In recent years, the notion of time has been studied in different ways in Dung-style Argumentation Frameworks. For example, time intervals of availability have been added to arguments and relations. As a result, the output of Dung semantics varies over time. In this paper, we consider the situation in which arguments hold with a certain probability distribution during a given interval. To model the uncertain character of events, we propose different notions of temporal coniflct between arguments according to the type of availabilities intersection (partial, inclusive or total). Then, we refine these notions of conflict by a defeat relation, using criterion functions that evaluate an attack's significance according to the probability over time. After extending Dung's semantics with these defeat notions, we present a new temporal acceptability of arguments based on the concept of defence, allowing for finer results in time.
Argumentation Theory studies how conclusions can be drawn starting from a given set of facts
or premises, and, in the field of Artificial Intelligence, it provides tools for modelling
humanfashioned logical reasoning where the available information may be discordant. A simple yet
powerful representation of conflicting information is provided by the Abstract Argumentation
Frameworks [
In general, the AFs considering the first notion use time to know when arguments or attacks
are available [
The second concept that interests us is the consideration of probability in arguments and
relations; see [
In this paper, we take a further step towards a more expressive AF and consider that the arguments are certain on the interval but uncertain on their occurrence. In particular, we assume only to know the probability distribution of the events associated with the arguments.
Please note that this paper is an extension of a previous short article [
{2, . . . , 7}); • argument : A surveillance Camera recorded the victim walking at 3 pm (i.e. {3}); • argument : According to the Doctor, the victim died between 6 pm and 10 pm (i.e.
{6, . . . , 10}).
The attacks between arguments , , , and are given in Figure 1, which provides a static representation of the events. The probability distribution over time for the arguments is then represented in Figure 2. In this example, we use a uniform distribution for arguments and , while argument , which is more likely to occur around 8 pm, follows a normal distribution. Finally, argument holds with probability 1 at 3 pm. Note that one can choose different probability distributions to represent various types of uncertainty.
Since events can be uncertain over time, the notion of conflict between arguments also needs to be revised. For example, two contradictory arguments, such as “the victim was fighting” and “the victim was walking”, may not be in conflict if they hold at different times.
To deal with temporal and probabilistic aspects of argumentation, we first introduce Temporal Probabilistic Argumentation Frameworks (TPAFs), an extension to classical AFs, and propose a method for deriving conflict between arguments. Then, to evaluate the acceptability of arguments, 1Notice that, we consider events happening at a time point. Therefore, intervals are represented as sets of time points. we provide a set of semantics based on the notion of defence over time. We also study the concept of minimal defence to investigate the conditions under which an argument can be accepted with respect to a time interval.
The remainder of this paper is organised as follows. Section 2 summarises the basic definitions of AF and extension-based semantics; in Section 3 we formalise TPAFs, providing conflict and defence notions and a set of semantics that take into account the probabilistic nature of event occurrences; Section 4, then, presents the idea of temporal Δ-acceptability; finally, Section 5 concludes the paper with final remarks on possible future work.
In this section, we recall the formal definition of an Abstract Argumentation Framework and the
related extension-based semantic [
Definition 1 (AF). An Abstract Argumentation Framework (AF) is a pair ⟨, ℛ⟩ where is a set of arguments, and ℛ is a binary relation on .
Consider two arguments , belonging to an AF. We denote with (, ) ∈ ℛ an attack from to ; we can also say that is defeated by . In order for to be acceptable, we require that every argument that defeats is defeated in turn by some other argument of the AF. Definition 2 (Acceptable argument). Given an AF ⟨, ℛ⟩, an argument ∈ is acceptable with respect to ⊆ if and only if ∀ ∈ such that is attacking , ∃ ∈ such that is attacking and we say that is defended by .
Using the notion of defence as a criterion for distinguishing acceptable arguments in the framework, one can further refine the set of selected arguments.
Definition 3 (Extension-based semantics). Let ⟨, ℛ⟩ be an AF. A set ⊆ is conflict-free if and only if ∄, ∈ such that (, ) ∈ ℛ. A conflict-free subset is then admissible, if each ∈ is defended by ; complete, if it is admissible and ∀ ∈ defended by , ∈ ; stable, if it is admissible and attacks every argument in ∖ ; preferred, if it is complete and ⊆ -maximal; grounded, if it is complete and ⊆ -minimal.
