[Paper Review] A simplicial complex model of dynamic epistemic logic for fault-tolerant distributed computing

Éric Goubault, Sergio Rajsbaum|arXiv (Cornell University)|Mar 31, 2017

Distributed systems and fault tolerance27 references3 citations

TL;DR

This paper introduces a simplicial complex model for dynamic epistemic logic (DEL) in fault-tolerant distributed computing, representing epistemic states as chromatic simplicial complexes dual to Kripke graphs. It shows that under reliable communication, topological invariants—particularly connectivity—of the initial epistemic complex are preserved after action model updates, directly linking topological structure to knowledge evolution and solvability of distributed tasks like binary consensus.

ABSTRACT

The usual epistemic S5 model for multi-agent systems is a Kripke graph, whose edges are labeled with the agents that do not distinguish between two states. We propose to uncover the higher dimensional information implicit in the Kripke graph, by using as a model its dual, a chromatic simplicial complex. For each state of the Kripke model there is a facet in the complex, with one vertex per agent. If an edge (u,v) is labeled with a set of agents S, the facets corresponding to u and v intersect in a simplex consisting of one vertex for each agent of S. Then we use dynamic epistemic logic to study how the simplicial complex epistemic model changes after the agents communicate with each other. We show that there are topological invariants preserved from the initial epistemic complex to the epistemic complex after an action model is applied, that depend on how reliable the communication is. In turn these topological properties determine the knowledge that the agents may gain after the communication happens.

Motivation & Objective

• To bridge dynamic epistemic logic (DEL) and combinatorial topology in distributed computing by modeling epistemic states as simplicial complexes.
• To formalize how knowledge evolves in multi-agent systems during communication events using topological invariants.
• To establish a categorical equivalence between Kripke models and simplicial models for distributed tasks.
• To investigate how communication reliability affects the preservation of epistemic topology and knowledge gain.
• To demonstrate that common knowledge corresponds to connected components in the simplicial model, enabling solvability analysis of distributed tasks.

Proposed method

• Dualize the Kripke graph into a chromatic simplicial complex, where each state corresponds to a facet and agent indistinguishability defines shared simplices.
• Model communication events via action models that induce a product update on the epistemic complex, preserving topological structure under reliable communication.
• Use morphisms between protocol complexes and task complexes to formalize knowledge transfer and solvability conditions.
• Apply the notion of common knowledge via connectivity in the simplicial model: $C_B\varphi$ holds iff $\varphi$ is true on all facets in the same connected component under agent group $B$.
• Establish categorical equivalence between Kripke models and simplicial models, enabling translation of epistemic logic semantics into topological terms.
• Analyze one-round and multi-round communication models, showing full topological preservation when message delivery is reliable.

Experimental results

Research questions

• RQ1 How can epistemic states in distributed systems be represented using higher-dimensional topological structures like simplicial complexes?
• RQ2 What topological invariants are preserved in the epistemic complex after communication events, and how do they relate to knowledge gain?
• RQ3 How does the reliability of communication affect the preservation of epistemic topology and the emergence of common knowledge?
• RQ4 Can the solvability of distributed tasks like binary consensus be characterized through topological connectivity in the protocol complex?
• RQ5 What is the categorical relationship between Kripke models and simplicial models in the context of dynamic epistemic logic for distributed computing?

Key findings

• The simplicial complex model provides a dual representation of Kripke models, where facets correspond to states and shared simplices encode agent indistinguishability.
• Common knowledge for a group of agents $B$ corresponds to truth of a formula on all facets within the same connected component of the simplicial model restricted to $B$.
• In the binary consensus task, the simplicial model $\Delta$ has two connected components, each encoding a different common knowledge: $\neg l_0 \vee \neg l_1$ and $l_0 \vee l_1$, respectively.
• The protocol complex for binary consensus fails to support common knowledge of input values, proving that no morphism exists from the protocol complex to the task complex, thus showing binary consensus is unsolvable.
• For one-round and multi-round communication with reliable message delivery, the topological structure of the initial epistemic complex is fully preserved in the updated complex.
• The framework establishes a precise categorical equivalence between dynamic epistemic Kripke models and simplicial models, enabling topological analysis of knowledge evolution in distributed systems.