Support Argumentation Frameworks (SAFs) are a type of the Abstract Argumentation Framework, where the interactions between arguments have a positive nature. A quantitative way of evaluating the arguments in a SAF is by applying a gradual semantics, which assigns a numerical value to each argument with the aim of ranking or evaluate them. In the literature, studied gradual semantics determine precise probability values; however, in many applications there is the necessity of imprecise evaluations which consider a range of values for assessing an argument. Thus, the first contribution of this article is an imprecise gradual semantics (IGS) based on credal networks theory. The second contribution is a set of properties for evaluating IGSs, which extend some properties proposed for precise gradual semantics. Besides, we suggest a classification of semantics considering the set of properties and evaluate our proposed IGS according to the extended properties. Finally, the practical application of the results is discussed by using an example from Network Science, i.e, PageRank. We also discuss how gradual semantics benefit PageRank research by allowing to generate contrastive explanations about the scores in a more natural way.
engine. In PageRank, the web can be seen as a directed are used for comparing the intervals. These are the locagraph, where nodes represent pages and edges repre- tion, the precision of the interval, and the combination sent the links between pages and the method measures of both. Thus, our approach allows a three dimensional each node’s influence. PageRank counts the number and comparison, which generates a ranking that can be inquality of links to a page to determine a rough estimate terpreted depending on the application and also can be of how important that page is. The directed graph that used for explaining the reasons an argument is stronger represents the web can be seen as a SAF and a gradual than another. In this sense, we propose the generation of semantics as a method for calculating the PageRank. Be- explanations for contrastive questions. Bouwel and Wesides, the influence between nodes can be seen as a sort ber [17] distinguish three types of contrastive questions: of causal relation where the positive quality of a page (i) P-contrast: why does object have property , rather causes the positive quality of a page . than property ′?, (ii) O-contrast: why does object have
The PR is a precise value fitted between 0 and 10 that property , while object ′ has property ′?, and (iii)
is calculated based on more than one criterion such as T-contrast: why does object have property at time ,
the number and the quality of links to a page, the update but property ′ at time ′?. In this work, we model the
frequency or the internal coherence of the page, or even ifrst two types and consider that arguments can be seen
on design issues. In [
To the best of our knowledge, there is no a gradual and SAFs.
semantics evaluation method that returns imprecise
values for ranking arguments in SAFs. Hence, we have our 2.1. Credal Networks
ifrst research questions: (i) How to model a SAF in the Before presenting credal networks, let us define credal
settings of imprecise probability?, (ii) how to calculate the sets (from Levi’s credal sets [
In addressing the former question, we use credal sets sets of the variables of X. In this work, we assume that
[
A credal network is a graphical model that associates the page rank, the probabilities in a credal set represent
nodes and variables with sets of probability measures the values of the criteria used to calculate the PageRank.
[19]. A credal network consists of a directed acyclic
graph, where each node in the graph is associated with a 3.1. Credal SAF
random variable and the parents (i.e., the variables
corresponding to the immediate predecessors of accord- Before presenting the concept of credal SAF, we present
ing to the graph) of are denoted by (). Each vari- the imprecise strength definition. Like in precise SAFs,
able is associated with a (conditional) credal set ( | in the imprecise context, an imprecise gradual semantics
()) = {1( | ()), ..., ( | ())}. In- is in charge of calculating the strength of each argument
ference is performed by applying Bayes rule to each mea- in the SAF from their support relations. Thus, for any
sure in a joint credal set. The goal is to combine these ∈ ARG, the imprecise strength of is given by the
credal sets into a set of joint distributions. Next, let us function (), where : ARG → [
Given a random variable and its credal set (), Let us recall that the support relation in our approach the lower and upper bounds for variable are deter- can be interpreted as a causality relation that exists bemined as follows: tween arguments. Thus, an argument in a causality rela () = {() | () ∈ ()} (1) tion can play two diferent roles, it can either be caused () = {() | () ∈ ()} or be the cause, this means that we can have caused arguments (this set is denoted by ARG← ), arguments that 2.2. Support Argumentation Framework cause other ones (this set is denoted by ARG→), and arguments that have no causality relation with the rest (SAF) (this set is denoted by ARG∘ ). We characterize these sets as follows. Given a set of arguments ARG and a support relation R+:
score expressed by a numerical value which is generally
in the interval [
S = ⟨ARG, R+, ⟩, where ARG is a non empty finite set of arguments, is a base score function on ARG and We can now define a credal SAF, where arguments are R+ ⊆ ARG × ARG is a support relation. For , ∈ ARG, assigned with credal sets, from which the imprecise base the notation (, ) ∈ R+ means that supports . score of each argument can be obtained.
signs to each argument in a SAF a value between 0 and
1. Thus, for all ∈ ARG, () denotes the image of
argument and it is called the strength degree of .
