Categorial Grammar and Lexical-Functional Grammar

¤I wish to thank the participants of LFG2001, especially Mary Dalrymple and Ron Kaplan, for valuable feedback. All comments are very welcome. This paper introduces ¸-grammar, a form of categorial grammar that has much in common with LFG. Like other forms of categorial grammar, ¸-grammars are multi-dimensional and their components are combined in a strictly parallel fashion. Grammatical representa-tions are combined with the help of linear combinators, closed pure ¸-terms in which each abstractor binds exactly one variable. Mathe-matically this is equivalent to employing linear logic, in use in LFG for semantic composition, but the method seems more practicable. While ¸-grammars could be used to formalize many approaches to grammatical theory, they are certainly natural as a basis for the formalization of LFG. This leads to a theory I would like to call ¸-LFG. In this paper it will be shown how the standard components of LFG can be set up in the framework. We will have descriptions of c-structure, descriptions of f-structure, and semantics. The di®erence between de¯ning and constraining information will be explained in terms of entailment, and requirements on long-distance paths in f-structure will be explained in terms of entailment in the presence of a simple set of axioms.