Subjective Logic provides a standard set of logical operators intended for use in domains containing uncertainty. At the same time, the motivations behind the adoption of Argumentation in AI are rooted into reasoning and explanation in presence of incomplete and uncertain information. This work uses Subjective Logic as a means to represent the beliefs of different agents towards arguments and attacks, and to aggregate them with the purpose to have an overall reputation from all the considered agents in the considered community. Agents are also allowed to form their opinion from others' opinions by exploiting trust paths. Finally, the obtained beliefs can be used to compute the community-biased expectation that a set of (abstract) arguments satisfies a given semantics.
An Abstract Argumentation Framework (AAF) [
Given a framework, it is possible to examine the question on which set(s) of
arguments can be accepted, hence collectively surviving the conflict defined by R.
Answering this question corresponds to defining an argumentation semantics [
Subjective Logic (SL) [
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Since arguments are often uncertain, it can be useful to quantify the uncertainty
associated with each argument, as previously explored in other works in the literature [
Afterwards, with the purpose to find a framework to assign opinions to arguments
and attacks, we introduce agents on top of slAAFs. In this scenario, new with respect to
[
This work extends the results in [
The paper is organised as follows: in Section 2 we summarise the background notions behind SL. Section 3 proposes Subjective Logic-based AAFs and how to work with them by computing the expectations of semantics and argument acceptance by using opinions instead of probability values. Then, in Section 4 we embed a trust model in top of slAAFs: we describe how trust paths among agents can be used to compute an indirect opinion on arguments and attacks. Section 5 and Section 6 ends the paper with related work and conclusions respectively. 2
A subjective opinion expresses belief about states of a state space called a “frame of
discernment”, or “frame” for short. In practice, a state in a frame can be regarded
where P (X ) denotes the powerset of X and |P (X )| = 2k. All proper subsets of X are
states of R(X ), but the frame X and the empty set 0/ are not states of R(X ), in line with
the hyper-Dirichlet model [
An opinion is a composite function consisting of belief masses, uncertainty mass
and base rates. It applies to a frame, also called a state space, and can have an attribute
that identifies the belief owner. An opinion is a composite function that consists of a
belief vector b,1 an uncertainty parameter u, and base rate vector a,2 which take values
in the interval [
Belief additivity: uX + Â bX (xi) = 1.
xi2R(X) k Base rate additivity: Â i=1
aX (xi) = 1, where xi 2 X .
A subjective (hyper) opinion of user A over the frame X is denoted as w XA = (bX , uX , aX ), where bX is a belief vector over the states of R(X ), uX is the complementary uncertainty mass, and aX is a base rate vector over X , all seen from the viewpoint of belief owner A. The belief vector bX has (2k 2) parameters, whereas the base rate vector aX only has k parameters. The uncertainty parameter uX is a simple scalar. Thus, a general opinion contains (2k + k 1) parameters and hence it is a hyper opinion. However, given that Eq.(2) and Eq.(3) remove one degree of freedom each, opinions over a frame of cardinality k only have (2k + k 3) degrees of freedom. The probability projection of hyper opinions is the vector denoted as EX in Eq.(4). as a statement or proposition, so that a frame contains a set of statements. Let X = {x1, x2, . . . , xk} be a frame of cardinality k, where xi (1 i k) represents a specific state. An opinion distributes belief mass over the reduced power-set of the frame denoted as R(X ) defined as:
EX (xi) =
 x j2R(X)
aX (xi/x j) bX (x j) + aX (xi) uX , where xi 2 R(X ) and aX (xi/x j) denotes relative base rate, i.e. the base rate of subset xi relative to the base rate of (partially) overlapping subset x j, defined as follows: (1) (2) (3) (4) (5) aX (xi/x j) = aX (xi \ x j) aX (x j)
, 8 xi, x j 2 R(X ).
Equivalent probabilistic representations of opinions, e.g. as Beta pdf (probability
density function) or a Dirichlet pdf, offer an alternative interpretation of subjective
opinions in terms of traditional statistics [
Projector
Ex Expectation value ax Base rate
bx opinions, but simple visualisations can be used for binomial and multinomial opinions as explained below.
