Nonterminal complexity of programmed grammars

AbstractWe show that, in the case of context-free programmed grammars with appearance checking working under free derivations, three nonterminals are enough to generate every recursively enumerable language. This improves the previously published bound of eight for the nonterminal complexity of these grammars. This also yields an improved nonterminal complexity bound of four for context-free matrix grammars with appearance checking. Moreover, we establish an upperbound of four on the nonterminal complexity of context-free programmed grammars without appearance checking working under leftmost derivations of type 2. We derive nonterminal complexity bounds for context-free programmed and matrix grammars with appearance checking or with unconditional transfer working under leftmost derivations of types 2 and 3, as well. More specifically, a first nonterminal complexity bound for context-free programmed grammars with unconditional transfer (working under leftmost derivations of type 3) which depends on the size of the terminal alphabet is proved