Language classes generated by tree controlled grammars with bounded nonterminal complexity

A tree controlled grammar is specified as a pair (G,G′) where G is a context-free grammar and G′ is a regular grammar. Its language consists of all terminal words with a derivation in G such that all levels of the corresponding derivation tree–except the last level–belong to L(G′). We define the nonterminal complexity of H=(G,G′) as the sum of the numbers of nonterminals of G and G′. In Turaev et al. (2011) [23] it is shown that tree controlled grammars H with are sufficient to generate all recursively enumerable languages. In this paper, we improve the bound to seven. Moreover, we show that all linear and regular simple matrix languages can be generated by tree controlled grammars with a nonterminal complexity bounded by three, and we prove that this bound is optimal for the mentioned language families. Furthermore, we show that any context-free language can be generated by a tree controlled grammar (G,G′) where the number of nonterminals of G and G′ is at most four.