Interpolation via Generalized Splitting

2026-07-26

AI Summary

This work addresses the problem of providing a unified proof-theoretic account of Lyndon interpolation across multiple logical systems—including linear logic, classical logic, and modal logics—by introducing a novel approach that does not rely on sequent calculi. Building on deep inference, the method generalizes the splitting lemma to reformulate interpolation as a decomposition of derivations into upper and lower fragments, thereby strengthening and unifying the treatment of interpolation properties. The proposed framework not only extends the applicability of cut-elimination results but also yields new cut-free deep-inference systems for several modal logics. Its successful application across diverse logical systems demonstrates both its generality and effectiveness.

Abstract

We propose a new proof theoretical method for proving Lyndon interpolation. Our proof does not use the sequent calculus but is based on a generalization of the splitting lemma in deep inference. We then formulate the interpolation theorem as a decomposition of a derivation into an up-fragment and a down-fragment. This can be seen as (i) a strengthening of the standard formulation of the interpolation theorem, and (ii) a generalization of the cut elimination theorem. We demonstrate the flexibility of our approach by applying it to linear logic, classical logic, and modal logics. For this, we also introduce novel cut-free proof systems for several modal logics in deep inference.

Problem

• Lyndon interpolation
• deep inference
• splitting lemma
• modal logics
• proof theory

Innovation

• deep inference
• Lyndon interpolation
• splitting lemma
• cut elimination
• modal logics