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| Paper | |
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 edit
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| description | scientific paper published in CEUR-WS Volume 3197 |
| id | Vol-3197/paper5 |
| wikidataid | Q117341804→Q117341804 |
| title | Bipolar Argumentation Frameworks with Explicit Conclusions: Connecting Argumentation and Logic Programming |
| pdfUrl | https://ceur-ws.org/Vol-3197/paper5.pdf |
| dblpUrl | https://dblp.org/rec/conf/nmr/RochaC22 |
| volume | Vol-3197→Vol-3197 |
| session | → |
Bipolar Argumentation Frameworks with Explicit Conclusions: Connecting Argumentation and Logic Programming
Bipolar Argumentation Frameworks with Explicit
Conclusions: Connecting Argumentation and Logic
Programming
Victor Hugo Nascimento Rocha1,*,† , Fabio Gagliardi Cozman1,*,†
1
Escola Politécnica, Universidade de São Paulo, Av. Prof. Luciano Gualberto, 380 - Butantã, São Paulo - SP, 05508-010, Brazil
Abstract
We introduce a formalism for bipolar argumentation frameworks that combines different proposals from the literature and
results in a one-to-one correspondence with logic programming. We derive the correspondence by presenting translation
algorithms from one formalism to the other and by evaluating the semantic equivalences between them. We also show that
the bipolar model encapsulates distinct interpretations of the support relations studied the literature.
Keywords
Argumentation, Logic Programming, Bipolar Argumentation
1. Introduction ment explicitly associated with a claim. This apparently
minor change to AAFs leads to a nice translation to logic
The ability to argue is essential to humans, as discussed programming and allows for further semantic equiva-
in philosophy since ancient times, in contexts ranging lences.
from politics and law to science and arts [1, 2]. Within In a different direction, Dung’s abstract argumentation
artificial intelligence, argumentation has been boosted frameworks have been extended with support relations
by the seminal work of Dung (1995) on Abstract Argu- [7]. And the similarities between logic programming and
mentation Frameworks (AAFs), where each argument Bipolar Argumentation Frameworks, where supports in-
is understood as an abstract entity whose acceptance teract with attacks, have also been noted, for instance
depends only on its attack relations to other arguments. by Alfano et al. (2020). Those authors have proposed al-
Since Dung’s paper, the connections between argumen- gorithms that translate different kinds of argumentation
tation frameworks and other non-monotonic reasoning frameworks, including bipolar ones, to logic programs
formalisms has been investigated at length. One such in order to evaluate their semantic differences. In addi-
connection, put forward by Dung himself, is to logic tion to the steps previously proposed by Caminada et al.
programming. That research agenda was further devel- (2015), Alfano et al. (2020) interpret the support relation
oped by Caminada et al. (2015), who managed to prove through non-negative atoms of the logic program.
equivalences between several of the semantics used by Other relevant proposals have studied connections be-
both formalisms and to present translation algorithms tween logic programming and various argumentation
between them — not all correspondences were obtained formalisms, for instance assumption-based ones [9, 10].
by them, however; in particular, the connection between However, to the best of our knowledge, none of these pre-
logic programming and AAFs breaks down in the con- vious proposals reaches a one-to-one correspondence be-
text of the latter semi-stable semantics. Proposals for tween some family of argumentation frameworks expres-
enlarged AAFs have been made and their equivalence to sive enough to convey bipolarity and some well-known
logic programming has been explored. Particularly in- logic programming formalism.
teresting here are the Claim-augmented Argumentation In this work we will address the relationship between
Frameworks [5, 6]. Such frameworks have each argu- logic programming and argumentation frameworks by
combining existing proposals, in particular the ones by
NMR 2022: 20th International Workshop on Non-Monotonic Reasoning, Dvorák and Woltran (2019) and by Alfano et al. (2020).
August 07–09, 2022, Haifa, Israel In doing so, we reach the Bipolar Conclusion-augmented
*
Corresponding author. Argumentation Framework and prove (for well-formed
†
These authors contributed equally. and non-redundant frameworks) its one-to-one equiv-
$ victor.hugo.rocha@usp.br (V. H. N. Rocha); fgcozman@usp.br
alence to normal logic programming. The translation
(F. G. Cozman)
http://www.poli.usp.br/p/fabio.cozman (F. G. Cozman) algorithms between both formalisms are also introduced.
� 0000-0002-8984-7982 (V. H. N. Rocha); 0000-0003-4077-4935 The proposed framework is able to encapsulate different
(F. G. Cozman) versions of the support relation in the literature.
© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0). In short, we show that a large family of bipolar argu-
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
CEUR Workshop Proceedings (CEUR-WS.org)
49
�mentation frameworks is normal logic programming, and
vice-versa. 𝐴3 𝐴1 𝐴0 𝐴2 𝐴4
Section 2 briefly goes over needed background: ab-
stract argumentation frameworks and their bipolar ex-
tension; logic programs and their semantics. Section 3 Figure 1: An Argumentation Graph as proposed by Dung.
starts with relevant results from the literature on the rela-
tionship between Dung’s framework and logic program-
ming and then introduces the conclusion-augmented ar- Definition 4. A labeling ℒ of the arguments in an AFF is
gumentation frameworks, showing that they improve conflict-free iff there are no arguments 𝐴 and 𝐵 in the set
on previous results by refining the equivalence between of arguments labeled In for which 𝐴 → 𝐵.
various formalisms. The bipolar conclusion-augmented
argumentation model is later used to obtain a one-to-one Definition 5. A conflict-free labeling ℒ of 𝒜 is admissi-
equivalence with logic programming while also offering ble iff for every argument 𝐴 labeled Out, there exists an
different interpretations of support. A novel discussion argument B labeled In such that 𝐵 → 𝐴.
about the framework and its relationship with other pro-
posals in the literature is developed in Section 4; finally, We can define several semantics using these concepts:
Section 5 concludes and describes possible future work. Definition 6. An admissible labeling ℒ of an AAF is com-
plete iff for every argument 𝐴 with the label Undecided,
2. Argumentation Frameworks there are no arguments 𝐵 with the label In that attack 𝐴
and every acceptable argument with respect to the set of
and Logic Programming arguments In is also labeled In.
In this section we review argumentation frameworks, Definition 7. A complete labeling ℒ of an AAF is pre-
bipolar frameworks, logic programming, and some of ferred iff the argument set In is maximal.
their semantics.
Definition 8. A complete labeling ℒ of an AAF is stable
iff the argument set labeled Undecided is empty.