Definition 2. (Gradual semantics) Let
S = ⟨ARG, R+, ⟩ be a SAF. A gradual semantics
is a function () : ARG →− [
tics based on credal networks theory. We present a credal SAF, in which, we use credal sets to model the degrees of belief about arguments. This means that each argument in a SAF has associated a credal set, which contains a set
credal sets is a tuple S = ⟨ARG, R+, , ⟩ where (i)
ARG = ARG← ∪ ARG→ ∪ ARG∘ ; (ii) R+ ⊆ ARG × ARG is the
support relation between arguments; (iii) : ARG → K
is a function that attributes a credal set to each argument,
where K is the set of all possible credal sets; and (iv)
: ARG → [
Example 1. Let S be the credal SAF for the scenario of
= ⟨ARG, R+, , ⟩ , where ARG = the PageRank: S {, , , , , }, R+ = {(, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, )}, and both and are shown in Figure 1 next to each argument. is represented by a vector and by an interval. Figure 1 shows the graph of the credal SAF S. The nodes represent the pages and the edges the support relation. The probability values of the credal sets correspond to (i) the quantity and quality of the supporting links, (ii) the update frequency, (iii) the internal coherence of the page, and (iv) the design issues. The imprecise base score are obtained from these credal sets by applying Equation (1).
0.62 [0.52,0.71] 00..5528
0.71 0.0 [0.0,0.7] 000...575 PaAge 0.65 0.7 [0.65,0.75] 0.75 0.75
Page B Page E
Page C Page F
Let us now 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 an initial credal set and one conditional credal set that has to be calculated based on the supporters. Finally, some arguments in ARG→ have associated only one credal set and the others have also a conditional credal set. This happens because some causing arguments are also caused ones. Formally: 1. ∀ ∈ ARG∘ , there is a credal set (), that is, a
non conditional credal set; 2. ∀ ∈ ARG← , there is a credal set () and a
conditional credal set ( | ()); 3. ∀ ∈ ARG→ − (ARG→ ∩ ARG← ), there is a credal
set ().
Example 2. (Cont. Example 1) In the credal SAF S, we have that ARG∘ = ∅, ARG← = {, , , , }, and ARG→ = {, , , , }. Every argument has a credal set () (for ∈ {, , , , , }) that can be used to obtain its imprecise base score. Those arguments that besides have a conditional credal set are , , , , and . Thus, ( | , ) is the conditional credal set of , ( | , ) is for the conditional credal set of , ( | ) is the conditional credal set of , ( | , , ) is the conditional credal set of , and ( | ) is the conditional credal set of . 3.2. Calculating the Imprecise Strength This section shows how the imprecise strength of arguments is calculated. Let us recall that such strength is represented by an interval. In the same manner as the imprecise base score, the interval is the result of assessing the credal set associated to each argument, which is calculated considering the support relations. In the case of causality, these relations are expressed as conditional ones between arguments.
Definition 4. (Calculation of the imprecise strength) Let S = ⟨ARG, R+, , ⟩ be a credal SAF and ∈ ARG an argument, the imprecise strength of , that is () = [ (), ()], is obtained as follows: - If () = ∅, then () and () are obtained by applying Equation (1) to () - If () ̸= ∅, then
1. Obtain the conditional credal set for (that is, ( | ())) by applying the Bayes rule in the following way:
() × (() | ) ( | ()) =
(()) 2. Calculate () and () by applying Equation (1) to the resultant conditional credal set ( | ()).
Once we have the interval that represents the imprecise strength, we need a way to compare such intervals. For evaluating the ordering of the intervals we will base on the approach of [21], which considers the precision of the intervals (denoted by PREC), the location of the intervals (denoted by LOCA), or the combination of both (denoted by COMB). Thus, given an argument whose associated interval is = [ (), ()], the evaluating criteria are calculated as follows: PREC() = ()+ () , and
2 1 − ( () − ()), LOCA() = COMB() = PREC() × LOCA().
