## 5 Trust Model  
We have used advanced sensing logic framework defined in preliminary section as our trust model. Following are the major definitions in sensing logic. Definition 1: \"Trust representation\" An opinion metric in sensing logic can be represented as \(T=[B,D,N,E]\), where \((B,D,N,E)\in[0,1]\) and \(B+D+N+E=1\) also \(B,D\) showing probability of belief and disbelief and \(N,E\) correspond posterior and prior uncertainty of said event. An opinion metric \(T_{1}=[0.7,0.2,0,0.1]\) and \(T_{2}=[0.4,0.5,0,0.1]\), shows high and low trust values respectively. Definition 2: \"Mapping\" Let's \(T_{Y}^{X}=[B,D,N,E]\) be vehicle \(Y\)'s opinion about vehicle \(X\)'s trustworthiness in a VANET, and let \(r,s\) and \(o\) denote the observed piece of evidence that a vehicle is reliable, fake or neutral. Using Equation (5), given in preliminary section, we can map evidence space to opinion spaces. Definition 3: \"Trust Combination\" In our trust model, A vehicle will make a relative judgment about the neighbor vehicle by means of first and second hand observation. First hand observation comes from direct or self experience and second hand observation comes as advise by other neighbors or friends. These two observation are combined through consensus and discounting operations, and a final opinion is computed towards target vehicle. Consensus combination: Let \(S_{X}\), \(S_{Y}\) and \(S_{Z}\) be three vehicles. Then \(T_{Y}^{X}=[B_{Y}^{X},D_{Y}^{X},N_{Y}^{X},E_{Y}^{X}]\) and \(T_{Z}^{X}=[B_{Z}^{X},D_{Z}^{X},N_{Z}^{X},E_{Z}^{X}]\), shows opinions of vehicle \(S_{Y}\) and \(S_{Z}\) about truthfulness of vehicle \(S_{X}\). Their consensus opinion space is defined as  
\[ \begin{split} B_{Y,Z}^{X}=\frac{E_{Z}^{X}B_{Y}^{X}+E_{Y}^{X}B_{Z}^ {X}}{E_{Y}^{X}+E_{Z}^{X}-E_{Y}^{X}E_{Z}^{X}},& D_{Y,Z}^{X}=\frac{E_{Z}^{X}D_{Y}^ {X}+E_{Y}^{X}D_{Z}^{X}}{E_{Y}^{X}+E_{Z}^{X}-E_{Y}^{X}E_{Z}^{X}}\\ N_{Y,Z}^{X}=\frac{E_{Z}^{X}N_{Y}^{X}+E_{Y}^{X}N_{Z}^{X}}{E_{Y}^{X}+E_ {Z}^{X}-E_{Y}^{X}E_{Z}^{X}},& E_{Y,Z}^{X}=\frac{E_{Y}^{X}E_{Z}^{X }}{E_{Y}^{X}+E_{Z}^{X}-E_{Y}^{X}E_{Z}^{X}}\end{split}  \tag{9}\]  
The trust value using subjective opinions with consensus combining provide more flexible trust model of the real world. By referring to Equation (9), the consensus of trust opinions generated by vehicles \(s_{i=1}^{n_{j}}\) in time interval \(t\) about vehicle \(s_{j}\) is  
\[T_{1}^{j,t}\oplus\ldots\oplus T_{i}^{j,t}\oplus\ldots\oplus T_{n_{j}}^{j,t}=T_ {1,\ldots,j,\ldots,n_{j}}^{j,t}\]  
Discounting combination: Let \(S_{X}\), \(S_{Y}\), and \(S_{Z}\) be three vehicles. Then, we can present algebraically as \(T_{X}^{Y}=[B_{X}^{Y},D_{X}^{Y},N_{X}^{Y},E_{X}^{Y}]\) and \(T_{Y}^{Z}=[B_{Y}^{Z},D_{Y}^{Z},N_{Y}^{Z},E_{Y}^{Z}]\) shows opinions of \(S_{X}\) about \(S_{Y}\) trustworthiness and \(S_{Y}\) about truthfulness of vehicle \(S_{Z}\). Their discounting opinion is defined as \(T_{X,Y}^{Z}=T_{X}^{Y}\otimes T_{Y}^{Z}=[B_{X,Y}^{Z},D_{X,Y}^{Z},N_{X,Y}^{Z},E _{X,Y}^{Z}]\), shows \(S_{X}\) opinion on vehicle \(S_{Z}\) as advised by vehicle \(S_{Y}\). \[ \begin{split} B_{X,Y}^{Z}=B_{X}^{Y}B_{Y}^{Z},\quad D_{X,Y}^{Z}=B_{ X}^{Y}D_{Y}^{Z}\\ N_{X,Y}^{Z}=1-B_{X,Y}^{Z}-D_{X,Y}^{Z}-E_{Y}^{Z},\quad E_{X,Y}^{Z}=E _{Y}^{Z}\end{split}  \tag{10}\]The discounting operation is used along a recommendation path of multiple vehicles about the particular one. Definition 4: Let \(S_{X}\) and \(S_{Y}\) be two vehicles. Then \([T_{Y}^{X,t_{1}},\ldots,T_{Y}^{X,t_{n}}]\) shows opinions of vehicle \(S_{Y}\) about trustworthiness of \(S_{X}\) for time intervals \([t_{1},\ldots,t_{n}]\) respectively, where \(T_{Y}^{X,t_{n}}=[B_{Y}^{X,t_{n}},D_{Y}^{X,t_{n}},N_{Y}^{X,t_{n}},E_{Y}^{X,t_{n}}]\). Vehicle \(S_{Y}\)'s opinion on \(S_{X}\) trustworthiness in different time intervals can be combined as \([t_{1},\cup\ldots\cup,t_{n}]\) is defined as  
\[