2.1. Abstract Argumentation Frameworks
(AAFs) Definition 9. A complete labeling ℒ of an AAF is
grounded iff the argument set In is minimal.
Dung’s argumentation frameworks are based on argu-
ments and attacks between them. Arguments are under- Definition 10. A complete labeling ℒ of an AAF is semi-
stood as abstract entities whose internal structure is not stable iff the argument set Undecided is minimal.
relevant.
There are still other possible semantics [11], but in this
Definition 1. An AAF is a tuple (𝒜, ℛ), where 𝒜 is the paper we restrict ourselves to the previous ones.
set of arguments and ℛ is an attack relationship on 𝒜 × 𝒜.
An attack from an argument 𝐴 to another argument 2.2. Bipolar Argumentation Frameworks
𝐵, represented by 𝐴 → 𝐵, intuitively means that if 𝐴 is (BAFs)
accepted, 𝐵 cannot be. An AAF can be represented as a An argumentation scenario seems to require not only
graph structure, where nodes stand for arguments and attacks but also “positive” support relations between ar-
edges as the attack relation between them (see example guments [12]. The definition of support relations, unlike
in Figure 1). the attack relation, varies quite a bit in the literature
Dung defined semantics through extensions. The latter
[13, 14]. In this text we stick to the three interpretations
represent sets of arguments that are acceptable according
explained by Cayrol and Lagasquie-Schiex (2013): the nec-
to some criterion. In this text, however, we will use essary support [15, 16], the deductive support [17], and
labelings to define semantics [11]. the evidential support [18, 19]. A necessary support from
Definition 2. A labeling ℒ of an AFF is a function ℒ : one argument 𝐴 to another 𝐵, represented by 𝐴 ⇒𝑛 𝐵,
𝒜 → {In, Out, Undecided}. means that if 𝐵 is accepted (received the label In), 𝐴 must
also be accepted. A deductive support from one argu-
Some concepts needed later are: ment 𝐴 to another 𝐵, on the other hand, is represented
by 𝐴 ⇒𝑑 𝐵, and means that if 𝐴 is accepted (received
Definition 3. An argument A ∈ 𝒜 is acceptable iff all the label In), 𝐵 must also be accepted. Finally, there are
arguments 𝐵 such that 𝐵 → 𝐴 are not acceptable. two types of arguments in a BAF that contains evidential
support: prima-facie arguments, which do not need any
50
� Definition 15. Let 𝒮 ⊆ 𝒜. The set 𝒮 is conflict-free iff
𝐴3 𝐴1 𝐴0 𝐴2 𝐴4 there are no 𝐴, 𝐵 ∈ 𝒮 such that 𝐴 defeats (directly or
indirectly) 𝐵.
Definition 16. Let 𝒮 ⊆ 𝒜. The 𝒮 set is admissible iff it
Figure 2: A Bipolar Argumentation Graph.
is conflict-free and defends all its elements.
We can then redefine the complete semantics for BAF:
support to be accepted, and common arguments, which Definition 17. A set 𝒮 ⊆ 𝒜 of arguments is complete iff
need to be supported by an accepted argument of the it is admissible and every argument 𝐴 that can be accepted
first type to be accepted. together with 𝒮 is part of 𝒮.
Regardless of the interpretation of the support relation,
a BAF can be defined as [7]: From the definition of complete semantics, the pre-
ferred, grounded, stable and semi-stable semantics can
Definition 11. A Bipolar Argumentation Framework be adopted as before (as they are all special cases of the
(BAF) is a tuple (𝒜, ℛ− , ℛ+ ), where 𝒜 represents the set complete semantics in which some label is maximized or
of arguments, ℛ− the attack relation and ℛ+ the support minimized).
relation.
As AFFs, BAFs can also be represented by graphs. Fig- 2.3. Logic Programming and its Semantics
ure 2 depicts a BAF where nodes are arguments, edges
In this work we focus on propositional normal logic pro-
encode the attack relation and double edges encode the
grams [4]:
support relation (for instance, 𝐴2 supports 𝐴4 ).
To handle support relations, the semantics, the label- Definition 18. A (normal) logic program 𝑃 is composed
ing types and the acceptability criteria must be adapted. by a finite set of rules. Each rule 𝑟 is an expression of the
Several proposals have been made in order to achieve form 𝑟 : 𝐻 :− 𝐴1 , ..., 𝐴𝑛 , not 𝐵1 , ..., not 𝐵𝑛 , where
this, with differences in the way the relative strength 𝐻, 𝐴𝑖 and 𝐵𝑖 represent atoms and not is the classical
between attack and support relations is taken into ac- negation. 𝐻 represents the head of the formula, while the
count [20, 21]. We will follow the proposal by Cayrol and others are the body. A formula without the body is called a
Lagasquie-Schiex (2005). fact and is written as 𝐻. The Herbrand Base of the program
We assume for now that support is of the necessary 𝑃 is the set 𝐻𝐵𝑃 of all atoms that appear in the program.
type; one can proceed analogously for the other types of
support. Example 1. The following is an example of a logic pro-
Once supports are taken into account, in addition to de- gram with Herbrand Base consisting of the atoms 𝑎, 𝑏, 𝑐,
feat by a traditional attack, an argument can be defeated 𝑑:
indirectly. 𝑟0 : 𝑎 :− not 𝑐, 𝑟1 : 𝑏 :− not 𝑎, not 𝑏,
Definition 12. An argument 𝐵 can be defeated indirectly 𝑟2 : 𝑐 :− not 𝑎, 𝑟3 : 𝑏 :− not 𝑏,
by a sequence 𝐴1 𝑅1 ... 𝑅𝑛−1 𝐵, where 𝑖 = 1, ..., 𝑛 − 2, 𝑟4 : 𝑑 :− 𝑐, not 𝑎.
𝑅𝑖 = 𝑅+ and 𝑅𝑛−1 = 𝑅− or where 𝑖 = 2, ..., 𝑛 − 1,
𝑅𝑖 = 𝑅+ and 𝑅1 = 𝑅− .
