Non-Terminal Complexity of Simple Semi-Conditional Grammars

We study the complexity of simple semi-conditional grammars (SSCGs) in terms of the number of their terminals and non-terminals. We show that SSCGs with three non-terminals can generate all RE languages; two non-terminals suffice for linear languages; and one suffices for unary regular languages. We also determine both upper bounds and fundamental limitations of SSCGs: while certain one-non-terminal SSCGs can generate non-context-free languages, every language generated by a one-non-terminal SSCG is in non-deterministic linear space. We also prove that there exists a regular language of alphabet size at least 3 which cannot be generated by any one-non-terminal SSCG. Finally, we prove that membership testing for SSCGs of two non-terminals is already NP-hard.