# Applying Infinite Petri Nets to the Cybersecurity of Intelligent Networks, Grids and Clouds
## 6 Specifying and Analysing Infinite Petri Nets
### Finite Specification of Infinite Petri Net  
**A finite specification of an infinite Petri net has been introduced in the form of a parametric multi-set rewriting system [47] called for brevity a parametric expression (PE).** It comes from a traditional way of specifying a Petri net enumerating its transitions, a transition is specified by a pair of places' lists, separated by \"\(\rightarrow\)\" symbol, --for input and output places, respectively, the arc weight specified as well. A parametric expression (5) specifies the cellular automaton model [41] shown in Figure 2a. A usual arc is represented by mentioning the corresponding place name, for instance \(c_{i}\); an inhibitor and a read arc is specified by the corresponding inequality, for instance \(c_{i}=0\) for inhibitor arc and \(c_{i+1}>0\) for read arc.  
\[\left(\begin{array}{c}t01_{i}:c_{i}=0,c_{i+1}>0\to c_{i},\\ t10_{i}:c_{i-1}>0,c_{i},c_{i+1}>0.\rightarrow\end{array}\right) \tag{5}\]  
When studying computing and communication grids on plane (2D) [45], the parametric expression (6) specifies a switching device model having \(np\) ports, a triangle, square, and hexagon are obtained at the parameter values \(np=3,4,6\), respectively. At \(np=4\), we obtain a formal specification of a square grid cell shown in Figure 4.  
\[\left(\left(\begin{array}{c}(to_{u}:pb_{u},pol_{u}\to p o_{u},\, pbl),\\ (ti_{u,v}:pi_{u},pbl\to p\bar{v},\,pil_{u}),1\leq v\leq np,v\
eq u \end{array}\right),1\leq u\leq np.\right) \tag{6}\]  
Infinitely repeating a cell model, in correspondence with a chosen lattice, and supplying it with a definite border condition, we come to the models of grid structures. To specify infinite nets, we use parameters having countable infinite range. Since our basic  
Figure 4: Model of switching device with 4 ports situated on sides of a square—a cell model for square grids composition; upper indices specify the cell location within a grid.  
parameters represent the grid size and the number of dimensions, we use natural numbers in the range from unit to (plus) infinity. Let us employ the square model (6), \(np=4\) to compose a closed infinite Petri net of the second kind [48] that represents a surface of a torus (1). Cells are connected by merging contact places; to avoid double names, for port 2 we use contact places of the neighboring cell to the right (port 4) and for port 3 we use contact places of the neighboring cell to the bottom (port 1). Actually, we connect the opposite borders using a function (7) which gives the next cell index for the inside cells and the first cell index for the border cells (to the right - port 2 and to the bottom--port 3).  
\[nx(x)=\begin{cases}x+1,&x<k\\ 1,&x=k.\end{cases} \tag{7}\]  
In PE (1), the current cell location within the grid is specified as an upper index. According to (6), each port specification by a pair of lines (corresponding to its input and output tract, respectively) is written in an explicit form. Ports 1 (upper) and 4 (left) use elements having the same cell index. Ports 2 (right) and 3 (bottom) use names of contact places for neighboring cells, in the corresponding dimension, given by the function \(nx\) (7).