Hence it makes sense to define the defeat/support of Definition 19. A three-valued interpretation of a logic
an argument by a set. program P is a pair 𝐼 = (𝒯 , ℱ) such that 𝒯 ∩ ℱ = ∅
and that both 𝒯 and ℱ contain elements from the Her-
Definition 13. Let 𝒮 ⊆ 𝒜 and 𝐴 ∈ 𝒜. The set 𝒮 brand base of P. 𝒯 is understood as true, ℱ as false and
defeats 𝐴 iff there is a direct or indirect defeat for 𝐴 from 𝐻𝐵𝑃 \(𝒯 ∪ ℱ) as undecided.
some element of 𝒮. The set 𝒮 supports 𝐴 iff there is a A three-valued model of P is an interpretation such that
sequence of the form 𝐴1 𝑅1 ... 𝑅𝑛−1 𝐴𝑛 , 𝑛 ≥ 2, such that for each 𝑎 ∈ 𝐻𝐵𝑃 we have:
𝑖 = 1, ..., 𝑛 − 1, 𝑅𝑖 = 𝑅+ with 𝐴𝑛 = 𝐴 and 𝐴1 ∈ 𝒮.
• 𝑎 is in 𝒯 if there is a rule whose head is 𝑎 = 𝐻
A set of arguments can also defend other arguments: and where each 𝐴𝑖 is in 𝒯 ;
Definition 14. Let 𝒮 ⊆ 𝒜 and 𝐴 ∈ 𝒜. The set 𝒮 • 𝑎 is in ℱ if every rule whose head is 𝑎 = 𝐻 is such
collectively defends 𝐴 iff for some set ℬ ⊆ 𝒜, if ℬ defeats that there is some 𝐴𝑖 in ℱ.
𝐴 then there is a 𝐶 ∈ 𝒮 such that 𝐶 defeats ℬ.
The reduct of P with respect to a three-valued interpreta-
Given this, a conflict-free set and an admissible set can tion ℐ, denoted P/ℐ, is a logic program constructed using
be redefined: the following steps:
51
� • First, remove from P every rule that contains • Starting with a set of rules, process one rule at a
not 𝐵 in its body for some 𝐵 ∈ 𝒯 ; time;
• Then, for each remaining rule, remove not 𝐵 from • If a rule of the form 𝐻 :− not 𝐵1 , . . . , not 𝐵𝑛
the rule body if 𝐵 ∈ ℱ ; is found, then generate an argument 𝐴 with rules
• Finally, replace any remaining occurrences of {𝐻 :− not 𝐵1 , . . . , not 𝐵𝑛 }, vulnerabilities
not 𝐵 ′ with a new 𝑢 symbol representing “un- Vul(𝐴) {𝐵1 , . . . , 𝐵𝑛 }, conclusion 𝐻 and a set of
decided”. sub-arguments that contain only 𝐴 itself ;
So P/ℐ has a unique three-valued model with 𝒯 mini- • If a rule of the form 𝐻 :− 𝐴1 , . . . , 𝐴𝑚 , not 𝐵1 ,
mum and ℱ maximum (with respect to the set inclusion). . . . , not 𝐵𝑛 is found and assuming that each
We denote this model ΦP (ℐ). 𝐴𝑖 has an associated argument 𝐴𝑟𝑔𝑖 , then gener-
ate an argument 𝐴 with a set of sub-arguments
It is then possible to define the semantics of a logic pro- 𝐴𝑟𝑔𝑖 , conclusion 𝐻, rules composed by the union
gram 𝑃 , given an interpretation 𝐼 = (𝒯 , ℱ), in several of {𝐻 :− 𝐴1 , . . . , 𝐴𝑚 , not 𝐵1 , . . . , not 𝐵𝑛 }
ways [4]: with the rules of each sub-argument and vulnera-
Definition 20. A partial stable (P-stable) model of 𝑃 is bilities Vul(𝐴) as the union of {𝐵1 , . . . , 𝐵𝑛 } with
an interpretation 𝐼 such that ΦP (ℐ) = 𝐼. the vulnerabilities of the sub-arguments;
• After going through all the rules, the relations
Definition 21. A model of 𝑃 is well-founded iff it is P-
between arguments are established. If an argu-
stable and 𝒯 is minimal.
ment 𝐴 has a conclusion that is present in the
Definition 22. A model of 𝑃 is regular iff it is P-stable vulnerabilities of another argument 𝐵, then 𝐴
and 𝒯 is maximal. attacks 𝐵. With this, the AAF is created.
Definition 23. A model of 𝑃 is stable iff it is P-stable and
Example 2. If we apply the WCG algorithm described
𝒯 ∪ ℱ = 𝐻𝐵𝑃 .
above to the rules in Example 1, we obtain the AAF shown
Definition 24. A model of 𝑃 is L-stable iff it is P-stable in Figure 1.
and 𝒯 ∪ ℱ is maximal.
Once the logic program is translated into an argumen-
The definitions above were crafted by Caminada et al. tation framework, we can apply any semantics to the lat-
(2015) to emphasize connections with the corresponding ter and obtain a labeling ℒ at the argument level. To then
argumentation semantics. obtain the atom level (or “conclusion” level) labeling of
𝑃 , the following mapping [4] can be used: the labeling of
3. Correspondences between a conclusion is the one with the highest value among the
arguments that are associated with it. The order of values
Argumentation Frameworks between the labels is given by In > Undecided > Out
and Logic Programs and the idea behind it is that each conclusion is repre-
sented by the argument that best defends it.
In this section we look at equivalences between argu- We then ask: if we apply some semantics at the argu-
mentation frameworks and logic programming. We start ment level and map the labeling to the conclusions, is the
by presenting results in the literature and then introduce result equivalent to applying some other semantics di-
the Conclusion-augmented Argumentation Frameworks rectly to the logic program? The answer to this question
(CAF), where a small change in the representation sig- is positive as was shown in [4].
nificantly improves the relationship between both for-
malisms. Furthermore, we expand CAFs by adding vari- Theorem 1. (Theorems 19, 20, 21 and 22 of [4]). The
ous interpretations of the support relation between argu- labels of conclusions obtained by the semantics P-stable,
ments and discuss how those relations translate to logic regular, well-founded and stable in a logic program are
programming. equivalent to those obtained by the complete, preferred,
grounded and stable semantics in an AFF respectively, with
the subsequent transformation to the conclusions level.
3.1. The Connection between Abstract
Argumentation Frameworks (AAFs) The attentive reader may have noticed that there is
a notable exception in Theorem 1. There is no equiva-
and Logic Programs
lence between the semi-stable and L-stable semantics.
