Similarity notions in bipolar abstract argumentation

1.Introduction

Human commonsense reasoning involves in many occasions a process of analysis over a set of (potentially contradictory) alternatives and the evaluation of their support. The study of such cognitive process has led to the development of several formalisms that were introduced in an attempt to provide a formal model for this mechanism. In this direction, argumentation has become a topic of significant impact in the field of Artificial Intelligence (AI). The basic idea is to identify arguments in favor and against an affirmation and then select which arguments are acceptable among them, with the goal of resolving whether the reasoner can accept the assertion. Thus, Argumentation Theory provides reasoning mechanisms able to handle contradictory information concerning specific issues.

In the course of the research on the argumentation field, several argument-based formalisms have emerged to study the various possible relations among arguments. In [32], Dung proposes Abstract Argumentation Frameworks (AF) to model real-world situations by representing attack relations between abstract entities called arguments, providing different acceptability semantics for determining which sets of arguments are acceptable. Subsequently, Cayrol and Lagasquie-Schiex in [22] extended Dung’s framework [32] taking into account two independent types of interaction between arguments by adding the relation of support to the original relation of attack in abstract argumentation frameworks. The resulting Bipolar Argumentation Frameworks (BAFs) allow to model situations in which an argument can, for instance, reinforce another argument providing more reasons to believe in it; moreover, they adapt Dung’s acceptability semantics adding the consideration of the support relationship. Also, there exist some argumentative formalisms that represent the attributes associated to arguments providing more information to determine arguments acceptability [15,19,20,23], while others consider relations between the arguments to calculate the acceptability and safety of a given set of arguments [10].

Leaving aside the advances in argumentation frameworks [15,20,23], several representational aspects of the argumentation process still require further study. For example, a very natural tool for argument-based reasoning is the notion of similarity among arguments: during an argumentation process we sometimes tend to group arguments according to their shared characteristics or to the topics to which they refer. It can be argued that any comparison process requires the definition of a context in which such comparison can be meaningful [25,43,73]. The same applies to arguments: two arguments may be similar in a given context, but they may be entirely unrelated (or even incomparable) under different circumstances of analysis. Argumentative reasoning that gives importance to the similarities between pieces of knowledge represents a natural form of everyday human reasoning [76,77,79]. As computational argumentation aims to the definition of useful systems based on common sense, it seems reasonable, and desirable, to formalize the notion of similarity between arguments. As it happens with the similarity analysis in other settings where entities are compared provided they have some essential aspects in common, similarities between arguments also require the consideration of the context where they are issued identifying the aspects that are relevant in the comparison and the assessment of their importance in the similarity analysis. Notwithstanding the usefulness of exploring these problems concerning the argumentation process, similarity relations between arguments have not been deeply explored in the argumentation literature.

Notice that the similarity between a pair of arguments can be judged differently by each user, and consequently, the argumentation process where the relations between arguments are analyzed should reflect this important aspect. Although the perceived similarity among arguments is not the only tool that can be used to define the strength of the relations of support and attack it provides a natural alternative that can help to weight the relationships based on the (dis)similarity of the different situations in which the arguments are issued.

We will propose a Similarity-based Bipolar Argumentation Framework, which provides mechanisms for considering the context of the comparison between arguments, based on a set of descriptors that are common to the arguments which are being analyzed. Thus, we can represent and determine similarities between arguments introducing means to enrich the representation of the relationships between them and to be able to distinguish among arguments that are weakly related from those whose relationship is stronger. In this direction, we use an arguments’ similarity degree, computed from the descriptors that arguments have in common, combined with the weight those descriptors have in the process comparison. Thus, we determine a cohesion value between supporting arguments and a controversy value between conflicting arguments. Based on this analysis, we can refine the acceptability process provided by a BAF to obtain a new family of argumentation semantics. It is important to remark that the descriptors attached to arguments are additional knowledge representation devices added by this proposal. The basic assignment of the particular values to the descriptors is a task that can be performed in different ways. On the one hand, this can be performed by hand by a knowledge engineer; on the other hand, several tools have been introduced in the computational argumentation field by the research of argument mining techniques for which we will provide references below since the topic is outside the reach of this work.

