Encoding MILL grammars as linear-time ht-systems with non-injective rules

Prove that every hypergraph MILL grammar can be converted into a linear-time hypergraph transformation system with non-injective rules (i.e., rules H → H′ where ext_{H′} is allowed to be non-injective) by a direct encoding of MILL inference rules as hypergraph transformations.

Background

After establishing that linear-time ht-systems map into hypergraph MILL grammars, the authors discuss the converse direction. They claim it should be possible to encode each hypergraph MILL grammar into a linear-time ht-system if non-injective rules are permitted, via a straightforward encoding of MILL inference rules.

However, they explicitly state that they leave proving this encoding for future work, making it an unresolved constructive problem in the paper.