There are two major approaches to formal accounts of the syntax of natural language, the proof-theoretic and the model-theoretic approach. Both aim at providing frameworks for answering the question whether a given natural language expression is grammatical. Their methodology, however, is rather different: In a proof-theoretic framework, one tries to set up a system of derivation rules (such as the rules in a context-free grammar) so that each well-formed natural language expression stands in correspondence with a derivation in that system. In contrast, in a model-theoretic framework, one attempts to specify a class of models for natural language expressions and a set of constraints on these models such that an expression is well-formed iff it has a model satisfying all the constraints. The main contribution of this paper is the characterisation of a class of structures that provides a new model-based perspective on Tree Adjoining Grammar (Joshi and Schabes (1997)), a well-known proof-theoretic syntactic framework.
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