### Rewriting with Linear Inferences in Propositional Logic

#### Abstract

Linear inferences are sound implications of propositional logic where each variable appears exactly once in the premiss and conclusion. We consider a specific set of these inferences, MS, first studied by Strassburger, corresponding to the logical rules in deep inference proof theory. Despite previous results characterising the individual rules of MS, we show that there is no polynomial-time characterisation of MS, assuming that integers cannot be factorised in polynomial time.
We also examine the length of rewrite paths in MS, utilising a notion of trivialisation to reduce the case with units to the case without, amongst other observations on MS-rewriting and the set of linear inferences in general.