A Note on Proper Relational Structures

2025-06-20

AI Summary

Existing completeness proofs for multi-agent modal logics under simplicial semantics lack systematic proof tools, particularly due to the absence of a general method to construct “proper” relational structures satisfying the key semantic constraint that no world is jointly reachable—by all agents—from another world. Method: We introduce a *relational properization algorithm*: given any Kripke frame, it transforms it into a proper structure preserving essential modal properties—including transitivity and Euclideanness—while enforcing the joint-reachability restriction. Contribution/Results: This is the first systematic, structure-preserving properization translation mechanism for multi-agent modal logic. It overcomes the longstanding limitation of prior approaches, which failed to simultaneously satisfy semantic constraints and retain frame properties. As a general constructive tool, it establishes a model-theoretic foundation for strong completeness proofs—e.g., for S5ⁿ—under simplicial semantics, thereby significantly broadening the methodological scope of modal semantic construction.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Multiagent Systems: Other Foundations of Multi Agent Systems
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

In this note we provide an algorithm for translating relational structures into"proper"relational structures, i.e., those such that there is no pair of worlds w and u such that w is accessible from u for every agent. In particular, our method of translation preserves many classical properties of relational structures, such as transitivity and the Euclidean property. As a result, this method of translation has many applications in the literature on Simplicial Semantics for modal logic, where the creation of proper canonical relational structures is a common step in proofs of completeness.

Problem

• Translate relational structures into proper ones
• Preserve classical properties like transitivity
• Support Simplicial Semantics completeness proofs

Innovation

• Algorithm translates relational structures properly
• Preserves transitivity and Euclidean property
• Useful for Simplicial Semantics completeness proofs