What are the limits of control in formal language theory? This paper explores the intricacies of leftmost derivation in matrix grammars, a classical topic in regulated rewriting. While matrix grammars have been studied for decades, this research systematically investigates various possibilities for defining leftmost derivation. Twelve types of such a restriction are defined, only four of which being discussed in literature. For seven of them, we find a proof of a characterization of recursively enumerable languages (by matrix grammars with arbitrary context-free rules but without appearance checking). Other three cases characterize the recursively enumerable languages modulo a morphism and an intersection with a regular language. Ultimately, this study solves nearly all problems listed as open on page 67 of the monograph [7], significantly advancing the field. Additionally, a characterization of recursively enumerable languages is found for matrix grammars with the leftmost restriction defined on classes of a given partition of the nonterminal alphabet.
This article aligns with the International Journal of Foundations of Computer Science's focus on theoretical computer science. By systematically investigating leftmost derivation in matrix grammars and characterizing recursively enumerable languages, the paper contributes to the journal's scope. The findings on restrictions and classifications in formal language theory are relevant to the journal's readership.