To study possible equivalences between AAFs and logic Intuitively, this happens because all other semantics are
programs, one must consider translations from each other. special cases of the P-stable models in logic programs and
A suitable starting point is the WCG (Wu, Caminada and of the complete semantics in argumentation, with one of
Gabbay) algorithm [4], which is summarized below: the labels maximized or minimized. For the labels In and
52
�Out this last process is equivalent both at the argument
and at the conclusion levels, but the same is not true for 𝐴3 (𝑏) 𝐴1 (𝑏) 𝐴0 (𝑎) 𝐴2 (𝑐) 𝐴4 (𝑑)
the label Undecided. Consider the following example:
Example 3. If we apply the complete semantics to the AAF
Figure 3: A Conclusion-Augmented Argumentation Graph
in Figure 1, we get the following three solutions (expressed
as (In set, Undecided set, Out set)): (∅, {𝐴0 , 𝐴1 , 𝐴2 , 𝐴3 ,
𝐴4 }, ∅), ({𝐴0 }, {𝐴3 }, {𝐴1 , 𝐴2 ,𝐴4 }) and ({𝐴2 , 𝐴4 }, {𝐴1 , 𝐴3 },
{𝐴0 }). From this, we obtain the conclusion labelings: (∅, seems to take a more complicated route than ours to ar-
{𝑎, 𝑏, 𝑐, 𝑑}, ∅), ({𝑎}, {𝑏}, {𝑐, 𝑑}) and ({𝑐, 𝑑}, {𝑏}, {𝑎}). As it rive at them and to study them; we thus describe our
can be seem, if the label In is maximized/minimized at own route in some detail in this section, even though
both argument and conclusion levels, we obtain the same we acknowledge that the results are equivalent to previ-
result. The same applies for the label Out. However, for the ous ones by Dvorák and Woltran, Dvorák et al., Dvořák
label Undecided, at the argument level we obtain only one et al., Rapberger (2019, 2020, 2020, 2020). We also note
solution by minimizing the label, while at the conclusion that CAFs have been employed in recent work on proba-
level we get two solutions. bilistic argumentation frameworks [24].
CAFs are defined as follows:
Despite this unfortunate feature of the semi-stable
semantics, it is our position that the semi-stable seman- Definition 25. A Conclusion-augmented Argumentation
tics is the most appropriate semantics for argumentation. Framework (CAF) is a tuple (𝒜, 𝒮, 𝑓, ℛ), with 𝒜 being
Compare with the other semantics. The grounded se- a finite set of arguments, 𝒮 a set of conclusions, 𝑓 is a
mantics, in situations of mutual attacks, does not reach function of arguments to conclusions, and ℛ ⊆ 𝒜 × 𝒜 is
any decisions, while the preferred semantics is very per- the attack relation as previously defined.
missive. And the stable semantics may fail to produce a We then say that every conclusion 𝑐 ∈ 𝒮 is associated
labeling when an Undecided label is unavoidable — in to a finite set of arguments 𝑓 −1 (𝑐). Analogously to AAFs,
an argumentative scenario however, we believe that not a conclusion-augmented argumentation graph is a direc-
arriving at some labeling is undesirable. We would thus tional graph in which each node represents an argument
like to choose the semi-stable semantics, but this seems to associated with its conclusion and each arrow represents
clash with our desire to obtain a correspondence between an attack. It is important to note that the CAF maintains
argumentation frameworks and logic programs. In the a high degree of abstraction as the internal structure of
next subsection we show how to enlarge argumentation the arguments is not explicit. The only change is that
frameworks to obtain our desired correspondences. each argument 𝐴 is represented along with its conclusion
𝑎.
3.2. The Connection between To each conclusion 𝑐 ∈ 𝒮 we can assign the labels
Conclusion-Augmented In, Undecided, and Out. Naturally, this labeling of con-
clusions is related to the labeling of arguments through
Argumentation Frameworks (CAFs) 𝑓 (𝐴), and the procedure described previously obtains a
and Logic Programs labeling of conclusions from one of arguments.
The first step in our pursuit of a complete correspondence A CAF can be generated by a slightly altered version
between argumentation frameworks and logic programs of the WGU algorithm presented in Section 3.1, which
is to augment arguments with their associated conclu- associates each argument with a conclusion at the end of
sions. We do so to guarantee that semi-stable semantics the process. We show in Figure 3 the result of applying
does have a correspondence in logic programming. this altered algorithm to the logic program of Example 1.
To do so, we adopt recent work on Claim-augmented One can easily adapt the definition of complete label-
Argumentation Frameworks (CAFs), where each argument ing for arguments to an analogous concept of complete
is augmented with its associated claim [5, 22, 23, 6]. We labeling of conclusions:
prefer to use “conclusion” rather than “claim” as the for- Definition 26. Let (𝒜, 𝒮, 𝑓, ℛ) be a CAF and ℒ be a
mer term is employed in most of the previous literature conclusion labeling of 𝒮. So ℒ is a complete conclusion
dealing with connections between argumentation frame- labeling if:
works and logic programs (for instance, by Caminada (i) a conclusion 𝑎 is Out then for each argument 𝐴 ∈
et al. (2015)). So we will use Conclusion-augmented Ar- 𝑓 −1 (𝑎) there is an argument 𝐵 that attacks 𝐴 with a con-
gumentation Frameworks (CAFs), but we emphasize that, clusion 𝑓 (𝐵) labeled In;
despite the slight nomenclature change, the latter are (ii) a conclusion 𝑎 is In then for some argument 𝐴 ∈
equivalent to Claim-augmented Argumentation Frame- 𝑓 −1 (𝑎), every argument 𝐵 that attacks 𝐴 must have its
works. However we note that the previous work on CAFs conclusion 𝑓 (𝐵) labeled Out.
53
�(iii) a conclusion 𝑎 is Undecided then there is no argument Example 5. Applying the algorithm in Definition 28 to
𝐴 ∈ 𝑓 −1 (𝑎) for which all arguments 𝐵 that attack 𝐴 the CAF in Figure 3, we get the following set of rules:
have 𝑓 (𝐵) as Out; and for some argument 𝐴 ∈ 𝑓 −1 (𝑎),
there is no argument 𝐵 that attacks 𝐴 with conclusion 𝑟0 : 𝑎 :− not 𝑐, 𝑟1 : 𝑏 :− not 𝑎, not 𝑏,
𝑓 (𝐵) In, and there is an argument 𝐶 that attacks that 𝑟2 : 𝑐 :− not 𝑎, 𝑟3 : 𝑏 :− not 𝑏,
same 𝐴 and whose conclusion 𝑓 (𝐶) is not Out. 𝑟4 : 𝑑 :− not 𝑎.
