Section: New Results

Intuitionistic Logic in the Calculus of Structures

Participants : Nicolas Guenot, Lutz Straßburger.

The calculus of structures has mainly be used for “classical” logics that come with a De Morgan duality. The reason is that all normalization procedures developed so far for the calculus of structures rely on this De Morgan duality.

In this work, we give two proof systems for implication-only intuitionistic logic in the calculus of structures. The first is a direct adaptation of the standard sequent calculus to the deep inference setting. It comes with a cut elimination procedure that is similar to the one from the sequent calculus, using a non-local rewriting. The second system is the symmetric completion of the first, as normally given in deep inference for logics with a De Morgan duality: all inference rules have duals, as cut is dual to the identity axiom. For this symmetric system we prove a generalization of cut elimination, that we call symmetric normalization, where all rules dual to standard ones are permuted up in the derivation. The result is a decomposition theorem having cut elimination and interpolation as corollaries. This work has been presented at the CSL-LICS 2014 conference [22] .