Binomial opinions, which be extensively used in the remainder of the paper, apply to binary frames and have a special notation as described below. Let X = {x, x} be a binary frame, then a binomial opinion about the truth of state x is the ordered quadruple w x = hb, d, u, ai where: b, belief: belief mass in support of x being true; d, disbelief: belief mass in support of x (NOT x); u, uncertainty: uncertainty about probability of x; a, base rate: non-informative prior probability of x.
The special case of Eq.(2) in case of binomial opinions is expressed by Eq.(6). b + d + u = 1.
Ex = b + au.
Similarly, the special case of the probability expectation value of Eq.(4) in case of binomial opinions is expressed by Eq.(7).
A binomial opinion can be visualised as a point inside an equal sided triangle as shown in Figure 1, where the belief, disbelief and uncertainty axes go perpendicularly from each edge to the opposite vertex indicated by bx, dx and ux. The base rate ax shows on the base line, and the probability expectation value Ex is determined by projecting the opinion point to the base line in parallel with the base rate director.
In case the opinion point is located at the left or right corner of the triangle, i.e. with d = 1 or b = 1 and u = 0, the opinion is equivalent to boolean TRUE or FALSE, then SL becomes equivalent to binary logic. Moreover, where b + d = 1 a binomial opinion is equivalent to a traditional probability, where b + d < 1 it expresses degrees of uncertainty, and where b + d = 0 it expresses total uncertainty.
Most operators in Table 1 are generalisations of binary logic and probability operators. For example, addition is simply a generalisation of addition/union of probabilities, (6) (7) Subjective Logic operators
Addition Subtraction Multiplication
Division Comultiplication
Codivision Complement Deduction
Abduction
Transitivity / discounting Cumulative fusion / consensus
Averaging fusion Constraint fusion
Operator notation w xA[ y = w xA + w yA w xA\y = w xA w yA w xA^ y = w xA · w yA w xA^ ¯ y = w xA\w yA w xA_ y = w xA t w yA w xA_ ¯ y = w xA t¯w yA
w xA¯ = ¬w xA
w yAkx = w xA (w yA|x, w yA|x¯)
w yAk¯x w=xAw :BxA=¯ (ww BAyA|⌦x, ww xxAB|y¯, ay)
w xA⇧B = w xA w xB
w xA⇧B = w xA w xB
w xA&B = w xA w xB
while multiplication is conjunction/and. Other operators, e.g., deduction, abduction,
discounting, are not related to logic instead. For the mathematical details of the
operators in Table 1, refer to [
In this section we redefine the constellations approach in probabilistic argumentation
(see Section 5) by using SL instead of plain probability values on arguments and attacks
(as accomplished in the standard definition of the constellations approach instead). All
the results in this section are background information taken from [
tion Framework (AAF) is a pair hA, Ri of a set A of arguments and a binary relation R on A, called attack relation. 8 ai, a j 2 A, R(ai, a j) means that ai attacks a j (R is asymmetric).
A semantics specifies how to derive a set of extensions from an AAF, where an extension B ✓ A is a subset of “collectively” acceptable arguments.
Definition 2 (Semantics [
The acceptance state of a single argument can be conceived in terms of its extension membership.
Definition 3 (Argument acceptance [
A SL-based AAF extends Dung’s argument framework by associating an opinion with each argument and attacks in the original AAF.
mentation framework (slAAF) is a tuple hA , R, OA , ORi where hA , Ri is a Dung’s AAF (Definition 1), OA : A ! W A and OR : R ! W R , where W A and W R respectively are the set of binomial opinions on each argument and each attack.
