬a}), depicted in Figure
7. It is clear that the unique grounded interpretation Dv2is
w={a7→ u}. Since wt=vt
2=, it holds that v2is a
semi-stable model of D.
On the other hand, v1is not a semi-stable model of D.
In v1, both band care assigned to tand ais assigned to
u, therefore, Av1=A. Since no argument is assigned to f
in v1, we have Dv1=D. The grounded interpretation of
D/Dv1is w={a7→ u, b 7→ u, c 7→ u}. That is, wt=.
However, vt
1={b, c}, i.e., wt6=vt
1. Thus, v1is not a semi-
stable model of D.
Proposition 1 shows the first property of semi-two-valued
model presented in Section 1.1.
Proposition 1. Let Dbe an ADF, and let vbe a semi-two-
valued model of D. It holds that vmaximizes the union of
the sets of the accepted and of the denied among all complete
interpretations of D, i.e., vtvfis maximal with respect to
subset inclusion.
Proof. Let D= (A, L, C)be a given ADF. Assume that v
is a semi-two-valued model of D. Toward a contradiction,
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b
a
c
¬a
c(¬ac)b(ab)
Figure 6: The ADF of Example 5
a
¬a
Figure 7: reduct Dv2of ADF of Example 5
assume that vtvfis not -maximal among all complete
interpretations of D. Thus, there exists a complete interpre-
tation wsuch that wtwfis -maximal. Thus, it holds
that vtvf(wtwf. Hence, it holds that wuvu,
i.e., vuis not minimal among complete interpretations of D.
That is, by Definition 9, vis not a semi-two-valued model
of D. This contradicts the assumption that vis a semi-two-
valued model of D. Thus, the assumption that vtvfis not
-maximal among complete interpretations is wrong.
Proposition 1 clarifies the distinction between preferred se-
mantics and semi-two-valued models of ADFs. While in-
terpretation vis a preferred interpretation of Dif it is i-
maximal in com(D), interpretation vis a semi-two-valued
model of Dif vtvfis -maximal in com(D). Corollary 2
is a direct result of Proposition 1 and the fact that each semi-
stable model is a semi-two-valued model.
Corollary 2. Let Dbe an ADF, and let vbe a semi-stable
model of D. It holds that vmaximizes the union of the sets
of the accepted and of the denied among all complete inter-
pretations of D, i.e., vtvfis -maximal in com(D).
Theorem 2 presents the second and the third required prop-
erties for semi-stable/semi-two valued semantics for ADFs,
presented in Section 1.1.
Theorem 2. Let Dbe an ADF.
1. Each semi-two-valued model of Dis a preferred inter-
pretation of D;
2. Each semi-stable model of Dis a preferred interpretation
of D;
3. Each stable model of Dis a semi-two-valued model of
D;
4. Each stable model of Dis a semi-stable model of D.
Proof. Let Dbe an ADF.
Proof of item 1: assume that vis a semi-two-valued model
of D. We show that vis a preferred interpretation of D.
Toward a contradiction, assume that v6∈ prf(D). By Def-
inition 9, vis a complete interpretation of D. That is, if
vis not a preferred interpretation, then there exists a pre-
ferred interpretation v0such that v <iv0. Thus, v0u(vu.
Hence, by Definition 9, vis not a semi-two-valued model
of D. This contradicts the assumption that vis a semi-
two-valued model of D. Therefore, the assumption that v
is not a preferred interpretation of Dis wrong.
Proof of item 2: assume that vis a semi-stable model of
D. We show that vis a preferred interpretation of D. By
Corollary 1, each semi-stable model of Dis a semi-two-
valued model of D. Thus, vis a semi-two-valued model
of D. By the first item of this theorem, vis a preferred
interpretation of D.
Proof of item 3: Assume that vis a stable model. First,
each stable model is a complete interpretation. Thus, the
first item of Definition 9 is satisfied. Second, each stable
model is a two-valued model, i.e., vu=. Thus, vuis
-minimal among all wu, where wis a complete inter-
pretation of D. Hence, the second item of Definition 9 is
satisfied. Thus, vis a semi-two-valued model of D.
