Mathematical Structures in Computer Science,
2018 · doi: 10.1017/S0960129518000117
Linear logical frameworks with subexponentials have been used for the specification of,among other systems, proof systems, concurrent programming languages and linearauthorisation logics. In these frameworks, subexponentials can be configured to allow or notfor the application of the contraction and weakening rules while the exchange rule canalways be applied. This means that formulae in such frameworks can only be organised assets and multisets of formulae not being possible to organise formulae as lists of formulae.This paper investigates the proof theory of linear logic proof systems in thenon-commutative variant. These systems can disallow the application of exchange rule onsome subexponentials. We investigate conditions for when cut elimination is admissible inthe presence of non-commutative subexponentials, investigating the interaction of theexchange rule with the local and non-local contraction rules. We also obtain some newundecidability and decidability results on non-commutative linear logic with subexponentials.