We also need the notion of time intervals for reasoning with temporal aspects of arguments.
For example, in Timed AFs [
Definition 4 (Temporal interval). Let T be the discrete universe of time points. A temporal
interval is a subset = {1, 2, . . . , } of T where 1 and are respectively the minimum and
maximum bounds of . Moreover, = {} denotes the instant and = {} is not allowed.
Definition 5 (TAFs). A Timed Abstract Argumentation Framework (TAF) is a tuple ⟨, ℛ, ⟩
where is a set of arguments, ℛ is a binary relation on , and : → ℘(T) is the availability
function for arguments.2
3. Temporal Probabilistic Argumentation Frameworks
To reason about uncertain events in time using argumentation-based tools, we must first be able
to represent the probabilistic and temporal aspects of arguments in a single framework. To this
end, we instantiate the generic framework proposed in [
Note that depending on the user’s needs, for example, if the TPAF occurs over a long period of time, it is useful to be able to restrict the study of a TPAF to a specific time interval. Thus, in the rest of the article, we will specify the time interval we are working on.
When an argument has a probability of occurring equal to zero, it should not be considered in the reasoning process. Therefore, we extract, for each argument, the instants in which its probability is positive, i.e. when the argument can occur. 2We use ℘(T) to indicate the powerset of T.
Definition 7 (Positive probability over time). Let = ⟨, ℛ, ⟩ be a TPAF, ∈ an argument, and a time interval. We define the set of non-null probability of in by () = { ∈ | {}() > 0}.
Example 3. Consider the TPAF of Example 2. We have that {1,...,4}() = {1,...,10}() = {1, . . . , 4}.
Given the probability over time of arguments, the conflicts are not sure and can be interpreted in different ways according to various notions.3 In particular, we propose three notions of conflict based on the availability of involved arguments and three criterion functions defining when the conflict is significant, i.e. it is a defeat.
Definition 8 (Temporal probabilistic conflicts). Let = ⟨, ℛ, ⟩ be a TPAF, , ∈ two arguments and a time interval. We define a boolean conflict function CFIx : × → {⊤ , ⊥}, with x ∈ {p, i, t} (where p, i and t stand for partial, included and total, respectively), which determines a conflict from to within when (, ) ∈ ℛ and: • Partial conflict: CFIp(, ) = ⊤ if and only if () ∩ () ̸= ∅; • Included conflict: CFIi(, ) = ⊤ if and only if () ∖ () = ∅; • Total conflict: CFIt(, ) = ⊤ if and only if () = ().
Otherwise for any x ∈ {p, i, t}, CFIx(, ) = ⊥. while CFi{1,...,4}(, ) = ⊥; CFt{2,3,4}(, ) = ⊤ while CF{t1,...,4}(, ) = ⊥.
Example 3. (Continued) In the following, we illustrate the different temporal probabilistic conflicts of Definition 8: CFp{1,...,5}(, ) = ⊤ while CFp{6,7}(, ) = ⊥; CFi{1,...,4}(, ) = ⊤
Note that partial conflict and total conflict are symmetric, while the included conflict is not. Moreover, the notion of CFIt implies the notion of CFIi which implies, in turn, CFIp.
(, ) ∈ ℛ and a time interval.
• If CFIt(, ) = ⊤ then CFIi(, ) = ⊤. • If CFIi(, ) = ⊤ then CFIp(, ) = ⊤.
Let = ⟨, ℛ, ⟩ be a TPAF, , ∈ such that
The notion of conflict only considers the positive probability over time of the arguments, i.e. we only check if the probability of arguments involved in an attack is positive. We can refine the concept of conflict by using the probability values attached to arguments to establish whether a conflict is significant according to a criterion function. In addition, we use the term defeat to refer to a significant conflict.
Definition 9 (Criterion functions). Let = ⟨, ℛ, ⟩ be a TPAF, , ∈ such that (, ) ∈
ℛ and a time interval. We define a boolean criterion function CTIx : × → {⊤ , ⊥} where
x ∈ {Sg, Wg, A} as follows:
3For example, in [
• Weak greater: CTIWg(, ) = ⊤ if and only if ∀ ∈ such that {}() × {}() > 0, {}() > {}(); {}() > {}(); • Strong greater: CTISg(, ) = ⊤ if and only if ∃ ∈ such that {}() × • Aggressive: CTIA(, ) = ⊤ if and only if ∃ ∈ such that {}() × {}() < 1. {}() > 0 and {}() > 0 and Otherwise ∀x ∈ {Wg, Sg, A}, CTIx(, ) = ⊥.
