On Two Hierarchies of Subregularly Tree Controlled Languages

Tree controlled grammars are context-free grammars where the associated language only contains those terminal words which have a derivation where the word of any level of the corresponding derivation tree belongs to a given regular language. In this paper, we consider first the case that we take only such regular languages as the control set which can be represented by finite unions of monoids. We show that the corresponding hierarchy of tree controlled languages collapses already at the level 2. Moreover, we present a characterization of both levels by well-known language families generated by extended Lindenmayer systems. Furthermore, we give some comments on the hierarchy of tree controlled languages if one restricts the number of states allowed in the accepting automaton of the regular control language.