Capacity Bounded Grammars and Petri Nets

The close relationship between Petri nets and language theory has been extensively studied for a long time [1, 2]. Results from the theory of Petri nets have been applied successfully to provide elegant solutions to complicated problems from language theory [4, 7]. A context-free grammar can be associated with a context-free (communica-tion-free) Petri net, whose places and transitions, correspond to the nonterminals and the rules of the grammar, respectively, and whose arcs and weights reflect the change in the number of nonterminals when applying a rule. In some recent papers, context-free Petri nets enriched by additional components have been used to define regulation mechanisms for the defining grammar [3, 10]. Our paper continues the research in this direction by restricting the (context-free or extended) Petri nets with place capacity. Quite obviously, a context-free Petri net with place capacity regulates the defining grammar by permitting only those derivations where the number of each nonterminal in each sentential form is bounded by its capacity. A similar mechanism was discussed in [5] where the total number of nonterminals in each sentential form is bounded by a fixed integer. There it was shown that grammars regulated in this way generate the family of context-free languages of finite index, even if arbitrary nonterminal strings are allowed as left-hand sides. The main result of this paper is that, somewhat surprisingly, grammars with capacity bounds have a greater generative power. This paper is organized as follows. Section 2 contains some necessary definitions and notations from language and Petri net theory. The concepts of grammars with capacities and grammars controlled by Petri nets with place capacities are introduced in section 3. The generative power and closure properties of capacity-bounded grammars are investigated in sections 4 and 5. Results on grammars controlled by Petri nets with place capacities are given in section 6.