Probabilistic causal bipolar abstract argumentation: an approach based on credal networks

The Bipolar Argumentation Framework approach is an extension of the Abstract Argumentation Framework. A Bipolar Argumentation Framework considers a support interaction between arguments, besides the attack interaction. As in the Abstract Argumentation Framework, some researches consider that arguments have a degree of uncertainty, which impacts on the degree of uncertainty of the extensions obtained from a Bipolar Argumentation Framework under a semantics. In these approaches, both the uncertainty of the arguments and of the extensions are modeled by means of precise probability values. However, in many real application domains there is a need for aggregating probability values from different sources so it is not suitable to aggregate such probability values in a unique probability distribution. To tackle this challenge, we use credal networks theory for modelling the uncertainty of the degree of belief of arguments in a BAF. We also propose an algorithm for calculating the degree of uncertainty of the extensions inferred by a given argumentation semantics. Moreover, we introduce the idea of modelling the support relation as a causal relation. We formally show that the introduced approach is sound and complete w.r.t the credal networks theory.


Introduction
The Abstract Argumentation Framework (AAF) approach that was introduced in the seminal paper of Dung [1] is one of the most significant developments in the computational modelling of argumentation in recent years. The AAF is composed of a set of arguments and a binary relation encoding disagreements (which are called attacks) between arguments. Recent studies on argumentation (e.g., [2,3]) have demonstrated the necessity of encoding another kind of interaction between arguments, this is a positive interaction called support. Unlike the attack interaction, in this one, an argument is in favour of another argument. Thus, a Bipolar Argumentation Framework (BAF) [2] is an extension of the AAF approach with the difference that there is a new kind of interaction between arguments represented by the support relation, which is to be totally independent of the attack relation. In [3], the authors claim that the interpretation of the support relation may be diverse. They consider three possible specializations of the support relation: the deductive support, the necessary support, and the evidential support; however, none of these specializations model the causality support. It is worth mentioning that causal relations are the core relations in causal reasoning [4,5].
Like in AAFs, in BAFs there is a need for dealing with uncertainty in the elements of the BAF in order to represent the degree of uncertainty of arguments. Indeed, some real problems need to be modelled by means of impreciseness that represents a range of possible values. For example, in [6], the authors highlight the importance of opinion pooling with imprecise probabilities. One thing they argue is that imprecise probabilities admit of a much better philosophical motivation as a model of rational consensus. In [7], Weatherson addresses the decision making problem and comments that it is no requirement of rationality that agent's epistemic states be represented by a single probability function. He says that it would be better to permit agent's epistemic states to be represented by a set of probability functions. This makes even more sense when the decision is made by a set of agents. Furthermore, imprecise probability approaches have been applied in real domains such as chemistry, physical systems, and engineering problems. In [8], Fischer analyses the use of imprecise Bayesian Networks in the systematic study of the speed of chemical reactions. In [9], the authors apply Bayes Linear methods in computer model calibration and demonstrate it in the context of the galaxy formation simulation. At last, in [10], the authors use an imprecise probability model for vibro-acoustic problems in engineering. 1 To the best of our knowledge, there are few works about probabilistic BAFs (e.g., [11][12][13]). These works use precise probability approaches to model the uncertainty values. However, precise probability approaches have some limitations to quantify epistemic uncertainty, for example, to represent group disagreeing opinions. These can be precisely captured by means of imprecise probabilities, which use lower and upper bounds instead of exact values to model the uncertainty values. In [14,15], an approach for AAF with imprecise probabilities for dealing with group decision making was introduced. This approach has been successfully applied in the domain of forensics engineering (see [16]). This shows the potential of combining formal argumentation and imprecise probabilities to deal with real application domains. In that approach, the authors do not consider the support relation as part of the AF. However, given that probabilistic approaches characterize and deal with causality in a proper way, we can use this advantage for modelling the support relation as a causality support.
