From Trust Among Agents to Reputation of Abstract Arguments by Using Subjective Logic Francesco Santini Department of Mathematics and Computer Science, University of Perugia, Italy francesco.santini@dmi.unipg.it Abstract. 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 ex- planation 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. 1 Introduction An Abstract Argumentation Framework (AAF) [10] is an abstract structure consisting of a set arguments, whose origin, nature and possible internal organisation is not spec- ified, and by a binary relation of attack on the set of arguments, whose meaning is not specified either: that is, an AAF can be represented as a pair hA , Ri, which in turn can be represented as a directed graph where nodes are arguments and a ! b if (a, b) 2 R. As a classical example, argument a may stand for “Tomorrow will rain be- cause the national weather forecast says so”, while argument b for “Tomorrow will not rain because the regional weather forecast says so”; the corresponding framework is hA = {a, b}, R = {(a, b), (b, a)}i. Given a framework, it is possible to examine the question on which set(s) of argu- ments can be accepted, hence collectively surviving the conflict defined by R. Answer- ing this question corresponds to defining an argumentation semantics [10]. Considering the previous example, either {a} or {b} alone can be accepted, while {a, b} cannot be accepted because of the internal conflict. Subjective Logic (SL) [16] is a calculus for subjective opinions which in turn repre- sent probabilities affected by degrees of uncertainty. In general, SL is suitable for mod- elling and analysing situations involving uncertainty and relatively unreliable sources. A subjective opinion can express trust in a source or it can express belief about events and propositions. A binomial opinion applies to a binary state variable, and can be rep- resented as a Beta PDF (Probability Density Function) [16]. A multinomial opinion ap- plies to a state variable of multiple possible values, and can be represented as a Dirichlet PDF [16]. SL has been already used for modelling subjective trust networks [17] and structured Argumentation [20]. Copyright c 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). Since arguments are often uncertain, it can be useful to quantify the uncertainty as- sociated with each argument, as previously explored in other works in the literature [14, 13, 19, 26]. Do we believe more in national or regional weather forecast? How much are we certain about our belief? For this reason, we define Subjective Logic-based AAFs (slAAFs), where both arguments and attacks are associated with a binomial opinion defined in SL, i.e., described in terms of belief, disbelief, and uncertainty values, i.e., hb, d, ui. As shown in [24], slAAFs can be straightforwardly reconnected to the constel- lations approach proposed in [19], but information is more granular due to the fact that a probability value can be derived from a triple hb, d, ui. A dogmatic opinion, that is with u = 0, is equivalent to probabilities. An absolute opinion, that is b = 1, is equivalent to true. A vacuous opinion, that is u = 1 is equivalent to undefined. 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 [24], different opinions related to arguments and attacks between arguments come from different agents. Consequently, SL operators can be used to aggregate these subjective opinions together in a resulting opinion, which describes the belief/disbelief/uncertainty of the whole group of agents. This represents the reputation of an argument (or attack) in the considered community, which consist of individuals bounds together by social relationships. This reputation comes from all the direct subjective-opinions of agents, but also from (indirect) opinions of other agents in the same community, by considering transitive trust-relationships: if A trusts B who strongly believes in argument a (i.e., high belief and low uncertainty rating), then the direct opinion waA can be aggregated with waB through the opinion of A towards B: wBA . If A has no opinion about a, then she can make one as just explained. By aggregating the beliefs of all the agents w.r.t. the same argument/attack, e.g., waA -waB -waC , then we compute the reputation of a. The same example can be rephrased by computing the reputation towards an attack from a to b, A -w B -w C . Finally, these opinions can be used in the same way as i.e., (a, b): w(a,b) (a,b) (a,b) in the constellations approach (Section 5), for instance