The strong greater criterion leads to more frequently identifying a defeat, whereas the weak greater criterion will be more cautious in indicating that a conflict is significant. Note that for the aggressive criterion, there is no need to differentiate between a strong and a weak version: since we consider a probability distribution with a sum of 1 over the entire interval if there exists a non-zero probability strictly less than a 1, then there is no instant at which the probability is 1.
We show next that, as usual, the universal quantifier (weak) implies the existential (strong) one, and the greater criteria imply the aggressive criterion.
Proposition 2 (Relation between criterion functions). Let = ⟨, ℛ, ⟩ be a TPAF, , ∈ two arguments such that (, ) ∈ ℛ and consider a time interval . We have that: • If CTIWg(, ) = ⊤ then CTISg(, ) = ⊤; • If CTIWg(, ) = ⊤ then CTIA(, ) = ⊤; • If CTISg(, ) = ⊤ then CTIA(, ) = ⊤.
We define a temporal probabilistic defeat function by combining a notion of conflict and a criterion function.
Definition 10 (Temporal probabilistic defeat function). Let = ⟨, ℛ, ⟩ be a TPAF, , ∈ two arguments, and consider a criterion function CT and a conflict function CF. We define ΔCT,CF : × → {⊤ , ⊥} the defeat function determining that defeats in the interval , with respect to CT and CF. In particular, ΔCT,CF(, ) = ⊤, if and only if CTI(, ) = CFI(, ) = ⊤. Otherwise ΔCT,CF(, ) = ⊥. Δ{Sg1,,.p..,10}(, ) = ΔA{,1p,...,10}(, ) = ⊤. instead, we obtain Δ{A,1p,...,7}(, ) = ⊤, meaning that defeats in .
Example 3. (Continued) We show below how different temporal probabilistic defeat functions behave according to the partial conflict of Definition 8. First, consider arguments and of G and the interval {1, . . . , 7}. We have that does not defeat within according to the greater criteria. In fact, ΔW{g1,,.p..,7}(, ) = ΔS{g1,,.p..,7}(, ) = ⊥. If we consider the aggressive criterion, Then, for arguments and in the interval {1, . . . , 10} we have Δ{Wg1,,.p..,10}(, ) = ⊥ and For a better understanding of the impact of the defeat functions and the restriction of the time intervals, it is worthwhile to define the resulting TPAF according to these parameters. Definition 11 (Restricted TPAF). Let = ⟨, ℛ, ⟩ be a TPAF, a time interval and Δ a temporal probabilistic defeat function (on ). We denote the restricted graph by Δ - = ⟨′, ℛ′, ⟩, where: • the arguments are restricted to the time interval , such that ′ ⊆
if and only if () ̸= ∅; • the attacks are restricted according to Δ such that ℛ′ ⊆ ℛ if and only if , ∈ ′ and Δ(, ) = ⊤.
and ∀ ∈ , ∈ ′ and ∀(, ) ∈ ℛ, (, ) ∈ ℛ′
Let us visualise (see Figure 3) the result of a restricted TPAF according to these parameters. Example 3. (Continued) We can see in the following the restricted graphs Δ -G.
ΔW{g4,,p...,10}-G ΔS{g4,,p...,10}-G ΔA{,4p,...,10}-G
The implications between the different defeat functions can be derived by analysing together the relations between conflict functions (Proposition 1) and criterion functions (Proposition 2). We show in Figure 4 the relations between all the defeat functions. In particular, we observe that the strongest (most conflicting) defeat is ΔA,p and the weakest (less conflicting) defeat is ΔWg,t. ΔWg,t ΔSg,t ΔA,t ΔWg,i ΔSg,i ΔA,i ΔWg,p ΔSg,p ΔA,p
Until the end, we will use Δ to refer to a generic temporal probabilistic defeat function. We now extend the notion of conflict-freeness to TPAFs on the basis of a defeat function Δ. Definition 12 ( Δ-conflict-free). Let = ⟨, ℛ, ⟩ be a TPAF, ⊆ a set of arguments and Δ a defeat function. is Δ-conflict-free if and only if ∄, ∈ such that Δ(, ) = ⊤. Example 3. (Continued) We refer again to the TPAF of Example 2 and check if the set = {, , } is Δ-conflict-free. We can verify that is not Δ{Sg1,,.p..,4}-conflict-free since ⊤. Δ{Sg1,,.t..,4}-conflict-free. Δ{Sg1,,.p..,4}(, ) =
⊤. Then, is not even Δ{Sg1,,.i..,4}-conflict-free since We also observe that ∀, ∈ {, , }, Δ{Sg1,,.t..,4}(, ) = Δ{Sg1,,.i..,4}(, ) = ⊥ and thus is
According to a Δ-conflict-free notion we define the notion of a defence of an argument against another one according to a set of arguments. This function thus returns the instants at which the argument is defended by the set against this single attacker.