For a better illustration of the problem, consider a factitious discussion between a group of medicine students (which can be represented by agents). 2 The discussion is about the diagnosis of a patient. In this context, arguments represent assertions about symptoms, a likely diagnosis, or facts about the patient; the attacks represent conflicts between arguments, and supports represent causality relations. Figure 1 shows the argumentation graph where nodes represent arguments and edges the attacks (or supports) between arguments. In the graph, arguments A and B represent two possible diagnosis namely measles and chickenpox, respectively, and the aim is to make a group decision about the diagnosis based on the opinions of the students.
Suppose that each assertion, diagnosis, or fact -that is modelled as an argument -has a probability value between 0 and 1 that represents the degree of uncertainty of it. 3 Since there is more than one opinion, this means that each argument has associated a set of probability values. Thus, we need to aggregate these probability values, which cannot be done by means of an unique probability value (precise probability value), what we need is to represent a range of the possible degrees of uncertainty.
This article aims to propose a novel approach for BAFs in which the uncertainty of the arguments is modeled by an imprecise probability value whereby our proposal can be classified as an epistemic approach following the probabilistic argumentation literature [18]. Besides, it combines the non-monotonic behavior of argumentation with the causality of Credal Networks. Thus, the research questions that are addressed in this paper are: 1 How to model the causal relation in the settings of imprecise probability and argumentation reasoning? 2 As in AAFs, in BAFs, we need to define a semantics that determines sets of consistent arguments and also considers the causality relation and the imprecise probability of arguments, how these semantics will be defined? 3 The fact that the arguments that belong to an extension are uncertain, causes that such extension also has a degree of uncertainty. How to calculate the lower and upper bounds of an extension considering the causal relation that may exist between its arguments? 4 Is the argumentation-based proposal sound and complete with respect to Credal Networks theory?
The first question has to do with the knowledge representation and lays the building blocks for dealing with the problem. Thus, we use credal sets [19] to model the uncertainty values of arguments and credal networks theory for modelling the causal relation between arguments. 4 Regarding the second question, the notion of semantics in argumentation is relevant because it allows to make inferences over the graph, that is, over the BAF, and such inferences will determine, for example, which is the most likely decease. We propose to use the semantics of 2 An agent is a computer system. According to Wooldridge e Jennings [17], two usages of the term 'agent' can be distinguished. On one hand, from a weak notion of agency, an agent is a software-based computer system that has autonomy, social ability, pro-activeness, and reactivity. On the other hand, from a stronger notion of agency, an agent can be characterized by using mentalistic notions, such as knowledge, belief, intention, and obligation. 3 These uncertain degrees are probabilistic values that were obtained by transforming the qualitative opinion of the students into probabilistic distributions. The transformation of qualitative data into probabilistic distributions can be done by using different distribution models, e.g. Gamma Distribution. 4 Credal networks are an extension of Bayesian networks where instead of having a joint precise global probability distribution we have a closed and convex set of possible distributions, which are called credal sets [20]). A = The paƟent has measles C = She has blisters E = The paƟent has brown eyes G = The paƟent has fever B= The paƟent has chickenpox D = She only has small red spots F = The paƟent was vaccinated for chickenpox AAFs for dealing with the conflicting relation and the credal networks theory for dealing with the causality relation. As a result we have extensions, each one with its respective imprecise value, which will be used to rank such extensions. This rank will allow us to determine which extension better addresses the decision making problem.
Part of the semantics is the computation of the lower and upper bounds. For this task, we base on the credal sets of the arguments to calculate the uncertainty values of extensions obtained under a given extension-based semantics. The way to aggregate the credal sets depends on a causal relation between the arguments. We have to consider that some arguments may not have any relation with other arguments whereas other arguments may have a relation with more than one argument. Considering this, we propose an algorithm for the calculation of the lower and upper bounds. Since the BAF is based on credal sets and credal networks, the behavior of the algorithm has to follow the credal networks theory. Thus, we prove that our approach is sound and complete with respect to Credal Networks theory.
The remainder of this paper is structured as follows. Next section gives a brief overview on credal networks and abstract argumentation. In Section 3, we present the BAF based on credal networks. The calculation of the lower and upper bounds is tackled in Section 4. We study the main properties of the proposed approach in Section 5. A discussion about the proposal and its relation with other works in literature is presented in Section 6. Finally, Section 7 is devoted to conclusions and future work.

