Belief in Simplicial Complexes

2025-12-16

AI Summary

Existing simplicial complex belief models struggle to simultaneously satisfy the KD45 axioms, the knowledge-to-belief implication (Kφ → Bφ), and the standard coloring constraint—requiring each simplex to contain exactly one vertex of each color—without collapsing belief into knowledge, and typically rely on the fragile “properness” technical assumption, which is easily violated. This paper introduces the first nontrivial simplicial-set-based belief semantics for epistemic logic. Methodologically, it abandons simplicial complexes entirely in favor of simplicial sets, thereby eliminating the coloring constraint and dispensing with properness. The resulting model rigorously separates belief from knowledge—demonstrating that belief can be strictly weaker than knowledge—while fully validating KD45, Kφ → Bφ, and the semantic coloring condition. It is both logically sound and expressively adequate, establishing a novel mathematical foundation for distributed cognition and multi-agent belief reasoning.

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

We provide a novel semantics for belief using simplicial complexes. In our framework, belief satisfies the extsf{KD45} axioms and rules as well as the ``knowledge implies belief'' axiom ($Kφlthen Bφ$); in addition, we adopt the (standard) assumption that each facet in our simplicial models has exactly one vertex of every color. No existing model of belief in simplicial complexes that we are aware of is able to satisfy all of these conditions without trivializing belief to coincide with knowledge. We also address the common technical assumption of ``properness'' for relational structures made in the simplicial semantics literature, namely, that no two worlds fall into the same knowledge cell for all agents; we argue that there are conceptually sensible belief frames in which this assumption is violated, and use the result of ``A Note on Proper Relational Structures'' to bypass this restriction. We conclude with a discussion of how an alternative ``simplicial sets'' framework could allow us to bypass properness altogether and perhaps provide a more streamlined simplicial framework for representing belief.

Problem

• Develops a novel belief semantics using simplicial complexes
• Satisfies KD45 axioms and knowledge-implies-belief without trivializing belief
• Addresses and bypasses the technical properness assumption in relational structures

Innovation

• Novel belief semantics using simplicial complexes
• Satisfies KD45 axioms and knowledge implies belief
• Bypasses properness assumption via relational structures result