Bi-Intuitionistic Linear Logic (BILL) is an extension of Intuitionistic Linear Logic with a par, dual to the tensor, and subtraction, dual to linear implication. It is the logic of categories with a monoidal closed and a monoidal co-closed structure that are related by linear distributivity, a strength of the tensor over the par. It conservatively extends Full Intuitionistic Linear Logic (FILL), which includes only the par.

We give proof nets for the multiplicative, unit-free fragment MBILL-. Correctness is by local rewriting in the style of Danos contractibility. This rewrite relation yields sequentialization into a relational sequent calculus that extends the existing one for FILL. We give a second, geometric correctness condition via Danos-Regnier switching, and demonstrate composition both inductively and as a one-off global operation.