## Abstract

Various synchronization algorithms have been introduced in literature during the last decades to deal with the firing squad synchronization problem on cellular automata (CA). Among others defective CA algorithms, where the CA cell is able to transmit information without previous processing, have been also presented. In our case, originating from the classical Mazoyer’s paper, where a minimum-time solution is presented with 6 states, a one-dimensional CA where one cell may permanently fail is presented. In the proposed algorithm, the defective cell can neither process nor transmit information any longer, while it is considered that such dynamic defects may become apparent in any time step of computation. A thorough analysis of the synchronization, in terms of location and time at which cell fails, for the cells found in both sides of defective cell is delivered to decipher the corresponding maximal possible number of synchronized cells in each part of the cut, due to defect, CA array. The proposed algorithm is properly extended to consider more than one defective cells that may occur in the under study one-dimensional CA. Based on the aforementioned analysis, we provide the generalization of synchronization with multiple totally defective cells, while application examples of the generalized CA algorithm in case of two defective cells are also presented. Finally, another intriguing aspect refers to handling of states that could be tentatively characterized as unknown, in a confrontation similar to the previous defective state but also different, since now this(these) cell(s) are not stated as faulty but unknown. As a result, a new one-dimensional CA with less states, compared to the previous CA defective algorithms, able to synchronize the maximal possible number of cells in each part occurs.

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## References

Balzer RM (1967) An 8-state minimal time solution to the firing squad synchronization problem. Inf Control 10:22–42

Čulik K II (1989) Variations of the firing squad problem and applications. Inf Process Lett 30:153–157

Čulik K II, Dube S (1991) An efficient solution of the firing mob problem. Theor Comput Sci 91:57–69

Dimitriadis A, Kutrib M, Sirakoulis G Ch (2016) Cutting the firing squad synchronization. In: El Yacoubi S, Was J, Bandini S (eds) Cellular automata—12th international conference on cellular automata for research and industry, ACRI 2016. Lecture notes in computer science, vol 9863. Springer, pp 123–133

Fay B, Kutrib M (2004) The fault-tolerant early bird problem. IEICE Trans Inf Syst E87–D:687–693

Gács P (1986) Reliable computation with cellular automata. J Comput Syst Sci 32(1):15–78

Goto E (1962) A minimal time solution of the firing squad problem. In: Course notes for applied mathematics, vol 298. Harvard University, Cambridge, MA

Grasselli A (1975) Synchronization of cellular arrays: the firing squad problem in two dimensions. Inf Control 28:113–124

Harao M, Noguchi S (1975) Fault tolerant cellular automata. J Comput Syst Sci 11:171–185

Herman GT, Liu WH, Rowland S, Walker A (1974) Synchronization of growing cellular arrays. Inf Control 25:103–122

Imai K, Morita K (1996) Firing squad synchronization problem in reversible cellular automata. Theor Comput Sci 165(2):475–482

Jiang T (1992) The synchronization of nonuniform networks of finite automata. Inf Comput 97:234–261

Kobayashi K (1978a) The firing squad synchronization problem for a class of polyautomata networks. J Comput Syst Sci 17:300–318

Kobayashi K (1978b) On the minimal firing time of the firing squad synchronization problem for polyautomata networks. Theor Comput Sci 7:149–167

Kutrib M, Löwe JT (2002) Massively parallel fault tolerant computations on syntactical patterns. Future Gener Comput Syst 18:905–919

Kutrib M, Vollmar R (1991) Minimal time synchronization in restricted defective cellular automata. J Inf Process Cybern EIK 27:179–196

Kutrib M, Vollmar R (1995) The firing squad synchronization problem in defective cellular automata. IEICE Trans Inf Syst E78–D(7):895–900

Mazoyer J (1987) A six-state minimal time solution to the firing squad synchronization problem. Theor Comput Sci 50:183–238

Mazoyer J (1989) A minimal time solution to the firing squad synchronization problem with only one bit of information exchanged. Technical Report TR 89-03, Ecole Normale Supérieure de Lyon, Lyon

Moore EF (1964) The firing squad synchronization problem. In: Moore EF (ed) Sequential machines—selected papers. Addison-Wesley, Reading, pp 213–214

Moore FR, Langdon GC (1968) A generalized firing squad problem. Inf Control 12:17–33

Nishio H, Kobuchi Y (1975) Fault tolerant cellular spaces. J Comput Syst Sci 11:150–170

Romani F (1978) The parallelism principle: speeding up the cellular automata synchronization. Inf Control 36:245–255

Rosenstiehl P, Fiksel JR, Holliger A (1972) Intelligent graphs: networks of finite automata capable of solving graph problems. In: Read RC (ed) Graph theory and computing. Academic Press, New York, pp 219–265

Shinahr I (1974) Two- and three-dimensional firing-squad synchronization problems. Inf Control 24:163–180

Szwerinski H (1982) Time optimal solution of the firing squad synchronization problem for n-dimensional rectangles with the general at an arbitrary position. Theor Comput Sci 19:305–320

Umeo H (1994) A fault-tolerant scheme for optimum-time firing squad synchronization. In: Joubert GR, Trystram D, Peters FJ, Evans DJ (eds) Parallel computing: trends and applications. North-Holland, Amsterdam, pp 223–230

Umeo H (1996) A note on firing squad synchronization algorithms. In: Kutrib M, Thomas W (eds) IFIP cellular automata workshop 1996. Universität Giessen, Giessen, p 95

Umeo H (2004) A simple design of time-efficient firing squad synchronization algorithms with fault-tolerance. IEICE Trans Inf Syst E87–D(3):733–739

Umeo H (2009) Firing squad synchronization problem in cellular automata. In: Meyers R (ed) Encyclopedia of complexity and systems science. Springer, Berlin, pp 3537–3574

Umeo H, Maeda M, Hisaoka M, Teraoka M (2006) A state-efficient mapping scheme for designing two-dimensional firing squad synchronization algorithms. Fundam. Inform. 74(4):603–623

von Neumann J (1956) Probabilistic logics and the synthesis of reliable organisms from unreliable components. In: Shannon CE, McCarthy J (eds) Automata studies. Princeton University Press, Princeton, pp 43–98

Vollmar R (1991) FSSP for cellular automata with busses. Trans IEICE E 74(9):2965–2968

Waksman A (1966) An optimum solution to the firing squad synchronization problem. Inf Control 9:66–78

Yunès JB (1996) Fault tolerant solutions to the firing squad synchronization problem. Technical Report LITP 96/06, Institut Blaise Pascal, Paris

Yunès JB (2009) Goto’s construction and Pascal’s triangle: new insights into cellular automata synchronization. In: Durand B (ed) Symposium on cellular automata—journées automates cellulaires (JAC 2008). MCCME Publishing House, Moscow, pp 195–203

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Dimitriadis, A., Kutrib, M. & Sirakoulis, G.C. Revisiting the cutting of the firing squad synchronization.
*Nat Comput* **17, **455–465 (2018). https://doi.org/10.1007/s11047-017-9628-z

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### Keywords

- Defective cellular automata
- Firing squad synchronization problem
- Fault tolerance
- Synchronization algorithm
- Multiple totally defective cells
- Unknown states