Given the equivalence between the complete labels at
the conclusion level and at the arguments level proved by
Caminada et al. (2015), we can define the other semantics Thus, the following theorem proves the desired equiv-
for the CAFs. From the results of complete semantics alences:
for the arguments, it is possible to obtain the equivalent
labels at the conclusion level by applying the criteria Theorem 3. Let C = (𝒜, 𝒮, 𝑓, ℛ) be a CAF and PC
described in section 3.1. The latter are the complete con- the corresponding logic program . So the partial stable, reg-
clusion labeling solutions and, from them, we can max- ular, well-founded, stable, L-stable models of P assign the
imize/minimize any desired label. With this, we obtain same labels to the conclusions as respectively the complete,
the other semantics mentioned in this work, which are preferred, grounded, stable, semi-stable semantics of C.
defined as: With this, a correspondence between the semantics of
Definition 27. Assume 𝐿 is a conclusion labeling. logic programs and argumentation frameworks in both
𝐿 is grounded if it is complete and the conclusion set In is directions is specified. Due to space constraints we refer
minimal. to our previous work [24] to prove both Theorems 2 and
𝐿 is preferred if it is complete and the conclusion set In is 3.
maximal. However, it should be clear that the move to CAFs is
𝐿 is stable if it is complete and no conclusion remains not enough to achieve a completely satisfactory one-to-
Undecided. one equivalence with logic programming. That can be
𝐿 is semi-stable if it is complete and the conclusion set seem if we compare the rules of Examples 1 and 5. Rule
Undecided is minimal. 𝑟4 differs by the omission/inclusion of the conclusion 𝑐
in those examples. That means that repeated translations
Given that the maximization/minimization process between formalisms loses some information. To solve
was done directly at the conclusion level, the desired that problem, in the next section we take the support
equivalences are obtained: relations into account.
Theorem 2. Let P be a logic program and C =
(𝒜, 𝒮, 𝑓, ℛ) the CAF generated by the modified WCG al- 3.3. Bipolar Conclusion-augmented
gorithm. Then the complete, preferred, grounded, stable Argumentation Framework (BCAF)
and semi-stable conclusion labels of C are identical to the
This section introduces Bipolar Conclusion-augmented
labels assigned respectively by the partial stable, regular,
Argumentation Frameworks (BCAFs). Most concepts re-
well-founded, stable and L-stable models of P
lated to bipolarity are based on previous work by Cayrol
Example 4. (continuing Example 3). Since in CAFs we and Lagasquie-Schiex (2005).
apply the maximization/minimization of labels directly at We thus introduce:
the conclusion level, the semi-stable semantics will yield
Definition 29. A Bipolar Conclusion-augmented
two solutions. Thus it is equivalent to applying the L-stable
Argumentation Framework (BCAF) is a tuple
semantics to the logic program in Example 1.
(𝒜, 𝒮, 𝑓, ℛ− , ℛ+ ), with 𝒜 being a finite set of ar-
Consequently, the semantic correspondence between guments, 𝒮 a set of conclusions, 𝑓 being a function of
a logic program and its derived CAF is proved, and it is arguments to conclusions, ℛ− ⊆ 𝒜 × 𝒜 is the attack
also possible to demonstrate it in the opposite direction. relation and ℛ+ ⊆ 𝒜 × 𝒜 is the support relation.
That is, starting from an CAF and generating a logical
program, the correspondences are maintained. We use Each argument 𝐴 ∈ 𝒜 can be defined as a
the following translation process: rule 𝐻 :− 𝐴1 , . . . , 𝐴𝑚 , not 𝐵1 , . . . , not 𝐵𝑛 .
We say 𝐴 has conclusion 𝐻, rules {𝐻 :−
Definition 28. Let C = (𝒜, 𝒮, 𝑓, ℛ) be a CAF. 𝐴1 , . . . , 𝐴𝑚 , not 𝐵1 , . . . , not 𝐵𝑛 }, vulnerabili-
For each argument 𝐴, generate a rule 𝑓 (𝐴) :− not ties Vul(𝐴) {𝐵1 , . . . , 𝐵𝑛 } and necessities Nec(𝐴)
𝑓 (𝐵1 ), ..., not 𝑓 (𝐵𝑚 ) where 𝐵𝑖 are the arguments that {𝐴1 , . . . , 𝐴𝑚 }.
attack 𝐴. We denote PC the logic program that consists of The attack relation is defined in the usual sense of
the generated rules. Dung’s work, that is, an argument attacked by another
54
�that is accepted, must be rejected. In a BCAF, one ar-
gument attacks another when its conclusion is one of 𝐴3 (𝑏) 𝐴1 (𝑏) 𝐴0 (𝑎) 𝐴2 (𝑐) 𝐴4 (𝑑)
the other’s vulnerabilities. The support relation, for now
based on the discussed necessary support and on the
work by Alfano et al. (2020), is defined as follows: Figure 4: A Bipolar Conclusion-augmented Argumentation
Graph
Definition 30. An argument 𝐴 supports another argu-
ment 𝐵 iff 𝑓 (𝐴) = 𝑎 is a necessity of 𝐵. Thus, for 𝐵 to be
accepted, at least one of the supports 𝐴𝑖 with 𝑓 (𝐴𝑖 ) = 𝑎
must also be accepted. Figure 4 shows a bipolar conclusion-augmented graph,
where the adapted WCG algorithm was applied to the
Given the ways of how an argument can relate to an- rules in Example 1.
other in a BCAF, the definition of a redundant BCAF is One should ask whether, like the original WCG algo-
relevant: rithm, the adapted version guarantees the equivalence
Definition 31. Let 𝐴𝐹 = (𝒜, 𝒮, 𝑓, ℛ− , ℛ+ ) be a of the P-stable semantics in logic programming and the
BCAF. 𝐴𝐹 is said to be redundant if there is at least pair complete semantics in the bipolar conclusion-augmented
of arguments 𝐴, 𝐵 ∈ 𝒜 such that 𝑓 (𝐴) = 𝑓 (𝐵), Vul(𝐴) argumentation framework. The answer to this question
= Vul(𝐵) and Nec(𝐴) = Nec(𝐵). is once again yes, but to prove it a few definitions must
be stated.
Also of interest is the definition of the well-formed First, the rule for translating labels from arguments
BCAF as adapted from Dvorák and Woltran (2019): to conclusions is the same as the one discussed earlier,
Definition 32. Let 𝐴𝐹 = (𝒜, 𝒮, 𝑓, ℛ− , ℛ+ ) be a where each conclusion is represented by the argument
BCAF. 𝐴𝐹 is said to be well-formed if all the arguments that best defends it.
