**Abstract**

This chapter is dedicated to several structure features of Petri nets. There is detailed description of appropriate access in Petri nets and reachable tree mechanism construction. There is an algorithm that describes the minimum sequence of possible transitions. The algorithm developed by us finds the shortest possible sequence for the network promotion state, which transfers the mentioned network state to the coverage state. The corresponding theorem is proven, which states that due to the describing algorithm, the number of transitions in the covering state is minimal. This chapter studies the interrelation of languages of colored Petri nets and traditional formal languages. The Venn diagram, modified by the author, is presented, which shows the relationship between the languages of the colored Petri nets and some traditional languages. As a result, it is shown that the language class of colored Petri nets includes an entire class of context-free languages and some other classes. The results obtained show that it is not possible to model the Patil problem using the well-known semaphores P and V or classical Petri nets, so the mentioned systems have limited properties.

**Keywords:** petri nets, colored petri nets, traditional languages, transition, position

## **1. Introduction**

Modeling and designing systems cannot be imagined without the use of computer technology. When creating automated systems and designing them, the problem of choosing a formal model for representing systems first arises. From the model through the algorithmic to the software—this is the way of modern modeling and system design. When considering lumped physical systems, a convenient model is a linear graph, each vertex of which corresponds to a functional or constructive component, and an arc to a causal relationship.

Petri nets are a mathematical apparatus for modeling dynamic discrete systems. Their feature is the ability to display parallelism, asynchrony, and hierarchy. They were first described by Karl Petri in 1962.

The Petri net is a bipartite oriented graph consisting of two types of vertices positions and transitions—connected by arcs between each other; vertices of the same type cannot be directly connected. Positions can be placed tags (markers) that can move around the network [1].

Petri net—a tool for modeling dynamic systems. The theory of Petri nets makes it possible to model a system with a mathematical representation of it in the form of a Petri net, the analysis of which helps to obtain important information about the structure and dynamic behavior of the simulated system.

The language generated by CPN allows to represent a model that is a collection of modules, allowing you to hierarchically represent complex networks or systems. In classical Petri nets, the tokens do not differ from each other; they are colorless. In colored Petri nets, a position can contain chips that are of arbitrary complexity, such as lists, etc., that allow you to simulate more reliable models [8–10, 13].

To build models of discrete systems, it needs various components of the system with abstract operations: switching the transition from one state to another; the action of a program operator, machine, or conveyor; interruptions in the operating system; phase completion in the project; etc. The same system can work differently under different conditions, generating many processes that will bring nondeterministic work. In real systems, cases occur at certain periods and last a certain time. In synchronous models of discrete systems, events are correctly associated with certain pauses, moments during which all components simultaneously change the

The modeling approach has several drawbacks when dealing with large systems. To make the model look impressive, first of all, with every change, the system

Secondly, with the above approach, information in systems disappears between

Thirdly, the so-called asynchronous systems can cause undefined events at time

Causal relationships make it possible to more clearly describe the structural

Asynchronous models of nonformal description of the case, in particular, Petri nets, must involve relationships of time (early, late, not at the same time, etc.), when it is convenient or accepted, but they represent a causal relationship. Great interplay of asynchronous systems, typically, has a complex dynamic structure. The relationship between the two will be described more clearly if not immediate contacts are marked, or cases and situations in which the case can be realized. In this case, the conditions of implementation of the system of global situations are

The term has its capacity. The term is not fulfilled (capacity is equal to 0), the term is fulfilled (capacity is equal to 1), and the term is fulfilled in n times (capacity

Most systems are suitable as discrete structures that consist of two elements: type of events and terms. Cases and terms in Petri nets, sets that do not intersect with each other, respectively, are called positions and transitions. Transitions are vertical lines and places with circles in a graphical representation of Petri nets [1, 2].

**2.1 The relationship of petri nets, reachable states, and reachable trees**

**Definition 1:** Petri nets are *M C*ð Þ , *μ* , where *C* ¼ ð Þ *P*, *T*,*I*, *O* is the network structure and *μ* is the network condition. *P* is positions and *T* is transitions, which are finite sets. *<sup>I</sup>* : *<sup>T</sup>* ! *<sup>P</sup>*<sup>∞</sup>, *<sup>O</sup>* : *<sup>T</sup>* ! *<sup>P</sup>*<sup>∞</sup> are input and output functions, respectively, where *<sup>P</sup>*<sup>∞</sup> are all possible multisets (repetitive elements) of *<sup>P</sup>*. *<sup>μ</sup>* : *<sup>P</sup>* ! *<sup>N</sup>*<sup>0</sup> is the function of condition, where *N*<sup>0</sup> ¼ f g 0, 1, ⋯ is the set of integers and included 0.

**2. The algorithm description of the shortest possible sequence of**

**transitions in petri nets**

random links.

features of the system.

formed in the named local operations.

is equal to n, where n- is a positive integer).

intervals.