to find the community-biased expectation that a set of arguments satisfies a given semantics. This work extends the results in [24] by introducing Trust Network-based slAAF as a way to connect trust in agents with trust in arguments and attacks. Concisely, the paper links a trust network among agents with the belief the same agents have in the components of the considered AAF. The paper is organised as follows: in Section 2 we summarise the background no- tions 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 us- ing 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 Subjective Logic 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 66 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 de- noted as R(X) defined as: R(X) = P(X) \ {X, 0} / , (1) 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 [12]. R(X) has cardinality k = 2k 2. 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 [0, 1]. An opinion satisfies the following additivity requirements. Belief additivity: uX + Â b X (xi ) = 1. (2) xi 2R(X) k Base rate additivity: Â a X (xi ) = 1, where xi 2 X. (3) i=1 A subjective (hyper) opinion of user A over the frame X is denoted as wXA = (bbX , uX , a X ), where b X is a belief vector over the states of R(X), uX is the complementary uncertainty mass, and a X is a base rate vector over X, all seen from the viewpoint of belief owner A. The belief vector b X has (2k 2) parameters, whereas the base rate vector a X 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). EX (xi ) = Â a X (xi /x j ) b X (x j ) + a X (xi ) uX , (4) x j 2R(X) where xi 2 R(X) and a X (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: a X (xi \ x j ) aX (xi /x j ) = , 8 xi , x j 2 R(X). (5) a X (x j ) 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 [18]. There is no simple visualisation of hyper 1 A belief vector b specifies the distribution of belief masses over the elements of R(X). 2 Base rate generally refers to the (base) class probabilities unconditioned on featural evidence, frequently also known as prior probabilities. The concept of base rates is central in the theory of probability. Base rates are for example useful for default and for conditional reasoning. 67 ux Opinion Zx Projector dx bx Ex ax Expectation value Base rate Fig. 1. Binomial opinion point in triangle. 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 wx = 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. (6) Similarly, the special case of the probability expectation value of Eq.(4) in case of bi- nomial opinions is expressed by Eq.(7). Ex = b + au. (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 opera- tors. For example, addition is simply a generalisation of addition/union of probabilities, 68 Subjective Logic operators Operator notation Addition A = wA + wA wx[y x y Subtraction A = wA wA wx\y x y Multiplication A = wA · wA wx^y x y Division wxA^y A ¯ = wx \wy A Comultiplication wx_y = wx t wyA A A Codivision wxA_y A ¯ = wx t̄wy A Complement wx̄A = ¬wxA Deduction A = w A (w A , w A ) wykx x y|x y|x̄ Abduction wyk̄x = wxA ¯ (wy|x A A , wA , a ) x|ȳ y Transitivity / discounting wxA:B = wBA ⌦ wxB Cumulative fusion / consensus wxA⇧B = wxA wxB Averaging fusion wxA⇧B = wxA wxB Constraint fusion wxA&B = wxA wxB Table 1. Some SL operators. For a more detailed explanation of them refer to [16]. The superscripts A and B are attributes that identify the respective belief sources or belief owners (e.g., two agents uttering arguments); x is a state in the considered frame of discernment (e.g., an argument or an attack). while multiplication is conjunction/and. Other operators, e.g., deduction, abduction, discounting, are not related to logic instead. For the mathematical details of the opera- tors in Table 1, refer to [16]. Some of the operators are only meaningful for combining binomial opinions, but some also apply to multinomial opinions. Most of the operators in Table 1 are binary, but complement is unary, deduction is ternary and abduction is quaternary. 3 SL-based Abstract Argumentation Frameworks 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 [10] (for what con- cerning AAFs) and [24] (concerning slAAFs). We start by recalling the classical defi- nitions behind AAFs: Definition 1 (Abstract Argumentation Frameworks [10]). An Abstract Argumenta- tion Framework (AAF) is a pair hA, Ri of a set A of arguments and a binary relation R on A, called attack relation. 8ai , a j 2 A, R(ai , a j ) means that ai attacks a j (R is asym- metric). 