## 3 Generalized Possibilistic Decision Process  
In the design of practical systems, there are systems that have both nondeterministic choices of actions and possibility distributions of states. We give the notion of the generalized possibilistic decision process as the models for uncertainty systems. Definition 1: A generalized possibilistic decision process (GPDP) [24] is a tuple \(M=(S,I,Act,R,AP,L),\) where \(S\) is a finite, nonempty set of states; \(I:S\rightarrow[0,1]\) is the possibilistic initial distribution function; \(Act\) is a set of actions; \(R:S\times Act\times S\rightarrow[0,1]\) is the possibilistic transition distribution function, and for every state \(s\in S\) and every action \(\alpha\in Act,\) there is a state \(s^{\prime}\) such that \(R(s,\alpha,s^{\prime})>0;\) AP is a set of atomic propositions; \(L:S\times AP\rightarrow[0,1]\) is a labeling function, \(L(s,a)\) denotes the possibility that the atomic proposition a holds on state \(s.\)_If \(S\) and \(AP\) are both finite, we call \(M\) a finite GPDP. \(R(s,\alpha,t)\) represents the possibility that the system evolves from state \(s\) into state \(t\) by action \(\alpha\). If \(R(s,\alpha,t)>0\), state \(t\) is called the successor of state \(s\), and state \(s\) is the predecessor of state \(t\). The set of direct \(\alpha\)-successors of \(s\) is defined as: \(\text{Post}(s,\alpha)=\left\{t\in S\big{|}R(s,\alpha,t)>0\right\}\). The set of predecessors of \(t\) is defined by: \(\text{Pre}(t)=\left\{s\in S\big{|}R(s,\alpha,t)>0\right\}\). The set of actions that state \(s\) can trigger is defined as: \(\text{Act}(s)=\left\{\alpha\in\text{Act}\right|\exists s^{\prime}\in S,R(s, \alpha,s^{\prime})>0\right\}\)._  
_The paths in the GPDP \(M\) is denoted as \(\pi\), and \(\pi=s_{0}\alpha_{0}s_{1}\alpha_{1}s_{2}\cdots\text{Paths}(s)\) and \(\text{Paths}_{fin}(s)\) denote the set of infinite paths and the set of finite paths starting from state \(s\), respectively. \(\text{Paths}(M)\) and \(\text{Paths}_{fin}(M)\) represent the set of all infinite paths and the set of all finite paths in \(M\), respectively._  
_In \(M=(S,I,Act,R,AP,L)\), for any \(\alpha\in Act\), we can use the fuzzy matrix to express the possibilistic transition distribution function \(R:S\times\alpha\times S\rightarrow\left[0,1\right],\) which is denoted as \(R_{\alpha}\),\(R_{\alpha}=\left(R(s,\alpha,s^{\prime})\right)_{s,s^{\prime}\in S}\), in which \(R_{\alpha}\) is the fuzzy transition matrix corresponding to action \(\alpha\) in \(M.\) The maximum possibilistic transition matrix is expressed as \(R_{max}=\vee_{i=0}^{\pi}R_{ai}\), which is also expressed by_  
\[\left(R_{max}(s,t)\right)_{s,t\in S}=\left(\underset{\alpha\in Act(s)}{\vee} R(s,\alpha,t)\right)_{s,t\in S}.\]  
_The minimum possibilistic transition matrix is expressed as \(R_{min}=\wedge_{i=0}^{n}R_{ai}\), that is,_  
\[\left(R_{min}(s,t)\right)_{s,t\in S}=\left(\underset{\alpha\in Act(s)}{\wedge }R(s,\alpha,t)\right)_{s,t\in S}.\]  
_The transition matrix closure of a fuzzy matrix \(R\) is denoted as \(R^{+}\), \(R^{+}=R\lor R^{2}\lor R^{3}\vee\cdots R^{\left[S\right]}\), where \(R^{k+1}=R^{k}\circ R\). The reflexive transition closure of the fuzzy matrix \(R\) is denoted as \(R^{*}\), \(R^{*}=R^{0}\lor R^{+}\), where \(R^{0}\) denotes the identity matrix._  
_Let us take the generalized possibilistic decision process (GPDP) in Figure 1 as an example. The set of system states is denoted as \(S=\{s_{0},s_{1},s_{2},s_{3}\}\); \(I=\{s_{0}\}\) represents the initial state of the system; \(Act=\{\alpha,\beta\}\) denotes the set of actions; \(R(s_{0},\beta,s_{1})=0.72\) denotes that the possibility of the system changing from state \(s_{0}\) to \(s_{1}\) by action \(\beta\) is 0.72; the set of atomic propositions is \(AP=\{a,b,c\}\).