Non-associative, Non-commutative Multi-modal Linear Logic

Adding multi-modalities (called subexponentials) to linear logic enhances its power as a logical framework, which has been extensively used in the specification of e.g. proof systems, programming languages and bigraphs. Initially, subexponentials allowed for classical, linear, affine or relevant behaviors. Recently, this framework was enhanced so to allow for commutativity as well. In this work, we close the cycle by considering associativity. We show that the resulting system (acLLΣ ) admits the (multi)cut rule, and we prove two undecidability results for fragments/variations of acLLΣ