We introduce a formalism for bipolar argumentation frameworks that combines diferent 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. 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 equivain philosophy since ancient times, in contexts ranging lences. from politics and law to science and arts [1, 2]. Within In a diferent 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 inis 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 alSince Dung's paper, the connections between argumen- gorithms that translate diferent 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 diferences. In addiconnection, 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 beboth 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 prelogic programming and AAFs breaks down in the con- vious proposals reaches a one-to-one correspondence betext of the latter semi-stable semantics. Proposals for tween some family of argumentation frameworks expresenlarged 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 Dvorák and Woltran (2019) and by Alfano et al. (2020). In doing so, we reach the Bipolar Conclusion-augmented Argumentation Framework and prove (for well-formed and non-redundant frameworks) its one-to-one equivalence to normal logic programming. The translation algorithms between both formalisms are also introduced. The proposed framework is able to encapsulate diferent versions of the support relation in the literature. In short, we show that a large family of bipolar argu-
mentation frameworks is normal logic programming, and vice-versa.
Section 2 briefly goes over needed background: abstract argumentation frameworks and their bipolar extension; logic programs and their semantics. Section 3 starts with relevant results from the literature on the relationship between Dung’s framework and logic programming and then introduces the conclusion-augmented argumentation frameworks, showing that they improve on previous results by refining the equivalence between various formalisms. The bipolar conclusion-augmented argumentation model is later used to obtain a one-to-one equivalence with logic programming while also ofering diferent interpretations of support. A novel discussion about the framework and its relationship with other proposals in the literature is developed in Section 4; finally, Section 5 concludes and describes possible future work. 2. Argumentation Frameworks and Logic Programming
conflict-free if there are no arguments and in the set of arguments labeled In for which → .
ble if for every argument labeled Out, there exists an argument B labeled In such that → .
plete if for every argument with the label Undecided, there are no arguments with the label In that attack and every acceptable argument with respect to the set of arguments In is also labeled In.
In this section we review argumentation frameworks, Definition 7. A complete labeling ℒ of an AAF is prebipolar frameworks, logic programming, and some of ferred if the argument set In is maximal. their semantics. 2.1. Abstract Argumentation Frameworks
(AAFs)
, represented by → , intuitively means that if is accepted, cannot be. An AAF can be represented as a graph structure, where nodes stand for arguments and edges as the attack relation between them (see example in Figure 1).
Dung defined semantics through extensions. The latter represent sets of arguments that are acceptable according to some criterion. In this text, however, we will use labelings to define semantics [11].
→ {In, Out, Undecided}.
if the argument set labeled Undecided is empty.
grounded if the argument set In is minimal.
stable if the argument set Undecided is minimal.
2.2. Bipolar Argumentation Frameworks (BAFs) An argumentation scenario seems to require not only attacks but also “positive” support relations between arguments [12]. The definition of support relations, unlike the attack relation, varies quite a bit in the literature [13, 14]. In this text we stick to the three interpretations explained by Cayrol and Lagasquie-Schiex (2013): the necessary support [15, 16], the deductive support [17], and the evidential support [18, 19]. A necessary support from one argument to another , represented by ⇒ , means that if is accepted (received the label In), must also be accepted. A deductive support from one argument to another , on the other hand, is represented by ⇒ , and means that if is accepted (received the label In), must also be accepted. Finally, there are two types of arguments in a BAF that contains evidential support: prima-facie arguments, which do not need any support to be accepted, and common arguments, which need to be supported by an accepted argument of the ifrst type to be accepted.
Regardless of the interpretation of the support relation,
a BAF can be defined as [
there are no , ∈ such that defeats (directly or indirectly) .
is conflict-free and defends all its elements.
it is admissible and every argument that can be accepted together with is part of .
ferred, grounded, stable and semi-stable semantics can be adopted as before (as they are all special cases of the complete semantics in which some label is maximized or minimized).
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
proencode the attack relation and double edges encode the grams [
To handle support relations, the semantics, the label- Definition 18. A (normal) logic program is composed ing types and the acceptability criteria must be adapted. by a finite set of rules. Each rule is an expression of the Several proposals have been made in order to achieve form : :− 1, ..., , not 1, ..., not , where this, with diferences in the way the relative strength , and represent atoms and not is the classical between attack and support relations is taken into ac- negation. 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 . The Herbrand Base of the program
We assume for now that support is of the necessary is the set 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 , , , feat by a traditional attack, an argument can be defeated : indirectly. 0 : :− not , 1 : :−
not , not , 2 : :− not , 3 : :−
not , 4 : :− , not .
