# Applying Infinite Petri Nets to the Cybersecurity of Intelligent Networks, Grids and Clouds

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## Abstract

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## 1. Introduction

## 2. Modern Trends in Verification of Networking Protocols

## 3. Getting Familiar with Petri Nets

## 4. Infinite Petri Net of First Kind: A Single Infinite Structure

## 5. Infinite Petri Net of Second Kind: An Infinite Set of Finite Structures

## 6. Specifying and Analysing Infinite Petri Nets

#### 6.1. 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 “→” 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.

#### 6.2. Solving Infinite Linear Systems in Parametric Form

**To analyze the properties of infinite Petri nets, we compose and solve infinite systems of linear algebraic equations.**The peculiarity of the process consists in the fact we work with Diophantine systems and it is required to solve them in non-negative numbers. For finding place invariants (p-invariants), a homogeneous system represents balance of terms corresponding to the incoming and outgoing arcs of transitions; it is constructed directly on a given parametric expression [45,48]. For example, an infinite system (2) for finding p-invariants is composed on parametric expression (1), unknowns traditionally have prefix “x”. Saying plainly, to obtain (2) from (1) we replaced commas by pluses and the arrow by equality symbol, then we moved all the variables to the left side of the system. p-invariants are applied to prove the net conservativeness and boundedness. To find transition invariants (t-invariants), a dual parametric specification, which enumerates the net places, is applied [12]. t-invariants play important role when investigating a net liveness—one of the most significant properties.

#### 6.3. Complex Deadlocks within Computing Grid Models

**Special graphs have been introduced and studied to prove other properties of grids’ models.**A graph of packet transmissions has been introduced to prove t-invariance via explicit composition of stationary repeated sequences of transitions’ firings [45]. A graph of mutual blockings have been introduced to classify complex deadlocks within grid structures, a three causes of deadlocks have been revealed: (i) a circle of blockings; (ii) a chain of blockings ending at an already blocked cell; (iii) isolation by blocked nodes. Afterwards, it has been proven that complex deadlocks can be induced by ill-intentioned traffic and appear in avalanche-like way imposing a serious thread to the grid security [49]. For this purpose colored Petri nets have been applied which allow hierarchical composition of a model and specification of timed parameters. Guns of traffic have been attached to the grid borders, the following concise and the most dangerous configurations involving two guns have been revealed: (i) a traffic duel; (ii) crossfire, and (iii) side shot. An example of a complete deadlock of an 8 × 8 grid is shown in Figure 5. Inscriptions on arcs specify the number of packets in the internal buffer, forwarded to the corresponding device, and the number of packets in the port buffer, respectively; the internal buffer size is 100 packets. Within a real-life network, a deadlock is overcome by drop packet and timeout techniques but repeated deadlicks decrease the network performance and QoS considerably.

#### 6.4. Generalization of Obtained Results

**The results obtained for square grids have been further generalized**on triangular and hexagonal grids on plane [13,34] and hyper-cube in multidimensional space [12,50], closing opposite edges of a hyper-cube, a hyper-torus have been composed and studied [44,48,51]. Generators of the mentioned models have been programmed in C language and uploaded to GitHub [12]. An example of a generated hexagonal grid of size 6, to study cellular phone systems [34], is shown in Figure 6.

## 7. Open Problems

**Among the exciting open problems encouraging further research, we could enumerate the following:**

- A general method for solving infinite systems of Diophantine linear algebraic equations, especially in non-negative numbers.
- Methods to find siphons and traps of infinite Petri nets to solve tasks of liveness and liveness-enforcing.
- Composition methods on infinite Petri nets, say composition of clans.
- Representation and application of reachability and coverability tree for infinite Petri nets.
- Composing and analyzing infinite Petri nets built of a few repeated components.
- An algorithm of mutual transformation for direct and dual specification of infinite Petri nets.
- Recognition of disguised attacks via induced deadlocks and corresponding countermeasures.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 2.**Modeling elementary cellular automaton Rule 110 by infinite Petri net, seven cells fragment.

**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.

**Figure 6.**Modeling hexagonal communication grid with plug devices on edges; an example of grid having size 6.

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**MDPI and ACS Style**

Zaitsev, D.A.; Shmeleva, T.R.; Probert, D.E. Applying Infinite Petri Nets to the Cybersecurity of Intelligent Networks, Grids and Clouds. *Appl. Sci.* **2021**, *11*, 11870.
https://doi.org/10.3390/app112411870

**AMA Style**

Zaitsev DA, Shmeleva TR, Probert DE. Applying Infinite Petri Nets to the Cybersecurity of Intelligent Networks, Grids and Clouds. *Applied Sciences*. 2021; 11(24):11870.
https://doi.org/10.3390/app112411870

**Chicago/Turabian Style**

Zaitsev, Dmitry A., Tatiana R. Shmeleva, and David E. Probert. 2021. "Applying Infinite Petri Nets to the Cybersecurity of Intelligent Networks, Grids and Clouds" *Applied Sciences* 11, no. 24: 11870.
https://doi.org/10.3390/app112411870