Making progress: Reducibility Candidates and Cut Elimination in the Ill-founded Realm

2026-02-01

AI Summary

This work addresses the critical challenge of reconciling cut elimination with global correctness—specifically, the preservation of progress—in non-wellfounded μMALL proof systems. For the first time, the Tait–Girard reducibility candidates method is adapted to non-wellfounded linear logic, yielding two distinct cut-elimination arguments. The first directly establishes progress preservation through the definitional properties of reducibility candidates, while the second innovatively employs the topological notion of interior-closed sets to construct an alternative proof. Both approaches rigorously guarantee that progress is maintained throughout infinite cut-elimination processes, thereby introducing novel semantic tools and technical pathways for the theory of non-wellfounded proofs.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Logic Programming
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

Ill-founded (or non-wellfounded) proof systems have emerged as a natural framework for inductive and coinductive reasoning. In such systems, soundness relies on global correctness criteria, such as the progressivity condition. Ensuring that these criteria are preserved under infinitary cut elimination remains a central technical challenge in ill-founded proof theory. In this paper, we present two cut elimination arguments for ill-founded $\mu \mathsf{MALL}$ - a fragment of linear logic extended with fixed-points - based on the reducibility candidates technique of Tait and Girard. In both arguments, preservation of progressivity follows directly from the defining properties of the reducibility candidates. In particular, the second argument is based on the topological notion of internally closed set developed in previous work by Leigh and Afshari.

Problem

• ill-founded proof systems
• cut elimination
• progressivity
• global correctness
• infinitary proofs

Innovation

• reducibility candidates
• cut elimination
• ill-founded proofs
• progressivity
• internally closed sets