The presentation will be structured as follows. We begin with an introduction to the BAFs is presented in Section 2. In Section 3 a brief, general description of the similarity concept will be addressed. The development of the tools for similarity comparison between arguments in the argumentation domains is presented in Section 4. Then, Section 5 contains the proposed argumentation formalism, the Similarity-based Bipolar Argumentation Framework, where an interpretation of supporting and attacking arguments is given based on the notion of similarity, also studying how this interpretation affects the acceptability process. In Section 6, we offer a case study by using an example in the food and nutrition domain. Finally, in Section 7 and 8, we present related works and offer conclusions, respectively.

2.Bipolar Argumentation Framework

When using arguments to reason, different types of relationships between them can be considered. One possible view is that arguments exhibit a “bipolar” behavior since reasons in favor of an affirmation can be considered as being positive while reasons against it can be viewed as negative. This approach was taken in the presentation of Bipolar Argumentation Frameworks (BAFs) that was proposed by Cayrol and Lagasquie-Schiex in [22], extending Dung’s notion of acceptability by distinguishing two independent forms of interaction between arguments: support and attack. We will assume that the reader is familiar with the formalism of Abstract Argumentation Frameworks [32].

To provide a graphical representation of BAFs, Cayrol and Lagasquie-Schiex also extended argumentation graphs presented by Dung in [32] by adding the representation of the support relation between arguments. This argumentation model introduced a starting point to analyze human reasoning adding the consideration of the bipolar aspect that naturally occurs in any debate to our modeling toolbox. Furthermore, they introduced the notions of supported and secondary attack that combine a sequence of supports with a direct attack considering the interaction between supporting and attacking arguments. These relations are defined as follows:

Cayrol and Lagasquie-Schiex argued in [22] that a set of arguments must keep in some sense a minimum of coherence to be able to model one side of any reasonable dispute adequately. They propose that the coherence of an acceptable set of arguments can be kept internally by requiring the set not to contain an argument that attacks another in the same set, and externally by requiring the set not to include both a supporter and an attacker of the same argument. Internal coherence can be obtained by extending the definition of conflict free set proposed in [32] and external coherence can be captured by the notion of a safe set.

The notion of conflict-freeness requires taking in consideration the direct, supported, and secondary attacks. Additionally, Cayrol and Lagasquie-Schiex show that the notion of safe set is powerful enough to encompass the concept of conflict-freeness, i.e., if a set is safe, it is also conflict-free. The closure under Rs was introduced in [22] is a requirement that only concerns the support relation.

Using these ideas, Cayrol and Lagasquie-Schiex in [22] extended the notions of defense for an argument with respect to a given set, where they take into account the relations of support and conflict.

The authors proposed three different definitions of admissibility which reflect three different levels of generality. The most general, called d-admissibility, is based on Dung’s admissibility capturing the internal coherence requirement through the conflict-free property. Then, the authors refine d-admissibility in two directions: one that captures the external coherence notion through the safe property, called s-admissibility; and the other, that considers the closure property over a conflict-free set, called c-admissibility. Formally:

Note that, a c-admissible extension is a set of arguments that is conflict-free and closed under support. Naturally, this extension is also a d-admissible extension since it is conflict-free and defends all its elements. Furthermore, the arguments that belong to the extension cannot attack an argument that they support, because this action violates the conflict-freeness property. Thus, this extension satisfies the external coherence property obtaining that a c-admissible extension is also a s-admissible extension. On the other hand, the s-admissible extension satisfies the conflict-freeness property, since a safe set of arguments is a conflict-free set that satisfies the external coherence. Roughly speaking, we can note that a d-admissible extension is refined to consider: external coherence in a s-admissible set, and closure under support condition in a c-admissible set.

Cayrol and Lagasquie-Schiex in [22] proposed different semantics for computing acceptability. These semantics consider the previous admissibility notions, redefining the classical ones proposed in [32] by Dung.

Example 2.

Our running example can be represented by a BAF, characterized by Θ=⟨Arg,Ra,Rs⟩, where:

Arg={A,B,C,D,E,F,G,H},Ra={(F,B),(G,F),(H,D),(D,C)},Rs={(A,B),(B,C),(C,G)}.

Fig. 1.

Representation of attack and support relations in BAF.

Figure 1 represents the attack and support relation between arguments, using solid arrows and dashed arrows, respectively. Observe that there is a supported attack from A to F, a secondary attack from F to G and a secondary attack from D to G. Besides, the set S1={A,B,D,E} is conflict-free but is not safe due to the fact that B supports C, that is attacked by D; also S1 is not closed under Rs; furthermore S1 is not d-admissible because there is no argument that defends D from H attacks.

