A Simplicial Complex Model for Dynamic Epistemic Logic to study Distributed Task Computability

Abstract

The usual epistemic S5 n 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.

A Dynamic Epistemic Logic Analysis of the Equality Negation Task

“…Organization. Section 2 recalls the DEL framework introduced in [10], and defines the layered message-passing model in this context. In Section 3 we study the equality negation task using DEL.…”

“…We recap here the new kind of model for epistemic logic based on chromatic simplicial complexes, introduced in [10]. The geometric nature of simplicial complexes allows us to consider higher-dimensional topological properties of our models, and investigate their meaning in terms of knowledge.…”

“…The agents communicate to each other via the layered messagepassing model described above. The layered message-passing action model described next is equivalent to the immediate snapshot action model of [10] in the case of two processes.…”

“…We now describe how the set up of [10] is used to prove impossibility results in distributed computing. It is closely related to the usual topological approach to distributed computability [11], except that the input complex, output complex and protocol complex are now viewed as simplicial models for epistemic logic.…”

Tasks in Modular Proofs of Concurrent Algorithms

“…A third direction is to study if the duality between the epistemic logic approach and the topological approach shown in [15] might be useful in verifying concurrent algorithms. Generally speaking, it is shown in [15] that a task can be represented as a Kripke model with an action model, specifying the knowledge obtained by processes when solving the task. It could be interesting to explore how this knowledge could be reflected in our set/get construction and if it could be useful in proving correctness.…”