A New Class of the Smallest Four-State Partial FSSP Solutions for One-Dimensional Ring Cellular Automata

The synchronization 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 Yunès [9] answered partially by introducing a concept of partial FSSP solutions and proposing 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 [7] 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 smallest one in the class of FSSP protocols proposed so far. A question remained is that “are there any other four-state partial solutions?”. In this paper, we answer to the question by proposing a new class of the smallest 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 solutions and 132 asymmetric ones, 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, a duality between them, inclusion of powers-of-2 solutions, reflected solutions and so on.