Abstract
We survey a sequence of results relating model-theoretic and language-theoreticdefinability over an infinite hierarchy of multi-dimensional tree-like structures and explore their applications to a corresponding range of theoriesof 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 sameway that GPSG extends CFLs. In addition, we look at a hierarchy oflanguage classes, Weir's version of the Control Language Hierarchy, which ischaracterized by definability in our hierarchy and speculate on possiblelinguistic significance of higher levels of these hierarchies.
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