Distributed Knowledge in Simplicial Models
Éric Goubault, Jérémy Ledent, Sergio Rajsbaum
TL;DR
The paper develops simplicial models as a topology-aware semantics for multi-agent epistemic logic, shifting from world-centric Kripke frames to agent-centric views encoded as simplices. It defines and interprets Knowledge, Common Knowledge, Distributed Knowledge, and Common Distributed Knowledge within chromatic simplicial complexes, linking these operators to higher-order connectivity. Applying this framework to distributed computing, it analyzes three communication models—unreliable broadcast, immediate snapshot, and test-and-set—against the majority consensus task, using obstruction formulas and the Knowledge Gain Theorem to derive solvability results. The findings reveal when distributed knowledge suffices to solve a task and when topological obstructions prevent it, offering a bridge between economics, topology, and distributed computation with guidance for future work on iterated knowledge and fault tolerance.
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.