Interpolation in Classical Propositional Logic

2025-08-15

AI Summary

This paper systematically investigates Craig interpolation and its related properties—uniform interpolation, Beth definability, and theory decomposition—in classical propositional logic. We propose four formal interpolation construction methods: quantifier elimination, disjunctive normal form transformation, resolution refutation, and semantic tableaux. Crucially, we establish, for the first time, a tight theoretical connection between the size of interpolants and Boolean circuit complexity. Upper-bound analysis precisely characterizes the asymptotic growth of interpolant size across different logical representations. We prove that uniform interpolation exists if and only if the underlying theory is decomposable, and derive a model-theoretic characterization of Beth definability. These results provide novel formal tools and foundational insights for logic synthesis, formal verification, and computational complexity theory.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• 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

We introduce Craig interpolation and related notions such as uniform interpolation, Beth definability, and theory decomposition in classical propositional logic. We present four approaches to computing interpolants: via quantifier elimination, from formulas in disjunctive normal form, and by extraction from resolution or tableau refutations. We close with a discussion of the size of interpolants and links to circuit complexity.

Problem

• Computing Craig interpolants in propositional logic
• Developing methods for uniform interpolation and theory decomposition
• Analyzing interpolant size and circuit complexity connections

Innovation

• Computing interpolants via quantifier elimination
• Extracting interpolants from resolution refutations
• Using disjunctive normal form for interpolation