On Constraint-Based Lambek Calculi

We explore the consequences of layering a Lambek proof system over an arbitrary (constraint) logic. A simple model-theoretic semantics for our hybrid language is provided for which a particularly simple combination of Lambek's and the proof system of the base logic is complete. Furthermore the proof system for the underlying base logic can be assumed to be a black box. The essential reasoning needed to be performed by the black box is that of entailment checking. Assuming feature logic as the base logic entailment checking amounts to a subsumption test which is a well-known quasi-linear time decidable problem. 1 Background In recent years there has been a growing awareness of a need to design grammar logics that incorporate both the resource-mindedness of categorial grammars and the typed constraint-based approach of HPSG 15] 16]. We believe that the long-term goal of this enterprise is to provide an incremental and largely deterministic model of sentence comprehension within a constraint-based setting | something that current HPSG lacks. The constraint-based basis is important, since this provides an excellent knowledge representation, engineering and structuring environment for NLP. Although, there has been a lot of work on ner systems of categorial grammars 11] 12] none directly build on a uniication based framework. In a system such as that of Moortgat's 10] categories can be built using tuples containing type, syntax and structure in the spirit of UCG 23]. However, feature descriptions cannot be employed for instance to describe the syntax. Thus the consequences of a more direct and perhaps more pragmatic approach remains unexplored. In this paper, this is precisely what we do. UCG 23] and CUG 21] provided the initial integration of such an approach. However, both lacked a rigorous model-theory. Even worse, their underlying uniication-based proof system is incompatible with the straightforward model theory of feature-based categorial types (where feature terms take the place of basic categories like in (cat: s)=(cat: np & case: acc)), when making the following