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 [10]). Let F = hA, Ri be an AAF. A set B ✓ A is conflict-free, denoted B 2 cf (F), iff there are no a, b 2 B, such that R(a, b). An argument a 2 A is 69 defended by a set B ✓ A if for each b 2 A, such that R(b, a), there is c 2 B s.t. R(c, b). A conflict-free set is also admissible, that is S 2 adm(F), if each a 2 B is defended by B. Given a conflict-free B, the semantics originally defined in [10] are: complete: B 2 com(F), if B 2 adm(F) and for each a 2 A defended by B, a 2 B holds; preferred: B 2 prf (F), if B 2 adm(F) and there is no C 2 adm(F) with B ⇢ C; stable: B 2 stb(F), if for each a 2 A\B, 9b 2 B s.t. R(b, a); grounded: B = grd(F) if B 2 com(F) and there is no C 2 com(F) with C ⇢ B. The acceptance state of a single argument can be conceived in terms of its extension membership. Definition 3 (Argument acceptance [23]). Given one of the semantics s 2 {com, stb, prf } and a framework F, an argument a is i) justified iff 8B 2 s (F), a 2 B, ii) a is defensible if 9B 2 s (F), a 2 B and a is not justified, iii) a is overruled iff 6 9B 2 s (F), a 2 B. A SL-based AAF extends Dung’s argument framework by associating an opinion with each argument and attacks in the original AAF. Definition 4 (SL-based Argumentation Frameworks). A SL-based Abstract Argu- mentation framework (slAAF) is a tuple hA , R, OA , OR i where hA , Ri is a Dung’s AAF (Definition 1), OA : A ! WA and OR : R ! WR , where WA and WR respec- tively 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 } rep- resents 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 con- siderations 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 wai = 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(a , a ). i i 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”. 70 w(b,c) w(c,d) a b c d wa wb wc wd Fig. 2. An example of Subjective Logic-based Abstract Argumentation Framework (slAAF). We use the same example proposed in [19] in order to better show the dif- ferences between the two approaches (i.e., probability values and SL). Table 2. Opinons for the slAAF in Figure 3. Table 3. Opinions on arguments. Table 4. Opinions on attacks. opinion b d u wa 1 0 0 opinion b d u wb 0.6 0.2 0.2 w(b,c) 1 0 0 wc 1 0 0 w(c,d) 1 0 0 wd 0.5 0.2 0.3 In Figure 2 we show an example of slAAF; we use the same example used in [19], in order to better show the differences between the original constellations approach in [19], and by using SL instead. In Table 2 we provide the values for the tuples wai = hb, d, u, ai and ww(a ,a ) = hb, d, u, ai with respect to the AAF in Figure 2. i j As a reminder from Section 2, the base rate a is the prior probability of the proposi- tion in the absence of specific belief or disbelief. The default value is the relative atom- icity, 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., wb , is not reported be- cause it can be obtained easily from wb = 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 [19], we call this creation process the inducement of an AAF from a slAAF. All arguments and attacks with a probability expectation of 1 will be found in the induced AAF, which can also contain additional arguments and attacks, as specified in Definition 5. 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 , OR i iff the remainder holds: – A✓A, – R ✓ (R \ (A ⇥ A)), – 8a 2 A such that wa = h1, 0, 0i, then a 2 A, – 8(ai , a j ) 2 R such that w(ai ,a j ) = h1, 0, 0i and wai = wa 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. 71 Given Definition 5, an AAF induced from a slAAF contains a subset of the argu- ments 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}, 0i / F2 = h{a, b, c}, {(b, c)}i F3 = h{a, c, d}, {(c, d)}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 [19]. We call this set RA : 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 , OR i, the expectation of F = hA, Ri 2 I(slAAF) can be computed as in Eq. 8: EFI = ’ Eai ’ (1 Eai ) ’ E(ai ,a j ) ’ (1 E(ai ,a j ) ) (8) ai 2A ai 2(A \A) (ai ,a j )2R (ai ,a j )2(RA \R) We can then list the expectation value for all the four induced AAFs: EFI 1 = 0.105, EF2 = 0.245, EFI 3 = 0.195, EFI 4 = 0.455. For example, EFI 1 = (1 ⇥ 1) ⇥ ((1 0.7) ⇥ (1 I 0.65)) = 0.105; no attack is considered in the computation because F1 = h{a, c}, 0i./ 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 [19], we can derive the following property: Proposition 1. The sum of all the expectation values of all the AAFs that can be in- duced from a slAAF is 1: Â EFI i = 1 Fi 2I(slAAF) . 