Some questions on polynomially computable representations for generating grammars and Backus-Naur forms

In the present article, we consider the question on modeling Backus-Naur forms (BNF-systems) and generating grammars in GNF-systems. GNF-systems serve as the base for construction of monotone operators whose least fixed points are polynomially computable. We obtain our results by construction of GNF-systems and application of a generalized polynomial analog of Gandy's fixed point theorem. This allows us to answer some questions on existence of a polynomially computable representation for the set of derivations in generating grammars. Moreover, we show that, for each GNF-system modeling a BNF-system and every nonterminal symbol in the BNF-system, the set of preimages in the GNF-system of representations of this symbol is polynomially computable. This result allows us to encode all definable constructions of the BNF-system, including the syntax of programs in high-level programming languages, so that they become recognizable in polynomial time.