The firing squad synchronization problem with sub-generals

The Firing Squad Synchronization Problem (FSSP), one of the most well-known problems related to cellular automata, was originally proposed by Myhill in 1957 and became famous through the work of Moore. The first solution to this problem was given by Minsky and McCarthy, and a minimal time solution was given by Goto. A considerable amount of research has also dealt with variants of this problem. In this paper, we introduce a new state called the sub-general to the original problem and propose the FSSP with sub-generals. In particular, we consider the case of one sub-general and determine the position of the sub-general in the array that minimizes the synchronization time. Moreover, we determine the minimal time to solve this problem and show that there exists no minimal time solution for any length of array. However, we show that the total time of our algorithm approaches arbitrarily close to the minimal time.