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.

翻译：多智能体认知逻辑的常规语义基于克里普克模型，该模型通过可能世界集合上的二元关系定义。近年来，利用单纯复形而非图结构作为多智能体认知逻辑模型的研究日益受到关注。该方法以智能体的视角而非世界作为基本对象，不同智能体对同一世界的视角集合构成单纯形，单纯形的集合则定义出可作为多智能体认知逻辑模型的单纯复形。这种新方法揭示了克里普克模型中隐含的拓扑信息，因为二元不可区分关系在单纯复形中更清晰地呈现为n元关系。本文面向经济学研究者，向非专业读者介绍单纯形模型，并建立分布式计算、认知逻辑与拓扑学之间的关联。我们重点关注分布式知识及其不动点——公共分布式知识。这些概念产生于思考智能体群体在能够完美交流局部知识时将获得何种知识的情形。虽然公共知识已被证明与共识问题相关，但我们阐释了分布式知识如何与弱于共识的多数共识任务相关联。我们描述了三种通信模型，包括广为人知的即时快照模型，以及较少研究的广播模型与测试-置位模型。当多数共识可解时，我们描述了用于解决该问题的分布式知识；当不可解时，我们提出了一种逻辑障碍——即根据任务规范应当始终被知晓，但参与者实际无法获知的逻辑公式。