Control of variable movement through modalities

Develop a modality-based extension of the linear calculus that controls movement, including associativity and commutativity, of variables in contexts, and establish its corresponding semantic and syntactic treatment.

Background

The calculus uses modalities to control duplication and disposal of variables, while the authors note that modalities could also regulate structural operations governing the movement of variables in contexts, such as associativity and commutativity.

The paper suggests that the treatment might parallel its handling of optional distributivity over products, but does not develop the extension or analyze its consequences.

References

Support for this even more general case could work in a similar way as our treatment of (optional) distributivity over products, but is left for future work.

— Arbitrary Functors as Modalities in a Linear Calculus  (2610.23023 - Bernardy et al., 16 Oct 2026) in Section 11, “Even fewer structural rules,” p. 31

The main reason for us leaving it out is that we could not make sense of it with regard to substitution. Indeed, it demands that two substitutions Γ1 ↦ → Δ and Γ2 ↦ → Δ can be merged, which appears impossible in general.

— Arbitrary Functors as Modalities in a Linear Calculus  (2610.23023 - Bernardy et al., 16 Oct 2026) in Section 10, Related work, p. 30