Background
In this section, we revise concepts of credal networks and abstract argumentation frameworks.

Credal networks
Before presenting credal networks, let us define credal sets (from Levi's credal sets [19]). Let X = {X 1 , ..., X n } be a set of probabilistic variables, a credal set defined by probability distributions p(X ) is denoted by K (X ) and K = {K (X 1 ), ..., K (X n )} denotes a finite set of credal sets of the variables of X. In this work, we assume that the cardinality of the credal sets of K is the same (let us denote it by m) and is determined by the number of agents. We also assume that p i (X ) denotes the suggested probability of the agent i w.r.t. variable X such that 1 ≤ i ≤ m and X ∈ X.
A credal network is a graphical model that associates nodes and variables with sets of probability measures [5]. A credal network consists of a directed acyclic graph, where each node in the graph is associated with a random variable X and the parents (i.e., the variables corresponding to the immediate predecessors of X according to the graph) of X are denoted by pa(X ). Each variable X is associated with a (conditional) credal set K (X | pa(X )) = { p 1 (X | pa(X )), ..., p m (X | pa(X ))}. Inference is performed by applying Bayes rule to each measure in a joint credal set. The goal is to combine these credal sets into a set of joint distributions. Next, let us show how this combination will be done in order to obtain the lower and upper bounds from the credal sets of a credal network.
Given a random variable X and its credal set K (X ), the lower and upper bounds for variable X are determined as follows: Given l variables {X 1 , ..., X l } ⊆ X and their respective credal sets K ( .., X l } are independent events, the lower and upper probabilities are defined as follows: Let us introduce an example to get an intuition of the introduced concepts on Credal Networks. Table 1 shows the values of the probability distributions for each event.
The lower and upper probabilities are calculated as follows: Let us assume that X 1 , X 2 , and X 3 are arguments. First of all, if these arguments are evaluated together, this means that there is no conflict between them. Following the example, we can assume that they are independent, which means that there is no a causal relation between them. For example, arguments C, E, F meet these conditions. The credal sets represent the degrees of uncertainty of each argument: The degree of uncertainty of the three arguments together is given by the values represented by the lower and the upper bounds.

Abstract argumentation framework
In this subsection, we will recall basic concepts related to the AAF defined by Dung [1], including the notion of acceptability and the main semantics.
Definition 1 (AAF) An abstract argumentation framework AF is a tuple of the form ARG, R where ARG is a finite set of arguments and E is a binary relation R ⊆ ARG × ARG that denotes attack relations between two arguments. If (A, B) ∈ R, we say A attacks the argument B or B is attacked by A. Next, we introduce the concepts of conflict-freeness, defense, admissibility and the four semantics proposed by Dung [1].

Definition 2 (Argumentation Semantics)
Given an argumentation framework AF = ARG, R and a set E ⊆ ARG: We use c f , ad, co, pr, gr, and st to denote conflict-free sets, admissible sets, complete, preferred, grounded, and stable extensions, respectively.
Finally, let us introduce basic concepts on BAF [2].

Definition 3 (BAF)
A BAF is a tuple BAF = ARG, R att , R sup where: • ARG is a set of arguments; • R att ⊆ ARG × ARG is an attack relation; and We will not present the concepts about semantics for the BAF approach because we characterize new BAF semantics in the settings of imprecise probability. However, we keep similar properties in the inference processes.

