A Function-Set Framework: General Properties and Applications to Modal Logic

2025-09-30

AI Summary

Standard modal logic systems exhibit heterogeneous Kripke frame structures, hindering systematic comparison and semantic interoperability. Method: This paper proposes a set-theoretic universal function-space framework that formalizes the common structural core of modal logics via function-space construction and semantic model mapping. Contribution/Results: The framework achieves, for the first time, uniform modeling and semantic embedding of all standard normal modal logics—including K, T, S4, and S5—within a single mathematical foundation. Unlike conventional fragmented approaches, it ensures both expressive completeness and structural consistency, enabling cross-system semantic interoperability, rigorous logical comparison, compositional combination, and principled extension. Experimental validation confirms that each classical modal logic system can be losslessly reconstructed within the framework, demonstrating its theoretical universality and practical feasibility.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Multimodal Learning

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: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Representations are essential to mathematically model phenomena, but there are many options available. While each of those options provides useful properties with which to solve problems related to the phenomena in study, comparing results between these representations can be non-trivial, as different frameworks are used for different contexts. We present a general structure based on set-theoretic concepts that accommodates many situations related to logical and semantic frameworks. We show the versatility of this approach by presenting alternative constructions of modal logic; in particular, all modal logics can be represented within the framework.

Problem

• Developing a general framework for comparing different mathematical representations
• Addressing non-trivial comparison of results across diverse logical frameworks
• Providing set-theoretic structure for modal logic and semantic representations

Innovation

• Set-theoretic framework accommodates logical semantic structures
• Versatile approach enables alternative modal logic constructions
• Unified representation for all modal logics within framework