On another hand, the set S2={A,B,C,G,H,E} is conflict-free, safe, and closed under Rs; moreover, S2 is d-admissible, s-admissible, and c-admissible, given the characteristics already analyzed and adding that H defends C from D attacks, and G defends B from F attacks. So, S2 is a stable, d-preferred, s-preferred, and c-preferred extension of Θ. Furthermore, the set S3={A,F,H,E} is conflict-free, but it is not safe and not closed under Rs since F attacks B which is supported by A. Thus, S3 is only d-admissible. So, S3 is a stable and d-preferred extension of Θ.

A bipolar argumentation framework obtains an acceptable set of arguments based on a specific analysis that considers the support and conflict relations between the arguments involved in a dispute. However, this formalism does not provide with tools to adequately analyze how cohesive or controversial are the arguments in a discussion. In the following sections, we extend the basic BAF to refine the argumentation analysis in this sense.

3.Preliminaries on similarity and similarity measures

The concept of similarity has been studied in many areas of Computer Science, in general sense, that is considering the similarity between two pieces of text. Thus, the textual semantic similarity is a measure of the equivalence of meaning between these two texts, which expresses the similarity or resemblance between them [1]. However, similarity can refer to syntactical or semantical aspects, and therefore the concept has been studied by various areas of AI such as natural language processing and argumentation. From the perspective of natural language processing, there exists, for example, several well-known techniques for disambiguation of words [2,28,57] or induction of word meaning [3,27,72]. From the field of argumentation, numerous advances have occurred in the area of argumentation mining such as the development of techniques linked to the processing of natural language [41,56,64]. However, the use of a measurement of similarity between arguments for reasoning that involves uncertain or potentially contradictory information is an area where, to the best of our knowledge, no previous work addresses the issue in depth.

Lin et al. in [54] present an independent definition of similarity, which compares the information included in the description of two objects A and B definitions, based on the information Theory, comparing the amount of information contained in the definition of the two objects. They argue that it is also essential to establish a measure of the difference between the objects to be compared because the more differences there are between the objects, the less similar they are. In that work, the authors propose that the similarity between two objects A and B be calculated from the relationship between the average of the information held in the description of the objects and the average of the information referring to their common aspects. Formally:

On the other hand, numerous studies have put the focus on finding a measure of similarity between objects, according to the domain of application to which these objects belong. One of the best-known measures is the cosine similarity [44] from the representation of the entities in the Bag-Of-Words (BOW) format [50]. The BOW format consists of finding a set of words that describe an entity, e.g., those that are present in a text, and each word has associated a weight representing the importance of the word in that text. This format allows to calculate the similarity between two entities using the measure of cosine similarity, and it is especially useful for information retrieval [44]. More clearly, in each text, it is possible to identify the words that will be part of the BOW and the weight assigned to each word may be the frequency of its appearance in the text; so, the terms that appear most frequently in the text are those that will have a higher weight.

In [74] the authors suggest a practical use of the BOW format to compare two entities using an ontology and to determine the semantic similarity between these entities. In this specific case, the process is carried out to find the semantic similarity between domains, using two sets of words obtained from the application of web mining techniques in search and recommendation systems. From the ontology, it is feasible to find concepts related to the entities under consideration by applying a spreading process.

In a similar direction, Rusu et al. [71] presents a summary of different possible metrics to determine the similarity between concepts in an ontology, highlighting the theoretical models proposed by cognitive psychology. Among them are mentioned: measures based on definitions like Lesk algorithm [53]; measures based on structures, for example distance Rada [69], the Leacock and Chodorow Similarity [51] and the Wu and Palmer Similarity [80]; and measures based on the information content, such as Resnik Measure [70] or Jiang Distance [48]. These measurements have their origin in a geometric model [26], or in the coincidence of features or characteristics models [75]. The first represents the concepts and relationships between them, stored in a computer memory that simulates human memory, using three hierarchical levels of storage: the concept, its category, and its properties. The second one mentions the comparison between objects of any nature to find similarities and dissimilarities between them, beyond the distance between points, by comparing characteristic features [75]. The proposal of Rusu et al. in [71] is also based on the distance between concepts defined on an ontology, but concepts and relationships are weighted; then, the similarity between the concepts according to the weight of the shortest path between them. Finally, Amgoud et al. in [11] explore several similarity measures between logical arguments and define a very general function denoted as a similarity measure. Then, they define a set of basic principles that a similarity measure should satisfy, such as syntax independence, maximality, symmetry, substitution, monotony, and dominance. While in [12], the authors work with logical entities representing the structures of the arguments and propose a mechanism to calculate similarity measures using concise refinements of arguments based on the arguments that only contain useful information in their premises to infer the conclusion. Furthermore, in [8], the authors use the similarity between arguments as the means to analyze an individual argument and its attackers and the concept to introduce a new semantics. Also, they present properties that this semantics should satisfy. This brief description summarizes some of the results that can be found in the literature on the subject, and which are the basis for the approach taken in our proposal.