Hence, given A = {a1, a2, . . . , an}, for each ai 2 A we have that Xai = {ai, ai} represents a binary frame where, with an abuse of notation, ai indicates that “argument ai is trustworthy” and where ai states that “argument ai is not trustworthy”. The same considerations hold for R = {(ai, a j), . . . , (al , ak)}: X(ai,a j) = {(ai, a j), (ai, a j)} represents a binary frame where (ai, a j) indicates “attack (ai, a j) is trustworthy”, and (ai, a j) states “ (ai, a j) is not trustworthy”.3 Therefore, our framework collects a binomial opinion w ai = hb, d, u, ai for each ai 2 A , and w (ai,a j) = hb, d, u, ai for each (ai, a j) 2 R. Remark 1. In this paper, we suppose agents trust an argument if they generically believe in that argument: for instance, if the believe its premises are true, and if they believe the consequence of the claim is logically sound. Therefore, an agent trusts an argument if it believes it is both valid and sound. Indeed this evaluation is subjective: some agents might not catch a statement is a fallacy instead,4 or the fact some of the premises are just false instead of true. Different agents possess different knowledge about the same facts. Similarly, agents can differently judge whether two arguments are in conflict or not, if such arguments do not exactly negate each other: for example, “doing a” or “doing b” in the same time interval are not in conflict if they there is time to do both of them in sequence. 3 Note that i can be equal to j in case we have a self attack R(ai, ai). 4 A fallacy is the use of invalid or otherwise faulty reasoning in the construction of an argument.
For instance, hasty generalization is making assumptions about a whole group or range of cases based on a sample, e.g., “graduated students are nerd”. c w c d w d
In Figure 2 we show an example of slAAF; we use the same example used in [
As a reminder from Section 2, the base rate a is the prior probability of the proposition in the absence of specific belief or disbelief. The default value is the relative atomicity, i.e., 0.5 for a binary state space containing the proposition and its negation. For this reason, a is not reported in Table 2, and quadruples are in the following simplified as triples hb, d, ui. The opinion related to the complement, e.g., w b, is not reported because it can be obtained easily from w b = h0.6, 0.2, 0.2i as h0.2, 0.6, 0.2i, by exchanging belief with disbelief (see complement operator in Table 1).
A slAAF represents the set of all Dung’s classical frameworks that can potentially
be created from it. Similarly to [
Definition 5 (Inducing an AAF from a slAAF). A Dung’s framework AAF = hA, Ri is said to be induced from a slAAF = hA , R, OA , ORi iff the remainder holds: – A ✓ A , – R ✓ (R \ (A ⇥ A)), – 8 a 2 A such that w a = h1, 0, 0i, then a 2 A, – 8 (ai, a j) 2 R such that w (ai,a j) = h1, 0, 0i and w ai = w a j = h1, 0, 0i, then (ai, a j) 2
R.
Moreover, we write I(slAAF) to represent the set of all AAFs that can be induced from a slAAF.
Given Definition 5, an AAF induced from a slAAF contains a subset of the arguments found in the source slAAF, together with a subset of attacks in the slAAF, subject to these defeats containing only arguments found within the induced AAF.
In practice, the process described in Definition 5 splits the uncertainty expresses in a slAAF into constellations (see Section 5) of different possible worlds, each with a different probability. For instance, given the slAAF in Figure 2 and Table 2, then I(slAAF) is equivalent to the following set of four derived frameworks: F1 = h{a, c}, 0/i F3 = h{a, c, d}, {(c, d)}i
F2 = h{a, b, c}, {(b, c)}i
F4 = h{a, b, c, d}, {(b, c), (c, d)}i
This allows us to compute the expectation of some AAF being induced from a
slAAF. Informally, such expectation value can be computed via the joint expectations
of the arguments and attack relations appearing in the considered slAAF. In order to
formalise such a concept compactly, we first need to identify the set of attacks that may
appear in an induced AAF, as accomplished in [
RA = {(ai, a j) | ai, a j 2 A and (ai, a j) 2 R}
Hence, it is possible to compute the expectation of some AAF being induced from a slAAF, as defined in Definition 6. The expectations Eai and E(ai,a j) are computed from the opinions returned by OA (ai) and OR (ai, a j) respectively, for each ai 2 A and (ai, a j) 2 R. As a remainder from Section 2, expectations for binomial opinion is given by b + au from hb, d, ui, with a = 0.5 (see Eq. 7 in Section 2).
Definition 6 (Expectation of an induced AAF). Given slAAF = hA , R, OA , ORi, the expectation of F = hA, Ri 2 I(slAAF) can be computed as in Eq. 8:
EFI = ’ ai2A
Eai ’ ai2(A \A) (1
Eai ) ’ (ai,a j)2R
E(ai,a j)
’ (ai,a j)2(RA \R) (1
E(ai,a j)) (8)
We can then list the expectation value for all the four induced AAFs: EFI1 = 0.105, EFI2 = 0.245, EFI3 = 0.195, EFI4 = 0.455. For example, EFI1 = (1 ⇥ 1) ⇥ ((1 0.7) ⇥ (1 0.65)) = 0.105; no attack is considered in the computation because F1 = h{a, c}, 0/ i.
