Ordered Adjoint Logic (Extended Version)

2026-05-18

AI Summary

This work proposes a unified framework based on adjoint modalities to overcome the limitations of traditional ordered logic, which enforces linear resource usage and thus struggles to accommodate structural rules such as weakening, contraction, and exchange. For the first time, the framework systematically integrates a range of fine-grained structural properties—including weakening, left and right contraction, and commutativity—into ordered logic. By employing a sequent calculus augmented with adjoint modalities and a natural deduction system featuring implicit structural rules, the approach transcends strict linearity while preserving high expressivity. The resulting system enjoys cut elimination and admits decidable proof checking, thereby establishing a robust theoretical foundation for higher-order, resource-sensitive programming languages and logical systems.

Abstract

Ordered logics and type systems have been used in a variety of applications including computational linguistics, memory allocation, stream processing, logical frameworks, parametricity, and enforcing security protocols. In most formulations, ordered types are also linear, requiring each resource to be used exactly once. Prior work by Kanovich et al. has investigated calculi that relax this constraint through subexponentials within a linear ordered logic. We generalize their work by using adjoint modalities to combine logics with varying fine-grained structural properties, including weakening, left contraction, right contraction, left mobility, and right mobility. We show that the resulting sequent calculus admits cut elimination. We further provide a natural deduction formulation in which structural rules are implicit, and show that proof checking for this system is decidable. This makes it a suitable foundation for an expressive adjoint programming language or logical framework.

Problem

• ordered logic
• linear types
• structural rules
• adjoint modalities
• subexponentials

Innovation

• Ordered Logic
• Adjoint Modalities
• Cut Elimination
• Natural Deduction
• Structural Properties