A note on undecidability of propositional non-associative linear logics

Abstract: We introduce a non-associative and non-commutative version of propositional intuitionistic linear logic, called propositional non-associative non-commutative intuitionistic linear logic (NACILL for short). We prove that NACILL and any of its extensions by the rules of exchange and/or contraction are undecidable. Furthermore, we introduce two types of classical versions of NACILL, i.e., an involutive version of NACILL and a cyclic and involutive version of NACILL. We show that both of these logics are also undecidable.