Hence, also the semantics change over these two AAFs: in F1 the set {a, c} satisfies the grounded and stable semantics (no attack is present), while F2 returns different extensions: stb(F2) = {{a, b}}, and grd(F2) = {a, b}.
Similarly to [
The proof simply derives from exhaustively considering all the possible worlds; in our running example, 0.105 + 0.245 + 0.195 + 0.455 = 1.
We can now define the expectation of some set of arguments satisfying one of the semantics s in the literature, for example the properties introduced in Definition 2, i.e, s 2 {com, prf , stb, grd} (notice that other semantics have been successively in the literature [2, Ch. 2]). For this reason, we define a function v : (s , B, F) ! { f alse, true} that returns true if and only if the set of arguments B represents one of the extensions satisfying s given a framework F: that is, v(s , B, F) is true if and only if B 2 s (F), false otherwise.
Definition 7 (Semantics expectation). Given a slAAF = hA , R, OA , ORi, the expectation that a given set of arguments B 2 A satisfies a semantics s is:
EsI (B, slAAF) =
For instance, the expectation EgIrd({a, c}, slAAF) = 0.105 + 0.195 = 0.3: the set {a, c} represents a grounded extension in F1 and F3, whose expectation is respectively 0.105 and 0.195. EsItb({a, b, d}, slAAF) = 0.455 since the set {a, b, d} is a stable extension only in F4, whose expectation is 0.455.
In the same way, we can compute the expectation of acceptance of an argument w.r.t. I(slAAF) and s : the same argument can be justified/defensible/overruled (i.e., j/d/o, see Definition 3) in multiple generated worlds. We take advantage of a function z : (s , acpt, ai, F) ! { f alse, true}, which returns true if argument ai is accepted as requested (acpt 2 {j, d, o}) in F, given a semantics s .
Definition 8 (Acceptance expectation). Given a slAAF = hA , R, OA , ORi, the expectation that an argument a 2 A is justified/defensible/overruled (acpt 2 {j, d, o}) w.r.t. semantics s is:
EsI ,acpt(a, slAAF) = F3, and F4), while EaIdm,d(c, slAAF) = 0.105 + 0.195 = 0.3: argument c is accepted in F1 (expectation 0.105) and F3 (expectation 0.195).
Note that generating all the possible worlds in the constellations and then enumerate
all the extensions for each of them can lead to computational issues: the number of
worlds exponentially grows in the size of the considered slAAF. Even if the state of the
art of argumentation solvers is quite advanced [
The work in [
In this section we outline a computational framework where we use TNA-SL to
compute the reputation in the attacks and arguments uttered in a public (for a
community) debate, for which we suppose not all of the agents in that community have
had the opportunity to attend to. Alternatively, some agents could have attended but
have not been able to form an opinion because of impediments as, for instance,
cultural differences in the audience, the education level, or cognitive limitations in
general [
We first define Trust Network-based slAAFs.
viated to TN-slAAF) is formed by a slAAF hA , R, OA , ORi (see Definition 4), and a Trust Network represented as hP, T i, where P is the set of agents (we require A \ P = 0/ ) and T is a binary trust relation on P: 8 pi, p j 2 P, T (pi, p j) means that pi trusts p j (T is asymmetric). Moreover, there is a further binary relation N of direct and derived binomial opinions (see Sec. 2), where each element (p, x) 2 N relates an agent p 2 P with x 2 A or x 2 R.
The next definition is used to describe trust paths in a TN-slAAF.
Definition 10 (Trust path in TN-slAAF). Given a TN-slAAF, a trust path is always rooted in p 2 P and either ends in a 2 A or (a, b) 2 R.
