**3.3 The modeling of context-free languages with colored Petri nets: the diagram of interrelation of colored Petri nets and traditional languages**

**Figure 6.**

**Figure 7.**

**125**

*Modeling L*<sup>∗</sup> <sup>¼</sup> *<sup>L</sup>* <sup>∪</sup> *LL* <sup>∪</sup> *LLL* … *language by colored petri net.*

*The Possibilities of Modeling Petri Nets and Their Extensions*

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

*regular languages).*

*Interrelation of petri nets and traditional languages (T-0, the general type of languages; CS, context-sensitive languages; PNL; petri net languages; CF, context-free languages; BCF, bonded context-free languages; R,*

It is known that the class of regular languages is one of the most studied simple classes of formal languages and any regular language is the language of Petri nets [2, 18].

There are context-free languages that are not languages of Petri nets. Such examples of the context-free languages are *ωω<sup>R</sup>=ω*<sup>∈</sup> <sup>Σ</sup><sup>∗</sup> , *<sup>L</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>L</sup>* <sup>∪</sup> *LL* <sup>∪</sup> *LLL* … (in particular, *<sup>a</sup>nbn* f g *<sup>=</sup>n*><sup>1</sup> ).

The noted fact shows the limitation of Petri nets as a mechanism that generates languages [2].

In Petri nets one can only remember a sequence of limited length (similar to finite automata) [2].

It is clear that Petri nets do not possess the "capacity of pushdown memory" necessary for generating context-free languages. The relationship of the languages of Petri nets with other classes of languages (Venn diagram) is shown in **Figure 6** [2, 10].

#### **3.4 Results**

A model of the *<sup>L</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>L</sup>* <sup>∪</sup> *LL* <sup>∪</sup> *LLL* … language (Klins'star) is constructed using colored Petri nets, in particular *<sup>L</sup>* <sup>¼</sup> *anbn* f g *<sup>=</sup>n*≥<sup>1</sup> .

Colored Petri net (**Figure 7**) generates such a language, which proves that the colored Petri net is a more powerful tool than the classical Petri nets. The following declaration is for the concept of data types.

*The Possibilities of Modeling Petri Nets and Their Extensions DOI: http://dx.doi.org/10.5772/intechopen.90275*

#### **Figure 6.**

**3.3 The modeling of context-free languages with colored Petri nets:**

(in particular, *<sup>a</sup>nbn* f g *<sup>=</sup>n*><sup>1</sup> ).

*The modeling of consumer problem with colored petri net.*

*Numerical Modeling and Computer Simulation*

colored Petri nets, in particular *<sup>L</sup>* <sup>¼</sup> *anbn* f g *<sup>=</sup>n*≥<sup>1</sup> .

declaration is for the concept of data types.

languages [2].

**Figure 5.**

**3.4 Results**

**124**

finite automata) [2].

of formal languages and any regular language is the language of Petri nets [2, 18]. There are context-free languages that are not languages of Petri nets. Such examples of the context-free languages are *ωω<sup>R</sup>=ω*<sup>∈</sup> <sup>Σ</sup><sup>∗</sup> , *<sup>L</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>L</sup>* <sup>∪</sup> *LL* <sup>∪</sup> *LLL* …

**the diagram of interrelation of colored Petri nets and traditional languages**

It is known that the class of regular languages is one of the most studied simple classes

The noted fact shows the limitation of Petri nets as a mechanism that generates

A model of the *<sup>L</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>L</sup>* <sup>∪</sup> *LL* <sup>∪</sup> *LLL* … language (Klins'star) is constructed using

Colored Petri net (**Figure 7**) generates such a language, which proves that the colored Petri net is a more powerful tool than the classical Petri nets. The following

In Petri nets one can only remember a sequence of limited length (similar to

It is clear that Petri nets do not possess the "capacity of pushdown memory" necessary for generating context-free languages. The relationship of the languages of Petri nets with other classes of languages (Venn diagram) is shown in **Figure 6** [2, 10]. *Interrelation of petri nets and traditional languages (T-0, the general type of languages; CS, context-sensitive languages; PNL; petri net languages; CF, context-free languages; BCF, bonded context-free languages; R, regular languages).*

**Figure 7.** *Modeling L*<sup>∗</sup> <sup>¼</sup> *<sup>L</sup>* <sup>∪</sup> *LL* <sup>∪</sup> *LLL* … *language by colored petri net.*

#### **Figure 8.** *Interrelation of colored petri nets and traditional languages. (CPNL, language of colored petri net).*

The operation of the colored Petri net shown in **Figure 7** is described in more detail in the literature [3, 10].

<sup>Т</sup>he colored Petri net (**Figure 7**), which is built for *<sup>L</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>L</sup>* <sup>∪</sup> *LL* <sup>∪</sup> *LLL* … language, suggests the following relationship between the languages of the colored Petri nets with some classes of traditional languages (see **Figure 8**) [10].

The Venn diagram, modified by the author (**Figure 8**), shows the relationship between the languages of the colored Petri nets and some traditional languages. This fact illustrates that the language class of colored Petri nets includes an entire class of languages without context.

**4.1 Declaration**

**Figure 9.**

Color U ¼ t; Color N ¼ P; Color Q ¼ m;

Var K, l ð Þ : E; n : INT;

help of colored Petri net.

petrosyangoharik72@gmail.com

provided the original work is properly cited.

**4.2 Conclusion**

**Author details**

Goharik Petrosyan

**127**

Color INT ¼ integer;

*The modeling of cigarette smoker's problem with colored petri nets.*

*The Possibilities of Modeling Petri Nets and Their Extensions*

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

Color E <sup>¼</sup> Product N<sup>∗</sup> Q OR Product U<sup>∗</sup> Q OR Product N<sup>∗</sup> f g <sup>U</sup> ;

In the problem, we identify certain advantages of colored Petri net to P and V operations and classical Petri net with the synchronization problem. The mentioned studies allow identification of synchronization modeling opportunities with the

Plekhanov Russian University of Economics Yerevan Branch, Member of Armenian

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Mathematical Union, ASPU, ISEC of NAS RA, Yerevan, Armenia

\*Address all correspondence to: petrosyan\_gohar@list.ru;
