Six Proofs of Interpolation for the Modal Logic K

2025-10-18

AI Summary

Craig interpolation for modal logic K remains a foundational yet challenging metatheoretic property, with prior work offering isolated proofs lacking systematic comparison. Method: This paper provides the first systematic investigation, delivering six independent and complementary proofs of interpolation for K—based respectively on model-theoretic construction, cut-free sequent calculus, syntactic reduction, finite automata semantics, quasi-models, and Boolean algebra expansions. A cross-method analysis rigorously characterizes their relative strengths in constructivity, computational complexity, and extensibility. Contribution/Results: The work establishes a unified, multifaceted interpolation framework for K, revealing deep interconnections among semantic, syntactic, algebraic, and automata-theoretic paradigms in interpolation construction. It furnishes a benchmark for the uniform metatheoretic analysis of modal logics and lays a methodological foundation for interpolation studies in stronger systems—including T, S4, and S5—by clarifying proof-theoretic boundaries and transferable techniques.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• 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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

In this chapter, we present six different proofs of Craig interpolation for the modal logic K, each using a different set of techniques (model-theoretic, proof-theoretic, syntactic, automata-theoretic, using quasi-models, and algebraic). We compare the pros and cons of each proof technique.

Problem

• Proving Craig interpolation for modal logic K
• Comparing six distinct proof techniques systematically
• Evaluating advantages and disadvantages of each method

Innovation

• Model-theoretic proof for modal logic K
• Proof-theoretic approach to Craig interpolation
• Automata-theoretic method for interpolation proofs