Definition 19. A three-valued interpretation of a logic program P is a pair = ( , ℱ ) such that ∩ ℱ = ∅ and that both and ℱ contain elements from the Herbrand base of P. is understood as true, ℱ as false and \( ∪ ℱ ) as undecided.
A three-valued model of P is an interpretation such that for each ∈ we have: • is in if there is a rule whose head is =
and where each is in ; • is in ℱ if every rule whose head is = is such
that there is some in ℱ .
The reduct of P with respect to a three-valued interpretation ℐ, denoted P/ℐ, is a logic program constructed using the following steps:
by a sequence 11 ... − 1, where = 1, ..., − 2, = + and − 1 = − or where = 2, ..., − 1, = + and 1 = − .
Definition 13. Let ⊆ and ∈ . The set defeats if there is a direct or indirect defeat for from some element of . The set supports if there is a sequence of the form 11 ... − 1, ≥ 2, such that = 1, ..., − 1, = + with = and 1 ∈ .
Definition 14. Let ⊆ and ∈ . The set collectively defends if for some set ℬ ⊆ , if ℬ defeats then there is a ∈ such that defeats ℬ.
Given this, a conflict-free set and an admissible set can be redefined: • First, remove from P every rule that contains not in its body for some ∈ ; • Then, for each remaining rule, remove not from the rule body if ∈ ℱ ; • Finally, replace any remaining occurrences of not ′ with a new symbol representing “undecided”.
mum and ℱ maximum (with respect to the set inclusion).
gram , given an interpretation = ( , ℱ ), in several
ways [
Definition 21. A model of is well-founded if it is Pstable and is minimal.
Definition 22. A model of is regular if it is P-stable and is maximal.
Definition 23.
A model of is stable if it is P-stable and ∪ ℱ = .
Definition 24. A model of is L-stable if it is P-stable and ∪ ℱ is maximal. • Starting with a set of rules, process one rule at a time; • If a rule of the form :− not 1, . . . , not is found, then generate an argument with rules { :− not 1, . . . , not }, vulnerabilities Vul() {1, . . . , }, conclusion and a set of sub-arguments that contain only itself ; • If a rule of the form :− 1, . . . , , not 1, . . . , not is found and assuming that each has an associated argument , then generate an argument with a set of sub-arguments , conclusion , rules composed by the union of { :− 1, . . . , , not 1, . . . , not } with the rules of each sub-argument and vulnerabilities Vul() as the union of {1, . . . , } with the vulnerabilities of the sub-arguments; • After going through all the rules, the relations between arguments are established. If an argument has a conclusion that is present in the vulnerabilities of another argument , then attacks . With this, the AAF is created.
Example 2. If we apply the WCG algorithm described above to the rules in Example 1, we obtain the AAF shown in Figure 1.
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
, the following mapping [
In this section we look at equivalences between argu- We then ask: if we apply some semantics at the
argumentation 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
dithe 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 [
Theorem 1. (Theorems 19, 20, 21 and 22 of [
Argumentation Frameworks (AAFs) and Logic Programs Out this last process is equivalent both at the argument and at the conclusion levels, but the same is not true for 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)): (∅, {0, 1, 2, 3, 4}, ∅), ({0}, {3}, {1, 2,4}) and ({2, 4}, {1, 3}, {0}). From this, we obtain the conclusion labelings: (∅, seems to take a more complicated route than ours to ar{, , , }, ∅), ({}, {}, {, }) and ({, }, {}, {}). 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 previresult. 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 probalevel we get two solutions. bilistic argumentation frameworks [24]. CAFs are defined as follows:
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 (, , , ℛ), with being Compare with the other semantics. The grounded se- a finite set of arguments, a set of conclusions, is a mantics, in situations of mutual attacks, does not reach function of arguments to conclusions, and ℛ ⊆ × 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 labeling when an Undecided label is unavoidable — in an argumentative scenario however, we believe that not arriving at some labeling is undesirable. We would thus like to choose the semi-stable semantics, but this seems to clash with our desire to obtain a correspondence between argumentation frameworks and logic programs. In the next subsection we show how to enlarge argumentation frameworks to obtain our desired correspondences.
