Completeness of MILL with respect to hypergraph language models

Establish whether full first-order multiplicative intuitionistic linear logic (MILL) is complete with respect to the hypergraph language models introduced in Definition 5.1, i.e., show that every sequent true in all such models is derivable in MILL.

Background

The second part of the paper introduces hypergraph LLMs grounded in HR-algebras, interpreting MILL formulas as sets of hypergraphs with tensor as parallel composition and quantifiers as set operations. The authors prove soundness for MILL and completeness only for the {⊗,∀}-fragment, leaving the full logic’s completeness unresolved.

In the conclusion, they explicitly pose the completeness of full MILL with respect to these hypergraph LLMs as an open question.