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State-efficient realization of fault-tolerant FSSP algorithms

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Abstract

The firing squad synchronization problem (FSSP, for short) on cellular automata has been studied extensively for more than fifty years, and a rich variety of FSSP algorithms has been proposed. Here we study the classical FSSP on a model of fault-tolerant cellular automata that might have possibly some defective cells and present the first state-efficient implementations of fault-tolerant FSSP algorithms for one-dimensional (1D) and two-dimensional (2D) cellular arrays. It is shown that, under some constraints on the length and distribution of defective cells, any 1D cellular array of length n with p defective cell segments can be synchronized in \(2n-2+p\) steps and the algorithm is realized on a 1D cellular automaton of length \(n, 2 \le n \le 50\), having 164 states and 4792 transition rules. In addition, we give by far a smaller-state implementation of a 2D FSSP algorithm that can synchronize any 2D rectangular array of size \(m \times n\), possibly including at most O(mn) isolated defective zones, exactly in \(2(m+n)-4\) steps on a cellular automaton with only 6 states and 935 transition rules.

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Acknowledgements

A part of this work is supported by JSPS 16K00026. The authors would like to thank reviewers for many helpful comments and suggestions to improve the paper.

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  1. University of Osaka Electro-Communication, Neyagawa-shi, Hastu-cho, 18-8, Osaka, 572-8530, Japan

    Hiroshi Umeo, Naoki Kamikawa, Masashi Maeda & Gen Fujita

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  1. Hiroshi Umeo
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  2. Naoki Kamikawa
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  4. Gen Fujita
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Correspondence to Hiroshi Umeo.

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A preliminary version of this work appeared at 13th International Conference on Cellular Automata for Research and Industry, ACRI 2018, held on 17–21, September, 2018, in Como, Italy.

Appendix

Appendix

Transition rule set for the 2D 6-state fault-tolerant FSSP algorithm

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Umeo, H., Kamikawa, N., Maeda, M. et al. State-efficient realization of fault-tolerant FSSP algorithms. Nat Comput 18, 827–844 (2019). https://doi.org/10.1007/s11047-019-09765-3

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