When we compare two intervals, we can use their precision, their location, or the combination of both. Even though these criteria are represented by a precise value; actually, an argument is stronger if its precision is high or its location is close to 1. This allows to compare arguments in more than one way and gives flexibility to the approach, which is important depending on the context or domain of the application. For example, in the PageRank scenario, let us suppose that (′) = [0.3, 0.35] and (′) = [0.6, 1.0]. Thus, we have PREC( (′)) = 0.95, PREC( (′)) = 0.6, LOCA( (′)) = 0.325, LOCA( (′)) = 0.8, COMB( (′)) = 0.31, and COMB( (′)) = 0.48. We can notice that ′ has better precision than ′ whereas ′ has better location than ′. If we only consider the precision criteria, we can say that ′ has a better PR score than ′ and if we consider location or the combined measure, we can say that ′ has a better PR score than ′. The question is: which is a better criteria to be used in this context? even with a better location, the range of values of ′ makes dificult to classify it in a given PR. This means that a good location with low imprecision shows a high uncertainty degree, which in this context is not desirable. Thus, although Table 1 the location of ′ is not good, its high precision helps to Values for calculating the imprecise strength. The names of better determine its PR. the probabilities are shown in the top row. The rest of rows
Now, imagine another scenario where the impre- show the values of the corresponding credal sets. cise strength of an argument ′ is the interval [0.4, 0.8] and the imprecise strength of another argu- P(A, C | B) P(A, C) P(B, D | C) P(B, D) P(F | D) P(A, C, F | E)P(A, C, F) P(C | F) ment ′ is [0.5, 0.7]. We have that PREC( (′)) = 0.6, PREC( (′)) = 0.8, LOCA( (′)) = 0.5, 0.3 0.45 0.5 0.74 0.55 0.55 0.57 0.5 LOCA( (′)) = 0.5, COMB( (′)) = 0.3, and 0.48 0.45 0.55 0.66 0.5 0.5 0.55 0.65 COMB( (′)) = 0.4. If only location is considered, 0.45 0.55 0.65 0.59 0.62 0.6 0.62 0.6 we could say that both have the same strength; however, 0.6 0.52 0.7 0.79 0.4 0.7 0.65 0.62 we can use precision for breaking the tie and determine which is stronger. The point is that although the criteria for comparing intervals (other criteria can also be considered) are represented by precise values, this does not diminish the quality of the information when expressed with intervals.
Example 3. (Cont. Example 2). Let us recall that in the credal SAF S, we have that ARG∘ = ∅, which means PREC 0.3 0.42 0.54 0.73 0.84 0.76 that there is no argument that has no support or does not LOCA 0.35 0.52 0.43 0.81 0.72 0.62 support another argument. Also, notice that only argument COMB 0.105 0.22 0.23 0.59 0.61 0.47 does not have any parent, that is, it is not supported by any other argument. Regarding the rest of arguments, all of them have at least one parent, which means that 4. Generating Contrastive the conditional credal sets for them have to be calculated.
Table 1 presents the values necessary for calculating the Explanations imprecise strength of the arguments.
