Hybrid Type-Logical Grammars, First-Order Linear Logic and the Descriptive Inadequacy of Lambda Grammars

Hybrid type-logical grammars (Kubota & Levine 2012, Kubota & Levine 2013c, Kubota & Levine 2013a) are a relatively new framework in computational linguistics, which combines insights from the Lambek calculus (Lambek 1958) and lambda grammars (Oehrle 1994, Muskens 2001, Muskens 2003)— lambda grammars are also called, depending on the authors, abstract categorial grammars (de Groote 2001) and linear grammars (Pollard 2011), though with somewhat different notational conventions. The resulting combined system solves some know problems of both the Lambek calculus and of lambda grammars and the additional expressiveness of hybrid type-logical grammars permits the treatment of linguistic phenomena such as gapping which have no satisfactory solution in either subsystem. The goal of this paper is to prove that hybrid type-logical grammars are a fragment of first-order linear logic. This embedding result has several important consequences: it not only provides a simple new proof theory for the calculus, thereby clarifying the proof-theoretic foundations of hybrid type-logical grammars, but, since the translation is simple and direct, it also provides several new parsing strategies for hybrid type-logical grammars. Second, NP-completeness of hybrid type-logical grammars follows immediately. The main embedding result also sheds new light on problems with lambda grammars, which are a subsystem of hybrid type-logical grammars and hence a special case of the translation into first-order linear logic. Abstract categorial grammars are attractive both because of their simplicity — they use the simply