**115**

state of the system, changing the state of the system.

*The Possibilities of Modeling Petri Nets and Their Extensions*

*DOI: http://dx.doi.org/10.5772/intechopen.90275*

must take into account all the components of its general condition.

Petri nets and the above types of models are called asynchronous.

There are several ways of practical application of Petri nets in the design and analysis of systems. In one of the approaches, the Petri nets are considered as an auxiliary analysis tool. Here, to build the system, generally accepted design methods are used, then the constructed system is modeled by the Petri net, and the constructed model is analyzed.

In another approach, the entire process of design and characterization is carried out in terms of Petri nets. In this case, the task is to transform the representation of the Petri net into a real information system [2].

The undoubted advantage of Petri nets is a mathematically rigorous description of the model. This allows their analysis with the help of modern computing techniques (including those with a massively parallel architecture) [1].

In modern society, reliable transmission and protection of information are of wide use and are topical tasks. The main task of Petri nets is the modeling of realistic systems from the point of view of optimization. Systematic study of the properties of Petri nets and the possibility of using them for solving applied problems, mainly problems related to models and means of parallel processing of information.

The following issues can serve as examples of those problems that often arise in the design and study of discrete systems:


These tasks are basically "qualitative" not quantitative.

The goal of in-depth study of various extensions of Petri nets (from the point of view of optimization) for modeling real-time systems brings to the design of such technical equipment where one has to minimize resource costs and time and maximize speed.

Colored petri net (CPN) modeling mechanisms are a convenient graphic language for designing, modeling, and testing systems [3–7]. They are well suited for systems that discuss interaction issues and synchronize. The colored Petri nets are well suited for modeling distributed systems, automated production systems, and for the design of VLSI circuit chips [8–10].

Colored Petri nets are called if the chips are the values of some types of data, which are usually called color sets. Expressions are assigned to arcs in such a network. When transitions are triggered, the values of expressions on arcs are calculated. The results of the calculations are extracted from the markup of the input transition points and placed in the marking of the output points. Transitions may be assigned with security expressions. If the guard expression assumes the value "false," the transition is prohibited [3–6, 11, 12].

a Petri net, the analysis of which helps to obtain important information about the

There are several ways of practical application of Petri nets in the design and analysis of systems. In one of the approaches, the Petri nets are considered as an auxiliary analysis tool. Here, to build the system, generally accepted design methods

In another approach, the entire process of design and characterization is carried out in terms of Petri nets. In this case, the task is to transform the representation of

The undoubted advantage of Petri nets is a mathematically rigorous description of the model. This allows their analysis with the help of modern computing tech-

In modern society, reliable transmission and protection of information are of wide use and are topical tasks. The main task of Petri nets is the modeling of realistic systems from the point of view of optimization. Systematic study of the properties of Petri nets and the possibility of using them for solving applied problems, mainly problems related to models and means of parallel processing of

The following issues can serve as examples of those problems that often arise in

• Is it possible to simplify the system or replace its individual components and subsystems with more perfect ones, without disturbing its overall functioning?

The goal of in-depth study of various extensions of Petri nets (from the point of view of optimization) for modeling real-time systems brings to the design of such technical equipment where one has to minimize resource costs and time and max-

Colored petri net (CPN) modeling mechanisms are a convenient graphic language for designing, modeling, and testing systems [3–7]. They are well suited for systems that discuss interaction issues and synchronize. The colored Petri nets are well suited for modeling distributed systems, automated production systems, and

Colored Petri nets are called if the chips are the values of some types of data, which are usually called color sets. Expressions are assigned to arcs in such a network. When transitions are triggered, the values of expressions on arcs are calculated. The results of the calculations are extracted from the markup of the input transition points and placed in the marking of the output points. Transitions may be assigned with security expressions. If the guard expression assumes the

• Is it possible to design more complex systems that meet the specified

are used, then the constructed system is modeled by the Petri net, and the

niques (including those with a massively parallel architecture) [1].

• Does the system perform the functions for which it is intended?

structure and dynamic behavior of the simulated system.

the Petri net into a real information system [2].

the design and study of discrete systems:

• Can mistakes and emergencies occur in it?

requirements from these systems, etc.?

for the design of VLSI circuit chips [8–10].

value "false," the transition is prohibited [3–6, 11, 12].

These tasks are basically "qualitative" not quantitative.

• Does it have potential bottlenecks?

• Does it function effectively?

constructed model is analyzed.

*Numerical Modeling and Computer Simulation*

information.

imize speed.

**114**

The language generated by CPN allows to represent a model that is a collection of modules, allowing you to hierarchically represent complex networks or systems.

In classical Petri nets, the tokens do not differ from each other; they are colorless. In colored Petri nets, a position can contain chips that are of arbitrary complexity, such as lists, etc., that allow you to simulate more reliable models [8–10, 13].