4.Introducing similarity measures for arguments

A crucial component in the determination of the similarity between arguments is the definition of the conditions under which such comparison is performed [77]. In [54], D. Lin states that the main problem with existent similarity measures is that they assume a particular domain model, i.e., the conditions over which the similarity is calculated are considered preexistent. Although the similarity is related to the properties shared between the two entities being compared, the comparison of two arguments largely depends on an agent’s perception which can be influenced by the mental status of the agent, e.g., her beliefs, goals, or by external, possible unknown external variables. All these factors are part of a context that affects the assessment of the similarities between two arguments. The intuition that we have just presented is essential to define a similarity measure between arguments.

In this section, we present a method that allows us to determine the similarity between arguments, according to a given context. In general, this method consists of the following three stages: (i) the specification of argument’s descriptors; (ii) the setting of the context based on argument’s descriptors; and (iii) the computation of the similarity degree among the arguments being compared. In what follows, we present and develop these three stages.

4.3.Computing similarity degree between arguments

Some recent works have studied the problem of similarity between arguments. In [62], the authors propose to find similar arguments, or argument facets, by detecting propositions paraphrased or labels that refer to a particular aspect of an argument, limiting to political and social dialogues. Rather than providing a mechanism to decide whether two arguments are similar or not, we propose to evaluate the values of argument’s descriptors in a given context C to specify a degree of similarity between the arguments. For this, we examine how well the different descriptors in context C match: the higher the number of matching values per descriptor is, the higher the similarity degree will be, weighted by the relevance of each descriptor provided in the context definition. We establish that two arguments are incomparable given a context C when no descriptor associated with them belong to C, note that a pair of arguments can be incomparable in a given C, but could be compared in a different one.

In the following, given an enriched argument A=⟨A,δA⟩, we denote with DA the set of descriptors that appear in δA, i.e., DA={d∣(d,VA)∈δA} and with VdA the set of values associated to descriptor d∈DA.

Definition 12

Definition 12(Similarity coefficient for a descriptor).

Let 𝔸rg be a set of enriched arguments, A=⟨A,δA⟩ and B=⟨B,δB⟩ two enriched arguments in 𝔸rg, and C a context. We define the similarity coefficient for each descriptor d∈DA∩DB∩DC, denoted Coefd(A,B), as follows:

Coefd(A,B)=∣VdA∩VdB∣∣VdA∩VdB‾∣·wdif |VdA∩VdB‾|≠0,wdotherwise.

Definition 13

Definition 13(Similarity degree between arguments).

Let 𝔸rg be a set of enriched arguments, A=⟨A,δA⟩ and B=⟨B,δB⟩ two enriched arguments in 𝔸rg, and C be a context. We define the similarity degree between arguments in a context C, denoted SimC, as SimC:𝔸rg×𝔸rg→[0,1], such that:

SimC(A,B)=αnif DA∩DB∩DC={d1,…,dn},0otherwise,

where α1=Coefd1(A,B) and αi=⊙(αi−1,Coefdi(A,B)) with 2⩽i⩽n, and, ⊙ should be either a t-norm satisfying the following properties: commutative, associative, monotony increasing, and having 1 as the neutral element; or ⊙ should be a c-norm, satisfying the following properties: commutative, associative, monotony increasing, and having 0 as the neutral element.

Remark

Remark(Similarity coefficient for a descriptor).

The coefficient function operates considering the number of descriptors specified in the context. For this reason, the lower bound of the function is zero, while the upper bound is not possible to determine a priori but is finite and positive.

Remark

Remark(Similarity degree between arguments).

