The Smallest FSSP Partial Solutions for One-Dimensional Ring Cellular Automata: Symmetric and Asymmetric Synchronizers

A synchronization problem in cellular automata has been known as the Firing Squad Synchronization Problem (FSSP) since its development, where the FSSP gives a finite-state protocol for synchronizing a large scale of cellular automata. A quest for smaller state FSSP solutions has been an interesting problem for a long time. Umeo, Kamikawa and Yunes (2009) answered partially by introducing a concept of partial FSSP solutions and proposed a full list of the smallest four-state symmetric powers-of-2 FSSP protocols that can synchronize any one-dimensional (1D) ring cellular automata of length \(n=2^{k}\) for any positive integer \(k \ge 1\). Afterwards, Ng (2011) also added a list of asymmetric FSSP partial solutions, thus completing the four-state powers-of-2 FSSP partial solutions. The number four is the lower bound in the class of FSSP protocols. A question: are there any four-state partial solutions other than powers-of-2? has remained open. In this paper, we answer the question by proposing a new class of the smallest symmetric and asymmetric four-state FSSP protocols that can synchronize any 1D ring of length \(n=2^{k}-1\) for any positive integer \(k \ge 2\). We show that the class includes a rich variety of FSSP protocols that consists of 39 symmetric and 132 asymmetric solutions, ranging from minimum-time to linear-time in synchronization steps. In addition, we make an investigation into several interesting properties of these partial solutions, such as swapping general states, reversal protocols, and a duality property between them.