TY - GEN
T1 - The Structure of Configurations in One-Dimensional Majority Cellular Automata
T2 - 15th International Conference on Cellular Automata for Research and Industry, ACRI 2022
AU - Nakar, Yonatan
AU - Ron, Dana
N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.
PY - 2022
Y1 - 2022
N2 - We study the dynamics of (synchronous) one-dimensional cellular automata with cyclical boundary conditions that evolve according to the majority rule with radius r. We introduce a notion that we term cell stability with which we express the structure of the possible configurations that could emerge in this setting. Our main finding is that apart from the configurations of the form (0r+10∗+1r+11∗)∗, which are always fixed-points, the other configurations that the automata could possibly converge to, which are known to be either fixed-points or 2-cycles, have a particular spatially periodic structure. Namely, each of these configurations is of the form s∗ where s consists of O(r2) consecutive sequences of cells with the same state, each such sequence is of length at most r, and the total length of s is O(r2) as well. We show that an analogous result also holds for the minority rule.
AB - We study the dynamics of (synchronous) one-dimensional cellular automata with cyclical boundary conditions that evolve according to the majority rule with radius r. We introduce a notion that we term cell stability with which we express the structure of the possible configurations that could emerge in this setting. Our main finding is that apart from the configurations of the form (0r+10∗+1r+11∗)∗, which are always fixed-points, the other configurations that the automata could possibly converge to, which are known to be either fixed-points or 2-cycles, have a particular spatially periodic structure. Namely, each of these configurations is of the form s∗ where s consists of O(r2) consecutive sequences of cells with the same state, each such sequence is of length at most r, and the total length of s is O(r2) as well. We show that an analogous result also holds for the minority rule.
UR - http://www.scopus.com/inward/record.url?scp=85137004024&partnerID=8YFLogxK
U2 - 10.1007/978-3-031-14926-9_6
DO - 10.1007/978-3-031-14926-9_6
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AN - SCOPUS:85137004024
SN - 9783031149252
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 63
EP - 72
BT - Cellular Automata - 15th International Conference on Cellular Automata for Research and Industry, ACRI 2022, Proceedings
A2 - Chopard, Bastien
A2 - Arabi Haddad, Mira
A2 - Bandini, Stefania
A2 - Dennunzio, Alberto
PB - Springer Science and Business Media Deutschland GmbH
Y2 - 12 September 2022 through 15 September 2022
ER -