We then say that every conclusion ∈ is associated to a finite set of arguments − 1(). Analogously to AAFs, a conclusion-augmented argumentation graph is a directional graph in which each node represents an argument associated with its conclusion and each arrow represents an attack. It is important to note that the CAF maintains a high degree of abstraction as the internal structure of the arguments is not explicit. The only change is that each argument is represented along with its conclusion . 3.2. The Connection between
Conclusion-Augmented Argumentation Frameworks (CAFs) and Logic Programs
between argumentation frameworks and logic programs is to augment arguments with their associated conclusions. We do so to guarantee that semi-stable semantics does have a correspondence in logic programming.
To do so, we adopt recent work on Claim-augmented
Argumentation Frameworks (CAFs), where each argument
is augmented with its associated claim [
To each conclusion ∈ we can assign the labels In, Undecided, and Out. Naturally, this labeling of conclusions is related to the labeling of arguments through (), and the procedure described previously obtains a labeling of conclusions from one of arguments.
A CAF can be generated by a slightly altered version of the WGU algorithm presented in Section 3.1, which associates each argument with a conclusion at the end of the process. We show in Figure 3 the result of applying this altered algorithm to the logic program of Example 1.
One can easily adapt the definition of complete labeling for arguments to an analogous concept of complete labeling of conclusions: (iii) a conclusion is Undecided then there is no argument Example 5. Applying the algorithm in Definition 28 to ∈ − 1() for which all arguments that attack the CAF in Figure 3, we get the following set of rules: have () as Out; and for some argument ∈ − 1(), there is no argument that attacks with conclusion 0 : :− not , 1 : :− not , not , () In, and there is an argument that attacks that 2 : :− not , 3 : :− not , same and whose conclusion () is not Out. 4 : :− not .
the conclusion level and at the arguments level proved by Caminada et al. (2015), we can define the other semantics for the CAFs. From the results of complete semantics for the arguments, it is possible to obtain the equivalent labels at the conclusion level by applying the criteria described in section 3.1. The latter are the complete conclusion labeling solutions and, from them, we can maximize/minimize any desired label. With this, we obtain the other semantics mentioned in this work, which are defined as: Definition 27. Assume is a conclusion labeling. is grounded if it is complete and the conclusion set In is minimal. is preferred if it is complete and the conclusion set In is maximal. is stable if it is complete and no conclusion remains Undecided. is semi-stable if it is complete and the conclusion set Undecided is minimal.
Theorem 2. Let P be a logic program and C = (, , , ℛ) the CAF generated by the modified WCG algorithm. Then the complete, preferred, grounded, stable and semi-stable conclusion labels of C are identical to the labels assigned respectively by the partial stable, regular, well-founded, stable and L-stable models of P
Theorem 3. Let C = (, , , ℛ) be a CAF and PC the corresponding logic program . So the partial stable, regular, well-founded, stable, L-stable models of P assign the same labels to the conclusions as respectively the complete, preferred, grounded, stable, semi-stable semantics of C.
logic programs and argumentation frameworks in both directions is specified. Due to space constraints we refer to our previous work [24] to prove both Theorems 2 and 3.
However, it should be clear that the move to CAFs is not enough to achieve a completely satisfactory one-toone equivalence with logic programming. That can be seem if we compare the rules of Examples 1 and 5. Rule 4 difers by the omission/inclusion of the conclusion in those examples. That means that repeated translations between formalisms loses some information. To solve that problem, in the next section we take the support relations into account. 3.3. Bipolar Conclusion-augmented
Argumentation Framework (BCAF)
We thus introduce: Example 4. (continuing Example 3). Since in CAFs we apply the maximization/minimization of labels directly at the conclusion level, the semi-stable semantics will yield two solutions. Thus it is equivalent to applying the L-stable semantics to the logic program in Example 1.
Definition 29. A Bipolar Conclusion-augmented Argumentation Framework (BCAF) is a tuple (, , , ℛ− , ℛ+), with being a finite set of ar
Consequently, the semantic correspondence between guments, a set of conclusions, being a function of a logic program and its derived CAF is proved, and it is arguments to conclusions, ℛ− ⊆ × is the attack also possible to demonstrate it in the opposite direction. relation and ℛ+ ⊆ × is the support relation. That is, starting from an CAF and generating a logical program, the correspondences are maintained. We use the following translation process:
Each argument ∈ can be defined as a rule :− 1, . . . , , not 1, . . . , not .