Definition 13 ( Δ-SingleDefence of from by ). Given = ⟨, ℛ, ⟩ be a TPAF, a time interval, , ∈ and ⊆ be a Δ-conflict-free set of arguments within . According to the defeat notion Δ used for the Δ-conflict-freeness, the Δ single defence of from with respect to within , is defined as follows: Δ -1def(, , ) = () ∩
⋃︁ ∈{|∈,Δ (,)=⊤} () ∩ () Δ{Sg2,,.p..,7}-1def(, , ) = {6, 7}.
Example 3. (Continued) From the TPAF G, let us see what is the Δ{Sg2,,.p..,7} single defence of from and with respect to the set of arguments = {, }: Δ{Sg2,,.p..,7}-1def(, , ) = {3} and
Thanks to the previous definition we can now define when an argument is Δ defended by a set of arguments in a TPAF against a single attacker. In the following, we generalise this notion of defence against a set of attackers.
Definition 14 ( Δ-Defence of with respect to ). Let = ⟨, ℛ, ⟩ be a TPAF, a time interval and be a Δ-conflict-free set of arguments within . The Δ-defence for with respect to , is defined as follows: Δ -def(, ) =
⋂︁ ∈{|Δ (,)=⊤}
( () ∖ ()) ∪ Δ -1def(, , )
Note that, an argument is defended on the instants that it is not attacked or when an argument defends it on this time.
We now define Δ-admissible, Δ-complete, Δ-preferred, Δ-stable and Δ-grounded semantics for TPAFs.
Definition 15 ( Δ-Semantics). Let = ⟨, ℛ, ⟩ be a TPAF, a time interval, Δ a defeat function and consider a set of arguments ⊆ . We say that: • is a Δ -admissible extension of within , denoted by ∈ Δ -ad() if and only if for all ∈ it holds that () = Δ -def(, ); • is a Δ -complete extension of within , denoted by ∈ Δ -co() if and only if is a Δ -admissible extension of and contains all the arguments such that () = Δ -def(, ); • is a Δ -preferred extension of within , denoted by ∈ Δ -pr() if and only if is a ⊆ -maximal Δ -complete extension; • is a Δ -stable extension of within , denoted by ∈ Δ -st() if and only if is Δ -admissible and for all ∈ ∖ , there exists ∈ such that defeats , i.e., Δ (, ) = ⊤; • is the Δ -grounded extension of within , denoted by ∈ Δ -gr() if and only if is the ⊆ -minimal Δ -complete extension.
We show in Table 1 a comparison between the different semantics with respect to ΔWg,p, ΔSg,p and ΔA,p. For the remaining examples in this paper, we will assume = {4, . . . , 10}. In this interval, each semantics returns the same sets of extensions for criteria Wg and Sg. However, with the criterion A, argument is able to defend itself and thus is accepted by all semantics except for the grounded one, which is empty since there are no undefeated arguments.
ΔWg,p-ad(G) = ΔSg,p-ad(G) ΔWg,p-co(G) = ΔSg,p-co(G) ΔWg,p-pr(G) = ΔSg,p-pr(G) ΔWg,p-st(G) = ΔSg,p-st(G) ΔWg,p-gr(G) = ΔSg,p-gr(G) ΔA,p-ad(G) ΔA,p-co(G) ΔA,p-pr(G) ΔA,p-st(G) ΔA,p-gr(G) {∅, {}, {}, {, }} {{, }} {{, }} {{, }} {{, }} {∅, {}, {}, {, }, {}} {∅, {, }, {}} {{, }, {}} {{, }, {}} {∅}
The two following propositions show that the Δ semantics satisfy the classical properties that we have in non-temporal frameworks.