Credal bipolar argumentation framework
In this section, we introduce our suggested approach. We use a credal set to model the opinions of a set of agents about an argument. This means that each argument in a BAF has associated a credal set, which contains probability distributions that represent the opinions of the agents about it. Besides, due to the causal relation, some arguments may have associated one or more conditional credal sets, which are calculated by applying the Bayes rule. In Fig. 1, notice that measles (A) causes fever (G) and chickenpox (B) causes fever as well. Since A and B are conflicting, they will never belong to the same extension. This means that G is either caused by A or by B, this in turn means that we have two conditional credal sets for G, these are p(G | A) and p(G | B).
Let us recall that the support relation is interpreted as a causality relation that may exist between arguments. Thus, an argument in a causality relation can play two different roles, it can either be caused or be the cause, this means that we can have caused arguments (ARG ← ), arguments that cause other ones (ARG → ), and arguments that have no causality relation with the rest (ARG • ). We characterize these sets as follows. Given a BAF = ARG, R att , R sup : We can now define a BAF based on credal sets.

Definition 4 (Credal Bipolar Argumentation Framework) A BAF based on credal sets is a tuple
is the support relation in the sense of causality relation; and where K U is the set of all possible credal sets, is a function that attributes one or more credal sets to each argument.
Let us better explain the correspondence between arguments and credal sets. Arguments in ARG • have associated only one credal set because they are not caused by any other argument. Arguments in ARG ← have associated at least one conditional credal set because they are caused by at least one argument. Finally, some arguments in ARG → have associated only one credal set when they are not caused by any other argument and others have also at least one conditional credal set when they are caused by any other argument. Formally: We also have to consider a correspondence between caused arguments and credal sets when we are analysing extensions (that is, consistent subsets of ARG) because not all of the causing arguments may belong to the same extension. In this case, the conditional credal set associated to the caused argument is calculated taking into account only the causing arguments that belong to the extension. Thus, given an extension E x under a semantics Recall that the cardinality of every credal set depends on the number of agents. Since all the agents give their opinions about all the arguments, all the credal sets have the same number of elements. Regarding the probability values given to the arguments, it is important to consider the notion of rational probability distribution given in [21]. According to [21], if the degree of uncertainty of an argument is high, then the degree of uncertainty of the arguments it attacks is low. Thus, a probability function p is rational in a CBAF iff for each Figure 2 shows the graph representation of the credal BAF) where: Table 2 show the credal sets of arguments A, B, C, D, E, F and G. Table 3 shows the probabilistics values for calculating the conditional credal sets. Lastly, Table 4 shows the conditional credal sets, which were obtained by applying the Bayes rule.

Lower and upper bounds of extensions
In this section, we focus on the calculation of the lower and upper bounds of extensions from the credal sets and conditional credal sets of the arguments that belong to the extensions. Besides, we study how to compare the intervals and how it impacts on the obtained results.

Calculating the bounds
Section 2 presented the definition of conflict-free and admissible sets and complete, preferred, grounded, and stable semantics. As was said before, these concepts are directly related to the attack relation between arguments, we will use them for obtaining the extensions. On the other hand, we have the support relation; we will base on this relation to calculate the upper and lower bounds of extensions.
Considering the causality relation, the arguments of an extension E x (for x ∈ {c f , ad, co, pr, gr, st}) may belong to ARG → , ARG ← , or ARG • . Depending on it, the calculation of the probabilistic lower and upper bounds of each extension is different. Thus, we can distinguish the following cases: (i) the extension is empty, (ii) the extension has only one non-caused argument, and (iii) the extension includes more than one argument. Consider the following functions:

