Approaches in the context of underspecified semantics that deal with scope ambiguities have the common property that they all use a partial order to represent scope. For scope ambiguities, this partial order is not fully specified. First a general definition of TDGs is presented, and afterwards a restricted variant called local TDGs is proposed. Since the elements of a local TDG are tree descriptions, an extended domain of locality as in TAGs is provided by this formalism. Consequently, local TDGs can be lexicalized, and local dependencies such as filler gap dependencies can be expressed in the descriptions occurring in the grammar.
Dominance is a partial order and can be used to represent scope relations.
The generative capacity of local TDGs is greater than the one of TAGs. However, the generative capacity of local TDGs is restricted in such a way that only semilinear languages are generated. Therefore these languages are of constant growth, a property generally ascribed to natural languages. In order to describe the relation between two languages, in particular the relation between syntax and semantics, I introduce synchronous local TDGs. Using synchronous local TDGs, I illustrate the derivation of underspecified representations for quantifier scope ambiguities with local TDGs.