Interpolation in Proof Theory

2026-02-18

AI Summary

This work presents a systematic investigation of the interpolation property across classical, intuitionistic, modal, and substructural logics, proposing a unified, constructive, modular, and syntax-driven methodology that integrates Maehara’s and Pitts’ classical techniques into a general proof-theoretic framework. By uncovering a structural correspondence between interpolation and well-behaved proof systems, the study not only establishes the existence of interpolation theorems for a wide range of logics but also delivers a reusable blueprint for constructing interpolation proofs within modern formal proof systems. This approach emphasizes the role of syntactic structure in enabling modular and scalable interpolation results, thereby advancing the theoretical understanding and practical applicability of interpolation in diverse logical settings.

Technology Category

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

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: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

This chapter provides a comprehensive overview of proof-theoretic methods for establishing interpolation properties across a range of logics, including classical, intuitionistic, modal, and substructural logics. Central to the discussion are two foundational techniques: Maehara's method for Craig interpolation and Pitts' method for uniform interpolation. The chapter demonstrates how these methods lead to results on the existence of well-behaved proof systems in the contemporary framework of universal proof theory and how they provide a road map for constructing interpolation proofs using modern proof formalisms. The emphasis of the chapter is on constructive, modular, and syntax-driven techniques that illuminate deeper connections between interpolation properties and proof systems.

Problem

• interpolation
• Craig interpolation
• uniform interpolation
• proof theory
• logical systems

Innovation

• proof theory
• interpolation
• Maehara's method
• Pitts' method
• uniform interpolation