After the calculations, we have the following imprecise In this section, we present how to generate explanations strengths: () = [0.0, 0.7], () = [0.23, 0.81], for contrastive questions. These kinds of questions can be () = [0.2, 0.66], () = [0.67, 0.94], () = answered with a contrastive explanation that compares [0.64, 0.8], and ( ) = [0.5, 0.74]. The precision, loca- the properties of the intervals associated to arguments. tion, and combined values for each interval are presented Producing this kind of explanation benefits from our in Table 2. These evaluation criteria give us three ways approach by enriching the returned information to the for comparing pages. We can assume that the value of the user. decimal gives the PR of a page. Thus, if we use precision For generating the explanations, we will consider the for assigning the PR to the pages, E is the page with the ranking based on the combined value; thus, we generate highest PR (that is 8). We can also observe that D and F the explanations based the criteria precision and locashare the same PR (that is, 7); however, since F is more tion. Given an argument , the contrastive questions are precise then it is more relevant than D, which may impact expressed in the following way: on which page will be showed first and therefore on the -P-contrast: WHY(, (), pos) (Why is argument in visits to such pages. If we use location, D is the page with position (), rather than in position pos?) the highest PR (that is 8), and if we use the combined value, -O-contrast: WHY(, (), , ()) (Why is arguE has PR 8 and it is the best ranked page. Let us note that ment in position () whereas argument in posithe benefit of using imprecise evaluation for PR gives the tion ()?) option of obtaining diferent rankings which may reflect where () and () are functions that return a the preferences of users. For example demanding users may position of argument and argument , respectively, use the combined value because they want to obtain pages under an imprecise strength function and pos is an with both good location and precision. Other users may expected position. This position can be based on the rankwant to get well located pages disregarding the precision ing constructed using the COMB. The resultant contrastive or vice-versa. Besides, precision and location are not the explanations can be seen as sequences of observations only ways for comparing intervals, so this gives a range of that constitute beliefs for the agent. possibilities for modelling users preferences and therefore For contrastive question WHY(, (), pos), we turn PR result more customizable. consider the case when () > pos. Algorithm 1 shows how the explanation is generated. The algorithm takes as input a credal SAF S and an imprecise strength function and returns a set of be- WHY(, (), , ()) liefs EXP. The beliefs that can be generated are: (i) __(, ), which means that argument is more precise and better located than argument ; (ii) _(, ), which means that argument is better located than argument ; and (iii) _(, ), which means that argument is more precise than argument .
For contrastive question WHY(, (), , ()), we consider the case when () < (). Algorithm 2 shows how the explanation is generated.
Require: S := ⟨ARG, R+, , ⟩, Ensure: EXP 1: if PREC( ()) > PREC( ())
LOCA( ()) > LOCA( ()) then 2: EXP := __(, ) 3: else 4: if PREC() > PREC() then 5: EXP := _(, ) 6: end if 7: else 8: EXP := _(, ) 9: end if Algorithm 2 Explanation for the O-contrast question - When we say that two intervals are equal, we mean that both the lower and the upper bounds are the same. Formally, given two intervals [ (), ()] and [ (), ()] for arguments A and B, respectively.
When we say that [ (), ()] = [ (), ()], it
means that () = () and () = ().
- We use ⊤ and ⊥ for denoting [
We can now begin with the axioms. The first one is about minimality. For the precise case, this axiom ensures that if an argument does not have any support, its strength is equal to its base score. In the imprecise case, we compare the interval of the imprecise base score with the interval of the imprecise strength. When there is no support for an argument both its lower and the upper bounds have to remain the same to satisfy minimality. Regarding precision and location, these are not considered because two diferent intervals may result in the same precision (or location) value, which does not mean that minimality was satisfied. Since intervals are compared element by element, we call this axiom of absolute.
Axiom 1. (Absolute Minimality) An imprecise gradual semantics satisfies absolute minimality if for any imprecise SAF S = ⟨ARG, R+, ⟩, for any argument ∈ ARG, if R+() = ∅ then () = ().
Example 4. (Cont. Example 3) Let’s consider Table 2, which shows the values for precision, location, and the combination of both for the scenario of PageRank. The ranking based on the combined value is the following: , , , , , , where is the best ranked argument and the worst one. Let us now show two explanations: before the P-contrast question WHY(, 6, 3), we have EXP = {__(, ), __(, ), __(, )} and before the O-contrast question WHY(, 1, , 2), we have EXP = _(, ).
