Firing Squad Synchronization Problem on Bidimensional Cellular Automata with Communication Constraints

Abstract

We are given a network of identical processors that work synchronously at discrete time steps, a Cellular Automaton. Processors are arranged as an array of m rows and n columns with the constraint that they can exchange only one bit of information with their neighbours. We study the problem to synchronize the cellular automata, that is to let all the processors enter the same state for the first time at the very same instant. This problem is also known as “The Firing Squad Synchronization Problem”, introduced by Moore in 1964. We give algorithms to synchronize two dimensional Cellular Automata of (nxn) cells at some given times: n ², n⌈log n⌉, n⌈vn ⌉ and 2ⁿ. Moreover we also give some general constructions to synchronize Cellular Automata of (mxn) cells. Our approach is a modular description of synchronizing algorithms in terms of “fragments” of cellular automata that are called signals.