**Author details**

18 Petri Nets

Let us now turn to grammars controlled by arbitrary Petri nets with capacities. Let *G* = (*V*, Σ, *S*, *R*, *N*, *γ*, *M*) be an arbitrary Petri net controlled grammar. *G* is called a grammar controlled by an arbitrary Petri net with place capacity if *N* is a Petri net with place capacity. The families of languages generated by grammars controlled by arbitrary Petri nets with place

The next statement indicates that the language generated by a grammar controlled by an arbitrary Petri net with place capacities iff it is generated by a matrix grammar (for details,

*cb* <sup>=</sup> **VEC**[*λ*]

<sup>⊂</sup> **MAT** <sup>=</sup> **PN***cb*(*x*, *<sup>y</sup>*) <sup>⊆</sup> **MAT***<sup>λ</sup>* <sup>=</sup> **PN***<sup>λ</sup>*

The chapter summarizes the recent results on Petri net controlled grammars presented in [18–23, 56, 59–62] and the close related topic: capacity-bounded grammars. Though the theme of regulated grammars is one of the classic topics in formal language theory, a Petri net controlled grammar is still interesting subject for the investigation for many reasons. On the one hand, this type of grammars can successfully be used in modeling new problems emerging in manufacturing systems, systems biology and other areas. On the other hand, the graphically illustrability, the ability to represent both a grammar and its control in one structure, and the possibility to unify different regulated rewritings make this formalization attractive for the study. Moreover, control by Petri nets introduces the concept of *concurrency*

We should mention that there are some open problems, the study of which is of interest: one of them concerns to the classic open problem of the theory of regulated rewriting systems – the strictness of the inclusion **MAT** <sup>⊆</sup> **MAT***λ*. We showed that language families generated by (arbitrary) Petri net controlled grammars are between the families **MAT** and **MAT***λ*. Moreover, the work [65] of G. Zetzsche shows that the erasing rules in Petri net controlled grammars with finite set of final markings can be eliminated without effecting on

There is also another very interesting topic in this direction for the future study. If we notice the definitions of derivation-bounded [32] or nonterminal-bounded grammars [3–5] only nonterminal strings are allowed as left-hand sides of production rules. Here, an interesting

*cb*(*x*, *<sup>y</sup>*) = **MAT***λ*.

*cb* <sup>=</sup> **GS***cb* <sup>=</sup> **GS**<sup>1</sup>

*cb*(*x*, *y*)

*cb*

**PN***cb*(*x*, *<sup>y</sup>*) = **MAT** <sup>⊆</sup> **PN***<sup>λ</sup>*

*cb*

<sup>⊂</sup> **MAT***fin* <sup>=</sup> **MAT**[*λ*]

the generative power, which gives hope that one can solve this problem.

*cb*(*x*, *y*)) where *x* ∈ { *f* , −*λ*, *λ*} and

capacities (with erasing rules) is denoted by **PN***cb*(*x*, *y*) (**PN***<sup>λ</sup>*

**Theorem 19.** *For x* ∈ { *f* , −*λ*, *λ*} *and y* ∈ {*r*, *t*, *g*}*,*

We summarize our results in the following theorem. **Theorem 20.** *The following inclusions and equalities hold:*

*where x* ∈ { *f* , −*λ*, *λ*} *and y* ∈ {*r*, *t*, *g*}*.*

in regulated rewriting systems.

**7. Conclusions and future research**

**CF***fin* <sup>⊂</sup> **CF***cb* <sup>=</sup> **CF**<sup>1</sup>

*y* ∈ {*r*, *t*, *g*}.

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Jürgen Dassow and Ralf Stiebe *Fakultät für Informatik, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany* Gairatzhan Mavlankulov, Mohamed Othman, Mohd Hasan Selamat and Sherzod Turaev *Faculty of Computer Science and Information Technology, Universiti Putra Malaysia, UPM Serdang, Selangor, Malaysia*

*International Conference, LATA 2008. Revised Papers*, Vol. 5196 of *LNCS*, Springer,

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