Connectivity at the crossroad of intuitionistic and classical polarizations in linear logic

2026-07-22

AI Summary

This work addresses a longstanding limitation in Multiplicative Exponential Linear Logic (MELL), where existing correctness criteria are merely necessary and thus insufficient for effectively reconstructing sequent calculus proofs. We introduce a geometric constraint that strengthens connectivity-based correctness conditions to become both sufficient and efficiently decidable, leading to the definition of a new fragment, VMELL, which unifies classical and intuitionistic polarizations. By integrating proof nets, Danos–Regnier correctness graphs, and geometric analysis, we provide an explicit translation from bang-calculus terms into VMELL and precisely characterize their structure at the proof-net level. This framework successfully simulates the cut-elimination dynamics of bang reduction and uniformly accommodates the standard translations of both call-by-name and call-by-value λ-calculi.

Abstract

We investigate a property that extends the Danos-Regnier correctness criterion for linear logic proof-structures. The property applies to the correctness graphs of a proof-structure: it states that any such graph is acyclic and the number of its connected components is exactly one more than the number of nodes bottom or weakening. This is known to be necessary but not sufficient in multiplicative exponential linear logic (MELL) to recover a sequent calculus proof from a proof-structure. We present a geometric restriction on proof-structures allowing us to turn this necessary property into a sufficient one, computationally efficient: we can thus introduce the notable fragment VMELL of MELL for which the property is indeed a correctness criterion. The fragment VMELL brings together the classical and intuitionistic polarizations. We translate the bang calculus terms into proof-nets of VMELL, factorize the usual translations in linear logic of the call-by-name and call-by-value lambda-calculi, prove that cut elimination simulates bang reduction, and provide an explicit characterization of the bang calculus terms as proof-nets.

Problem

• linear logic
• proof-structures
• correctness criterion
• MELL
• polarization

Innovation

• proof-nets
• linear logic
• polarization
• bang calculus
• correctness criterion