𝐴 ∈ 𝒜 that hold the same conclusion 𝑓 (𝐴) attack and Second, we define the functions that translate labelings
support the same arguments. from arguments to conclusions and vice versa. These
definitions are adapted from Caminada et al. (2015) to
From this point on we will assume the BCAFs to be the context of BCAFs:
well-formed and non-redundant unless stated otherwise.
As previously mentioned, Caminada et al. (2015) stud- Definition 33. Let 𝑃 be a logical program and 𝐴𝐹𝑃 =
ied the WCG algorithm for the translation of logic pro- (𝒜, 𝒮, 𝑓, ℛ− , ℛ+ ) its associated bipolar conclusion-
grams into Dung’s argumentation frameworks. Here, augmented argumentation structure. Let ArgLabs be the
this translation is adapted for BCAFs and the semantic set of all argument labels from 𝐴𝐹𝑃 and let ConcLabs be
equivalences are once again proved. We thus propose an the set of all conclusion labels from 𝑃 and 𝐴𝐹𝑃
adapted WCG algorithm for BCAFs:
1. We define an ArgLab2ConcLab function: ArgLabs
1. Starting with a rule set, process each rule at a
→ ConcLabs such that for each ArgLab ∈ ArgLabs,
time;
it is true that ArgLab2ConcLab(ArgLab) is the as-
2. If a rule of the form 𝐻 :− not 𝐵1 , . . . , not 𝐵𝑛
sociated conclusion labeling of ArgLab;
is found, then generate an argument 𝐴 with con-
clusion 𝐻, rules {𝐻 :− not 𝐵1 , . . . , not 𝐵𝑛 } 2. We define an ConcLab2ArgLab function: ConcLabs
and vulnerabilities Vul(𝐴) {𝐵1 , . . . , 𝐵𝑛 }; → ArgLabs such that for each ConcLab ∈ ConcLabs
and each 𝐴 ∈ 𝒜 it is true that:
3. If a rule of the form 𝐻 :− 𝐴1 , . . . , 𝐴𝑚 , not 𝐵1 ,
. . . , not 𝐵𝑛 is found, then generate an ar- a) ConcLab2ArgLab(ConcLab)(𝐴) = In if for
gument 𝐴 with conclusion 𝐻, rules {𝐻 :− each 𝑣 ∈ Vul(𝐴) it is true that ConcLab(𝑣) =
𝐴1 , . . . , 𝐴𝑚 , not 𝐵1 , . . . , not 𝐵𝑛 }, vulnerabil- Out and for each 𝑤 ∈ Nec(𝐴) it is true that
ities Vul(𝐴) {𝐵1 , . . . , 𝐵𝑛 } and necessities Nec(𝐴) ConcLab(𝑤) = In;
{𝐴1 , . . . , 𝐴𝑚 }; b) ConcLab2ArgLab(ConcLab)(𝐴) = Out if
4. After going through all the rules, the relations there is a 𝑣 ∈ Vul(𝐴) such that ConcLab(𝑣)
between arguments are established. If an argu- = In and/or if there is a 𝑤 ∈ Nec(𝐴) such
ment 𝐴 has a conclusion that is present in the that ConcLab(w) = Out;
vulnerabilities of another argument 𝐵, then 𝐴 c) ConcLab2ArgLab(ConcLab)(𝐴) =
attacks 𝐵. On the other hand, if the conclusion Undecided if not for all 𝑣 ∈ Vul(𝐴)
of 𝐴 is present in the necessities of 𝐵, then 𝐴 ConcLab(𝑣) = Out; if there is no 𝑣 ∈ Vul(𝐴)
supports 𝐵. With this, and keeping that each such that ConcLab(𝑣) = In; if it is not true
argument is linked to a conclusion, the bipolar that for all 𝑤 ∈ Nec(𝐴) ConcLab(𝑤) =
conclusion-augmented argumentation graph is In and there is no 𝑤 ∈ Nec(𝐴) such that
created. ConcLab(𝑤) = Out.
55
� Given the definitions above we can state the following argument 𝐸 with 𝑓 (𝐸) = 𝑒 for each con-
theorem, inspired by Theorem 19 from [4]: clusion 𝑒 that supports 𝐴 has ArgLab(𝐸)
= In (i). There is also no attacker 𝐷 of 𝐴
Theorem 4. In the case of complete argument labellings that has ArgLab(𝐷) = In and/or no conclu-
and complete conclusion labellings, the functions Ar- sion 𝑒 ∈ Nec(𝐴) such that all arguments
gLab2ConcLab and ConcLab2ArgLab are bijections and 𝐸 with 𝑓 (𝐸) = 𝑒 have ArgLab(𝐸) =
each other’s inverse Out (ii). From (i) together with (ii) it fol-
Proof. The proof is inspired by the proof of Theorem 19 lows that there is an attacker 𝐵 of 𝐴 with
from [4]. It is enough to prove two things: ArgLab(𝐵) = Undecided and/or a group
𝐸 of arguments with the conclusion 𝑒 ∈
1. ConcLab2ArgLab(ArgLab2ConcLab(ArgLab)) = Nec(𝐴) with ArgLab(𝐸) = Undecided. Let
ArgLab. 𝑏 = 𝑓 (𝐵) and 𝑒 = 𝑓 (𝐸). From (ii) to-
Let ArgLab be a complete argument labeling and gether with the definition of attack, it fol-
let 𝐴 be an argument. Three cases are distin- lows that there is no argument 𝐵 ′ with
guished. 𝑓 (𝐵 ′ ) = 𝑏 such that ArgLab(𝐵 ′ ) = In. So
a) ArgLab(𝐴) = In. From the fact that ArgLab ArgLab2ConcLab(ArgLab)(𝑏) = Undecided.