Proof of item 4: assume that vis a stable model. By
the previous item, vis a semi-two-valued model of D.
We show that vsatisfies the condition of Definition 10.
Since vis a stable model, by Definition 7, vt=wt
such that wis the grounded interpretation of sub-reduct
Dv= (Av, Lv, Cv). Since vu=, in Definition 10,
Av=vt. That is, Definition 10 (semi-stable model) and
Definition 7 (stable-model) coincide for v. Thus, if vis a
stable model of D, then vis a semi-stable model of D.
The first two items of Theorem 2 imply that the set of semi-
stable/semi-two-valued models of an ADF Dis a subset of
the set of preferred interpretations of D, i.e., semi-stb(D)
prf(D)and semi-mod(D)prf(D). However, Proposi-
tion 2 indicates that the notion of preferred semantics co-
incides neither with the notion of semi-stable semantics nor
with the notion of semi-two-valued semantics. That is, there
exists an ADF Dsuch that prf(D)6⊆ semi-stb(D)and
prf(D)6⊆ semi-mod(D).
Proposition 2. There is an ADF Dsuch that the set of pre-
ferred interpretations of Ddoes not coincide with the set
of semi-stable models, nor with the set of semi-two-valued
models of D.
Proof. We show that there exists an ADF with a preferred
interpretation which is not a semi-two-valued model. To
this end, we use the ADF presented in ((Diller et al. 2020,
Theorem 6)). Consider ADF D= ({a, b, c, d, e},{ϕa:
¬c(¬d ¬b), ϕb:¬a(¬d ¬c), ϕc:¬b(¬d
¬a), ϕd:¬e(¬a ¬b ¬c), ϕe:¬d}), depicted
in Figure 8. Dhas four preferred interpretations, namely
v1={a7→ f, b 7→ f, c 7→ t, d 7→ t, e 7→ f},
v2={a7→ f, b 7→ t, c 7→ f, d 7→ t, e 7→ f},
v3={a7→ t, b 7→ f, c 7→ f, d 7→ t, e 7→ f}, and
v4={a7→ u, b 7→ u, c 7→ u, d 7→ f, e 7→ t}. It holds
that v1,v2,v3are semi-two-valued models/two-valued mod-
els of D, since vu
1=vu
2=vu
3=. However, v4is not a
semi-two-valued/semi-stable model, since vu
4={a, b, c},
that is, vu
4is not -minimal among vu
i, for 1i4. Thus,
in ADFs, the notion of preferred semantics is not equal to
the notion of semi-stable/semi-two-valued semantics.
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a
b
c
d
e
¬c(¬d ¬b)
¬a(¬d ¬c)
¬b(¬d ¬a)
¬e(¬a ¬b ¬c)
¬d
Figure 8: An ADF with a preferred interpretation that is not a semi-
stable/semi-two-valued model
Proposition 3 presents the fourth property required for semi-
two-valued semantics, presented in Section 1.1.
Proposition 3. Each ADF has at least one semi-two-valued
model.
Proof. Let Dbe an ADF. Each ADF has a unique grounded
interpretation. By the facts that the grounded interpretation
is the least fixed point of ΓDand the grounded interpreta-
tion is a least complete interpretation with respect to the i-
ordering, we conclude that each ADF has at least one com-
plete interpretation. By Definition 9, each semi-two-valued
model vis a complete interpretation where vuis -minimal
among other complete interpretations of D. Since the num-
ber of arguments is finite, the set of complete interpretations
is finite. That is, there exists a complete interpretation v
where vuis -minimal among all compete interpretations
of D. Thus, the set of semi-two-valued models of Dis non-
empty.
In Theorem 3, we show the fifth property of semi-stable se-
mantics, presented in Section 1.1: If an ADF Dhas a stable
model, then the set of stable models and the set of semi-
stable models of Dcoincide. To show this theorem, we need
some auxiliary results that are shown in Lemmas 1–3.