72 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 , OR i, the expec- tation that a given set of arguments B 2 A satisfies a semantics s is: EsI (B, slAAF) = Â EFI i where v(s , B, Fi ) = true (9) Fi 2I(slAAF) For instance, the expectation Egrd I ({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. Estb I ({a, b, d}, slAAF) = 0.455 since the set {a, b, d} is a stable exten- sion 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 , OR i, the expec- tation 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) = Â EFI i where z(s , acpt, ai , Fi ) = true (10) Fi 2I(slAAF) I For instance, the expectation Eadm,j (a, slAAF) = 1 (argument a is accepted in F1 , F2 , I F3 , and F4 ), while Eadm,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 [4], the exact expectation value need to be approximated [19]. 4 From Trust Between Agents to Belief in Arguments The work in [15] describes a method for trust network analysis using subjective logic (TNA-SL). It provides a simple notation for expressing transitive trust relationships, and 73 defines a method for simplifying complex trust networks so that they can be expressed in a concise form and be computationally analysed. Trust measures are expressed as beliefs, and Subjective Logic operators are used to compute trust between arbitrary parties in the network. 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 com- munity) 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, cul- tural differences in the audience, the education level, or cognitive limitations in gen- eral [25]. Hence, some of the agents form their derived opinion from friends and ac- quaintances by using trust relationships (derived opinions represent recommendations). At the same time, also agents who have a direct opinion are influenced by other parties they know [6]. The proposed approach follows these steps: 1. direct and indirect opinions of the same agent can be aggregated in order to produce a single belief for the same argument or attack, and 2. aggregated opinions of single agents can be further aggregated with the purpose to produce a reputation for an argument or attack, which reflects the belief of all the considered community; 3. finally, the obtained slAAF, where each argument and attack is weighed with an opinion as derived from items 1 and 2, can be studied by using the constellations approach proposed in Section 3. We first define Trust Network-based slAAFs. Definition 9 (Trust Network-based slAAF). A Trust Network-based slAAF (abbre- viated to TN-slAAF) is formed by a slAAF hA , R, OA , OR i (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: 8pi , 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 lower- case 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 [15]. With no restrictions on the possible trust arcs, trust paths from a given source X to a given target y can contain cycles, which could result in inconsistent calculative results. Cycles in the trust graph must therefore be controlled when applying calculative methods to derive measures of trust between 74 wBA wDB B wED wCA wDC A D E E w(c,b) A w(c,b) waA C waC w(d,c) w(d,e) a b c d e wa w(a,b) wb w(c,b) wc wd we w(c,d) slAAF Fig. 3. An example of Trust Network-based slAAF, with a community of agents inter- acting on it through trust paths. Dashed edges represent derived opinions. 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 [15]. For the sake of brevity, we point the reader to [15] for a more ex- haustive explanation. An example of TN-slAAF is reported in Figure 3: the upper part of the figure rep- resents 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. 75 Table 5. Some opinions for the TN-slAAF in Figure 3. wBA wCA wDB wDC wED waC E w(c,b) h0.9, 0, 0.1i h0.9, 0, 0.1i h0.9, 0, 0.1i h0.9, 0, 0.1i h0.3, 0, 0.7i h0.3, 0.5, 0.2i h0.5, 0.1, 0.4i – e: To overcome prison overcrowding, most of the people think new prisons need to be built. Then, we should do that. 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 [16] operator wxA:B = wBA ⌦ wxB in Table 1 can be used to compute transitive trust along a trust path, while the consensus [16] operator wxA⇧B = wxA wxB in Table 1 can be used to fuse two beliefs into one, thus composing parallel paths together. Formally, wxA:B = hbAB bBx , dBA dxB , dBA + uAB + bAB uBx i, and wxA⇧B = hbAB bBx , dBA dxB , dBA + uAB + bAB uBx i. With the cumulative fusion operator, i.e. ⌦, the observations are supposed as independent; the cumulative rule is equivalent to a posteriori updating of Dirichlet dis- tributions.5 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 waA and w(c,b) A as waA = wCA ⌦ waC A w(c,b) = ((wBA ⌦ wDB ) (wCA ⌦ wDC )) ⌦ wED ⌦ w(c,b) E . Given the beliefs in Table 5, we can compute waA = h0.28, 0.48, 0.24i and w(c,b)A = 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, waA = w̄aA w eaA w baA . In Figure 5 the reputation of argument a can be computed via the consensus operator, i.e., wa = waA waB waC ; wa can be then directly used in the computational framework presented in Section 3. 