Undecidability of Linear Logics without Weakening

2025-08-30

AI Summary

This paper investigates the decidability of two variants of Classical Linear Logic (CLL) obtained by removing the weakening rule for exponential modalities: $mathbf{CLLR}$ (with units) and $mathbf{CLLRR}$ (unit-free). Using a precise simulation of two-counter machines, we establish, for the first time, the undecidability of sequent provability in $mathbf{CLLR}$. We then reduce this undecidability to $mathbf{CLLRR}$ by exploiting implicit encodings of weakening via the constants $1$ and $ot$, which remain available even without explicit weakening or unit structural rules. Our results demonstrate that linear logic retains Turing-complete computational power even in the complete absence of weakening and unit-structural rules—challenging the intuition that restricting structural rules necessarily reduces computational complexity. This work provides a sharp characterization of the metatheoretic boundary of linear logic, showing that essential expressive power persists under extreme structural restriction.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

The goal of this paper is to establish that it remains undecidable whether a sequent is provable in two systems in which a weakening rule for an exponential modality is completely omitted from classical propositional linear logic $mathbf{CLL}$ introduced by Girard (1987), which is shown to be undecidable by Lincoln et al. (1992). We introduce two logical systems, $mathbf{CLLR}$ and $mathbf{CLLRR}$. The first system, $mathbf{CLLR}$, is obtained by omitting the weakening rule for the exponential modality of $mathbf{CLL}$. The system $mathbf{CLLR}$ has been studied by several authors, including Meliès-Tabareau (2010), but its undecidability was unknown. This paper shows the undecidability of $mathbf{CLLR}$ by reducing it to the undecidability of $mathbf{CLL}$, where the units $mathbf{1}$ and $ot$ play a crucial role in simulating the weakening rule. We also omit these units from the syntax and inference rules of $mathbf{CLLR}$ in order to define the second system, $mathbf{CLLRR}$. The undecidability of $mathbf{CLLRR}$ is established by showing that the system can simulate any two-counter machine proposed by Minsky (1961).

Problem

• Proving undecidability of linear logic without weakening rule
• Establishing undecidability for CLLR system without exponential modality
• Showing undecidability of CLLRR system without logical units

Innovation

• Omit weakening rule for exponential modality
• Simulate weakening using units 1 and ⊥
• Simulate two-counter machines for undecidability