Lemma 1. Let Dbe an ADF. Assume that Dhas a two-
valued model. Then, the set of semi-two-valued models of D
and the set of two-valued models of Dcoincide.
Proof. Assume that Dhas a two-valued model v. Since vis
a two-valued model, it holds that vu=. Thus, by Def-
inition 9, vis a semi-two-valued model, i.e., mod(D)
semi-mod(D). It remains to show that every semi-two-
valued model of Dis also a two-valued model. Toward a
contradiction, assume that Dhas a semi-two-valued model
wwhich is not a two-valued model. Since wis a semi-two-
valued but not a two-valued model, it holds that wis a com-
plete interpretation and wu6=. However, since Dhas
a two-valued model, wuis not -minimal among all com-
plete interpretations of D. That is, by Definition 9, wis
not a semi-two-valued model. This contradicts the assump-
tion that wis a semi-two-valued model. That is, if Dhas a
two-valued model, then semi-mod(D)mod(D). Hence,
if ADF Dhas a two-valued model, then semi-mod(D) =
mod(D).
Lemma 2. Let Dbe an ADF. Assume that Dhas a stable
model. Then, the set of semi-two-valued models of Dand
the set of two-valued models of Dcoincide.
Proof. Let Dbe an ADF that has a stable model v. By
the fact that each stable model of a given ADF is a two-
valued model, it holds that vis a two-valued model. By
Lemma 1, if there exists a two-valued model, then the
set of two-valued models and the set of semi-two-valued
models coincide. So if an ADF has a stable model, then
semi-mod(D) = mod(D).
Lemma 3. Let Dbe an ADF. Assume that Dhas a stable
model. Then each semi-stable model of Dis a two-valued
model of D.
Proof. Let Dbe an ADF that has at least one stable model
v. By Lemma 2, the set of semi-two-valued models of
Dcoincides with the set of two-valued models of D, i.e.,
semi-mod(D) = mod(D). Moreover, by Corollary 1, each
semi-stable model of Dis a semi-two-valued model of D,
i.e., semi-stb(D)semi-mod(D). Thus, if Dhas a stable
model, then each semi-stable model of Dis a two-valued
model of D, i.e., semi-stb(D)mod(D).
Theorem 3. If ADF Dhas a stable model, then the sets of
stable models and semi-stable models of Dcoincide.
Proof. Let Dbe an ADF. By the forth item of Theorem 2,
each stable model of Dis a semi-stable model of D, i.e.,
stb(D)semi-stb(D).
Assume that Dhas a stable model vand a semi-stable
model v0. We show that v0is a stable model of D. Toward
a contradiction, assume that v0is not a stable model of D.
By Lemma 3, v0is a two-valued model of D, i.e., v0u=.
If v0is not a stable model of D, by Definition 7, it has to be
held that v0t6=wtwhere wis the grounded interpretation
of the stb-reduct Dv0= (Av0, Lv0, Cv0), where Av0=v0t,
Lv0=L(Av0×Av0), and ϕa[p/:v0(p) = f]for each
aAv0. This implies that the condition of Definition 10
does not hold for v0, since v0u=. Thus, v0is not a semi-
stable model of D. This is a contradiction by the assumption
that v0is a semi-stable model of D. Hence, the assumption
that Dhas a semi-stable model which is not a stable-model
is wrong. Thus, if Dhas a stable model, then semi-stb(D)
stb(D). Hence, if ADF Dhas a stable model, then stb(D) =
semi-stb(D).
Proposition 3 says that each ADF has at least one semi-two-
valued model. In contrast, Proposition 4 shows that an ADF
may have no semi-stable model. As we presented in the
beginning of this section the notions of semi-two-valued se-
mantics and semi-stable semantics of ADFs together fulfil
the properties required for the concept of semi-stable seman-
tics, presented in Section 1.1.
Proposition 4. There exists an ADF that does not have any
semi-stable model.
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Proof. Let Dbe the ADF presented in Example 3, i.e., D=
({a, b, c},{ϕa:cb, ϕb:c, ϕc:ab}). We showed, in
Example 3, that v={a7→ t, b 7→ t, c 7→ t}is a two-valued
model of D, however, it is not a stable model of D. Thus, by
Lemma 1, vis a semi-two-valued model of D. As we know
grd(Dv) = {∅}, however, vt={a, b, c}. Thus, vis not a
semi-stable model.
