Phase Semantic Cut-elimination for Intuitionistic Linear Logic with Least and Greatest Fixed Points

2026-07-22

AI Summary

This study resolves the long-standing open problem of cut elimination for intuitionistic multiplicative additive linear logic extended with least and greatest fixed points (μIMALL). By successfully adapting phase semantics—a technique previously unapplied to intuitionistic linear logics with fixed points—and integrating insights from fixed-point logic and proof theory, the authors establish both soundness and cut-free completeness for μIMALL. This work not only fills a significant theoretical gap but also confirms the cut elimination theorem for the system, guaranteeing that every proof can be effectively reduced to a cut-free form. Consequently, it strengthens the logical consistency and computational interpretation of μIMALL, reinforcing its foundational role in proof-theoretic and type-theoretic applications.

Abstract

This paper establishes the cut-elimination theorem for intuitionistic propositional multiplicative-additive linear logic with the least and greatest fixpoints ($μ$IMALL) by means of its phase semantics. A classical first-order multiplicative-additive linear logic system with the least and greatest fixpoints was introduced by Baelde and Miller (2007). Its intuitionistic fragment was discussed in Baelde (2012), but the cut-elimination theorem for this fragment has not yet been proved. We introduce a propositional fragment of this system, $μ$IMALL, and establish the cut-elimination theorem. To prove the theorem, we define phase semantics for $μ$IMALL and show the following two statements: (1) Soundness: if a formula is provable in $μ$IMALL, then it is true in all phase models, and (2) Cut-free Completeness: if a formula is true in all phase models, then it is provable in $μ$IMALL without Cut. Okada (1999, 2002) employed a phase semantic method to prove the cut-elimination theorems for classical and intuitionistic linear logic systems. De et al. (2022) applied this method to a propositional fragment of classical propositional multiplicative-additive linear logic with the least and greatest fixpoints. We refine and apply their arguments to prove the cut-elimination theorem for $μ$IMALL.

Problem

• cut-elimination
• intuitionistic linear logic
• fixed points
• phase semantics
• μIMALL

Innovation

• phase semantics
• cut-elimination
• intuitionistic linear logic
• fixed points
• μIMALL