WHY(, (), pos) Require: S := ⟨ARG, R+, , ⟩, Ensure: EXP 1: ARG_PREV := { ∈ ARG | () < () and () ≥ pos} 2: EXP := ∅ 3: for all ∈ ARG_PREV do 4: if PREC( ()) > PREC( ()) and
LOCA( ()) > LOCA( ()) then 5: EXP′ := __(, ) 6: else 7: if PREC( ()) > PREC( ()) then 8: EXP′ := _(, ) 9: end if 10: else 11: EXP′ := _(, ) 12: end if 13: EXP := EXP ∪ EXP′ 14: end for
with the role of supports. It states that a support strength5. Axioms for IGSs ens its target by increasing its strength. In this case, we can use the evaluation criteria in order to compare the In this section, we extend some properties studied in [18] intervals. Thus, in terms of intervals, we can say that the for the imprecise context, that is, for IGSs. We study the more precise the interval of an argument, the stronger behaviour of these properties considering the intervals the argument and the closer to 1 the location of the inand the evaluation criteria: precision and location. terval is the stronger its argument is. Besides, when the
Before presenting the axioms, let us make some asser- interval of an argument is already [
Axiom 2. (Strengthening) An imprecise gradual se- they support. In the context of impreciseness, we can mantics satisfies strengthening if for any imprecise SAF have two evaluations for dummy: (i) the first one considS = ⟨ARG, R+, ⟩, for any argument ∈ ARG, ers that a dummy argument has associated the interval if crit( ()) < 1 and ∃ ∈ R+() such that [0, 0] and (ii) the second one considers that the value of crit( ()) > crit( ()) then crit( ()) < precision (location or measure) of the dummy argument crit( ()) (for crit ∈ {PREC, LOCA, COMB}). is zero. As in the case of equivalence, we call the former absolute dummy and the latter just dummy.
that the only way of increasing the strength of an argu- Axiom 5. ((Absolute) dummy) An imprecise gradual ment is by supporting it with an acceptable argument. semantics satisfies absolute dummy (resp. dummy) if for In this case, we also use precision and location because any imprecise SAF S = ⟨ARG, R+, ⟩, for all argument these are criteria for evaluating the behavior of intervals, , ∈ ARG, if () = () (resp. crit( ()) = this means, that they allow us to measure if an interval crit( ())), R+() = R+() ∖ {} and ∈ is stronger than other. R+() with () = [0, 0] (resp. crit( ()) = 0) then () = () (resp. crit( ()) = crit( ()), for crit ∈ {PREC, LOCA, COMB}).
The following axiom has to do with the number of supporters and their quality. It states that the more quantity of acceptable supporters an argument has, the stronger the argument is. In the context of impreciseness, the quality is related to the precision and location of the supporters. Thus, only when a supporter is more precise or has a better location, it has a positive impact on the strength of the supported argument.
The next axiom is about equivalence, the idea is that arguments with equal conditions in terms of supporters and base score have the same strength. In the imprecise context, we consider that we can have two types of equivalence: (i) the first type considers that two arguments Axiom 6. (Counting) An imprecise gradual semanhave the same interval as imprecise base score and (ii) tics satisfies counting if for any imprecise SAF S = the second one considers that two arguments have the ⟨ARG, R+, ⟩, for all argument , ∈ ARG, if () = same precision or/and location values, which not nec- (), () < ⊤ and R+() = R+() ∪ {} with essarily means that both arguments have the same im- i.e., () > ⊥, then crit( ()) > crit( ()) (for precise base score. For example, assume that [0.3, 0.35] crit ∈ {PREC, LOCA, COMB}). and [0.5, 0.55] be the imprecise base score of arguments and , respectively. The precision value is the same for both intervals: it is 0.95. Now, assume that [0.6, 1] 6. Semantics Classification and [0.7, 0.9] be the imprecise base score of arguments ′ and ′, respectively. The location value is the same of both intervals: it is 0.8. Thus, even diferent intervals may have the same precision or location value, which has to be reflected in the equivalence property. We call the case (i) absolute equivalence and the case (ii) just equivalence.
the fulfilment of the axioms.
