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

  • Hiroshi UmeoEmail author
  • Naoki Kamikawa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10421)


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.


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of Osaka Electro-CommunicationNeyagawa-shi, OsakaJapan

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