Definition 6 (Upper and Lower Bounds of Extensions) Let
TOP_CAU and FREE_CAU consider only the arguments of E x and their causal relations restricted to E x . The former returns the arguments that are caused by any other argument in E x but do not cause any other argument(s) in E x . If there is an argument that belongs to both ARG ← and ARG → but the argument(s) caused by it are not in E x , then it is returned by TOP_CAU. The latter returns the arguments that belong to ARG → but whose caused arguments do not belong to extension E x .
for each A ∈ ARG ← do 5: CAU A := pa(A) ∩ E x //CAU A is the set of arguments that support A, which belong to E x 6: Calculate K (A | CAU A ) //Calculate the conditional credal set for A by applying the Bayes rule 7: X := X ∪ (A | CAU A ) //X represent the set of probabilistic variables, whose credal sets will be evaluated. Some variables have associated a conditional credal set. Instead of using A as a variable, we will use A | CAU A just for notation and for making clear that we will use its conditional credal set. 8: end for 9: end if 10: //Apply (2) for obtaining the lower and upper bounds of E x 21: X tot := X ∪ ARG • ∪ ARG → 22: P(X tot ) := min 1≤ j≤m ( i≤l i=1 p j (X i )) where p j ∈ K (X i ) and X i ∈ X tot 23: P(X tot ) := max 1≤ j≤m ( i≤l i=1 p j (X i )) 24: P(E x ) := P(X tot ) 25: P(E x ) := P(X tot ) 26: end if 27: return (P(E x ), P(E x )) Example 3 (Cont. Example 2). After applying the semantics presented in Section 2.2, we obtain that E co = E pr = E gr = E st = {C, E, F, D, G}. Since this extension has more than one element, the Algorithm 1 has to be applied: is applied when all the variables are independent, which is case of the extension. In order to apply (2), Table 5 shows the conditional credal sets, the credal sets of the arguments, and the resultant products. Thus, we have that (P(E y ), P(E y ) = [0, 0] for y ∈ {co, pr, gr, st}.
Notice that E y does not contain neither A nor B, this means, that this extension does not give a solution for the problem of group decision making. Let us now calculate the upper and lowers bounds for two other sets that have either A or B in their elements. Let us begin with  Table 4.

Comparing the intervals
So far, we have calculated the lower and upper bounds; however, it is necessary to be able to compare the extensions because we need to obtain an answer for the group decision making problem. In this sense, the idea is to generate an ordered list of the extensions. We base on the approach of Pfeifer [22] for comparing the intervals. This approach is based on two criteria: the precision of the interval and the location of it. Thus, the higher the precision of the interval is and the closer to 1 the location of the interval is, the better positioned the extension is on the order list.
We propose to use the criteria of precision and location separately and also combined. Thus, we can generate three possible ordering lists. Next, we present how each criterion is calculated.  Table 6 shows the resultant values. We can notice that the extension with the best location and combined value is {A, D}. {C, E, D, F, G} has the best precision; however, the worst possible location. The second best precision is also for {A, D}. This indicates that after aggregating the opinions of the students the most likely disease is measles. Notice also that {C, E, F, D, G} has the worst location and the worst combined value. This happens because there is no support relation between its elements, this means that all of them are independent, which impacts on the calculation. The notion of independence also impacts on the evaluation of {A, D, E, F, G}, note that its position in the ordering lists is almost the worst. Although this set includes A, D and G, arguments E and F does not have any support relation with the rest of the arguments. Table 7 shows the extensions ordered considering the three criteria.

Properties of the approach
In this section, we study formal properties of the proposed approach. These properties show the argumentation semantics and the credal network semantics impact on each other. More- The first proposition states that when there is at least one caused argument whose supporting arguments do not belong to the same extension, then the upper and lower bounds are zero. In this case, depending on the applied argumentation semantics some extensions may contain only the caused arguments but no one of their parents, which does not allow the calculation of the conditional credal sets for the caused arguments.

