Conjunctive categorial grammars and Lambek grammars with additives

TL;DR

This paper introduces a new family of categorial grammars with conjunction, demonstrating their expressive equivalence to conjunctive grammars and embedding into Lambek calculus, with implications for computational complexity.

Contribution

It defines a new family of categorial grammars with conjunction, proves their expressive power matches conjunctive grammars, and embeds them into Lambek calculus with disjunction.

Findings

Equivalent expressive power to conjunctive grammars

Embedding into Lambek calculus with conjunction and disjunction

NP-completeness result for certain sets

Abstract

A new family of categorial grammars is proposed, defined by enriching basic categorial grammars with a conjunction operation. It is proved that the formalism obtained in this way has the same expressive power as conjunctive grammars, that is, context-free grammars enhanced with conjunction. It is also shown that categorial grammars with conjunction can be naturally embedded into the Lambek calculus with conjunction and disjunction operations. This further implies that a certain NP-complete set can be defined in the Lambek calculus with conjunction. We also show how to handle some subtle issues connected with the empty string. Finally, we prove that a language generated by a conjunctive grammar can be described by a Lambek grammar with disjunction (but without conjunction).