Proposition 3 (Unicity of the Δ-grounded extension). For any = ⟨, ℛ, ⟩ be a TPAF, there always exists one and only one Δ-grounded extension.
The relations between Δ-admissible, Δ-preferred, Δ-stable, and Δ-complete extensions are given below.
Proposition 4 (Relation between semantics). Let = ⟨, ℛ, ⟩ be a TPAF. Then: 1. Let ⊆ . Then, is ⊆ -maximal Δ-admissible if and only if is a ⊆ -maximal Δ-complete extension; 2. A Δ-preferred extension is also a Δ-complete extension; Let us now define the notion of sceptical acceptability according to a Δ semantics.
Definition 16 ( Δ-Skeptical acceptability). Let = ⟨, ℛ, ⟩ be a TPAF, an interval, and let {1, . . . , } be the set of Δ -extensions of , with respect to a semantics between: admissible (ad), complete (co), preferred (pr), stable (st) and grounded (gr). An argument ∈ , is Δ skeptical acceptable under Δ -s, denoted by ∈ Δ -sk-s() where s ∈ {, , , , }, if and only if ∀ ∈ {1, . . . , }, ∈ .
Example 3. (Continued) We compare in Table 2, the different semantics using ΔWg,p , ΔSg,p and ΔA,p on G between 4 and 10. As seen in Table 1, since semantics based on the Wg and Sg criteria have the same extensions, their sceptical arguments are also identical. Finally for the semantics using the criterion A, since each argument can defend itself (because each argument in defeat, defeats its attackers also), there is no sceptically accepted argument.
ΔWg,p-sk-ad(G) = ΔSg,p-sk-ad(G) ΔWg,p-sk-co(G) = ΔSg,p-sk-co(G) ΔWg,p-sk-pr(G) = ΔSg,p-sk-pr(G) ΔWg,p-sk-st(G) = ΔSg,p-sk-st(G) ΔWg,p-sk-gr(G) = ΔSg,p-sk-gr(G)
∅ {, } {, } {, } {, } ΔA,p-sk-ad(G) ΔA,p-sk-co(G) ΔA,p-sk-pr(G) ΔA,p-sk-st(G) ΔA,p-sk-gr(G) ∅ ∅ ∅ ∅ ∅
We finally show that the Δ-semantics (Definition 15) obtain the same extension as the classical semantics (Definition 3) on the AFs coming from the restricted TPAFs according to a time interval and a defeat function Δ.
Theorem 1 (Link between the Dung’s semantics and the Δ-semantics). Let = ⟨, ℛ, ⟩ be a TPAF, a time interval and Δ a defeat function.
• If is a Δ -admissible extension of , then is an admissible extension of Δ -. • If is a Δ -complete extension of , then is a complete extension of Δ -. • If is a Δ -preferred extension of , then is a preferred extension of Δ -. • If is a Δ -stable extension of , then is a stable extension of Δ -.
• If is a Δ -grounded extension of , then is a grounded extension of Δ -.
Note that the previous result is not a characterisation (if and only if) since if we only look at the graph typology (as in classical AF) we can have a defence between arguments but depending on their temporality (as in TPAF), this defence may not be over the whole duration of the availability of an argument. Therefore, constructing a restricted graph and then applying classical semantics is a relaxed version of Δ-semantics. Indeed the Δ-semantics is required to accept only the totally defended arguments whereas if we apply the Dung semantics on a restricted graph, it will be enough that an argument has a defended instant to be accepted. 4. Consistent Temporal Δ-Acceptability As we saw in the previous section, the notion of sceptically acceptable argument only identifies arguments that are acceptable in each instant of the studied interval. However, it is also interesting to know if a given argument is acceptable in some instants and defeated in others. For this purpose, we introduce a finer-grained notion of acceptability over time which extracts the instants in which an argument is defended. We first define the notion of minimal temporal Δ-acceptability which extracts the instants of an argument that are always defended according to all maximal consistent sets of arguments containing the evaluated argument.
