A Minimal Time Solution to the Firing Squad Synchronization Problem with Von Neumann Neighborhood of Extent 2

Cellular automata provide a simple environment in which to study global behaviors. One example of a problem that utilizes cellular automata is the Firing Squad Synchronization Problem, first proposed in 1957. This paper provides an overview of the standard Firing Squad Synchronization Problem and a commonly used technique in solving it. This paper also provides a statement of a new extension of the Standard Firing Squad Synchronization Problem to a different neighborhood definition - a Von Neumann neighborhood of extent 2. An 8 state 651 rule minimal time solution to the extended problem is described, presented and proven, along with Python code used in running simulations of the solution

A 4-states Algebraic Solution to Linear Cellular Automata Synchronization

International audienceIn this paper, we aim to present a completely new solution to the firing squad synchronization problem based on Wolfram's rule 60. This solution solves the problem on an infinite number of lines but not all possible lines. The two remarkable properties are that the state complexity of it is the lowest possible, say 4 states and 32 transitions (we prove that no line of length n5 can be synchronized with only 3 states) and that the algorithm is no more based on geometric constructions but relies on some algebraic properties of the transition function. The solution is almost in minimal time: up to one unit of time