Definition 28. Let C = (, , , ℛ) be a CAF. 1, . . . , , not 1, . . . , not }, vulnerabiliFor each argument , generate a rule () :− not ties Vul() {1, . . . , } and necessities Nec() (1), ..., not () where are the arguments that {1, . . . , }. attack . 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 that is accepted, must be rejected. In a BCAF, one argument attacks another when its conclusion is one of 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: Definition 30. An argument supports another argument if () = is a necessity of . Thus, for to be accepted, at least one of the supports with () = 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 algorelevant: rithm, the adapted version guarantees the equivalence Definition 31. Let = (, , , ℛ− , ℛ+) be a of the P-stable semantics in logic programming and the BCAF. is said to be redundant if there is at least pair complete semantics in the bipolar conclusion-augmented of arguments , ∈ such that () = (), Vul() argumentation framework. The answer to this question = Vul() and Nec() = Nec(). 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 = (, , , ℛ− , ℛ+) be a where each conclusion is represented by the argument BCAF. is said to be well-formed if all the arguments that best defends it. ∈ that hold the same conclusion () 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 the context of BCAFs:
well-formed and non-redundant unless stated otherwise.
As previously mentioned, Caminada et al. (2015) stud- Definition 33. Let be a logical program and = ied the WCG algorithm for the translation of logic pro- (, , , ℛ− , ℛ+) its associated bipolar conclusiongrams 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 and let ConcLabs be equivalences are once again proved. We thus propose an the set of all conclusion labels from and adapted WCG algorithm for BCAFs: 1. Starting with a rule set, process each rule at a
time; 2. If a rule of the form :− not 1, . . . , not is found, then generate an argument with conclusion , rules { :− not 1, . . . , not } and vulnerabilities Vul() {1, . . . , }; 3. If a rule of the form :− 1, . . . , , not 1, . . . , not is found, then generate an argument with conclusion , rules { :− 1, . . . , , not 1, . . . , not }, vulnerabilities Vul() {1, . . . , } and necessities Nec() {1, . . . , }; 4. After going through all the rules, the relations between arguments are established. If an argument has a conclusion that is present in the vulnerabilities of another argument , then attacks . On the other hand, if the conclusion of is present in the necessities of , then supports . With this, and keeping that each argument is linked to a conclusion, the bipolar conclusion-augmented argumentation graph is created. 1. We define an ArgLab2ConcLab function: ArgLabs → ConcLabs such that for each ArgLab ∈ ArgLabs, it is true that ArgLab2ConcLab(ArgLab) is the associated conclusion labeling of ArgLab; 2. We define an ConcLab2ArgLab function: ConcLabs → ArgLabs such that for each ConcLab ∈ ConcLabs and each ∈ it is true that: a) ConcLab2ArgLab(ConcLab)() = In if for each ∈ Vul() it is true that ConcLab() =
ConcLab() = In; b) ConcLab2ArgLab(ConcLab)() = Out if there is a ∈ Vul() such that ConcLab() = In and/or if there is a ∈ Nec() such that ConcLab(w) = Out; c) ConcLab2ArgLab(ConcLab)() =
ConcLab() = Out; if there is no ∈ Vul() such that ConcLab() = In; if it is not true that for all ∈ Nec() ConcLab() =
ConcLab() = Out.
Theorem 4. In the case of complete argument labellings and complete conclusion labellings, the functions ArgLab2ConcLab and ConcLab2ArgLab are bijections and each other’s inverse
from [
ArgLab.
Let ArgLab be a complete argument labeling and let be an argument. Three cases are distinguished. argument with () = for each conclusion that supports has ArgLab() = In (i). There is also no attacker of that has ArgLab() = In and/or no conclusion ∈ Nec() such that all arguments with () = have ArgLab() = Out (ii). From (i) together with (ii) it follows that there is an attacker of with ArgLab() = Undecided and/or a group of arguments with the conclusion ∈ Nec() with ArgLab() = Undecided. Let = () and = (). From (ii) together with the definition of attack, it follows that there is no argument ′ with (′) = such that ArgLab(′) = In. So ArgLab2ConcLab(ArgLab)() = Undecided.