In the remainder we will use capital letters A, B, . . . for agent names in P, and
lowercase letters for arguments (i.e., a, b, . . . ). Moreover, when describing trust paths, the
symbol “:” will be used to denote the transitive connection of two consecutive trust arcs
to form a transitive trust path. The “⇧” symbol visually resembles a simple graph of two
parallel paths between a pair of agents [
w ED D
E
w (d,e)
e
w e
slAAF
two parties. Normalisation and simplification are two different control approaches, and
the trust model presented in this paper can take advantage from these techniques as
introduced in [
An example of TN-slAAF is reported in Figure 3: the upper part of the figure represents all the agents in a community, while the lower part shows the considered debate in the form of a slAAF as described in Section 3. A Trust Network is thus tied to an AAF, and the opinions related to arguments and attacks are represented and aggregated in SL. Arguments are detailed in Example 1.
Example 1. We detail the slAAF arguments in Figure 3, taking into consideration a discussion in favour/against the legalisation of Marijuana. A, B, C, D, E in Figure 3 are the audience of a debate concerning this topic.
– a: Official report from rating agencies say the financial crisis dramatically impacted on the overall financial budget. – b: The budget allocated to healthcare for light drugs needs to be increased, because statistics say the number of light drugs users suffering from effects is increasing and treatments are expensive. – c: Marijuana should not be legalised because it would rise healthcare expenses. – d: Marijuana should be legalised because prisons are overcrowded and a large part of prisoners is in custody because they are marijuana users.
In this example, we focus on agent A in Figure 3 (the complete network can be more complex than Figure 3), and we derive an indirect opinion towards argument a along the trust path [A,C] : [C, a], and an indirect opinion towards attack (c, b) along the path ([A, E]) = ([A, B] : [B, D]) ⇧ ([A,C] : [C, D]) : [D, E] : [E, (c, b)].
The discounting [
The effect of discounting in a transitive path is to increase uncertainty, that is to reduce the confidence in the expectation value. The effect of the consensus operator is to reduce uncertainty, that is to increase the confidence in the expectation value. Then, we can compute w aA and w (Ac,b) as
w aA = w CA ⌦ w aC w (Ac,b) = ((w BA ⌦ w DB) (w CA ⌦ w DC)) ⌦ w ED ⌦ w (Ec,b).
Given the beliefs in Table 5, we can compute w aA = h0.28, 0.48, 0.24i and w (Ac,b) = h0.44, 0.09, 0.48i. Finally, these beliefs can be assigned to dashed edges in Figure 3.
As previously advance, we can use SL to aggregate direct and derived beliefs of the same agent, and to aggregate beliefs of different agents. This is visually described in two small TN-slAAF examples, respectively in Figure 4 and in Figure 5. In Figure 4, agent A can aggregate its direct opinion with two derived ones, which are obtained by two trust paths (not shown in the figure) as previously introduced in this section: hence, w aA = w ¯aA w eaA w baA. In Figure 5 the reputation of argument a can be computed via the consensus operator, i.e., w a = w aA w aB w aC; w a can be then directly used in the computational framework presented in Section 3. 5
We revise the literature about probabilistic argumentation (i.e., constellations and
epistemic approach), and also from the point of view of trust sources and systems.
5 More details on this operator, and how to compute it, can be found in [
a wC a a
Fig. 5. An example of multiple opinions from different agents to be aggregated and reach a final reputation for a.
Constellations. In the constellations approach, uncertainty in the topology of the graph
(probabilities on arguments and attacks) is used to make probabilistic assessments on
the acceptance of arguments. The authors of [
Epistemic. In the epistemic approach instead, the topology of the graph is fixed but
probabilistic assessments on the acceptance of arguments are evaluated w.r.t. the
relations of the arguments in the graph. For instance, in [
The aim of this paper is to encompass Trust Network Analysis [
In the future, we would like to extend this study along two different lines. The fist
one concerns the argumentation side of our proposal: for instance, we are interested
in deal with slAAFs from the point of view of the epistemic approach (see Section 5).
The second line concerns the trust analysis of the network among agents. Future goals
are to enrich the framework by taking into consideration ageing factors: agents (and in
particular human agents) may change their behaviour over time, so it is desirable to give
greater weight to more recent ratings using longevity factors. In addition, we would like
to enrich the picture with distrust besides trust, also by exploring other computational
frameworks as [