Two collapsing 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 as control sets such regular languages which can be represented by finite unions of monoids. We show that the corresponding hierarchy of tree controlled languages collapses already at the second level. Second, we restrict the number of states allowed in the accepting automaton of the regular control language. We prove that the associated hierarchy has at most five levels.