Multimodality in the Hypergraph Lambek Calculus

Abstract

The multimodal Lambek calculus NL♦ is an extension of the Lambek calculus that includes several product operations (some of them being commutative or/and associative), unary modalities, and corresponding residual implications. In this work, we relate this calculus to the hypergraph Lambek calculus HL. The latter is a general pure logic of residuation defined in a sequent form; antecedents of its sequents are hypergraphs, and the rules of HL involve hypergraph transformation. Our main result is the embedding of the multimodal Lambek calculus (with at most one associative product) in HL. It justifies that HL is a very general Lambek-style logic and also provides a novel syntactic interface for NL♦: antecedents of sequents of NL♦ are represented as tree-like hypergraphs in HL, and they are derived from each other by means of hyperedge replacement. The advantage of this embedding is that commutativity and associativity are incorporated in the sequent structure rather than added as separate rules. Besides, modalities of NL♦ are represented in HL using the product and the division of HL, which explicitizes their residual nature.