On the Expressivity of Two Refinements of Multiplicative Exponential Linear Logic

The decidability of multiplicative exponential linear logic (MELL) is currently open. I show that two independently interesting refinements of MELL that alter only the syntax of proofs—leaving the underlying truth untouched— are undecidable. The first refinement uses new modal connectives between the linear and the unrestricted judgments, and the second is based on focusing with priority assignments that conforms to a staging discipline. Both refinements can adequately encode the transitions of a two-register Minsky machine. While neither refinement is weak enough to entail the undecidability of MELL, they show that no additive connectives are necessary for undecidability.