Grammars Based on a Logic of Hypergraph Languages

TL;DR

This paper introduces hypergraph Lambek grammars, a more powerful formalism than hyperedge replacement grammars, capable of generating complex hypergraph languages with the same computational complexity.

Contribution

It presents hypergraph Lambek grammars based on a new logic, expanding the class of hypergraph languages that can be generated beyond HRGs.

Findings

Hypergraph Lambek grammars can generate languages of unbounded connectivity.

They include languages like all graphs, bipartite graphs, and regular graphs.

The formalism maintains NP-complete complexity.

Abstract

The hyperedge replacement grammar (HRG) formalism is a natural and well-known generalization of context-free grammars. HRGs inherit a number of properties of context-free grammars, e.g. the pumping lemma. This lemma turns out to be a strong restriction in the hypergraph case: it implies that languages of unbounded connectivity cannot be generated by HRGs. We introduce a formalism that turns out to be more powerful than HRGs while having the same algorithmic complexity (NP-complete). Namely, we introduce hypergraph Lambek grammars; they are based on the hypergraph Lambek calculus, which may be considered as a logic of hypergraph languages. We explain the underlying principles of hypergraph Lambek grammars, establish their basic properties, and show some languages of unbounded connectivity that can be generated by them (e.g. the language of all graphs, the language of all bipartite graphs, the language of all regular graphs).