is a complete labeling, it follows that Likewise, (ii) together with the definition
ArgLab(𝐵) = Out for every attacker 𝐵 of of support implies that for some conclusion
𝐴 and that ArgLab(𝐶) = In for at least 𝑒 ∈ Nec(𝐴) ArgLab2ConcLab(ArgLab)(𝑒)
one argument 𝐶 with 𝑓 (𝐶) = 𝑐 for each = Undecided (iii). Furthermore, from (ii)
conclusion 𝑐 that supports 𝐴. From the together with the definitions of attack and
definition of attack it follows that for every support, it follows that for each argument
𝑏 ∈ Vul(𝐴) and for every argument 𝐵 𝐷 with 𝑓 (𝐷) ∈ Vul(𝐴) it is valid that
with f(𝐵) = 𝑏, ArgLab(𝐵) = Out. This then ArgLab(𝐷) ̸= In and for at least one ar-
implies that for every 𝑏 ∈ Vul(𝐴) it is true gument 𝐹 with a conclusion 𝑓 ∈ Nec(𝐴),
that ConcLab2ArgLab(ArgLab2ConcLab( ArgLab(𝐹 ) ̸= Out. Therefore, for every
ArgLab))(𝑏) = Out. Similarly, from the def- 𝑑 ∈ Vul(𝐴), ArgLab2ConcLab(ArgLab)(𝑑)
inition of support, it follows that for every ̸= In and for at least one 𝑓 ∈ Nec(𝐴),
𝑐 ∈ Nec(𝐴), for at least one argument 𝐶 ArgLab2ConcLab(ArgLab)(𝑓 ) ̸= Out (iv).
with f(𝐶) = 𝑐, ArgLab(𝐶) = In, that is, Con- Finally, From (iii) and (iv), and the def-
cLab2ArgLab(ArgLab2ConcLab(ArgLab) inition of ConcLab2ArgLab, it follows
)(𝑐) = In. By the definition of that ConcLab2ArgLab(ArgLab2ConcLab(
ConcLab2ArgLab, we finally ArgLab))(𝐴) = Undecided.
obtain that ConcLab2ArgLab( 2. ArgLab2ConcLab(ConcLab2ArgLab(ConcLab)) =
ArgLab2ConcLab(ArgLab))(𝐴) = In; ConcLab.
b) ArgLab(𝐴) = Out. From the fact that Let ConcLab be a complete conclusion la-
ArgLab is a complete argument labeling, beling. This, by definition, implies that
it follows that there is an attacker 𝐵 of there is a complete labeling of ArgLab ar-
𝐴 such that ArgLab(𝐵) = In or there is a guments with ArgLab2ConcLab(ArgLab) =
set of supporters 𝐶 of 𝐴 with the same ConcLab. As noted earlier, it is true that
conclusion 𝑐 such that ArgLab(𝐶) = Out. ConcLab2ArgLab(ArgLab2ConcLab(ArgLab)) =
Let 𝑏 = f(𝐵) and 𝑐 = f(𝐶). From the defini- ArgLab. It then follows that ConcLab2ArgLab(
tion of attack, it follows that 𝑏 ∈ Vul(𝐴). ConcLab) = ArgLab. This implies that Ar-
From the definition of ArgLab2ConcLab, it gLab2ConcLab(ConcLab2ArgLab( ConcLab))
follows that ArgLab2ConcLab(ArgLab)(𝑏) = ArgLab2ConcLab(ArgLab). Combining
= In. Likewise, from the definition of these observations, we finally get Ar-
support, it follows that 𝑐 ∈ Nec(𝐴) and gLab2ConcLab(ConcLab2ArgLab(ConcLab)) =
ArgLab2ConcLab(ArgLab)(𝑐) = Out. ConcLab.
From the definition of ConcLab2ArgLab,
it follows that ConcLab2ArgLab( We thus obtain the desired result.
ArgLab2ConcLab(ArgLab))(𝐴) = Out; Given the proof above, the desired equivalence can be
c) ArgLab(𝐴) = Undecided. From the fact
stated:
that ArgLab is a complete argument label-
ing, it follows that not every attacker 𝐶 of Theorem 5. Let P be a logic program and 𝐴𝐹𝑃 =
𝐴 has ArgLab(𝐶) = Out and/or not every (𝒜, 𝒮, 𝑓, ℛ− , ℛ+ ) the BCAF generated by the modified
56
�WCG algorithm. Then the complete, preferred, grounded, In short, by introducing the support relation first pro-
stable and semi-stable conclusion labels of 𝐴𝐹𝑃 are identi- posed by Alfano et al. (2020) to well-formed and non-
cal to the labels assigned respectively by the partial stable, redundant CAFs and proving the equivalence between
regular, well-founded, stable and L-stable models of P semantics, we have shown that both formalims yield the
same results and can be translated from one to the other
Proof. Given the proof of theorem 4, the results obtained without any loss of information.
by the P-stable semantics in the logic program and by the
complete semantics in the BCAF are equivalent. From
that, we can prove the equivalences between the pre- 3.4. Modeling Different Types of Support
ferred, grounded, stable and semi-stable semantics of As discussed in Section 2.2, there is, in the literature,
the BCAF with respectively the regular, well-founded, more than one possible interpretation for the meaning
stable and L-stable semantics from logic programming. of the support relation in argumentation frameworks.
This is due to the fact that the latter are special cases of So far, BCAF has been dealt with using a specific inter-
complete/P-stable solutions in which there is a maximiza- pretation of the support relation which, in the special
tion or minimization of a given label at the conclusion case where each conclusion is uniquely associated with
level. an argument, converges to the necessary support (For
more general cases, the definition changes a little, since
Example 6. (Continuing Examples 3 and 4.) The results
in BCAFs support is given from the conclusions and if
obtained by the complete semantics when applied to the
an argument 𝐴 is accepted and another 𝐴′ is not, where
BCAF in Figure 4 are the same as the one shown in Example
𝑎 = 𝑓 (𝐴) = 𝑓 (𝐴′ ), an argument supported by 𝑎 can
3. Since the maximizing/minimizing of labels is done at
still be accepted).
the conclusion level, the BCAF semi-stable semantics pro-
However it seems natural that the BCAFs should be
duces the same results as the logic programming L-stable
able to model different types of support. The results by
semantics.
Cayrol and Lagasquie-Schiex (2013) that show that there
The translation of BCAFs to a logic program is slightly is a translation between the deductive and the necessary
changed from the version shown for CAFs: supports and by Polberg and Oren (2014) that deduced
the same for the evidential and the necessary supports
Definition 34. Let C = (𝒜, 𝒮, 𝑓, ℛ− , ℛ+ ) be a reinforces the intuition behind the previous sentence.
BCAF. For each argument 𝐴, a rule is generated as We now show that in addition to the similarity with the
𝑓 (𝐴) :− 𝑓 (𝐴1 ), ..., 𝑓 (𝐴𝑚 ), not 𝑓 (𝐵1 ), ..., not 𝑓 (𝐵𝑛 ) necessary support, it is possible to redefine the BCAF in
where 𝐴1 ...𝐴𝑚 are the arguments that support 𝐴 and order to encompass other interpretations of support.