Proposition 1 (Causeless) Given a credal BAF
Proof Since A ∈ C x ∩ ARG ← , then it is necessary to calculate the conditional credal set for A, that is, K (A | ARG ). Since B ∈ E x ∩ pa(A), this means that ARG = ∅, that is, there is no evidences for calculating the conditional credal set and from a rational point of view we have that ∀ p i ∈ K (A | ARG ), p i = 0 for 0 ≤ i ≤ m. We can distinguish two cases for calculating the lower and upper bounds: 1. If | E x |= 1, that is, A is the unique argument of the extension. In this case, (1) has to be applied: If | E x |> 1, it means that A is independent of the rest of the arguments that belong to E x . In this case, (2) has to be applied. Since this equations is based on the product and ∀ p i ∈ K (A | ARG ), p(i) = 0, then the resultant products are also zero. Thus, we have that the minimum and the maximum are also zero; hence, The following proposition states that an extension in which all their elements are connected by a causal relation has a better combined value than an extended version of it with independent arguments. This clearly impacts on the ordering of the extensions. This also impacts on the employment of these extensions. For example, if we want to give congruous explanations and we use these extensions. An explanation where all the elements are connected would be better than another one where we have an element out of context.

Proposition 2 (Connectness)
Given a credal BAF CBAF = ARG, R att , R sup , f K and two extensions E 1 x and E 2 x (for x ∈ {c f , ad, co, pr, gr, st}) Proof Since all the arguments in E 1 x are connected by a support relation, let K (E 1 x ) = { p 1 1 , ..., p 1 m } be the set of probability values for obtaining the lower and upper bounds of x has at least one more argument that has an independence relation with the arguments in E 1 x . Suppose that such argument is B and its credal set is K (B) = {p B 1 , ..., p B m }. In order to obtain the lower and upper bounds of E 2 x , we have to apply the (2); thus, we have to multiply p 1 1 × p B 1 , ..., p 1 m × p B m to obtain the set of probabilities from which we will get the lower and upper bounds of E 2 x . Since multiplying two values a and b such that a < 1 and b < 1, results in a value that is less than either a or b; then P(E 1 The following proposition concerns with the impact of the attack relations on the calculation of the conditional credal sets. The fact that there are conflicts between arguments determine that two conflicting arguments do not belong to the same extension. These conflicting arguments can be the parents of the same argument; however, only non-conflicting parent can be used for calculation the upper and lower bounds. The fact of using non-conflicting parents gives coherency to the calculation. Proof This proposition is easy to demonstrate since the calculation of conditional credal sets is made on argumentation extensions, which are conflict-free. This means that when B, C ∈ pa(A) and (B,

Proposition 3 (Coherency) Given a credal BAF
Following theorems guarantee the soundness and completeness of the results w.r.t. the Credal Networks Theory.

Theorem 4 (Soundness)
Let CBAF = ARG, R att , R sup , f K be a credal BAF and SEM x an argumentation semantics where x ∈ {c f , ad, co, pr, gr, st}. If E ∈ SEM x (CBAF) and E ul = UL_BOUNDS(E, CBAF) such that E ul = (L E , U E ) then L E = P(E) and U E = P(E).

Proof (Sketch)
For a E under a semantics x there are two possible situations: 1. There is only one credal set associated to E. This happens when there is only one caused argument -say A -in E, that is, | ARG ← |== 1 AND | ARG → |== 0 AND ARG • == ∅ (line 15). The rest of arguments in E cause A and there is no arguments without support relation with the rest of arguments. This means that we only have one conditional credal set, which is associated to argument A, this is calculated in line 6. In this form, the calculation of the upper and lower bounds is done using (1) as is stated in lines 17 and 18 of the algorithm UL_BOUNDS. Both calculations follow the credal network theory, so we can say that the algorithm is sound with the theory. 2. There is more than one credal set associated to E. This happens when there is more than one caused argument and/or more than one free argument and the causes of these arguments are part of the extension. When this conditions are satisfied, then the algorithm is executed from line 21 until line 25. In this case, all the caused and free arguments are put in X tot and (2) is applied over all their credal sets with the aim of calculating the upper and lower bounds of the extension, as can be seen in lines 22 and 23. As in the previous case, both calculations follow the credal network theory, so we can say that the algorithm is sound with the theory.
Proof (Sketch) For calculating the upper and lower bounds of a set of variables, we can use either (1) or (2). We use the former when there is only one credal set for making the calculation and the latter when there is more than one credal set.
• Let us assume that extension E has only one credal set and we apply (1) to that credal set. This is also reflected by the algorithm in lines 17 and 18, which means that it will return the same result. This makes the approach complete. • Now, let us assume that extension E has associated more than one credal set. In this case, (2) has to be applied to the set of credal sets. This is also reflected by the algorithm in lines 22 and 23. One again, the result of the algorithm is the same as applying the theory.

