**7. Conclusions and future research**

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* in regulated rewriting systems.

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

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 question is emerged, what kind of languages can be generated if we derestrict this condition, i.e., allow any string in the left-hand side of the rules?

In all investigated types of Petri net controlled grammars, we only used the sequential firing mode of transitions. The consideration of simultaneous firing of transitions, another fundamental feature of Petri nets, opens a new direction for the future research: one can study *grammars controlled by Petri nets under parallel firing strategy*, which introduces concurrently parallelism in formal language theory.

Grammar systems can be considered as a formal model for a phenomenon of solving a given problem by dividing it into subproblems (grammars) to be solved by several parts in turn (CD grammar systems) or in parallel (PC grammar systems). The control of derivations in grammar systems also allows increasing computational power grammar systems. We can extend the regulation of a rule by a transition to the regulation a set of rules by a transition, which defines a new type of grammar systems: the firing of a transition allows applying several (assigned) rules in a derivation step parallelly and different modes.

In [19–22, 41, 62] it was shown that by adding places and arcs which satisfy some structural requirements one can generate well-known families of languages as random context languages, valence languages, vector languages and matrix languages. Thus, the control by Petri nets can be considered as a unifying approach to different types of control. On the other hand, Petri nets can be transformed into *occurrence nets*, i.e., usually an infinite, tree-like structure whose nodes have the same labels as those of the places and transitions of the Petri net preserving the relationship of adjacency, using *unfolding technique* introduced in [43] and given in [24] in detail under the name of *branching processes*. Any *finite initial* part, i.e., *prefix* of the occurrence net of a cf Petri net can be considered as a derivation tree for the corresponding context-free grammar as it has the same structure as a usual derivation tree, here we can also accept the rule of reading "leaf"-places with tokens from the left to the right as in usual derivation trees. We can also generalize this idea for regulated grammars considering prefixes of the occurrences nets obtained from cf Petri nets with additional places. Hence, we can take into consideration the grammar as well as its control, and construct (Petri net) derivation trees for regulated grammars, which help to construct effective parsing algorithms for regulated rewriting systems. Though the preliminary results (general parsing algorithms, Early-like parsing algorithm for deterministic extended context-free Petri net controlled grammars, etc.) were obtained in [63, 64], the problem of the development of the effective parsing algorithms for regulated grammars remain open.
