A Non-Elementary Complexities for Branching VASS, MELL, and Extensions

— the undecidability of provability in propositional linear logic (LL), thanks to a reduction from the halting problem in Minsky machines proved by Lincoln, Mitchell, Scedrov, and Shankar [1992], who initiated much of this line of work, — the decidability of the !-Horn fragment of multiplicative exponential linear logic, proved by Kanovich [1995] by reduction to reachability in vector addition systems, — the decidability of provability in affine linear logic, first shown by Kopylov using a notion of vector addition games [Kopylov 2001], — the ACKERMANN-completeness of provability in the conjunctive implicative fragment of relevance logic, proved by Urquhart [1999], using reductions to and from expansive alternating vector addition systems, and