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Belief in Simplicial Complexes

Philip Sink, Adam Bjorndahl

TL;DR

This paper develops a simplicial semantics for belief in multi-agent settings by introducing agent-specific belief subcomplexes and proving a soundness and completeness theorem for a KD45+K axiom; it also analyzes the relation to relational semantics, the role of properness, and prospects for simplicial sets as a way to bypass restrictive assumptions. The approach resolves the non-factivity of belief in simplicial frameworks and provides a modular way to model inter-agent belief, with connections to existing works and future directions. By showing how knowledge implies belief (via $K_a\varphi\rightarrow B_a\varphi$) while avoiding belief trivialization, the framework offers a robust semantic foundation for integrating belief into simplicial epistemic logic and suggests concrete avenues for extending to more general combinatorial structures.

Abstract

We provide a novel semantics for belief using simplicial complexes. In our framework, belief satisfies the \textsf{KD45} axioms and rules as well as the ``knowledge implies belief'' axiom ($Kφ\lthen Bφ$); 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.

Belief in Simplicial Complexes

TL;DR

This paper develops a simplicial semantics for belief in multi-agent settings by introducing agent-specific belief subcomplexes and proving a soundness and completeness theorem for a KD45+K axiom; it also analyzes the relation to relational semantics, the role of properness, and prospects for simplicial sets as a way to bypass restrictive assumptions. The approach resolves the non-factivity of belief in simplicial frameworks and provides a modular way to model inter-agent belief, with connections to existing works and future directions. By showing how knowledge implies belief (via ) while avoiding belief trivialization, the framework offers a robust semantic foundation for integrating belief into simplicial epistemic logic and suggests concrete avenues for extending to more general combinatorial structures.

Abstract

We provide a novel semantics for belief using simplicial complexes. In our framework, belief satisfies the \textsf{KD45} axioms and rules as well as the ``knowledge implies belief'' axiom (); 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.

Paper Structure

This paper contains 6 sections, 6 theorems, 16 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Semantics for Belief
  3. Connection to Relational Semantics
  4. Soundness and Completeness
  5. Related and Future Work
  6. Proofs

Key Result

Proposition 2.1

Let $\mathcal{M}$ be a simplicial belief model with $S_{a} = S_{b}$. Then $\mathcal{M} \models B_{a}(B_{b} \varphi \rightarrow \varphi)$.

Figures (4)

  • Figure 1: A 3-agent relational frame for introspective knowledge and belief
  • Figure 2: A translation of the model from Figure \ref{['fgr:3ag']} into a proper model
  • Figure 3: A translation of the model from Figure \ref{['fgr:3agprop']} into a simplicial model (purple facets belong to $S_a$ and $S_b$, orange facets belong to $S_a$ and $S_c$, while black facets belong to only $S$)
  • Figure 4: A translation of the model from Figure \ref{['fgr:3ag']} into a simplicial simplicial set $S$ together with subsimplicial sets $S_{a}$, $S_{b}$, and $S_{c}$ (purple facets belong to $S_a$ and $S_b$, orange facets belong to $S_a$ and $S_c$, while black facets belong to only $S$)

Theorems & Definitions (12)

  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3: Note1, Proposition 2.4
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • ...and 2 more