On Symbol Elimination and Uniform Interpolation in Theory Extensions

2025-06-02

AI Summary

This paper addresses the construction problem of generalized uniform interpolation in theory extensions. To overcome the limited applicability of existing approaches, we first formally define generalized uniform interpolation and establish its systematic connection to symbol elimination. We propose a novel paradigm that leverages symbol elimination as a bridge, utilizing quantifier-free uniform interpolation algorithms to construct generalized uniform interpolants. We precisely characterize the reducibility conditions and applicability boundaries of this method, and prove that generalized uniform interpolation is constructively decidable over theory extensions that support quantifier-free uniform interpolation. This result significantly broadens the class of theories amenable to uniform interpolation, thereby providing a more general logical foundation for modular reasoning and formal verification.

Technology Category

Constraint Satisfaction and Optimization: Satisfiability Modulo TheoriesKnowledge Representation and Reasoning: Automated Reasoning and Theorem ProvingReasoning 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 semanticsGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsUser Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

We define a notion of general uniform interpolant, generalizing the notions of cover and of uniform interpolant and identify situations in which symbol elimination can be used for computing general uniform interpolants. We investigate the limitations of the method we propose, and identify theory extensions for which the computation of general uniform interpolants can be reduced to symbol elimination followed by the computation of uniform quantifier-free interpolants in extensions with uninterpreted function symbols of theories allowing uniform quantifier-free interpolation.

Problem

• Defining general uniform interpolant in theory extensions
• Using symbol elimination for computing uniform interpolants
• Identifying limitations in theory extensions for interpolation

Innovation

• General uniform interpolant extends cover concept
• Symbol elimination computes general uniform interpolants
• Reduction to quantifier-free interpolation in extensions