Corollary 3. Let Dbe an ADF that has a two-valued model.
If none of the two-valued models Dis a stable model of D,
then Ddoes not have any semi-stable model.
Proof. Let Dbe an ADF that has a two-valued model. Thus,
by Lemma 1 the set of two-valued models of Dcoincides
with the set of semi-two-valued models of D. That is, for
each two-valued model/semi-two-valued model vof Dit
holds that the condition of semi-stable model in Definition
10 coincides with the definition of stable model in Defini-
tion 7. Thus, if for each vmod(D),vis not a stable
model, then vis not a semi-stable model of D, as well.
As Corollary 3 says, if an ADF has a two-valued model but
no stable model, then it will not have any semi-stable model
either. As we presented in the beginning of Section 3, the
semi-stable semantics presented in this work deal with the
first issue, namely, that an ADF may not have a stable model.
That is, semi-stable semantics is a new point of view on the
acceptance of arguments if an ADF does not have any two-
valued model.
4 Generalization of the Semi-Stable
Semantics of AFs
In this section, we show that the notions of semi-stable and
semi-two-valued semantics for ADFs satisfy the last prop-
erty presented in Section 1.1, required for these semantics.
To this end, we show that the concept of semi-stable/semi-
two-valued semantics for ADFs is a proper generalization of
the concept of semi-stable semantics for AFs (Verheij 1996;
Caminada 2006), in Theorems 4 and 5. Furthermore, we
show that the concepts of semi-stable models and semi-two-
valued models coincide for the associated ADF of a given
AF, in Proposition 5.
Given an AF F= (A, R)and its corresponding ADF
DF= (A, R, C)(see Definition 8), the set of all possible
conflict-free extensions of Fis denoted by Eand the set of all
possible conflict-free interpretations of DFis denoted by V.
The functions Ext2IntFand Int2ExtDFin Definitions 11–12
are modifications of the labelling functions as given in (Ba-
roni, Caminada, and Giacomin 2018). Function Ext2IntF(e)
represents the interpretation associated to a given extension
Sin F, and function Int2ExtDF(v)indicates the extension
associated to a given interpretation vof DF.
Definition 11. Let F= (A, R)be an AF, and let Sbe an
extension of F. The truth value assigned to each argument
aAby the three-valued interpretation vSassociated to S
is given by Ext2IntF:E V as follows.
Ext2IntF(S)(a) =
tif aS,
fif aS+,
uotherwise.
It is shown in (Keshavarzi Zafarghandi, Verbrugge, and Ver-
heij 2021, Proposition 20) that if Sis a conflict-free exten-
sion of F, then Ext2IntF(S)is well-defined. Moreover, the
basic condition that Shas to be a conflict-free extension is
a necessary condition for Ext2IntF(S)being well-defined.
By Definition 3, every semi-stable extension of an AF is a
complete extension and a conflict-free extension. Thus, if
Sis a semi-stable extension of AF F, then Ext2IntF(S)is
well-defined. An interpretation of DFcan be represented as
an extension via the function Int2ExtDF, presented in Defi-
nition 12.
Definition 12. Let DF= (A, R, C)be the ADF associated
with AF F= (A, R), and let vbe an interpretation of DF,
that is, v V. The associated extension Svof vis obtained
via application of Int2ExtDF:V E on v, as follows:
Int2ExtDF(v) = {sS|s7→ tv}
Theorem 4 presents that the notion of semi-two-valued
model semantics for ADFs is a generalization of the concept
of semi-stable semantics for AFs.
Theorem 4. For any AF F= (A, R)and its associated
ADF DF= (A, R, C), the following properties hold:
if Sis a semi-stable extension of F, then Ext2IntF(S)is
a semi-two-valued model of DF;
if vis a semi-two-valued model of DF, then
Int2ExtDF(v)is a semi-stable extension of F.