The similarity function is operated under a t-norm or c-norm, satisfying the commutative and associative properties. For this reason, the order in which the descriptors in the set {d1,…,dn} are operated does not affect the final similarity measure; therefore, we can assume an arbitrary order to calculate this measure.

Note that SimC is parameterized by the context C and the operator ⊙. The rationale behind such a general definition for SimC is to provide the possibility of customizing the operator by taking into consideration the application’s requirements, according to the goal of each particular argument comparison. For instance, it may be required to apply SimC under several different contexts, or with different assumptions of how the combination of values for each descriptor in the context should be done, and these goals can be achieved independently from each other. In the following, sometimes we will not explicitly include one or both parameters whenever they are clear from the text. Furthermore, the proposed similarity function satisfies the following properties, as a consequence of being based on a t-norm or a c-norm function.

Let C be a context, then for all A=⟨A,δA⟩,B=⟨B,δB⟩∈𝔸rg, it holds that:

• – Positivity: SimC(A,B)⩾0.

• – Symmetry: SimC(A,B)=SimC(B,A).

• – Maximality: SimC(A,A)⩾SimC(A,B).

Example 5.

In our running example, we focus on the value that each of the descriptors takes for every argument, according to the following context:

C={(weather_conditions,0.5),(general_activity,0.3),(frame_of_mind,0.2)}.

In Fig. 2, we summarize the similarity degrees corresponding to the set of enriched arguments shown in Example 3.

Fig. 2.

Similarity degrees for the set of enriched arguments from Example 3.

In this case, based on the argument descriptors established in the Example 3, an optimistic instantiation of the similarity function, i.e., a maximum t-conorm is used to define SimC. Figure 2 shows how the similarity degree between pairs of arguments is computed; For instance, in the sixth row, to calculate SimC(D,C) we have:

• Coefweather_conditions(D,C)=1×0.5=0.5, due to the set of values associated to this descriptor is unique (rainy_day) for both arguments.

• Coefgeneral_activity(D,C)=03×0.3=0, since the set of values associated to this descriptor are different for the involved arguments (for the argument D the value is keeping_house while for the argument C are go_out and shopping).

• Coefframe_of_mind(D,C)=03×0.2=0, because the set of values associated to this descriptor are different for the involved arguments (for the argument D the value is depresing while for the argument C are happy and good_mood).

Finally, we instantiate the similarity function with the maximum t-conorm as follows:

SimC(A,B)=αnif DA∩DB∩DC={d1,…,dn} with n⩾1,0otherwise,

where α1=Coefd1(A,B) and αi=max(αi−1,Coefdi(A,B)). Then, by successively computing the max between the three values obtained, we have SimC(D,C)=0.5.

In the next section, we analyze how this similarity function can be applied to characterize the conflict and support relations in a BAF. In particular, we instantiate the similarity function to present an optimistic (or pessimistic) posture in the interpretation process that we perform over a BAF.

From the elements presented in this section, we address in the following section a mechanism to interpret the support and attack relationships in a BAF. Our leading purpose is to determine the cohesion of a set of supporting arguments and the controversy associated with a set of attacking arguments.

5.A similarity valued argumentation framework

We are now ready to extend the notions of support and attack from abstract argumentation frameworks to enriched arguments. Thus, two enriched arguments ⟨A,δA⟩ and ⟨B,δB⟩ are related through a support (attack) relation if and only if the underlying abstract arguments A and B are related through a support (attack) relation. Formally, this is as follows:

Note that, these notions provide an extension associated with an underlying BAF Θ. Consequently, we use the bipolar argumentation graph associated to Θ to represent the enriched arguments and the relations between them.

In [21], the authors introduce a formalism to compute the similarity between two arguments, as long as they possess descriptors in common. The idea behind this proposal is that to determine the similarity degree between entities it is necessary that these entities have common and comparable aspects in the context where the comparison is made. However, it is possible to use the similarity between arguments to analyze the relations between them more effectively, by specifying a controversy evaluation of an attack relation, and a cohesiveness assessment of a support relation, respectively providing a measure of the strength of conflicts and supports. That is to say that the controversy or cohesion values associated with two arguments which are in an attack or support relationship should be high when these arguments both refer to the same aspect of the debate, in a specific sense. Following these intuitions, we examine these issues in the context of a BAF, introducing similarities between arguments as a device to enrich the argumentation analysis needed to distinguish those entities weakly related from those whose relationship is stronger.