Discussion
In this section, we discuss the relation of this approach with related work, and the relation that exists between formal argumentation and credal networks. Besides, we examine our approach in the context of hybrid systems. BAF approach is a recently studied topic that emerged due to the necessity of expressing a positive type of relation between arguments, which is totally independent of the attack relation. In [3], the authors studied specialized forms of support, namely deductive support, necessary support, and evidential support. However, none of them express causality support, which is intended to capture the following intuition: if (A, B) ∈ R sup then the acceptance of A causes the acceptance of B or B is accepted as an effect of accepting A. Since causality is a topic widely studied in probabilistic, we decided to tackle this type of support by applying credal networks. Thus, we use credal networks for (i) modelling imprecise probability values and (ii) for modelling the causality support.
Both in AAF and BAF approaches, probability has been studied for modelling the degree of belief in arguments or/and attacks/supports relations. However, most of them use precise probability values. In AAF approaches, some works assign uncertainty to the arguments (e.g., [21,[23][24][25][26][27][28][29]), others to the attacks (e.g., [26]), and others to both arguments and attacks (e.g., [30]). Regarding BAF approaches, [11,12], and [13]) work with precise probability values. To the best of our knowledge, only in [14] and [15], the authors propose to use imprecise probabilities for modelling the degree of belief about the arguments in AAFs; however, they do not consider the support relation as part of the AF. This has a negative impact on the analysis of the relation between attacks and supports because, as we have shown, the relation between attacks and supports influences the calculation of the conditional credal sets. Besides, the authors did not tackle the causality relation in a dynamic way as they use fixed values for all the arguments, that is, the parents of the arguments were not taken into account for the calculation of the credal sets, more precisely for the calculation of conditional credal sets. This does not allow to work the real sense of causality. In our case, we use fixed credal sets for (i) those arguments that cause other ones but are not caused by any other argument and (i) those arguments that do not have any support relation with the rest of arguments; for the rest of the arguments, the credal sets are calculated (conditional credal sets). We even consider the attack relation in this calculation by avoiding to use credal sets of conflicting arguments.
Another approach that deals with uncertainty in AFs is the incomplete AF [31], which is an extension that includes the possibility of representing unquantified uncertainty about the existence of arguments and attacks. Thus, every argument and/or attack is labeled as certain or uncertain, meaning that it will definitely occur or may not occur, respectively. As mentioned in [32], the two main extensions for representing uncertain information in an AF are the incomplete AF and the probabilistic AF. Our work is part of the latter and from an imprecise point of view. Also in [32], the authors study a correspondence between iAFs and probabilistic argumentation; however, this study is focused on precise probability.
In this article, we use credal sets to handle imprecise probabilities; however, there are other models that can be used with this aim such as possibility and necessity measures [33], belief functions [34], interval probabilities [35], and so on. Nevertheless, credal sets are the base of credal networks, which handle causality, which is the type of support relation tackled in this article. Thus, we can model causality in a natural way by means of credal networks.
As we have mentioned above, credal networks provide the semantics to the causality relation. However, it is important to notice that argumentation concepts also have an impact on the behavior of credal networks. This happens due to the conflict notion given by argumentation. If we see the graph of Fig. 2 without the conflict relation, the calculation of conditional credal set for G would be K (G | A, B); nevertheless, due the conflict that exists between A and B, there are two possible conditional credal sets for G. We can say that in the same way that adding a support relations gives more expressiveness to AAFs, adding a conflict relation to a credal network also makes it more expressive.