𝐵1 ...𝐵𝑛 are the ones that attack it. We denote as PC the Let us take the deductive support as an example. This
logical program constructed by this method. support means that if an argument 𝐴 supports 𝐵, if 𝐴
Interestingly, two-way translations ensure that there is accepted, 𝐵 must also be accepted. We can adapt
is no loss of information, so the logic programs and non- Definition 30 to reflect this different interpretation [8]:
redundant BCAFs generated in repeated translations are Definition 35. An argument 𝐴 supports another 𝐵, if
always the same. That can be seem if we apply the transla- 𝑓 (𝐵) = 𝑏 is a necessity of 𝐴. Thus, for 𝐴 to be accepted,
tion algorithm to the graph in Figure 4. We obtain exactly at least one of the arguments 𝐵 with 𝑓 (𝐵 ) = 𝑏 must
𝑖 𝑖
the rules of Example 1, which in turn are the same rules also be accepted.
that were translated to the BCAF. It is thus clear that
well-formed and non-redundant BCAFs guarantee cor- Given this, the other definitions for the BCAF remain
respondences with logic programming in translations in mostly the same, but a different interpretation of the
both directions. support relation is modeled. However, this definition only
Hence we now have the desired one-to-one equiv- corresponds to the deductive support in the special case
alence between argumentation (well-formed and non- that each argument has an unique conclusion. For the
redundant BCAFs) and logic programs. more general case, there is a similar divergence. Figure 5
We also argue that, despite not being a one-to-one shows the BCAF generated once again from the rules in
equivalence, the results are relevant even for the redun- Example 1.
dant BCAF case. It is so because even if the BCAF graph The same procedure can be run in order to adapt the
generated by multiple translations is not the same, the definition of support to the evidential support: A BCAF
conclusions and the relations they hold to each other that contains the evidential support differentiates be-
will remain the same. That means that the adapted WCG tween two types of arguments: the prima-facie argu-
algorithm eliminates the redundancies, keeps the desired ments, which do not need any support to be accepted,
relations between conclusions and maintains the equiva- that is, they do not have a set of necessities; and common
lence of semantics. arguments, which need to be supported by an accepted
57
� (in the well-formed and non-redundant case). In the orig-
𝐴3 (𝑏) 𝐴1 (𝑏) 𝐴0 (𝑎) 𝐴2 (𝑐) 𝐴4 (𝑑) inal model, if we start from an logic program, translate
to a CAF and then go back to the logic program, we may
lose relationships between the conclusions expressed in
Figure 5: The Bipolar Conclusion-augmented Argumentation the original logic program. With the proposed model
Graph generated from the rules in example 1 using Definition and its translations, this no longer happens. This sug-
35. It differs from the BCAF in Figure 4 only by the direction gests that BCAFs and logic programs can be understood
of the support. as two different but equivalent formalisms, so that the
properties of one can be properly translated to the other.
Despite the several proposals in the literature about
argument of the first type in order to be accepted as well, possible translations between logic programs and some
that is, they have a set of necessities. form of argumentation framework, to the best of our
Again, the remaining definitions for the BCAF are knowledge, ours is the only one that generates an one-
mostly retained when considering the different support to-one translation between the two formalisms, where
interpretation and, as before, the definition given above translation runs in both directions, for some sizeable
is broader than the original definition of evidential sup- class of argumentation frameworks — moreover, with
port. That is, the two converge only in the special cases the additional degree of argumentative expressiveness
where either each conclusion is linked to a single argu- provided by the bipolarity.
ment and/or no conclusion is linked to both types of Alfano et al. (2020) propose methods of translation be-
arguments. tween logic programs and BAFs that are very similar to
those proposed in this text in definitions 30 and 35. How-
Example 7. The graph in Figure 4 is also the representa- ever, our BCAFs model places conclusions as the main
tion of the BCAF created from the rules in example 1 using goal of the argumentative process; this also changes the
the support definition based on the evidential support. The translations. In addition, Alfano et al. (2020) focus their
arguments 𝐴0 , 𝐴1 , 𝐴2 , 𝐴3 are considered prima-facie and work on the translations from Dung’s argumentation
𝐴4 a normal argument graphs to logic programs, but not in the reverse case.
In general, therefore, it is clear that BCAF have pow- Our model, on the other hand, deals with translations in
erful tools for different forms of support, and in spe- both directions and guarantees semantic equivalences by
cial cases, they can correspond exactly to the definitions focusing on conclusions.
given for BAFs, and in others, to a more general version Kawasaki et al. (2019) also propose a similar translation
of the same principle. Thus, for all interpretations of sup- between logic programs (a variation for legal settings
port, the translation algorithm and the proofs presented called PROLEG) and BAFs in order to develop a system
can be adapted. That is, regardless of which form the capable of aiding in legal reasoning. The authors however
support takes, the properties of well-formed and non- only deal with the translations from PROLEG to BAFs
redundant BCAF remain the same, including its one-to- and not the other way around. In addition to that, we feel
one correspondence to logic programming. our approach is more straightforward and encompasses
different interpretations of support.
Other notable proposals include various translation
4. Discussion and Related Work schemes that employ Assumption-Based Argumentation
Frameworks [10] and are able to prove equivalences to
BCAFs have all the advantages that their version without logic programming [9]. A similar proposal was put forth
bipolarity (the CAFs) obtains. One gets correspondence by Pisano et al. (2020), in which the author created the
between semi-stable and L-stable semantics and, due to Arg-tuProlog, a tool capable of dealing with the 𝐴𝑆𝑃 𝐼𝐶 +
their equivalence with the logic program, BCAFs main- formalism [28] for logic programming and the Dung style
tain the same computational complexity as the original AAFs with preferences and weights [27, 29]. None of
model. In addition to that, BCAFs have two additional them obtain both the properties our model introduces.
benefits over CAFs. The first is that the argumentation
model now has some form of positive relationship, that
is, a form of support. This makes it more expressive (in 5. Conclusion
argumentation terms) than the CAFs and closer to the
human way of arguing, as discussed in [7, 13]. It is well-known that both abstract argumentation frame-
The second advantage over CAFs is the one-to-one cor- works and logic programs capture broad elements of non-
respondence with logic programming, i.e. no information monotonic reasoning. It is also well-known that several
is lost in repeated translations between the formalisms semantics for abstract argumentation frameworks corre-
spond to semantics for logic programs and vice-versa, but
58
�not all popular semantics satisfy this property. In addi- our paper.
tion, it is also known that conclusion(claim)-augmented
arguments lead to a satisfactory set of semantic corre-
spondences; we have rehearsed here those recent re- References
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