As with the similarity function SimC, depending on the user modeling intentions, the operators ⊕ and ⊗ can be instantiated in different ways. These operators must satisfy commutativity, monotonicity, and associativity; for example, if we have a pessimistic posture to analyze the cohesiveness associated to a BAF, then ⊕ should be instantiated through a t-norm modeling the weak-link principle, while a c-norm implements an optimistic posture modeling a strong-composition principle. Of course, the inverse analysis can be performed over the operator ⊗ to compute the controversy value. Note that the use of t-norms is widely recognized in different application domains; for example, Dubois and Prade [30] use t-norms to model the uncertainty associated with subjective evidence in the analysis of legal cases. In another vein, Lukasiewicz and Straccia [58] analyze the efficiency of t-norms to model the uncertainty and the precision of the information in the Semantic Web domain. On the other hand, the c-norm operators are used in different application domains; for example, Grabisch et al. [30] present different ways to perform aggregation of arguments based on user preference through t-conorms. Another example is the work of Krause et al. [58], which introduces a series of criteria to perform aggregation of arguments supporting a particular conclusion in decision-making support systems, taking into account the uncertainty level associated with these arguments. The authors highlight the use of t-conorms as a sensible way to obtain the uncertainty level of a conclusion supported by multiple arguments. Furthermore, in a general direction, Budán et al. in [17,18,65] shows the application of these operations (c-norm and t-norm) within argumentation as a sensitive tool to diverse domains, such as decision support systems, recommendation systems, and legal systems.

In a given context, cohesiveness and controversy values are two distinct outcomes arising from the similarity function. They can be interpreted independently or jointly, depending on the domain in which they are applied. In this paper, cohesiveness represents the degree of support that an argument receives from a set of supporting arguments, while controversy expresses how much a particular set of arguments resists an argument according to pre-established comparison conditions. Thus, in a bipolar analysis, these notions are used, for example, to examine supported attacks where the support and conflict relations play a role together (see Fig. 5). Furthermore, when internal and external coherency is analyzed, we will use both concepts to identify the cohesion and controversy level associated with the argument set (see Fig. 7). Besides, the information provided by the enriched arguments is fundamental in the examination of the argumentative process.

We now define a S-BAF, which extends a BAF [10], incorporating the tools to interpret the relationship between the arguments in the argumentation process under a different light. In particular, this framework provides the means to compute the similarity between related arguments to decide which attacks (supports) must be considered in the discussion, given their degree of controversy (cohesion, respectively).

The definition of a S-BAF is quite general so that we can use, in theory, different similarity functions for the comparisons among arguments. This proposal is based on the use of Sim as proposed in Definition 13 instantiated as necessary for specific contexts in which the function is applied and the ⊙ operator. These parameters are left to be set according to the needs of the application user. In the rest of this paper, we assume that a single (but arbitrary) context is fixed beforehand for all comparisons performed in the analysis of the S-BAF, i.e., users select the set of descriptors by which they want to analyze all relationships between arguments. From now on, given a context C, the definition of a S-BAF Φ will be expanded to ⟨Θ‾,SimC,CohCΘ‾,ContCΘ‾⟩, unpacking Θ‾ when necessary to its components ⟨𝔸rg,Rs,Ra⟩.

It is important to note that, in the argumentation framework, once we determine (or fix) the similarity function that is to be used to compute the similarity between related arguments, we can affirm that the similarity values associated to these relations do not change even if the discussion is extended. That is, if the number of arguments that participate in the discussion increases, the computed similarity values calculated for the original set of arguments do not change giving a fixed similarity function. We formalize this idea in the next proposition.

The result in Proposition 1 states that we can analyze an argumentation framework once, and if in the future we want to consider an expansion of it there will not be necessary to recompute all similarity values between the arguments in the previously existent relationships, i.e., as long as the similarity function does not change, it will suffice to compute the new ones. This result potentially gives us a way to work in the argumentation process dynamically and incrementally when necessary.

Under this new framework, we analyze how the attacks between the arguments must be considered in a Enriched BAF taking into account the similarity function defined in the previous section. Next, we discuss a possible way to classify attacks based on similarity, controversy, and cohesion. Informally at first, we use notions such as “high” and “weak” to refer to these degrees but we formally establish their proper meaning later on this section.