In the presented axioms, we can notice that there are diferent criteria that are considered for comparing the intervals. Let us recall that the fact that one interval is more precise than other does not mean that it is better located and vice-versa. What we can say is that these criAxiom 4. ((Absolute) equivalence) An imprecise grad- teria have an impact on how a semantics fulfils an axiom. ual semantics satisfies absolute equivalence (resp. equiv- We can have that a semantics fulfils an axiom when the alence) if for any imprecise SAF S = ⟨ARG, R+, ⟩, criteria precision is used and when the criteria location (froerspa.ll acrrgiutm(en(t)), =∈ crAiRtG(, if( )()),) th=ere ex(ists) ifsulnfiloatnuasxeidomandwvhiecne-tvheersceo.mWbeincaatniohnaovfepsreemciasniotincsatnhdat a bijective function from R+() to R+() such location is used. Furthermore, there may be semantics tchraitt(∀()∈) =Rcr+i(t(), ((()))) =then ((()=)) (r(esp). tahpaptlifeudlfil. aCnoanxsiiodmerwinhgenthbeostehapsrpeeccitssio,nwaencdalnoccalatisosnifyaraen (resp. crit( ()) = crit( ()), for crit ∈ IGS as follows: {PREC, LOCA, COMB}). 1. Absolute semantics: An IGS is absolute when the semantics satisfies absolute minimality, absolute equiva
The following axiom, called dummy, states that argu- lence, and absolute dummy. ments with strength 0 have no impact on the arguments Theorem 1. Given an IGS : 1. If satisfies absolute equivalence (resp. absolute dummy), then also satisfies equivalence (resp. dummy). 2. If is an absolute semantics, then will be also a combined semantics. 2. One-criterion semantics: An IGS is one-criterion Proof 2. Let S = ⟨ARG, R+, , ⟩ be a credal SAF. when the all the axioms that it fulfils are satisfied in only - For absolute minimality: Let S = ⟨ARG, R+, ⟩ one criterion: either precision or location. be the respective imprecise SAF of S . This means that 3. Two-criteria semantics: An IGS is two-criteria when ∀ ∈ ARG, is obtained by applying Equation (1) over the axioms that it fulfils are satisfied in both precision . When we say that R+() = ∅, this means that and location. ∈ ARG∘ or () = ∅. According to Definition 4, for 4. Combined semantics: An IGS is combined when the obtaining (), Equation (1) has to be applied to . axioms it fulfils are satisfied in the combination of both Since Equation (1) is applied to the same credal set, this criteria, that is, precision and location. means that () = ().
Note that absolute, one-criterion, and two-criteria IGSs - For strengthening soundness: By Reductio ad abare disjoint sets. From the above classification, we can surdum. Let us assume that ∄ ∈ R+() such that conclude the following theorem. crit( ()) > 0 (for crit ∈ {PREC, LOCA, COMB}). This means that Equation (1) has to be applied to , which in turn means that crit( ()) = crit( ()). This contradicts the premise of the axiom.
calculations on credal sets, which contain the probability values necessary for the inference by applying Proof 1. Let S = ⟨ARG, R+, , ⟩ be a credal SAF. Bayes rule. We can notice that when we apply Equa1. If satisfies absolute equivalence, this means that tion (1) to two diferent credal sets we can obtain the ∀, ∈ ARG, () = (). This in turn means same imprecise base score or imprecise strength; howthat () = () and () = (). Since the in- ever, it does not ensure that after the inference with tervals are the same, when we apply the precision, loca- another credal set, the resultant intervals will be the tion, or combination criterion, the result is the same. Thus, same. For example, let () = {0.4, 0.76, 0.56, 0.87} we have crit( ()) = crit( ()) (for crit ∈ and () = {0.75, 0.4, 0.87, 0.6} be two credal sets {PREC, LOCA, COMB}), which means that satisfies equiv- whose lower and upper bounds are [0.4, 0.87]; however, alence. The same reasoning applies for absolute dummy. if we aggregate each of them with a third credal set by 2. If is absolute, this means that () = () and applying Bayes rule, the result will be diferent, even () = (). Since the intervals are the same, when we considering that this third credal set and the conditionals apply the combination criterion, the result is the same. So, are the same for both. Thus, the only way to guarantee is a combined semantics. the same result is by using the same credal sets in all the inference process. Therefore, we can say that considering that all the credal sets have the same values, some axioms 7. Theoretical Evaluation can be fulfilled. In the case of absolute equivalence, the credal sets of the equivalente arguments, the credal sets of their parents, and their conditional credal sets have to be the same. In the case of absolute dummy, the credal sets of one or more of their parents are the same because they all have zero as probability values.
ual semantics by checking which properties it fulfils and which it does not.
The first theorem states that the properties that are fulfilled by the proposed gradual semantics are absolute minimality and strengthening soundness. In the case of absolute minimality, since no equation has to be applied for calculating a conditional credal set, the credal sets for calculating the imprecise base score and the imprecise strength are the same, so the lower and upper bounds are also the same. Regarding strengthening soundness, the only way for an interval become more precise or better located is by improving the associated credal set and this can only happen when it has supporters.
Theorem 2. Given an imprecise SAF S = ⟨ARG, R+, ⟩. The imprecise gradual semantics based on credal network theory fulfils absolute minimality and strengthening soundness.
