Converse to the linear-time ht-system lower bound

Determine whether every hypergraph MILL grammar, as defined in Definition 4.2, can be converted into an equivalent linear-time hypergraph transformation system (Definition 2.15) that generates the same hypergraph language, i.e., establish the converse of Theorem MILL1G>LTHTS.

Background

The paper proves in Theorem MILL1G>LTHTS that each linear-time hypergraph transformation system (ht-system) can be converted into an equivalent hypergraph MILL grammar, showing a strong lower bound on the expressive power of hypergraph MILL grammars. However, whether this correspondence holds in the reverse direction remains unresolved. Clarifying the converse would precisely delimit the relation between hypergraph MILL grammars and ht-systems and help characterise the class of hypergraph languages definable by MILL grammars.

In the conclusion, the authors explicitly identify this as one of the open questions for future work, highlighting the need to establish if hypergraph MILL grammars can be fully captured by linear-time ht-systems as defined earlier in the paper.