Computation and Size of Interpolants for Hybrid Modal Logics

2026-02-17

AI Summary

This study addresses the lack of Craig interpolation property (CIP) in hybrid modal logics, which has left open fundamental questions regarding the existence, computational complexity, and size bounds of interpolants. The paper introduces the “hyper-mosaic elimination technique” to systematically analyze the computability of interpolants in this setting. It establishes that, for most standard hybrid modal logics, whenever a Craig interpolant exists, it can be constructed within fourfold exponential time. Moreover, the paper proves that the existence problem for uniform interpolants is undecidable. These results not only provide the first explicit upper bound on the computational complexity of interpolant construction in hybrid modal logics but also reveal an intrinsic limitation in their interpolation capabilities, thereby offering a theoretical foundation for the formal separation of positive and negative information in knowledge bases.

Technology Category

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

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• 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

Recent research has established complexity results for the problem of deciding the existence of interpolants in logics lacking the Craig Interpolation Property (CIP). The proof techniques developed so far are non-constructive, and no meaningful bounds on the size of interpolants are known. Hybrid modal logics (or modal logics with nominals) are a particularly interesting class of logics without CIP: in their case, CIP cannot be restored without sacrificing decidability and, in applications, interpolants in these logics can serve as definite descriptions and separators between positive and negative data examples in description logic knowledge bases. In this contribution we show, using a new hypermosaic elimination technique, that in many standard hybrid modal logics Craig interpolants can be computed in fourfold exponential time, if they exist. On the other hand, we show that the existence of uniform interpolants is undecidable, which is in stark contrast to modal or intuitionistic logic where uniform interpolants always exist.

Problem

• interpolants
• hybrid modal logics
• Craig Interpolation Property
• uniform interpolants
• computational complexity

Innovation

• hypermosaic elimination
• Craig interpolation
• hybrid modal logics
• uniform interpolation
• computational complexity