Grammars Controlled by Petri Nets with Place Capacities Mohd Hasan Selamat and Sherzod Turaev Faculty of Computer Science and Information Technology University Putra Malaysia 43400 UPM Serdang, Selangor, Malaysia {hasan, sherzod}@fsktm.upm.edu.my Abstract—A Petri net controlled grammar is a grammar equipped with a Petri net whose transitions are labeled with production rules and nonterminals of the grammar, and the associated language consists of all terminal strings which can be derived in the grammar and the the sequence of rules in every terminal derivation corresponds to some occurrence sequence of transitions of the Petri net which is enabled at the initial marking and finished at a final marking of the net. In the paper we investigate the generative power of grammars controlled by Petri nets with place capacities. Keywords-Formal languages and grammars, grammars with regulated rewriting, Petri nets, Petri net controlled grammars I. I NTRODUCTION Formal language theory has also been widely involved in modeling and investigating phenomena appearing in com- puter science, biology, linguistics, mathematics and other related fields. A model for a phenomenon is usually con- structed by representing it as a language (a set of words) over a certain alphabet, and defining a grammar (a genera- tive mechanism) which identifies exactly the words of this set. Context-free grammars, which are the most investigated type in formal language theory, are widely used in many applications of formal languages. However, they cannot cover all aspects which occur in modeling of phenomena. Thus, different types of grammars with regulated rewriting have been introduced in order to supplement shortcomings of context-free grammars in applications preserving their elegant mathematical properties (see [1]). The rapid developments in present day industry, biology, and other areas challenge to deal with various tasks which need new suitable tools for their modeling and investiga- tion. For instance, biochemical processes in living cells are needed to be accurately modeled and investigated by such methods that they should not only describe biochemical processes but also represent the structure and communication in biochemical networks. Petri net controlled grammars can be used as models for representing and analyzing of such systems where Petri nets are responsible for the structure and communication, and grammars represent generative pro- cesses. In [2]–[4] we introduced different variants of Petri net controlled grammars. Resource limitations, production device bounds and other related problems have been always made important to in- vestigate “economical” models: in formal language theory, rewriting mechanisms with different restrictions have also been studied. For instance, grammars with bounded compo- nents in [5], [6], and grammars with bounded derivations were investigated in [7], [8]. In our recent paper [9] we introduced and studied capacity-bounded grammars and their extended context-free Petri net counterparts by permitting only those derivations where the number of each nonterminal in each sentential form is bounded by its capacity. This paper continues the research in this direction by considering k-Petri nets and arbitrary Petri nets with bounded place capacities. The paper is organized as follows. Section II contains some necessary definitions and notations from language and Petri net theories. The concept of grammars controlled by k-Petri nets with place capacities are introduced and their computational power are studied in Section III. The generative powers of the families of languages generated by grammars controlled by arbitrary Petri nets with place capacities are investigated in Section IV. II. PRELIMINARIES The reader is assumed to be familiar with basic concepts of formal language and Petri net theories, for details we refer to [1], [10], [11]. A. Grammars Let Σ ∗ denote the set of all strings over a finite alphabet Σ. A subset L of Σ ∗ is called a language. A context-free grammar is a quadruple G =(V, Σ, S, R) where V and Σ are disjoint finite sets of nonterminals and terminals, respectively, S ∈ V is the start symbol and a finite set R ⊆ V × (V ∪ Σ) ∗ is a set of (production) rules. Usually, a rule (A, x) is written as A → x. A rule of the form A → λ is called an erasing rule. The string x ∈ (V ∪ Σ) + directly derives y ∈ (V ∪Σ) ∗ , written as x ⇒ y, iff there is a rule r = A → α ∈ R such that x = x 1 Ax 2 and y = x 1 αx 2 . The reflexive and transitive closure of ⇒ is denoted by ⇒ ∗ . A derivation using the sequence of rules π = r 1 r 2 ··· r n is denoted by π = ⇒. The language generated by G is defined by L(G)= {w ∈ Σ ∗ | S ⇒ ∗ w}. The family of context-free languages is denoted by CF. Second International Conference on Computer Research and Development 978-0-7695-4043-6/10 $26.00 © 2010 IEEE DOI 10.1109/ICCRD.2010.43 51