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Simplicial Belief

Christian Cachin, David Lehnherr, Thomas Studer

TL;DR

The paper addresses modeling belief within simplicial epistemic models by introducing polychromatic complexes where adjacent vertices may share colors, enabling a plausibility-based interpretation of belief. It develops a homeomorphy between topology and epistemic logic through a well-founded plausibility order, defining safe belief and most-plausible belief, and proving that belief need not align with truth while knowledge can entail belief. It also proves a no-knowledge-gain result along model morphisms for positive formulas, while showing that belief gain can occur, underscoring non-monotonic reasoning. The work compares polychromatic models to alternative formalisms (multisets and simplicial sets), discusses their advantages and limitations, and outlines directions for future work, including group belief and neighborhood semantics. Overall, this work provides the first topological semantics for belief in simplicial models and lays groundwork for further integration with distributed computing and epistemic reasoning.

Abstract

Recently, much work has been carried out to study simplicial interpretations of modal logic. While notions of (distributed) knowledge have been well investigated in this context, it has been open how to model belief in simplicial models. We introduce polychromatic simplicial complexes, which naturally impose a plausibility relation on states. From this, we can define various notions of belief.

Simplicial Belief

TL;DR

The paper addresses modeling belief within simplicial epistemic models by introducing polychromatic complexes where adjacent vertices may share colors, enabling a plausibility-based interpretation of belief. It develops a homeomorphy between topology and epistemic logic through a well-founded plausibility order, defining safe belief and most-plausible belief, and proving that belief need not align with truth while knowledge can entail belief. It also proves a no-knowledge-gain result along model morphisms for positive formulas, while showing that belief gain can occur, underscoring non-monotonic reasoning. The work compares polychromatic models to alternative formalisms (multisets and simplicial sets), discusses their advantages and limitations, and outlines directions for future work, including group belief and neighborhood semantics. Overall, this work provides the first topological semantics for belief in simplicial models and lays groundwork for further integration with distributed computing and epistemic reasoning.

Abstract

Recently, much work has been carried out to study simplicial interpretations of modal logic. While notions of (distributed) knowledge have been well investigated in this context, it has been open how to model belief in simplicial models. We introduce polychromatic simplicial complexes, which naturally impose a plausibility relation on states. From this, we can define various notions of belief.
Paper Structure (6 sections, 10 theorems, 50 equations, 7 figures)

This paper contains 6 sections, 10 theorems, 50 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Simplicial Knowledge
  3. Simplicial Belief
  4. Knowledge Gain
  5. Alternative Formalizations
  6. Conclusion

Key Result

Lemma 1

Let $\mathcal{C}\xspace = (C,\chi,W, \ell)$ be a simplicial model. For each $G\subseteq \mathsf{Ag} \xspace$, the relation $\sim_G$ is an equivalence relation on $\mathsf{Alive} \xspace_\mathcal{C}\xspace(G)$ and empty otherwise.

Figures (7)

  • Figure 1: A simplicial complex $\mathcal{C}\xspace_1$ in which the agents $a$ and $b$ cannot distinguish between the worlds $X$ and $Y$.
  • Figure 2: Despite being in $X$, agent $a$ thinks that the actual world is $Y$. Thus, $a$ wrongly believes that $c$ died.
  • Figure 3: A polychromatic simplicial complex that models the same situation as $\mathcal{C}\xspace_2$ but without a belief function.
  • Figure 4: Unicolored edges are used to represent a distance between worlds. For example, when in $X$, $a$ considers $Y$ to be more plausible than $Z$ because the $a$-path from $X$ to $Y$ is shorter than the $a$-path from $X$ to $Z$.
  • Figure 5: A polychromatic model with two minimal worlds.
  • ...and 2 more figures

Theorems & Definitions (35)

  • Definition 1
  • Remark 1
  • Definition 2
  • Remark 2
  • Definition 3
  • Lemma 1
  • Definition 4
  • Definition 5
  • Lemma 2
  • proof
  • ...and 25 more