Once the attacks are appropriately evaluated, we analyze the coherence of an enriched argument set. As in [10], the internal coherence is captured by the definition of conflict free set, and external coherence is captured with the notion of safe set. In our case, the classical conflict-free notion can be weakened by the existence of weak attacks, which allows a certain degree of tolerance to inconsistent information (see Fig. 6). So, we can specify a strong-conflict-free set (a set with no conflict), a τ-conflict-free set (only admitting weak-conflicts whereas the controversy value associated with this set does not exceed the limit τ), and a weak-conflict-free set (the more general set, since it is possible to admit any weak attacks in the domain). Furthermore, the classical notion of safety can also be weakened considering weak attacks and supports to an external argument. Note that, in Fig. 7, S2′ and S3′ are composed by the arguments used to perform a support for an external argument, while S2″ and S3″ have just the arguments involved in an attack (supported, secondary, or direct attack) over the external argument. Thus, we can specify a strong-safe set (there not exist arguments in the set attacking and supporting an external argument), τ-safe set (there are no arguments in the set performing both a strong attacking and a strong supporting, i.e., the cohesion value associated to the support chain does not exceed the threshold τ, whereas the controversy value associated to the set does not exceed the threshold τ, and a weak-safe set, i.e., there are no arguments in the set performing both a strong-attack and strong-support over an external argument. Formally:

Fig. 6.

Interpretation of conflict-free set (conflict-free strength).

Fig. 7.

Interpretation of safe set (safe strength).

Next, in Proposition 2, we show the relationships between the strong conflict-freeness and safety versions and the classical ones, while Proposition 3, shows the connection between the safety, conflict-freeness, and closure (under Rs) properties.

Note that, conflict-freeness and safety sets are expanded from a classical one to be more “flexible” sets. In this sense, we can introduce weak attackers into the argument sets; consequently, these new arguments can be strong attackers of the previous ones (see Fig. 8). Thus, there is no strict inclusion relationship between the different types of conflict-free sets. The intuition is that, if we raise the tolerance to attack up to a threshold τ big enough, strong attackers may appear and eliminate arguments that are weakly defended.

Fig. 8.

Reinstatement of arguments with strong attacks in τ and weak-conflict-free sets.

Next, we present an example that will make clear the intuitions of this approach.

Example 6.

Continuing with the Example 3 which is represented in Fig. 9, where the similarity between the involved arguments are computed in the Example 5 considering the context C={(weather_conditions,0.5),(general_activity,0.3),(frame_of_mind,0.2)}. We now introduce for this special case a possible instantiation of the Cohesion and Controversial operators through the Einstein sum c-norm and the Probabilistic sum c-norm respectively, to analyze the relations between the presented arguments.

CohCΘ‾(S)=βnif Rs={(A1,B1),…,(An,Bn)},0otherwise,

where β1=SimC(A1,B1) and βi=βi−1+SimC(Ai,Bi)1−βi−1×SimC(Ai,Bi) with n⩾2.

ContCΘ‾(S)=γnif Ra={(A1,B1),…,(An,Bn)},0otherwise,

where γ1=SimC(A1,B1) and γi=γi−1+SimC(Ai,Bi)−γi−1×SimC(Ai,Bi) with n⩾2.

To facilitate the practical interpretation of the enunciated concepts, Fig. 9 will be useful:

Fig. 9.

Representation of attack and support relations in S-BAF.

Thus, for this example and considering a setting where a threshold τ=0.4, we identify the following relations:

• H is a weak direct attacker for D, since ContCΘ‾({(H,D)})=0.3;

• G is a weak direct attacker for F, since ContCΘ‾({(G,F)})=0.2;

• D is a strong direct attacker for C, since ContCΘ‾({(D,C)})=0.5;

• C is a weak supported attacker for F, since CohCΘ‾({(C,G)})=0.3;

• F is a strong direct attacker for B, since ContCΘ‾({(F,B)})=0.6;

• B is a weak supported attacker for F, since CohCΘ‾({(B,C),(C,G)})=0.53, however ContCΘ‾({(G,F)})=0.2;

• A is a weak supported attacker for F, since CohCΘ‾({(A,B),(B,C)(C,G)})=0.82, but ContCΘ‾({(G,F)})=0.2;

• D is a weak supported attacker for G, since ContCΘ‾({(D,C)})=0.5, however CohCΘ‾({(C,G)})=0.3;

• F is a weak secondary attacker for C, since CohCΘ‾({(B,C)})=0.2;

• F is a strong secondary attacker for G, since CohCΘ‾({(B,C),(C,G)})=0.53, and ContCΘ‾({(F,B)})=0.6.

