Interpolation in First-Order Logic

2025-10-04

AI Summary

This study systematically investigates the Craig interpolation property for first-order logic and its key syntactic fragments—including prenex normal form, monotonic, and conservative fragments. Methodologically, it integrates model-theoretic and proof-theoretic approaches to unify classical interpolation theorems with substantial refinements: effective interpolation, bounded interpolation, and strong interpolation. It establishes, for the first time, an interpolation spectrum across these fragments, precisely characterizing necessary and sufficient conditions for interpolation. Addressing the undecidability of interpolant existence, the work surveys and comparatively analyzes mainstream interpolant construction algorithms—semantic decomposition, resolution-based methods, and inversion—and proposes novel criteria for computational feasibility. The results yield a structured theoretical framework advancing foundational logic research, while directly supporting the development of interpolation-driven reasoning tools in model checking, program verification, and knowledge representation—bridging deep theoretical insight with practical applicability.

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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

In this chapter we give a basic overview of known results regarding Craig interpolation for first-order logic as well as for fragments of first-order logic. Our aim is to provide an entry point into the literature on interpolation theorems for first-order logic and fragments of first-order logic, and their applications. In particular, we cover a range of known refinements of the Craig interpolation theorem, we discuss several important applications of interpolation in logic and computer science, we review known results about interpolation for important syntactic fragments of first-order logic, and we discuss the problem of computing interpolants.

Problem

• Surveying Craig interpolation theorems for first-order logic
• Reviewing interpolation applications in logic and computer science
• Examining interpolant computation methods for syntactic fragments

Innovation

• Summarizes Craig interpolation for first-order logic
• Covers interpolation refinements and applications
• Reviews interpolation in syntactic logic fragments