Syntactic Structures as Multi-Dimensional Trees
                                 James Rogers
                                Earlham College

  We survey a sequence of results relating model-theoretic and
language-theoretic definability over an infinite hierarchy of multi-dimensional
tree-like structures and explore their applications to a corresponding range of
theories of syntax.  We discuss, in particular, results for Government and
Binding Theory (GB), Tree-Adjoining Grammar (TAG) and Generalized
Phrase-Structure Grammar (GPSG) along with a generalized version of TAG
extending TAG in much the same way that GPSG extends CFLs. In addition, we look
at a hierarchy of language classes, Weir's version of the Control Language
Hierarchy, which is characterized by definability in our hierarchy and
speculate on possible linguistic significance of higher levels of these
hierarchies.