A synchronization problem in cellular automata has been known as the Firing Squad Synchronization Problem (FSSP), 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. It has been shown by Balzer (Inf Control 10:22–42, 1967), Sanders (in: Jesshope, Jossifov, Wilhelmi (eds) Proceedings of the VI international workshop on parallel processing by cellular automata and arrays, Akademie, Berlin, 1994) and Berthiaume et al. (Theoret Comput Sci 320:213–228, 2004) that there exists no 4-state FSSP solution in arrays and rings. The number four is the state lower bound in the class of FSSP protocols. Umeo et al. (Parallel Process Lett 19(2):299–313, 2009), by introducing a concept of full versus partial FSSP solutions, provided a list of the smallest 4-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 (in: Partial solutions for the firing squad synchronization problem on rings, ProQuest Publications, Ann Arbor, MI, 2011) also added a list of asymmetric FSSP partial solutions, thus completing the 4-state powers-of-2 FSSP partial solutions. A question whether there are any 4-state partial solutions for ring lengths other than powers-of-2 has remained open. In this paper, we answer the question by providing a new class of the smallest symmetric and asymmetric 4-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 to linear synchronization time. In addition, we make an investigation into several interesting properties of those partial solutions, such as swapping general states, transposed protocols, a duality property between them, and an inclusive property of powers-of-2 solutions.

元胞自动机中的同步问题被称为射击小队同步问题 (FSSP)，其中 FSSP 给出了用于同步大规模元胞自动机的有限状态协议。长期以来，寻求较小状态的 FSSP 解决方案一直是一个有趣的问题。Balzer (Inf Control 10:22–42, 1967), Sanders (in: Jesshope, Jossifov, Wilhelmi (eds) Proceedings of the VI International Workshop on parallel processing by cell automata and arrays, Akademie, Berlin, 1994 ) 和 Berthiaume 等人。（Theoret Comput Sci 320:213–228, 2004）在阵列和环中不存在 4 态 FSSP 解决方案。数字四是 FSSP 协议类中的状态下限。梅奥等人。（Parallel Process Lett 19(2):299–313, 2009），通过引入完整的概念与局部FSSP解决方案，提供的4状态的最小列表对称功率为2的FSSP协议可以同步长度的任何一维（1D）环胞自动机\（N = 2 ^ {K} \）对于任何正整数\(k \ge 1\)。之后，Ng（在：环上射击小队同步问题的部分解决方案，ProQuest 出版物，安娜堡，密歇根州，2011 年）还添加了一个非对称列表FSSP 偏解，从而完成 2 次幂的 4 态 FSSP 偏解。对于除 2 的幂以外的环长度是否存在任何 4 状态部分解的问题仍然存在。在本文中，我们通过提供一类新的最小对称和非对称 4 态 FSSP 协议来回答这个问题，该协议可以同步任何长度为\(n=2^{k}-1\) 的一维环对于任何正整数\ (k \ge 2\)。我们表明该类包括丰富多样的 FSSP 协议，其中包括 39 个对称和 132 个非对称解决方案，范围从最小到线性同步时间。此外，我们对这些部分解的几个有趣的属性进行了调查，例如交换一般状态、转置协议、它们之间的对偶属性以及 2 次幂解决方案的包含属性。