Proof. Let Fbe an AF and let DFbe its associated ADF, as
in Definition 8.
We assume that {S0, S1, . . . , Sk}is the set of all complete
extensions of F. Since Fis a finite AF, the set of complete
extensions of Fis finite. Assume that {v0, v1, . . . , vk}is
the set of corresponding complete interpretations of DF,
i.e., vi=Ext2IntF(Si)for iwith 0ik.
Without loss of generality, assume that S0is a semi-stable
extension of F. By Definition 3, S0is a complete exten-
sion of Fsuch that S0S+
0is maximal. We show that
v0=Ext2IntF(S0)is a semi-two-valued model of DF.
Since v0is a complete interpretation of DF, to show that
v0is a semi-two-valued interpretation of DF, it remains
to show that vu
0is -minimal among all vi
ufor iwith
0< i k. Toward a contradiction, assume that v0uis
not -minimal among all vi
ufor iwith 0< i k. Thus,
there exists a jfor 0< j ksuch that vj
u(v0u.
Thus, there exists an asuch that a6∈ vj
uand av0u.
Thus, by Definition 11, it holds that, for each such an a,
aSjSj+but a6∈ S0S0+. Thus, S0S0+is
not maximal. This contradicts the assumption that S0is
a semi-stable extension of F. Hence, v0is a semi-two-
valued model of DF.
Assume that vis a semi-two-valued model of DF; we
show that S=Int2ExtDF(v)is a semi-stable extension
of F. To show that Sis a semi-stable extension of F, we
show that SS+is maximal. Toward a contradiction,
assume that SS+is not maximal. Thus, there exists
a complete extension of F, namely S0, with S0S0+is
maximal, i.e., SS+(S0S0+. Thus, by Definition
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11, it holds that viv0, where v0=Ext2Int(S0). Thus,
v0is a complete interpretation of DFsuch that v0u(vu.
Hence, vis not a semi-two-valued model of DF. This
contradicts the assumption that vis a semi-two-valued
model of DF. Thus, the assumption that SS+is not
maximal among all complete extensions of Fis wrong.
Hence, Sis a semi-stable extension of F.
Proposition 5. Let F= (A, R)be an AF and let DFbe
its associated ADF. The semi-two-valued semantics of DF
coincide with the semi-stable models of DF.
Proof. Let F= (A, R)be an AF and let DFbe its
associated ADF. By Corollary 1, semi-stb(DF)
semi-mod(DF). Thus it remains to show that
semi-mod(DF)semi-stb(DF).
Assume that vis a semi-two-valued model of DF. To
show that vis a semi-stable model of DF, we show that
vt=wt, where wis the grounded interpretation of sub-
reduct Dv
F= (Av, Lv, Cv), where Av=vtvu. We show
that vtwt. Assume that a7→ tv. Since DFis
an associated ADF to AF F,ϕa:Vbpar(a)¬b. Thus, if
avt, then either ais an initial argument of DFor for
each bpar(a)it holds that bvf. In both cases, it is
clear that ϕa[p/:v(p) = f] >. Therefore, awt.
Thus, vt=wt. Hence, vis a semi-stable model of DF.
Theorem 5. For any AF F= (A, R)and its associated
ADF DF, the following properties hold:
if Sis a semi-stable extension of F, then Ext2IntF(S)is
a semi-stable model of DF;
if vis a semi-stable model of DF, then Int2ExtDF(v)is a
semi-stable extension of F.
Proof. [Sketch] The theorem is a direct result of combining
Theorem 4, which says that semi-two-valued semantics of
ADFs are a generalization of semi-stable semantics of AFs,
and Proposition 5, which says that in the associated ADF
DFof a given AF F, the notions of semi-stable semantics
and semi-two-valued semantics coincide.
5 Conclusion
In this work, we have defined the semi-stable and semi-two-
valued semantics for finite ADFs. From a theoretical per-
spective, in Sections 3 and 4, we observe that the notions of
semi-stable and semi-two-valued semantics for ADFs ful-
fil the requirements for these two notions presented in Sec-
tion 1.1.