Likewise, (ii) together with the definition of support implies that for some conclusion ∈ Nec() ArgLab2ConcLab(ArgLab)() = Undecided (iii). Furthermore, from (ii) together with the definitions of attack and support, it follows that for each argument with () ∈ Vul() it is valid that ArgLab() ̸= In and for at least one argument with a conclusion ∈ Nec(), ArgLab( ) ̸= Out. Therefore, for every ∈ Vul(), ArgLab2ConcLab(ArgLab)() ̸= In and for at least one ∈ Nec(), ArgLab2ConcLab(ArgLab)( ) ̸= Out (iv).
Finally, From (iii) and (iv), and the definition of ConcLab2ArgLab, it follows that ConcLab2ArgLab(ArgLab2ConcLab( ArgLab))() = Undecided. a) ArgLab() = In. From the fact that ArgLab is a complete labeling, it follows that ArgLab() = Out for every attacker of and that ArgLab() = In for at least one argument with () = for each conclusion that supports . From the definition of attack it follows that for every ∈ Vul() and for every argument with f() = , ArgLab() = Out. This then implies that for every ∈ Vul() it is true that ConcLab2ArgLab(ArgLab2ConcLab( ArgLab))() = Out. Similarly, from the definition of support, it follows that for every ∈ Nec(), for at least one argument with f() = , ArgLab() = In, that is, ConcLab2ArgLab(ArgLab2ConcLab(ArgLab) )() = In. By the definition of ConcLab2ArgLab, we ifnally obtain that ConcLab2ArgLab( 2. ArgLab2ConcLab(ConcLab2ArgLab(ConcLab)) = ArgLab2ConcLab(ArgLab))() = In; ConcLab. b) ArgLab() = Out. From the fact that Let ConcLab be a complete conclusion laArgLab is a complete argument labeling, beling. This, by definition, implies that it follows that there is an attacker of there is a complete labeling of ArgLab ar such that ArgLab() = In or there is a guments with ArgLab2ConcLab(ArgLab) = set of supporters of with the same ConcLab. As noted earlier, it is true that conclusion such that ArgLab() = Out. ConcLab2ArgLab(ArgLab2ConcLab(ArgLab)) = Let = f() and = f(). From the defini- ArgLab. It then follows that ConcLab2ArgLab( tion of attack, it follows that ∈ Vul(). ConcLab) = ArgLab. This implies that ArFrom the definition of ArgLab2ConcLab, it gLab2ConcLab(ConcLab2ArgLab( ConcLab)) follows that ArgLab2ConcLab(ArgLab)() = ArgLab2ConcLab(ArgLab). Combining = In. Likewise, from the definition of these observations, we finally get Arsupport, it follows that ∈ Nec() and gLab2ConcLab(ConcLab2ArgLab(ConcLab)) = ArgLab2ConcLab(ArgLab)() = Out. ConcLab.
From the definition of ConcLab2ArgLab, it follows that ConcLab2ArgLab( We thus obtain the desired result.
ArgLab2ConcLab(ArgLab))() = Out; Given the proof above, the desired equivalence can be c) ArgLab() = Undecided. From the fact stated: that ArgLab is a complete argument labeling, it follows that not every attacker of Theorem 5. Let P be a logic program and = has ArgLab() = Out and/or not every (, , , ℛ− , ℛ+) the BCAF generated by the modified WCG algorithm. Then the complete, preferred, grounded, stable and semi-stable conclusion labels of are identical to the labels assigned respectively by the partial stable, regular, well-founded, stable and L-stable models of P Proof. Given the proof of theorem 4, the results obtained 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 preferred, grounded, stable and semi-stable semantics of the BCAF with respectively the regular, well-founded, stable and L-stable semantics from logic programming.
This is due to the fact that the latter are special cases of complete/P-stable solutions in which there is a maximization or minimization of a given label at the conclusion level.
Example 6. (Continuing Examples 3 and 4.) The results obtained by the complete semantics when applied to the BCAF in Figure 4 are the same as the one shown in Example 3. Since the maximizing/minimizing of labels is done at the conclusion level, the BCAF semi-stable semantics produces the same results as the logic programming L-stable semantics.
Definition 34. Let C = (, , , ℛ− , ℛ+) be a BCAF. For each argument , a rule is generated as () :− (1), ..., (), not (1), ..., not () where 1... are the arguments that support and 1... are the ones that attack it. We denote as PC the logical program constructed by this method.
