A System of Interaction and Structure II: The Need for Deep Inference

This paper studies properties of the logic , which is an extension of multiplicative linear logic (MLL) with a self-dual non-commutative operator. is presented in the calculus of structures, a proof theoretic formalism that supports deep inference, in which inference rules can be applied anywhere inside logical expressions. The use of deep inference results in a simple logical system for MLL extended with the self-dual non-commutative operator, which has been to date not known to be expressible in sequent calculus. In this paper, deep inference is shown to be crucial for the logic , that is, any restriction on the “depth” of the inference rules of would result in a strictly less expressive logical system.