1 Generating Control Languages with

We show that the class of level-k control languages, as defined by Weir, is properly included in the class of 2 k−1-multiple context-free languages for each k ≥ 2. The proof of inclusion uses a representation of the set of derivation trees for a level-k control language in terms of a second-order abstract categorial grammar. 1.1 Introduction Weir (1992a) introduced a hierarchy of language classes known as the control language hierarchy, starting with the context-free languages at level one. Control languages at level two and above are each generated by a context-free grammar coupled with a language at the previous level, which controls derivation trees of the grammar in a certain " linear " fashion. Thus, for k ≥ 2, level-k control languages are defined in terms of non-regular sets of derivation trees and the usual simple yield function, which merely concatenates yields of subtrees. Even though it is known that the class of level-2 control languages coincides with the class of tree-adjoining languages or head languages (Weir 1992a, Vijay-Shanker and Weir 1994), it is not immediately clear how the higher levels are related to language classes generated by grammar formalisms that have regular sets of derivation trees and complex yield