Definition 5. (Equality in credal sets) Let () and () be two credal sets. We say that () and () are equal when ∀() = () for 1 ≤ ≤ , where is the total number of elements of the credal sets.
Theorem 3. Given an imprecise SAF S = ⟨ARG, R+, ⟩. Let , ∈ ARG be two arguments that are absolute equivalent. The IGS based on credal network theory fulfils absolute equivalence when ∀, : 1. R+() = R+(); 2. () and () are equal; 3. () and () are the same; 4. () | and () | are the same. Proof 3. Let S = ⟨ARG, R+, , ⟩ be a credal SAF. ory to calculate the strength of arguments; however, the
Let () and () be the credal sets for arguments intended meaning of gradual semantics complements the and , respectively. Let us assume that () = (). interpretability of the resultant values. In the case of After applying Equation (1) to both, we can say that SAFs, the calculated strength may reflect how supported () = (). Following the premise of the axiom, we an argument is. This can be interpreted in two ways: know that both arguments have the same supports. Let us (i) an argument is strong because it has many supports also assume that (i) the credal sets of such supports are also and/or (ii) an argument, even with few supports, is strong equal and (ii) the calculated conditional credal have the because its supports are strong enough. This interpretasame values. This means that after applying Equation (1) tion can be the base for generating explanations about the to credal sets and conditional credal sets, the value of the behaviour of the elements of the graph and for analyse imprecise strength will be the same: () = (). such behaviour in the light of other interactions. Our approach also complement the calculations by
Regarding the other axioms, their fulfilment can not be applying the location and precision criteria in order to
guaranteed due to nature of the inference, which is based rank the resultant intervals. The use of more than one
on the Bayes rule. For instance, for counting axiom, the dimension for comparing the intervals, gives flexibility
amount of supporters do not mean that the supported to the approach and allows to use one or more of them
will increase its strength, it depends on the quality of the depending on the domain of application. We have
provalues of all the supporters together with the conditional posed to use such criteria; however, any other criteria
credal set of the parents of the supported argument given can be used with this aim.
the supported one. To the best of our knowledge there are few works that
study explainability in gradual semantics. Albini et al.
8. Discussion [23] generate three types of explications for PR. They
use a QBAF for modelling the problem and generate
conIn this section, we discuss our approach by comparing it trastive explanations as well. In their case, they focus on
with credal networks. We make such comparison because answering ‘What are the links that make pages A and
we base on credal networks for the calculation of the B have diferent scores?’. We can note that the contrast
imprecise strength and it is important to highlight what focus is diferent and they do not consider the ranking
diferences gradual semantics from it. We are not going to generated by the gradual semantics. In [
A credal network is a method for information fusion where the inference derives the probability of one or 9. Conclusions and Future Work more random variables taking a specific value or set of values. On the other hand, gradual semantics aims to This work presented an IGS for SAFs considering that calculate the strength value of the arguments of an AAF the support relation is causality. We use credal sets to with the aim of ranking or ordering them. The strength model the probability values of each argument and credal value and the ranking can be used to make decisions or networks theory for calculating the imprecise strength, determine how acceptable each argument is. For example, which is represented by an interval. For ranking the inimagine a scenario of a negotiation persuasive dialogue terval, we use two criteria, the location and the precision where two agents (proponent and opponent) want to of the interval. From the resultant ranking, we propose convince the other to accept a proposal [22]. In this to generate contrastive explanations about the positions scenario, the agents exchange rhetorical arguments that of the arguments. represent threats and rewards. Both agents generate In order to evaluate our proposed IGS, we study and such arguments and have to decide which of them to propose a set of axioms that describe the behavior of send to his respective opponent. The calculation of the IGSs. We demonstrated that our approach fulfils absostrength of such arguments can help the agents to make lute minimality and strengthening soundness and fulfils such decision. On the other hand, in the scenario of the absolute equivalence and absolute dummy under some PageRank, the strength value of each page can be seen circumstances. Besides, we propose a classification of as a measure of how acceptable each page is. The more IGSs based on the fulfilment of axioms. acceptable, the more PR value the page has. In general, in formal argumentation explanations have
Since, the support relation that we are tackling in our a qualitative nature, that is, they are based on sets of approach is the causal one, we use credal networks the