The Linear Logic of Multisets

We consider a cumulative hierarchy of multisets over some set of urelements, equipped with additive union ] and a transform relation ¤, and investigate the Horn fragments of Intuitionistic Linear Logic (ILL) that are interpretable in it. The operator ! is defined in an asymptotic way which causes some deviations from the linear-theoretic behavior. Soundness, completeness and partial completeness results are proved for the various fragments. Certain processes of multisets suggest rules for the multiplicatives not compatible with full ILL. One such rule added to the Horn fragment makes the system sound and complete with respect to “coherent processes”.