5 Related Work We revise the literature about probabilistic argumentation (i.e., constellations and epis- temic 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 [16]. 76 baA w wAa A wBa a a A eaA w B wCa w̄aA C Fig. 4. An example of one direct and Fig. 5. An example of multiple opin- two derived opinions (through one trust ions from different agents to be aggre- path each). gated 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 [11] provided the first proposal to extend abstract argumentation with a probability distribution over sets of arguments which they use with a version of assumption-based argumentation in which a subset of the rules are probabilistic rules. In [19] a probability distribution over the sub-graphs of the argument graph is introduced, and this can then be used to give a probability assignment for a set of arguments being an admissible set or extension of the argument graph. In [7] the authors characterise the different semantics from the approach of [19] in terms of probabilistic logic with the purpose of providing an uniform logical formalisation and also pave the way for future implementations. 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 rela- tions of the arguments in the graph. For instance, in [3] the authors cast epistemic prob- abilities in the context of de Finetti’s theory of subjective probability, and they analyse and revise the relevant rationality properties in relation with de Finettis notion of co- herence. However, most of the work in this directions is authored by M. Thimm [26] and A. Hunter [14]. In the first work, the authors proposes a probabilistic semantics for Abstract Argumentation is proposed in order to assign probabilities or degrees of be- lief to individual arguments. The presented semantics generalise the classical notions of semantics [10]. In the second work, the author starts from considering logic-based ar- gumentation with uncertain arguments, but ends showing how this formalisation relates to uncertainty of abstract arguments. The two authors join their efforts in [13]. Trust and Argumentation. Trust and Argumentation are two strictly related concepts, as the florid literature of the last years proves. In [22] the authors investigate the com- bination of trust measures on agents and the use of argumentation for reasoning about belief, thus combining an existing system for reasoning about trust and an existing sys- tem of argumentation. In [1] the authors study how the different arguments interact and how an agent may decide to trust another source and thus to accept information coming 77 from that source. The system also deals with graded trust (like agent i trusts to some extent agent j). Trust of sources is also studied in [27], together with a model to trust in a trustworthy way. In [21] the authors identify two types of argumentative relevance: internal relevance, i.e. the extent to which a premise has a bearing on its purported con- clusion (thus considering structured arguments), and external relevance, i.e. a measure of how much a whole argument is pertinent to the matter under discussion. Two more works on Trust and Argumentation are [8] and [9]. 6 Conclusion The aim of this paper is to encompass Trust Network Analysis [15] and to irradiate the effect of direct and derived trust among the agents in a network towards a Subjective Logic-based AAF. Hence, entities can form their opinion by considering their direct be- lief, and the beliefs of parties through trust paths linking them together. In addition, all these subjective opinions can be fused into a reputation score related to each argument and attack. Such a score represents how much the studied community of agents evalu- ates their credibility. Finally, the resulting slAAF can be studied using the constellations approach as in related works [19]. 