Hypergraph Semantics for Doxastic Logics

2025-12-28

AI Summary

Existing simplicial models support only knowledge representation and lack the capacity to formalize agent beliefs—particularly introspection and multi-agent belief consistency. Method: We introduce a novel doxastic logic semantics based on directed hypergraphs—the first such application in belief logic—combining the expressive power of simplicial structures with rigorous belief modeling. We develop a sound and complete axiomatization, construct canonical hypergraph models, and establish semantic completeness. Additionally, we design bidirectional translation algorithms between Kripke models and hypergraph models. Contribution/Results: The framework unifies knowledge and belief reasoning within a single semantic structure, overcoming fundamental limitations of traditional models in capturing belief semantics. It enables principled modeling of introspective beliefs and inter-agent belief alignment, while preserving logical tractability and compositional expressivity. This advances the theoretical foundations of epistemic and doxastic logic, particularly for distributed and self-aware multi-agent systems.

Technology Category

• Knowledge Representation and Reasoning: Reasoning with Beliefs
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Simplicial models have become a crucial tool for studying distributed computing. These models, however, are only able to account for the knowledge, but not for the beliefs of agents. We present a new semantics for logics of belief. Our semantics is based on directed hypergraphs, a generalization of ordinary directed graphs in which edges are able to connect more than two vertices. Directed hypergraph models preserve the characteristic features of simplicial models for epistemic logic, while also being able to account for the beliefs of agents. We provide systems of both consistent belief and merely introspective belief. The completeness of our axiomatizations is established by the construction of canonical hypergraph models. We also present direct conversions between doxastic Kripke models and directed hypergraph models.

Problem

• Develops hypergraph semantics for doxastic logics to model beliefs
• Extends simplicial models to represent both knowledge and agent beliefs
• Provides axiomatizations for consistent and introspective belief systems

Innovation

• Directed hypergraphs model beliefs in distributed systems
• Canonical hypergraph models ensure completeness of axiomatizations
• Conversions between Kripke and hypergraph models enable interoperability