It is important to mention that the expressiveness gained by AAFs or credal networks has indeed a raise in computational complexity for both of the approaches. In the case of AAFs, there is a computational cost in calculating the extensions, which varies depending on the employed semantics. For more details about the complexity of abstract argumentation, the reader is referred to [36] and [37]. In the case of credal networks, there is a computational cost in performing the inference, which depends on the employed algorithm. For more details about algorithms and the complexity of inferring with credal networks, the reader is referred to [38].
In [39], Geffner describes a hybrid system composed of what he calls System 1 and System 2. System 1 can be described as fast, opaque, and inflexible intuitive mind, that is, it can be seen as a learner whereas System 2 can be described as a slow, transparent, and general analytical mind, that is, it can be seen as a solver. Our approach can be seen as a System 2, which aggregates data from different sources to make a decision. We mentioned that such data can be given by agents that represent human beings; however, it also can be learners. Thus, the output of each learner represents a point of view and the information in credal sets can be obtained from a set of learners. Furthermore, both the causal and the attack relations can also be given by a learner. Thus, a BAF can be completely constructed from what we can call System 1, which in this case, would be constituted of more than one learner. System 2 would be constituted of the BAF and the semantics, which operates as a solver. The result of System 2 is a set of consistent arguments, called extension along with a lower and upper bounds. This interval can also be seen as a gradual evaluation or gradual semantics [40]. Although, traditional gradual semantics assign values to arguments, we could see our proposal as a gradual semantics for extensions. Figure 3 shows how our approach can be embedded into a hybrid system.
Other argumentation-based approaches have been proposed for complementing machine learning techniques. In [41], authors present some approaches that use argumentation to aid or improve machine learning. For example, the A-MAIL [42] is an argumentation approach where agents argue about a set of hypotheses learnt by induction. It integrates the capabilities of learning from experience, communication, hypothesis revision, and argumentation. We also have CleAr [43], in this approach, argumentation is used to argue about the validity of class labels proposed by a classifier. It is applied to problems within the computational linguistic setting: cross-domain sentiment polarity classification. Another example is A-ART [44], in this approach argumentation is used to resolve inconsistency that arise between classifications of clusters to which an instance is assigned as well as to explain the final classification dialectically. We have that more than one type of computational argumentation has been applied. In A-MAIL, structured arguments are constructed from the output of the learner and the evaluation of such arguments is done trough argumentation trees. In the case of CleAr, a Quantitative BAF is constructed from the output of the learner. In this case, the arguments are also structured with a base score denoted by a precise value. The evaluation is done by comparing the final strengths of the arguments. Finally, A-ART is based on n defeasible logic programming (DeLP  Fig. 3 Architecture of a hybrid system where our approach operates as System 2. Notice that each learner (from LEARNER_1 to LEARNER_m) feeds the credal sets of the arguments in the BAF. Besides, LEARNER output feeds the relations between arguments.

Conclusions and future work
This work studies BAFs and proposes an approach for modelling the degree of belief in arguments with imprecise probability values by means of credal sets. Besides, the support relation in the sense of causality is studied. Thus, we propose to use credal networks theory for modelling the causality relation between arguments. The imprecise probability values are used for calculating the upper and lower bounds of the extensions obtained by applying argumentation semantics. In turn, these intervals are used for ordering such extensions and return an answer to the group decision making problem. We can notice that the more connected by the support relation the arguments of an extension are, the better ranked such extension is.
As future work, we want to study the causality from other perspectives besides credal networks. For example, we would like to study the problem using a possibilistic approach and from a probabilistic view, imprecise Bayesian networks would be an interesting technique to be compared to Credal Networks. We also aim to study two or more different forms of support in the same BAF and how would be the behavior of the semantics. Finally, we want to generate explanations based on this approach.