Hypergraph Lambek Calculus

TL;DR

This paper introduces the hypergraph Lambek calculus (HL), a logical framework extending the Lambek calculus to graphs, enabling more powerful graph grammars with properties like cut elimination and NP-complete derivability.

Contribution

It extends the Lambek calculus to graphs, creating the hypergraph Lambek calculus (HL), which surpasses hyperedge replacement grammars in expressive power and retains key logical properties.

Findings

HL can generate all graphs without isolated nodes

HL can generate bipartite graphs

Membership problems are NP-complete

Abstract

It is known that context-free grammars can be extended to generating graphs resulting in graph grammars; one of such fundamental approaches is hyperedge replacement grammars. On the other hand there are type-logical grammars which also serve to describe string languages. In this paper, we investigate how to extend the Lambek calculus ($\mathrm{L}$) and grammars based on it to graphs. The resulting approach is called hypergraph Lambek calculus ($\mathrm{HL}$). It is a logical sequential calculus whose sequents are graphs; it naturally extends the Lambek calculus and also allows one to embed its variants (commutative $\mathrm{L}$, $\mathrm{NL\diamondsuit}$, $\mathrm{L}_{\mathbf{1}}^\ast$). Besides, many properties of the Lambek calculus (cut elimination, counters, models) can be lifted to $\mathrm{HL}$. However, while Lambek grammars are equivalent to context-free grammars in the string case, hypergraph Lambek grammars are much more powerful than hyperedge replacement grammars. Particularly, the former can generate the language of all graphs without isolated nodes; the language of all bipartite graphs; finite intersections of languages generated by hyperedge replacement grammars. Nevertheless, the derivability problem in $\mathrm{HL}$ and the membership problem for grammars based on $\mathrm{HL}$ are NP-complete as well as the membership problem for hyperedge replacement grammars.