https://doi.org/10.1016/j.physrep.2020.05.004

Networks beyond pairwise interactions: Structure and dynamics

Simplicial complexes are a versatile and convenient paradigm on which to build all the tools and techniques of the logic of knowledge, on the assumption that initial epistemic models can be described in a distributed fashion. Thus, we can define: knowledge, belief, bisimulation, the group notions of mutual, distributed and common knowledge, and also dynamics in the shape of simplicial action models. We give a survey on how to interpret all such notions on simplicial complexes, building upon the foundations laid in prior work by Goubault and others.

pdf/knowledge-and-simplicial-complexes-6zv468i5l5.pdf

Knowledge and simplicial complexes.

A dynamic epistemic logic analysis of equality negation and other epistemic covering tasks

We propose a logic of knowledge for impure simplicial complexes. Impure simplicial complexes represent distributed systems under uncertainty over which processes are still active (are alive) and which processes have failed or crashed (are dead). Our work generalizes the logic of knowledge for pure simplicial complexes, where all processes are alive, by Goubault et al. Our logical semantics has a satisfaction relation defined simultaneously with a definability relation. The latter restricts which formulas are allowed to have a truth value: dead processes cannot know or be uncertain about any proposition, and live processes cannot know or be uncertain about propositions involving processes they know to be dead. The logic satisfies some but not all axioms and rules of the modal logic S5. Impure simplicial complexes correspond to Kripke models where each agent’s accessibility relation is an equivalence relation on a subset of the domain only. We also propose a notion of bisimulation for impure simplicial complexes and show bisimulation correspondence.

pdf/wanted-dead-or-alive-epistemic-logic-for-impure-simplicial-2cuacvnhmk.pdf

Wanted Dead or Alive: Epistemic Logic for Impure Simplicial Complexes

We study the topological µ-calculus, based on both Cantor derivative and closure modalities, proving completeness, decidability and FMP over general topological spaces, as well as over T 0 and T D spaces. We also investigate relational µ-calculus, providing general completeness results for all natural fragments of µ-calculus over many different classes of relational frames. Unlike most other such proofs for µ-calculus, ours is modeltheoretic, making an innovative use of a known Modal Logic method (–the ’final’ submodel of the canonical model), that has the twin advantages of great generality and essential simplicity.

pdf/the-topological-mu-calculus-completeness-and-decidability-2szzhceawm.pdf

The topological mu-calculus: completeness and decidability

The usual S 5 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.

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

/pdf/a-simplicial-complex-model-for-dynamic-epistemic-logic-to-srxuiuvyo3.pdf

The usual epistemic model S5n 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

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

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

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