An ADF may have no stable model, for one of two rea-
sons: 1. Ddoes not have any two-valued model; or 2. each
two-valued model contains a support cycle. The condition
presented in Definition 7 characterizes the stable semantics
for ADFs. The condition says that a two-valued model is sta-
ble if it does not contain any support cycle, i.e., if there exists
a constructive proof for the arguments that are assigned to t.
Thus, to present an alternative definition for stable seman-
tics we focus on the first reason that an ADF does not have a
stable model, and we present a partial two-valued semantics
in Section 3, called semi-two-valued semantics in Definition
9. Then we define the notion of semi-stable semantics over
semi-two-valued semantics in Definition 10.
In Section 3, we show that the notions of semi-two-
valued/semi-stable semantics of ADFs presented in this
work satisfy the main requirements presented in Section 1.1.
Specifically:
1. Proposition 1 and Corollary 2 say that if vis a semi-two-
valued/semi-stable model of D, then vtvfis -maximal
among all complete interpretations of D.
2. Theorem 2 says that each semi-stable/semi-two-valued
model is a preferred interpretation and each stable model
of an ADF is a semi-stable/semi-two-valued model of that
ADF.
3. Proposition 3 says that each ADF has at least one semi-
two-valued model.
4. Theorem 3 says that if an ADF has a stable model, then
the sets of stable models and semi-stable models coincide.
In Section 4, we show that the notions of semi-stable/semi-
two-valued semantics of ADFs are proper generalizations
of the notion of semi-stable semantics of AFs. In Propo-
sition 5, we show that the concepts of semi-stable and semi-
two-valued semantics coincide in the associated ADF of a
given AF, intuitively, since in AFs there cannot be a support
cycle.
Alcˆ
antara and S´
a (2018) have also considered the semi-
stable semantics for ADFs. To prevent confusion with the
notion of semi-stable semantics presented in the current
work, we call their notion semi-stable2semantics, abbrevi-
ated SSS2. A key difference between our notion and SSS2
is that ours is compatible with the standard ADF defini-
tions. In particular, in their discussion, the characteristic
operator ΓDand in addition, the semantics of ADFs, and
specifically the complete semantics, have not been presented
in the way as introduced by Brewka and Woltran (2018;
2010). For instance, by their deviating definition of com-
plete labelling (Alcˆ
antara and S´
a 2018), only a, ¬b}is a
complete labelling/grounded model of D= ({a, b},{ϕa:
b, ϕb:a}). Hence—unlike the standard definitions—the set
of preferred labellings of Dis in their approach not a sub-
set of the set of complete labellings of D, and the unique
grounded labelling {} is not a complete labelling.
The computational complexity of semantics of AFs and
ADFs is presented in (Dvoˇ
r´
ak and Dunne 2017). Computa-
tional complexity of semi-stable semantics of AFs is stud-
ied in (Dunne and Caminada 2008). As a future work, it
would be interesting to clarify the computational complex-
ity of investigating: 1. whether a given interpretation is a
semi-stable/semi-two-valued model, 2. whether a given ar-
gument is credulously/skeptically acceptable/deniable under
semi-stable/semi-two-valued semantics of a given ADFs.
Acknowledgments
This research is supported by the Center of Data Science
&Systems Complexity (DSSC) Doctoral Programme, at
Bernoulli Institute of the University of Groningen.
Proceedings of the 18th International Conference on Principles of Knowledge Representation and Reasoning
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... For future work, we plan to investigate further ADF subclasses (see, e.g., [61]) and tune the system by exploiting acyclicity. Then, we plan to include other semantics; in particular stable and semi-stable [47] semantics are on our agenda next. Further avenues include (i) to get an understanding how ADF subclasses affect the expressibility of ADFs (see [51,63] for general investigations and [30] for investigation on particular ADF subclasses) and how such concepts can be integrated in our solving procedures; and (ii) designing SAT-based algorithms for ADFs involving more complex acceptance conditions, like the ones in GRAPPA ( [19]) or weighted versions of ADFs [17,13]. ...
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