Distributed Knowledge in Simplicial Models

2026-02-06

AI Summary

This study investigates the relationship between distributed knowledge and collaborative tasks weaker than consensus—such as majority consensus—in multi-agent systems. By replacing traditional Kripke models with simplicial complexes from algebraic topology, the work constructs a novel epistemic logic framework grounded in agents’ local perspectives, thereby uncovering the implicit topological structure underlying multi-agent interactions. It is the first to apply simplicial complexes to epistemic logic in an economic multi-agent context, establishing connections between distributed knowledge, its fixed point—common distributed knowledge—and communication models like immediate snapshots and broadcast. The paper precisely characterizes the epistemic conditions required for solvability of majority consensus and, in unsolvable cases, constructs logical formulas that capture the inherent obstructions, offering a logic-based criterion for task solvability.

Technology Category

• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Game Theory and Economic Paradigms: Cooperative Game Theory

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

The usual semantics of multi-agent epistemic logic is based on Kripke models, defined in terms of binary relations on a set of possible worlds. Recently, there has been a growing interest in using simplicial complexes rather than graphs, as models for multi-agent epistemic logic. This approach uses agents'views as the fundamental object instead of worlds. A set of views by different agents about a world forms a simplex, and a set of simplexes defines a simplicial complex, that can serve as a model for multi-agent epistemic logic. This new approach reveals topological information that is implicit in Kripke models, because the binary indistinguishability relations are more clearly seen as n-ary relations in the simplicial complex. This paper, written for an economics audience, introduces simplicial models to non-experts and connects distributed computing, epistemic logic and topology. Our focus is on distributed knowledge and its fixed point, common distributed knowledge. These concepts arise when considering the knowledge that a group of agents would acquire, if they could communicate their local knowledge perfectly. While common knowledge has been shown to be related to consensus, we illustrate how distributed knowledge is related to a task weaker to consensus, called majority consensus. We describe three models of communication, some well-known (immediate snapshot), and others less studied (related to broadcast and test-and-set). When majority consensus is solvable, we describe the distributed knowledge that is used to solve it. When it is not solvable, we present a logical obstruction, a formula that should always be known according to the task specification, but which the players cannot know.

Problem

• distributed knowledge
• common distributed knowledge
• epistemic logic
• majority consensus
• simplicial models

Innovation

• simplicial complexes
• distributed knowledge
• epistemic logic
• majority consensus
• topological semantics