Definition 17 (Minimal temporal Δ-acceptability). Let = ⟨, ℛ, ⟩ be a TPAF, ∈ and a time interval. We define the minimal temporal Δ-acceptability of within according to Δ by min-ΔI-acc() = ⋂︀∈maxFree-Δ () Δ -def(, ), where maxFree-Δ () is the set of all ⊆ -maximal Δ -conflict-free set of arguments over containing .
The minimal temporal Δ-acceptability time of arguments in G is reported in Table 3. In this example we consider the ΔWg,p and ΔSg,p defeats.
min-ΔIWg,p-acc = min-ΔISg,p-acc = {4}, = {5, 6, 7}, = ∅, = {6, . . . , 10}
As we saw in Figure 3 and Table 3, arguments and defeat each other (e.g. in ΔSg,p-G) altough they share some minimal temporal Δ-acceptability time. We then propose to compute an argument’s consistent temporal Δ-acceptability by excluding the instants in which it is defeated. Definition 18 (Consistent temporal Δ-acceptability). Let = ⟨, ℛ, ⟩ be a TPAF, ∈ and a time interval. We define the consistent temporal Δ-acceptability of according to Δ by con-ΔI-acc() = min-ΔI-acc() ∖ ⋃︀∈ such that Δ (,)=⊤ min-ΔI-acc().
Figures 5 and 6 illustrate the consistent temporal Δ-acceptability time of arguments in G within the interval {4, . . . , 10}.
We can notice that in contrast to the results of the semantics and, in particular, for the sceptically accepted arguments, we have here different results between the criteria Wg and Sg. According to the Wg criterion, arguments and are never defeated and thus sceptically accepted. On the other hand, for the Sg criterion, argument is never defeated, and defends ; thus, they are sceptically accepted.
It is interesting to note that defending differs from not being defeated. Indeed, if we look at the instants 6 and 7, in Figure 6 these times are not consistent temporal Δ-acceptable for 4 5 6 because does not defeat in {6, 7}, hence is defeated in {6, 7}; which is not the case in Figure 5 where is not defeated.
Moreover, it is also important to note that thanks to this notion, even if an argument does not have all its time acceptable (as for sceptically accepted arguments), we can extract the subsets of times that are consistent temporal Δ-acceptable, as for the argument at time 5.
Finally, as can be seen in Figure 5, with the criterion function Wg, it is possible that arguments which attack each other (e.g. and where (, ), (, ) ∈ ℛ) are not considered defeated. Therefore, it is up to the user to determine when these arguments are acceptable in overlapping time intervals. We now show that when using the criteria functions Sg and A with the conflict function p, it is, however, not possible to have arguments that attack each other and have consistent temporal Δ-acceptability times in common when they have a different probability distribution. Proposition 5 (Consistency with the attack relation). Let = ⟨, ℛ, ⟩ be a TPAF, , ∈ two arguments such that (, ), (, ) ∈ ℛ and a time interval. If ∃ ∈ such that {}() ̸= {}(), {}() > 0, and {}() > 0; and x ∈ {Sg, A}, then con-ΔIx,p-acc() ∩ con-ΔIx,p-acc() = ∅.
The ability to model and reason with probability on events occurrence is crucial for addressing realworld argumentation problems. The framework we propose captures the temporal probabilistic nature of arguments and provides a tool for drawing conclusions starting from a set of conflicting facts/events for which the placement in time is uncertain. The probability associated with timed arguments is subject to interpretations which vary according to context and use case. For this reason, we propose different criteria for establishing whether arguments are in conflict and if the conflict is significant enough to represent a defeat. Based on this graph restriction process (as shown in Figure 3), we can then apply different semantics to calculate the acceptability of the arguments, for instance, those of Dung (Section 3) or news more adapted to the notion of time, as the consistent temporal acceptability which allows assessing the acceptability of arguments over ifner time scales (Section 4).
In the future, we plan to carry on this work by examining other aspects of argumentation that
relate uncertainty to the notion of time. The current proposal considers events lasting only a
single instant (e.g. “the victim died between 6 pm and 10 pm”). A natural extension to that is
allowing events with a duration in time (e.g. “the victim has been walking between 2 pm and
7 pm”). In this case, we could use a probability measure to express the likelihood of an event
taking place over a time interval. Finally, in addition to uncertainty about the time when a given
argument is valid, we may also consider probability associated with arguments and attacks (i.e.
they are uncertain), as in Probabilistic Argumentation Frameworks [