Generalized Tree-Adjoining Grammar

This paper continues a program extending results related to the descriptive characterization of the CFLs in terms of deenability in the weak monadic second-order theory of trees to the TALs and the entire hierarchy of Weir's Control Languages. Previously, we have shown that the languages in this hierarchy can be characterized by deenability in the weak monadic second-order theories of a generalization of trees to increasingly higher dimensions. Here, we explore the eeect of admitting models with arbitrary nite branching. In the two-dimensional case we have shown previously that this admits tree sets recognized by innnite, but regular , tree-automata. As these sets correspond, in a very strong sense, to the sets of trees licensed by GPSG grammars we have referred to these automata as Generalized Finite-State Tree Automata and the sets they accept as Generalized Recognizable Sets of trees. In lifting this result to the third and higher dimension, we show that the deenable sets of structures are exactly those recognized by d-dimensional tree automata which are, in essence, generalized recognizable sets of (d ? 1)-dimensional structures. In three-dimensions, these correspond to Tree-Adjoining Grammars in which the set of initial trees can be any generalized recognizable set of trees|a variation we refer to, by analogy, as Generalized Tree Adjoining Grammars. We show that this permits adoption of GPSG-style accounts of coordination which can potentially provide a conceptually clean account of coordinate constructions that have been problematic for TAG in the past. Over the last few years we have been exploring the generalization of long standing descriptive characterizations of the regular (B uchi, 1960; Elgot, 1961) and context-free (Doner, 1970; Thatcher and Wright, 1968) languages| characterizations in terms of model-theoretic deenability over certain classes of structures| to the Tree-Adjoining Languages and beyond. The perspective that underlies this generalization is an analogy between TAGs as a tree-rewriting formalism and CFGs as a string rewriting formalism. Just as CFGs, in a standard conception, build trees by expanding non-terminal nodes into strings of child nodes|in essence attaching a depth-one tree at the frontier of a tree|TAGs can be conceived as building three-dimensional tree-like structures by expanding nodes (at which adjunction is permitted) into trees of child nodes (the adjoined auxiliary tree)|in essence attaching a depth-one pyramidal structure at the frontier of a similar structure of arbitrary depth. There is a direct and easy mapping between these three-dimensional derivation structures and the TAG derivation …