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 [5]. References 1. Amgoud, L., Demolombe, R.: An argumentation-based approach for reasoning about trust in information sources. Argument & Computation 5(2-3), 191–215 (2014) 2. Arrow, K.J., Sen, A., Suzumura, K.: Handbook of Social Choice and Welfare. North-Holland (2002) 3. Baroni, P., Giacomin, M., Vicig, P.: On rationality conditions for epistemic probabilities in abstract argumentation. In: Computational Models of Argument - Proceedings of COMMA. FAIA, vol. 266, pp. 121–132. IOS Press (2014) 4. Bistarelli, S., Ross, F., Santini, F.: Not only size, but also shape counts: abstract argumenta- tion solvers are benchmark-sensitive. J. Log. Comput. 28(1), 85–117 (2018) 5. Bistarelli, S., Santini, F.: On merging two trust-networks in one with bipolar preferences. Mathematical Structures in Computer Science 27(2), 215–233 (2017) 6. Campbell-Meiklejohn, D.K., Bach, D.R., Roepstorff, A., Dolan, R.J., Frith, C.D.: How the opinion of others affects our valuation of objects. Current Biology 20(13), 1165–1170 (2010) 7. Doder, D., Woltran, S.: Probabilistic argumentation frameworks - A logical approach. In: Scalable Uncertainty Management - 8th International Conference, SUM. Lecture Notes in Computer Science, vol. 8720, pp. 134–147. Springer (2014) 78 8. Dondio, P., Longo, L.: Trust-based techniques for collective intelligence in social search systems. In: Next Generation Data Technologies for Collective Computational Intelligence, Studies in Computational Intelligence, vol. 352, pp. 113–135. Springer (2011) 9. Dondio, P., Longo, L.: Computing trust as a form of presumptive reasoning. In: 2014 IEEE/WIC/ACM International Joint Conferences on Web Intelligence (WI) and Intelligent Agent Technologies (IAT. pp. 274–281. IEEE Computer Society (2014) 10. Dung, P.M.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence 77(2), 321–357 (1995) 11. Dung, P.M., Thang, P.M.: Towards (probabilistic) argumentation for jury-based dispute res- olution. In: Computational Models of Argument: Proceedings of COMMA. FAIA, vol. 216, pp. 171–182. IOS Press (2010) 12. Hankin, R.K.: A Generalization of the Dirichlet Distribution. Journal of Statistical Software 33(11), 1–18 (February 2010) 13. Hunter, A., Thimm, M.: Probabilistic reasoning with abstract argumentation frameworks. J. Artif. Intell. Res. 59, 565–611 (2017) 14. Hunter, A.: A probabilistic approach to modelling uncertain logical arguments. Int. J. Ap- prox. Reasoning 54(1), 47–81 (2013) 15. Jøsang, A., Hayward, R., Pope, S.: Trust network analysis with subjective logic. In: Com- puter Science 2006, Twenty-Nineth Australasian Computer Science Conference (ACSC). CRPIT, vol. 48, pp. 85–94. Australian Computer Society (2006) 16. Jøsang, A.: Subjective Logic - A Formalism for Reasoning Under Uncertainty. Artificial Intelligence: Foundations, Theory, and Algorithms, Springer (2016) 17. Jøsang, A., Bhuiyan, T.: Optimal trust network analysis with subjective logic. In: Proceed- ings of the Second International Conference on Emerging Security Information, Systems and Technologies, SECURWARE. pp. 179–184. IEEE Computer Society (2008) 18. Jøsang, A., Hankin, R.K.: Interpretation and Fusion of Hyper Opinions in Subjective Logic. In: Proceedings of the 15th International Conference on Information Fusion (FUSION 2012) (2012) 19. Li, H., Oren, N., Norman, T.J.: Probabilistic argumentation frameworks. In: Theorie and Applications of Formal Argumentation - First International Workshop, TAFA. LNCS, vol. 7132, pp. 1–16. Springer (2011) 20. Oren, N., Norman, T.J., Preece, A.D.: Subjective logic and arguing with evidence. Artif. Intell. 171(10-15), 838–854 (2007) 21. Paglieri, F., Castelfranchi, C.: Trust, relevance, and arguments. Argument & Computation 5(2-3), 216–236 (2014) 22. Parsons, S., T., Y., Sklar, E., McBurney, P., Cai, K.: Argumentation-based reasoning in agents with varying degrees of trust. In: 10th International Conference on Autonomous Agents and Multiagent Systems (AAMAS). pp. 879–886. IFAAMAS (2011) 23. Prakken, H., Vreeswijk, G.: Logics for defeasible argumentation. In: Handbook of philo- sophical logic, pp. 219–318. Springer (2002) 24. Santini, F., Jøsang, A., Pini, M.S.: Are my arguments trustworthy? abstract argumentation with subjective logic. In: 21st International Conference on Information Fusion, FUSION. pp. 1982–1989. IEEE (2018) 25. Tajfel, H.: Social and cultural factors in perception. Handbook of social psychology 3, 315– 394 (1969) 26. Thimm, M.: A probabilistic semantics for abstract argumentation. In: ECAI - 20th European Conference on Artificial Intelligence. FAIA, vol. 242, pp. 750–755. IOS Press (2012) 27. Villata, S., Boella, G., Gabbay, D.M., van der Torre, L.W.N.: A socio-cognitive model of trust using argumentation theory. Int. J. Approx. Reasoning 54(4), 541–559 (2013) 79