# A Program Model of Fuzzy Interpreted Petri Net to Control Discrete Event Systems
## 3 The Formal Basis and the Conception of FIPN  
Three definitions describe the formal basis of FIPN. The first shows the construction of the net. **Definition 1**. : _The fuzzy interpreted Petri net is the system [51]:_  
\[\text{FIPN}=(P,\,T,\,\Omega,\,\Psi,\,R,\,\Delta,\,K,\,W,\,\Gamma,\,\Theta,M_{0}, \,e),\]  
_where: \(P=P^{\prime}\cup P^{\prime\prime}\)--is a nonempty finite set of places, where: \(P^{\prime}=\{{p^{\prime}}_{1},{p^{\prime}}_{2},\dots,{p^{\prime}}_{a^{\prime}}\}\)--is a set of places for processes modelling, and_  
\(P^{\prime\prime}=\{{p^{\prime\prime}}_{1},{p^{\prime\prime}}_{2},\dots,{p^{ \prime\prime}}_{a^{\prime\prime}}\}\)--_a set of places for resources modelling;_  
\(T=\{t_{1},t_{2},\dots,t_{b}\}\)--is a nonempty finite set of transitions;  
\(\Omega\)= \(\{\omega_{1},\omega_{2},\dots,\omega_{a^{\prime}+a^{\prime\prime}}\}\)--is a nonempty finite set of statements;  
\(\Psi=\{\psi_{1},\psi_{2},\dots,\psi_{b}\}\)--is a nonempty finite set of conditions;  
\(P,\,T,\,\Omega,\,\Psi\)--where none of these sets have common elements;  
\(R\subseteq(P\times T)\cup(T\times P)\)--is the incidence relation that assigns a place to each transition \(t_{i}\in T(1\leq i\leq b)\),  
_where there is the place \(p^{\prime}\in P^{\prime}\) such that \((p^{\prime},t_{i})\in R\) or \((t_{i},p^{\prime})\in R\) ;_  
\(\Delta\)_: \(P\to\Omega\)--is the function that assigns a statement to each place;_  
\(K:P^{\prime}\to 1\) _and \(P^{\prime\prime}\to\aleph\{1\}\)--is the function that assigns a capacity to each place,_  
_where: \(\aleph=\{1,2,\,\dots\}\);_  
\(\Gamma\)_: \(T\,\Psi\)--is the function that assigns a condition to each transition;_  
\(\Theta\)_: \(T\to[0,\,1]\)--is the function that defines the degree to which the conditions corresponding to the transitions \(t\) are satisfied;_  
_W: \(R\to\aleph\)--is the weight function that meets two conditions,_  
_where: \(W(p,t)\leq K(p)\), and_  
\(W(t,p)\leq K(p)\) _(_\(p\) _means_ \(p^{\prime}\) _or_ \(p^{\prime\prime}\)_);_  
\(M_{0}:P^{\prime}\to\{0,1\}\) _and \(P^{\prime\prime}\to W_{+}\)--is the initial marking function,_  
_where: \(M_{0}({p^{\prime\prime}}_{j})=z_{j}/K({p^{\prime\prime}}_{j})\),_  
\(z_{j}\in\aleph\cup\{0\}\)_,_  
\(z_{j}\leq K({p^{\prime\prime}}_{j})\)_,_  
\(j=1,2,\dots,a^{\prime\prime}\)_, and_  
\(W_{+}\)_--is a set of non-negative rational numbers;_  
_e_--is an event that synchronizes the work of all transitions._  
FIPN can be represented as a bipartite graph. An exemplary net is shown in Figure 1. There are two types of places in the net: \(p^{\prime}\)-type (for processes modelling) and \(p^{\prime\prime}\)-type (for resources modelling). They are drawn as circles. For both types, the marking is a real number from the range \([0,1]\) located inside the circle. However, the marking of \(p^{\prime\prime}\)-type places is presented as a fraction and can store a number of tokens greater than one. The capacity of these places \(K\,(p)>1\) arranged in the denominator is a normalization coefficient by which the marking value is bounded into the interval \([0,1]\). Moreover, statements can be assigned to \(p^{\prime}\)-type places to set a value of output variables. Transitions are represented by rectangles, and can be related to binary and analogue sensors. Additionally, some logic conditions can be assigned to synchronized transitions and the arcs which link places with transitions are labelled with weighting factors. The conception of using statements, sensors, and logic conditions is presented at the end of this section and in Section 4.2. The transfer of markers from input to output places across the transition can begin when the conditions (1) and (2) of Definition 2 are satisfied and ends when conditions (3) and (4) are fulfilled. **Definition 2**.