A simplicial complex model for dynamic epistemic logic to study distributed task computability

Abstract

The usual S5 n epistemic model for a multi-agent system is based on a Kripke frame, which is a graph whose edges are labeled with agents that do not distinguish between two states. We propose to uncover the higher dimensional information implicit in this structure, by considering a dual, simplicial complex model. We use dynamic epistemic logic (DEL) to study how an epistemic simplicial complex model changes after a set of agents communicate with each other. We concentrate on an action model that represents the so-called immediate snapshot communication patterns of asynchronous agents, because it is central to distributed computability (but our setting works for other communication patterns). There are topological invariants preserved from the initial epistemic complex to the one after the action model is applied, which determine the knowledge that the agents gain after communication. Finally, we describe how a distributed task specification can be modeled as a DEL action model, and show that the topological invariants determine whether the task is solvable. We thus provide a bridge between DEL and the topological theory of distributed computability, which studies task solvability in a shared memory or message passing architecture.

The McKinsey–tarski Theorem for Locally Compact Ordered Spaces

“…As noticed by Vickers [29] and Abramsky [1], the notion of observability is best modeled using topology. More recently, developments in formal learning theory [12,3], distributed computing [18], and epistemic logic in multi-agent systems [2,25,17] have been based on topological semantics. In particular, topological methods have been employed in epistemic studies of modeling and reasoning about evidence and knowability [2,25].…”

A Simplicial Model for $KB4_n$: Epistemic Logic with Agents that May Die

“…The realization that distributed computability is of a topological nature motivated us to give a formal semantics to epistemic multi-agent formulas in terms of simplicial models [13]. We derived a new class of models, based on simplicial complexes, which is equivalent to the usual Kripke model semantics for S5 n .…”

“…We derived a new class of models, based on simplicial complexes, which is equivalent to the usual Kripke model semantics for S5 n . We were able to provide tools to reason about solvability of distributed tasks such as consensus, approximate agreement and equality negation [13,32], and explore bisimilarity of simplicial models [8] and connections with covering spaces [32]. The simplicial model semantics led to a logical obstruction to the solvability of set agreement by Yagi and Nishimura [35] using the notion of distributed knowledge [14], in a sense a higher dimensional version of knowledge.…”

“…The categorical equivalence of [13] between S5 n Kripke models and simplicial models associates each world of the Kripke model with a facet of the corresponding simplicial model. A core assumption of these models is that the same set of n agents always participate in every possible world.…”

“…

The standard semantics of multi-agent epistemic logic S5n is based on Kripke models whose accessibility relations are reflexive, symmetric and transitive. This one dimensional structure contains implicit higher-dimensional information beyond pairwise interactions, that has been formalized as pure simplicial models in [13]. Here we extend the theory to encompass all simplicial models -including the ones that are not pure.

…”