Signal Machine And Cellular Automaton Time-Optimal Quasi-Solutions Of The Firing Squad/Mob Synchronisation Problem On Connected Graphs

TL;DR

This paper presents a time-optimal quasi-solution for the firing mob synchronisation problem on finite connected graphs, utilizing signal machines and cellular automata to achieve synchronization with minimal time.

Contribution

It introduces a novel approach using signal machines over continuum graph representations to construct a quasi-solution with optimal timing.

Findings

Achieves time-optimal synchronization on bounded-degree graphs.

Uses signal machines to bridge continuum and discrete automata.

Provides a quasi-solution with countably infinite states.

Abstract

We construct a time-optimal quasi-solution of the firing mob synchronisation problem over finite, connected, and undirected multigraphs whose maximum degrees are uniformly bounded by a constant. It is only a quasi-solution because its number of states depends on the graph or, from another perspective, does not depend on the graph but is countably infinite. To construct this quasi-solution we introduce signal machines over continuum representations of such multigraphs and construct a signal machine whose discretisation is a cellular automaton that quasi-solves the problem. This automaton uses a time-optimal solution of the firing squad synchronisation problem in dimension one with one general at one end to synchronise edges, and freezes and thaws the synchronisation of edges in such a way that all edges synchronise at the same time.