Annotation-Free Sequent Calculi for Full Intuitionistic Linear Logic -- Extended Version

TL;DR

This paper introduces a simple, annotation-free display calculus for full intuitionistic linear logic (FILL) and its extension BiILL, providing cut-elimination, deep inference, and proof search properties, advancing proof theory for linear logic.

Contribution

It presents the first annotation-free display calculus for FILL and BiILL, with cut-elimination, deep inference, and conservativity over FILL, simplifying proof-theoretic analysis.

Findings

The calculus satisfies Belnap's cut-elimination theorem.

Proofs of FILL formulas contain no exclusion trace.

The calculus is NP-complete, with subformula property and terminating proof search.

Abstract

Full Intuitionistic Linear Logic (FILL) is multiplicative intuitionistic linear logic extended with par. Its proof theory has been notoriously difficult to get right, and existing sequent calculi all involve inference rules with complex annotations to guarantee soundness and cut-elimination. We give a simple and annotation-free display calculus for FILL which satisfies Belnap's generic cut-elimination theorem. To do so, our display calculus actually handles an extension of FILL, called Bi-Intuitionistic Linear Logic (BiILL), with an `exclusion' connective defined via an adjunction with par. We refine our display calculus for BiILL into a cut-free nested sequent calculus with deep inference in which the explicit structural rules of the display calculus become admissible. A separation property guarantees that proofs of FILL formulae in the deep inference calculus contain no trace of exclusion. Each such rule is sound for the semantics of FILL, thus our deep inference calculus and display calculus are conservative over FILL. The deep inference calculus also enjoys the subformula property and terminating backward proof search, which gives the NP-completeness of BiILL and FILL.