Next, we examine the different conflict-free sets that are possible to obtain analyzing the presented framework: The set S1={A,B,C,E,G,H} is strong-conflict-free since the controversy value associated to the set is zero. That is, there are not conflicting arguments in S1. On another hand, the set S2={A,B,D,E,H,G} is τ-conflict-free since the controversy associated to this set is ContCΘ‾(S2)=0.3<τ (there exists a weak direct attack from H to D with ContCΘ‾({(H,D)})=0.3), while the set S3={A,D,E,F,G,H} is weak-conflict-free with a controversy value associated to S3 equal to 0.44 (H is a weak direct attacker for D and G is a weak direct attacker for F with CohCΘ‾({(G,F)})=0).

On the other hand, analyzing the external coherence of the sets, we have that: S1 is a strong-safe set, while S3 is a weak-safe set since there exist arguments that belong to S3 supporting and attacking an external argument (there exists a strong direct attack from F to B with the controversy value is equal to 0.6 and a weak support from A to B with a cohesion value equal to 0.2). Also, S2 is a weak-safe set since there exist arguments that belong to S2 supporting and attacking an external argument (there exists a strong direct attack from D to C with the controversy value equal to 0.5 and a strong support from A to C through B with a cohesion value equal to 0.41).

Next, we extend the notions of defense for an argument with respect to a set of arguments, where we consider the attack relations introduced in Definition 17 instead.

We present different definitions for admissibility, from the most general and strong to the most specific and weak. The most general is based on the classical notion of admissibility where only the attack relations are considered (both the strong and the weak ones). Then, we extended this notion taking into account external coherence considering the different attack and support degrees among arguments. Finally, external coherence is strengthened by requiring the closure of the Rs relation.

In this manner, admissibility becomes a characteristic of a set of arguments that can be evaluated from different perspectives. The most restrictive admissible sets are those that do not admit conflicts and that defend all their elements with values of controversy greater than the given threshold. A more flexible admissibility property is one in which a certain level of controversy associated with the set, limited by a threshold, is acceptable. In this case, the defense related to the arguments can oscillate between strong and weak. Finally, the most flexible set is one that allows the incorporation of conflicts where the controversy associated with them is strictly less than the threshold; i.e., the controversy is analyzed in each conflict between arguments. Note that, in the two last cases, the defense related to the arguments can oscillate between strong and weak.

Next, in Proposition 4, we show the relationships between the strong versions of admissibility and the classical ones. Furthermore, in Proposition 5, we identify how the conflict-freeness, safety, and closure property are related in the admissibility notion.

From the notions of coherence (internal and external) and admissibility, it is possible to introduce different acceptability semantics. First, we define strong-stable, τ-stable, and weak-stable extensions. The strong-stable extension asks for a strong-conflict-free set and the existence of a strong attack for every element that does not belong to the extension. Meanwhile a weak-stable extension would a be maximal weak-conflict-free set, and the attack produced from the extension to each element that does not belong to it must be strong (direct, supported, or secondary attack) or, alternatively, be a strong-conflict-free set for which there must exist at least one element that does not belong to the set and it is only weak attacked (direct, supported, or secondary attack) by the set. In other words, a strong-stable extension guarantees inconsistency with all external arguments. A τ-stable extension is reasonably destabilized since it allows some degree of inconsistency as long as it does not exceed the determined threshold.

In general, an extension will be stable when it can ensure its internal and external coherence and the defense of its elements; this may have different implications as we explain in what follows. First, it is clear that the strong-stable extension is equivalent to a BAF’s stable extension. Since the controversy of the set cannot exceed the τ-value, in the τ-stable extension, it may be null (when the value of τ is very close to zero, or it is zero). If this is the case, there must be at least a weak attack over each external element. Being the conditions more flexible, it is possible that there exist no internal attacks in the weak-stable extension, in which case the set must weakly attack each external element. However, if the extension has internal attacks, where these individual attacks do not exceed the τ-value, it is sufficient that the set strongly attacks each external element. Next, in Proposition 6, we show the relationships between the strong versions of stable extension and the classical one. Furthermore, in Proposition 8 we establish the case in which a stable extension combined with a safety property results in a set that is closed under support, and the circumstances in which a stable extension combined with a closed under support condition result in a safety set.