Simple Matrix Grammars and Their Leftmost Variants

In essence, simple matrix grammars can be seen as sequences of context-free grammars, referred to as their components, which work in parallel. The present paper demonstrates that two-component simple matrix grammars are as powerful as ordinary matrix grammars. Then, it places three leftmost derivation restrictions upon these grammars and demonstrates that under two of these restrictions, simple matrix grammars are computational complete — that is, they are equivalent with Turing machines. From a historical perspective, concerning simple matrix grammars, the paper also makes several remarks that correct false statements published about them in the past.

ON ALTERNATING PHRASE-STRUCTURE GRAMMARS

The concepts of alternation and of state alternation are extended from context-free grammars to context-sensitive and arbitrary phrase-structure grammars. For the resulting classes of alternating grammars the expressive power is investigated with respect to the leftmost derivation mode and with respect to the unrestricted derivation mode. In particular new grammatical characterizations for the class of languages that are accepted by alternating pushdown automata are obtained in this way.

ON THE LEFTMOST DERVIATION IN MATRIX GRAMMARS

Matrix grammars are one of the classical topics of formal languages, more specifically, regulated rewriting. Although this type of control on the work of context-free grammars is one of the earliest, matrix grammars still raise interesting questions (not to speak about old open problems in this area). One such class of problems concerns the leftmost derivation (in grammars without appearance checking). The main point of this paper is the systematic study of all possibilities of defining leftmost derivation in matrix grammars. 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. In this way, we solve nearly all problems listed as open on page 67 of the monograph [7], which can be seen as the main contribution of this paper. Moreover, we find a characterization of the recursively enumerable languages for matrix grammars with the leftmost restriction defined on classes of a given partition of the nonterminal alphabet.