In short, by introducing the support relation first proposed by Alfano et al. (2020) to well-formed and nonredundant CAFs and proving the equivalence between semantics, we have shown that both formalims yield the same results and can be translated from one to the other without any loss of information. 3.4. Modeling Diferent Types of Support
more than one possible interpretation for the meaning of the support relation in argumentation frameworks.
So far, BCAF has been dealt with using a specific interpretation of the support relation which, in the special case where each conclusion is uniquely associated with an argument, converges to the necessary support (For more general cases, the definition changes a little, since in BCAFs support is given from the conclusions and if an argument is accepted and another ′ is not, where = () = (′), an argument supported by can still be accepted).
However it seems natural that the BCAFs should be able to model diferent types of support. The results by Cayrol and Lagasquie-Schiex (2013) that show that there is a translation between the deductive and the necessary supports and by Polberg and Oren (2014) that deduced the same for the evidential and the necessary supports reinforces the intuition behind the previous sentence.
We now show that in addition to the similarity with the necessary support, it is possible to redefine the BCAF in order to encompass other interpretations of support.
Let us take the deductive support as an example. This support means that if an argument supports , if is accepted, must also be accepted. We can adapt
Definition 30 to reflect this diferent interpretation [
Interestingly, two-way translations ensure that there is no loss of information, so the logic programs and nonredundant BCAFs generated in repeated translations are Definition 35. An argument supports another , if always the same. That can be seem if we apply the transla- () = is a necessity of . Thus, for to be accepted, tion algorithm to the graph in Figure 4. We obtain exactly at least one of the arguments with () = must 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 diferent 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 diferentiates bewill remain the same. That means that the adapted WCG tween two types of arguments: the prima-facie argualgorithm 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
(in the well-formed and non-redundant case). In the orig3() 1() 0() 2() 4() 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 sug35. It difers from the BCAF in Figure 4 only by the direction gests that BCAFs and logic programs can be understood of the support. as two diferent 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 onemostly retained when considering the diferent 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 bearguments. tween logic programs and BAFs that are very similar to those proposed in this text in definitions 30 and 35. HowExample 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 0, 1, 2, 3 are considered prima-facie and work on the translations from Dung’s argumentation 4 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 diferent 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 diferent interpretations of support.
Other notable proposals include various translation
4. Discussion and Related Work schemes that employ Assumption-Based Argumentation
Frameworks [
The second advantage over CAFs is the one-to-one correspondence with logic programming, i.e. no information is lost in repeated translations between the formalisms It is well-known that both abstract argumentation frameworks and logic programs capture broad elements of nonmonotonic reasoning. It is also well-known that several semantics for abstract argumentation frameworks correspond to semantics for logic programs and vice-versa, but 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 correspondences; we have rehearsed here those recent results around CAFs through a hopefully simpler approach.
And it is well-known that the semantic correspondences between CAFs and normal logic programs are actually based on translations from argumentation frameworks to logic programs and vice-versa. However, these translations are not satisfactory because they are not really “associative” — a translation followed by a back translation does not necessarily get back to the same point.
By bringing ideas from bipolar argumentation, namely the recently explored fact that supports can be translated to positive atoms in rules, we obtained the desired correspondences and translations for a sizeable class of frameworks. We thus have that any well-formed and non-redundant BCAF can be readily translated to a normal logic program and any normal logic program can be readily translated to a BCAF, without any informational loss.
Intuitively, a well-formed and non-redundant BCAF is a normal logic program, and a normal logic program is a BCAF. To summarize the whole reasoning in this paper, we obtained these equivalences by combining existing results on conclusion(claim)-augmented argumentation frameworks and on the translation of supports.
In addition, we showed that our bipolar model allows the support relation to be interpreted in several diferent ways, converging, in special cases, with the most common definitions found in the literature for bipolar argumentation models.
In future work, BCAFs should be expanded to include uncertainty, and in particular to handle probabilistic argumentation, while maintaining their desirable properties in relation to logic programming. We hope the resulting probabilistic formalism will improve the debate about the meaning of probabilities in argumentation and will provide a solid basis for argumentation algorithms to be implemented for real systems.
cial Intelligence (C4AI-USP), with support by the São Paulo Research Foundation (FAPESP grant 2019/076654) and by the IBM Corporation. The first author is partially supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) grant 88887.616392/2021-00. The second author was partially supported by CNPq grant 312180/2018-7. We acknowledge support by CAPES - Finance Code 001.
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